Interaction Analysis and Control of Bioreactors for Nitrogen Removal ¨ B

Interaction Analysis and Control of Bioreactors for Nitrogen Removal ¨ B
IT Licentiate theses
2007-006
Interaction Analysis and Control of
Bioreactors for Nitrogen Removal
B J ÖRN H ALVARSSON
UPPSALA UNIVERSITY
Department of Information Technology
Interaction Analysis and Control of Bioreactors for
Nitrogen Removal
BY
B J ÖRN H ALVARSSON
December 2007
D IVISION
S YSTEMS AND C ONTROL
D EPARTMENT OF I NFORMATION T ECHNOLOGY
U PPSALA U NIVERSITY
U PPSALA
S WEDEN
OF
Dissertation for the degree of Licentiate of Philosophy in Electrical Engineering with
Specialization in Automatic Control
at Uppsala University 2007
Interaction Analysis and Control of Bioreactors for Nitrogen Removal
Björn Halvarsson
[email protected]
Division of Systems and Control
Department of Information Technology
Uppsala University
Box 337
SE-751 05 Uppsala
Sweden
http://www.it.uu.se/
c Björn Halvarsson 2007
ISSN 1404-5117
Printed by the Department of Information Technology, Uppsala University, Sweden
Abstract
Efficient control of wastewater treatment processes are of great importance.
The requirements on the treated water (effluent standards) have to be met
at a feasible cost. This motivates the use of advanced control strategies. In
this thesis the activated sludge process, commonly found in the biological
wastewater treatment step for nitrogen removal, was considered. Multivariable interactions present in this process were analysed. Furthermore, control
strategies were suggested and tested in simulation studies.
The relative gain array (RGA), Gramian based interaction measures and
an interaction measure based on the H2 norm were considered and compared.
Properties of the H2 norm based measure were derived. It was found that
the Gramian based measures, and particularly the H2 norm based measure,
in most of the considered cases were able to properly indicate the interactions. The information was used in the design of multivariable controllers.
These were found to be less sensitive to disturbances compared to controllers
designed on the basis of information from the RGA.
The conditions for cost-efficient operation of the activated sludge process
were investigated. Different fee functions for the effluent discharges were
considered. It was found that the economic difference between operation
in optimal and non-optimal set points may be significant even though the
treatment performance was the same. This was illustrated graphically in
operational maps. Strategies for efficient control were also discussed.
Finally, the importance of proper aeration in the activated sludge process was illustrated. Strategies for control of a variable aeration volume were
compared. These performed overall well in terms of treatment efficiency, disturbance rejection and process economy.
Keywords: activated sludge process; biological nitrogen removal; bioreactor
models; cost-efficient operation; interaction measures; multivariable control;
wastewater treatment.
Acknowledgements
First of all, I would like to express my sincere gratitude to my supervisor,
Professor Bengt Carlsson, for all his help and encouragement during my
research so far.
Special thanks also go to my co-author and “mentor” Dr. Pär Samuelsson
at Dalarna University (formerly with us here at the Division of Systems
and Control) for all fruitful discussions and good advice. He read previous
versions of this thesis and his suggestions certainly improved the quality.
Furthermore, I wish to thank all colleagues at the Division of Systems
and Control and the Division of Scientific Computing for providing such a
pleasant working atmosphere.
Part of this work has been financially supported by the EC 6th Framework programme as a Specific Targeted Research or Innovation Project
(HIPCON, Contract number NMP2-CT-2003-505467). Furthermore, I would
like to thank Stiftelsen J. Gust. Richerts Minne for financial support.
I am also grateful to Dr. Ulf Jeppsson for letting me use his Simulink
implementation of BSM1 and to Assistant Professor Torsten Wik, Chalmers
University of Technology, Göteborg, for taking the time of being my licentiate opponent.
Finally, very special thanks go to my friends and to my family.
Contents
1 Introduction
1.1 Interaction measures . . . . . . . . . . . . .
1.1.1 Motivational example . . . . . . . .
1.2 Wastewater treatment systems . . . . . . .
1.2.1 The activated sludge process (ASP)
1.2.2 The benchmark model BSM1 . . . .
1.2.3 Control of WWTPs . . . . . . . . .
1.3 Thesis outline . . . . . . . . . . . . . . . . .
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2 Controllability and Interaction Measures
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2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Systems description . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.1 State controllability for continuous-time systems . . . 21
2.3.2 State controllability for discrete-time systems . . . . . 22
2.3.3 Output controllability . . . . . . . . . . . . . . . . . . 23
2.4 The Relative Gain Array (RGA) . . . . . . . . . . . . . . . . 24
2.5 Gramian based interaction measures . . . . . . . . . . . . . . 25
2.5.1 The Hankel norm . . . . . . . . . . . . . . . . . . . . . 25
2.5.2 Energy interpretations of the controllability and observability Gramians for discrete-time systems . . . . 28
2.5.3 The Hankel Interaction Index Array (HIIA) . . . . . . 28
2.5.4 The Participation Matrix (PM) . . . . . . . . . . . . . 29
2.5.5 The selection of proper scaling . . . . . . . . . . . . . 30
2.6 An interaction measure based on the H2 norm . . . . . . . . 30
2.6.1 The Σ2 interaction measure . . . . . . . . . . . . . . . 30
2.6.2 The H2 norm . . . . . . . . . . . . . . . . . . . . . . . 31
2.6.3 Calculation of the H2 norm . . . . . . . . . . . . . . . 32
2.6.4 Energy interpretation for discrete-time systems . . . . 33
2.6.5 Properties of the H2 norm based interaction measure
Σ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6.6 The H2 norm and induced norms . . . . . . . . . . . . 36
2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3
2.8
2.7.1 Example
2.7.2 Example
2.7.3 Example
2.7.4 Example
2.7.5 Example
Conclusions . .
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3 Interaction Analysis in a
3.1 Introduction . . . . . .
3.2 The bioreactor model
3.3 RGA analysis . . . . .
3.4 HIIA analysis . . . .
3.5 Discussion . . . . . . .
3.6 Conclusions . . . . . .
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Bioreactor Model
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4 Interaction Analysis and Control of the Denitrification Process
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The bioreactor model . . . . . . . . . . . . . . . . . . . . . .
4.3 Analysis of the model . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Linearization and scaling of the model . . . . . . . . .
4.3.2 RGA analysis of the model . . . . . . . . . . . . . . .
4.3.3 HIIA analysis of the model . . . . . . . . . . . . . . .
4.4 Control simulations . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Decentralized control . . . . . . . . . . . . . . . . . . .
4.4.2 Multivariable control . . . . . . . . . . . . . . . . . . .
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Economic Efficient Operation of a Pre-denitrifying Activated Sludge Process
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The model and the operational cost functions . . . . . . . . .
5.2.1 The nitrate fee . . . . . . . . . . . . . . . . . . . . . .
5.2.2 The ammonium fee . . . . . . . . . . . . . . . . . . . .
5.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Simulation results for the denitrification process . . .
5.3.2 Simulation results for the combined denitrification and
nitrification process . . . . . . . . . . . . . . . . . . .
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Aeration Volume Control in an Activated Sludge
– Discussion of Some Strategies Involving On-Line
nium Measurements
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
6.2 The simulation setup . . . . . . . . . . . . . . . . . .
6.3 Description of the proposed control strategies . . . .
6.3.1 The reference aeration control strategies . . .
6.3.2 Feedforward aeration volume control I and II
6.3.3 Supervisory feedback aeration volume control
6.4 Simulation results . . . . . . . . . . . . . . . . . . .
6.4.1 The reference aeration control strategies . . .
6.4.2 Feedforward aeration volume control I . . . .
6.4.3 Feedforward aeration volume control II . . .
6.4.4 Supervisory feedback aeration volume control
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . .
A The minimized condition number
5
Process
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6
Chapter 1
Introduction
This thesis concerns the interaction analysis and the control of bioreactors
for nitrogen removal. More precisely, models of the activated sludge process
commonly found in the biological treatment of wastewater are considered.
The interactions present in these processes will be analysed and different
controller structures will be compared in simulation studies. The influence
of various input signals on the treatment efficiency, both in terms of the
treatment performance and in terms of the process economy, will also be
investigated.
1.1
Interaction measures
Many control systems of today are multivariable. This means that they have
multiple inputs and multiple outputs. Such systems are called multiple-input
multiple-output (MIMO) systems. Compared to single-input single-output
(SISO) systems, the control design for MIMO systems is more elaborate.
One reason for this is that different parts of a multivariable system may
interact and cause couplings in the system. As an example, consider a
shower with separate flow control for hot and cold water. This is a MIMO
system since the two inputs, the flow of hot water and the flow of cold water,
are utilized to control the two outputs which are the flow from the tap and
the temperature of the effluent water. Evidently, when changing one of the
inputs, both of the outputs will be affected. This means that there are
significant couplings in the system. In other words, interaction occurs if a
change in one input affects several outputs.
Often, an easy way to control a fairly decoupled MIMO system is to
use a multi-loop strategy, i.e. to separate the control problem into several
single-loop SISO systems and then use conventional SISO control on each
of the loops, see Kinnaert (1995) and Wittenmark et al. (1995). This gives
rise to the pairing problem:
Which input signal should be selected to control which output signal to get
7
the most efficient control with a low degree of interaction?
In real-life applications the considered MIMO system could be rather
complex: In the chemical process industry a complexity of several hundred
control loops is not unusual, see Wittenmark et al. (1995). The proper
pairing selection is thus often not at all obvious. Also, the choice of pairing
is crucial since a bad choice may give unstable systems even though each loop
separately is stable. This problem could arise due to interaction between
the different loops. Generally, the stronger the interactions, the harder
it is to obtain satisfactory control performance using a multi-loop strategy.
Evidently, there is a need for a measure that can both give some advise when
solving the pairing problem and that also quantifies the level of interaction
occurring in the system.
One such measure is the Relative Gain Array (RGA) developed by Bristol
(1966). The RGA considers steady-state properties of the plant and gives a
suggestion on how to solve the pairing problem in the case of a decoupled
(decentralized) control structure. Such structure will be diagonal. It also
indicates which pairings that should be avoided due to possible stability and
performance problems.
Later, a dynamic extension of the RGA was proposed in the literature,
see e.g. Kinnaert (1995) for a survey. With the extension, the RGA could be
used to analyse the considered plant at any frequency but still only at one
single frequency at a time. A recent approach to define a dynamic relative
gain array was made by Mc Avoy et al. (2003). Moreover, the RGA can be
generalized for non-square plants and be employed as a screening tool to get
a suggestion on what inputs or outputs that should be removed in the case
of excess signals, see Skogestad and Postlethwaite (1996).
Over the years, several resembling tools have been developed. One such
example is the Partial Relative Gain (PRG) suggested by Häggblom (1997)
that is intended to handle the pairing problem for larger systems in a more
reliable way than the conventional RGA. Other examples are the µ interaction index (Grosdidier and Morari, 1987) and the Performance Relative Gain
Array (PRGA) (Hovd and Skogestad, 1992). An interesting novel approach
is found in (He and Cai, 2004) where pairings are found by minimizing the
loop interaction energy characterized by the General Interaction (GI) measure. This measure is used in combination with the pairing rules of the
RGA and of the Niederlinski Index (NI) (Niederlinski, 1971). The NI can
be used as an indicator of possible instability issues when solving the pairing
problem. In the Effective RGA (ERGA) proposed by Xiong et al. (2005)
the steady state gain and the bandwidth of the process are utilized to form
a dynamic interaction measure. He et al. (2006) suggest an algorithm for
control structure selection where the ideas by He and Cai (2004) are further
developed. Other examples are given by Kinnaert (1995) where a survey of
interaction measures for MIMO systems can be found.
8
The RGA provides only limited knowledge about when to use multivariable controllers and gives no indication of how to choose multivariable
controller structures. A somewhat different approach for investigating channel interaction was therefore employed by Conley and Salgado (2000) and
Salgado and Conley (2004) when considering observability and controllability Gramians in so called Participation Matrices (PM). In a similar approach Wittenmark and Salgado (2002) introduced the Hankel Interaction
Index Array (HIIA). These Gramian based interaction measures seem to
overcome most of the disadvantages of the RGA. One key property of these
is that the whole frequency range is taken into account in one single measure. Furthermore, these measures seem to give appropriate suggestions for
controller structures both when a decentralized structure is desired as well
as when a full multivariable structure is needed. The use of the system H2
norm as a base for an interaction measure has been proposed by Birk and
Medvedev (2003) as an alternative to the HIIA.
1.1.1
Motivational example
As a first motivational example, consider a system previously analysed by
Goodwin et al. (2005). The system has a transfer function



G(s) = 

−10(s+0.4)
(s+4)(s+1)
2
s+2
−2.1
s+3
0.5
s+1
20(s−0.4)
(s+4)(s+2)
3
s+3
−1
s+1
1
s+2
30(s+0.4)
(s+4)(s+3)
and a steady-state gain of


−1.0000 0.5000 −1.0000
G(0) =  1.0000 −1.0000 0.5000  .
−0.7000 1.0000
1.0000
The interaction measures are:

Λ(G(0)) =
ΣH
=
Φ =
Σ2 =




2.8571 −1.2857 −0.5714
 −2.8571 3.2381
0.6190  ,
1.0000 −0.9524 0.9524


0.1330 0.0324 0.0648
 0.0648 0.2827 0.0324  ,
0.0454 0.0648 0.2798


0.0768 0.0036 0.0144
 0.0144 0.4377 0.0036  ,
0.0071 0.0144 0.4279


0.0915 0.0011 0.0044
 0.0088 0.2992 0.0022 
0.0065 0.0132 0.5732
9
where Λ is the RGA, ΣH is the HIIA, Φ is the PM and finally, Σ2 is an H2
norm based interaction measure. All of these will be defined in Chapter 2.
The aim in this example is to find the decentralized pairing recommendation
so that each input signal is paired uniquely with one output signal. In the
case of the RGA input-output pairings corresponding to elements close to
one should be selected and negative elements should be avoided. The other
of the considered interaction measures recommend the input-output pairings
that result in the largest sum when adding the corresponding elements in the
measure. Evidently, all interaction measures suggest the diagonal pairing:
input 1 – output 1, input 2 – output 2 and input 3 – output 3. However, no
useful pairing information can be found by inspecting G(0) or G(0). This
demonstrates the need of dedicated interaction measures even for pairing
suggestions relevant for operation in steady state. Even though, the considered interaction measures are rather similar in this particular example,
this is not generally the case. Further examples and theoretical differences
between these will be examined in Chapter 2–4.
1.2
Wastewater treatment systems
Until some time during the 19:th century, the activity of man had not affected the environment to any appreciable extent. When the industrial revolution came, a rapidly increased standard of living as well as a substantially
population growth followed. The society became more and more urbanized
and the problem of taking care of the human waste products and waste disposal became a serious (hygienic) problem. The introduction of the water
closet solved the problem locally, but only locally, since the problem was
instead moved to the surrounding environment with an increased load on
the recipients (e.g. lakes and rivers). This could not be handled by the
recipients without heavily disturbed local ecosystems. The degradation of
organic material present in the wastewater, consumes oxygen and the recipient will thus suffer from lack of oxygen after some while. Even if most
of the organic matter is removed before the wastewater reaches the recipient, chemical compounds such as phosphorous and nitrogen are still present,
and may cause eutrophication (i.e. over-fertilization). Eventually, this will
also result in a lack of oxygen. Therefore, the aim of wastewater treatment
should be to remove both the content of organic matter and suspended solids
as well as the content of nitrogen and phosphorous to a reasonable extent.
In the beginning of the 20:th century, the first wastewater treatment
plants were introduced in Sweden. They were simple plants using only a
mechanical treatment step. This step could consist of a grid and a sand
filter to remove larger objects and particles. In the late 1950’s the biological
treatment step, was introduced. Hereby, microorganisms (e.g. bacteria) are
used to remove organic matter present in the incoming wastewater. Later, in
10
the 1970’s, the chemical treatment step, was employed to reduce the content
of phosphorous. Nowadays, the biological step is also utilized to reduce the
content of nitrogen and phosphorous. A general wastewater treatment plant
(WWTP), consisting of the above mentioned steps, is given in Figure 1.1.
The sludge also needs to be treated. The main procedures are depicted
in Figure 1.1. In the thickening procedure, the sludge is concentrated. Then,
the sludge is stabilized in order to reduce odor and pathogenic content. Finally, the moisture content of sludge can be reduced by the use of dewatering.
For a description of how to practically realize these steps, see e.g. Hammer
and Hammer Jr. (2008).
1
Mechanical treatment
2 Biological treatment
Activated sludge
Grid
Sand
filter
Primary
Sedimentation
3 Chemical treatment
Chemicals
Effluent
water
Preciptation
4
Sludge treatment
Dewatering
Dewatered sludge
Sludge
thickening Stabilization
Supernatants + Backwashing
Figure 1.1: A general WWTP (Kommunförbundet, 1988).
In the complex process of wastewater treatment, many different causeeffect relationships exist, and therefore, there are many possible choices of
input and output signals, see Olsson and Jeppsson (1994). This makes
the WWTP models particularly interesting to study with respect to the
interactions present and the selection of proper control structures.
When treating wastewater, the aim is to reduce as much as possible
of the undesired constituents such as organic matter, nitrogen and phosphorous. This is commonly done using wastewater treatment plants. In a
WWTP several biological processes occur simultaneously. These processes
need to be properly controlled in order to maintain the concentrations of
undesired constituents in the outlet water within the legislated limits. As
the public awareness of environmental issues increases, the environmental
legislation becomes stricter, and thus, the requirements on WWTPs become
even harder to fulfill. The used control strategies need then to be as efficient as possible, see e.g. Olsson and Newell (1999). Therefore, models of
the WWTP processes are interesting to study with respect to the choice of
e.g. control structure. An example of such models are the bioreactor models.
From a theoretical point of view, the bioreactor models are non-linear
11
multivariable systems that may contain a significant degree of coupling.
Hence, this also gives an interesting opportunity to test the performance
of the methods for input-output pairing selection mentioned in the previous
section. The aim of Section 1.2 is to give a brief description of the bioreactor
models that will be analysed in the forthcoming chapters.
1.2.1
The activated sludge process (ASP)
The biological treatment step can be realized in several different ways. One
of the most common is the activated sludge process where activated sludge,
i.e. microorganisms (mainly bacteria), is employed to degrade (i.e. oxidize)
organic material. The basic set-up consists of an aerated basin where oxygen
is added by blowing air into the water, and a settler tank, see Figure 1.2.
In the aerated basin, the bacteria degrade the incoming organic material
while consuming oxygen. In this way the microorganisms fulfill their need
of energy and as a result bacterial growth will occur. Together with decayed
microorganisms and other particulate material, the living microorganisms
form sludge. To separate the sludge from the purified water a settler, where
the sludge settles, can be used directly after the aerated tank. Since the
amount of microorganisms needs to be kept at a high level, some sludge
is recirculated as shown in Figure 1.2, while the rest is removed as excess
sludge. With the excess sludge, some nitrogen (and phosphorus) is removed,
but still far too much remains.
Influent
Effluent
Aerobic
Settler
Sludge recirculation
Excess sludge
Figure 1.2: A basic activated sludge process with an aerated basin and a
settler.
However, if the activated sludge process is extended to consist of both
aerated and non-aerated (anoxic) basins, then bacteria may be employed for
efficient nitrogen removal. In the aerated basins, bacteria oxidize ammonium
to nitrate in a two-step process called nitrification:
−
+
NH+
4 + 1.5O2 → NO2 + H2 O + 2H ,
−
NO−
2 + 0.5O2 → NO3 .
For these processes to occur, the concentration of dissolved oxygen (DO)
12
must be sufficiently high and a long sludge age (the average time each particle stays in the system) is required due to slow bacteria growth.
In the anoxic tanks, another type of bacteria is employed in the denitrification process, described by
+
2NO−
3 + 2H → N2 (g) + H2 O + 2.5O2
i.e., the bacteria convert nitrate into nitrogen gas using the oxygen in the
nitrate ions. However, no dissolved oxygen should be present for this process
to take place, instead, a sufficient amount of readily biodegradable substrate
is needed. Hence, together, nitrification and denitrification convert ammonium into nitrogen gas which is harmless to the environment. For further
descriptions of these processes, see Henze et al. (1995).
Nitrogen removal can be performed in several different types of WWTPs.
One of the most popular is the pre-denitrification system (ibid.). In this
design, the anoxic tanks are placed before the aerated basins, and thus,
denitrification is performed before the nitrification process, see Figure 1.3.
Influent
Effluent
Anoxic
Anoxic
Aerobic Aerobic Aerobic
Internal recirculation
Settler
Sludge recirculation
Excess sludge
Figure 1.3: An activated sludge process configured for nitrogen removal (predenitrification).
To supply the denitrification process with nitrate, there is a feedback
flow from the last tank as shown in Figure 1.3. In some cases, when the
influent water has a low content of carbon, the bacteria in the anoxic tank
need to be fed with an external carbon source. For this purpose, methanol
or ethanol is often used.
For a further discussion about the ASP, see e.g. Olsson and Newell (1999)
and Hammer and Hammer Jr. (2008).
1.2.2
The benchmark model BSM1
The comparison between different control strategies for a WWTP is often
difficult due to the variable influent conditions and the high complexity
of a WWTP. Therefore, to enable objective comparisons between different
control strategies, a simulation benchmark activated sludge process, Benchmark Simulation Model No.1 (BSM1), has been developed by the COST 682
Working Group No.2, see Copp (2002) and IWA (November 19, 2007).
13
In the BSM1 a typical activated sludge process with pre-denitrification is
implemented. It consists of five biological reactor tanks configured in-series.
The first two tanks have a volume of 1000 m3 each, and are anoxic and
assumed to be fully mixed. The remaining three tanks are aerated and have
a volume of 1333 m3 each. All biological reactors are modelled according to
the ASM1 model. Finally, there is a secondary settler modelled using the
double-exponential settling velocity function of Takács et al. (1991).
To get an objective view of the performance of the applied control strategy, it is important to run the BSM1 simulation with different influent disturbances. Therefore, influent input files for three different weather conditions – dry, stormy and rainy weather – are available together with the
benchmark implementation. A number of different performance criteria are
defined, such as various quality indices and formulas for calculating different
operational costs.
1.2.3
Control of WWTPs
As previously stated, WWTPs may be seen as complex multivariable systems. Therefore, to obtain satisfactory control performance, it is often necessary to use more advanced control strategies. However, since wastewater
treatment traditionally has been seen as non-productive compared to the
industry, the extra investments needed to employ such advanced control
strategies have been hard to justify economically. Nowadays, as the effluent demands get tighter, the interest for more advanced control strategies is
again awakening, see Olsson and Newell (1999).
The plant has to be run economically and at the same time the discharges to the recipient should be kept at a low level. The control problem
is hence twofold. The economical aspect involves minimizing operational
costs such as pumping energy, aeration energy and dosage of different chemicals. Consequently, the main problem is how to keep the effluent discharges
below a certain pre-specified limit to the lowest possible cost (ibid.). One
way of solving this conflict of interest is to design the control algorithms in
such a way that the overall operational costs are minimized. To make sure
that also the wastewater treatment performance demands are fulfilled, the
effluent discharges can be economically penalized. The corresponding cost
can then be included together with the actual costs (energy and chemicals)
in the calculation of the overall cost.
Control handles for nitrogen removal
In the nitrogen removal process, there are several variables that can be
used as actuators, or control handles, to control the outputs. In a predenitrification system, there are five main control handles, as stated by
Ingildsen (2002):
14
1. The airflow rate (in the aerated compartments);
2. The internal recirculation flow rate;
3. The external carbon dosage;
4. The sludge outtake flow rate (excess sludge);
5. The sludge recirculation flow rate.
In this thesis, only the three first of these are considered. The last two
control handles are described by for example Yuan et al. (2001) and Yuan
et al. (2002).
The first control handle, the airflow rate, is employed to affect the DO
concentration in the aerated compartments. Hereby, the performance of the
autotrophic nitrification bacteria will be influenced. Most common today
is to control the airflow rate to maintain a specific DO level. Another way
is to make use of online-measurements of the ammonium concentration in
the last aerated compartment, and let these control the time-varying DO
set point, see e.g. Lindberg (1997).
The internal recirculation flow rate affects the supply of nitrate for the
denitrification process but also the DO concentration in the anoxic compartments since some DO may be transported from the last aerated compartment. The DO transportation between the processes, can however, be
reduced by introducing an anoxic tank after the last aerated basin.
External carbon dosage can be applied when the influent water does
not have enough readily biodegradable substrate to feed the denitrification
bacteria.
Controlled output signals for nitrogen removal
The primary outputs from a WWTP are the effluent ammonium concentration, the organic matter, the nitrate concentration and the suspended
solids, see Ingildsen (2002). For a more thorough discussion on cause-effect
relationships in activated sludge plants, see Olsson and Jeppsson (1994).
1.3
Thesis outline
Chapter 2
In Chapter 2 different interaction measures are reviewed and compared for
some MIMO plants. In addition a simulation study is performed where the
influence of a time delay on the coupling is examined. State controllability
and output controllability are also discussed and further motivations for
incorporating the concept of output controllability in an interaction measure
are given. A H2 norm based interaction measure is investigated.
15
Chapter 3
In Chapter 3 the interactions in a multivariable ASP model configured for nitrogen removal are studied. The RGA and the HIIA are utilized to quantify
the degree of coupling present in the system. Both the nitrification and the
denitrification process are studied since the output signals (the controlled
signals) are the effluent concentration of ammonium and the effluent concentration of nitrate. The input signals (control handles) are the dissolved
oxygen concentration set point in the aerobic compartment and the internal
recirculation flow rate.
The material is based on:
Halvarsson, B., P. Samuelsson and B. Carlsson (2005). Applications of
Coupling Analysis on Bioreactor Models. In: Proceedings of the 16th
IFAC World Congress, Prague, Czech Republic, July 4-8.
Chapter 4
Chapter 4 once again considers the interactions present in an ASP. Here, the
focus is on controlling the denitrification process when an external carbon
source is added. Thus, one of the two considered control handles (input
signals) is the readily biodegradable organic substrate in the influent water
(which has the same influence as an external carbon source would have). The
other input signal is the internal recirculation flow rate. The output signals
(controlled signals) are the nitrate concentration in the anoxic compartment
and the nitrate concentration in the effluent. The model is analysed using
the RGA and the HIIA. The results are discussed from a process knowledge
point of view, and are also illustrated with some control experiments.
The chapter is based on:
Samuelsson, P., B. Halvarsson and B. Carlsson (2005). Interaction
Analysis and Control Structure Selection in a Wastewater Treatment
Plant Model. IEEE Transactions on Control Systems Technology 13(6).
Samuelsson, P., B. Halvarsson and B. Carlsson (2004). Analysis of
the Input-Output Couplings in a Wastewater Treatment Plant Model.
Technical Report 2004-014. Division of Systems and Control, Department of Information Technology, Uppsala University, Uppsala, Sweden.
Chapter 5
In this chapter, the focus is on finding optimal set-points and cost minimizing
control strategies for the activated sludge process. Both the denitrification
and the nitrification process are considered. In order to compare different
16
criterion functions, simulations utilizing the COST/IWA simulation benchmark (BSM1) are considered. By means of operational maps the results
are visualized. It is also discussed how efficient control strategies may be
accomplished.
The material is based on:
Halvarsson, B. and B. Carlsson (2006). Economic Efficient Operation
of a Predenitrifying Activated Sludge Process. HIPCON Report number HIP06-86-v1-R Deliverable D6.5. Uppsala University, Uppsala,
Sweden.1
which is an extended version of:
Samuelsson, P., B. Halvarsson and B. Carlsson (2007). Cost-Efficient
Operation of a Denitrifying Activated Sludge Process. Water Research
41(2007) 2325-2332.
Samuelsson, P., B. Halvarsson and B. Carlsson (2005). Cost Efficient Operation of a Denitrifying Activated Sludge Plant – An Initial
Study. Technical report 2005-010. Division of Systems and Control,
Department of Information Technology, Uppsala University, Uppsala,
Sweden.
In these two references only the denitrification process is studied.
Chapter 6
In the concluding chapter the influence of the aeration on the efficiency of the
nitrogen removal in an ASP is studied. Different strategies for controlling
the DO set point as well as the aerated volume are compared in terms of
efficiency in a simulation study.
Chapter 6 is based on:
Halvarsson, B. and B. Carlsson (2006). Aeration Volume control in
an activated sludge process – Discussion of some strategies involving
on-line ammonia measurements. HIPCON Report number HIP06-86v1-R Deliverable D6.5. Uppsala University, Uppsala, Sweden.1
1
This paper is an internal EU project report which is available from the author.
17
18
Chapter 2
Controllability and
Interaction Measures
In this chapter the concept of controllability is discussed and different interaction measures are reviewed and compared. In particular, the two Gramian
based interaction measures the Hankel Interaction Index Array (HIIA) and
the Participation Mtarix (PM) are considered. Moreover, motivations for
incorporating the concept of output controllability in an interaction measure
are given and a H2 norm based interaction measure is investigated.
2.1
Introduction
There are today several different measures for quantifying the level of inputoutput interactions in multivariable systems. The perhaps most commonly
used is the Relative Gain Array (RGA) introduced by Bristol (1966). The
RGA is a measure that can be employed in order to decide a suitable inputoutput pairing when applying a decentralized control structure. It can also
be used to decide whether a certain pairing should be avoided. This measure, however, suffers from some major disadvantages. For instance it only
considers the plant in one frequency at the time and it often provides limited knowledge about when to use multivariable controllers. Neither is the
RGA able to give advice on how to select an appropriate multivariable controller structure. The RGA is also unable to suggest a proper pairing in the
case of plants with triangular structure or large off-diagonal elements (this
particular situation is further investigated in Chapter 3).
A different approach for investigating channel interaction was employed
by Conley and Salgado (2000) when considering observability and controllability Gramians in so called Participation Matrices (PM). In a similar
approach Wittenmark and Salgado (2002) introduced the Hankel Interaction Index Array (HIIA). These Gramian based interaction measures seem
to overcome most of the disadvantages of the RGA. One key property of
19
these is that the whole frequency range is taken into account in one single
measure. Furthermore, these measures seem to give appropriate suggestions
for controller structures both when a decentralized structure is desired and
when a full multivariable structure is needed. For applications and comparisons between the RGA and various types of Gramian based interaction
measures, see for instance Salgado and Conley (2004), Birk and Medvedev
(2003), Samuelsson et al. (2005c) and Halvarsson et al. (2005).
The use of the system H2 norm as a base for an interaction measure has
been proposed by Birk and Medvedev (2003) as an alternative to the HIIA.
In HIIA the Hankel norm forms the basis. In this chapter this H2 norm based
interaction measure is investigated. Further motivations for incorporating
the concept of output controllability in an interaction measure are also given.
This chapter is organized in the following way: Section 2.2 gives a brief
description of the systems that will be analysed and some general assumptions. In Section 2.3 state controllability and output controllability are
defined. Section 2.4 introduces the reader to the RGA. Section 2.5 presents
the Gramian based interaction measures the Hankel Interaction Index Array
and the Participation Matrix and their theoretical foundations. Section 2.6
defines the H2 norm based interaction measure and investigates its relation
to the concept of output controllability. In Section 2.7 different interaction measures are compared in the analysis of the interactions present in
some MIMO systems. A simulation study is also performed. Finally, the
conclusions are drawn in Section 2.8.
2.2
Systems description
Consider a continuous-time linear time-invariant system, with inputs at time
t given by the N × 1 vector u(t) and outputs at time t given by the p × 1
vector y(t). The system can be described as a state-space realization
ẋ(t) = Ax(t) + Bu(t),
y(t) = Cx(t) + Du(t)
(2.1)
where A, B, C and D are matrices of dimension N × N , N × q, p × N and
p × q, respectively. x(t) is the state vector.
Furthermore, a discrete-time system
x(t + 1) = Ax(t) + Bu(t),
y(t) = Cx(t) + Du(t)
(2.2)
will be considered as well. Note that (A, B, C, D) both denote the continuoustime system matrices and the discrete-time system matrices. These do not
generally coincide; what quadruple of matrices that are referred to will be
clear from the context.
20
2.3
2.3.1
Controllability
State controllability for continuous-time systems
Most often, the term controllability refers to the property of a system as
being state controllable. The concepts of state controllability and state
observability were introduced by Kalman, see for example Kalman et al.
(1963), Kreindler and Sarachik (1964) and the references therein.
With an initial state x(t0 ) and an input u(t), the solution of (2.1) for
t ≥ t0 is given by
Z t
x(t) = eA(t−t0 ) x(t0 ) +
eA(t−τ ) Bu(τ )dτ.
(2.3)
t0
This is a standard result found in many text books such as (Skogestad and
Postlethwaite, 1996; Zhou et al., 1996). Since the system is time-invariant
t0 can be set to 0. A system with an arbitrary initial state x(0) = x0 is
said to be state controllable if there exists a piecewise continuous input u(t)
such that x(t1 ) = x1 for any final state x1 and t1 > 0. Equivalently, a state
controllable system can be transferred from any initial state x(t0 ) to any
final state x(t1 ) in finite time. It can be verified using (2.3) that one input
that satisfies this criterion is given by (ibid.)
u(t) = −B T eA
T (t −t)
1
Wc (t1 )−1 (eAt1 x0 − x1 )
where Wc (t) is a Gramian matrix defined as
Z t
T
Wc (t) =
eAτ BB T eA τ dτ.
(2.4)
(2.5)
0
Clearly, for the solution in (2.4) to exist, the inverse of Wc (t) needs to
exist, i.e. Wc (t) must have full rank for every t > 0. For a stable timeinvariant system it is enough to require Wc (∞) to have full rank. Hence,
state controllability can be investigated by considering the controllability
Gramian, P , defined for stable time-invariant systems as
Z ∞
T
P ,
eAτ BB T eA τ dτ.
(2.6)
0
If P has full rank the system is state controllable. Similarly, a stable system
will be state observable if the observability Gramian, Q, defined as
Z ∞
T
Q,
eA τ C T CeAτ dτ
(2.7)
0
has full rank. These Gramians can be obtained by solving the following
continuous-time Lyapunov equations (Skogestad and Postlethwaite, 1996):
AP + P AT + BB T = 0,
21
(2.8a)
AT Q + QA + C T C = 0.
(2.8b)
The rank of P is the dimension of the controllable subspace corresponding
to the given system, and correspondingly, the rank of Q is the dimension of
the observable subspace of the same system.
State controllability and state observability can also be examined by
considering the matrices1
Wc , [B
AB
...
AN −1 B],
(2.9a)

