Robust Modelling Using Bi-Lateral Delay Lines for High Speed

Robust Modelling Using Bi-Lateral Delay Lines for High Speed
Robust Modelling Using Bi-Lateral Delay Lines for High Speed
Simulation of Complex Systems
Petter Krus
Divsion of Fluid and Mechatronic Systems, Linköping University, SE-581 83 Linköping, Sweden
Abstract: A very effective method for modelling and simulation of large complex dynamic systems is represented by
distributed modelling using transmission line elements (or bi-lateral delay lines). This method evolves naturally for
calculation of pressures when hydraulic pipelines are modelled with distributed parameters, and it can be used to
effectively partition the model to use local solvers for the differential equations in each component or subsystem. It is
also applicable to other physical systems, such as mechanical, electrical, gas etc.
One interesting application for distributed solvers using bi-lateral delay lines is in real time simulation, since they are
very robust and usually quite large simulation times steps can be used. Modelling for real-time applications puts
special requirements on robustness in the numerical methods used. In real-time applications there is no room for
decreasing time step in numerically critical stages. Furthermore, if a system is relaying on a real-time simulation for
its functionality, failure in the numerical properties is unacceptable. It is also in many applications possible to
simulate the system faster than real time, which means that high fidelity system simulation can be used to plan ahead
in control applications, and for simulation based optimisation.
Since solvers can be embedded in components or subsystems, it is very straightforward to implement parallel
processing using the multi-core processors which now is the standard for desk top computers. There is also an
increasing need in many situations to resolve the time scale to finer details, where the effect of wave propagation
needs to be modelled.
Keywords: System simulation, transmission lines
INTRODUCTION
The interest in real-time simulation (RTS) has increased in recent years. There are several reasons for this; more
functionality in products is realized through embedded software, and real-time simulation is needed in hardware-in-theloop simulation (HWIL), which is a key technology for validation of such control systems. Also, real-time simulation is
needed in human in the loop simulation (HIL), which is used both is product design and in training simulators. There
are also application in the operation phase of a product, e.g. model based diagnostic systems and in model-predictive
control systems. Currently, specialized, highly simplified, and often linear models are used for real-time simulation.
As a result there is increasing demands for real-time in a range of industrial applications. However, since
commercial platforms for multi-domain system simulation, use architectures that are designed for off-line simulation, it
is difficult if not impossible, to use anything but, highly simplified, often linear models, for real-time simulation. With
the rapid development in hardware performance, real-time simulation, should no longer be a problem, but methods and
tools for generating real-time simulation of high-fidelity, fully non-linear multi-domain system, is lacking. The large
application for off-line simulation models in today’s product development has promoted the application of central
numerical solution methods. With the advent of real-time simulation models it becomes increasingly obvious that these
models will differ greatly from off-line simulation models. Still, it is desirable to keep the two sets of models as close as
possible.
In many real-time applications it is an advantage if the system model can be composed by distributed simulations
of subsystems on different processors. Analyzing systems of components originating from different vendors may even
make distributed simulations an absolute necessity, in order to limit disclosure of information. Partitioning of simulation
models for parallel simulation is also useful in order to take advantage of multi-core processor architectures rapidly
becoming the norm for desk top computing. High speed simulation also enables, faster than real-time simulation
(FRTS), which can be used in e.g. control systems based on prediction of future responses to alternative actions
(Anagnostopoulos et.al 2003). Furthermore, it also enables the use of simulation based optimisation also for larger
systems.
In the recent decade, system simulation has had something of a breakthrough, where large systems can be
modelled with a high degree of fidelity, i.e. Lantto et al.. However, the technologies used in commercial software are
decidedly off-line technologies that do not allow for real time simulation, this includes centralized solver as the normal
approach to simulation. Although great advances have been made in the development of algorithms and software, this
approach suffers from inherently poor scaling. I.e. execution time grows more than linear with system size.
In contrast, distributed modelling, where solvers are embedded in subsystem, and even component models, has
almost linear scaling properties. Special considerations are needed, however, to connect the subsystems to each other in
a way that maintains stability properties, and do not introduce unwanted numerical effects. Technologies based on
Robust Modelling Using Bi-Lateral Delay Lines
bilateral delay lines, see Auslander 1968, (or transmission line modelling, TLM) have been developed and used for a
long time at Linköping University, and has successfully been implemented in the HOPSAN simulation package, which
at time of writing is the only simulation package that utilise the technology, within mechanical engineering, and fluid
power. It has also been demonstrated for parallel simulation in Krus et al 1990 and subsequently by Burton et al 1994.
Although the method has its roots already in the sixties, it has never been widely adopted, probably because its
advantages are not evident for small systems, and that wave-propagation is regarded as a marginal phenomena in most
areas, and thus not generally well understood.
Simulation using bilateral delay lines is also highly suited to simulate systems where wave propagation is an issue.
One particular area is in rock drill equipment, where a high fidelity representation of wave propagation phenomena in
both mechanical, and hydraulic parts, is fundamental for describing the functionality of the system.
Using distributed solvers with bi-lateral delay lines as connection elements, gives a physically motivated
partitioning of the system. In this way component models can be numerically insulated from each other, which provide
highly robust numerical properties. This technique is also useful for high speed simulation of systems, and has been
used successfully for simulation based optimization, where the system is simulated a large number of times with
different parameter sets. The use of transmission line elements for partitioning of systems is a non-exclusive approach.
Conventional simulation techniques can still be used within the subsystems. This means that transmission line elements
can be used to connect simulation models developed in different simulation packages. Using distributed solvers also has
the advantage that it allows a model to be assembled from precompiled modules. This can be highly valuable in
collaborative system design, since it does not require disclosure of the source code, when providing a module to
partners.
Differential Algebraic Systems
A general approach to represent a system is to represent it as a differential algebraic system. This also allows for
algebraic loops. The simulation language Modelica [6] is a language that is based on this form.
F ( x, x , u , t )  0
(1)
where x is the variable vector, u is an input vector, and t is time.
However, Eq. (1) implies that the system essentially has to be written in state space form, something that may be
considered as too limited. Many relationships are usually given in transfer function form, which makes it more natural
to allow for higher derivatives. The system can then instead be expressed as
 dy d 2 y
d2y 
F  y , , 2 , , 2 , t   0
dt
 dt dt

