Robust reduced order H-infinity control via an unbiased observer

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Robust reduced order H-infinity control via an unbiased observer

Michel Zasadzinski, Harouna Souley Ali, Mohamed Darouach

To cite this version:

Michel Zasadzinski, Harouna Souley Ali, Mohamed Darouach. Robust reduced order H-infinity control

via an unbiased observer. International Journal on Science and Techniques of Automatic Control and

Computer Engineering, 2007, 1 (special issue), pp.261-275. �hal-00344159�

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Robust Reduced Order H

_∞

Control via an Unbiased Observer

M. Zasadzinski, H. Souley Ali, M. Darouach

Centre de Recherche en Automatique de Nancy (CRAN), Nancy-Université, CNRS, 186 rue de Lorraine, 54400 Cosnes-et-Romain, France Michel.Zasadzinski@uhp-nancy.fr, Harouna.Souley@uhp-nancy.fr,

Mohamed.Darouach@uhp-nancy.fr

Résumé. This paper investigates the robustH∞reduced order observer- based control. The control gain is designed using standardH∞techniques.

This gain is then used to resolve unbiasedness condition on the estimation error. Finally, the observer matrices are derived through the computation of a unique gain by means of Linear Matrix Inequalities.

Keywords. Observer-based control, Reduced-order controller, H∞ optimization, LMI.

1 Introduction

The observer-based control is usually applied (see [1–8]) when we do not have access to all the states of a system. In the case of linear systems without uncertainty, it is well established that the observer-based control problem can be resolved into two separate problems: the observer synthesis and the control law design.

Thus, this separation principle permits to decrease the computation time and to reduce the complexity of the synthesis problem because this one is divided into two separate subproblems.

But in many practical situations, there are uncertainties which affect the nominal system. Here, we will consider the observer-based control problem for a system subjected to norm-bounded uncertainties as in [9–14].

In this paper, a method is proposed to design a robust reduced order control law for uncertain systems into two steps. First, we search for a linear control law which ensures anH_∞ performance. Second a functional filtering techniques developed in [15, 16] is used to determine the observer-controller matrices. We prove that the obtained observer-based controller guarantees the quadratic stability and an H_∞ performance. We use a functional filter which estimates the control law without estimating the all state of the system contrary to “standard”

observer-based control approach (see [1]). Our approach allows to reduce the order of the controller since the designed functional filter is of the same order than the functional to be estimated,i.e.the dimension of the control inputu(t).

In the sequel,A^† is a generalized inverse of matrix AsatisfyingA=AA^†A [17].

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2 Preliminary results

Let us consider the following uncertain system

˙

x= (A+∆A(t))x+ (B+∆B(t))w (1a)

z=Cx+Dw. (1b)

where x(t)∈IRⁿ is the state vector, z(t)∈IR^q is the output, andw(t)∈IR^r is the disturbance vector.

The matrices∆_A(t)and ∆_B(t) represent the parametric uncertainties and satisfy the following relation

∆A(t)∆B(t)

=M ∆(t)

NxNw

(2) with∆^T(t)∆(t)6Ik and∆(t)∈IR^ℓ×k.

In this paper the usualH_∞performance indexJzwwill be considered i.e.

Jzw= Z ∞

0

z^Tz−γ²w^Tw

dt (3)

for a givenγ >0.

2.1 First formulation of the preliminary result

The following lemma is given to ensure the stability and theH_∞perfomance of system (1) and is used to prove theorem 2 in the sequel.

Lemma 1. The uncertain system (1) is quadratically stable and satisfies the H_∞ performanceJzw<0for a givenγ >0if there exist matricesP =P^T >0, Gand a real µ >0such that







GA+A^TG^T +µN_x^TN_xP−G+A^TG^T GB+µN_x^TN_w C^T GM P−G^T +GA −G−G^T GB 0 GM B^TG^T +µN_w^TN_x B^TG^T −γI+µN_w^TN_w D^T 0

C 0 D −γI 0

M^TG^T M^TG^T 0 0 −µI







<0. (4)

■ Note that if LMI (4) is feasible, thenG+G^T >0, soGis nonsingular.

