On Indirect Input Measurements Jonas Linder , Martin Enqvist 13th March 2015

On Indirect Input Measurements Jonas Linder , Martin Enqvist 13th March 2015
Technical report from Automatic Control at Linköpings universitet
On Indirect Input Measurements
Jonas Linder? , Martin Enqvist?
Division of Automatic Control
E-mail: [email protected], [email protected]
13th March 2015
Report no.: LiTH-ISY-R-3080
Submitted to the 17th IFAC Symposium on System Identification
Address:
?
Department of Electrical Engineering
Linköpings universitet
SE-581 83 Linköping, Sweden
WWW: http://www.control.isy.liu.se
AUTOMATIC CONTROL
REGLERTEKNIK
LINKÖPINGS UNIVERSITET
Technical reports from the Automatic Control group in Linköping are available from
http://www.control.isy.liu.se/publications.
Abstract
A common issue with many system identification problems is that the true
input to the system is unknown. In this paper, a framework, based on indirect input measurements, is proposed to solve the problem when the input is
partially or fully unknown, and cannot be measured directly. The approach
relies on measurements that indirectly contain information about the unknown input. The resulting indirect model formulation, with both directand indirect input measurements as inputs, can be used to estimate the desired model of the original system. Due to the similarities with closed-loop
system identification, an iterative instrumental variable method is proposed
to estimate the indirect model. To show the applicability of the proposed
method, it is applied to data from an inverted pendulum experiment with
good results.
Keywords: System identification, Model structure, Physical models, Instrumental Variable, Closed-loop
1
Introduction
System identification is the science of building dynamical models from system
data, i.e. signals. The objective is to estimate a model between an input and the
resulting output. A common assumption is that the relevant input and output
signals are available or known. One of the major challenges is to separate the
effects of disturbances, present in the in- and output signals, from the relevant
input-output behavior that would be preferable to use for estimating the model.
A first example is the classical open-loop system identification problem, where
the input is assumed to be known while the output is corrupted by an unknown disturbance (Ljung, 1999). Other examples are the classical closed-loop
and errors-in-variables (EIV) problems, where the disturbances are of different
characters and thus present different challenges. In the closed-loop problem,
the input is typically known but is correlated with the same disturbance as
the output (Forssell, 1999), and in the EIV problem, the input is assumed to be
measured with noise and the focus is on the input’s dependency on disturbances
(Söderström, 2007).
In this paper, we are presenting a framework where the input vector u is
assumed to be partially unknown but the effects of the inputs are visible in
several measurements. The goal is to estimate a model GO (p) from the partially
unknown input vector u to a set of the available measurements yO . Two type
of external signals are assumed to affect the system, user-controlled signals
affecting the input, and unknown disturbances affecting both the in- and output.
It is assumed that all elements of the input vector can be correlated with the
external user-controlled signals.
The input is divided into known inputs uK , inputs uD that are directly
measured with noise through a known transfer function and finally, inputs uI
that are indirectly measured with noise. An indirect input measurement is a
signal that is affected by the input uI and thus, indirectly contains information
about uI . An example is of course the output yO , but here an indirect input
measurements should be thought of as a measurement in addition to the output
yO . By formulating an indirect model, where the known input, the direct and
the indirect input measurements are used as inputs, in terms of the original
model GO (p), it is possible to estimate the dynamical model of interest.
The resulting estimation problem is similar to closed-loop system identification due to the correlation of the disturbance to both the input and output of
the indirect model. In fact, due to the assumed generality of the framework,
many classical and recently proposed models are recovered as special cases of
the indirect model. For instance, if all inputs are assumed to be known, the
open-loop or closed-loop case is recovered depending on the relations with the
external disturbances. An approach based on the external user-controlled signal and an iterative instrumental variable method is proposed to estimate the
wanted dynamical model.
The remainder of this paper is organized as follows: In Section 2 indirect
input measurements are introduced, the indirect model is derived, a couple
of examples of models with indirect input measurements are presented and finally, identifiability issues are discussed. In Section 3, an iterative instrumental
variable method is proposed to estimate the indirect model. It is complemented
with a discussion about discretization of the indirect model and how to overcome
certain identifiability issues by using several datasets. The proposed method is
1
applied to one of the examples from Section 2 together with experimental data
in Section 4 and finally, in Section 5, the paper is summarized with conclusions.
2
Indirect input measurements
Figure 1 shows a system GO (p) with input u ∈ Rnu and output yO ∈ RnO .
The goal is to estimate a model of the transfer function matrix GO (p) from the
relation
yO = GO (p)u + HO (p)τ + eO
(1)
where p is the differential operator, HO (q) is a transfer function matrix, τ is
a vector of unknown external disturbances and eO is measurement noise. The
input u is assumed to be given by
u = Fδ (p)δ + Fτ (p)τ
(2)
where Fδ (p) is a possibly unknown transfer function matrix, δ is a known vector
of user-controllable external signals and Fτ (p) is a transfer function matrix.
Note that (1) can be part of a bigger system since both u and yO are dependent
on τ , and that both HO (p) and Fτ (p) can have elements equal to zero. For
instance, the input u and the output yO can be signals in a system operating in
closed-loop.
It is assumed that the input is partially unknown and that it can be divided
into
 
