Robust H-infinity reduced order filtering for uncertain bilinear systems

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Robust H-infinity reduced order filtering for uncertain bilinear systems

Harouna Souley Ali, Michel Zasadzinski, Hugues Rafaralahy, Mohamed Darouach

To cite this version:

Harouna Souley Ali, Michel Zasadzinski, Hugues Rafaralahy, Mohamed Darouach. Robust H-infinity reduced order filtering for uncertain bilinear systems. Automatica, Elsevier, 2006, 42 (3), pp.405-413.

�hal-00089755�

Robust H

reduced order filtering for uncertain bilinear systems

H. Souley Ali, M. Zasadzinski^∗, H. Rafaralahy and M. Darouach

CRAN₋CNRS

IUT de Longwy, Universit´e Henri Poincar´e₋Nancy I 186, rue de Lorraine, 54400 Cosnes et Romain, FRANCE

Tel : +33 +3 82 39 62 22₋ Fax : +33 +3 82 39 62 91 E-mail : {souley,mzasad,hrafa,darouach}@iut-longwy.uhp-nancy.fr

Abstract

This paper investigates both the _H∞ and robust _H∞ reduced order unbiased filtering problems for respectively a nominal bilinear system and a bilinear system affected by norm-bounded structured un- certainties in all the system matrices. First, an algebraic framework is used to solve the unbiasedness condition and second, a change of variable is introduced on the inputs of the system to reduce the con- servatism inherent to the requirement of exponential convergence of the filter. Then the reduced order filtering solution is obtained through LMI with an equality constraint by transforming the problem into a robust state feedback in the nominal case and a robust static output feedback in the presence of uncertainties. In the last case, an additional bilinear matrix equality must also be solved.

Keywords : Bilinear systems, Structured uncertainties, Robust reduced order filtering, Unbiasedness, H∞ optimization, LMI.

1 Introduction

In the last decades, due to the fact that many physical processes may be appropriately modeled as bilin- ear systems when linear models are inadequate, great interest has been accorded to the state estimation of bilinear systems. Most work on state estimation of bilinear systems has been done on observers design (Funahashi, 1979; Tibken and Hofer, 1989; Derese et al., 1979; Wang and Kao, 1991). TheH∞ filtering for linear systems has been deeply visited in the full order case (Nagpal and Khargonekar, 1991; Shaked and Theodor, 1992b; Shaked and Theodor, 1992a) and even in the reduced order one (Grigoriadis and Watson, 1997; Watson and Grigoriadis, 1998). The reduced order filtering problem presented by (Watson and Grigoriadis, 1998) is an unbiased one and considers only the nominal case. The full order H∞ fil- tering for nonlinear systems has been studied by (de Souza et al., 1993; Reif et al., 1999) and is also shown to be equivalent to an L2 gain attenuation problem for the mapping between the disturbances and the estimation error. The robust full order filtering for uncertain linear systems has been treated in (Fu et al., 1992; Li and Fu, 1997; P˜alhares and Peres, 2000; Sayed, 2001). The robust filtering gets its importance from the necessity to still keep good performances even if parameter uncertainties affect the system.

In this paper, a reduced order H∞functional filtering method is proposed to reconstruct a linear combi- nation of the states of a bilinear system by exploiting the nonlinearities in the nominal and the robust cases as ‘structured uncertainties’. This is achieved through the design of a filter whose dynamics has the same dimension as this linear combination. In addition to the exponential convergence and L2 gain attenuation requirements, the filter must also be unbiased, i.e. the estimation error does not depend explicitely on the states of the system. The proposed approach is based on the resolution of algebraic

∗Corresponding author : E-mail: mzasad@iut-longwy.uhp-nancy.fr−Fax : +33 +3 82 39 62 91

Sylvester equations to find conditions for the existence of the unbiased reduced order filter. Then the exponential convergence andL2 gain attenuation problems are reduced to a robust state feedback in the nominal case. It is shown that the robust functional unbiased filtering problem for uncertain bilinear systems subjected to time-varying norm-bounded uncertainties can be seen as a particular case of a static output feedback one under some conditions. This problem requires to solve LinearMatrix Inequalities (LMI) with an additional non convexBilinearMatrixEquality (BME) constraint.

The paper is organized as follows. The conditions for the unbiasedness, exponential convergence and L2

gain attenuation of a reduced order H∞ functional filter for continuous-time nominal bilinear systems are studied in section 2. It is shown through section 3 that the robust filtering problem for bilinear systems affected by structured norm-bounded time-varying uncertainties can be solved as a static output feedback problem. An illustrative example is given in section 4. Then, some conclusions are presented in section 5.

Notations. "x" = √

x^Tx and "A" = !