Proof. As in [18–20], let us consider the following descriptor system (using the relationφ= ˙x)

I_n0 0 0

˙ x φ˙

=

0 I_n A_∆−I_n

x φ

+ 0

B_∆

w (5a)

z= C0

x φ

+Dw (5b)

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where

A∆= (A+∆A(t)) and B∆= (B+∆B(t)). (6) Consider the Lyapunov function candidate given by (P =P^T >0))

V(x) =x^TP x=

x^T x˙^T I_n0

0 0

P 0 G^T G^T

x

˙ x

(7) whereGis a given matrix with appropriate dimensions. Then

V˙(x) = 2

x^T x˙^T P G

0 G

˙ x 0

= 2

x^T x˙^T P G

0 G

˙ x

−x+A˙ ∆x+B∆w

. (8) Using (5) into (8) and from the fact thatV(0) = 0 andV(∞)>0, we have the following inequality

Jzw6 Z ∞

0

z^Tz−γ²w^Tw+ ˙V(x)

dt (9)

whereJ_zwis the usualH_∞ performance index.

Inequality (9) is equivalent to (see (8)) J_zw6

Z ∞ 0





x^T x˙^T w^T

Θ11 Θ12

Θ^T₁₂−γ²I+D^TD 

 x

˙ x w







dt (10)

where

Θ₁₁= 0 I

A_∆−I T

P 0 G^T G^T

+

P G 0 G

0 I A_∆−I

Θ12= P G

0 G 0 B_∆

+

C^TD 0

So, if the relation 



Θ11 Θ12 0 Θ₁₂^T −γ²I D^T

0 D −I_q



<0 (11) holds, thenJ_zw<0 (V˙(x)<0also).

Moreover using the Schur lemma, inequality (11) is equivalent to







GA_∆+A^T_∆G^T P−G+A^T_∆G^T GB_∆ C^T P−G^T +GA_∆ −G−G^T GB_∆ 0

B_∆^TG^T B^T_∆G^T −γI D^T

C 0 D −γI





<0 which can be written as







GA+A^TG^T P−G+A^TG^T GB C^T P−G^T+GA −G−G^T GB 0

B^TG^T B^TG^T −γI D^T

C 0 D −γI





+H1∆(t)H2+H₂^T∆^T(t)H₁^T <0 (12)

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withH1=

(GM)^T (GM)^T 0 0T

, H2= [N_x 0 N_w 0 ].

Finally it is well-known that if there existsµ >0 such that







GA+A^TG^T P−G+A^TG^T GB C^T P−G^T+GA −G−G^T GB 0

B^TG^T B^TG^T −γI D^T

C 0 D −γI





+µ⁻¹H1H₁^T +µH₂^TH2<0 (13)

then inequality (12) is verified (see [21]). So, applying Schur lemma again on

(13) leads to the LMI (4) of lemma 1.

2.2 Second formulation of the preliminary result

An alternative formulation of lemma 1 is given in a second lemma to ensure the stability and theH_∞perfomance of system (1). This second lemma is used to prove theorem 1.

Lemma 2. The uncertain system (1) is quadratically stable and satisfies the H_∞ performanceJ_zw<0for a givenγ >0if there exist matricesP =P^T >0, Gandν >0such that







AG+G^TA^T +νM M^T P−G+G^TA^T +νM M^T B G^TC^T G^TN_x^T P−G^T+AG+νM M^T −G−G^T +νM M^T B 0 0

B^T B^T −γI D^T N_w^T

CG 0 D −γI 0

NxG 0 Nw 0 −νI







<0.