(2)
This also has the advantage that the variable vector is reduced, since y is shorter than x, y contains a subset of the states
in x. It should, however, be pointed out that it is only possible to impose strong non-linearities (such as limitations on
the state variables) represented in the y vector. Also all variables that are of any interest must be included in the y vector
otherwise they will not be computed explicitly. Finally high order differentials should be avoided since the equations
becomes numerically ill conditioned if the word length is limited.
In order to solve the dynamic part of the system in a numerically stable way, the trapezoidal rule can be used.
Using the trapezoidal rule the time differential is solved as:
1
x(t  h)  x (t )  h  x(t )  x(h  t ) 
2
A more effective way of using the trapezoidal rule is to reformulate it in the form known as the bilinear transform.
1
d 2 1 q

dt h 1  q 1




(3)
(4)
where q in this context represents the time displacement operator such that:
qy  y (t  h)
(5)
Using the bilinear transform in Eq. (4) means that it can be rewritten as a function G of y and old states.
G ( y (t ), y (t  h),  , y (t  nh), u (t ), u (t  h),  , u (t  nh), t )  0
(6)
P. Krus
When solving the system all the old values y(t  h) y(t  nh) can be regarded as constants since they have already
been established in previous time steps. Likewise the input vector is also known. Equation (6) is therefore rewritten as:
G ( y (t ), t )  0
(7)
In order to solve this system of equations in a numerically stable way, the Jacobian matrix is needed, which is defined
as:
J ijk 
Gi ( yk (t ))
y j
(8)
The equation can then be solved numerically using Newton-Raphson iteration.
yk 1  yk (t )  J k (t ) 1 G ( yk (t ))
(9)
Since an iterative procedure is used, there is a potential for performance loss due to the number of iterations needed to
solve the system. However, the values from the previous time step can be used as start values.
y0 (t )  y (t  h)
J 0 (t )  J (t  h)
(10)
If the system is linear, the system can be solved in only one iteration, and it is usually sufficient with only one iteration
even for non-linear systems, especially if a small time step is used. There are, however, situations when input signals
changes suddenly, e.g. a valve is changed step wise during one time step, that requires more than one iteration. In
practice, however, it has been found that two iterations increase the tolerance against non-linearities dramatically, while
a further increase to three iteration gives only minor improvement. Two iterations have therefore been found to be
something near to an optimum for almost all situations. For implementation it is better to use LU-decomposition rather
than using the matrix inverse of the Jacobian.
Provided the system is reasonably linear (slow variation of J, Eq. (9) is an A-stable method. However, in reality,
rather large variations of J can be tolerated. Even pure discontinuities can also be handled satisfactory using the above
approach, when fixed-time step is used (as in real-time simulation).
Eq. (9) also illustrates a dilemma associated with all numerically stable methods. They need knowledge of the
Jacobian, and if the system is stiff and highly non-linear this must be updated very often. We also realize that the
computational burden is much more than linearly dependent on the size of the system. This makes these methods
unsuitable for large problems. Eq. (9) is, however, very effective for solving small systems which makes it very suitable
for solving subsystems in a distributed modelling context.
Example
As an example the modelling of a simple hydrostatic transmission consisting of a pump and motor with an inertia
load is considered.
Figure 1. Hydraulic pump-motor example.
The vector F for the system equations becomes
Robust Modelling Using Bi-Lateral Delay Lines
pa  pb