λ_max(A^TA) are the Euclidean vector norm and the spectral matrix norm respectively whereλmax(A^TA) is the maximal eigenvalue of the symmetric matrixA^TA. A^† is a generalized inverse of matrix A satisfying A = AA^†A. bdiag(A₁, . . . , A_k) denotes a block-diagonal matrix with A₁, . . ., A_k as block-diagonal ‘elements’ and herm(A) = A+A^T. L2[0,∞) is the space of signals with bounded energy.

2 Reduced order unbiased H

filtering in the nominal case

2.1 Problem Formulation

Consider the nominal bilinear system described by

˙

x = A₀x+

"m i=1

A_iu_ix+Bw (1a)

y = Cx+Dw (1b)

z = Lx (1c)

whereu(t) = [u₁(t) . . . u_m(t) ]^T ∈IR^m is the known control input vector,x(t)∈IRⁿ is the state vector, y(t)∈IR^p is the measured output and z(t) ∈IR^r is the vector to be estimated wherer !n. The vector w(t) ∈ IR^q represents the disturbance vector. A₀, A_i, B, C, D and L are known constant matrices.

Without loss of generality, it is assumed that rankL=r. The problem is to estimate the vector z(t) from the measurements y(t) and the inputs u(t). For many physical processes, disturbances are continuous and inputs are continuous and bounded, which is reflected in the following assumption for bilinear system (1).

Assumption 1. The disturbance signalw(t)is continuous. The control inputs,u(t), are continuous and bounded, i.e. u(t)∈Ω⊂IR^m, where

Ω:={u(t)∈IR^m|ui,min!ui(t)!ui,max, i= 1, . . . , m}. (2) The proposed reduced order functional filter is given by

˙

η = H₀η+

"m i=1

H_iu_iη+J₀y+

"m i=1

J_iu_iy (3a)

#

z = η+Ey (3b)

where #z(t)∈IR^r is the estimate ofz(t).

The estimation error is given by

e=z−#z=Lx−#z=e−EDw (4)

e= Ψx−η (5)

Ψ =L−EC. (6)

The generalization of the well-known H∞ attenuation in the linear case to the nonlinear systems is the L2 gain attenuation defined as follows (van der Schaft, 1992).

Definition 1. Let γ >0, the mapping from w(t) toe(t) is said to haveL2 gain less than or equal toγ if

$ _∞

0 "e(t)"² dt!γ²

$ _∞

0 "w(t)"² dt (7)

∀w(t)∈ L2[0,∞), and with zero initial conditions.

In this paper, the problem of the filter design is to determine H₀,H_i,J₀,J_i and E such that :

(i) the filter (3) is unbiased ifw(t) = 0 (see (Seron et al., 1997), p. 176), i.e. the estimation error is independent of x,

(ii) the filter (3) is exponentially convergent foru(t)∈Ω andw(t) = 0,

(iii) the mapping from the disturbance inputw(t) to the estimation errore(t) has L2 gain less than a given scalarγ foru(t)∈Ω.

2.2 Unbiasedness condition fulfillment

From (4), notice that the time derivative of the erroreis function of the time derivative of the disturbances w. To avoid the use of ˙w in the dynamics of the errore, consider eas a new ‘state vector’. Then theL2

gain fromw toehas the following state space realization e˙ = (H0+

"m i=1

u_iH_i)e+ (ΨA0−H₀Ψ−J₀C)x (8a)

+

"m i=1

(ΨA_i−H_iΨ−J_iC)u_ix+ (ΨB−J₀D−

"m i=1

J_iDu_i)w (8b)

e=e−EDw (8c)

and the unbiasedness of the filter is achieved if and only if the following Sylvester equations

ΨAi−H_iΨ−J_iC = 0 i= 0, . . . , m (9) hold. As L is of full row rank, (9) is equivalent to

(ΨA_i−H_iΨ−J_iC)%

L^† I_n−L^†L&

=0 i= 0, . . . , m (10)

and since rankL=r, one hasLL^†=Ir. Using the definition of Ψ, (10) is equivalent to

0 = ΨA_iL^†−H_iΨL^†−J_iCL^† i= 0, . . . , m (11a) 0 = ΨA_i+H_iEC−J_iC i= 0, . . . , m (11b) where

Ai = Ai(In−L^†L), i= 0, . . . , m (12a)

C = C(In−L^†L). (12b)

Using (6), relation (11a) can be rewritten as

H_i=A_i− KiC_i i= 0, . . . , m (13)

where, for i= 0, . . . , m,

A_i = LA_iL^†, C_i=

'CA_iL^† CL^†

(

, (14a)

Ki = % E K_i

&

withK_i=J_i−H_iE (14b)

Then relation (11b) can be expressed in the following compact form

KΣ =LA (15)

where

A = %

A₀ . . . A_m

&

(16a) Σ =

' CA

bdiag(C, ..., C) (

(16b) K = %

E K₀ . . . K_m&

, (16c)

and a general solution to (15), if it exists, is given by K=%

E K₀ . . . K_m

&

=LAΣ^†+Z(I_(m+2)p−ΣΣ^†) (17)

where Z= [Z_E Z0 . . . Zm ] is an arbitrary matrix.