uK } known
(3)
u =  uI 
unknown
uD
with the transfer function matrix divided into
GO (p) = GOK (p) GOI (p) GOD (p)
(4)
The unknown inputs are here divided into inputs uD ∈ RnD directly measured
with noise and inputs uI ∈ RnI that are indirectly measured with noise. Here,
indirect refers to that a measurement of the input uI is not available, but that
there is an extra measurement that is affected by the input uI and thus, indirectly contains information about uI .
Assuming that these direct and indirect measurements are available, (1)
can be extended with the outputs yD ∈ RnD corresponding to the directly
δ
τ
Fδ
Fτ
+
u
τ
GO
HO
+
yO
eO
Figure 1: The system GO (p) represented as a block diagram showing the dependencies on the external user-controllable signal δ and external disturbance
τ . Note that both the input u and the output yO are assumed to be correlated
with τ .
2
measured inputs and the outputs yI ∈ RnI
measurements, which gives
  
GOK GOI GOD
yO
 yI  = 
 GIK GII GID
yD
0
0
GDD
| {z } |
{z
y
corresponding to the indirect input
G(p)
   
u
H
  K  O
u
HI τ + e
+

I
uD
0
} | {z } | {z }
u
(5)
H(p)
where the transfer functions are of suitable dimensions, GII is assumed to be
invertible, GDD is assumed to be known and invertible, and e is a vector of
measurement noises. Here, the dependencies on p have been dropped for brevity.
2.1
The indirect model
The unknown inputs of (5), and thus of (1), can now be eliminated by algebraic
manipulation. Firstly, the noise-free output is given by
     
