System Identification of Linear Parameter Varying

State-space Models

Adrian Wills∗and Brett Ninness

Abstract

This chapter examines the estimation of multivariable linear modelsfor which the parameters vary in a time-varying manner that dependsin an affine fashion on a known or otherwise measured signal. Theselocally linear models which depend on a measurable operating point areknown as linear parameter varying (LPV) models. The contributionhere relative to previous work on the topic is that in the Gaussian case,an expectation-maximisation (EM) algorithm-based solution is derivedand profiled.

1 Introduction

This chapter considers the problem of estimating the parameters in linearparameter varying (LPV) model structures on the basis of observed input-output data records. LPV models are linear time varying structures whereinthe time dependence is affinely related to a known “scheduling” signal. Theyhave proven to be of importance in aerospace [18], engine control [2], andcompressor control applications [8] amongst others [20].

Due to their relevance, the problem of estimating LPV models has at-tracted significant research attention. The contribution [16] was the firstto not require state measurements, and work only with input-output mea-surements. It examined least squares prediction error (PE) estimation ofmultiple-input multiple-output (MIMO) state space structures with pos-sible noise corruptions on the output measurements via a gradient-basedsearch technique. This approach has been further examined and appliedin [7, 23, 24].

∗Both authors are with the School of Electrical Engineering and Computer Science,University of Newcastle, Callaghan, NSW 2308, Australia. Email: {Adrian.Wills,Brett.Ninness}@newcastle.edu.au

1

The estimation of single-input single-output (SISO) transfer functionLPV model structure has also been addressed. The work in [3] has consid-ered estimation of a ARX-type LPV models via least mean square (LMS)and recursive least squares (RLS) algorithms. The recent paper [15] ad-dresses more general Box–Jenkins type LPV models via an instrumentalvariable method.

A final key strand of research has been the development of subspace-based methods, again for the estimation of MIMO state-space LPV mod-els [6, 21, 22, 25]. A main advantage of these methods is that they are non-iterative in nature, and their effectiveness has been demonstrated empiri-cally.

This chapter addresses the estimation of MIMO state-space LPV sys-tems that are affected by both state and measurement noise. Among thatmany possibilities, here a maximum-likelihood (ML) criteria is used. Akey rationale for the consideration is the availability of underpinning theoryassuring their general consistency, even in the presence of quite arbitrarystable closed loop control [17]. The particular importance of this for LPVsystems has been emphasised and explained in [21].

There are several possible choices of algorithm for maximising the like-lihood function. This chapter will develop and illustrate a new estimationapproach using the expectation-maximisation (EM) algorithm. The perfor-mance of this technique for other linear time invariant, bilinear,frequency do-main and other identification problems has been previously demonstrated [9,10, 12–14, 26]. As will be illustrated here, it also delivers an effective andnumerically robust solution for LPV estimation. Furthermore, it has rea-sonable computational requirements which scale modestly with increasingsystem state and input-ouput dimension.

Having developed the EM method for MIMO LPV systems, the chapterprofiles it empirically via a simulation study.

2 Problem Formulation

This chapter considers the following linear time-varying state-space modelstructure

xt+1 = Atxt + Btut + wt, (1a)yt = Ctxt + Dtut + et. (1b)

Here xt ∈ Rn is the system state, ut ∈ Rm is the system input, and yt ∈ R`

is the system output. The noise terms wt ∈ Rn and et ∈ R` are assumed to

2

be zero mean i.i.d. process for which

Cov{[

wt

et

]}=

[Q SST R

], Π � 0. (1c)

The time variations in the system matrices in (1a), (1b) are assumed to beof the following linear forms

At =nα∑i=0

αt(i)A(i), A(i) ∈ Rn×n ∀i, (1d)

Bt =nβ∑i=0

βt(i)B(i), B(i) ∈ Rn×m ∀i (1e)

Ct =nα∑i=0

αt(i)C(i), C(i) ∈ R`×n ∀i, (1f)

Dt =nβ∑i=0

βt(i)D(i), D(i) ∈ R`×m ∀i (1g)

where the signals

αt =

αt(0)...