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■ Proof. Notice that from (4), by pre and post-multiplying it bybdiag G⁻¹, G⁻¹,0,0 and takingP =G⁻¹P G^−T,G=G^−T, we get







AG+G^TA^T P−G+G^TA^T B G^TC^T P−G^T +AG −G−G^T B 0

B^T B^T −γI D^T

CG 0 D −γI





+F1∆(t)F2+F₂^T∆^T(t)F₁^T <0 (15) withF1=

M^T M^T 0 0T

, F2= [NxG 0 Nw 0 ].

The rest of the proof is similar to that of lemma 1.

3 Robust observer-based controller design

In this part, we consider the following uncertain linear system

˙

x= (A+∆A(t))x+(B1+∆B₁(t))w+B2u (16a)

z=C1x+D11w+D12u (16b)

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y= (C2+∆C₂(t))x+ (D21+∆D₂₁(t))w (16c) wherex(t)∈IRⁿis the state vector,z(t)∈IR^qis the controlled output,y(t)∈IR^p is the measured output, u(t) ∈IR^m is the control input and w(t)∈ IR^r is the disturbance vector. Without loss of generality, we assume thatm < n.

The matrices ∆A(t), ∆B₁(t), ∆C₂(t) and ∆D₂₁(t)represent the parametric uncertainties and satisfy the following relation

∆A(t) ∆B₁(t)

∆C₂(t)∆D₂₁(t)

= Mx

My

∆(t)

NxNw

(17) with∆^T(t)∆(t)6I_k and∆(t)∈IR^ℓ×k.

We aim to design an observer-based controller with the following structure

˙

η=Hη+J1y+J2u (18a)

u=η+Ey (18b)

whereη(t)∈IR^m(withm6n) is the observer state and matricesH,J1,J2and E are to be designed.

For this purpose, we first introduce the following definition.

Definition 1. The uncertain system (16)is said to be robustly stabilizable based on function observer if there exist a gain matrix L, a function observer η˙ = Hη+J1y+J2uand a control lawu=η+Ey such that

(i) limt→∞u(t)−Lx(t) = 0if w(t) = 0,

(ii) the designed controller must be an observer-based one for the nominal system (i.e. a separation principle like condition is satisfied : the eigenvalues of A+B2L and H are those of the state matrix of the nominal closed-loop system),

(iii) the closed-loop system (16)-(18)is quadratically stable. ■ Now, the problem to be treated can be stated as follows.

Problem 1. The objective is to establish a function observer (18a) and a control law (18b) such that

(i) limt→∞u(t)−Lx(t) = 0ifw(t) = 0,

(ii) the designed controller must be an observer-based one for the nominal system (see item (ii) in definition 1).

(iii) the resulting closed-loop system (16)-(18) is quadratically stable and satisfies theH_∞ performanceJ_zw<0 for a givenγ >0. ■ The approach used in this paper is to design the controller into two steps. The first one consists to use the item (i) of problem 1 i.e. to search for a state feedback gainLverifyingu(t)−Lx(t)for subsystem (16a)-(16b). Once this gain is found we then search in a second step, an observer-based controller (18) permitting to construct this gainL.

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3.1 Synthesis of the robust state-feedback gain

In this part, we replaceubyLxin system (16a)-(16b) and get the following subsystem

˙

x= (A+B2L+∆_A(t))x+(B1+∆B₁(t))w (19a)

z= (C1+D₁₂L)x+D₁₁w (19b)

The above system is similar to system (1) and the result of section 2.2 can then be applied to build the gainLthrough the following theorem.

Theorem 1. The uncertain system (19)is quadratically stable and satisfies the H_∞ performanceJzw<0for a givenγ >0if there exist matricesP =P^T >0, G,Y andν >0 such that







(1,1) (1,2) B1 G^TC₁^T+Y^TD₁₂^T G^TN_x^T (1,2)^T −G−G^T+νMxM_x^T B1 0 0

B₁^T B^T₁ −γI D^T₁₁ N_w^T

C1G+D12Y 0 D11 −γI 0

N_xG 0 N_w 0 −νI







<0 (20)

where

(1,1) =AG+G^TA^T +B2Y +Y^TB₂^T +νMxM_x^T (1,2) =P−G+G^TA^T +Y^TB₂^T +νMxM_x^T Then the gain Lis given by

L=Y G⁻¹. (21)

■ Proof. Note thatGis invertible once the LMI (20) is satisfied, then the gainL can be computed.