ẙO
yO
HO
 ẙI  =  yI  −  HI τ − e
(6)
ẙD
yD
0
Secondly, solving the third row for uD gives
uD = G−1
ẙ ,
DD D
(7)
Thirdly, the second row is solved for uI and combined with (7) which gives
ẙI − GIK uK − GID G−1
ẙ
uI = G−1
DD D
II
(8)
and finally, the first row is combined with (7) and (8) giving
ẙO = G̃OK uK + G̃OI ẙI + G̃OD ẙD
(9)
where
G̃OI = GOI G−1
,
II
G̃OK = GOK − G̃OI GIK
(10)
and
h
i
G̃OD = GOD − G̃OI GID G−1
DD
(11)
In reality, the noise free outputs are not available and instead the noise-corrupted
outputs need to be used. Equation (6) can be substituted into (9) giving the
indirect model
yO = G̃OK uK + G̃OI yI + G̃OD yD
− G̃OI (HI τ + eI ) − G̃OD eD + HO τ + eO
= G̃OK uK + G̃OI yI + G̃OD yD + τ̄
= G̃O ũ + τ̄
3
(12)
T
where G̃O = G̃OK G̃OI G̃OD , ũ = uK yI yD
and τ̄ is a vector of aggregated disturbances. Note that in this framework, there is some flexibility in
the choice of the outputs since both yO and yI in (5) can act as the output in
(12) (if GOI , i.e. the new GII , has the correct properties). Which measurements
that are suitable to be selected as outputs depends on the application and many
choices can lead to satisfactory results.
Note that even if the original problem (1) is open loop, i.e. that u is uncorrelated with τ and eO , the indirect model might have correlation between
τ̄ and ũ since the direct and indirect input measurements are used as inputs.
The indirect model formulation can be seen as having artificial loops and is in
this sense similar to a closed-loop system identification problem. Hence, even
if the original system is open-loop, it is sensible to treat the estimation using
(12) as a closed-loop problem. Furthermore, in contrast to a system with an
actual controller in the loop, the indirect model might have a direct term in this
artificial loop and some closed-loop identification methods might fail if this is
not considered (Linder et al., 2014; Dankers, 2014).
2.2
Connection to existing system identification problems
The indirect model (12) can in some sense be seen as a generalization of a
number of classical problems considered in system identification literature and
the connection to these are discussed below.
The most straight-forward cases are when the input is uncorrelated with
the external disturbance, i.e. Fτ = 0. The transfer function matrix Fδ can in
these cases, for example, be seen as either some actuator dynamics or can be
set to identity if the inputs are controlled directly. If all inputs are known,
i.e. nK = nu , the classical open-loop case is recovered (Ljung, 1999) and the
classic open-loop errors-in-variable case is recovered if GDD = 1 and nD = nu
(Söderström, 2007).
The positive feedback closed-loop cases are recovered if Fδ = S and Fτ =
F SHO where F is the system in the feedback and S = (1 − F GO )−1 is the
sensitivity function. In the same way as the open-loop cases, the classic closedloop case is recovered if nK = nu (Forssell, 1999) and the closed-loop errors-invariable case is recovered if GDD = 1 and nD = nu (Söderström et al., 2013).
In some system identification problems, the input to the system is unknown.
Blind, or sensor-only, system identification relies on knowledge of properties of
the input signal. An alternative is the sensor-to-sensor system identification
(S2SID) framework that is recovered if Fτ τ = 0 and nu = nI . In this case. the
resulting G̃OI would correspond to the pseudo transfer function in the S2SID
framework (D’Amato et al., 2009).
Another related and interesting model family concerns dynamic networks. In
the framework of Dankers (2014), a dynamic network has a number of internal
variables that are dynamically related to each other and a set of these variables is
assumed to be measurable. There are also two types of external signals assumed
to affect the network, firstly, variables that are known and can be manipulated
by the user called external variables and, secondly, unmeasured disturbances,
see Dankers (2014) for more details.
4
vK
rK
vI
rI
vD
rD
uK
GOK
uI
GOI
yO
vO
GIK
GII
uD
GOD
GID
v̄I
yI
eD
yD
GDD
Figure 2: The model (5) seen as a dynamic network where circle corresponds to
sums and blocks corresponds to the transfer functions. The black parts corresponds to the model of interest, red corresponds the indirect input measurements
and blue corresponds to the direct input measurements.
By introducing the notation
 
 
vK
rK
 r  = Fδ (p)δ,  v  = Fτ (p)τ and vO = HO (p) τ + eO ,
I
I
HI (p)
v̄I
eI
vD
rD
(13)
then according to the framework of Dankers (2014), the model (5) can be written
as the network model
  