 Tp
 p Bp  D

p


  D  q

p p
p1



 p Dp  q p 2



qm1  q p1


 p1  C

s1




qm 2  q p 2
F ( y, y , u , t )   p 2 

Cs 2


  D  q

m
m
m
1


 m Dm  qm 2



p
p

  B  a

b
 Tm
m
m


Dm


B
p
p

T
 
m 
a
b
L 





m
m

Jm
J m Dm
J m 


(11)
The variable vector y is
 Tp 


 q p1 
q 
 p2 
 p1 


y   p2 
q 
 m1 
 qm 2 
T 
 m 
m 


(12)
The variable vector u is
  
u  p
T 
 L
(13)
Using bilinear transform to convert the time continuous differential algebraic system into a discrete time system yields
(14)
where DS is the delay step function defined as
DS ( n, x(t ))  x (t  nh)
The Jacobian of this system is:
(15)
P. Krus
(16)
Eq. (9) can then be used to solve this system in each time step. Although it is possible to use Eq. (9) directly it is wise to
replace the inverse of the Jacobian by using LU-decomposition instead, there are also a few other actions that can be
done in order to further enhance the efficiency of the solver. In general, however, the effort to solve the system
increases more than linear with the system size.
Figure 2. Scheme of transformations
Figure 2. shows a scheme of the transformations involved. The differential algebraic system DAE is transformed
into time discrete form using bilinear transform. In Elmqvist et al 1995 the concept of inline integration is introduced,
where extra equations are introduced to perform the integration. The scheme here is related but would be more
appropriately called inline transformation, and inline integration can be viewed as a special case of that. The Jacobian J
is obtained by symbolic partial differentiation of the time discrete system G. G and the Jacobian J are used to solve the
system in each time step using the Newton-Raphson method for solving the system in each time step.
Distributed Modelling For Simulation of Fluid Power Components and Systems
Distributed parameters
Simulation of fluid power systems are characterized by difficulties such as very strong nonlinearities, stiff
differential equations and a high degree of complexity. Using conventional integration techniques it is often necessary
to use very small time steps in order to be able to deal with numerically stiff problems, and strong nonlinearities.
If the state variables in the system are to be unique for each subsystem and not shared by other subsystems,
distributed parameters (variables) must be introduced. This can be accomplished if the propagation of waves in
connecting components, such pipes, in the system is considered.
A very suitable method for modelling and simulation of large complex dynamic systems is represented by
distributed modelling using transmission line elements. The origin of this concept goes back at least to Auslander 1968
[1] who first introduced transmission lines (or bi-lateral delay lines). This method evolves naturally for calculation of
pressures when pipelines are modelled with distributed parameters. This approach was adopted for simulation of fluid
power systems with long lines in the HYTRAN program already in the seventies.
A related method is the transmission line modelling method (TLM) presented by Johns and O'Brien (Ref. [2]) for
simulation of electrical networks.
Johns and O'Brien pointed out that an important aspect of modelling using transmission line elements is that most
of the numerical errors introduced by an ordinary solver are avoided. The errors made due to the introduction of
transmission line elements, are better described as modelling errors.
An attractive feature with this is that laws of conservation of mass and energy still hold for the solution, since there
always exist a plausible physical system for the model, although the line lengths may vary compared to the original
system. This also implies that the user may tolerate a larger numerical error since, generally, quite large modelling