C
CA
..
.

(2.9b)




Wo , 
.


N
−1
CA
The system (A, B) is then state controllable if Wc has full rank N where N
is the number of states. Similarly, the system (A, C) is state observable if
Wo has full rank N .
Even though a system is state controllable, it should be noted that there
is no guarantee that the system can remain in its final state x1 as t → ∞.
Furthermore, nothing is said about the behaviour of the required inputs.
These can both be very large and change suddenly. Therefore, state controllability is rather a result of theoretical interest than a result of practical
importance.
2.3.2
State controllability for discrete-time systems
The discrete-time case can be treated similarly. The discrete controllability
Gramian is given by (Weber, 1994)
P = Wc WcT
(2.10)
and the discrete observability Gramian by
Q = WoT Wo .
(2.11)
Similarly to the continuous-time case, these Gramians can also be obtained
as the solutions to the (discrete-time) Lyapunov equations
AP AT − P + BB T = 0,
1
(2.12a)
It can be verified that the controllable states can be expressed as linear combinations
of the matrices B, AB, . . . , An−1 B by considering the solution to (2.1) given in (2.3) (let
x0 = 0) and expressing eAt as a power series and using the Cayley-Hamilton theorem.
This motivates the introduction of Wc . The Cayley-Hamilton theorem says that every
quadratic matrix satisfies its own characteristic equation. For details, see for instance
Glad and Ljung (1989) where also further motivations for the definition of Wo are given.
22
AT QA − Q + C T C = 0.
(2.12b)
Once again, the same symbols are used for both the continuous-time quantities and the discrete-time counterparts. Note also that in the continuoustime case, P and Q cannot be obtained from Wc and Wo as in (2.10) and
(2.11) for discrete-time systems.
2.3.3
Output controllability
Whereas state controllability considers the ability of affecting the states
of a given system by manipulating the inputs, output controllability rather
considers the situation of affecting the outputs by means of manipulating the
inputs. In practical control problems it is often more relevant to be able to
control the outputs rather than the states (see e.g. Kreindler and Sarachik
(1964)). State controllability is “neither necessary nor sufficient” to be able
to control the outputs as pointed out by Kreindler and Sarachik (1964).
According to Skogestad and Postlethwaite (1996), state controllability is
rather a “system theoretical concept” and it “does not imply that the system
is controllable from a practical point of view.” For this reason the concept
of output controllability was introduced.
Kreindler and Sarachik (1964) discuss time-varying plants of the form
given in (2.1) and defines a plant as being “completely output-controllable on
[to , tf ] if for given t0 and tf any final output y(tf ) can be attained starting
with arbitrary initial conditions in the plant at t = t0 .” For a plant without
a direct term, i.e. D(t) = 0, this holds if and only if the Gramian
Poc (t0 , tf ) ,
Z
tf
t0
Hy (tf , τ )HyT (tf , τ )dτ
(2.13)
is non-singular (Kreindler and Sarachik, 1964) where Hy (t, τ ) is the impulse response matrix (Skogestad and Postlethwaite, 1996), For linear timeinvariant stable plants with t0 set to 0 the Gramian in (2.13) transforms to
the output controllability Gramian given by
Z ∞
T
Poc =
CeAτ BB T eA τ C T dτ
0
Z ∞
T
= C
eAτ BB T eA τ dτ C T
0
= CP C T .
(2.14)
For plants including direct transmission (i.e. D 6= 0) the matrix DDT has
to be added to the Gramian in (2.14) for the output controllability criteria
to be valid.
In contrast to the state controllability Gramian, P , Poc is independent of
the selected state-space realization. To see this, change the state coordinates
23
by multiplying the state vector x(t) with a linear non-singular transformation matrix T . This is a similar transformation that transforms the state
vector x(t) to z(t) = T x(t). The plant can now be described by
ż(t) = T AT −1 z(t) + T Bu(t),
y(t) = CT −1 z(t) + Du(t).
(2.15)
For the new realization, the output controllability Gramian becomes
Z ∞
−1
−1 ∗ ∗ ∗
′
Poc =
CT −1 eT AT t T BB ∗ T ∗ e(T ) A T t (T −1 )∗ C ∗ dt
Z0 ∞
∗
=
CT −1 T eAt T −1 T BB ∗ T ∗ (T −1 )∗ eA t T ∗ (T −1 )∗ C ∗ dt
0
Z ∞
∗
= C
eAt BB ∗ eA t dt C ∗
0
= CP C ∗
= Poc
(2.16)
where it has been utilized that
n
o
−1
A2 t2
eT AT t = eAt = I + At +
+ . . . = . . . = T eAt T −1
2!
and that the plant is assumed to be time-invariant so that C is independent
′ = P
of time. Clearly, Poc
oc and thus Poc is independent of the selected
state-space realization.
2.4
The Relative Gain Array (RGA)
The static RGA for a quadratic plant is given by
RGA(G) = G(0). ∗ (G(0)−1 )T
(2.17)
where G(0) is the steady-state transfer function matrix and “.∗” denotes the
Hadamard or Schur product (i.e. elementwise multiplication). Each element
in the RGA can be regarded as the quotient between the open-loop gain and
the closed-loop gain. Hence, the RGA element (i, j) is the quotient between
the gain in the loop between input j and output i when all other loops are
open and the gain in the same loop when all other loops are closed. For a full
derivation of the RGA, see e.g. Bristol (1966), Kinnaert (1995) or Skogestad
and Postlethwaite (1996).
In the case of a 2×2 system, the following RGA matrix is obtained:
λ
1−λ
RGA(G) =
.
(2.18)
1−λ
λ
Depending on the value of λ, five different cases occur (Kinnaert, 1995):
24
λ = 1: This is the ideal case when no interaction between the loops is
present. The pairing should be along the diagonal, i.e. u1 − y1 and
u2 − y2 ;
λ = 0: This is the same situation as above, except that now the suggested
pairing is along the anti-diagonal, i.e. u1 − y2 , u2 − y1
0 < λ < 1: This case is not desirable since the gain increases (i.e. ĝij increases) when the loops are closed, hence, there is interaction;
λ > 1: Now, the gain decreases when the loops are closed. This situation is
therefore also undesirable.
λ < 0: This situation corresponds to the worst case scenario since now, even
the sign changes when the loops are closed and this is highly undesirable.
The conclusion is that u1 should only be paired with y1 when λ > 0.5,
otherwise it should be paired with y2 . For the higher-dimensional case, the
rule should be to choose pairings that have an RGA-element close to one.
Negative pairings should definitely be avoided.
2.5
2.5.1
Gramian based interaction measures
The Hankel norm
The controllability and observability Gramians as defined in (2.6) and (2.7)
can be seen as measures of how hard it is to control and to observe the states
of the given system. Unfortunately, both of these Gramians depend on the
chosen state-space realization. However, as can be verified, the eigenvalues
of the product of these will not.
The Hankel norm for a system with transfer function G (continuous-time
or discrete-time) can be calculated as
p
kGkH = λmax (P Q) = σ1H
(2.19)
where σ1H is the maximum Hankel singular value (HSV). Clearly, this measure is invariant with respect to the state-space realization and it is therefore
well suited as a combined measure for controllability and observability. In
fact, the Hankel singular values can be interpreted as a measure of the joint
controllability and observability of the states of the considered system, see
for instance Farsangi et al. (2004), Skogestad and Postlethwaite (1996) and
Lu and Balas (1998). Furthermore, the HSV:s of G can be regarded as
measures of the gain between past inputs and future outputs since these
are the singular values of the Hankel matrix (defined below) for discretetime systems, or equivalently, for the Hankel operator (defined below) for
25
continuous-time systems (Zhou et al., 1996; Skogestad and Postlethwaite,
1996; Wilson, 1989; Antoulas, 2001; Weber, 1994; Glover, 1984; Wittenmark
and Salgado, 2002). To see this, consider the discrete-time time-invariant
system given in (2.2) and let D = 0. Assume starting from zero initial state,
x(−L) = 0, the influence of the L past inputs on the state x(0) is given by
(Weber, 1994; Glover, 1984)
x(0) = [B
AB

u(−1)
u(−2)
..
.


... AL−1 B] 

u(−L)






 = Wc 


u(−1)
u(−2)
..
.
u(−L)





(2.20)
and the influence of the initial state x(0) on the L future outputs is given
by (Weber, 1994; Glover, 1984)






y(0)
y(1)
..
.
y(L − 1)

 
 
=
 

C
CA
..
.
CAL−1


 x(0) = Wo x(0)

(2.21)
where it is assumed that u(t) = 0 for t ≥ 0. When L = N , Wc and Wo
are the controllability and observability matrices, respectively. For L > N
these are the extended controllability matrix and the extended observability
matrix. To be able to reconstruct all of the states in the state vector at time
0, i.e. x(0), from the past inputs according to (2.20), Wc must have full rank
N . Similarly, Wo must have full rank N so that the outputs can be found
from (2.21). For a derivation of (2.20) and (2.21), see Weber (1994).
Combining (2.20) and (2.21) the result is the following expression that
links the past inputs to the future outputs via the state x(0) at time zero
(Weber, 1994; Antoulas, 2001)





y(0)
y(1)
..
.
y(L − 1)






=
W
x(0)
=
W
W

o
o c


u(−1)
u(−2)
..
.
u(−L)






=
Γ




u(−1)
u(−2)
..
.
u(−L)



.

(2.22)
Γ is the Hankel matrix which in the considered time-invariant case is defined
as (Antoulas, 2001; Weber, 1994)



Γ=

S1
S2
..
.
S2
S3
..
.
...
...
SL
SL+1
..
.
SL SL+1 . . . S2L−1
26





(2.23)
where {Sk } are the Markov2 parameters of the system. For multiple-input
multiple-output (MIMO) systems the Markov parameters are matrices and
consequently, the Hankel matrix is a block matrix. The impulse response in
the discrete-time case for the system given in (2.2) with D = 0 is given by
(Antoulas, 2001)
CAt−1 B t > 0
h(t) =
(2.24)
0
t < 0.
For this reason the Hankel matrix can be expressed as

CB
CAB . . . CAL−1 B
 CAB
CA2 B . . .
CAL B

Γ=
..
..
..

.
.
.
CAL−1 B CAL B . . . CA2L−2 B



.

(2.25)
Clearly, it follows that Γ = Wo Wc as stated in (2.22). The Hankel singular
values equal the non-zero singular values of the Hankel matrix.
In the continuous-time case, the counterpart to the Hankel matrix is the
Hankel operator Γ given by (see e.g. Antoulas (2001))
Z ∞
y(t) = (Γu)(t) =
g(t + τ )u(−τ )dτ
t≥0
(2.26)
0
where g(t) is the continuous-time impulse response matrix given by
0
t<0
g(t) =
(2.27)
At
Ce B
t≥0
when D in (2.2) is assumed to be 0.
Similarly to the discrete-time case, the Hankel operator relates the past
inputs to the future outputs. The Hankel singular values are the same as
the singular values of the Hankel operator. However, note that these do not
coincide with the singular values of the corresponding Markov parameters
as in the discrete-time case.
The Hankel norm can also be regarded as an induced norm3 . In fact, it
is the induced operator norm of the Hankel operator. For a stable system
2
A strictly proper continuous transfer
function G(s) can be expressed as a power series
P∞
−k
in the Laplace variable s as G(s) =
where {gk } = {Sk } are the Markov
k=1 gk s
parameters. In the discrete-time case, the Markov parameters are the impulse response.
(Weber, 1994)
3
Let || · || be some vector norm. Then the norm
||A||i = max
x6=0
||Ax||
||x||
is said to be an induced norm of the current vector norm. ||A||i can be interpreted as the
maximum gain for all possible input directions of a system with amplification A and input
x. See Skogestad and Postlethwaite (1996), Zhou et al. (1996) and Horn and Johnson
(1985) for a more detailed description of induced norms.
27
G(s), the Hankel norm is given by (Skogestad and Postlethwaite, 1996)
qR
∞
2
0 ||y(τ )||2 dτ
||G(s)||H = max qR
.
(2.28)
0
u(t)
2 dτ
||u(τ
)||
2
−∞
This expression can be interpreted as follows (Skogestad and Postlethwaite,
1996, p.155): Apply an input u(t) up to time t = 0 and then measure the
resulting output y(t) for t > 0 and maximize the 2-norm ratio between these
signals.
For a more thorough description of the continuous-time Hankel operator,
see for example Glover (1984), Zhou et al. (1996), Antoulas (2001), Wilson
(1989), Birk and Medvedev (2003) and Weber (1994).
2.5.2
Energy interpretations of the controllability and observability Gramians for discrete-time systems
The controllability and observability Gramians can also be interpreted in
terms of energy (Weber, 1994). One way of expressing the energy in a signal
is to calculate its square-sum. From Equation (2.21) the energy released
from a given state x(0) is
L−1
X
|y(n)|2 = xT (0)WoT Wo x(0) = xT (0)Qx(0)
(2.29)
n=0
where Q is the discrete-time observability Gramian. Hence, a small Q
(i.e. the eigenvalues of Q are small) corresponding to low observability implies that the state variables release a small amount of energy in the outputs.
Similarly, the controllability Gramian may be seen as a measure of the
amount of energy that is needed in the inputs to obtain a given state x(0).
This energy can be expressed as
L−1
X
|u(−n)|2 = . . . = xT (0)P −1 x(0).
(2.30)
n=0
If the plant is hard to control, P will have small eigenvalues and the eigenvalues of P −1 will be large. Therefore a large amount of energy is needed
in the inputs to reach the desired state x(0). Similar interpretations can be
made in the continuous-time case but are omitted here. For further details
see Weber (1994) and Glover (1984).
2.5.3
The Hankel Interaction Index Array (HIIA)
A stable MIMO system represented by (A, B, C, 0) can be split into fundamental SISO subsystems (A, Bj , Ci , 0) with one input uj and one output yi
28
each, where Bj is the j:th column in B, Ci is the i:th row in C (Conley
and Salgado, 2000; Salgado and Conley, 2004). For each of these, the controllability and the observability Gramians can be calculated. Furthermore,
the controllability and observability Gramians for the full system will be the
sum of the Gramians for the subsystems. If the Hankel norm is calculated
for each fundamental subsystem and arranged in a matrix Σ̃H given by
[Σ̃H ]ij = kGij kH
(2.31)
this matrix can be used as an interaction measure. A normalized version
is the Hankel Interaction Index Array (HIIA) proposed by Wittenmark and
Salgado (2002):
kGij kH
[ΣH ]ij = P
.
(2.32)
kl kGkl kH
With the normalization, the sum of the elements in ΣH is one. The larger
the element, the larger the impact of the corresponding input signal on the
specific output signal. Hence, expected performance for different controller
structures can be compared by summing the corresponding elements in ΣH .
Clearly, due to the normalization, the aim is to find the simplest controller
structure that corresponds to a sum as near one as possible. Of course, a big
difficulty could be to decide whether an entry in the HIIA matrix is large
enough to be relevant or not, and there are currently no clear rules for this.
If the intention is to find a decentralized controller, the HIIA can be used
and interpreted in a similar way to the RGA.
When Gij = 0 the Gramian product, P (j) Q(i) , will be zero and so will the
corresponding element in the matrix ΣH . This implies that the structure of
ΣH will be the same as the structure of G and thus, non-diagonal elements
will not be hidden as in the case of the RGA (see for instance Halvarsson
et al. (2005) or Chapter 3). Hence, the HIIA can also be used to evaluate
other controller structures than just the diagonal, decentralized, ones.
2.5.4
The Participation Matrix (PM)
The Hankel norm is given by the largest HSV (see Section 2.5.1). For elementary (SISO) subsystems with only one HSV this is no issue. However,
for subsystems with several HSV:s it can be argued that a more relevant
way of quantifying the interactions is to take into account all of the HSV:s,
at least if there are several HSV:s that are of magnitudes close to the maximum HSV. One way of doing this is to calculate the trace of the Hankel
matrix (for discrete-time systems) or Hankel operator (for continuous-time
systems) – or equivalently of the Gramian product P Q. This is what is
done in the participation matrix (PM) approach, proposed by Conley and
Salgado (2000). Each element in the PM is defined as
φij =
tr(Pj Qi )
tr(P Q)
29
(2.33)
where tr denotes the trace. tr(Pj Qi ) is then the sum of the squared HSV:s
of the subsystem with input uj and output yi . The measure tr(P Q) is,
however, in general not an induced norm such as the Hankel norm. Only
when the system has rank one (so that only one eigenvalue exists) tr(P Q)
is an induced norm (it then coincides with the Hankel norm). Note that
tr(P Q) equals the sum of all tr(Pj Qi ), i.e.
X
tr(P Q) =
tr(Pj Qi ).
(2.34)
i,j
See Salgado and Conley (2004) and Salgado and Oyarzún (2005) for a further
discussion of PM theory and properties.
2.5.5
The selection of proper scaling
All of the considered Gramian based interaction measures depend on the
selected scaling of the system. This means that some effort must be spent
on finding proper scaling matrices. Salgado and Conley (2004) deal with this
issue by normalizing the ranges for the considered signals. However, what
seems to matter is that the scaled system has a fairly low condition number.
As a guidance what fairly low means, the minimized condition number (see
Appendix A) can be of interest.
For instance, in the interaction studies of bioreactor models performed
by Halvarsson et al. (2005) and Samuelsson et al. (2005c) the scaling matrices were selected so that the maximum deviations from the average point of
the considered signals lie in the interval [-1,1]. This scaling procedure significantly reduced the steady state condition number for the plants (i.e. the
condition number for G(0)): from between 5046 and 2.4·106 for the different
operating points to between 7 and 95, and from 2145 to 6.0, respectively.
The minimized condition numbers were 1 and 2.4.
2.6
An interaction measure based on the H2 norm
Birk and Medvedev (2003) suggest the use of the H2 norm and the H∞ norm
as bases for new interaction measures. The proposed interaction quantifiers
share the same form as the HIIA given in (2.32) but with the use of the H2
norm and the H∞ norm instead of the Hankel norm.
In this section the H2 norm based interaction measure proposed by Birk
and Medvedev (2003) will be defined, properties of the H2 norm will be
reviewed and some interpretations of the H2 norm will be given. Finally
some properties of the H2 norm based interaction measure will be derived.
2.6.1
The Σ2 interaction measure
Birk and Medvedev (2003) suggest a new interaction measure, here denoted
30
Σ2 , similar to the HIIA but with the Hankel norm interchanged by the H2
norm, i.e.
kGij k2
[Σ2 ]ij = P
.
(2.35)
kl kGkl k2
This measure is normalized in the same way as the HIIA and the PM and
should be used in the same manner as these to analyse the interactions
present in MIMO systems.
2.6.2
The H2 norm
The system H2 norm for a stable and strictly proper (i.e. D = 0) system
with transfer function G(s) is given by (Skogestad and Postlethwaite, 1996)
s Z
∞
1
||G(s)||2 =
tr G∗ (jω)G(jω) dω.
(2.36)
2π −∞
By the use of Parseval’s relation, the above equation can be expressed as
(ibid.)
sZ
∞ tr gT (τ )g(τ ) dτ
||G(s)||2 = ||g(t)||2 =
0
v
uX Z
u
= t
i,j
∞
|gij (τ )|2 dτ
(2.37)
0
where g is the impulse response matrix. Hence, the H2 norm can be interpreted as the energy of the impulse response, see for example Zuo and
Nayfeh (2003) and Zhou et al. (1996). Furthermore, the H2 norm of a given
stable system can be seen as the sum of the H2 norm of the outputs that are
produced if a unit impulse is applied to each input, one after another. This
interpretation follows from (2.37) (Skogestad and Postlethwaite, 1996).
For a SISO system (2.36) becomes
s Z
∞
1
|G(jω)|2 dω
(2.38)
||G(s)||2 =
2π −∞
and hence, the H2 norm is proportional to the integral of the magnitudes in
the Bode diagram. Clearly, the H2 norm can be regarded as a measure of
energy.
In the case of (continuous) unit variance white noise input the H2 norm
is the power, or root-mean-square (RMS), of the output signal y(t). To see
this, consider the power semi-norm of y(t) given by (Zhou et al., 1996)
Z T
1
||y||2RM S = lim
||y(τ )||2 dτ = tr Ryy (0)
(2.39)
T →∞ 2T −T
31
where Ryy is the autocorrelation matrix of y(t) defined as
Z T
1
Ryy (τ ) = lim
y(t + τ )y ∗ (t)dt
T →∞ 2T −T
(2.40)
where y ∗ (t) is the transposed complex conjugate of y(t). The Fourier transform of Ryy (τ ) is the spectral density Syy (jω). If this function is known,
Ryy can be obtained as
Z ∞
1
Ryy (τ ) =
Syy (jω)ejωτ dω.
(2.41)
2π −∞
If the stable and strictly proper transfer function G relates the input u(t)
with the output y(t) then
Syy (jω) = G(jω)Suu (jω)G∗ (jω).
(2.42)
When the input is unit variance white noise, i.e. the spectral density Suu (jω) =
I, where I is the identity matrix, the RMS norm of the output is given by
||y||2RM S = tr Ryy (0)
Z ∞ 1
=
tr Syy (jω) dω
2π −∞
Z ∞ 1
=
tr G(jω)G∗ (jω) dω
2π −∞
Z ∞ 1
=
tr G∗ (jω)G(jω) dω
2π −∞
= ||G||22
(2.43)
where Equations
(2.39, 2.41, 2.42, 2.36) and the fact that tr GG∗ =
tr G∗ G have been used. This shows that ||G||2 = ||y||RM S in the case
of unit variance white noise input.
2.6.3
Calculation of the H2 norm
For a continuous-time strictly proper system the H2 norm can be calculated
as
Z ∞ 2
2
||G(s)||2 = ||g(t)||2 =
tr gT (τ )g(τ ) dτ
Z0 ∞ T
T
=
tr (CeA τ B)T CeA τ B dτ
0
Z ∞
T
T
= tr B
eA τ C T CeAτ dτ B
0
T
= tr B QB
(2.44)
32
where the impulse response matrix for a continuous-time stable system given
in (2.27) has been utilized with D =0 and the definition
of the observability
T
T
Gramian, Q, in (2.7). Since tr xx = tr x x holds, where x is a vector,
the derivation in (2.44) can also be written as
Z ∞ 2
2
T
||G(s)||2 = ||g(t)||2 =
tr g (τ )g(τ ) dτ
Z0 ∞ =
tr g(τ )gT (τ ) dτ
Z0 ∞ =
tr CeAτ B(CeAτ B)T dτ
0
Z ∞
T
= tr C
eAτ BB T eA τ dτ C T
0
= tr CP C T .
(2.45)
For a stable strictly proper system G given by the state space description
(A, B, C, 0) the H2 norm may therefore be calculated as
r r T
(2.46)
||G||2 = tr B QB = tr CP C T .
Hence, [Σ2 ]ij in (2.35) can be calculated as
q
tr Ci Pj CiT
[Σ2 ]ij = P q
.
T
tr
C
P
C
k l k
kl
(2.47)
For SISO systems CP C T reduces to a scalar.
Therefore, for the considered
fundamental subsystems in Σ2 , tr CP C T = CP C T and consequently, in
this particular case, CP C T = B T QB.
2.6.4
Energy interpretation for discrete-time systems
Since tr CP C T = tr B T QB it is of interest to interpret the quantity
B T QB in terms of energy. Consider the discrete-time system given in (2.2)
and let D = 0 and recall that the observability Gramian is given by Q =
WoT Wo (Equation (2.11)). Thus B T QB can be expressed as (use (2.9b) and
(2.25))