αt(nα)

∈ Rnα , βt =

βt(0)...

βt(nβ)

∈ Rnβ (1h)

are termed “scheduling parameters” that are assumed known or measurable.A model of this form (1a)-(1h) is an example of a linear parameter varying(LPV) system, and in practice the values taken by the scheduling parameterspecify a given system operating point.

In what follows, it will prove important to note the equivalent formula-tion [

xt+1

yt

]=

[A BC D

] [(αt ⊗ xt)(βt ⊗ ut)

]+

[wt

et

](2)

whereA =

[A(0) . . . A(nα)

], B =

[B(0) . . . B(nβ)

](3)

C =[C(0) . . . C(nα)

], D =

[D(0) . . . D(nβ)

](4)

and ⊗ denotes the Kronecker tensor product [4].Assume that the system matrices A,B, C,D, the disturbance covariance

Π and the initial state x1 are appropriately parametrized by the elements

3

of a vector θ ∈ Rnθ . Details on how this may be achieved are provided inwhat follows.

Then the problem considered in this chapter is the estimation of a valueof θ that best explains (in a manner soon to be made precise) an observedlength N data record

U = [u1, · · · , uN ], Y = [y1, · · · , yN ] (5)

of the input-output response.

3 Maximum-Likelihood Estimation

One approach taken in this chapter to address this estimation problem isthe employment of the classical Maximum-Likelihood (ML) technique. Thisrequires the formulation of the joint density pθ(Y ) of the observation, whichmay be expressed using Bayes’ rule as

pθ(Y ) = pθ(y0)N∏

t=1

pθ(yt | yt−1, · · · , y1). (6)

A maximum likelihood estimate θML of the parameter vector θ is then de-fined as any value θML satisfying

θML ∈{θ ∈ Θ : L(θ) ≤ L(θ′), ∀θ′ ∈ Θ

}, (7)

whereL(θ) , − log pθ(Y ) (8)

and Θ ⊆ Rnθ is a user chosen compact subset. In the above, the commonnotation pθ is used to refer to a range of different densities, with the argu-ments to pθ indicating what is intended. Furthermore, note that while theform of pθ will depend on U , it is not formally an argument since it is nota random variable.

In the case where stochastic elements wt, et are Gaussian distributed[wt

et

]∼ N (0,Π) (9)

then the density pθ(yt | Yt−1) will also have a Gaussian density and lead to(ignoring constant terms that do not affect the minimisation (7)) [1]

L(θ) =N∑

t=1

log det{Λt(θ)}+ εTt (θ)Λ−1

t εt(θ) (10)

4

whereεt(θ) = yt − yt|t−1(θ), Λt(θ) = Cov{yt|t−1(θ)} (11)

with yt|t−1(θ) being the mean square optimal one step ahead predictor of yt.This can be computed using a standard Kalman filter.

yt|t−1 = Ctxt|t−1 + Dtut, (12)xt+1|t = Atxt|t−1 + Btuk + Ktet, (13)

Kt =(AtPt|t−1C

Tt + S

)Λ−1

t , (14)

Pt+1|t = AtPt|t−1ATt + Q−KtΛtK

Tt , (15)

Λt = CtPt|t−1CTt + R. (16)

where xt|t−1 is the mean square optimal one step ahead state predictionwhich has associated covariance Pt|t−1.