errors are present anyway (errors of the order of 10\% are generally considered acceptable from an engineering point of
view).
Robust Modelling Using Bi-Lateral Delay Lines
The Unit Transmission Line Element
In transmission line modelling the basic dynamic element is the unit transmission line. In the HOPSAN package
this is used to connect different components to each other. In the general case it can be used to model both capacitances
and inductances. In the HOPSAN-package, however, it is used primarily to represent capacitances (oil volumes and
mechanical springs).
q1
q2
p1
p2
Figure 3. Transmission line
The complete set of equation that describes a lossless transmission line are:
p1 (t )  p2 (t  T )  Z c  q1 (t )  q2 (t  T ) 
p2 (t )  p1 (t  T )  Z c  q2 (t )  q1 (t  T ) 
(17)
Here Zc is the characteristic impedance of the line, p and q are pressures and flows respectively. T is the time delay in
the line. Note that the main property of these equations is the time delay they introduce in the communication between
the ends. Introducing
c1 (t )  p2 (t  T )  Z c q2 (t  T )
(18)
c2 (t )  p1 (t  T )  Z c q1 (t  T )
Here c is the wave variables that represent information that has been transmitted from the other side of the transmission
line. With these, the following set of equations is obtained.
p1 (t )  c1 (t )  Z c q1 (t )
p2 (t )  c2 (t )  Z c q2 (t )
(19)
Figure 4. Block diagram of transmission line.
An interesting observation is found if c2 in Eq. (18) is substituted with Eq. (19) and the outlet at 2 is blocked.
p1 (t )  p1 (t  2T )  Z c ( q1 (t  2T )  q1 (t ))
(20)
Compared to the trapezoidal method for integration
y (t  h )  y ( t ) 
h
 y (t )  y (t  h) 
2
(21)
Where h is the simulation time step. These equations are the same if T=h/2 !
The relationship between flow entering a volume and the pressure can be written as:
p 
q
C
(22)
Zc 
h
C
(23)
where C is the capacitance. Identification yields
P. Krus
The implication of this is that the trapezoidal method is used to integrate pressure in a volume (capacitance) between
two components; this corresponds to introducing a short pipe instead of a pure capacitance.
The introduction of a transmission line element in place of a capacitance can therefore be viewed as a kind of
integration method. In general it can be written such that the integration of an equation as (24) is performed by splitting
the variable y into the two variables y1 and y2 .
y  f  x1  x2 
(24)
These values are the same in steady state, and the difference between them can be regarded (and a measure of) a
numerical error. They are calculated using the following equations.
y1 (t )  y2 (t  h)  hf  x1 (t )  x2 (t  h) 
y2 (t )  y1 (t  h)  hf  x2 (t )  x1 (t  h) 
(25)
y1 is then of course used at the equations associated with x1 and y2 with x2.
It should be noted that in order to further improve the numerical properties of the transmission line element
damping can be introduced as in ref. [3]. This greatly improves the behaviour with no significant side effects. Since a
transmission line also has inductance, an unwanted parasitic inductance will result from using a transmission line to
represent a capacitance. The inductance L can be calculated as:
L  hZ c 
h2
C
(26)
As can be seen it rapidly diminishes with step size. It also means that a transmission line can be used to replace a
inductance with a resulting parasitic capacitance as a side effect.
System simulation using transmission line elements
Using this substitution for the pressures p1 and p2 in the hydraulic transmission example yields the new system
(instead of Eq. (11)) of equations as:
Here the variable p1 has been split into the variables p1a
p2 . The variable vector becomes
(27)
and p1b across the transmission line, and the same with
Robust Modelling Using Bi-Lateral Delay Lines
 Tp 