     
yO
yO
0
vO
0 0 0 GOK GOI GOD
 yI   0 0 0 GIK GII GID   yI   0   v̄I 
  
     
 yD   0 0 0
y   0   0 
0
0
GDD 
 =
 D +   +  
(14)
uK   0 0 0

    
0
0
0 
  
uK  rK  vK 
u   0 0 0
0
0
0   uI   rI   vI 
I
0 0 0
0
0
0
uD
uD
rD
vD
which can also be seen in Figure 2. Since there are no direct paths from yI and yD
to yO , they can not be used directly to estimate GO in this framework (Dankers,
2014). In some sense, the transformation from (5) to the indirect model (12)
create artificial paths which makes it possible to utilize these measurements.
The direct and indirect input measurements can with this point of view be
seen as an generalization of the additive measurement error of the framework
in Dankers (2014).
5
2.3
Examples
In this section two examples of parametric models with indirect input measurements will be presented. The first example will later be used in the experimental
verification in Section 4. Here, the models are only summarized and for a complete derivation and deeper discussion, see Linder (2014).
2.3.1
A modified inverted pendulum
Figure 3 shows a modified inverted pendulum hinged on a cart. The mass of the
pendulum is split into a nominal mass M and an additional load with mass m.
The nominal mass has its center of gravity (CG) a distance zg from the center
of rotation (CR) and with the inertia Ix around its CG. The additional mass
has the inertia Ix,m and is positioned a distance zm from the CR. There are
two torsional torques acting on the pendulum, one linearly dependent on the
angle φ and one linearly dependent on the angular velocity φ̇ corresponding to
damping. There is an external unknown force acting on the cart which results
in the acceleration ay . Finally, a torque disturbance τ is assumed to act on the
CR. A model of the system is given by
damping
restoring torque
disturbances
}|
{ z }| { z
}|
{
k + M gzg + mgzm
d
M zg + mzm
τ
φ − φ̇ +
ay +
φ̈ = −
I1
I1
I1
I1
z
(15)
2
. The transfer function from ay to the angle
where I1 = Ix + M zg2 + Ix,m + mzm
φ is given by
b0
G1 (p) = 2
(16)
p + a1 p + a2
where
b0 =
d
k + M gzg + mgzm
M zg + mzm
, a1 = , a2 =
I1
I1
I1
Ix,m
(17)
m
mg
Ix
M
Mg
zg
zm
as
zs
d ˙, k
ay
Figure 3: A sketch of the approximate inverted pendulum model showing the
relation of the important quantities.
6
There is an inertial measurement unit (IMU) mounted on the cart a distance zs
from the CR and it is assumed to measure
y1 = φ̇ + e1
(18)
y2 = as + e2 = zs φ̈ + gφ − ay + e2
where as is the tangential acceleration (assuming small angles) and ei , i = 1, 2
are measurement noise. In this example, there are no known or directly measured inputs and only the signals uI = ay , yI = y2 and yO = y1 are used. This
gives
GOI (p) = G1 (p)p, GII (p) = G1 (p)(zs p2 + g) − 1,
(19)
HO (p) =
1
G (p) and HI (p) = HO (p)(zs p2 + g)
b0 OI
(20)
which results in
(p) =
G̃OI (p) = GOI (p)G−1
II
where
β0 = −
M zg + mzm
,
I2
β0 p
p2 + α1 p + α2
α1 =
d
,
I2
α2 =
k
I2
(21)
(22)
and I2 = Ix + M zg (zg − zs ) + Ix,m + mzm (zm − zs ).
2.3.2
Ship’s roll dynamics
Figure 4 shows a ship performing a turning maneuver. Due to its rudder angle
δ, the ship’s CR is affected by a force resulting in the acceleration ay . There
are a multitude of models for a ship’s roll dynamics. Here, a nonlinear model
developed and discussed in Blanke and Christensen (1993) and Perez (2005),
and used in Linder (2014), is considered.
After linearization and a few manipulations, the model
A1 φ̈ = −dφ̇ − (k + M gzg + mgzm )φ
+ (Kv̇ + M zg + mzm )v̇ + (Kur + M zg + mzm )U r
(23)
+ Kδ δ + τ
2
, see Linder (2014) for a derivation.
is acquired, where A1 = Ax + M zg2 + mzm
T
The input is considered to be u = [δ, v̇, r] where v̇ is the sway acceleration
and r is the angular velocity around the yaw axis.
The transfer function matrix from u to φ is given by
G2 (p) = b1
b2
b3
1
p2 + a3 p + a4
(24)
where
Kδ
Kv̇ + M zg + mzm
Kur + M zg + mzm
, b2 =
, b3 =
U,
A1
A1
A1
k + M gzg + mgzm
d
a3 =
and a4 =
A1
A1
b1 =
7
(25)
, rudder angle
ay
,
X, Y
yn
yb
t
the fron
n from
Ship see
⌧t
m
xb
as
mg
xn
zs
M
IMU
as (tangential acceleration)
zg
Mg
ay
d ˙, k
r (yaw rate)
zm
˙ = p (roll rate)
Figure 4: A sketch of a ship performing a turning maneuver which results in
an unknown acceleration ay acting on the CR. The shift in displaced water
(green/red area) forces the ship back towards its equilibrium.
The motion of the ship is measured using an IMU. The measurements available
are
y1 = φ̇ + e1
y2 = as + e2 = zs φ̈ + gφ −v̇ − U r +e2
| {z }
(26)
−ay
y3 = −r + e3
where φ̇ is the angular velocity around the roll axis, as is the tangential acceleration and ei , i = 1, 2, 3 are measurement noise. In this example, the signals
are uK = δ, uI = v̇, uD = r, yO = y1 , yI = y2 and yD = y3 , which gives
GO (p) = G2 (p)p,
GI (p) = G2 (p)(zs p2 + g) + 0
GDD (p) = −1,
that results in
G̃(p) = β1
β2
β3
p2
−1
−U
and
1
+ α3 p + α4
(27)
(28)
where
Kδ
Kv̇ + M zg + mzm
Kr
, β2 = −
, β3 =
,
A2
A2
A2
d
k − Kv̇ g
α3 =
, α4 =
,
A2
A2
β1 =
(29)
A2 = Ax + M zg (zg − zs ) + mzm (zm − zs ) − Kv̇ zs and where HO and HI are
nonzero but quite complex due to the coupling in the original model.
8
2.4
Identifiability of indirect model
The subject of unique estimation of the parameters in a chosen model structure
is both connected to the model structure considered and the informativity of
the data set used for identification (Bellman and Åström, 1970; Ljung and Glad,
1994; Bazanella et al., 2010). Here, (1) is considered to be a linear system, but
note that the model structure might very well be nonlinearly parameterized. For
an arbitrary nonlinear parameterization of (1), it is difficult to give an explicit