CB
 CAB 

T
T
T
T
T 
B QB = B Wo Wo B = B Wo 
(2.48)
 = ΓT (:, 1)Γ(:, 1)
..


.
|
33
CAL−1 B
{z
}
Γ(:,1)
where Γ(:, 1) is the first column of the Hankel matrix given in (2.25). In the
SISO case, when B and C have dimensions N × 1 and 1 × N , respectively,
the first column of the Hankel matrix equals the transpose of the first row,
i.e. Γ(:, 1) = ΓT (1, :). Hence,
B T QB = ΓT (:, 1)Γ(:, 1) = Γ(1, :)ΓT (1, :).
(2.49)
In the MIMO case the Hankel matrix is a block matrix and for this reason
the property used in (2.49) does not hold. As can be seen from the first row
of Equations (2.22) and (2.25)


u(−1)
 u(−2) 


y(0) = Γ(1, :) 
(2.50)
.
..


.
u(−L)
|
{z
}
U
Therefore the power of y(0) may be calculated as
|y(0)|2 = y T (0)y(0) = U T ΓT (1, :)Γ(1, :) U = U T Γ̃U.
(2.51)
Equation (2.51) indicates that the matrix Γ̃ may be interpreted as the matrix
that transfer energy from the past inputs to the current output y(0). A
way of quantifying this energy transmission is to calculate the trace of this
matrix. In the SISO case (use (2.49))
tr Γ̃in
= tr ΓT (1, :)Γ(1, :) = tr Γ(1, :)ΓT (1, :)
= tr B T QB = B T QB.
(2.52)
Since B T QB = CP C T in the SISO case – as for the fundamental subsystems
considered in Σ2 – this implies that CP C T , or tr(CP C T ) the H2 norm,
of each fundamental subsystem can be seen as the coupling in terms of
the energy transmission rate (power) between the past inputs U and the
current output y(0). This view supports the previously discussed energy
interpretations of the H2 norm in Section 2.6.2.
One key difference in the interpretations of the HIIA and the PM compared to the H2 norm based interaction measure Σ2 is that the former quantify the interactions between (past) inputs and (future) outputs directly (see
Equation (2.22)) whereas the latter relates (past) inputs with (present) outputs in terms of energy (see e.g. Equation (2.51)), i.e. it relates squared
inputs and squared outputs.
Obviously, there are several interesting interpretations of the H2 norm
based interaction measure Σ2 . The definition of output controllability given
by Kreindler and Sarachik (1964) and Equations (2.13–2.14) indicate that
34
the considered measure in fact is a measure of the output controllability
of the plant. Recall that the HIIA and the PM rather are quantifiers of
the combined state controllability and state observability. Furthermore, the
measure Σ2 can be interpreted both in terms of energy and as the RMSvalue, depending on the type of inputs, as previously discussed.
2.6.5
Properties of the H2 norm based interaction measure
Σ2
Here, some of the basic properties of the considered interaction measure Σ2
will be derived.
Independence of realization
The Σ2 is independent of the selected realization. This follows from the
definition in (2.35). It is also a consequence of CP C T being independent of
realization as shown in Section 2.3.3.
Preservation of structure
The structure of the plant, G, is preserved in Σ2 . To see this, assume Gij = 0
for some i, j 6= 0. Then tr(Ci Pj CiT ) = ||Gij ||2 = 0 and the stated property
follows.
Frequency scaling
One advantage of the HIIA and the PM is that they are insensitive to
frequency scaling. It can be verified (Salgado and Conley, 2004) that for
a frequency scaled system with transfer function Gij (s/ξ) the corresponding controllability and observability Gramians transform to P̂j = Pj /ξ and
Q̂i = ξQi and clearly, P̂ Q̂ = P Q which explains why the PM and the
HIIA are preserved. However, the quantity Ci Pj CiT will be affected by the
frequency scaling since
Ĉi P̂j ĈiT = Ci P̂j CiT = Ci
Pj T
C .
ξ i
(2.53)
Fortunately, the influence of the frequency scaling on the interaction measure
Σ2 given in Equation (2.35) will be canceled in the normalization since it
1 P
1
P
corresponds to a division by kl tr(Ck P̂l CkT ) 2 = kl tr(Ck (Pl /ξ)CkT ) 2 .
35
Time delays
Since time delays may alter the process dynamics significantly, knowledge
of these are important when selecting the controller structure. Since only
discrete time systems are able to model time delays in finite-dimensional
state-space models the following discussion will address this type of systems.
Salgado and Conley (2004) show that the PM is able to indicate the
presence of a time delay. In fact, they prove that a pure time delay of
1/z ℓ , ℓ ∈ N, applied at the output of the non-delayed system, gives an extra
contribution to the expression tr(P Q) for the delayed system:
tr(Pℓ Qℓ ) = tr(P0 Q0 ) + ℓ · C0 P0 C0T = tr(P0 Q0 ) + ℓ||G0 ||22
(2.54)
where the index 0 is used for the non-delayed system and the index ℓ for
the delayed system. Following the proof therein, it can be verified that the
quantity CP C T will not be affected by the time delay, i.e.
Cℓ Pℓ CℓT = C0 P0 C0T
ℓ ∈ N.
(2.55)
Hence Σ2 is unaffected by time delays.
However, as will be illustrated later in Example 4 in Section 2.7, the
presence of a time delay is by itself not a sufficient reason to include – nor
exclude — that particular input-output pair in the controller.
2.6.6
The H2 norm and induced norms
The Hankel norm is the induced operator norm of the Hankel operator in
the continuous-time case. The same interpretation holds in the discrete-time
case if the Hankel matrix is seen as an operator.
It should be noted that the H2 norm is not an induced norm. For instance
it does not satisfy the multiplicative property (Skogestad and Postlethwaite,
1996)
||AB|| ≤ ||A|| ||B||.
(2.56)
In practice, this means that the behaviour of a connected system cannot be
judged from the H2 norm of the components of the system (ibid.). However,
as previously discussed, for SISO systems CP C T is a scalar. This implies
that
tr CP C T = λ(CP C T ) = λmax (CP C T )
(2.57)
where λ denotes the eigenvalues and λmax is the largest one. As pointed
out by Wilson (1989), Antoulas (2001), Chellaboina et al. (1999) and Dharmasanam et al. (1997) λmax (CP C T ) is in fact the 2-∞ induced operator
norm of the convolution operator:
q
||y||∞
= λmax (CP C T )
where ||y||∞ = sup max yi (t).
||G||2,∞ = sup
i
t
u6=0 ||u||2
(2.58)
36
The convolution operator G is defined as (Dharmasanam et al., 1997)
y(t) = (Gu)(t) =
Z
∞
g(t − τ )u(τ )dτ.
(2.59)
−∞
Therefore, for the considered H2 norm based interaction measure Σ2 , the
issue of the H2 norm of not being an induced norm in the general case
is of minor interest. Furthermore the way the interaction measure is supposed to be used does not mean that the norm necessarily has to satisfy the
multiplicative property (2.56).
As can be seen in Equation (2.58), the 2-∞ induced operator norm of the
convolution operator can be interpreted as the maximum output amplitude
that finite energy input signals give rise to (Antoulas, 2001). As pointed out
by Wilson (1989), this norm can be useful in a control design performance
criteria when the aim is to reduce the influence of finite energy input disturbances. Promising (simulation) results have been reported by Rotea (1993)
and by Dharmasanam et al. (1997) where various mixed performance criteria
based on the H2 norm, the 2-∞ induced operator norm of the convolution
operator and the ∞-∞ induced operator norm of the convolution operator
were used in the control design. This interpretation is also important for
the considered interaction measure.
2.7
Examples
The usefulness of the interaction measure Σ2 has previously been exemplified
for a linearized model of a coal injection vessel by Birk and Medvedev (2003).
In this section, further examples are given.
2.7.1
Example 1
In the first example the interactions present in a quadruple-tank system will
be examined (see Johansson (2000) for a general description of this process).
The considered linear minimum-phase model is given by the following state
space matrices:



B=


−0.0159
0
0.159
0

0
−0.0159
0
0.02651 
,
A=


0
0
−0.159
0
0
0
0
−0.02651

0.05459
0
0
0.07279 
 , C = 1 0 0 0 , D = 0 0 . (2.60)
0
0.0182 
0 1 0 0
0 0
0.03639
0
37
The steady-state transfer function is
3.4326 1.1442
G(0) =
.
2.2884 4.5768
(2.61)
If this matrix is normalized to make the sum of the magnitudes of all elements equal to 1 the following matrix is obtained:
0.3 0.1
G(0) =
.
(2.62)
0.2 0.4
G(0) can be seen as a rough measure of the relative importance of each
input-output channel in steady-state. The condition number for G(0) is 2.6
and the minimized condition number is around 2.4 so there is no need to
scale the system. The RGA, denoted Λ, for the system is
1.2 −0.2
Λ(G(0)) =
(2.63)
−0.2 1.2
and the Gramian based interaction matrices are
0.2866 0.1029
ΣH =
,
0.2285 0.3821
0.2809 0.0364
Φ =
,
0.1834 0.4994
0.3168 0.0320
Σ2 =
0.0880 0.5632
(2.64)
(2.65)
(2.66)
where the HIIA matrix is denoted ΣH , the PM Φ and Σ2 is the H2 norm
based interaction measure. To be able to make direct comparisons between
the three considered Gramian based interaction measures, it is beneficial to
calculate the square-root of the PM and of the Σ2 and then renormalize the
measures. The following matrices then result:
0.2856 0.1028
Φ =
,
(2.67)
0.2308 0.3808
0.3146 0.1000
Σ2 =
.
(2.68)
0.1658 0.4195
In this way the three matrices are expressed in the same units: Recall that
the HIIA can be interpreted as the gain between past inputs and future
outputs whereas Σ2 can be interpreted in terms of energy (hence the need
of the square root). Also recall that the HIIA and the PM differ both by the
number of eigenvalues considered and by a square root, see (2.19), (2.32)
and (2.33). This means that the PM rather is a measure that quantifies the
interaction in terms of energy.
38
As can be seen, the Gramian based interaction quantifiers all rank the
importance of the input-output channels in the same order and they advocate the same decentralized diagonal pairing as the RGA does. The product
P Q has four eigenvalues: 9.2133, 1.2668, 0.0050 and 0.0016. Since the first
one accounts for 88% of tr(P Q) there will only be minor numerical differences between ΣH and Φ as can be appreciated when comparing Equation
(2.64) with (2.67). Neither does Σ2 in Equation (2.68) differ much from ΣH
in Equation (2.64) nor Φ in Equation (2.67) with the exception of element
(2,1). Furthermore, note the resemblance of ΣH , Φ and Σ2 to G(0). This
is not surprising in view of the interpretation of the HIIA as being the gain
between old inputs and future outputs. If the system is low-pass filtered the
resemblance is even closer. Clearly, the considered system has most of its
process dynamics in the lower frequencies. Similar arguments can be applied
to explain the resemblance Σ2 shows.
2.7.2
Example 2
In the second example, the following 3 × 3-system is analysed


G(s) = 
0.4
(s+1)2
2
(s+2)(s+1)
6(−s+1)
(s+5)(s+4)
4(s+3)
(s+2)(s+5)
2
(s+2)2
4
(s+3)2
−2
s+4
1
s+2
8
(s+2)(s+5)


.
(2.69)
Salgado and Conley (2004) compare different control structures based on
the advice from the PM for this system. The steady-state gain and the
normalized magnitudes G(0) are

0.4000 1.2000

1.0000 0.5000
G(0) =
0.3000 0.4444

0.0709 0.2126

0.1772 0.0886
G(0) =
0.0531 0.0787

−0.5000
0.5000  ,
0.8000

0.0886
0.0886  .
0.1417
The interaction measures are


−0.0831 0.9111
0.1720
Λ(G(0)) =  1.3809 −0.2745 −0.1064  ,
−0.2979 0.3634
0.9345
ΣH


0.0703 0.1663 0.0728
=  0.1728 0.0878 0.0728  ,
0.1426 0.0781 0.1367
39
(2.70)
(2.71)
(2.72)
Bode Diagram
From: In(2)
From: In(1)
From: In(3)
To: Out(1)
0
−20
−40
−60
−80
To: Out(2)
Magnitude (dB)
0
−20
−40
−60
−80
To: Out(3)
0
−20
−40
−60
−80
−1
10
0
10
1
10
−1
10
0
1
10
10
Frequency (rad/sec)
−1
10
0
10
1
10
2
10
Figure 2.1: Bode magnitude diagram for the non-filtered plant in Example 2.


0.0370 0.2018 0.0385
Φ =  0.2226 0.0578 0.0385  ,
0.2193 0.0457 0.1389


0.0687 0.1604 0.0701
Φ =  0.1684 0.0858 0.0701  ,
0.1672 0.0763 0.1331
(2.73)




0.0065 0.3545 0.0816
0.0316 0.2331 0.1119
Σ2 =  0.0544 0.0204 0.0408  , Σ2 =  0.0913 0.0559 0.0791  .
0.3429 0.0242 0.0746
0.2292 0.0609 0.1070
(2.74)
The RGA, HIIA and PM all advocate the same decentralized pairing:
u1 – y2 , u2 – y1 , u3 – y3 . Φ is close to HIIA but it emphasizes the importance
of element (3,1) slightly more. Σ2 differs even more and suggests that u1
mostly affects y3 rather than y2 as indicated by the HIIA and the PM.
The maximum eigenvalue of P Q amounts to 62% of tr(P Q) explaining the
difference between HIIA and Φ.
Next, consider the original system low-pass filtered with
F LP (s) =
1
s+1
resulting in the interaction matrices


0.0740 0.1933 0.0797
 0.1825 0.0908 0.0830  ,
ΣLP
H =
0.0773 0.0787 0.1407
40
(2.75)
(2.76)
ΦLP




0.0429 0.2776 0.0472
0.0743 0.1889 0.0779
LP
=  0.2596 0.0642 0.0517  , Φ =  0.1827 0.0909 0.0815  ,
0.0574 0.0477 0.1517
0.0859 0.0783 0.1397
(2.77)




0.0215 0.3736 0.0718
0.0532 0.2216 0.0971
 0.1595 0.0498 0.0598  , ΣLP
 0.1448 0.0809 0.0887  .
ΣLP
2 =
2 =
0.0718 0.0465 0.1458
0.0971 0.0782 0.1384
(2.78)
Clearly, all of the Gramian based interaction measures now give the same
decentralized pairing recommendation as the RGA and the G(0) do: Σ2 has
aligned with the other measures.
A high-pass filtering of the original system with the filter
F HP (s) =
s
s+1
gives the following interaction measures:


0.0408 0.1873 0.0868
 0.1197 0.0751 0.0724  ,
ΣHP
H =
0.2058 0.0784 0.1336
ΦHP
(2.79)
(2.80)




0.0118 0.2861 0.0631
0.0400 0.1965 0.0923
HP
=  0.1022 0.0402 0.0438  , Φ
=  0.1175 0.0736 0.0769  ,
0.2803 0.0438 0.1287
0.1945 0.0769 0.1318
(2.81)




0.0021 0.3489 0.0845
0.0192 0.2470 0.1216
HP
ΣHP
=  0.0235 0.0117 0.0352  , Σ2 =  0.0641 0.0453 0.0785  .
2
0.4227 0.0176 0.0537
0.2719 0.0555 0.0969
(2.82)
Once again, all Gramian based interaction measures suggest the same decentralized control structure: u1 – y3 , u2 – y1 , u3 – y2 . This is the same pairing
as suggested by Σ2 in the analysis of the original non-filtered system, see
Equation (2.74). Recall that the aim is to find a pairing that yields a sum of
the elements in the considered Gramian based interaction measure as large
as possible. When a decentralized structure is sought-after, the pairing u3
– y2 has to be selected in favour of u3 – y1 and u3 – y3 (even though these
corresponding elements is larger than the selected one) since the elements
corresponding to u1 – y3 and u2 – y1 make a significant contribution to the
sum and should therefore be included.
To appreciate the dynamic behaviour of the plant a Bode magnitude
diagram is provided in Figure 2.1. Particularly, note the resonance top in
41
the Bode magnitude diagram for u1 to y3 . Since the H2 norm is proportional
to the integral of the magnitudes in the Bode diagram the shape of Σ2 is
outlined by the Bode diagram. In particular, note the relatively high gain
between u1 and y3 for high frequencies. This is reflected in Σ2 in (2.74) and
(2.82).
2.7.3
Example 3
In the third example a discrete-time system with time delay is considered and
the suggested pairing recommendations are tested in control simulations.
The system is given by
#
"
b
G(z) =
0.5
(z−0.5)
b21
(z−0.5)(z−0.8)
12
(z−0.8)z ℓ
0.3
(z−0.7)
.
(2.83)
When the nonnegative integer ℓ > 0 there is a time delay present in the
channel between u2 and y1 . The values of the parameters b12 and b21 are
varied so that the static gain in the anti-diagonal channels (including the
channel with time delay) are either relatively high or relatively low compared
to the static gain of the diagonal channels.
In the first case, b12 = 0.15 and b21 = 0.1 which corresponds to the low
gain scenario. This system has previously been analysed by Salgado and
Conley (2004) using the PM. Without the extra time delay in element G12 ,
i.e. with ℓ = 0 the following steady state transfer function G(0), normalized
magnitudes G(0) and interaction matrices result:
1.0000 0.7500
0.2667 0.2000
G(0) =
, G(0) =
,
(2.84)
1.0000 1.0000
0.2667 0.2667
4.0000 −3.0000
Λ(G(0)) =
−3.0000 4.0000
0.2882 0.1801
ΣH =
,
0.2774 0.2543
Φ=
Σ2 =
0.3171 0.1239
0.3122 0.2469
0.5060 0.0949
0.1312 0.2679
,
,
Φ=
Σ2 =
,
0.2857 0.1786
0.2835 0.2521
0.3746 0.1622
0.1907 0.2725
(2.85)
(2.86)
,
(2.87)
.
(2.88)
All of the interaction quantifiers favour the diagonal pairing (ui – yi ) for
decentralized control. The RGA also indicates that the anti-diagonal pairing
should be avoided due to stability issues (the RGA has negative anti-diagonal
elements). As pointed out by Salgado and Conley (2004) better control
42
performance could be expected if a sparse controller structure is designed
which also includes the coupling between u1 and y2 . This can for instance
be achieved by introducing a suitable feedforward.
When an extra time delay, ℓ = 10, is introduced between u1 and y2 the
HIIA and the PM change to
0.2631 0.2514
ΣH =
,
(2.89)
0.2533 0.2322
0.2193 0.3941
0.2372 0.3180
Φ=
, Φ=
.
(2.90)
0.2159 0.1707
0.2354 0.2093
Recall that the Σ2 (and the RGA) remains unaffected by the time delay. As
described in Section 2.6.5 the PM is able to detect time delays and this is
what is seen in (2.90). Now the PM recommends the anti-diagonal pairing
in contrast to both Σ2 and the RGA. It is hard to draw any clear conclusions
from the HIIA in (2.89).
To validate the relevance of these pairing recommendations decentralized, integrating, controllers were designed using a polynomial pole-placement methodology. The results are visualized in Figure 2.2. The upper plot
shows the plant outputs for the system without the extra time delay controlled by a diagonal controller. The sampling time was set to 0.5 s in all of
the control simulations. For each of the selected channels a SISO controller
were designed with poles in z = 0.4. Unit step changes were applied at
time 0 and 15 [s]. The lower plot shows control of the plant with the very
same controller but when the extra time delay of ℓ = 10 was introduced.
Furthermore, for the delayed system a controller with anti-diagonal pairing
as suggested by the PM were designed. The poles were placed in different
locations and the time delay was accounted for in the controller but stable
control of the plant could not be obtained unless the two input-output channels that were not included in the controller (i.e. u1 – y1 and u2 – y2 ) were
detached.
In the second system the parameters b12 and b21 were increased to 0.8 and
0.4, respectively, so that the importance of the anti-diagonal elements in G
in (2.83) increases. This way the influence of the time delay in G12 should
be more prominent compared to the previous case. The following steady
state transfer function G(0), normalized magnitudes G(0) and interaction
matrices were then obtained:
1.0000 4.0000
0.1000 0.4000
G(0) =
, G(0) =
,
(2.91)
4.0000 1.0000
0.4000 0.1000
−0.0667 1.0667
Λ(G(0)) =
,
(2.92)
1.0667 −0.0667
0.1103 0.3677
ΣH =
,
(2.93)
0.4247 0.0973
43
Diagonal control
1.4
Outputs
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
30
35
Time [s]
Diagonal control of the time delayed plant
1.4
Outputs
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
Time [s]
Figure 2.2: Plant outputs for decentralized diagonal control of the system
(2.83) with b12 = 0.15 and b21 = 0.1 in Example 3. The upper plot shows
the control of the plant with ℓ = 0 and the lower plot control of the delayed
one (ℓ = 10). The solid lines show output 1 and the dashed ones output 2.
Φ=
Σ2 =
0.0349 0.3879
0.5500 0.0272
0.0908 0.4844
0.3767 0.0481
,
,
Φ=
Σ2 =
0.1089 0.3629
0.4321 0.0961
0.1646 0.3802
0.3353 0.1198
,
(2.94)
.
(2.95)
Evidently, the higher gain in the anti-diagonal input-output channels made
all interaction measures to recommend the anti-diagonal pairing for decentralized control in contrast to the previous low gain case. With a time delay
of ℓ = 10 the HIIA and the PM become
0.0923 0.4706
,
(2.96)
ΣH =
0.3556 0.0815
0.0146 0.7446
0.0908 0.4844
Φ=
, Φ=
.
(2.97)
0.2295 0.0113
0.3767 0.0481
As seen there are no changes in the decentralized recommendation from
these measures when the time delay is introduced.
Controllers for these cases were also designed in a similar manner as for
the previously studied systems. Figure 2.3 shows the plant outputs for the
44
Anti−diagonal control
1.4
1.2
Outputs
1
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
60
70
80
90
100
90
100
Time [s]
Anti−diagonal control of the time delayed plant
1.4
1.2
Outputs
1
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
60
70
80
Time [s]
Figure 2.3: Plant outputs for decentralized anti-diagonal control of the system (2.83) with b12 = 0.8 and b21 = 0.4 in Example 3. The upper plot shows
the control of the plant with ℓ = 0 and the lower plot control of the delayed
one (ℓ = 10). The solid lines show output 1 and the dashed ones output 2.
Note that the time scale is different from the scale in Figure 2.2.
control simulations for the non-delayed system (upper plot) and the delayed
system (lower plot). In both of these cases the anti-diagonal input-output
pairing were selected with satisfactory control performance. Note that in
this particular case it was necessary to include the time delayed channel in
the controller. Due to the time delay the controllers had to be slow: The
poles were placed in z = 0.9. A working controller based on the diagonal
pairing was not found, and was not expected to be found due to the severe
cross-couplings that would then be present.
The control simulations indicate that the presence of a time delay by
itself is not a reason enough to say that this particular input-output pair
should be included in the controller when a decentralized controller structure
is desired. This is, for this particular example, in contrast to the indications
given by the PM, but in agreement with those of the Σ2 .
45
2.7.4
Example 4
Now consider the 2 × 2 process given by:
" −40s
G(s) =
5e
100s+1
−5e−4s
10s+1
e−4s
10s+1
5e−40s
100s+1
#
.
(2.98)
This process has been extensively analysed by Mc Avoy et al. (2003) and
Xiong et al. (2005) with the conclusion that the anti-diagonal pairing is
preferred for decentralized control. One reason for this is that the antidiagonal pairing corresponds to faster elements in G. Mc Avoy et al. (2003)
came to this conclusion using their dynamic relative gain array (DRGA) and
verified it in a simulation study. Xiong et al. (2005) used the ERGA with
the same result. The steady state transfer function G(0), the normalized
magnitudes G(0) and the interaction matrices are:
5.0000 1.0000
0.3125 0.0625
G(0) =
, G(0) =
,
(2.99)
−5.0000 5.0000
0.3125 0.3125
Φ=
Σ2 =
0.3289
0.3289
0.0806
0.8065
0.8333 0.1667
Λ(G(0)) =
,
0.1667 0.8333
0.3125 0.0625
ΣH =
,
0.3125 0.3125
0.0132
0.3125 0.0625
, Φ=
,
0.3125 0.3125
0.3289
0.0323
0.1726 0.1091
, Σ2 =
.
0.0806
0.5457 0.1726
(2.100)
(2.101)
(2.102)
(2.103)
The time delays have been approximated by third order Padé approximations. The RGA, the HIIA and the PM suggest the diagonal pairings for
decentralized control. However, the recommendation from the Σ2 is the diagonal pairing which is in agreement with the findings by Mc Avoy et al.
(2003) and Xiong et al. (2005).
2.7.5
Example 5
As a concluding example, consider the 3 × 3 process given by:


1
−4.19 −25.96
1−s 
G(s) =
6.19
1
−25.96  .
(1 + 5s)2
1
1
1
(2.104)
This process is used by Hovd and Skogestad (1992) as an example of when
the RGA does not recommend the most desirable pairing. The steady-state
46
gain and the normalized steady-state gain are:


1.0000 −4.1900 −25.9600
G(0) =  6.1900 1.0000 −25.96000  ,
1.0000 1.0000
1.0000


0.0149 0.0623 0.3857
G(0) =  0.0920 0.0149 0.3857  .
0.0149 0.0149 0.0149
The interaction measures are:


1.0009
5.0010 −5.0019
Λ(G(0)) =  −5.0028 1.0009
5.0019  ,
5.0019 −5.0019 1.0000
(2.105)
(2.106)


0.0149 0.0623 0.3857
ΣH =  0.0920 0.0149 0.3857  ,
(2.107)
0.0149 0.0149 0.0149




0.0149 0.0623 0.3857
0.0007 0.0125 0.4784
Φ =  0.0272 0.0007 0.4784  , Φ =  0.0920 0.0149 0.3857  ,
0.0149 0.0149 0.0149
0.0007 0.0007 0.0007
(2.108)




0.0007 0.0125 0.4784
0.0149 0.0623 0.3857
Σ2 =  0.0272 0.0007 0.4784  , Σ2 =  0.0920 0.0149 0.3857  .
0.0007 0.0007 0.0007
0.0149 0.0149 0.0149
(2.109)
The RGA recommends the diagonal pairing. However, as found by Hovd
and Skogestad (1992) this pairing is not suitable due to instability issues.
Instead, they recommend the pairing that corresponds to the RGA elements
with values near 5. The same pairing suggestion is found by He and Cai
(2004) when considering loop-by-loop interaction energy. The HIIA, the PM
and Σ2 all give the same pairing recommendation for decentralized control:
u1 – y2 , u2 – y3 and u3 – y1 . For the HIIA this gives a sum of 0.54. However,
if the rule of avoiding pairings corresponding to negative RGA elements is
obeyed, the HIIA, the PM and Σ2 suggest the very same pairing as the one
recommended by Hovd and Skogestad (1992) and He and Cai (2004). This
gives for the HIIA a sum of 0.46.
2.8
Conclusions
Theoretical arguments for including the H2 norm in an interaction measure
were given in Section 2.6. As seen, the H2 norm can be given various useful energy interpretations, for instance it is proportional to the integral of
the Bode magnitudes. Furthermore, as also investigated in Section 2.6, it
47
can be seen as a measure of the output controllability of the plant. Some
fundamental properties of the H2 norm based interaction measure Σ2 were
also derived. The Σ2 was found to be unaffected by time delays. The other
Gramian based interaction quantifiers, the Hankel Interaction Index Array (HIIA) and the Participation Matrix (PM), are not. All of the Gramian
based interaction measures (including the Σ2 ) are scaling dependent. Therefore a proper scaling of the considered systems is important. The condition
number can be compared with the minimized condition number as a guidance in the search for the scaling matrices.
In Section 2.7 different interaction measures, including the Σ2 , were compared in the analysis of different multivariable systems. It was found that
often the Σ2 is similar to the HIIA and the PM. In other examples it was
seen that Σ2 is potentially able to more clearly reveal interactions present
for high frequencies than the HIIA and the PM are. However, it should be
noted that this way important low frequency behaviour of the plant may
be less prominent in the Σ2 . It is therefore of vital importance that the
system is filtered in advance in order to focus on the interesting range of
frequencies.
Furthermore, it was found that the presence of a time delay in one of the
input-output channels does not necessarily imply that this channel should
be avoided – or selected – for decentralized control design. The impact of a
time delay has to be evaluated in each separate case.
It was also seen in one example that the Σ2 was able to select the proper
pairings in contrast to the RGA, the HIIA and the PM. In the final example, the HIIA, the PM and the Σ2 proposed the correct pairings if their
recommendations were combined with the use of the RGA rule of avoiding
pairings that correspond to negative RGA elements. This indicates that it
could be beneficial to consider several different interaction measures when
solving the pairing problem. This is the approach in e.g. (He et al., 2006).
However, to give general rules for how to design such a pairing algorithm is
out of the scope of this thesis.
48
Chapter 3
Interaction Analysis in a
Bioreactor Model
In this chapter the well-known Relative Gain Array (RGA) and the more recently proposed Hankel Interaction Index Array (HIIA) are utilized to quantify the degree of channel interaction in a multivariable bioreactor model, an
activated sludge process (ASP) configured for nitrogen removal. To be more
precise, both the nitrification and the denitrification process in an ASP are
studied. The considered model is a 2 × 2 system with the dissolved oxygen concentration set point in the aerobic compartment and the internal
recirculation flow rate as input signals (control handles). The effluent concentration of ammoniun and of nitrate are the output signals (the controlled
signals). The HIIA can deal with plant structures where the RGA fails and
can furthermore also be used to evaluate multivariable controller structures.
It was found that the RGA method was unable to give reasonable inputoutput pairing suggestions in some cases while the HIIA method provided
useful information in all of the considered cases.
3.1
Introduction
When comparing the HIIA to the RGA, there are some major differences.
The most important one from the authors point of view is that the RGA
assumes a decentralized control structure to be used, and therefore attempts
to suggest the best possible input-output pairing. This is not the case with
the HIIA, that rather considers the controllability and the observability of
every sub-system in the plant separately. This measure can therefore be
valuable when evaluating alternatives to decentralized control structures,
i.e. multivariable control structures with reduced complexity. In (Conley and
Salgado, 2000) and (Salgado and Conley, 2004), it is shown how to do this
when employing the similar interaction measure the Participation Matrix
(PM) (see Chapter 2 for a discussion of the PM). Another difference is that
49
when using the HIIA, the whole frequency range is taken into account, not
only one frequency at the time. As shown in examples given by Wittenmark
and Salgado (2002), the HIIA outperforms the RGA when dealing with
systems that have interactions with non-monotonic frequency behavior. The
reason for this is that the full dynamics of the system will be taken into
account when using Gramians. If the objective is to study the interactions
in a specific frequency range only, then the transfer functions can be filtered
before the HIIA is calculated, see Wittenmark and Salgado (2002). There are
also cases, see for instance Kinnaert (1995), where the RGA fails to suggest
a proper pairing due to large off-diagonal elements or triangular structure
in the plant. On the other hand, a drawback of the HIIA compared to the
RGA is that it is scaling dependent. It is therefore of great importance
that the system has been scaled in a physically relevant way in order for the
HIIA to provide meaningful results. The Gramian based approach is further
discussed in Salgado and Conley (2004) and in Section 2.5 in Chapter 2.
In this chapter, the RGA and the HIIA will be employed in the selection
of input–output signal pairings for a part of a MIMO bioreactor system:
an activated sludge process configured for nitrogen removal. Modelling and
control of the activated sludge process have been an intense research area
in the last decade, see for example Olsson (1993), Lindberg and Carlsson
(1996), Alex et al. (1999), Vanrolleghem et al. (1999), Samuelsson and Carlsson (2001), Yuan et al. (2002) and Jeppsson and Pons (2004). The results
from the RGA analysis will be compared with those of the HIIA and with
results obtained from physical insights of the considered system. It is also
discussed what additional conclusions that can be drawn from the HIIA
analysis. For a theoretical discussion and definition of the employed interaction measures, i.e. the RGA and the HIIA, see Section 2.4 and Section 2.5
in Chapter 2.
3.2
The bioreactor model
In the complex process of wastewater treatment, many different cause-effect
relationships exist, and therefore, there are many possible choices of input
and output signals, see Olsson and Jeppsson (1994). Consequently, this can
motivate the study of wastewater treatment plant models with respect to
the selection of input and output signals.
From a theoretical point of view, the bioreactor models are non-linear
multivariable systems that may contain a significant degree of coupling.
Hence, this also gives an interesting opportunity to test the performance
of the methods for input-output pairing selection discussed in the previous
section.
The objective in this chapter is to find suitable control structures. If the
couplings between the different control handles in the system are sufficiently
50
Influent, Q
Effluent
Anoxic
Aerobic
Internal recirculation, Qi
Settler
Sludge recirculation
Excess sludge
Figure 3.1: A basic activated sludge process (ASP) configured for nitrogen
removal.
low, then a controller selection involving several decoupled SISO controllers
may be suitable. If this is not the case, a MIMO controller structure will
provide a better solution. The MIMO solution will, however, generally be
much more complex. Both the RGA and the HIIA method will be used in
the sequel.
The considered model is a simplified version of the IAWQ Activated
Sludge Model No. 1 (ASM1) that models an activated sludge process configured for nitrogen removal. ASM1 is thoroughly described by Henze et
al. (1987). In this study the bioreactor consists of two tanks of equal volume (one anoxic and one aerobic of 1000 m3 each) and a settler, see Figure
3.1. The influent flow rate, Q, is 18 446 m3 /day. The model is valid in
the medium time-scale (i.e. hours to days). For a discussion of the model
parameters, see Halvarsson (2003).
Two different processes, nitrification and denitrification, are simultaneously being performed. To get an indication of how well these processes are
being performed the effluent ammonium concentration (SNH,2 (t)) and the
nitrate concentration (SNO,2 (t)), respectively, can be considered. Hence,
these concentrations are selected as output signals. The considered input
signals are the concentration of dissolved oxygen (DO set point, SO,2 (t)) in
the aerobic compartment and the internal recirculation flow rate (Qi (t)).
According to Ingildsen (2002) the denitrification is mainly influenced by
Qi (t) (among the selected input signals) while the nitrification is mainly
influenced by SO,2 (t). Hence, if the couplings between Qi (t) and SO,2 (t) are
low, then the denitrification and the nitrification process may be considered
separately when choosing controller structure and thus, SISO controllers
may be selected.
Three different operating points were selected1 . These correspond to the
input signals:
1
These operating points do not necessarily correspond to feasible choices concerning
an optimal operation of the plant. Instead, these are chosen in order to illustrate different
interaction points.
51
• u1 = [10 000 m3 /day
2 mg/l]T ,
• u2 = [36 892 m3 /day
2 mg/l]T ,
• u3 = [50 000 m3 /day
2 mg/l]T .
Since both the RGA and the HIIA are defined for linear models, the
simplified ASM1 model was linearized around each operating point using
the MATLAB function linmod. In a small neighbourhood of each operating
point the linearized model will mimic the characteristics of the nonlinear
system. Thus, the analysis in the following sections is strictly valid only in
the above mentioned neighbourhoods. However, as can be seen in the lower
part of Figure 3.2, the operational maps can be divided into two different
regions where the process shows different stationary characteristics. It is
therefore probable that each operating point describes the corresponding
area fairly well.
The obtained linear models can be represented in standard state-space
form as:
∆ẋ(t) = A∆x(t) + B∆u(t),
∆y(t) = C∆x(t).
(3.1)
where x(t) is the state vector given by
x(t) = [SNH,1 (t) SNH,2 (t) SNO,1 (t) SNO,2 (t) SS,1 (t) SS,2 (t)]T
(3.2a)
where the elements are the concentrations of ammonium (SNH,n ), nitrate
(SNO,n ) and readily biodegradable substrate (SS,n ) in compartment n in the
bioreactor. The operator ∆ refers to the deviation from the operating point.
For a more thorough description see Halvarsson (2003). The input signal
vector u(t) is given by:
Qi (t)
u(t) =
(3.2b)
SO,2 (t)
and the output signal vector is:
SNH,2(t)
y(t) =
(3.2c)
SNO,2 (t)
and
0 1 0 0 0 0
C=
.
(3.2d)
0 0 0 1 0 0
The steady-state operational maps for the model, are shown in Figure
3.2. The output signals, SNH,2 (t) and SNO,2 (t) are plotted against the two
input signals SO,2 (t) and Qi (t).
The operational maps in Figure 3.2 clearly indicate that different controller structures should be used in the different operating points, at least
in the lower operating point, u1 , compared to the upper operating points,
u2 and u3 . Note, however, that these operational maps can only be used to
give an indication of the interactions in the system.
52
4
6
x 10
NH,2
14
14.5
15
4
[mg/l]
3
15.5
i
S
16
Q [m3/day]
5
2
1
0
1
1.5
2
2.5
3
SO,2 [mg/l]
4
6
x 10
6
5
i
4
4.5
3
5
6
5.5
2
6
6.5
7
8
9
6.5
1
SNO,2 [mg/l]
5.5
Q [m3/day]
5
7
8
9
0
1
10
11
12
10
11
1.5
2
6.5
7
8
9
10
11
12
13
7
8
9
12
13
2.5
3
SO,2 [mg/l]
Figure 3.2: Steady-state operational maps for the considered bioreactor
model. The upper plot shows the level curves for the first output signal,
the effluent ammonium concentration, SNH,2 , and the lower one shows the
effluent nitrate concentration, SNO,2 . The operation points are indicated in
the plots.
3.3
RGA analysis
The steady-state RGA matrices for the linearized model in the three operating points are:
0.0055 0.9945
,
(3.3a)
RGA(Gu3 (0)) =
0.9945 0.0055
0.0051 0.9949
RGA(Gu2 (0)) =
,
(3.3b)
0.9949 0.0051
0.0041 0.9959
RGA(Gu1 (0)) =
.
(3.3c)
0.9959 0.0041
Apparently, the RGA suggests the anti-diagonal pairing SNH,2 (t)–SO,2 (t)
and SNO,2 (t)–Qi (t) for all of the three operating points. This contradicts
the results from the operational maps in Figure 3.2.
53
3.4
HIIA analysis
The HIIA is a scaling dependent tool. This motivates a scaling of the systems
before the HIIA is considered. A reasonable scaling procedure is to scale
the systems so that the maximum deviation from the average point of the
considered variables lies in the interval [−1, 1] (for a detailed description of
this scaling procedure, see Halvarsson (2003)). If all of the three operating
points are scaled in the same way the following steady state transfer function
matrices are obtained:
0.0004 −0.7048
scaled
,
(3.4a)
Gu3 (0) =
0.0748 0.6674
−0.0001 −0.7050
scaled
,
(3.4b)
Gu2 (0) =
−0.0135 0.6422
−0.0313 −0.7069
scaled
Gu1 (0) =
.
(3.4c)
−4.9176 0.4526
Furthermore, since the HIIA is a dynamic measure that considers all
possible frequencies while the considered model is only valid in a limited
frequency band it is also reasonable to perform a band-pass filtering before
calculating the HIIA. This was carried out using a simple first-order low-pass
filter, F (s), given by:
0.001
F (s) =
(3.5)
s + 0.001
where s is the Laplace-variable. This filter has a 3 dB cut-off frequency at
approximately 10−3 rad/s which is reasonable since the considered bioreactor
model is valid for frequencies ranging from approximately 10−5 rad/s up to
10−3 rad/s. Note also that this filter does not introduce any additional
scaling in the steady state. The filtering can be expressed as:
Gf iltered = GF.
(3.6)
If the systems are scaled in the suggested way and filtered using the lowpass filter F given in (3.5) before the HIIA is calculated, then the following
HIIA matrices, ΣH , are obtained for the three operating points:
0.0003 0.4869
u3
ΣH =
,
(3.7a)
0.0517 0.4611
0.0001 0.5181
u2
,
(3.7b)
ΣH =
0.0099 0.4719
0.0052 0.1157
u1
ΣH =
.
(3.7c)
0.8050 0.0741
If a decentralized controller structure is to be used, the HIIA analysis
suggests the same input-output pairings as the RGA, i.e the anti-diagonal
54
pairing in all of the considered operating points. However, since [ΣH ]22
is large for u2 and u3 this indicates that SO,2 affects both outputs, SNH,2
and SNO,2 . This in turn means that the suggested decentralized controller
structure could be insufficient to provide good control performance. Instead,
improved control performance can be expected if a (multivariable) triangular
controller structure that also includes the impact SO,2 has on SNO,2 is used.
In the lowest operating point, u1 , the HIIA also suggests a triangular
controller structure, even though not as strongly as for u2 and u3 . In fact,
in this operating point a decentralized controller may be good enough since
the sum of the anti-diagonal HIIA elements is 0.9207 which is close to one.
Concerning the scaling procedure, it was found that reasonable small
changes in the scaling matrices (for instance, ±40% in the element that
scales Qi ) do not alter the HIIA recommendations.
3.5
Discussion
In the RGA analysis of the bioreactor model it was seen that the RGA
method did not provide reasonable input-output pairings in all of the considered operating points. The reason for this can be found if the steady-state
gain matrices for the considered systems are studied. Triangular systems
will always give the same RGA, namely the identity matrix (under the assumption that the rows in the transfer function matrix are permuted to
get nonzero elements along the diagonal before calculating the RGA). The
transfer function matrices of the (scaled) model are almost right under triangular, see (3.4a)–(3.4c). Therefore, the structure of the RGA will be similar
for all of them: almost the anti-identity matrix. The RGA matrices are
given in equations (3.3a)–(3.3c), and evidently they are all very close to the
anti-identity matrix.
Obviously, the HIIA provides an interaction analysis that goes deeper
than the RGA is able to. When considering the information given by the
HIIA there is no longer any contradiction with the steady-state results in
the operational maps in Figure 3.2. This can also be seen as a confirmation
that the applied scaling procedure is reasonable. Note once again, that these
steady-state operational maps can merely be used to give an indication of
the interactions in the system, and what a reasonable controller structure
may look like.
Compared to the RGA, the HIIA possesses several advantages. Evidently, the HIIA is able to deal with special transfer function matrix structures such as the analysed nearly triangular ones. The HIIA does not require
decentralized (diagonal) controller structures as the RGA does. Instead, the
HIIA considers each subsystem in the model independently. Therefore, the
HIIA can be used to suggest MIMO controller structures as seen in Section 3.4. The RGA method is unable to do this.
55
It was also observed that the HIIA method is scaling-dependent. This
means that some effort must be spent on finding proper scaling matrices.
However, this is not necessarily a drawback, since this gives an opportunity
for the user to weight the considered signals according to his own choice.
The RGA method is scaling-independent and does not offer this possibility.
Based on the RGA results in this particular case, it should not be concluded that the couplings are low between the DO set point (SO,2 (t)) and the
internal recirculation flow rate (Qi (t)) independent of operation point. Instead, the operational maps indicate that there are some couplings between
the nitrification and the denitrification process. A MIMO controller structure can therefore be expected to give better control performance compared
to a solution involving decentralized control. The HIIA analysis supports
this view, and also suggests possible controller structure selections.
3.6
Conclusions
The RGA method provides a simple way to decide how a set of input signals
should be utilized to control a given set of output signals. Often this method
performs well, but in the analysis of the considered bioreactor model, it was
clearly seen that the RGA method does not work properly in all cases. The
reason for this was found to be the almost triangular structure of the transfer
function matrices. From this it can be concluded that the RGA should be
used with care. It is advisory to include an examination of the structure of
the considered transfer function matrices in the RGA analysis.
Furthermore, the more recently suggested HIIA method was employed to
quantify the level of interactions occurring between the inputs and outputs in
the considered bioreactor systems. It was noted that for the HIIA method to
give reasonable information in this particular case, the considered systems
had to be both scaled in a physically relevant way and low-pass filtered.
The filtering was performed to select the frequency band of interest. When
treating the systems according to this procedure, the HIIA method suggested
the same decentralized controller structure as the RGA, but it also gave
suggestions on other controller structures that may perform better. The
RGA is unable to give this extra information.
56
Chapter 4
Interaction Analysis and
Control of the Denitrification
Process
This chapter once again considers the problem of channel interaction in multivariable bioreactor systems. Similar to the previous chapter (Chapter 3)
nitrogen removal in the activated sludge process in a wastewater treatment
plant is studied. Here, the focus is on controlling the denitrification process
when an external carbon source is added. Thus, one of the two considered
control handles (input signals) is the readily biodegradable organic substrate
in the influent water (which has the same influence as an external carbon
source would have). The other input signal is the internal recirculation flow
rate. The nitrate concentration in the anoxic compartment and in the effluent are the two controlled signals (output signals). To evaluate the degree
of channel interaction, two different tools are compared: the well-known
Relative Gain Array (RGA) and the more recently proposed Hankel Interaction Index Array (HIIA). The results of the analysis are discussed from a
process knowledge point of view, and are also illustrated with some control
experiments. The main conclusion is that the HIIA gives a deeper insight
about the actual cross couplings in the system. This insight is also used in
order to design a suitable structured multivariable controller.
4.1
Introduction
In this chapter, both the RGA and the HIIA are calculated for different
operating points of a certain process in a bioreactor model. Cross couplings
in other processes in such systems have previously been studied by Ingildsen
(2002) and Halvarsson et al. (2005) (see Chapter 3). The bioreactor model
studied here models a pre-denitrifying wastewater treatment plant. The
results of the methods are compared and discussed. The aim is to illustrate
57
the different conclusions that can be drawn from the two measures. In
particular, it is shown how the HIIA can be used to determine more elaborate
control structures that may increase the closed loop performance compared
to the decentralized control case. The chosen control strategy as well as
a decentralized control strategy is also evaluated in a simulation study in
order to point out the validity of the analysis.
The structure of this chapter is as follows: The bioreactor model used to
illustrate the RGA and HIIA methods is discussed in Section 4.2, and the
presented model is analysed with respect to the RGA and the HIIA in Section 4.3. To illustrate the results of the analysis in Section 4.3, some control
experiments are performed in Section 4.4. In Section 4.5, the results in the
previous sections are discussed. The general conclusions are summarized in
Section 4.6. The used interaction measures, i.e. the RGA and the HIIA, are
defined and further discussed in Section 2.4 and in Section 2.5 in Chapter 2.
4.2
The bioreactor model
The RGA and the HIIA will be used on a bioreactor model describing reduction of nitrate in wastewater (conversion of nitrate to nitrous oxide, so
called denitrification). Generally, the bioreactor is connected to a clarifier,
and the process consisting of these two parts is commonly called an activated sludge process (ASP), see Henze et al. (1995). In recent years, the
control problems in this area have become more and more important due to
increased demands on the effluent water quality, see for instance Olsson and
Newell (1999), Ingildsen et al. (2002), Yuan et al. (2002) and Jeppsson et al.
(2002). In an ASP configured for nitrogen removal, ammonium is oxidized
to nitrate under aerobic conditions. This process is called nitrification. The
nitrate formed by the nitrification process, in turn, is converted into gaseous
nitrogen under anoxic conditions, this is the so called denitrification. Therefore, a multi-step configuration is generally needed in order to perform an
efficient nitrogen removal. If the anoxic part of the process is placed before
the aerobic, the process is said to be pre-denitrifying. In this chapter, a
pre-denitrifying process is considered, see Figure 4.1.
For the denitrification process to take place, a sufficient amount of organic substrate (readily biodegradable organic substrate) is needed as well
as an anoxic condition, i.e. absence of dissolved oxygen. In a pre-denitrifying
system, the access to nitrate in the anoxic part is achieved by recirculating
nitrate rich water from the aerobic to the anoxic part of the plant. To
ensure that enough readily biodegradable substrate is present, an external
carbon source (for example ethanol) is often added to the anoxic part. It
is thus natural to consider the flow rates of the internal recirculation and
the addition of an external carbon source as control signals (manipulated
variables) in the denitrification process, although the concentration of read58
ily biodegradable organic substrate in the influent water is here used as an
input signal instead of an external carbon source. The natural output signals are the nitrate concentration in the last anoxic compartment and the
nitrate concentration in the effluent water. Several papers in the literature
deal with the above described control problem, see for instance Carlsson and
Rehnström (2002).
The probably most used mathematical model describing biological nitrogen removal is the IAWQ Activated Sludge Model No. 1 (ASM1), see
Henze et al. (1987) for a full description. This is a rather complex nonlinear
model including eight different processes describing biomass growth and decay together with a number of hydrolysis processes. Due to the complexity
of the model, it is not very suitable for control purposes. Instead a somewhat simplified version of the ASM1 will be used in the analysis carried
out in this chapter. In the analysis, this simplified model is linearized for
different operating points, and the RGA and HIIA analyses are performed
on the linearized models. The model used here describes a pre-denitrifying
process with one anoxic and one aerobic compartment, see Figure 4.1. The
compartments are assumed to be completely mixed. For a full derivation of
this model and a discussion of the parameters therein, see Ingildsen (2002)
or Samuelsson et al. (2004).
Q
Q−Qw
Qi
Qr
Qw
Figure 4.1: An ASP with one anoxic and one aerobic compartment and a
clarifier. The influent flow rate is denoted Q, the internal recirculation flow
rate Qi , the flow rate of recirculated sludge Qr and the excess sludge flow
rate Qw .
In the sequel, the internal recirculation flow rate is denoted Qi [m3 /h]
and the influent concentration of readily biodegradable substrate SS,in [mg
(COD)/l]. The nitrate concentration in the anoxic compartment is denoted
an [mg (N)/l] and the nitrate concentration in the effluent water (aerobic
SNO
e
compartment) is denoted SNO
[mg (N)/l]. The input signal vector of the
system can thus be defined as
T
u = Qi SS,in
(4.1)
and the output signal vector as
an
T
e
y = SNO
SNO
.
59
(4.2)
4.3
Analysis of the model
In this section, the bioreactor model will be analysed using the RGA and
the HIIA described in Section 2.4 and Section 2.5.3 in Chapter 2. One
objective is to investigate the cross couplings in the system and to choose
suitable control structures. The desired control structure may change with
different operating points since the system is nonlinear. Another objective is
to illustrate the performance of both methods, i.e. what different conclusions
that can be drawn from each of the methods: For instance, how the HIIA
can be used to determine a suitable sparse multivariable control structure,
and what can be seen from the RGA in the corresponding case.
A first impression of the possible (stationary) cross couplings in the system can be obtained from the steady state operational maps of the nonlinear
model. Such operational maps are also used by, for instance, Ingildsen (2002)
and Galarza et al. (2001) in order to analyse the bahaviour of bioreactors. In
Figures 4.2 and 4.3 the level curves of the stationary nitrate concentrations
of the anoxic and aerobic compartments respectively are plotted against
constant values of the two inputs. From these operational maps it is clear
that the system behaves nonlinearly, i.e. the stationary characteristics are
different depending on the choice of operating point. The static gain in some
of the channels actually changes sign between the different operating ranges
(which could also be seen in a simple step response analysis). From Figure
e for
4.3 it can be seen that in order to accomplish a change stationary in SNO
low values of the input signals, the concentration of readily biodegradable
substrate in the influent, SS,in should be used. For larger values of the input
signal SS,in , the change seems to be best accomplished if the internal recire
for
culation flow rate Qi is altered. Note also the multiple equilibria of SNO
a given value of SS,in . In Figure 4.2 it is seen that both inputs affects the
an over the whole operating range, although the relative imporoutput SNO
tance of the inputs depends on the choice of operating point. A conclusion
that can be drawn is, however, that for high values of the input signal SS,in ,
an is low for both the input signals.
the gain of SNO
Although these plots contain only stationary values for the open loop
system and therefore cannot be assumed to fully describe all cross couplings
in the system, they will be used to roughly validate the results obtained
from the linear analysis utilizing the RGA and the HIIA.
As indicated above, Figure 4.3 implies that there are three different areas
in which the process may show different cross couplings. In order to analyse
this bahaviour, three different operating points will be considered. The three
operating points are indicated by stars “∗” in the operational maps in Figure
4.2 and Figure 4.3. These are the ones corresponding to the constant input
60
San [mg/l]
NO
4
1
2
3
0.5
7
6
10
12
5
9
13
5.5
11
8
6
x 10
0.1
4
5
7
0.
5
2
3
5
1
0.
5
0.0
8
2
05
0.
1
6
9
2.5
4
i
3
11
8
4
3.5
10
Q [m3/day]
4.5
7
1.5
1 5
20
6
3
1
2
4
40
0.03
0.5
1
0. 0.05
60
80
S
S,in
0.03
100
[mg/l]
120
140
160
Figure 4.2: Stationary operational map for the nitrate concentration in the
an . The stars show the locations of the three operatanoxic compartment, SNO
ing points.
signals given by
T
ū1 = 35 000 40 ,
T
ū2 = 26 000 100 ,
T
ū3 = 20 000 120
(4.3)
where the units are m3 /day for the first input signal and mg(COD)/l for
the other. The first operating point given by ū1 lies in the area where the
second input signal, the concentration of readily biodegradable substrate in
the influent water, SS,in , is low. The second operating point given by ū2 lies
in the transition phase between the areas, and the third point is in the area
where the concentration of readily biodegradable substrate in the influent
water is high.
4.3.1
Linearization and scaling of the model
Both the RGA and the HIIA are defined for linear models. In order to perform the analysis, the model needs to be linearized around each operating
point. Since the process characteristics are clearly different in the three operating points, three different linearizations are needed in order to properly
analyse the system, one for each operating point. In a small neighbourhood
of each point, the linearized model will mimic the characteristics of the nonlinear system. Thus, the analysis in the following section is strictly valid
only in the same operating points. However, as seen in Figure 4.3, the ope
erational map for SNO
can clearly be divided into three areas with the same
61
Se [mg/l]
4
5
7
12
10
8
13
15
14
17
18
5.5
16
6
6
NO
4
x 10
5
9
11
5
7
12
10
8
13
15
17
16
18
i
14
4
3.5
6
3
6
11
2.5
9
Q [m3/day]
4.5
7
2
1
20
12
13
17
16
1.5
14
15
8
10
8
9
11
40
60
80
S
S,in
10
100
[mg/l]
120
9
140
160
Figure 4.3: Stationary operational map for the nitrate concentration in the
e . The stars show the locations of
aerobic compartment (effluent water), SNO
the three operating points.
gain characteristics, which motivates that the results from the analysis hold
with good accuracy over larger neighbourhoods. This can also be verified
via further linearizations or simulations.
In practice, the linearizations were made using the MATLAB function
linmod. From the linearization procedure, standard linear state-space models of the form
∆ẋ = A∆x + B∆u,
∆y = C∆x
(4.4)
are obtained. Here x is a 7×1 state vector containing concentrations of
seven different compounds (for instance the output signals) and u is the
input signal vector defined in (4.1) as
T
u = Qi SS,in .
The symbol ∆ refers to deviation from the operating point so that ∆x = x−x̄
and ∆u = u − ū, where ū is the constant input signal vector that in steady
state renders the operating point x̄. The output signal vector is defined in
an S e ]T and ∆y = y − ȳ. The matrix A is a 7×7 matrix, B
(4.2) as y = [SNO
NO
is a 7×2 matrix and C is consequently given by a 2×7 matrix that is independent of the chosen operating point. The corresponding transfer function
matrix is then
G(s) = C(sI − A)−1 B.
62
(4.5)
As mentioned in Section 2.5.5, the HIIA is a scaling dependent tool. In
order to be able to compare the different elements of the HIIA directly, the
linearized model obtained by (4.4) and (4.5) must be properly scaled. A
standard procedure (as described by for instance Skogestad and Postlethwaite (1996)) is to scale the model according to