 q p1 
q 
 p2 
 p1a 


 p1b 
y   p2 a 


 p2 b 
q 
 m1 
 qm 2 


 Tm 
 
 m 
After bilinear transformation the system becomes
(28)
(29)
The corresponding Jacobian then becomes:
(30)
Note that this system can be partitioned into two uncoupled systems that consequently can be solved independently
from each other in each time step. This means that the system can be solved using two instances of Newton-Raphson,
equation (9), that are numerically insulated from each other. The price for this is that the Jacobian has becomes slightly
larger with the introduction of two variables for pa and pb respectively.
Modelling of Components
In order to demonstrate the principle of component modelling the very simple laminar orifice with the resistance
Rv is shown.
P. Krus
Figure 5. Laminar restrictor
The pressure-flow relationship is:
p1  p2
Rv
The following equations are solved at the component (connected to lines)
q2 
q2 
p1  p2
Rv
q1   q2
(31)
(32)
p1  c1  q1Z c1
p2  c2  q2 Z c 2
Here Rv is the resistance of the orifice, Zc1 and Zc2 are the characteristic impedances of the lines connected to the orifice.
Being a non-dynamic linear system, these equations can be solved algebraically for q1 and q2.
The equations used in the executable component model will thus be :
c1  c2
q2 
Rv  Z c1  Z c 2
(33)
q1  q2
p1  c1  q1Z c1
p2  c2  q2 Z c 2
A comparison with Eq.(32) and Eq.(36) shows that the adoption to transmission lines has the same effect on the
equations as adding restrictors with the resistance Zc. As a consequence, it is rather uncomplicated to modify any
component or simulator to adapt to transmission lines. The same principle is valid also for mechanical nodes. However,
since a resistor is a non-dynamic component, one algebraic state is usually introduced for each connector.
The figure below shows the block diagram of the restrictor
Figure 6. Block diagram of restrictor.
The laminar resistor is a simple linear model that can be derived analytically. The more general approach is using
Newton-Raphson Eq.(9).
Modelling of Systems
These components can be connected for system simulation. Here also a pressure source at the right end has been
added. Modelling of mechanical springs is performed in exactly the same way as the volume, since they also represent
pure capacitances. The only difference is that it handles speed, instead of flow, and force, instead of pressure. The
modelling of two dimensional mechanical systems is presented in detail in Krus 1995 [8].
Robust Modelling Using Bi-Lateral Delay Lines
Figure 7. Assembled block diagram, showing transmission line – laminar restrictor – pressure boundary condition.
Test Example
As a test a simple spring mass system is simulated.
Figure 8. Mass spring test system.
The spring is modelled as a bi-directional delay line and is connected to a fixed point to the left and to a pure
inertia load to the right. The mass is integrated using bilinear transform (equivalent to the trapezoidal rule). As can be
seen the time step can be increased to 0.1 seconds and the system still shows benign behaviour, only the frequency is
shifted slightly. If the time step is increased to one second the response is distorted, and much slower than the exact
solution due to the influence of parasitic inductance from equation (26), but the system is still stable, If this had been a
very fast component in a system, and had not been of prime interest, this behaviour could still be tolerated.
Figure 9. Simulated result of spring mass system with spring represented by transmission line. Dashed line is h=0.01 sec,
filled is h=0.1 sec and dotted line is h=1 sec.
As a reference the spring is modelled to simply yield a force to corresponding to the compression of the spring. For
slow variations, or very small time step, this should give the same result. However, since the spring is modelled as a
separate component there is a time delay of one time step from the force calculated in the spring is affecting the mass,
and an updated position of the mass is sent back to the spring. As a result, even with the smallest time step of one
millisecond, the system show considerable divergence, and is completely unstable for a time step of 10 milliseconds. To
be able to use larger time steps, it would be necessary to use one solver for the whole model.
P. Krus
Figure 10. Simulation result of mass spring system with spring force calculated as a function of compression. Time step h
= 0.001 seconds is dotted. h = 0.01 seconds is filled.
Event Free Modelling