4 The Expecation-Maximisation (EM) Algorithmfor ML Estimation

Consider for a moment that in addition to U , Y a record

X = [x1, x2, · · · , xN+1] (17)

of the state sequence for a system modeled by (1a), (1b) is also available.Via the definition of conditional probability, this would permit the negativelog-likelihood (8) to be expressed as

L(θ) , log pθ(Y ) = log pθ(X, Y )− log pθ(X|Y ). (18)

The reason for considering this idea is that, as will be seen presently, underthe LPV model (1a), (1b) the associated log-likelihood pθ(X, Y ) can bemaximised in closed form. However, it is unreasonable to assume that arecord of the state sequence X is available. To address this, the log-likelihoodlog pθ(X, Y ) can be approximated by its expected value

Q(θ, θk) , Ebθk{log pθ(X, Y ) | Y } (19)

conditional on the data Y observed, and an initial estimate θk. Due to theexpectation operator Ebθk

{· | Y } being linear, this approximation Q(θ, θk)can also be maximised in closed form.

5

Furthermore, via (18), this approximation is related to L(θ) accordingto

L(θ) = Q(θ, θk)− V(θ, θk) (20)

whereV(θ, θk) , Ebθk

{log pθ(X | Y ) | Y }. (21)

Therefore

L(θ)− L(θk) =[Q(θ, θk)−Q(θk, θk)

]+[

V(θk, θk)− V(θ, θk)]. (22)

The second term

V(θk, θk)− V(θ, θk) =∫

log

[pbθk

(X | Y )

pθ(X | Y )

]pbθk

(X | Y ) dX (23)

is the Kullback-Leibler divergence metric between pθ(X|Y ) and pbθk(X | Y ),

and hence is non-negative. As a result, any value of θ for which Q(θ, θk) >Q(θk, θk) implies that L(θ) > L(θk).

This suggests a strategy of maximising Q(θ, θk), which must increaseL(θ) via (22), and then setting θk+1 equal to this maximiser and repeatingthe process. This procedure is known as the expectation-maximisation (EM)algorithm, which can be stated in in abstract form as follows.

Algorithm 4.1 EM Algorithm

1. E-StepCalculate: Q(θ, θk); (24)

2. M-StepCompute: θk+1 = arg max

θ∈ΘQ(θ, θk). (25)

5 EM for LPV Models

Application of the EM Algorithm to the LPV model (1a), (1b) requiresdevelopment of how the E-Step and M-Step may be computed.

6

5.1 E-Step

The E-Step may be calculated by the following two lemmas.

Lemma 5.1 With regard to the LPV model structure (1a)-(1h) and thefollowing assumption on the initial state

x1 ∼ N (µ, P1) (26)

the joint log-likelihood approximation Q(θ, θk) defined via (17), (19) is givenby

−2Q(θ, θk) = log det P1 + N log det Π+

Tr{

P−11 Ebθk

{(x1 − µ)(x1 − µ)T | Y }}

+

Tr{Π−1

[Φ−ΨΓT − ΓΨT + ΓΣΓT

]}, (27)

where

Γ ,

[A BC D

](28)

Φ ,N∑

t=1

Ebθk{xt+1x

Tt+1 | Y } xt+1|NyT

t

ytxTt+1|N yty

Tt

(29)

Ψ ,N∑

t=1

αTt ⊗ Ebθk

{xt+1xTt | Y } βT

t ⊗ xt+1|NuTt

αTt ⊗ ytx

Tt|N βT

t ⊗ ytuTt

(30)

Σ ,N∑

t=1

αtαTt ⊗ Ebθk

{xtxTt | Y } αtβ

Tt ⊗ xt|NuT

t

βtαTt ⊗ utx

Tt|N βtβ

Tt ⊗ utu

Tt

(31)

withxt|N , Ebθk

{xt | Y }. (32)

7

Proof 1 The Markov property of the LPV model structure (2) together withBayes’ rule yields

pθ(Y, XN+1) = pθ(x1)N∏

t=1

pθ(xt+1, yt | xt). (33)

Furthermore, via (1c),(2)

pθ

([xt+1

yt

]∣∣∣∣ xt

)∼ N (Γzt,Π), (34)

where

ξt ,

[xt+1

yt

], zt ,

[αt ⊗ xt

βt ⊗ ut

](35)

and by the assumption (26) pθ(x1) = N (µ, P1). Therefore, using (33) andexcluding constant terms