G(s) = Dy−1 Go (s)Du
where the original input-output model is given by
y o (t) = Go (p)uo (t)
and the superscript “o ” denotes the original (or physical) variables. Thus,
Go (p) denotes the original transfer function matrix between output y o (t)
and input uo (t); Du and Dy are diagonal scaling matrices.
There exist many different possibilities for choosing the scaling matrices
Dy and Du depending on what the desired achievements are. In this chapter
the model is scaled in such a manner that the maximum deviation from the
average value of each signal lies in the interval [-1,1]. This is achieved here
by choosing
60000 0
,
Du =
0
160
3 0
Dy =
0 3
where the diagonal elements in Du are the maximum allowed value of each
input signal and the elements in Dy states that a maximum deviation in the
output of three units is accepted.
4.3.2
RGA analysis of the model
To test the ability of the RGA to provide reasonable pairing suggestions,
the stationary RGA was calculated for the linearized models for each of the
chosen operating points. The results were
1.1979 −0.1979
,
(4.6)
RGA(Gū1 (0)) =
−0.1979 1.1979
0.3327 0.6673
RGA(Gū2 (0)) =
,
0.6673 0.3327
(4.7)
0.3263 0.6737
RGA(Gū3 (0)) =
.
0.6737 0.3263
(4.8)
63
Clearly, since the anti-diagonal elements in the RGA matrix corresponding to the first operating point are negative and the diagonal elements are
fairly close to one, the RGA in this case suggests a diagonal controller,
an
i.e. that the first input signal Qi should be used to control the output SNO
e . The latter also
and the second input, SS,in , should control the output SNO
seems probable when comparing the results to the operational map in Figure
4.3, provided that a decentralized control structure is to be used.
For both the other operating points, anti-diagonal control structures are
suggested, although without any strong indication since the diagonal element
of the RGA matrix is quite far from one in both cases. For the second
operating point it is hard to evaluate the validity of this from the operational
maps, since the operating point lies in a transition phase. The result for
the third operating point, however, seems reasonable when considering the
operational maps.
4.3.3
HIIA analysis of the model
The HIIA is a measure that indicates the size of the impact of each input
signal on each output signal. It is therefore interesting to compare the results
from the HIIA analysis to the results from the RGA analysis. The HIIA as
defined in Section 2.5 was calculated for the linearized and scaled models
for each operating point. The obtained results were
0.1425 0.3596
ū1
ΣH =
,
(4.9)
0.0450 0.4530
ΣūH2
0.1252 0.1379
=
,
0.4009 0.3361
(4.10)
ΣūH3
0.0090 0.0053
=
.
0.7441 0.2415
(4.11)
For the first operating point, the HIIA indicates that the first output
an , is about equally dependent on both input signals since the elsignal, SNO
ements on the first row in the HIIA matrix have the same magnitude of
orders. Furthermore, it can be seen from the second row that the second
e , mostly depends on the second input signal, S
output, SNO
S,in . The corresponding HIIA element is also of the the same size as the elements in the
first row. The first element in the second row is however about ten times
smaller than the second element in this row. Thus, the second input should
definitely be employed in the control of the second output. Intuitively, the
first input should then be used in the control of the first output in order to
obtain a satisfying result. As pointed out in Section 2.5.3, it could be hard
to determine whether an entry in the HIIA matrix is large enough to be
64
relevant in the control design, and it is also beyond the scope of this thesis
to provide any general rules for such a decision. However, provided that a
reduction of control structure complexity is desirable together with the fact
that the first element on the second row is clearly the smallest element in
the matrix, it is most natural to ignore the interactions in the channel from
the first input to the second output in an eventual design of a reduced order
controller in this operating area. A summation of the elements of the HIIA
matrix as described in Section 2.5.3 also gives at hand that the contribution
of the first element on the second row is relatively small in the total sum.
The interpretation of the HIIA matrix in this case is thus that a good
option for controlling the process in this operating range may be a sparse
multivariable controller taking the analysis results into account. The control
structure selection would therefore in this case be
F1 (s) F3 (s)
U (s) =
E(s)
(4.12)
0
F2 (s)
where F1 (s), F2 (s) and F3 (s) are the transfer functions of each sub controller,
U (s) is the Laplace transform of the input signal vector as defined in (4.1)
and E(s) is the Laplace transform of the control error vector, i.e. a vector
an and S e . The validity of this choice of
containing the control errors of SNO
NO
control structure is also verified in the simulations in Section 4.4.
In the second operating point, corresponding to the transition phase
between the operating ranges, all elements in the HIIA matrix are of the
same magnitude of orders. This indicates that a full multivariable control
structure should be chosen here.
Finally, in the third operating point both elements in the first row of the
HIIA matrix are small compared to the elements in the second row. This
indicates that the first output signal could be difficult to control at all. A
physical interpretation of this will be given in Section 4.5.
Note that no pre-filtering of the transfer functions is performed before
calculating the HIIA. A pre-filtering procedure was evaluated but made no
difference in the results and was therefore omitted.
4.4
Control simulations
In order to illustrate the findings in the previous section, some control experiments were performed. As an example, control of the nonlinear system
in the neighbourhood of the first operating point given by the input signal
ū1 in (4.3) is considered here. Both a decentralized control structure and a
simple multivariable strategy are evaluated. The purpose is to compare the
results from the linear analysis in the previous section to the actual results
obtained when controlling the nonlinear system. In particular, it is shown
how the information extracted from the HIIA matrix can be used to design
a more elaborate control structure than decentralized control.
65
4.4.1
Decentralized control
A decentralized control law with the input-output pairing recommended
by both the RGA analysis in (4.6) and the HIIA in (4.9) was evaluated.
The results in (4.6) and (4.9) both suggested that if a decentralized control
structure was to be used, the natural pairing selection was to control the first
an , by manipulating the first input signal, Q . Consequently,
output signal, SNO
i
e , should be controlled by manipulating the second
the second output, SNO
input signal SS,in . Thus, the decentralized control law can be written as
Qi (s) = F1 (s)E1 (s),
(4.13)
SS,in (s) = F2 (s)E2 (s)
(4.14)
where E1 (s) is the Laplace transformed control error of the first loop, i.e.
an,sp
an (t), and E (s) is the Laplace transformed control
e1 (t) = SNO
(t) − SNO
2
e (t) − S e,sp (t) since this process is
error of the second loop, i.e. e2 (t) = SNO
NO
an,sp
e,sp
known to have negative gain. Here, SNO
and SNO
denote the set point
values of the output signals, respectively.
The controllers F1 (s) and F2 (s) were in this experiment chosen as ordinary PI-controllers and were tuned in order to achieve approximately the
same rise time in both control loops to make later comparisons meaningful.
The used PI-controllers were
F1 (s) = 8000 +
F2 (s) = 15 +
8
s
7000
,
s
(4.15)
(4.16)
where the large difference in size between the parameters in the controllers
are explained by the large gain differences in the open loop systems.
The decentralized control law (4.13)–(4.14) was then used to control the
nonlinear system. Figure 4.4 shows the output responses of the system when
an,sp
a 10 % step change in the set point of the first output, SNO
, is applied. In
the same way, Figure 4.5 shows the output responses for a 10 % step change
e,sp
in the set point value of the second output, SNO
. The conclusions from
an , is affected by both the
the HIIA in (4.9) were that the first output, SNO
e
input signals while the second output SNO
is mainly affected by the second
input signal, SS,in . Considering the control law (4.13)–(4.14) it is clear by
an,sp
definition that a change in the set point SNO
causes a direct change in the
first input signal, Qi while the second input signal is unaffected. In the same
e,sp
manner, a change in the other set point SNO
causes an immediate change
in the second input, SS,in , but leaves the first input signal unaffected.
Combining this reasoning with the results of the HIIA, it can thus be
an,sp
expected that a step change in SNO
will have a relatively small impact
e,sp
e
on the output SNO , while a step change in SNO
will have a larger impact
66
an
SNO [mg/l]
11
10
9
8
40
45
50
55
45
50
55
60
65
70
75
80
60
65
70
75
80
15.4
NO
Se [mg/l]
15.2
15
14.8
14.6
14.4
40
time [h]
Figure 4.4: Decentralized control output responses of the system for a step
an . Upper plot: Solid line shows
change in the set point of the first output, SNO
an . Dashed line shows the set point value. Lower
the response of the output SNO
e
plot: The response of the second output SNO
is plotted.
in the first output channel. This is also confirmed by the simulation results in Figures 4.4 and 4.5. The disturbance response of the first output is
e,sp
is changed than vice versa. It
considerably larger when the set point SNO
should be noted that like the stationary operational maps, these simulations
are merely strong indications that the HIIA in (4.9) provides a reasonable
result. Of course, the performance of the closed loop system depends also
on the choice of decentralized controller. However, using reasonable controllers tuned to achieve same rise time in both control loops should make
the comparison above relevant.
4.4.2
Multivariable control
Next, a simple multivariable control strategy is evaluated. The specific
structure of the controller is determined using the HIIA analysis results. In
Section 4.3.3, it was concluded from the HIIA analysis that in the neighbourhood of the first operating point given by the input signal ū1 in (4.3),
a structured multivariable controller might be preferable to decentralized
control. To choose a suitable control structure, first note that according to
(4.9), the dependence of the first input signal on the second output signal
is relatively low. One possibility to perform the control design could therefore be to approximate the nonlinear system with a triangular linear system
67
9.4
NO
San [mg/l]
9.2
9
8.8
8.6
8.4
40
45
50
55
45
50
55
60
65
70
75
80
60
65
70
75
80
SeNO [mg/l]
16
15
14
13
40
time [h]
Figure 4.5: Decentralized control output responses of the system for a step
e . Upper plot: The response
change in the set point of the second output, SNO
an
of the first output SNO is plotted. Lower plot: Solid line shows the response
e . Dashed line shows the set point value.
of the output SNO
according to
an
SNO (s)
G1 (s) G3 (s)
Qi (s)
e (s) =
SNO
0
G2 (s) SS,in (s)
(4.17)
where the elements in the transfer function matrix are obtained by linearizing the nonlinear system in the neighbourhood of the operating point given
by the input signal ū1 .
e , is assumed to depend only on the second
Since the second output, SNO
input, SS,in , it is convenient to choose SS,in according to
e,sp
e
SS,in (s) = F2 (s)(SNO
(s) − SNO
(s)).
(4.18)
an is affected by both input signals. A suitable
The first output signal, SNO
choice might be to take this into account in the control law, for instance by
letting
an,sp
an
Qi (s) = F1 (s)(SNO
(s) − SNO
(s)) + F3 (s)SS,in (s)
(4.19)
where the latter term can be seen as a feedforward part.
Inserting the control signals (4.18)–(4.19) into the expression for the
68
linearized system (4.17) directly yields
G1 (s)F1 (s)
S an,sp (s)+
1 + G1 (s)F1 (s) NO
(G1 (s)F3 (s) + G3 (s))
+
SS,in (s),
1 + G1 (s)F1 (s)
−G2 (s)F2 (s) e,sp
e
SNO
(s) =
S (s)
1 − G2 (s)F2 (s) NO
an
SNO
(s) =
(4.20)
(4.21)
and it is seen that the system will be completely decoupled if the feedforward
controller F3 (s) can be chosen as
F3 (s) =
−G3 (s)
.
G1 (s)
(4.22)
In the simulations, the controller F3 (s) was obtained in the way described
above. The nonlinear system was linearized in a neighbourhood correspondan
e ] = [9 15] which corresponds to the input signal values
ing to [SNO
SNO
u0 = [39400 43]. Using (4.22) on the obtained linear model resulted in
this case in a strictly proper linear feedforward controller. The controllers
F1 (s) and F2 (s) used in the experiment were the same as before, see (4.15)
and (4.16). The presented control law was applied to the nonlinear system.
Figures 4.6 and 4.7 show the output responses for step changes in the set
point of each output as in the previous simulations. Comparing Figures
4.5 and 4.7 it can be seen that the impact of the input signal SS,in on the
an is reduced when the feedforward controller is included,
output signal SNO
e
is applied, the magnitude of
i.e. when a step change in the set point of SNO
an
the disturbance response in SNO is much lower.
In the simulation example considered here, the decoupling above is only
approximate since the system is nonlinear. Thus, the disturbance in the
an in Figure 4.7 is not completely attenuated. Further, as menoutput SNO
tioned, in the true nonlinear system the impact of the first input signal on
the second output is not strictly zero as in the linear model example (4.17).
e in Figure 4.6 is not strictly zero.
Therefore, the disturbance response of SNO
This example, however, shows how the HIIA can be used to determine an
approximate decoupling control law for a nonlinear system in the neighbourhood of some operating point.
4.5
Discussion
Here, the results in the previous section are further discussed. The results
from the RGA and the HIIA analysis are also compared to each other and
the relevance of the results are discussed from a process knowledge point of
view. The validity of the results is also discussed from a more general point
of view.
69
an
SNO [mg/l]
11
10
9
8
40
45
50
55
45
50
55
60
65
70
75
80
60
65
70
75
80
15.4
NO
Se [mg/l]
15.2
15
14.8
14.6
14.4
40
time [h]
Figure 4.6: Feedforward control output responses of the system for a step
an . Upper plot: Solid line shows
change in the set point of the first output, SNO
an . Dashed line shows the set point value. Lower
the response of the output SNO
e
plot: The response of the second output SNO
is plotted.
For the first operating point, the RGA clearly suggested a diagonal pairan and that
ing, i.e. the input Qi should be used to control the output SNO
e . This seems reasonable when
the input SS,in should control the output SNO
considering the operational map in Figure 4.3 where the stationary value
e
of SNO
seem to depend mostly on the stationary values of SS,in . From the
other operational map in Figure 4.2 it is harder to draw any conclusions.
The HIIA analysis also shows that if a decentralized controller is to be used,
diagonal pairing is preferable. However, the HIIA also provides the inforan is also dependent on
mation that in this operating point, the output SNO
the input SS,in and thereby that a sparse multivariable controller according
to (4.12) may be a better option. Thus, the HIIA adds valuable information
about the cross couplings in this operating point. The results are also in line
with the conclusions that can be drawn from general process knowledge. In
the first operating point, SS,in is comparatively low, and there will be a lack
of readily biodegradable substrate available for denitrification. Since there
is not enough readily biodegradable substrate, the denitrification process in
the anoxic compartment will be incomplete, which means that all of the rean will
circulated nitrate is not denitrified. Thus, the nitrate concentration SNO
depend both on how much readily biodegradable substrate is added and on
how much nitrate that is recirculated from the aerobic compartment to the
anoxic, i.e. on both input signals. In this situation the nitrate concentration
e , depends mainly on the input S
in the aerobic compartment, SNO
S,in , because when the denitrification in the anoxic compartment is incomplete, the
70
9.4
NO
San [mg/l]
9.2
9
8.8
8.6
8.4
40
45
50
55
45
50
55
60
65
70
75
80
60
65
70
75
80
SeNO [mg/l]
16
15
14
13
40
time [h]
Figure 4.7: Feedforward control output responses of the system for a step
e . Upper plot: The response
change in the set point of the second output, SNO
an
of the first output SN O is plotted. Lower plot: Solid line shows the response
e . Dashed line shows the set point value.
of the output SNO
internal recirculation only leads to an internal transport of nitrate that does
not affect the effluent nitrate concentration of the system. In other words,
for low values of SS,in , there is no meaning in increasing Qi since it does not
e . To conclude, both the result
affect the effluent nitrate concentration SNO
from the RGA analysis and from the HIIA analysis therefore seem valid.
The HIIA gives more information about the actual cross couplings in the
system, and thereby gives an opportunity to design a better controller. The
control simulations in Section 4.4 also confirm these conclusions. The particular decentralized control structure suggested by the RGA is also suggested
by Ingildsen (2002), however not as a result of a cross coupling analysis but
from an economical point of view.
In the second operating point, the RGA gives an indication (however
not very strong) that the input-output pairing now should be the reversed,
i.e. an anti-diagonal pairing. The conclusion from the HIIA analysis is that
a full multivariable controller should be used. It is hard to evaluate the
relevance of the RGA analysis from the operational maps or from a physical
reasoning. It is clear from the operational maps that in this transition phase,
both outputs rely on both inputs, and the HIIA thus seems to provide a
reliable result.
In the third operating point, the RGA suggests the same pairing as in the
second operating point with approximately the same order of magnitude on
the RGA elements. Here, an interesting difference occur when considering
71
the HIIA analysis. Since the elements on the first row of the HIIA is close
an is difficult to affect at all using
to zero, it indicates that the first output SNO
e , is mainly affected
any of the two input signals. The second output, SNO
by the first input signal, Qi , but also to some extent by the other input
SS,in . A physical interpretation is that in this case the access of readily
an takes very
biodegradable substrate is sufficient, and the concentration SNO
low values. This means that the denitrification in the anoxic compartment
is complete. Since there is a good access to readily biodegradable substrate
an takes low values, S an will not decrease
and the nitrate concentration SNO
NO
further if more readily biodegradable substrate SS,in is added. If less readan will not increase as long as the
ily biodegradable substrate is added, SNO
addition is large enough for the denitrification to remain complete. In this
operating point, when more nitrate is recirculated through an increase of the
an will not be affected much as long as
internal recirculation flow rate, Qi , SNO
the transition phase is not passed, i.e. while the denitrification still is complete. If instead the internal recirculation flow rate is decreased, less nitrate
an remains relatively unchanged.
has to be denitrified and the stationary SNO
Thus, the gains from both input signals are low. As mentioned, the second
e , is in this operating point mainly affected by the internal reoutput, SNO
circulation flow rate, Qi . If, for instance, more nitrate is recirculated, more
e
nitrate will also be denitrified and the nitrate concentration SNO
is thus
reduced. This bahaviour is at least reflected in the HIIA, although these
conclusions are hard to draw directly from the HIIA without any prior process knowledge. In order to control the system in this operating range, i.e. so
that a low effluent nitrate concentration is obtained, one possible strategy
is to add sufficient amounts of readily biodegradable substrate to achieve
complete denitrification in the anoxic compartment. The nitrate concentration in the aerobic compartment can then be moderated by adjusting the
internal recirculation flow rate. Expressed in control terminology, the input
an at a low set point. The input
SS,in should be used to keep the output SNO
e
Qi should be used to control the output SNO
to some desired value. This
specific pairing is also what the RGA recommends, although as seen above,
additional useful information can be extracted from the HIIA.
The main conclusion that can be drawn from this study is that the HIIA
is a more powerful tool than the RGA when it comes to evaluating channel
interactions in general, and may therefore be used in order to determine more
elaborate control structures than just decentralized controllers. It should,
however, be noted that the RGA assumes a decentralized control law, while
the HIIA rather investigates the controllability and observability of each
partial input-output subsystem. The RGA certainly gives some indication
of how large the channel interactions may be, but does not preserve the
structure of the transfer function matrix. The HIIA gives a detailed information of the size of the interactions for each channel. One way of describing
it is that the HIIA investigates things that the RGA cannot because of the
72
needed preassumption. Another difference is that the HIIA takes the whole
frequency range into account. This could certainly increase the usefulness
for many applications with high energy content at higher frequencies. In
this particular study, however, this has very little to do with the different
conclusions that can be drawn since most of the energy is located at low
frequencies. For instance, the HIIA seemed insensitive to a a pre-filtering
procedure cutting off higher frequency components. The results of the HIIA
actually follows the conclusions that can be drawn directly from Figures
(4.2)–(4.3), although higher frequency components also are taken into account in the HIIA. The most obvious drawback of the HIIA method may
be that the results are scaling dependent. However, in the study in this
chapter, the results did not seem very sensitive to changes in the scaling
matrices, as long as the elements were chosen fairly reasonable in a physical
sense.
It should be noted that in reality, it might be sufficient to run the process
only in the operating range described by the first operating point depending
on the actual demands on effluent nitrate concentration. In the overall
consideration, process economy should also be weighted into the choice of
control structure and desired operating range, not just the goal to reduce
nitrate concentration as much as possible. In a case where the process
may run in different operating ranges, however, the use of different control
structures in the different operating ranges may provide better control of
the process.
4.6
Conclusions
In this chapter, the cross couplings in a bioreactor model describing a predenitrifying wastewater treatment plant have been studied. Two different
tools were used to evaluate the cross couplings, the Relative Gain Array
(RGA) and the Hankel Interaction Index Array (HIIA). A general conclusion from the presented analysis is that both the RGA and the HIIA give
reasonable results for the studied example. An important difference is, however, that the HIIA provides information that the RGA does not. The results
from the HIIA analysis gives an understanding of the actual cross couplings
in the system in terms of magnitude and character, and are thereby useful for suggesting suitable multivariable control structures. The validity of
the results is also illustrated by means of some control experiments where
the control structure suggested by the HIIA outperformed the decentralized
control structure.
73
74
Chapter 5
Economic Efficient Operation
of a Pre-denitrifying
Activated Sludge Process
In this chapter, the choice of optimum set points and cost minimizing control
strategies for the activated sludge process are treated. Both the denitrification and the nitrification process are considered. In order to compare
different criterion functions, simulations utilizing the COST/IWA simulation benchmark (BSM1) are considered. By means of operational maps the
results are visualized. It is found that it is easy to distinguish set point areas where the process can be said to be efficiently controlled in an economic
sense. The characteristics of these set point areas depend on the chosen
effluent nitrate and ammonium set point as well as the distribution of the
different operating costs. It is also discussed how efficient control strategies
may be accomplished.
5.1
Introduction
In recent years, cost minimization has become increasingly important in
the control and operation of wastewater treatment plants. In order to run
a plant economically, operational costs such as pumping energy, aeration
energy and dosage of different chemicals should be minimized. At the same
time, the discharges to the recipient should be kept at a low level. Of course,
minimizing the operational costs and at the same time treat the wastewater
properly may lead to a conflict of interest that must somehow be solved.
The main problem is how to keep the effluent discharges below a certain
pre-specified limit to the lowest possible cost, see Olsson and Newell (1999).
Part of the answer is to design the control algorithms in such a way that
the overall operational costs are minimized. This goal can be attained in
different ways. As an example, the controller set points could be separately
75
optimized or the cost could be minimized online by some control strategy,
for instance model predictive control (MPC). See Qin and Badgwell (2003)
for how MPC can be used in this context. In some countries, the authorities
charge according to effluent pollution. A possible way to formulate the
on-line minimization criterion in such a case is to use a cost function that
takes actual costs (energy and chemicals) into account and at the same time
economically penalizes the effluent discharges.
Over the years, much effort has been put in developing economically efficient control strategies for operation of wastewater treatment plants. An
interesting cost function is presented in Carstensen (1994) where the effluent
nitrogen is penalized using a piecewise linear discontinuous function. Effluent ammonium is penalized in a similar way. The papers (Yuan et al., 1997),
(Yuan et al., 2002) and (Yuan and Keller, 2003) all consider efficient control
of the denitrification process. The optimum set point for the nitrate concentration at the outlet of the anoxic zone is then found to be near 2 mg(N)/l
or, at least, in the interval 1–3 mg(N)/l. In (Ingildsen et al., 2002), an optimization of the dissolved oxygen (DO) and nitrate set points is made. In
(Galarza et al., 2001) steady-state operational maps are utilized to examine
the feasible operating area for two activated sludge processes with emphasis on sensitivity analysis. Fuzzy control evaluated using multi-criteria cost
functions is the subject of (Cadet et al., 2004). In (Vanrolleghem and Gillot,
2002), different multi-criteria control strategies are evaluated.
In this chapter, the choice of optimum set points and cost minimizing control strategies for an activated sludge process configured for predenitrification are evaluated. Both the denitrification and the nitrification
are treated. The manipulated variables (input signals) are the internal recirculation flow rate and the flow rate of an external carbon source and the controlled variables (output signals) are the effluent ammonium concentration,
the nitrate concentrations in the last anoxic compartment and in the effluent.
In order to compare the impact of different criterion functions, stationary
simulations utilizing the COST/IWA simulation benchmark (BSM1), see
Copp (2002), are considered. By means of operational maps the results are
visualized. It is also discussed how efficient control can be accomplished.
The organization of the chapter is as follows: In Section 5.2, the simulation model (BSM1) is briefly described together with the associated operational costs. In Section 5.3 simulation results are presented using operational
maps. The simulation results are discussed and interpreted in Section 5.4.
Finally, in Section 5.5 the general conclusions are drawn.
5.2
The model and the operational cost functions
In the simulation study presented in this chapter, the COST/IWA simulation
benchmark model number 1 (BSM1) is used, see Jeppsson and Pons (2004)
76
for a general survey and Copp (2002) for a more technical description. BSM1
is an important tool for simulation of the activated sludge process in various
realistic wastewater treatment scenarios. In BSM1, five biological reactors
are implemented using the IWA activated sludge model No. 1 (ASM1), see
Henze et al. (1987). Despite being a fairly complex model, ASM1 has some
limitations. For example, the pH does not affect the process rates. The pH
of the wastewater should hence be near neutrality. The alkalinity is, however, calculated in ASM1 and may be used to detect pH related problems,
see further Henze et al. (1987). The model plant is pre-denitrifying with two
anoxic and three aerated compartments. A secondary settler is also implemented. To allow for consistent experiment evaluation, three dynamic data
input files are defined, each describing different weather conditions. The
purpose of the simulation benchmark is to provide an objective and unbiased tool for performance assessment and evaluation of proposed automatic
control strategies. A great benefit of the benchmark is that it allows for
comparison of many automatic control strategies given the same conditions.
The aim here is to analyse the stationary operational costs of the activated sludge process, and in order to visualize the costs, these are presented
in stationary operational maps together with the considered output signals.
The output signals are the nitrate concentration in the last anoxic coman [mg(N)/l], the nitrate concentration in the effluent, S e
partment, SNO
NO
e
[mg(N)/l] and the ammonium concentration in the effluent, SNH
[mg(N)/l].
The available control handles considered in this chapter are the internal recirculation flow rate Qi [m3 /day], the flow rate of an external carbon source,
Qcar [m3 /day], and when the nitrification is studied, the concentration of
dissolved oxygen in the aerated compartments, DO.
To express the cost for controlling the denitrification process, a number
of partial costs are taken into account:
• Pumping costs due to the required pumping energy.
• Aeration costs due to the required aeration energy (which varies due
to the input load). Excessive use of an external carbon source has a
large impact on the required aeration energy, and thus on the total
cost.
• External carbon dosage costs.
• Possible fees for effluent nitrate discharge and for effluent ammonium
discharge.
In BSM1, the total average pumping energy over a certain period of time, T ,
depends directly on the internal recirculation flow rate Qi and is according
to Copp (2002) calculated as
Z
0.04 t0 +T
EP =
Qr (t) + Qi (t) + Qw (t) dt
(5.1)
T t0
77
expressed in units of kWh/day. In (5.1), Qr denotes the return sludge flow
rate and Qw the excess sludge flow rate, both in units of m3 /day.
The average energy in kWh/day required to aerate the last three compartments can in turn be written as
24
EA =
T
Z
t0 +T
t0
5
X
0.4032KL ai (t)2 + 7.8408KL ai (t) dt
i=3
(5.2)
where KL ai (t) is the oxygen transfer function in the aerated tank number i
in units of h−1 .
Further, assuming a prize kcar [EUR/m3 ] on the external carbon source
and that an external carbon flow rate, Qcar (t) [m3 /day], is fed into the
process during the time interval T , the cost per day of the external carbon
flow rate is
Z
1 t0 +T
kcar Qcar (t)dt
(5.3)
Ccar =
T t0
expressed in EUR/day.
Normally, when nitrogen concentrations in the effluent water are economically charged, the fees consider the total nitrogen discharges in the
effluent water. In this chapter, the first part only considers the denitrification process. Therefore, the effluent discharge fees used in this chapter
are separated into one fee for the effluent nitrate and one for the effluent
ammonium.
5.2.1
The nitrate fee
A reasonable way to describe a fee for the effluent nitrate discharge is to
let the fee depend on how large mass of nitrate that is discharged per time
unit. This depends, of course, on the effluent flow rate, Qe (t), and the
e (t). The nitrate discharge cost
nitrate concentration in the effluent, SNO
may be expressed as (in EUR/day)
CNO =
1
T
Z
t0 +T
t0
e
fNO (Qe (t), SNO
(t))dt
(5.4)
where fNO is some function describing the fee. Now, assuming a constant
energy prize, kE , the total cost expressed (in EUR/day) can be calculated
during a representative time interval T from (5.1)–(5.4) as
Ctot = kE (EP + EA ) + Ccar + CNO .
(5.5)
The fee functions for the discharge of nitrate can have different forms.
Normally, the fee functions are set-up by the legislative authorities. Here,
three typical fee functions are considered:
78
1. No charge is added for the disposal of nitrate, i.e.
e (t)) ≡ 0. In practice, even though not associated with
fNO (Qe (t), SNO
a direct fee, it is common to have legislative limits on the average
effluent nitrate concentration in such a case.
2. Effluent nitrate is charged with a constant cost, say ∆αNO per discharged kg. Such a fee function is
e
e
(t)) = ∆αNO SNO
(t)Qe (t).
fNO (Qe (t), SNO
(5.6)
3. Effluent nitrate is charged with a constant cost per discharged kg,
∆αNO , up to a legislative discharge limit for the effluent concentration,
αlimit,NO . Above this limit the cost of discharging additional nitrate
is ∆βNO . Exceeding the limit also imposes an additional charge of
β0,NO per volume effluent water. A mathematical description of this
cost function is given in (5.7):
e
fNO (Qe (t), SNO
(t)) =