statement of identifiability for the resulting model of the indirect model (12).
There are however a few points worth stressing.
It is not necessary that the chosen model structure of (1) is structurally
identifiable with respect to the parameters since the model of the indirect measurements might introduce more information. Consider, for instance, the model
)
yO = b1 uK + b22 uI
=⇒ yO = b1 uK + b2 yI
(30)
yI = b2 uI
where the original model is not identifiable, due to the square, but the indirect model is. If the chosen model structure of (1) is structurally identifiable
with respect to the parameters this does not necessarily imply identifiability
of the resulting model structure of the indirect model (12) with respect to the
parameters since cancellations might occur. Consider, for example, the model
)
yO = b1 uK + b2 uI
=⇒ yO = b1 uK + yI
(31)
yI = b2 uI
where the original model is identifiable but where b2 is eliminated in the indirect
model, which is thus not identifiable in the original parameters. Hence, the
resulting structure of the indirect model (12) has to be identifiable with respect
to the parameters and this has to be checked on a case to case basis.
Secondly, it is not sufficient that the input ũ of the indirect model is informative since it can be correlated with τ and the setup is thus similar to
identification in closed-loop. In the general case, i.e. when the input u is correlated with τ , a sufficient condition is that the external user-controlled signal
Fδ δ is informative enough. However, this is not a necessary condition since
the required informativity of Fδ δ depends on the complexity of Fτ (p), i.e. the
feedback structure, see Bazanella et al. (2010).
9
3
Estimation of the indirect model
There are several methods to estimate the indirect model (12) but here we will
focus on an instrumental variable (IV) method. The method is appealing due
to low requirements on prior knowledge about the disturbances. The method
is based on correlation between the signals in the estimation problem and the
instruments created from the vector of external signals δ and thus handles both
process and measurement disturbances.
3.1
Discretization using continuous-time parameters
All measurements are taken at discrete-time instances and a discrete-time model
is needed to relate the measurements to the parameters. Several different discretization methods can be used, but in this paper the transfer functions are
discretized with the bilinear transform p = T2 q−1
q+1 where T is the sample period
and q is the shift operator. The benefits of the bilinear transform are the simple algebraic relation between p and q while still being fairly accurate for short
sampling periods and that the resulting discrete-time model is stable for all sampling periods. Given that a parameterization with the parameters ϑ has been
chosen in continuous-time and that the transfer function is finite-dimensional,
the discrete time matrix fraction description can be written
−1
(32)
G̃O,d (q, ϑ) = G̃O T2 q−1
q+1 , ϑ = Ã O,d (q, ϑ)B̃O,d (q, ϑ)
where the matrices are normalized such that
à O,d (q, ϑ) = I + A 1 (ϑ)q−1 + . . . + A na (ϑ)q−na
B̃O,d (q, ϑ) = B0 (ϑ)q−nk + . . . + Bnb −1 (ϑ)q−nb −nk +1
(33)
which results in the difference equation
yO,t = − A 1 (ϑ)yO,t−1 − . . . − A na (ϑ)yO,t−na
+ B0 (ϑ)ũt−nk + . . . + Bnb −1 (ϑ)ũt−nb −nk +1 + τ̃
(34)
Assuming that the ith component can be written
(i)
yO,t
= ϕTi,t θi (ϑ) + τ̃i,t
(35)
then (34) can be cast as the linear regression
yO,t = ϕTt θ(ϑ) + τ̃
(36)
h
iT
θ(ϑ) = θ1T (ϑ) . . . θnTy (ϑ) and ϕt = blkdiag ϕ1,t . . . ϕny ,t
(37)
where
Note that τ̃ is a colored noise process even if τ̄ was white.
10
3.2
Identifiability issues and multiple datasets
In some situations it is possible to circumvent identifiability issues by introducing more information. A model can be made identifiable by introducing a priori
knowledge or by introducing more measurements. For instance, consider the
unidentifiable model
(38)
yO(1) = (a1 + b1 )uK
where both a1 and b1 are unknown. By introducing the second measurement
yO(2) = a1 uK
the extended model
yO =
(39)
(1) yO
a 1 + b1
=
uK
a1
yO(2)
(40)
becomes identifiable. If more measurements are not possible in the same experiment, measurements from several consecutive experiments can be used in
a similar way if the parameters change in a known fashion between the experiments. If the measurements from (38) and (39) were collected from different
experiments, the extended model becomes
(1) y
a + b1 0 u(1)
K
yO = O(2) = 1
(41)
0
a1 u(2)
yO
K
Hence, if we have several measurements which are related through the parameters, then (5) can be extended and cast into (36) and thus overcome certain
identifiability issues.
3.3
Extended instrumental variable method
There are two terms contributing to the output of (36), one containing information about the interesting input-output relation and the second containing
a contribution from disturbances. An instrumental variable method uses the
instrument vector ζt to extract the interesting information from the data. In
principle, the interesting information is estimated by requiring that
N
1 X
ζt (yO,t − ϕTt θ(ϑ)) = 0
N t=1
(42)
i.e. that the sample covariance between ζt and the prediction error should be
zero. A good instrument should be correlated with the regression vector ϕt but
be uncorrelated with the disturbance τ̃t . This idea is generalized in the extended
IV method, where the parameters are found by computing
2
ϑ̂ = argmin kYN − ΦN θ(ϑ)kQ
(43)
ϑ
2
where kxkQ = xT Qx, Q 0 is a weighting matrix,
ΦN
 T