In real time simulation it is not practical to handle events by finding zero-crossings and restart solvers when e.g.
limitations in variables are hit. Therefore other mechanisms needs to be used. Using the approach with differential
algebraic equations that are solved through bilinear transformation and Newton-Raphson using an analythical Jacobian,
it might seem that the Jacobian would be extremely difficult to derive, since most manually written models involves a
great deal of conditions and jumps as they are written in a procedural style. The introduction of an automated approach
means, that a functional programming style is imposed (the derivative of a function can always be defined, except in
singular points). Therefore all conditions have to be represented by functions. The algorithm
if(cond)then
a = a1
b = b1
c = c1
else
a = a2
b = b2
c = c2
end if
can be transformed into
a=If(cond,a1,a2)
b=If(cond,b1,b2)
c=If(cond,c1c2)
Since the If function is piecewise continuous function the differential is defined and can be used in the Jacobian .An
advantage with this style of programming is that the variables a, b, and c in the example are forced to be defined for
both cases, and cannot be forgotten as in normal procedural programming. This approach does not remove the events as
such but there is no notion of events in the code since everything is hidden in the functions.
Application
The technique described here can be used to connect large systems for high-speed simulation. One example is
given by the aircraft system model shown in the figure below. This particular model has been used to demonstrate
optimization of flight control system including actuator system [12].
Robust Modelling Using Bi-Lateral Delay Lines
Figure 11. Aircraft simulation model
Another application is to use transmission line elements for connecting different solvers to each other in cosimulation. An example of simulation of a complete wheel loader can be found in [3] and [9].
The HOPSAN simulation package has been developed at the division since the late 1970s, and has played an
important role in research projects over time. It has also been used widely in the industry. In recent years commercial
software has become increasingly available, and those have also been used at the division. In 2009 the development of
a new simulation platform, HOPSAN NG, was initiated. This is an object-oriented C++ application with focus on multicore support and compatibility.
Figure 12. The HOPSAN-NG simulation package
In this paper the method of using transmission lines for partitioning complex system models has been described.
Using such elements it is possible to use highly robust distributed solvers on small subsystems, which are then
connected to each other using the transmission lines, for system simulation. As a result highly robust models are
achieved that can be used also for real time simulation. Using transmission lines to numerically partition a system is a
non-exclusive approach that can be used together with conventional solvers. In this way it is possible to obtain a very
robust system modelling concept that that has been described here, which is also very useful also for real time
simulation. It is also a very useful technique for connecting different simulation software in co-simulation. The
transmission line modelling is implemented successfully in the HOPSAN-NG simulation software.
CONCLUSIONS
In this paper the method of using transmission lines for partitioning complex system models has been described.
Using such elements it is possible to use highly robust distributed solvers on small subsystems, which are then
connected to each other using the transmission lines, for system simulation. As a result highly robust models are
achieved that can be used also for real time simulation. Using transmission lines to numerically partition a system is a
non-exclusive approach that can be used together with conventional solvers. In this way it is possible to obtain a very
robust system modelling concept that that has been described here, which is also very useful also for real time
simulation. It is also a very useful technique for connecting different simulation software in co-simulation. The
transmission line modelling is implemented successfully in the HOPSAN-NG simulation software.
P. Krus
REFERENCES
[1] Auslander D. M., 'Distributed System Simulation with Bilateral Delay-Line Models' Journal of Basic Engineering,
Trans. ASME p195-p200, June 1968.
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