By insertingL=Y G⁻¹ in LMI (14), one obtains inequality (20) ifAandC are replaced byA+B2LandC1+D12Lrespectively (i.e.compare system (19) with system (1)). Then this theorem is proved using lemma 2.

3.2 Synthesis of the robust observer-based controller Assume that the following constraint [15, 16]

E

MyD21

= 0 (22)

holds. This assumption is justified in remark 2.

Using item (i) of definition 1, an observation error signal can be defined as e=Lx−u=Ψ x−η−E(D21w+My∆(t)(Nxx+Nww)) (23) where

Ψ =L−EC2. (24)

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Using (22), the observation error can be simplified as

e=Ψ x−η (25)

and has the following dynamics

˙

e=He+ (Ψ A−HΨ−J1C2)x+ (Ψ ∆A(t)−J1∆_C₂(t))x

+ (Ψ B1−J1D21)w+ (Ψ ∆B₁(t)−J1∆D₂₁(t))w+ (Ψ B2−J2)u (26) and the unbiasedness of the nominal part corresponding to item (1) of definition 1 is achieved if and only if the closed-loop is quadratically stable and the two following conditions hold

0 =Ψ A−HΨ−J1C2, (27a)

J2=Ψ B2. (27b)

IfrankL=qand using constraint (22), the nominal unbiasedness conditions (27a) is equivalent to [15, 16, 22]

E K Σ=

0 0LA

(28) and the general solution of (28), if it exists, is given by

E K

=

0 0LA

Σ^†+Z(I2p−Σ Σ^†) (29) where

Σ=

MyD21C2A

0 0 C

(30)

A=A(I−L^†L), (31)

C=C2(I−L^†L), (32)

K=J1−HE. (33)

andZ is an arbitrary matrix of appropriate dimensions.

Remark 1. As in [15, 16, 22], the matrixLmust be of full row rank. Notice that in case matrix L given by (21) is not full rank row, it suffices to add a small

perturbation to fulfill this condition. ■

Relation (29) is the general solution of (28) if and only if [17]

rank

M_yD21C2A

0 0 C

= rank



 0 0 LA MyD21C2A

0 0 C



 (34)

or if and only if [15, 16, 22]

rank





My D21LA 0 0 C2

0 0 L



= rank







0 0 LA MyD21LA 0 0 C2

0 0 L





. (35)

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Letξ^T(t) =

x^T(t) e^T(t)

be the augmented closed-loop state vector. Then, if (35) holds, the closed-loop system (16)-(18) is given by

ξ˙=

A+B2L−B2

0 H

!ξ+

∆_A(t) 0 Ψ ∆_A(t)−J1∆_C₂(t) 0

| {z }

∆_A(t)

ξ

+

B1

Ψ B1−J1D21

w+

∆_B₁(t) Ψ ∆_B₁(t)−J1∆_D₂₁(t)

| {z }

∆_B(t)

w (36a)

z=

C1+D12L−D12

ξ+D11w (36b)

Due to relations (27a) and (29), the following relations are obtained

H =Ab−ZCb (37) Ψ B1−J1D21=Bb−ZGb (38) where

Ab=LAL^†−

0 0LA Σ^†

C2AL^† C2L^†

, (39a)

Cb= (I2p−Σ Σ^†)

C2AL^† C2L^†

, (39b)

Bb=LB1−

0 0LA Σ^†

C2B1

D21

, (39c)

Gb= (I2p−Σ Σ^†) C2B1

D21

. (39d)

Remark 2. In system (36), the filtering errore(t)is bilinear in the gain parameter Z due to the product ZCE. This bilinearity is intrinsically linked to theb unbiasedness condition (27a) . Indeed, the “bilinearity” HΨ in (27a) yields a gain K containing the product HE (see (33)). In order to avoid this kind of