−2 log pθ(Y, XN+1) = log det P1 + N log det Π +(x1 − µ)T P−1

1 (x1 − µ) +N∑

t=1

(ξt − Γzt)T Π−1(ξt − Γzt). (36)

Applying the conditional expectation operator Ebθk{· | Y } to both sides of

equation (36) yields (27) with

Φ = Ebθk{ξtξ

Tt | Y }, Ψ = Ebθk

{ξtzTt | Y } (37)

Σ = Ebθk{ztz

Tt | Y }. (38)

Using the definitions (32),(35), the fact that αt, βt are deterministic, andelementary properties of the Kronecker product then completes the proof. �

This reduces much of the computation of Q(θ, θk) to that of calculatingthe Kalman smoothed state estimate xt|N together with its covariance Pt|N ,which then delivers Ebθk

{xtxTt | Y } = Pt|N + xt|N xT

t|N . What remains is theadditional computation of Ebθk

{xt+1xTt | Y }, which is not obtainable by a

standard Kalman smoother.The following lemma establishes how all these quantities my be obtained

via numerically robust “square root” recursions.

8

Lemma 5.2 The components

Ebθk{xT

t | Y } , xTt|N , (39)

Ebθk{xtx

Tt | Y } = xt|N xT

t|N + P1/2t|N P

T/2t|N , (40)

Ebθk{xtx

Tt−1 | Y } = xt|N xT

t−1|N + Mt|N (41)

required for the computation of (27) via (29)- (32) may be robustly computedas follows. The smoothed state estimate {xt|N} is calculated via the reverse-time recursion

xt|N = xt|t + Jt

[xt+1|N −Atxt|t −Btut − SR−1yt

], (42)

Jt , Pt|tATt P−1

t+1|t, (43)

whereAt , At − SR−1Ct, Bt , Bt − SR−1Dt. (44)

The associated state covariance matrices are computed from their squareroots, for example, Pt|t = P

1/2t|t P

T/2t|t , via recursions involving the following

QR-decompositionsP

T/2t|t A

Tt P

T/2t|t

QT/2 0

0 PT/2t+1|NJT

t

= Q1

R1

11 R112

0 R122

0 0

, (45)

PT/2t−1|t−1A

Tt−1

QT/2

= Q2

R21

0

, (46)

RT/2 0

PT/2t|t−1C

Tt P

T/2t|t−1

= Q3

R311 R3

12

0 R322

, (47)

where Q , Q− SR−1ST and then setting

PT/2t|N = R1

22, PT/2t|t−1 = R2

1, PT/2t|t = R3

22. (48)

9

The matrices MN |N and MN+1|N are calculated via the initialisation

MN |N = (I −KNCN )AN−1PN−1|N−1, (49)

MN+1|N = ANPN |N , (50)

followed by the the backwards recursion {Mt|N}N−1t=2 given by

Mt|N = Pt|tJTt−1 + Jt(Mt+1|N −AtPt|t)J

Tt−1. (51)

Finally, the reverse time recursion (42) is initialised by running to t = Nthe (robust) Kalman filter recursions

Kt = Pt|t−1CTt

(CtPt|t−1C

Tt + R

)−1= R3

12, (52)

xt|t−1 = At−1xt−1|t−1 + Btut−1 + SR−1yt−1, (53)xt|t = xt|t−1 + Kt

(yt − Ctxt|t−1 −Dtut

), (54)

for t = 1, . . . , N .

Proof 2 See Section 4.1 in [9].

5.2 M-Step

With the completion of the E-Step delivering Q(θ, θk), attention turns tomaximising it with respect to θ. At this point, the details of how θ parametrizesthe LPV model (1a)-(2) need to be established. For this purpose, we assumethat θ is defined in a partitioned manner according to

θT , [βT , ηT ]T (55)

where recalling (1c), (26)

βT ,[vec {Γ}T , µT

], ηT ,

[vec {Π}T , vec {P1}T

], (56)

and the vec{·} operator creates a vector from a matrix by stacking itscolumns on top of one another. We likewise partition the set Θ of can-didate parameter vectors as

Θ = Θ1 ×Θ2, β ∈ Θ1, η ∈ Θ2 (57)

where Θ1 ⊆ Rnβ , Θ2 ⊆ Rnη are both compact, and for which all η ∈ Θ2

imply symmetric positive definite Π, P1.With these definitions in place, local maximisers of Q(θ, θk) may be

directly computed via the results of the following lemma.