ϕ̄1
ȳO,1
1
1
.
.
=
[ζ1 . . . ζN ]  .. , YN =
[ζ1 . . . ζN ]  .. ,
N
N
T
ȳO,N
ϕ̄N
11
(44)
ȳO,t = L(q)yO,t , ϕ̄Tt = L(q)ϕTt and L(q) is a vector of stable prefilters. See, for
instance, Söderström and Stoica (1989) or Ljung (1999) for more details.
The estimator (43) will be consistent under fairly general conditions and
the real challenge is to choose the vector of instruments ζt and the vector of
prefilters L(q) such that the variance of the predictor is minimized. The covariance properties of the closed-loop IV methods have been investigated in, for
instance, Forssell and Chou (1998), Gilson and Van den Hof (2005) and Gilson
et al. (2011).
It is shown in (Gilson et al., 2011) that the optimal prefilter is given by
the true inverse noise model and that the optimal instruments are given by
the filtered noise-free regression vector. However, the optimal instruments are
not implementable in reality since knowledge of the true system and perfect
measurements would be required. Instead, a variant of the iterative approach
presented in Gilson et al. (2011) is used. In this iterative approach, the current
estimate of the parameters are used to create the instrument vector and prefilter
which are then used to get a new estimate of the parameters, and the iterations
are terminated when the parameters have converged. Here, the instrument
vector is given by
h
iT
T
T
T
T
ζt = L(q, η) −ŷO,t−1
. . . − ŷO,t−
µ̂
.
.
.
µ̂
t
t−n̂u +1
n̂y
(45)
where L(q, η) is a prefilter, F̂δ is an estimate of Fδ ,
  G (ϑ̂) G (ϑ̂) G (ϑ̂)



OK
OI
OD
ŷO,t
û  
ŷO,t

1
0
0
 =  K,t  = 

 ŷ  
I,t
G
(
ϑ̂)
G
(
ϑ̂)
G
(ϑ̂)
ˆ
IK
II
ID
ũt
ŷD,t
0
0
GDD (ϑ̂)