Lemma 1. The unbiasedness of the filter (3)is achieved if and only if the following rank condition holds

rank









LA CA bdiag(C, ..., C) bdiag(L, ..., L)









= rank







CA bdiag(C, ..., C) bdiag(L, ..., L)





 (18)

where

A=%

A₀ . . . A_m&

. (19)

Proof. Using the previous developments, filter (3) is unbiased, i.e. relation (9) holds, if and only if there exists a solution K to (15), that is if and only if

LA(I_n−Σ^†Σ) = 0, (20)

which is equivalent to

rank 'LA

Σ (

= rank Σ. (21)

The rest of the proof can be obtained from (Darouach et al., 2001).

2.3 Unbiasedness condition under ED= 0 The mapping from wtoe, given by (8a), becomes

e˙= /

H₀+

"m i=1

u_iH_i 0

e+ /

ΨB−K₀D−H₀ED−

"m i=1

u_i(KiD+H_iED) 0

w (22a)

e=e−EDw (22b)

or equivalently

e˙= /

A₀+

"m i=1

u_iA_i−1

LAΣ^†−Z(I_(m+2)p−ΣΣ^†)2 Λ₁(u)

0 e +1

LB−1

LAΣ^†−Z(I_(m+2)p−ΣΣ^†)2

Λ₂(u)2 w +

/ A₀+

"m i=1

u_iA_i−1

LAΣ^†−Z(I_(m+2)p−ΣΣ^†)2 Λ1(u)

0

EDw (23a)

e=e−EDw (23b)

where

Λ1(u) =



CA₀L^†+

"m i=1

u_iCA_iL^† ψ_C(u)



, Λ2(u) = ' CB

ψ_D(u) (

(24)

with ψ_C(u) =%

L^†TC^T u₁L^†TC^T ... u_mL^†TC^T &T

and ψ_D(u) =3

D^T u₁D^T ... u_mD^T 4T

.

Due to the termZ(I_(m+2)p−ΣΣ^†)Λ₁(u)E, the error is bilinear in the gain parameterZ in system (23).

This bilinearity is intrinsically linked to the unbiasedness condition (9). Indeed, the ‘bilinearity’ H_iΨ in (9) yields a gain K_i (see (14b)) containing the termH_iE. In order to avoid this bilinearity, we consider ED = 0 in the sequel; this allows to have an LMI tractable formulation for the problem instead of a bilinear matrix inequality intractable one. Adding the constraint ED= 0, relations (15), (16) and (17) become

KΣ =% 0 LA

&

(25) where

Σ =

'D CA

0 bdiag(C, ..., C) (

(26) and a general solution to (25), if it exists, is given by

K=% 0 LA

&

Σ^†+Z(I_(m+2)p−Σ Σ^†). (27)

Then, the following lemma is derived from lemma 1.

Lemma 2. The unbiasedness of the filter (3) is achieved under ED= 0 if and only if

rank









0 LA

D CA

0 bdiag(C, ..., C) 0 bdiag(L, ..., L)









= rank







D CA

0 bdiag(C, ..., C) 0 bdiag(L, ..., L)





. (28)

Now, assume that condition (28) in lemma 2 holds. Then relation (9) is verified with K given by (27) and, hence, e(t) =e(t) in (23), i.e.

˙ e=

/ A₀+

"m i=1

u_iA_i−1%

0 LA

&

Σ^†−Z(I_(m+2)p−Σ Σ^†)2 Λ₁(u)

0 e +1

LB−1%

0 LA&

Σ^†−Z(I_(m+2)p−Σ Σ^†)2

Λ₂(u)2

w. (29) As the item (i) of the design objectives has been solved, it remains to treat the points (ii) and (iii) of these objectives.