10

Lemma 5.3 Let Σ defined in (31) satisfy Σ > 0 and be used to define βaccording to (Ψ is defined in (30))

β ,

[vec

{Γ}T

, µT

]T

, Γ = ΨΣ−1, µ , x1|N . (58)

If Θ defined by (57) is such that β lies within Θ1, then for any fixed η ∈ Θ2,the point (58) is the unique maximiser

β = arg maxθ∈Θ1

Q([

βη

], θk

). (59)

Furthermore, η given by

η ,

[vec

{Π

}T, vec

{P1

}T]T

, Π = RT/222 R1/2

22 , (60)

P1 , P1/21|NP

T/21|N , (61)

forms a stationary point of Q(·, θk) with respect to η. Here, P1/21|N is defined

by (45) and R1/222 is defined by the Cholesky factorisation (see Algorithm

4.2.4 of [11]) [Σ ΨT

Ψ Φ

]=

[R11 R12

∅ R22

]T [R11 R12

∅ R22

]. (62)

Note that the right hand side of the expression for Π in (60) is

Π = Φ−ΨΣ−1ΨT (63)

realised in a numerically robust fashion that ensures essential properties ofsymmetry and non negative-definiteness of the result.

Proof 3 This follows by an argument identical to that used to establishLemma 4.3 of [10].

5.3 A Summary of the Algorithm

The preceding derivations are now summarised in the interests of clearlydefining the new algorithm proposed here.

Algorithm 5.1 (EM Algorithm for Bilinear Systems)

11

1. Initialise the algorithm by choosing a parameter vector θ0.

2. (E-Step) Employ the square-root implementation of the modified Kalmansmoother presented in Lemma 5.2 in conjunction with the parameterestimate θk to calculate the matrices Φ,Ψ and Σ as shown in equations(29) through (31).

3. (M-Step) In order to choose an updated parameter estimate θk+1,select Π, Γ, µ and P1 according to equations (58), (60) and (61).

4. If the algorithm has converged, terminate, otherwise return to step 2.

Regarding step 4, obvious strategies for gauging convergence involve copyingthose developed for gradient based search [5, 19]. In particular, this chaptersuggests a strategy of termination when relative likelihood increase on an it-eration drops below a predetermined threshold. This approach is supportedby empirical evidence. In the authors experience, once the rate of increasein the likelihood drops it rarely returns to higher levels.

6 Simulation Study

In this section we demonstrate the utility of the EM method detailed abovefor the estimation of LPV systems from input and output measurements.

6.1 SISO First Order Example

In order to gain some confidence that EM method is providing accurateestimates, we considered a first order SISO LPV system. More precisely,consider an LPV model in the form of (1) where n = 1, m = 1 and p = 1.Further nα = nβ = 2 delivering the LPV system

xt+1 = αt(1)A(1)xt + αt(2)A(2)xt + βt(1)B(1)ut + βt(2)B(2)ut + wt (64)yt = αt(1)C(1)xt + αt(2)C(2)xt + βt(1)D(1)ut + βt(2)D(2)ut + et (65)

For the purposes of this simulation, N = 1000 samples of the output yt

were generated using the following parameter values and signals. The inputwas generated as a square wave and is shown in the top plot of Figure 1.The scheduling parameters αt = βt were generated as sinusoidal waves thatvary between [0, 1] and so that αt(1) and αt(2) are π-radians out of phase,