 F̂δ δt

(46)
are the simulated output and input, µ̂t is a linear combination of delayed inputs
ˆt , for instance, µ̂t = ũ
ˆt − ũ
ˆt−2 , and ϑ̂ is the current estimate of the parameters.
ũ
The constants n̂y and n̂u are the number of time lags included in ζt for the
output and the input, respectively, i.e. n̂y = 0 means that ŷO,t is not included
in ζt . The prefilter is given by
L(q, η) = Ĥ −1 (q, η)
(47)
where Ĥ(q, η̂) is an ARMA model estimated from the residual εt = yO,t −
ϕTt θ(ϑ̂). Note that the simulated signals in (46) are only used as instruments
and thus, that an incorrect estimate of Fδ only affects the second-order properties of the estimator (Söderström and Stoica, 1989).
12
4
Experimental verification
To show the applicability of the suggested method, the modified inverted pendulum example was considered. Data, previously used in Linder et al. (2014),
collected from the pendulum seen in Figure 5 was used in the experiment. The
known physical quantities of the system, measured using a scale and caliper, and
calculated from the basic measurements, are listed in on the first row of Table
1. Note that only the major components were considered when calculating the
inertia and that screws and cables were not included.
Two IMUs were used, one mounted on the pendulum and one mounted
on the cart. The angular velocity was measured using the x-axis gyro and the
tangential acceleration was measured using the y-axis acceleration measurement
of the IMU mounted on the pendulum. The y-axis acceleration measurement
from the IMU on the cart was used as the reference signal δ which means that
δ is actually measured with noise. However, this should not have an impact on
the consistency since this measurement noise is uncorrelated with τ̃ .
The goals were to estimate the change in mass m and the change in center of
mass zm , but the inertia Ix , the damping coefficient d and the spring coefficient
k were also considered to be unknown. Due to identifiability issues, two datasets
were collected, one with the nominal mass and the other with the additional
mass, and the model was extended according to Section 3.2. Both datasets
were created by dragging the cart back and forth, are 10 minutes long and were
down-sampled to 50Hz with a total of 30 000 samples in each dataset.
zs,1
ys,1
xs,1
zs,2
ys,2
xs,2
Figure 5: A picture of the modified inverted pendulum. Uppermost in the
picture is the IMU (orange) measuring the rolling motion. In front of it is the
additional load fastened with white tape. To the left on the cart is the reference
IMU.
13
I x # 1000
True # 1000
Mean # 1000
7.02
7.04
7.06
7.08
7.1
7.12
7.14
#10 -3
m
True
Mean
0
0.02
0.04
0.06
0.08
0.1
zm
True
Mean
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
Figure 6: The result from the estimation of the parameters in Table 1. Here,
only the parameters with known true values are shown. Red corresponds to the
measured or calculated physical quantity in Table 1. Cyan is the histogram of
the samples drawn from N (ϑ̂, P̂ϑ ) and blue corresponds to the the sample mean.
Top: Ix , middle: m and bottom: zm .
The noise model
Ĥd (q, η) =
1 + c̄1 q−1 + c̄2 q−2 + c̄3 q−3 + c̄4 q−4
1 + ᾱ1 q−1 + ᾱ2 q−2
(48)
was used. Note that 1 + ᾱ1 q−1 + ᾱ2 q−2 is the denominator of (21) after discretization and hence, that the corresponding prefilter used in the estimator
is
1
L(q) =
(49)
−1
−2
1 + c̄1 q + c̄2 q + c̄3 q−3 + c̄4 q−4
Note that this noise model was flexible enough to capture the noise model of
(12) and also gives some flexibility for coloring in τ̄ , see Linder et al. (2014) for
a discussion.
The result of the estimation can be seen in Figure 6 and on row two and
three in Table 1. Figure 6 has been created by plotting the histogram of 100 000
samples from N (ϑ̂, P̂ϑ ), where P̂ϑ is the asymptotic covariance matrix. The
cyan bars are for the IV estimator, the blue bars correspond to the mean and
the red bars correspond to the true value. It can be seen that the true values
of m and zm are within the three standard deviations of the estimated value.
The true value of the inertia Ix is outside the three standard deviations of the
estimated value. This is partially due to sampling effects but also note that
smaller parts, such as, screws or the cable, were not included when the true
value was calculated. Also note that the relative error actually is quite small.
14
Table 1: The known physical quantities and the results from identification.
The first row corresponds to the known values. The second row contains the
estimated values and the third row contains the relative errors.
M
zg
Ix
d
k
m
zm
mzm
2
mzm
True
1.324
0.0211
0.0071
–
–
0.098
–
–
0.0071
0.0032
1.84
0.0922
−0.0119
0.0015