“bilinearity”, we consider the constraint (22). ■

Notice that the uncertain terms combinationsΨ ∆A(t)−J1∆C₂(t)andΨ ∆B₁(t)− J1∆D₂₁(t)are easily calculated by means of the system matrices. For example, using (24) and (29), the first term is given by

Ψ ∆_A(t)−J1∆_C₂(t) =

LM_x− E K

C2M_x M_y

∆(t)Nx

=

LM_x−

0 0LA Σ^†

C2Mx

My

−Z

I2p−Σ Σ^† C2Mx

My

∆(t)Nx. The matrix Ψ ∆B₁(t)−J1∆D₂₁(t) is calculated in a similar way and the closed-loop system (36) becomes

ξ˙= b

A+∆_A(t) ξ+

b

B+∆_B(t)

w (40a)

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z=Cbξ+D11w. (40b)

where

∆_A(t)∆_B(t)

=M∆(t)N_xN_w

(41) with

b A=

A+B2L −B2

0 Ab−ZCb

,Bb= B1

b B−ZGb

, (42a)

b C=

C1+D12L−D12 ,M=

M_x Mcy1−ZMcy2

, (42b)

Mcy1=LMx−

0 0LA Σ^†

C₂M_x M_y

, (42c)

c

M_y2= (I2p−Σ Σ^†) C2M_x

M_y

, (42d)

N_x= N_x0

,N_w=N_w. (42e)

Notice that the augmented system (36) is then rewritten in a similar form as system (1). It suffices to apply lemma 1 to get the unknown matrixZ.

Theorem 2. Assume that condition (35)holds. The robust H_∞ observer-based unbiased controller design problem 1 is solved underE[M_y D21] = 0and where L is given by (21), if, for some µ > 0, there exist P11 =P₁₁^T >0 ∈IR^n×n,P13 =P₁₃^T >0 ∈ IR^m×m, P12 ∈IR^n×m, G1 ∈ IR^n×n, G3∈IR^m×m andY3∈IR^m×2p such that the two following LMI hold







Φ₁₁ • • • • • • Φ21Φ22 • • • • • Φ31Φ32Φ33 • • • • Φ41Φ42Φ43Φ44 • • • Φ51Φ52Φ53Φ54Φ55 • • Φ61Φ62Φ63Φ64Φ65Φ66 • Φ71Φ72Φ73Φ74Φ75Φ76Φ77







<0 (43)

P11P12

P₁₂^T P13

>0 (44)

where “•” is the transpose of the off-diagonal part and

Φ11=G1A+G1B2L+A^TG^T₁ +L^TB₂^TG^T₁ +µN_x^TN_x Φ21=−B₂^TG^T₁

Φ22=G3Ab−Y3Cb+Ab^TG^T₃ −Cb^TY₃^T Φ₃₁=P₁₁−G^T₁ +G₁A+G₁B₂L Φ32=P12−G1B2

Φ33=−G1−G^T₁ Φ41=P₁₂^T

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Φ42=P13−G^T₃ +G3Ab−Y3Cb Φ₄₃= 0

Φ44=−G3−G^T₃ Φ51=B^T₁G^T₁ +µN_w^TN_x Φ₅₂=Bb^TG₃−Gb^TY₃^T Φ53=Φ51

Φ54=Φ52

Φ55=−γI+µN_w^TNw

Φ61=C1+D12L Φ62=−D12

Φ63= 0 Φ64= 0 Φ65=D11

Φ66=−γI Φ71=M_x^TG^T₁

Φ72=Mc_y1^TG₃^T−Mc_y2^TY₃^T Φ73=Φ71

Φ74=Φ72

Φ75= 0 Φ76= 0 Φ77=−µI

The gain matrixZ is then given by

Z =G⁻¹₃ Y3. (45)

■ Proof. Under constraint (22), if the rank condition (35) holds, then the filter (18) is unbiased and system (36) represents the closed-loop given by the connexion of the uncertain system (16) with the controller (18) where matricesH,J1, J2

andE are given by (37), (33), (27b) and (29).