e
e (t) ≤ α

if SNO
limit,NO
∆αNO SNO (t)Qe (t)
= ∆αNO αlimit,NO Qe (t) + β0,NO Qe (t)+


e (t) − α
e (t) > α
+∆βNO (SNO
if SNO
limit,NO )Qe (t)
limit,NO .
(5.7)
In the cost functions above, the effluent flow rate is calculated as Qe (t) =
Qin (t) − Qw (t). The difference between the cost functions presented in
(Carstensen, 1994) and the one in (5.7) is that here only the nitrate discharge is penalized, while in (Carstensen, 1994) the nitrate and ammonium
concentrations are lumped together and charged.
5.2.2
The ammonium fee
Effluent ammonium is penalized according a fee function that is similar to
the fee for effluent nitrate. The fee function is described by:
e
fNH (Qe (t), SNH
(t)) =

e
e (t) ≤ α

if SNH
limit,NH
∆αNH SNH (t)Qe (t)
= ∆αNH αlimit,NH Qe (t) + β0,NH Qe (t)+


e (t) − α
e (t) > α
+∆βNH (SNH
if SNH
limit,NH )Qe (t)
limit,NH .
79
(5.8)
5.3
Simulation results
The considered cost functions for penalizing nitrogen discharge are evaluated
utilizing BSM1. In the simulation study, the benchmark WWTP is fed with
constant influent flow rate of wastewater, 18 446 m3 /day. Basically, the
default values of BSM1 are used, with the following exceptions:
• The influent is assumed to have a concentration of readily biodegradable substrate, SS,in , of 60 mg(COD)/l.
• The external carbon source is ethanol with a COD of 1.2·106 mg
(COD)/l.
• No limit for the carbon dosage is assumed.
• The last three compartments are aerated utilizing DO controllers with
a fix set point of 2 mg/l in the first part of the study where only the
denitrification is considered. The DO control is fast compared to the
other control loops. In the simulations where also effluent ammonium
is charged, various fix DO set points in the range of 0.5 mg/l and 5.0
mg/l have been selected.
All of the other parameters adopt the standard values used in BSM1, these
values can be found in, for instance, (Copp, 2002) or on the benchmark webpage (IWA, November 19, 2007). See Table 5.1 and Table 5.2 for other values describing energy prices, carbon source prices and fee functions. These
values are in the sequel referred to as the nominal case.
Parameter
kE
kcar
Value
0.037
549
Unit
EUR/kWh
EUR/m3
Table 5.1: Nominal energy and carbon source prices used in the simulation
studies. The energy price, kE , is based on yearly average prices from a major
Swedish supplier.
In each experiment, the benchmark has been run for 150 simulation
days for a grid of constant input values. Only the last 100 simulation days
were considered when evaluating the cost functions to avoid the influence of
transients.
5.3.1
Simulation results for the denitrification process
In Figures 5.1–5.5 the benchmark has been run for a grid of different values
for the external carbon dosage, Qcar , and the internal recirculation flow rate,
80
Parameter
Value
Unit
αlimit,NO
β0,NO
∆αNO
∆βNO
8.0
1.4
2.7
8.2
mg(N)/l
EUR/1000 m3
EUR/kg NO
EUR/kg NO
αlimit,NH
β0,NH
∆αNH
∆βNH
1.5
2.7
4.1
12.3
mg(N)/l
EUR/1000 m3
EUR/kg NO
EUR/kg NO
Table 5.2: Default parameter values of the nitrogen fee functions used in the
simulation studies.
Qi , in order to obtain stationary values of the nitrate concentrations. Qcar
has been varied in steps of 0.1 m3 /day and Qi in steps of 2500 m3 /day.
In Figure 5.1 the stationary operational map depicting the total cost is
shown for the nominal case in Table 1 when no nitrate discharge fee is used.
The level curves for the nitrate concentrations in the anoxic and aerobic
an and S e , respectively, are plotted. From the figure, an
compartments, SNO
NO
optimum nitrate set point in the anoxic compartment is easily found for any
given effluent nitrate set point. Approximate locations for these optima are
marked by ’X’ in the figures. If, for instance, an effluent set point of 13 mg
(N)/l is desired it is seen from the figure that the corresponding optimum set
point of nitrate in the anoxic compartment is around 2.5 mg (N)/l and that
the corresponding optimum operational cost is around 400 EUR/day. With
an effluent nitrate set point of 7 mg(N)/l, the optimum set point of nitrate
in the anoxic compartment is between 1 and 1.5 mg(N)/l. If an operational
map for a wider operational area is studied it is seen more clearly that the
an at the cost optimal point decreases as S e
value of SNO
NO decreases. For
e
an near 0.3 mg(N)/l gives the cost-optimal
instance, if SNO
= 2 mg(N)/l, SNO
operating point. Note from Figure 5.1 that the difference in the operational
costs between using optimal and non-optimal set points of nitrate in the
anoxic compartment may be significant.
In order to illustrate the impact of changes in energy prices on the choice
of optimum nitrate set point in the last anoxic compartment, Figure 5.2
shows simulations with the same settings as in Figure 5.1 except that the
energy price is ten times as high (kE = 0.37 EUR/kWh). It is seen that the
external carbon dosage now is less dominant in the total cost and that the
optimum set point of nitrate in the anoxic compartment decreases to below
1 mg(N)/l for effluent nitrate set points less than 10 mg(N)/l, see Figure 5.2.
Another interesting case is illustrated in Figure 5.3, where the cost for dosing
an external carbon source is set to zero. This case is not unrealistic since
the carbon source may be provided free of charge if, for instance, industrial
81
4
6
0.0
5
4
8
6
8
0.0
3
0.5
1
1.5
0.3
7
3
2
0.1
12
4
11
9 14
11
7
X
2000
1800
1600
1400
1000
800
600
9
1200
10
8
x 10
10 15
10
0.3
5
5
X
5
12
01
0.
3
0.0
0.0
5
0.1
0.3
2
1
1.5
10
114
15
14
2000
1800
1600
1400
6
6
8
7
X
5
00
0.
5
4
3
0.
3
X
7
9
8
X
01
13
X
10
X
3
2
1200
1000
7
7
X
800
5
6
0.5
3
8
X
600
i
6
400
Q [m3/day]
13
9
7
5
1.
5
0.
1
11
2
3
8
2000
1.5
Qcar [m3/day]
11
10
12
13
3
0.00
9
03
1800
1
12
1
0.0
5
0.00
0.0
1600
0.5
15
3
0.0
0.0
11
1400
14
0.05
1200
1000
13
0.3
0.1
05
10
800
600
0
0
15
400
1
0.0
0
0.
.
12 0
14
03
0.
9
03
0.
5
1
0.
0.001
13
14
15
2
2.5
3
Figure 5.1: Stationary operational map for a grid of different values of Qcar
and Qi for the case with no nitrogen fee. Solid lines show the total cost, dashe , and dotted lines
dotted lines show the effluent nitrate concentration, SNO
an . Approxishow the nitrate concentration in the anoxic compartment, SNO
mate locations for the optimal nitrate set points in the anoxic compartment
for given desired effluent nitrate concentrations are marked by ’X’. Nominal
energy and carbon source prices according to Table 1 are used.
by-products are available. The principal behaviour of the cost function in
this case is the same as if a very high energy price is used. The conclusion
that can be drawn from Figure 5.3 is that even though provided for free, the
dosage of an external carbon source may have a large impact on the total
cost. This is due to the impact of external carbon addition on the aeration
costs. The cheaper the price of carbon source, the lower the optimum nitrate
set point in the anoxic compartment becomes.
Table 5.3 summarizes optimum set point values of nitrate in the anoxic
compartment for a number of effluent set point values and operational costs.
In the table, the tendencies described above is seen. The first part of the
table shows the impact of a decreasing effluent nitrate set point on the optimum value of the anoxic nitrate set point, the second part of the table shows
the impact of a higher energy price and the third part of the table shows
some optimum values when the carbon source is provided without cost.
82
4
1
5
0.0
0.25
0.5
5
6
8
9
00
55
5
6
12
00
60
00
65
0.0
25
1.5
2
00
8
0.1
3
10
11
4
10
7
8
15
45
00
50
9
7
x 10
11
10
40
00
9
14
13
X
7
01
0.
25
0.5
0.0
0.1
1
1.5
2
5
0.2
5
6
6
0
550
00
40
8
7
X
05
0
12
0.
8
X
01
0.
X1
11
5
0.
5
0
0.
8
9
5
02
0.
9
05
10
25
5000
2
1
.5
1
0.
0
450
10
X
3
2X
7
9
X
13
4 14
3
0
600
X
00
4
6
0.0
5
3
10
11
7
15 8
7
5
00
50
X
00
45
6
35
Qi [m3/day]
4
5
0.0
0.
2
0.00
11
12
10
13
3500
0.1
0
0
0.0
5
0.5
1
0.0
0.0
25
5500
14
11
12
4000
15
3000
1
13
14
15
5
0.00
2
0.00
15
1
1.5
3
12
131
0.00
2
14
2.5
3
Qcar [m /day]
Figure 5.2: Stationary operational map with the same settings as in Figure
5.1 except that the energy price is 10 times as high.
Next, the case with a constant cost per discharged kg effluent nitrate
according to (5.6) is investigated. Figure 5.4 shows this case with a fee of
5.5 EUR/kg effluent nitrate, i.e. ∆αNO = 5.5 EUR/kg. The nominal energy
and carbon source prices from Table 1 are used. The cost-optimum is now
found at Qcar = 0.7 m3 /day and Qi = 52 500 m3 /day with a cost of 1620
an = 1.7 mg(N)/l and S e
EUR/day corresponding to SNO
NO = 8.8 mg(N)/l.
The penalization of nitrate discharges creates a well-defined minimum in
the total cost function and makes it less desirable to discharge more nitrate
with the effluent. Consequently, the importance of the set point choice for
an has become larger. The overall optimum nitrate set point is marked by
SNO
a star.
The effluent nitrate fee certainly has an impact on the cost-optimal effluent nitrate set point. However, since there is no impact of the desired (set
point) effluent nitrate concentration and in the discharge fee, it may be hard
to relate the value of ∆αNO to the optimum operating point – or operating
region – at (or below) a certain effluent nitrate concentration.
The third investigated nitrate cost function given by (5.7) has a discontinuity at a certain predefined concentration of effluent nitrate, αlimit,NO .
If the jump at the discontinuity, β0 , is sufficiently large it is easy to find
83
4
x 10
3
0
0.2
5
0.5
0.
0
10
0.1
1
1.5
25
0.0
5
2
6
4
01
46
0
42
3
8
X
X
7
15
360
11
6
6
38
0.2
5
12
0
00
9
1
0
0.
0.
5
5
X
0
42
340
7
0
40
4
44
0
X8
5
0.
2
3
X
11
1
0.0
11
13
15 .5
0
2
0.00
380
360
5
0.2
10
12
14
11
300
13
5
0
0
1
0.0
9
05
10
12
0.1
8
0
X
X
9
5
02
0.
0.0
0.5
25
0.0
14
15
5
0.00
400
5
1.
05
0.
1
0.
34
2
10
X
320
14
13
3
1
7
8
X
2
0.00
15
1
1.5
3
Q
car
2
14
2.5
12
0.001
13
420
3
0.0
25
0.1
1.5
1
5
0
0.0
14
13
40
7
Qi [m /day]
5
0
380
5
48
44
7
6
0
0
9
9
12
8
6
X
400
6
8
50
11
11
420
10
9
7
4
0.0
5
15
2
46
5
10
4
44
0
7
8
10
3
[m /day]
Figure 5.3: Stationary operational map with the same settings as in Figure
5.1 except that the carbon source is assumed to be provided for free.
the optimum operating point. Also in this case, the nominal energy price
is considered. Figure 5.5 shows the stationary operational map when using
this cost function with parameter values given by Table 5.2 (nominal energy
and carbon source prices in Table 5.1 were used). The optimum point is
located at Qcar = 0.825 m3 /day and Qi = 63 000 m3 /day with a total cost
an = 1.7 mg(N)/l and S e
of 1227 EUR/day, SNO
NO equals 8 mg(N)/l.
In practice, when treating non-constant influents, it is of course advisory
to choose an operating point that is located slightly below the legislative
limit. The region of economic efficient operation, say operation with a total
cost below 1300 EUR/day, is relatively large. The larger β0 , the sharper the
limit at αlimit,NO becomes.
To check the robustness of the results, the sensitivity to changes in model
parameters on the location of the cost-optimal operational points was investigated. It was found that there were only minor differences between the
obtained operational maps, and in fact, for many changes of the model
parameters most of the optimum set points almost coincide. Since the conclusions from the sensitivity analysis were similar to the conclusions in the
previous cases, the results are omitted here, see Samuelsson et al. (2005a)
for details.
84
Case
kE =0.037,
kcar =549
kE =0.37,
kcar =549
kE =0.037,
kcar =0
e
SNO
[mg(N)/l]
10
7
5
2
10
7
10
7
an
SNO
[mg(N)/l]
1.8
1.3
0.9
0.35
1.0
0.6
0.6
0.4
Qi [m3 /d]
43
73
110
280
35
58
31
56
000
000
000
000
000
000
000
000
Qcar [m3 /d]
0.55
1.0
1.4
2.7
0.6
1.1
0.7
1.2
an , Q and Q
Table 5.3: optimum values of SNO
i
car for a number of given values
e
of SNO
are shown for the nominal case, the case in Figure 5.2 and the case
in Figure 5.3. The energy price kE are in units of EUR/kWh and the price
for carbon source in EUR/m3 .
Furthermore, in (Samuelsson et al., 2005a), the case with time varying
influents were also considered in order to study the effects of dynamic influents and different load situations. The time varying influents used were
the ones specified by BSM1 to describe different weather conditions. These
experiments did not lead to any essential changes in the results and are
therefore omitted in this chapter.
5.3.2
Simulation results for the combined denitrification and
nitrification process
The aeration is a key parameter when considering the nitrification. Therefore, the simulations discussed in this section have been run for a range of
different constant DO set points, from 0.5 mg/l up to 5.0 mg/l.
In Figure 5.6 operational maps for the effluent ammonium concentration
are shown for four different DO set points. A DO set point of 0.5 mg/l results
in very high effluent ammonium concentrations and is therefore clearly not
sufficient for good nitrification performance. The performance in terms of
effluent ammonium is much better with DO set points of 2.0 mg/l, 3.5 mg/l
and 5.0 mg/l, see Figure 5.6.
For a comparison, Figure 5.7 shows four operational maps for the effluent
nitrate concentration for the same DO set points [mg/l] as in Figure 5.6. As
expected, the denitrification benefits from low DO set points.
From Figure 5.8, where operational maps for the total cost are shown,
it is clear that the selection of the DO set point is crucial for the total cost.
When the DO set point is 0.5 mg/l the total cost is very high due to the
large effluent ammonium concentrations. For the other three considered DO
set points, note, for instance, how the operational area corresponding to a
85
4
0.0
3
2400
0.0
3
0.3
0.5
9
6
8
1700
16
50
8
1800
0.0
5
2200
1
1.5
0.1
2000
0.3
2
4
17
00
10
5
3
12
6
180
110
8
10
13
9
4
5
200
15 0
7
7
x 10
11
10
9
5
14
2600
0.
03
220
0
0.0
5
200
0
18
240
0
00
165
0
6
0.1
1
0.3
2
1.5
10
*
11
165
0
13
5
12
5
170
0
4
6
3
7
8
15
180
0
6
8
05
7
01
3
9
0.
03
0.5
17
00
0
0.
0.
0.
4
1650
3
6
7
Qi [m /day]
0.
01
7
7
3
10
2
2
1700
1
5
1.
3
0.
12
11
05
0
03
0.
0
20
0.
00
22
00
24
15
0
0
00
0
010
30
3
2200
5
3
0.0
0.5
280
14
5
0.00
0.00
1
3
1.5
Q
car
0.00
12
0
260
15
2400
11
0.00
13
0
2000
0.0
9
03
0.0
1
12
0.0
3
0.0
14
0.1
05
0.0
11
0.5
0.3
10
8
28
9
1
0.
13
1
26
00
3
14
8
0.001
13
2
[m3/day]
15
2
14
2.5
3
Figure 5.4: Stationary operational map for a grid of different values of Qcar
and Qi . Solid lines show the total cost including a constant nitrate-charge per
kg effluent nitrate, dash-dotted lines show the effluent nitrate concentration,
e , and dotted lines show the nitrate concentration in the anoxic compartSNO
an . The star indicates the minimum-cost point. Here ∆α
ment, SNO
NO = 5.5
EUR/kg.
total cost of less than 2000 EUR/day expands when the DO set point is
decreasing from 5 mg/l to 2 mg/l. A DO set point near 2 mg/l appears
to be close to optimal from a total cost point of view. Note also, that the
impact of the ammonium fee on the total cost is rather limited in the case
of sufficient aeration (i.e. in this study for the DO set points 2 mg/l, 3.5
mg/l and 5 mg/l). Therefore the optimum operating point does not change
much when also charging effluent ammonium. This is further illustrated in
Figure 5.9 that shows the operational map when the DO set point is 2.0
mg/l. The optimum point is located at approximately Qi = 58 000 m3 /day
and Qcar = 0.75 m3 /day.
5.4
Discussion
Cost efficient operation of the denitrification and nitrification processes was
studied using operational maps. Considering the case when no charge on
86
4
x 10
0.0
3
1300
9
2000
1400
0.3
4
1800
1600
0.5
8
2200
0.0
5
0.3
1
1.5
2
11
7
3
0. 0
1
1400
0.3
10
2
1.5
0.
0.5
11
01
6
0.
7
0.
3
9
00
5
00
7
1800
1600
8
4
1600
14
8
14
20
10
13
3
5
0.1
3
6
5
12
15
8
18
7 00
1300
4
0.0
13
6
*
6
05
1
0.
0.
1
2
12
600
11
.01
0
0
180
3
03 8
0.0
9
0.
0.
5
0.
00
03
1
5
1.
2
2200
1600
5
7
i
0.1
7
3
10
5
6
12
8
1800
15
2000
10
14
9
7
Q [m3/day]
5
1300
8
3
1400
9
5
4
6
11
10
10
2000
220
0
9
0
240
11
10
12
1
0
0
15
13
1800
0.3
2200
14
2000
0.1
2200
5
0.0
3
0.0
2400
0.5
15
0.01
1
2400
2600
13
14
5
2600
3
0.00
0.00
2800
1.5
Qcar [m3/day]
2800
12
0.001
3000
15
2
11
13
14
2.5
3
Figure 5.5: Stationary operational map for a grid of different values of Qcar
and Qi . Solid lines show the total cost including a nitrate-charge according
e , and
to (5.7), dash-dotted lines show the effluent nitrate concentration, SNO
dotted lines show the nitrate concentration in the last anoxic compartment,
an . The additional charge for exceeding the legislative discharge limit at
SNO
αlimit,NO = 8.0 mg(N)/l is here β0,NO = 1.4 EUR/1000 m3 . The star indicates the minimum-cost point.
effluent nitrate is imposed it is seen that, in the nominal case, the cost for
dosing an external carbon source dominates the total cost. It is also clear
that the operational area with respect to effluent nitrate is divided in two
parts with different gain characteristics, and that for each desired value (set
e , there is a cost-optimal point in the operational map correpoint) of SNO
sponding to a certain value of the nitrate concentration in the last anoxic
an . This point naturally also corresponds to certain stationcompartment, SNO
ary values of the input signals, Qcar and Qi . In contrast to what is described
in, for instance (Yuan and Keller, 2003), the findings in this chapter are that
the cost-optimal set point of nitrate in the anoxic compartment depends on
the choice of effluent nitrate set point, as well as the specific operational
an level decreases with decreasing S e
costs. The optimum SNO
NO levels and
with increasing energy costs (or decreasing costs for external carbon). The
location of the optimum set point choice is, however, not very sensitive to
87
SeNH with DOsp =0.5
4
10
34
34.5
Qi [m3/day]
31
32
30.5
30
6
4
1.2
6
0.9
4
0.8
1
0.9
31.5
0
0
0.5
1
1.5
3
2
2
0
0
2.5
3
1.1
1
0.7
0.8
33
35
33.5
2
x 10
8
32.5
8
SeNH with DOsp =2
4
Qi [m3/day]
10
x 10
0.6
0.7
0.5
1
Qcar [m /day]
e
10
sp
SNH with DO
4
x 10
e
=3.5
10
0.
x 10
85
Qi [m3/day]
Qi [m3/day]
0.7
0.65
0.5
2
0.5
1
0.
65
6
0.5
4
0.6
0.45
0.5
5
2
0.6
0.5
5
0.55
0.45
0
0
0.5
7
0.
0.6
0.4
1.5
2
3
2.5
=5
6
8
6
0.55
2.5
0.
8
0.7
5
4
sp
SNH with DO
4
0.7
0.6
5
2
Qcar [m /day]
0.
8
1.5
3
0
0
3
Qcar [m3/day]
0.5
0.5
0.4
0.35
0.45
1
1.5
2
2.5
3
Qcar [m3/day]
Figure 5.6: Stationary operational maps for a grid of different values of Qcar
and Qi showing the effluent ammonium concentrations for four different
constant DO set points [mg/l].
variations in the ASM1 parameters. Simulations showed that when an appropriate level of aeration is employed the effluent ammonium concentration
is kept low. Therefore, the location of the optimum operating region was
not significantly changed when also effluent ammonium was charged.
From the results in the previous section, some questions may arise. The
first is which control structures that can be expected to give a good performance in terms of disturbance rejection and set point tracking. In different
operating points different control structure selections may be suitable. As
indicated by e.g. Figure 5.1, in the area of the operational map corresponde is mostly affected by the input signal
ing to cost-efficient nitrate control, SNO
an
Qcar , while SNO is affected by both input signals in this area. This situation
in particular is further discussed by Samuelsson et al. (2005c) (see Chapter
4) and also to some extent by Yuan and Keller (2004).
The second question is how to design the actual control law in order to
minimize operational costs. Below, some possibilities for this control design
are discussed:
• One approach is to use two different control loops (for instance PI88
SeNO with DOsp =0.5
4
4
13
11
Qi [m3/day]
0.15
6
6
4
7
8
9
2
2
0
0
0
0
13
14
1
1.5
2
2.5
3
0.5
=3.5
10
19
8
9
2
0
0
18 17
11
12
13
0.5
1
1.5
14
15
16
2
2.5
9
5
11
10
6
7
4
8
9
10
2
10
14
18
17
8
10
4
16
Qi [m3/day]
9
7
17
15
16
13
12
11
6
7
15
3
6
6
8
5
12
Qi [m3/day]
8
2.5
SeNO with DOsp =5
4
x 10
14
10
sp
2
Qcar [m /day]
15
14
13
e
SNO with DO
1.5
3
Qcar [m /day]
4
13
14
1
3
x 10
12
15
16
17
8
0.5
10
11
12
7
0.2
Qi [m3/day]
6
5
7
8
6
8
8
10
14
10
SeNO with DOsp =2
4
x 10
10
9
4
x 10
0
0
3
Qcar [m3/day]
11
15
17
18
0.5
14
13
12
16
1
1.5
15
2
2.5
3
Qcar [m3/day]
Figure 5.7: Stationary operational maps for a grid of different values of Qcar
and Qi showing the effluent nitrate concentrations for four different constant
DO set points [mg/l].
e
an separately. Given the desired
controllers) to control SNO
and SNO
e
an can
nitrate effluent concentration, SNO , the optimum set point of SNO
be found from the operational maps depending on process conditions.
e is chosen as 8
For instance, in the nominal case, if the set point for SNO
an
mg(N)/l, the optimum set point for SNO is close to 1.5 mg(N)/l. If dee and Q S an .
centralized control is to be used, Qcar should control SNO
i NO
Other control structures may, however, yield a better performance, see
Samuelsson et al. (2005c) (see Chapter 4).
• Another possibility in the nominal case, see Figure 5.1, is to use a
constant high internal recirculation flow rate Qi and to use only Qcar in
order to control the effluent nitrate concentration. Since Qi has a much
smaller impact on the total cost than Qcar , this would render a close
to cost-optimal operation. This possibility has also been mentioned