12

100 200 300 400 500 600 700 800 900 1000

−1

0

1In

put

100 200 300 400 500 600 700 800 900 1000−0.5

0

0.5

1

1.5

α t(1)

and

α t(2)

0 100 200 300 400 500 600 700 800 900 1000−5

0

5

Out

put

Samples

Figure 1: Top: input signal; Middle: αt(1) (blue solid) and αt(2) (reddashed) signals; Bottom: output signal.

which is shown in the middle plot in Figure 1. The noise signals wt, et weregenerated as an i.i.d. normally distributed signal via[

wt

et

]∼ N

(0,

[Q SS R

])(66)

The “true” system parameters were chosen as[A(1) A(2) B(1) B(2)C(1) C(2) D(1) D(2)

]=

[0.9 0.6 1 10.5 0.7 0 0

](67)

with [Q, S, R

]=

[0.01, 0, 0.01

](68)

The above can be arranged into a parameter vector θ

θT = [A(1), A(2), B(1), B(2), C(1), C(2), D(1), D(2), Q, S, R]

13

The output for one noise realization is shown in the bottom plot of Figure 1.In order to examine the accuracy of the EM estimates θ of θ we performed

a Monte-Carlo simulation based on M = 1000 runs. In each run only thenoise realisation wt and et were changed. Moreover, we were also interestedin examining the robustness of the EM method to poor initial estimates ofθ. Therefore, the initial estimate θ0 was chosen as

θT0 =

[10−5[0.9, 0.6, 1, 1, 0.5, 0.7], 1, 0, 1

], (69)

so that the system matrices are almost zero and the noise covariance is 100times larger than the true values.

Recall that the state-space model (1) is not uniquely parametrized. Ingeneral this presents a difficulty when comparing true system values, likethose in (67), with estimated ones, since the estimates are likely to corre-spond to a different state coordinate system. However, for the first orderexample considered here, any similarity transformation of the state will can-cel for the A(1, 2) terms and will also cancel when considering the productof B(1) with C(1) and B(2) with C(2).

Based on the above Figure 2 presents a histogram of the estimatedA(1, 2), B(1)C(1), B(2)C(2) and D(1, 2) parameters for all the runs. Fromthe figure, it appears that estimates are accurate, even though the EMmethod is initialised far from the true values.

6.2 10’th Order MIMO Example

Inspired by the good performance of the EM algorithm above we appliedit to a more challenging situation where the system state was increasedto n = 10 and the number of system inputs and outputs were chosen asm = p = 2. Again nα = nβ = 2, but this time αt 6= βt.

In particular, the LPV system was formed by creating two random LTI-systems that were both ensured to be stable. This results in the matricesA(1, 2), B(1, 2), C(1, 2) and D(1, 2) to provide (recall from (28))

Γ =[A(1) A(2) B(1) B(2)C(1) C(2) D(1) D(2)

](70)

The noise terms were chosen according to (recall from (1c))[wt

et

]∼ N (0, Π) (71)

14

0.8 0.85 0.9 0.95 10

50

100

A(1) parameter estimates0.5 0.55 0.6 0.65 0.70

20

40

60

80

A(2) parameter estimates

0.4 0.45 0.5 0.55 0.60

50

100

B(1)C(1) parameter estimates0.65 0.7 0.75 0.8

0

20

40

60

80

B(2)C(2) parameter estimates

−0.1 −0.05 0 0.05 0.10

20

40

60

80

D(1) parameter estimates−0.1 −0.05 0 0.05 0.10

20

40

60

80

D(2) parameter estimates

Figure 2: Histograms of estimated parameter values.

15

with

Q = 10−2In, S = 0, R = 10−2Ip. (72)

The system was simulated using N = 1000 samples of the input (shown inthe top plot of Figure 3) together with the scheduling sginals αt(1, 2) andβt(1, 2) (shown in the middle and bottom plots of Figure 3, respectively).