Note thatG3is invertible once the LMI (43) is satisfied, then the gainZcan be computed.

Inserting Y3 = G3Z in the LMI (4), and taking P =

P11P12

P₁₂^T P13

and G = G1 0

0 G3

yield inequality (43). Then using lemma 1, the robust quadratic stability andH_∞ performance required in item (ii) of problem 1 is ensured. Item (i) of problem 1 holds since lim

t→∞ξ(t) = 0 if w(t) = 0 (and so lim

t→∞e(t) = 0) if the closed-loop is robustly quadratically stable. Item (ii) is verified since the

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eigenvalues ofA+B2LandH are those of the state matrix of the nominal part

of closed-loop system (36).

Remark 3. Notice that even if we are in the reduced-order case, we do not need to consider a block-diagonal matrix P (i.e. P12 = 0) as in [23] which can be conservative. This block-diagonal structure is then reported on matrixG. ■ Remark 4. Similarly to [24], it is easy to treat the multiobjective (and not mixed) control problem as it is possible to consider the same Lyapunov matrix P and

different matrixGfor each objective. ■

4 Numerical example

The different matrices of the uncertain system (16) are given by

A=





−1 0 1

−1 0.5 1 1 −2−5



, B1=



 1 1 0 0

−1 0



, B2=



1 0 0 1 0 0



, C1=

0 1 1

−1 0 2

, C2= 1 0 1

,

D11= 1 0

1 0

, D12= 1 0

0 0

, D21=−0.5 0 , Mx=





0.1 0 0 0 0.1 0 0.3 0 0.1



, My =

0 0.135 0 ,

Nx=





0 0 0.135 0 0.135 0 0.135 0 0



, Nw=





0.3 0 0 0.135 0.135 0



, ∆(t) =





a(t) 0 0 0 b(t) 0 0 0 c(t)



.

The nominal part of the uncertain system is unstable sinceAis non-Hurwitz.

The matrix ∆(t) is such that a(t) = cos(3t), b(t) = 0.5−0.3 sin(2t) and c(t) = 0.7 sin(4t).

We first apply theorem 1 and theorem 2 in a second time. The robust optimization givesγ= 8.2,ν= 8.3375andµ= 9.4793.

Then the gain matrices L = [L1 L2] (21) and

Z = [Z1 Z2](45) are given by L1=

−0.2546−0.3333 1.1273 −2.5091

, L2=

−0.3341

−1.9966

, Z1=

−0.0845 0

−1.0216 0

, Z2=

−0.0292−7.8692×10⁻⁷

−0.3516−5.1674×10⁻⁶

.

The filter-based controller matricesH,J1,J2 andEare given by (37), (33), (27b) and (29). This controller is of order2.

The perturbations are given in figure 1. The numerical simulation is illus- trated with figures 2 and 3.

Notice that, when the pertubationw(t)vanishes, the statex(t)and the estimation state errore(t)converge to zero even if the uncertainties in∆(t)are not null.

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5 Conclusion

In this paper, we have proposed an efficient method for robust reduced order observer based control for uncertain systems. Through the standard H_∞ techniques we first computed a robust state feedback gain and then used it to solve the nominal unbiasedness condition of the observer error dynamics. The observer matrices are then easily obtained by solving two LMI.

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Contr.46(2001) 1941–1946

AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA

AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA

0 2 4 6 8 10 12 14 16 18 20

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

Fig. 1.Perturbation signalw(t).

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AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA

AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA

0 2 4 6 8 10 12 14 16 18 20

-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Fig. 2.Closed-loop statesx(t).

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AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA

AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA

0 2 4 6 8 10 12 14 16 18 20

-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1

Fig. 3.Closed-loop error statese(t).