by Ingildsen (2002). This is, however, not a suitable strategy if the
carbon source is inexpensive, see Figure 5.3.
• To achieve a cost-optimal performance in the nominal case, the total
89
Total cost with DOsp =2
x 10
1
1.5
2
2.5
3
1600
2000
2200
0.5
1
Qcar [m3/day]
sp
Total cost with DO =3.5
2600
2200
1600
0
24
00
0
18
2
0
0
0.5
280
3000
2600
1
1.5
3
2.5
3
3200
2
3400
2.5
26
0
2600
00
20
4
0
0
0
3
24
00
00
6
2
0
2000
2200
2400
28
Qi [m3/day]
160
0
4
2
sp
x 10
8
2000
220
0
0
200
6
1.5
Total cost with DO =5
4
10
180
0
Qi [m3/day]
8
2800
3000
Qcar [m3/day]
0
10
0
260
2400
22
00
4
x 10
1800
0
1400
1400
1600
1800
0
0
2800
0.5
1600
2400
0
0
4
2
8000
2
6
150
4
1500
Qi [m3/day]
Qi [m3/day]
6
00
8
1800
13
7500
8
2400
4
10
2200
Total cost with DOsp =0.5
3000
x 10
2000
4
10
220
0.5
0
340
0
2400
2600
280
3000 00
32
1
1.5
3800
4000
3600
2
2.5
3
Qcar [m3/day]
Qcar [m /day]
Figure 5.8: Stationary operational map for a grid of different values of Qcar
and Qi showing the total cost including a nitrate-charge according to (5.7)
and an ammonium-charge according to (5.8) for four different constant DO
set points [mg/l].
cost could be minimized on-line using quadratic criteria yielding for
example LQG or MPC controllers. Such a criterion has the typical
form
Z
T
V =
eT (t)Q1 e(t) + uT (t)Q2 u(t)dt
(5.9)
0
where e is a column vector containing the control errors and u the input
signals. The weighting matrix Q2 can be chosen to reflect the costs for
the different input signals and Q1 can be seen as a performance weight.
The difficulty with this criterion is how to weight control performance
against cost minimization, i.e. how to choose the matrices Q1 and Q2 .
From the prior knowledge obtained from Figure 5.1, in the nominal
case, the elements of the matrix Q2 could be chosen in an intuitive
manner. A rule of thumb could be to choose Q2 as a diagonal matrix
with the element corresponding to Qcar significantly larger than the
element corresponding to Qi , since the external carbon source is much
more expensive than the pumping of the internal recirculation flow
90
rate in the nominal case and thereby dominates the total cost. Such
a choice clearly penalizes a large value of Qcar in the criterion.
• A simple grid search could be performed on-line until the optimum
point is reached. This method is simple and has the advantage that
no operational map and thereby no model is required. One such optimization algorithm is presented by Ayesa et al. (1998). This algorithm
is employed to minimize a global penalty function combining effluent
requirements and costs.
For the case when the nitrate discharge is penalized with a constant
charge per kg, see (5.6), it is seen from Figures 5.4 that this creates a
minimum in the total cost function (5.5). The main drawback using such a
cost function for automatic control is that it is hard to relate the location of
the cost-optimal set point to the nitrate discharge fee, ∆αNO , and thereby
hard to say which set point a certain fee results in. This also depends
on the effluent flow rate, Qe . Due to the discontinuity, this problem can
be overcome if instead the fee function according to Carstensen (1994) is
implemented. The location of the discontinuity of the fee function (5.7)
immediately coincides with the optimum set point for the effluent nitrate,
e , if the discontinuity is sufficiently large. Using this fee in the total
SNO
cost is a convenient way to achieve cost optimality for a certain set point of
e , see Figure 5.5. Minimizing this total cost function oneffluent nitrate, SNO
line using some automatic control strategy would be a good way to impose
the importance of good performance via penalizing the effluent discharge
into the control design. If the discharge of nitrogen over a certain legislative
limit is directly associated with a higher fee, this could clearly motivate
the use of more advanced control strategies. The impact of the fee in the
control design is also easy to understand even for people with a limited
knowledge in automatic control, compared to the related matter of choosing
a performance weight in some quadratic criterion.
91
4
1800
0.
01
01
5
00
0. 1
00
14
12
4
3
0.
15
00
7
2400
2200
130
i
11
10
9
1400
0
6
3
8
1
0.
5
5
.2
.1
0
1500
1
2000
14
2000
160.6
5
0.0
3
02
0.0
2
0.00
5
0.00
152800
2600 16
1
1.5
Q
car
[m3/day]
280011
2600
2400
15
2400
0.5
2
10
12
0.7
2200
14
0.0
400
11
13
1800
0.9
9
00
0522
0.00.
108
0.0
12
17
0.100
22
8
0.
0
180
11
0.7
03
0.
05
0
13
1600
10
00
20
9
0.8
5
1.
2
7
0.9
00
16
0
0
2200
0.0
0.0
0.1
0.2
5
5
1
0.9
0.8
1
0.0
3
2000
1600
3
0.5
1
1.5
2
5
1500
13 6
16
00
7
1237
X
8
2
5 1.
1
6
0.9
6
4
5
1800
1500
1400
4
1300
7
Q [m3/day]
0.1
0.25
0.5
1
1.5
1
3
12
8
8
5
0.0
9
150
10 0
5
1400
6
1600
14
1800
11
5
1.2
2400
13
8
2
6
1.1
0
9
4
130
1
9
7
7
x 10
10
10
13
14
2
0.7
3000
0.8
12 001
0.
3200
13
2.5
Figure 5.9: Stationary operational map for a grid of different values of Qcar
and Qi . Solid lines show the total cost including a nitrate-charge according to
(5.7) and an ammonium-charge according to (5.8), dash-dotted lines show
e , dotted lines show the nitrate conthe effluent nitrate concentration, SNO
an and dashed lines show the
centration in the last anoxic compartment, SNO
e
effluent ammonium concentration, SNH . The additional charge for exceeding
the legislative discharge limits at αlimit,NO = 8.0 mg(N)/l and αlimit,NH =
1.5 mg(N)/l is here β0,NO = 1.4 EUR/1000 m3 and β0,NH = 2.7 EUR/1000
m3 , respectively. X indicates the minimum-cost point.
92
3
5.5
Conclusions
In this chapter a bioreactor model describing a pre-denitrifying wastewater
treatment plant was studied from a process economic point of view. The
impact of different nitrate and ammonium cost functions on the location of
the cost-optimal operating point was examined. Given the desired value of
the effluent nitrate concentration, the energy price and the price of carbon
source, there is a corresponding optimum set point for the nitrate in the
anoxic compartment. For low effluent set points, the optimum anoxic set
point may be located well below 1 mg(N)/l. The simulations also show
that the difference in the operational costs between an optimum and nonoptimum anoxic set point may be large. The locations of the optimum
set point values are, however, not very sensitive to changes in the ASM1
parameters. In summary, it can be concluded that the approach presented
in this chapter may give a valuable tool towards running a WWTP in a
more cost effective way. Natural extensions and topics for further research
include:
• Evaluation of the suggested approach using live data from a full-scale
WWTP.
• Extending the criteria function with costs for the sludge handling. This
will indeed penalize an excessive carbon dosage. Another interesting
extension is to include chemical precipitation for phosphorous removal.
In a pre-denitrifying plant, there is an interesting trade-off between
removal of phosphorous and substrate where the latter is useful for
the denitrification process.
93
94
Chapter 6
Aeration Volume Control in
an Activated Sludge Process
– Discussion of Some
Strategies Involving On-Line
Ammonium Measurements
In this chapter the influence of the aeration on the efficiency of the nitrogen
removal in an activated sludge process is studied. Different strategies for
controlling the DO set point and the aerated volume are compared in terms
of treatment efficiency in a simulation study.
6.1
Introduction
In an activated sludge process (ASP) configured for nitrogen removal microorganisms (mainly bacteria) are employed to reduce the influent nitrogen.
Microorganisms in the aerated basins convert ammonium into nitrate while
consuming oxygen. This is the nitrification process. For this process to occur, the concentration of dissolved oxygen (DO) in the aerobic basins must
be sufficiently high. In the anoxic basins, another type of bacteria is employed in the denitrification process where nitrate is converted into nitrogen
gas.
Most common today is to control the airflow rate to maintain a specific
DO level. However, since the influent load situation may vary significantly
the ammonium in the influent may also undergo large variations. Improvements in controlling the nitrification rate could therefore be expected if
online-measurements of the ammonium concentration in, for instance, the
last aerated basin, are utilized in the control of a time-varying DO set point
95
(see e.g. Lindberg (1997)). Supervisory control of the DO set point could
then be implemented such that a specific concentration of ammonium in
the effluent is aimed at. Furthermore, in DO control, it is of great importance that the aerated volume is sufficiently large to ensure that the desired
DO concentration may be reached. If the aeration volume is too small, the
airflow rate may saturate before the desired DO concentration is achieved.
Also, a too high DO level is very expensive to maintain (regarding the energy cost for the aeration pumps) since the DO concentration in the water
does not increase linearly with the airflow rate. On the contrary, a too low
DO concentration not only reduces the nitrification but may also produce
sludge with poor settling capability and often requires basins with mixers.
In addition, nitrous oxide may be created.
The need of aeration control has been investigated by several authors.
Vrecko et al. (2006) present and compare different aeration controllers.
Meyer and Pöpel (2003) use fuzzy control in the determination of DO set
points and the ratio between aerated and non-aerated basins. Both of these
papers evaluate the performance of the suggested strategies in pilot plants.
Optimum aerobic volume control based on the oxygen uptake rate as an indicator of the nitrification performance is discussed by Svardal et al. (2003).
Gerksic et al. (2006) presents a method for on-line estimation of the respiration rate. Brouwer et al. (1998) and Samuelsson and Carlsson (2002)
discuss different strategies for aeration volume control.
Recently, two strategies for aeration volume control have been developed by Ekman (2005) (also presented in (Ekman and Carlsson, 2005a) and
(Ekman and Carlsson, 2005b)) and by Samuelsson (2005) (also presented in
(Samuelsson et al., 2005b)). A great advantage of the approach presented in
Ekman (2005) is that only standard DO sensors are needed. The key point
of this strategy is that the DO consumption reflects the load situation in
terms of influent ammonium. However, there may be situations when this is
not entirely true. These occasions may for instance occur during the winter
season when the water temperature is low. Therefore, if on-line ammonium
sensors are available some alternative solutions (involving volume control)
have been suggested to take care of this drawback. An example is the strategy presented by Samuelsson (2005) where the key idea is to feed back the
actual treatment efficiency (in terms of ammonium concentration) and use
supervisory DO set point control combined with a feedforward approach to
control the aeration volume. Instead of estimating the reaction rate a tuning
parameter is used.
In this chapter, an improved version of the strategy developed by Samuelsson (2005) is presented. Also, another strategy, supervisory feedback aeration volume control, involving feedback from the effluent ammonium concentration, that share some similarity with the strategy by Ekman (2005)
is discussed. For a more comprehensive background of the topic of aeration
96
volume control and DO set point control, see Samuelsson et al. (2005b) and
Samuelsson (2005) and the references therein.
6.2
The simulation setup
To illustrate and compare the suggested control strategies, a simulation
study was performed using the COST/IWA simulation benchmark model
number 1 (BSM1), see Jeppsson and Pons (2004) for a general survey and
Copp (2002) for a more technical description of this simulation platform.
The benchmark plant models pre-denitrifying wastewater treatment and
consists of five basins. The first two basins are always anoxic and have
a volume of 1000 m3 each. The volume of the last three basins is 1333 m3
each. The last two basins, number four and five, are aerated while the middle one, basin number three, may be aerated or anoxic, depending on the
selected strategy.
Three influent files based on the dry weather influent data, DRYINFLUENT, provided with BSM1, have been used to feed the simulation plant.
The first data file, here called D, is the DRYINFLUENT in original form,
whereas the other two data files, called D3 and D5, are the same as D except
that the influent flow rate is decreased to a third and to the half of the flow
rate in D (with an average value of 18 446 m3 /day), respectively. In this
way three different load scenarios are simulated.
For the aerated basins the maximum DO set point were limited to 5 mg/l
max , to 360 day−1 . In all simulations
and the maximum aeration intensity, KLa
sp
the set point for the effluent ammonium concentration, SNH
, was selected
to 3 mg/l. The DO controllers used in the experiment were ordinary PIcontrollers tuned to give a fast response in the DO concentration when the
set points were changed.
A further description of BSM1 and a schematic figure of the benchmark
plant layout are given in Chapter 1.
6.3
6.3.1
Description of the proposed control strategies
The reference aeration control strategies
The suggested volume control strategies were compared with two strategies
without volume control, the reference strategies. For both of these, the DO
set points for basin four and five are the same and controlled supervisory
such that the effluent ammonium concentration is kept at some pre-specified
level, here 3 mg/l. The third basin is in the first reference case always anoxic
while it is aerated in the same way as basin four and five in the second reference strategy. In short, the two reference strategies can be described as:
97
Reference strategy I:
• DOsp
3 = 0.
sp
e
• DOsp
4 =DO5 is controlled so that SNH = 3 mg/l.
Reference strategy II:
sp
• DOsp
3 =DO4,5 .
sp
e
• DOsp
4 =DO5 is controlled so that SNH = 3 mg/l.
6.3.2
Feedforward aeration volume control I and II
In the volume control strategy initially suggested by Samuelsson (2005),
feedforward control is combined with supervisory feedback control of the
DO set points. The basic idea is to calculate the desired aeration volume,
Vdes , from the simple relation
in
e
Vdes (t) = Kv Q(t) SNH
(t) − SNH
(t)
(6.1)
where Kv is some positive constant parameter tuned by the user, Q is the
in is the ammonium concentration in the influent and S e
flow rate, SNH
NH is
the ammonium concentration in the effluent. If the desired aeration volume
according to (6.1) is larger than the currently used aeration volume, then
more basins are aerated until the aeration volume is at least as large as the
calculated desired volume. Equation (6.1) is derived from the mass balance
consideration for SNH in one completely mixed basin given by
dSNH (t)
Q(t) in
e
= RSNH (t) +
SNH (t) − SNH
(t)
dt
V (t)
(6.2)
where RSNH is the reaction rate of ammonium.
The original strategy proposed by Samuelsson (2005) is here denoted
feedforward aeration volume control I. Feedforward aeration volume control
II is a slightly modified version of the previously discussed strategy where
sp
DOsp
3 is allowed to follow DO4,5 when basin three is aerated. In that way
DOsp
3 will certainly be more smooth. As a second improvement a simple
hysteresis rule is applied to Equation (6.1): Aeration of basin three with
sp
DOsp
3 =DO4,5 is switched on when the desired aerated volume exceeds 2666
3
m3 , but off, i.e. DOsp
3 =0, when this calculated volume is less than 2400 m .
The two feedforward aertaion volume control strategies can for this simulation study be summarized as:
98
Feedforward aeration volume control I:
• Calculate Vdes from (6.1).
– If Vdes > 2666 m3 aerate basin 3 with DOsp
3 =3 mg/l.
– Otherwise DOsp
3 =0.
sp
e
• DOsp
4 =DO5 is controlled so that SNH = 3 mg/l.
Feedforward aeration volume control II:
• Calculate Vdes from (6.1).
sp
– If Vdes > 2666 m3 aerate basin 3 with DOsp
3 =DO4,5 .
– If Vdes ≤ 2400 m3 and DOsp
3 > 0 switch off the aeration of basin
3, i.e. DOsp
=0.
3
– Otherwise DOsp
3 =0.
sp
e
• DOsp
4 =DO5 is controlled so that SNH = 3 mg/l.
6.3.3
Supervisory feedback aeration volume control
An alternative strategy where the measurement of the influent ammonium
concentration is not needed has also been developed. The idea is to aerate an
extra zone when the difference between the DO set point and the measured
DO concentration in the first aerated zone exceeds some selected threshold
and as long as the effluent ammonium concentration exceeds the set point.
A significant error here indicates that the current load situation is high and
therefore, that a larger aeration volume is needed.
In the strategy suggested by Ekman (2005) a similar idea is used to
decide the DO set points. However, in the strategy presented here this idea is
combined with supervisory feedback control of the DO set points for the last
two basins such that a specific effluent ammonium concentration is aimed
at. Hence, this strategy also involves on-line ammonium measurements.
In the simulations, basin four and five have the same DO set point that
is supervisory controlled by a PI controller with the effluent ammonium concentration as input and with a set point for this concentration of 3 mg/l.
Basin three is aerated when DOsp
4 − DO4 − estart ≥ 0.5, where DO4 is the
measured DO concentration in basin four and estart is a parameter tuned to
reduce the abrupt start of the aeration; DOsp
3 is controlled by a PI controller
sp
with DO4 − DO4 − estart as input. Here estart is 0.4. The aeration of basin
three is turned off when DOsp
4 − DO4 − estart ≤ 0.2. The strategy in this
simulation study can be summarized as:
99
Supervisory feedback aeration volume control:
sp
• When DOsp
4 − DO4 − estart ≥ 0.5: DO3 is controlled by a PI controller
sp
with DO4 − DO4 − estart as input.
sp
• When DOsp
4 − DO4 − estart ≤ 0.2 and DO3 > 0: The aeration of basin
sp
3 is turned off, i.e. DO3 = 0.
• Otherwise DOsp
3 = 0.
sp
e
• DOsp
4 =DO5 is controlled so that SNH = 3 mg/l.
6.4
6.4.1
Simulation results
The reference aeration control strategies
As can be seen in Figure 6.1 where the influent file is T, it is clear that
the influent ammonium load is too heavy for the first reference strategy:
Even though basin four and five are maximally aerated, i.e. DOsp
4,5 = 5
mg/l, the effluent ammonium concentration is high with a peak value of
24 mg/l and an average of 17 mg/l. It is obvious that a larger aeration
volume is needed in order to improve the nitrification. With the influent
files D3 and D5, the ammonium load is smaller compared to the previous
case, and consequently, the aeration volume needed to get good nitrification
performance is smaller and that is why the first reference strategy may
perform well in these particular cases, see Table 6.1 for numerical results.
In reference strategy II the aerated volume is larger since basin three is
sp
always aerated (DOsp
3 =DO4,5 ). As can be seen in Table 6.1 this strategy
performs better than the first reference strategy, compare for instance the
consumed energy per kg ammonium that is removed.
6.4.2
Feedforward aeration volume control I
In the simulation experiments the parameter Kv was tuned to 0.01. Since
basin three is aerated during periods of high influent load, this strategy
achieves much better nitrification in the case of influent T compared to the
first reference strategy, see Table 6.1. The peaks of the effluent ammonium
concentration are clearly reduced (maximum peak value is 12.9 mg/l) and
so is the average effluent concentration (4.6 mg/l). Compared to the second reference strategy, this strategy reduces the ammonium concentration
e
is 4.6 mg/l compared to 4.3 mg/l for reference strategy
slightly less (SNH
II) but at a higher efficiency – compare the consumed aeration energy per
kg removed ammonium. Also, the sum of the effluent ammonium and the
effluent nitrogen concentration is slightly lower.
However, the performance of this strategy may be sensitive to the value
of the selected parameter Kv . This can be reduced by introducing hysteresis.
100
Table 6.1: Numerical simulation results for the different strategies proposed.
e
e
SNH
is the effluent concentration of ammonium, and SNO
is the corresponding value of nitrate. AE is the aeration energy, i.e. the electrical energy
consumed in the aeration. The lowest values of effluent ammonium and nitrate concentrations as well as the most efficient strategies in terms of AE
for each influent are emphasized. DOsp
i is the DO concentration in basin i.
f.b. is feedback. All concentrations in the table are average values.
Strategy
Influent
Reference I
Reference I
Reference I
Reference II
Reference II
Reference II
Feedforward I
Feedforward I
Feedforward I
Feedforward II
Feedforward II
Feedforward II
Supervisory f.b.
Supervisory f.b.
Supervisory f.b.
D5
D3
D
D5
D3
D
D5
D3
D
D5
D3
D
D5
D3
D
e
SNH
[mg/l]
e
SNO
[mg/l]
e
e
SNH
+SNO
[mg/l]
AE/day
[kWh]
3.0
3.3
17.4
3.0
3.0
4.3
3.0
3.0
4.6
3.0
3.0
4.6
3.0
3.1
5.4
2.9
4.8
3.8
2.9
5.5
10.9
2.8
5.0
10.4
2.8
5.1
10.5
2.9
4.9
9.5
5.9
8.1
21.3
5.9
8.5
15.2
5.8
8.0
15.0
5.8
8.1
15.0
5.9
8.0
14.9
1869
4199
7849
1429
2592
8409
1645
2913
7429
1564
2629
7395
1890
4012
9539