Est.
−0.123
Err.
–
–
0.575%
–
–
5.41%
5.77%
10.9%
5
−0.115
−0.0126
0.0012
16.0%
Conclusions
In this paper, a framework using indirect input measurements has been proposed
for certain system identification problems where the input is partially unknown.
In this framework, the indirect model (12), where the unknown input was substituted with direct and indirect input measurements, have been proposed to
obtain an estimate of the desired model (1). It has been shown that due to
the generality of the framework, several already existing models are recovered
as special cases of the indirect model (12) and two examples of physical models
have been shown to fit into the framework.
An iterative instrumental variable method has been suggested to estimate
the indirect model due to its similarity to closed-loop system identification, i.e.
due to correlation between the disturbances and the input and output, and
the proposed method has been applied to data from a real inverted pendulum
experiment with good results.
In this paper, only the necessary number of signals are used to extend (1) and
thus, create the indirect model (12). If additional measurements are available,
the available information should probably be used for information fusion, for
instance, using an observer, to possibly reduce the resulting variance of the
estimator.
Acknowledgments
This work has been supported by the Vinnova Industry Excellence Center LINKSIC.
References
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Systems Symposium, Ottawa, Canada, 1993.
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Anthony M D’Amato, Adam J Brzezinski, Matthew S Holzel, Jun Ni, and Dennis S Bernstein. Sensor-only noncausal blind identification of pseudo transfer
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AG Dankers. System Identification in Dynamic Networks. PhD thesis, TU
Delft, Delft University of Technology, 2014.
Urban Forssell. Closed-loop identification: Methods, theory, and applications.
PhD thesis, Linköping University, Linköping, Sweden, 1999.
Urban Forssell and C.T. Chou. Efficiency of prediction error and instrumental
variable methods for closed-loop identification. In Proceedings of the 37th
IEEE Conference on Decision and Control, pages 1287–1288 vol.2, Tampa,
Florida, USA, 1998.
Marion Gilson and Paul Van den Hof. Instrumental variable methods for closedloop system identification. Automatica, 41(2):241–249, 2005.
Marion Gilson, Hugues Garnier, Peter C Young, and Paul MJ Van den Hof.
Optimal instrumental variable method for closed-loop identification. IET
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Jonas Linder. Graybox Modelling of Ships Using Indirect Input Measurements.
Linköping Studies in Science and Technology. Thesis 1681. 2014.
Jonas Linder, Martin Enqvist, and Fredrik Gustafsson. A closed-loop instrumental variable approach to mass and center of mass estimation using IMU
data. In Proceedings of the 53rd IEEE Conference on Decision & Control,
Los Angeles, CA, USA, December 2014.
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Hall, 1999. ISBN 0136566952.
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16
Avdelning, Institution
Division, Department
Datum
Date
Division of Automatic Control
Department of Electrical Engineering
2015-03-13
Språk
Language
Rapporttyp
Report category
ISBN
Svenska/Swedish
Licentiatavhandling
ISRN
Engelska/English
Examensarbete
C-uppsats
D-uppsats
—
—
Serietitel och serienummer
Title of series, numbering
Övrig rapport
ISSN
1400-3902
URL för elektronisk version
LiTH-ISY-R-3080
http://www.control.isy.liu.se
Titel
Title
On Indirect Input Measurements
Författare
Author
Jonas Linder, Martin Enqvist
Sammanfattning
Abstract
A common issue with many system identification problems is that the true input to the
system is unknown. In this paper, a framework, based on indirect input measurements, is
proposed to solve the problem when the input is partially or fully unknown, and cannot be
measured directly. The approach relies on measurements that indirectly contain information
about the unknown input. The resulting indirect model formulation, with both direct- and
indirect input measurements as inputs, can be used to estimate the desired model of the
original system. Due to the similarities with closed-loop system identification, an iterative
instrumental variable method is proposed to estimate the indirect model. To show the applicability of the proposed method, it is applied to data from an inverted pendulum experiment
with good results.
Nyckelord
Keywords
System identification, Model structure, Physical models, Instrumental Variable, Closed-loop
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