This resulted in N samples of the output yt according to (1). Basedon the input and output measurements, the EM algorithm 5.1 was usedto provide an estimate. The algorithm requires an initial estimate of theparameters and in an attempt to again demonstrate the robustness of theEM method to poor initial estimates, we chose

θT0 = [vec

{10−5Γ

}T, vec {100Π}T ] (73)

to ensure that the initial guess is far from the true values.Unlike the previous example, it is more difficult to present a comparison

of the true and estimated parameter values in the current case. Insteadwe have adopted the standard approach of providing a comparison of thetrue system output and the predicted one obtained from the EM estimatein Figure 4. This figure shows a close match between the two signals. Fur-thermore, we performed a standard whiteness test on the prediction errorsas shown in Fifgure 5. At least according to the confidence bounds, theerrors are uncorrelated. Additionally, and as a final test of the model wecomputed the cross-correlation between the prediction error and the input,shown in Figure 6. Again, according to the confidence intervals, the error isuncorrelated with the input signal.

7 Conclusion

In this chapter the expectation-maximisation (EM) algorithm has been pre-sented and examined for the purpose of finding Maximum-Likelihood esti-mates of state-space linear parameter varying models.

Advantages of the EM method for LPV model estimation are that it isrelatively straightforward to implement; it appears to be robust to bad initialparameter estimates; it scales linearly with the number of data points N ;and, it straighforwardly handles possibly high-order MIMO LPV systems.A disadvantage is that the EM method does not straightforwardly handlestructured LPV systems.

16

100 200 300 400 500 600 700 800 900 1000

−1

0

1In

puts

100 200 300 400 500 600 700 800 900 1000−0.5

0

0.5

1

1.5

α t(1)

and

α t(2)

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

β t(1)

and

β t(2)

Samples

Figure 3: Top: input signals ; Middle: αt(1) (blue solid) and αt(2) (reddashed) signals; Bottom: βt(1) (blue solid) and βt(2) (red dashed) signals.

References

[1] K.J. Astrom. Maximum likelihood and prediction error methods. Au-tomatica, 16:551–574, 1980.

[2] Gary J. Balas. Linear parameter-varying control and its applicationto a turbofan engine. International Journal of Robust and NonlinearControl, 12(9):763–796, 2002.

[3] Bassam Bamieh and Laura Giarre. Identification of linear parametervarying models. International Journal of Robust and Nonlinear Control,(12):841–853, 2002.

[4] Dennis S. Bernstein. Matrix Mathematics. Princeton University Press,2005.

17

0 100 200 300 400 500 600 700 800 900 1000−10

−5

0

5

10

15

20

Time (s)

Observed data vs predictions of model: Output 1

DataPrediction

0 100 200 300 400 500 600 700 800 900 1000−20

−10

0

10

20

Time (s)

Observed data vs predictions of model: Output 2

DataPrediction

Figure 4: Predicted (red-dashed) and actual (blue-solid) output responses.

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0 5 10 15 20 25−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Lag

Residual Correlation + Confidence Interval: Output 1

Residual Correlation99% Confidence on Whiteness

0 5 10 15 20 25−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Lag

Residual Correlation + Confidence Interval: Output 2

Residual Correlation99% Confidence on Whiteness

Figure 5: Autocorrelation of residual errors.

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−40 −20 0 20 40−0.1

−0.05

0

0.05

0.1

Lag

Cross Cov Rue + CI: Output 1, Input 1

Rue99% CI

−40 −20 0 20 40−0.1

−0.05

0

0.05

0.1

Lag

Cross Cov Rue + CI: Output 2, Input 1

Rue99% CI

−40 −20 0 20 40−0.1

−0.05

0

0.05

0.1

Lag

Cross Cov Rue + CI: Output 1, Input 2

Rue99% CI

−40 −20 0 20 40−0.1

−0.05

0

0.05

0.1

Lag

Cross Cov Rue + CI: Output 2, Input 2

Rue99% CI

Figure 6: Cross-correlation of residual error with input.

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[5] J.E. Dennis and Robert B. Schnabel. Numerical Methods for Uncon-strained Optimization and Nonlinear Equations. Prentice Hall, 1983.