101
AE/kg NH
removed
[kWh]
0.86
1.87
6.91
0.66
1.14
3.45
0.76
1.29
3.08
0.72
1.16
3.07
0.87
1.78
4.06
DOsp
3
[mg/l]
DOsp
4,5
[mg/l]
0.0
0.0
0.0
0.38
0.74
2.6
0.3
1.3
2.3
0.1
0.6
2.7
0.0
0.03
0.9
1.1
2.6
5.0
0.38
0.74
2.6
0.7
0.8
2.8
0.7
0.9
2.8
1.1
2.3
3.9
Se [mg/l]
30
NH
20
10
0
0
2
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
DO3 [mg/l]
1
sp
0
−1
0
sp
DO4,5 [mg/l]
6
5
4
3
0
Se
NO
[mg/l]
6
4
2
0
0
Figure 6.1: Effluent concentrations and DO set points for reference strategy
I with influent D.
Also, the aeration of basin three starts and ends rather abruptly since DOsp
3
is either 0 or 3 mg/l. As a typical illustration of this behaviour, consider
Figure 6.2 where the influent load is D3.
6.4.3
Feedforward aeration volume control II
Figures 6.3 and 6.4 show the performance of this strategy with influent
files D3 and D, respectively. Note particularly that now DOsp
3 is smoother
e
is slightly higher but
compared to the previous case in Figure 6.2. SNO
the aeration energy consumption is reduced and the efficiency in terms of
aeration energy consumed per kg ammonium removed is higher (i.e. lower
values).
6.4.4
Supervisory feedback aeration volume control
Figure 6.5 shows the performance of the strategy when the influent is D3.
The strategy works as expected since basin three is only aerated during the
load peaks. With input D basin three is aerated during longer periods, see
Figure 6.6. Note that in the case of influent D5, only basin four and five
102
Se [mg/l]
6
NH
4
2
0
0
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
3
DOsp [mg/l]
4
2
0
0
4
4,5
DOsp [mg/l]
2
2
0
0
5
e
SNO [mg/l]
10
0
0
Figure 6.2: Effluent concentrations and DO set points for the originally
proposed version of the feedforward aeration volume control strategy (feedforward aeration volume control I) with influent D3.
need to be aerated according the strategy, see Table 6.1.
As can be seen from the numerical results in Table 6.1, this strategy is
not as efficient in terms of aeration energy as most of the other strategies.
However, the reduction of nitrate and ammonium is very good for all of the
studied load situations, in particular for influent D3 and D5. Furthermore,
this strategy is the only of the considered volume control strategies that
achieves a totally anoxic third basin for influent D5. Note that a fine tuning
of the other strategies employing volume control may as well result in an
anoxic third basin for influent D5.
103
Se [mg/l]
6
NH
4
2
sp
DO3 [mg/l]
0
0
sp
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
4
2
0
0
DO4,5 [mg/l]
2
4
2
0
0
5
Se
NO
[mg/l]
10
0
0
Figure 6.3: Effluent concentrations and DO set points for the strategy feedforward aeration volume control II with influent D3.
104
10
5
e
SNH [mg/l]
15
0
0
2
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
DOsp [mg/l]
6
3
4
2
0
0
4
4,5
DOsp [mg/l]
6
2
0
0
10
e
SNO [mg/l]
20
0
0
Figure 6.4: Effluent concentrations and DO set points for the strategy feedforward aeration volume control II with influent D.
105
S
e
NH
[mg/l]
8
6
4
2
2
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
0.6
0.4
0.2
0
0
DO
sp
4,5
[mg/l]
6
4
2
0
0
e
NO
[mg/l]
8
S
sp
DO3 [mg/l]
0
0
6
4
2
0
0
Figure 6.5: Effluent concentrations and DO set points for the supervisory
feedback aeration volume control strategy with influent D3.
106
10
5
e
SNH [mg/l]
15
0
0
2
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
2
4
6
Time [days]
8
10
12
DOsp [mg/l]
3
3
2
1
0
0
4
4,5
DOsp [mg/l]
6
2
0
0
e
SNO [mg/l]
15
10
5
0
0
Figure 6.6: Effluent concentrations and DO set points for the supervisory
feedback aeration volume control strategy with influent D.
107
6.5
Conclusions
Five different aeration control strategies involving ammonium measurements
have been evaluated in a simulation study. Three strategies in the study involve aeration volume control. All of those were able to dampen the process
disturbances that occur in form of load variations in the influent wastewater. The suggested improved feedforward strategy performs overall very well
and is the most efficient strategy in terms of aeration energy for the high
load scenario (influent D). The second reference strategy is however more
cost efficient when considering the two low load scenarios (influent D5 and
D3). On the other hand, the reference strategies do not offer the feature of
volume control, and it is hence crucial that the operator selects to aerate an
appropriate number of basins. Otherwise, bad performance both in terms
of wastewater treatment and in terms of aeration energy consumption may
result. In this simulation study this is illustrated by the bad performance
of the first reference strategy when the influent load is high. With aeration volume control the selection of the number of aerated basins is decided
automatically. Clearly, this is an advantage both from a process economy
point of view as well as when considering the treatment efficiency.
108
Appendix A
The minimized condition
number
This appendix reviews the concept of minimized condition number. The
text is based on the description of condition numbers in Halvarsson (2003).
To obtain the minimized condition number the scaling matrices, S1 and
S2 , are chosen according to
γmin (G) = min γ(S1 GS2 )
S1 ,S2
(A.1)
As shown by Grosdidier et al. (1985) γmin are closely related to the RGA.
For the case of a 2 × 2 plant, G, Grosdidier et al. (1985) show that the
minimized condition number is given by
q
γmin = kΛ(G)k1 + kΛ(G)k21 − 1
(A.2)
where Γ is the RGA matrix and the 1-norm is defined as
kΛk1 = max
j
m
X
|λij |
(A.3)
i=1
i.e. “the maximum column sum”. It can also be shown that γmin is bounded
by kΛ(G)k1 according to
γmin ≤ 2kΛ(G)k1
(A.4)
with equality when kΛ(G)k1 → ∞.
For larger quadratic systems the following conjecture is valid (Grosdidier
et al., 1985):
γmin ≤ 2 max(kΛ(G)k1 , kΛ(G)k∞ )
(A.5)
where the ∞-norm is defined as
kΛk∞ = max
i
109
m
X
i=1
|λij |
(A.6)
i.e. “the maximum row sum”. The work of finding γmin by means of optimization theory is often rather tedious and therefore it is handy to first
calculate the RGA and then use (A.2), (A.4) or (A.5).
110
Bibliography
Alex, J., J.F. Beteau, C. Hellings, U. Jeppson, S. Marsili-Libelli, M.N. Pons,
H. Spanjers and H. Vanhooren (1999). Benchmark for evaluating control
strategies in wastewater treatment plants. In: Proceedings of the European
Control Conference, ECC99’. Karlsruhe, Germany.
Antoulas, A.C. (2001). Frequency domain representation and singular value
decomposition. UNESCO EOLSS (Encyclopedia for the Life Sciences),
Contribution 6.43.13.4.
Ayesa, E., B. Goya, A. Larrea, L. Larrea and A. Rivas (1998). Selection of
operational strategies in activated sludge processes based on optimization
algorithms. Water Science and Technology 37(12), 327–344.
Birk, W. and A. Medvedev (2003). A note on gramain-based interaction
measures. In: Proceedings of European Control Conference, Cambridge,
UK, September 2003.
Bristol, E. H. (1966). On a new measure of interaction for multivariable
process control. IEEE Trans. Automatic Control AC-11, 133–134.
Brouwer, H., M. Bloemen, B. Klapwijk and H. Spanjers (1998). Feedforward
control of nitrification by manipulating the aerobic volume in activated
sludge plants. Water Science and Technology 38(3), 245–254.
Cadet, C., J. F. Beteau and S. Carlos Hernandez (2004). Multicriteria control strategy for cost/quality compromise in wastewater treatment plants.
Control Engineering Practice 12(3), 335–347.
Carlsson, B. and A. Rehnström (2002). Control of an activated sludge process with nitrogen removal - a benchmark study. Water Science and Technology 45(4–5), 135–142.
Carstensen, J. (1994). Identification of Wastewater Processes. PhD thesis.
Institute of Mathematical Modelling, Technical University of Denmark.
Chellaboina, V., W.M. Haddard, D.S. Bernstein and D.A. Wilson (1999).
Induced convolution operator norms for discrete-time linear systems. In:
111
Proceedings of the 38:th Conference on Decision & Control. Phoenix, Arizona, USA.
Conley, A. and M. E. Salgado (2000). Gramian based interaction measure.
In: Proceedings of the 39th IEEE Conference on Decision and Control.
Sydney, Australia. pp. 5020–5022.
Copp, J. B., Ed.) (2002). EUR 19993 – COST Action 624 – The COST simulation benchmark – Description and simulator manual. European Communities. Luxembourg.
Dharmasanam, S., R. Scott Erwin and D. Bernstein (1997). Synthesis of
optimal generalized LQG and Hankel-norm controllers. In: Proceedings of
the 1997 American Control Conference.
Ekman, M. (2005). Modeling and Control of Bilinear Systems – Applications
to the Activated Sludge Process. PhD thesis. Uppsala University. Uppsala,
Sweden.
Ekman, M. and B. Carlsson (2005a). Control of the Aeration Volume in an
Activated Sludge Process using Supervisory Control Strategies. HIPCON
Report number HIP05-54-v1-r. Uppsala University.
Ekman, M. and B. Carlsson (2005b). Control of the Aeration Volume in an
Activated Sludge Process using Supervisory Control Strategies. HIPCON
Report number HIP05-49-v1-r. Uppsala University.
Farsangi, M.M., Y.H. Song and Kwang Y. Lee (2004). Choice of facts device
control inputs for damping interarea oscillations. IEEE Transactions on
Power Systems 19(2), 1135–1143.
Galarza, A., E. Ayesa, M. T. Linaza, A. Rivas and A. Salterain (2001).
Application of mathematical tools to improve the design and operation of
activated sludge plants. Case study: The new WWTP of Galindo-Bilbao,
part ii: Operational strategies and automatic controllers. Water Science
and Technology 43(7), 167–174.
Gerksic, S., D. Vrecko and N. Hvala (2006). Improving oxygen concentration control in activated sludge process with estimation of respiration and
scheduling control. Water Science & Technology 53(4-5), 283–291.
Glad, T. and L. Ljung (1989). Reglerteknik. Grundläggande teori. Studentlitteratur. In Swedish.
Glover, K (1984). All optimal Hankel-norm approximations of linear multivariable systems and their l∞ error bounds. Int. J. Control 39, 1115–1193.
112
Goodwin, G., M. Salgado and E. Silva (2005). Time-domain performance
limitations arisng from decentralized architectures and their relationship
to the RGA. International Journal of Control 78(13), 1045–1062.
Grosdidier, P. and M. Morari (1987). The µ interaction measure. Ind. Eng.
Chem. Res. 26(6), 1193–1202.
Grosdidier, P., M. Morari and B. R. Holt (1985). Closed-loop properties from
steady-state gain information. Ind. Eng. Chem. Fundam. 24, 221–235.
Häggblom, K. E. (1997). Control structure analysis by partial relative gains.
In: Proceedings of the 36th Conference on Decision and Control. San
Diego, California, USA. pp. 2623–2624.
Halvarsson, B. (2003). Applications of coupling analysis on bioreactor models. Master’s thesis. Uppsala University. Uppsala, Sweden.
Halvarsson, B., P. Samuelsson and B. Carlsson (2005). Application of coupling analysis on bioreactor models. In: Proceedings of the 16th IFAC
World Congress. Prague, Czech Republic.
Hammer, M. and M. Hammer Jr. (2008). Water and Wastewater Technology.
6 ed.. Pearson Prentice Hall.
He, M.-J. and W.-J. Cai (2004). New criterion for control-loop configuration
of multivariable processes. Ind. Eng. Chem. Res. 43, 7057–7064.
He, M.-J., W.-J. Cai and B.-F. Wu (2006). Control structure selection based
on relative interaction decomposition. International Journal of Control
79(10), 1285–1296.
Henze, M., C. P. L. Grady Jr., W. Gujer, G. v. R. Marais and T. Matsuo
(1987). Activated sludge model no. 1. Scientific and Technical Report No.
1. IAWPRC, London.
Henze, M., P. Harremoës, J. la Cour Jansen and E. Arvin (1995). Wastewater treatment, biological and chemical processes. Springer-Verlag, Berlin
Heidelberg.
Horn, R.A. and C.R. Johnson (1985). Matrix analysis. Cambridge University
Press.
Hovd, M. and S. Skogestad (1992). Simple frequency-dependent tools
for control system analysis, structure selection and design. Automatica
28(5), 989–996.
Ingildsen, P. (2002). Realising Full-Scale Control in Wastewater Treatment
Systems Using In Situ Nutrient Sensors. PhD thesis. Lund Institute of
113
Technology. Dept. of Industrial Electrical Engineering and Automation,
Lund, Sweden.
Ingildsen, P., G. Olsson and Z. Yuan (2002). A hedging point strategy – balancing effluent quality, economy and robustness in the control of wastewater treatment plants. Water Science and Technology 45(4–5), 317–324.
IWA (November 19, 2007). IWA Task Group on Benchmarking of Control
Strategies for WWTPs: BSM1. http://www.benchmarkwwtp.org/.
Jeppsson, U. and M.-N. Pons (2004). The COST benchmark simulation
model–Current status and future trends. Control Engineering Practice
12(3), 299–304. Ed.
Jeppsson, U., J. Alex, M. N. Pons, H. Spanjers and P. A. Vanrolleghem
(2002). Status and future trends of ICA in wastewater treatment - a european perspective. Water Science and Technology 45(4–5), 485–494.
Johansson, K.H. (2000). The quadruple-tank process: A multivariable laboratory process with an adjustable zero. IEEE Transactions on Control
Systems Technology 8(3), 456–465.
Kalman, R. E., Y. C. Ho and K. S. Narendra (1963). Controllability of
linear dynamical systems. Contributions to Differential Equations 1, No.
2, 189–213.
Kinnaert, M. (1995). Interaction measures and pairing of controlled and manipulated variables for multiple-input multiple-output systems: A survey.
Journal A 36(4), 15–23.
Kommunförbundet (1988). Introduktion till avloppstekniken. Sweden. In
Swedish.
Kreindler, E. and P.E. Sarachik (1964). On the concepts of controllability and observability of linear systems. IEEE Transactions on Automatic
Control Volume: 9, Issue: 2, 129– 136. (Correction: Vol. 10, No. 1, p.
118, 1965).
Lindberg, C-F. (1997). Control and Estimation Strategies Applied to the Activated Sludge Process. PhD thesis. Uppsala University. Dept. of Systems
and Control, Uppsala, Sweden.
Lindberg, C-F. and B. Carlsson (1996). Estimation of the respiration rate
and oxygen transfer function utilizing a slow DO sensor. Water Science
and Technology 33(1), 325–333.
Lu, W. and G. Balas (1998). A comparison between hankel norms
and induced system norms. IEEE Transcations on Automatic Control
43(11), 1658–1662.
114
Mc Avoy, T., Y. Arkun, R. Chen, D. Robinson and P. D. Schnelle (2003).
A new approach to defining a dynamic relative gain. Control Engineering
Practice 11(8), 907–914.
Meyer, U. and H. J. Pöpel (2003). Fuzzy-control for improved nitrogen removal and energy saving in WWT-plants with predenitrification. Water
Science and Technology 47(11), 69–76.
Niederlinski, A. (1971). A heuristic approach to the design of linear multivariable interacting control systems. Automatica 7(6), 691–701.
Olsson, G. (1993). Advancing ICA technology by eliminating the constraints.
Water Science and Technology 28(11–12), 1–7.
Olsson, G. and B. Newell (1999). Wastewater Treatment Systems. IWA publishing. London, UK.
Olsson, G. and U. Jeppsson (1994). Establishing cause-effect relationships in
activated sludge plants – what can be controlled. In: Proceedings of workshop modelling, monitoring and control of wastewater treatment plants.
pp. 2057–2070.
Qin, S. Joe and T. A. Badgwell (2003). A survey of industrial model predictive control technology. Control Engineering Practice 11(7), 733–764.
Rotea, M. (1993). The generalized h2 control problem. Automatica
29(2), 373–385.
Salgado, M. and D. Oyarzún (2005). MIMO interactions in sampled data
systems. In: Proceedings of the 16th IFAC World Congress. Prague, Czech
Republic.
Salgado, M. E. and A. Conley (2004). MIMO interaction measure and controller structure selection. Int. J. Control 77(4), 367–383.
Samuelsson, P. (2005). Control of Nitrogen Removal in Activated Sludge
Processes. PhD thesis. Uppsala University. Uppsala, Sweden.
Samuelsson, P. and B. Carlsson (2001). Feedforward control of the external carbon flow rate in an activated sludge process. Water Science and
Technology 43(1), 115 – 122.
Samuelsson, P. and B. Carlsson (2002). Control of the aeration volume in an
activated sludge process for nutrient removal. Water Science and Technology 45(4–5), 45–52.
Samuelsson, P., B. Halvarsson and B. Carlsson (2004). Analysis of the inputoutput couplings in a wastewater treatment plant model. Technical Report
115
2004-014. Div. of Systems and Control, Dept. of Information Technology,
Uppsala University. Uppsala, Sweden.
Samuelsson, P., B. Halvarsson and B. Carlsson (2005a). Cost–efficient operation of a denitrifying activated sludge process – an initial study. Technical Report 2005-010. Div. of Systems and Control, Dept. of Information
Technology, Uppsala University. Uppsala, Sweden.
Samuelsson, P., B. Halvarsson and B. Carlsson (2005b). Feedforward Aeration Volume Control in An Activated Sludge Process. HIPCON Report
number HIP05-51-v1-r. Uppsala University. Uppsala, Sweden.
Samuelsson, P., B. Halvarsson and B. Carlsson (2005c). Interaction analysis
and control structure selection in a wastewater treatment plant model.
IEEE Transactions on Control Systems Technology 13(6), 955–964.
Skogestad, S. and I. Postlethwaite (1996). Multivariable Feedback Control.
John Wiley & Sons. Chichester, UK.
Svardal, K., S. Lindtner and S. Winkler (2003). Optimum aerobic volume
control based on continuous in-line oxygen uptake monitoring. Water Science and Technology 47(11), 305–312.
Takács, I., G. G. Patry and D. Nolasco (1991). A dynamic model of the
clarification-thickening process. Wat. Res. 25(10), 1263–1271.
Vanrolleghem, P. A. and S. Gillot (2002). Robustness and economic measures
as control benchmark performance criteria. Water Science and Technology
45(4–5), 117–126.
Vanrolleghem, P., H. Spanjers, B. Petersen, P. Ginestet and I. Takács (1999).
Estimating (combinations of) activated sludge model no. 1 parameters and
components by respirometry. Water Science and Technology 39(1), 195–
214.
Vrecko, D., N. Hvala, Stare, Burica, Strazar, Levstek, Cerar and Podbevsek
(2006). Improvement of ammonia removal in activated sludge process with
feedforward-feedback aeration controllers. Water Science & Technology
53(4-5), 125–132.
Weber, B. (1994). Rational Transmitting Boundaries for Time-Domain
Analysis of Dam-Reservoir Interaction. PhD thesis. Swiss Federal Institute of Technology Zürich.
Wilson, D. (1989). Convolution and hankel operator norms for linear systems. IEEE Transactions on Automatic Control 34, 94–97.
116
Wittenmark, B. and M. E. Salgado (2002). Hankel-norm based interaction
measure for input-output pairing. In: Proc. of the 2002 IFAC World
Congress. Barcelona, Spain.
Wittenmark, B., K. J. Åström and S. B. Jörgensen (1995). Process Control. KFS i Lund AB. Dep of Automatic Control, Lund University, Lund,
Sweden.
Xiong, Q., W.-J. Cai and M.-J. He (2005). A practical loop pairing criterion
for multivariable processes. Journal of Process Control 15, 741–747.
Yuan, Z., A. Oehmen and P. Ingildsen (2002). Control of nitrate recirculation
flow in predenitrification systems. Water Science and Technology 45(4–
5), 29–36.
Yuan, Z. and J. Keller (2003). Integrated control of nitrate recirculation and
external carbon addition in a predenitrification system. Water Science and
Technology 48(11), 345–354.
Yuan, Z. and J. Keller (2004). Achieving optimal conditions for nitrogen
removal using on-line sensors and control. In: Proceedings of the 2nd IWA
Leading-Edge Conference on Water and Wastewater Treatment Technologies. Prague, Czech Republic.
Yuan, Z., H. Bogaert, C. Rosen and W. Verstraete (2001). Sludge blanket height control in secondary clarifiers. In: Proceedings of the 1st IWA
Conference on Instrumentation, Control and Automation. Malmö, Sweden. pp. 81–88.
Yuan, Z., H. Bogaert, P. Vanrolleghem, C. Thoeye, G. Vansteenkiste and
W. Verstraete (1997). Control of external carbon addition to predenitrifying systems. Journal of Environmental Engineering 123(11), 1080–1086.
Zhou, K., with J.C. Doyle and K. Glover (1996). Robust and Optimal Control. Prentice Hall.
Zuo, L. and S.A. Nayfeh (2003). Structured h2 optimization of vehicle suspensions based on multi-wheel models. Vehicle System Dynamics 40, 351–
371.
117
Recent licentiate theses from the Department of Information Technology
2007-005
Mahen Jayawardena: Parallel Algorithms and Implementations for Genetic
Analysis of Quantitative Traits
2007-004
Olof Rensfelt: Tools and Methods for Evaluation of Overlay Networks
2007-003
Thabotharan Kathiravelu: Towards Content Distribution in Opportunistic Networks
2007-002
Jonas Boustedt: Students Working with a Large Software System: Experiences
and Understandings
2007-001
Manivasakan Sabesan: Querying Mediated Web Services
2006-012
Stefan Blomkvist: User-Centred Design and Agile Development of IT Systems
2006-011
Åsa Cajander: Values and Perspectives Affecting IT Systems Development and
Usability Work
2006-010
Henrik Johansson: Performance Characterization and Evaluation of Parallel
PDE Solvers
2006-009
Eddie Wadbro: Topology Optimization for Acoustic Wave Propagation Problems
2006-008
Agnes Rensfelt: Nonparametric Identification of Viscoelastic Materials
2006-007
Stefan Engblom: Numerical Methods for the Chemical Master Equation
2006-006
Anna Eckerdal: Novice Students’ Learning of Object-Oriented Programming
2006-005
Arvid Kauppi: A Human-Computer Interaction Approach to Train Traffic Control
Department of Information Technology, Uppsala University, Sweden
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