[6] F. Felici, Jan-Willem van Wingerden, and Michel Verhaegen. Subspaceidentification of mimo lpv systems using a periodic scheduling sequence.Automatica, 43(10):1684–1697, 2007.

[7] A. Fujimori and Lennart Ljung. Model identification of linear parametervarying aircraft systems. Proceedings of the Institute of MechanicalEngineers, Part G Journal of Aerospace Engineering, 220(G4):337–346,2006.

[8] L. Giarre, D. Baruso, P. Falugi, and B. Bamieh. Lpv model identifi-cation for gain scheduling control:An application to rotating stall andsurge control problems. Control Engineering Practice, 14(4):351–361,2006.

[9] Stuart Gibson and Brett Ninness. Robust maximum-likelihood estima-tion of multivariable dynamic systems. Automatica, 41(10):1667–1682,2005.

[10] Stuart Gibson, Adrian Wills, and Brett Ninness. Maximum-likelihoodparameter estimation of bilinear systems. IEEE Transactions on Au-tomatic Control, 50(10):1581–1596, 2005.

[11] G. Golub and C. Van Loan. Matrix Computations. Johns HopkinsUniversity Press, 1989.

[12] G.C. Goodwin and A. Feuer. Estimation with missing data. Mathe-matical and Computer Modelling of Dynamical Systems, 5(3):220–244,1999.

[13] R. B. Gopaluni. A particle filter approach to identification of nonlin-ear processes under missing observations. The Canadian Journal ofChemical Engineering, 86(6):1081–1092, December 2008.

[14] Alf J. Isaksson. Identification of ARX-models subject to missing data.IEEE Trans. Automat. Control, 38(5):813–819, 1993.

[15] Vincent Laurain, Marion Gilson, Roland Toth, and Hugues Garnier.Refined instrumental variable methods for identification of lpv box–jenkins models. Automatica, pages 959–967, 2010.

21

[16] L. H. Lee and K. Poolla. Identification of linear parameter-varyingsystems using nonlinear programming. Journal of Dynamic Systems,Management, and Control, 121:71–78, March 1999.

[17] L.Ljung. Convergence analysis of parametric identification methods.IEEE Transactions on Automatic Control, AC-23(5):770–783, 1978.

[18] A. Marcos and G.J. Balas. Development of linear parameter vary-ing models for aircraft. Journal of Guidance, Control and Dynamics,27(2):218–228, 2004.

[19] Jorge Nocedal and Stephen Wright. Numerical Optimization. Springer-Verlag, New York, 1999.

[20] W. Rugh and J. Shamma. Research on gain scheduling. Automatica,36(10):1401–1425, 2000.

[21] Jan-Willem van Wingerden and Michel Verhaegen. Subspace identi-fication of bilinear and LPV systems for open- and closed-lopp data.Automatica, 45:372–381, 2009.

[22] V. Verdult and M. Verhaegen. Subspace identification of multivariablelinear parameter-varying systems. Automatica, 38(5):805–814, 2002.

[23] Vincent Verdult, Niek Bergboer, and Michel Verhaegen. Identificationof fully parametrized linear and nonlinear state-space systems by pro-jected gradient search. In Proceedings of the 13th IFAC Symposium onSystem Identification, Rotterdam, pages WeP07–3, 2003.

[24] Vincent Verdult, Marco Lovera, and Michel Verhaegen. Identificationof linear parameter varying state space models with application to heli-copter rotor dynamics. International Journal of Control, 77(13):1149–1159, 2004.

[25] Vincent Verdult and Michel Verhaegen. Kernel methods for subspaceidentification of multivariable lpv and bilinear systems. Automatica,41(9):1557–1565, 2005.

[26] A.G. Wills, B. Ninness, and S.H. Gibson. Maximum likelihood es-timation of state space models from frequency domain data. IEEETransactions on Automatic Control, 54(1):19–33, 2009.

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