2.4 Exponential convergence and L² gain attenuation

Here, a change of variables is introduced by considering each u_i(t) in equation (29) as a ‘structured uncertainty’. Notice that the definition of the ‘uncertainty set’ Ω in relation (2) can lead to some conservatism (see (Boyd et al., 1994)) since, in the general case, |u_i,min| (=|u_i,max|with |u_i,min| (= 1 and

|u_i,max| (= 1. To reduce this conservatism, each u_i(t) can be rewritten as follows

u_i(t) =α_i+σ_iε_i(t) (30)

where α_i ∈IR andσ_i ∈IR are given by αi=u_i,min+ui,max

2 ,σi=u_i,max−u_i,min

2 i= 1, . . . , m, (31)

α₀= 1 and σ₀ = 0. The new ‘uncertain’ variable isε(t)∈Ω⊂IR^m where the polytope Ω is defined as Ω :={ε(t)∈IR^m | ε_i,min=−1!ε_i(t)!ε_i,max= 1 fori= 1, . . . , m}. (32) By using relations (30)-(32), the dynamics of the errore(t) in (29) can be rewritten as follows

˙ e=1

A−ZC+1

A5−ZC52

∆e(ε)He

2 e+1

B−ZG+1

B5−ZG52

∆w(ε)Hw

2

w (33)

with

A=

"m i=0

α_iA_i−% 0 LA

&

Σ^† '6m

i=0α_iCA_iL^† α_C

(

,C= (I_(m+2)p−Σ Σ^†) '6m

i=0α_iCA_iL^† α_C

(

, (34a) A5 =%

σ₁A₁ . . . σ_mA_m&

−%

0 LA&

Σ^†Γ,B5=−%

0 LA&

Σ^†D, G5 = (I_(m+2)p−Σ Σ^†)D,(34b) C5 = (I_(m+2)p−Σ Σ^†)Γ,B=LB−%

0 LA&

Σ^†

'α₀CB αD

(

,G= (I_(m+2)p−Σ Σ^†)

'α₀CB αD

( .(34c) and

Γ =







σ₁CA₁L^† . . . σ_mCA_mL^†

0 . . . 0

bdiag(σ₁CL^†, ..., σ_mCL^†)





,D=

' 0

bdiag(σ1D, ..., σmD) (

, (35a)

α_C =%

α₀L^†TC^T ... α_mL^†TC^T

&T

, α_D =%

α₀D^T ... α_mD^T

&T

. (35b)

From (32), the ‘uncertain’ matrices ∆e(ε) and ∆w(ε) are bounded as

"∆e(ε)"!1 and "∆w(ε)"!1 (36) where ∆_e(ε) = bdiag(ε₁I_r, . . . , ε_mI_r ∈ IR^mr×mr, ∆_w(ε) = bdiag(ε₁I_q, . . . , ε_mI_q) ∈ IR^mq×mq, H_e = [I_r . . . I_r]^T ∈IR^mr×r and H_w = [I_q . . . I_q]^T ∈IR^mq×q.

According to the previous developments, the error dynamics (33) can be rewritten as the following system

˙

e=(A−ZC)e+1%

A5 B B5 &

−Z%

C5 G G5 &2

p_e p_w w



 (37a)



q_e qw

e



=



H_e 0 I_r



e+



 0 0 0 0 0 Hw

0 0 0







p_e pw

w



 (37b)

connected with ' p_e p_w (

7 89 :

e p

=

'∆e(ε) 0 0 ∆_w(ε)

( 'q_e q_w (

7 89 :

e q

. (38)

At this step, theH∞ reduced order unbiased filtering can be solved as a particular case of a dual robust state feedback problem with structured uncertainties. The following theorem ensures the exponential convergence of the filter (3) and the L2 gain attenuation from w(t) to e(t).

Theorem 1. Suppose that assumption 1 and condition (28) hold. If there exist P =P^T > 0, S_e = bdiag(µ1,eI_r, . . . , µ_m,eI_r)>0, S_w= bdiag(µ1,wI_q, . . . , µ_m,wI_q) >0, Y and a scalar ρ >0 such that (‘•’ is the transpose of the off-diagonal part)











Υ • • • • 0

A5^TP−C5^TY −S_e 0 0 0 0 B5^TP−G5^TY 0 −S_w 0 0 0 B^TP−G^TY 0 0 −γ²I_q 0 •

SeHe 0 0 0 −Se 0

0 0 0 S_wH_w 0 −S_w











<0 (39)

holds with Y =Z^TP and Υ = herm(PA−Y^TC) + (1 +ρ)I_r, then the reduced order unbiased filter (3) for the bilinear system (1) is exponentially convergent and has a L2 gain from w(t) to e(t) less than or equal to γ.

Proof. By considering system (37)-(38) as a diagonal norm-bounded linear differential inclusion, the following auxiliary system is derived from (37) (Boyd et al., 1994)

˙

e= (A−ZC)e+1%

A5S_e^−1/2 B5S_w^−1/2 γ⁻¹B&

−Z%

C5S_e^−1/2 G5S_w^−1/2 γ⁻¹G&2

pe

p_w w



 (40a)



q_e q_w e



=



S_e^1/2He

0 I_r



e+





0 0 0

0 0 γ⁻¹S_w^1/2H_w

0 0 0







p_e p_w w



 (40b)

where matrices S_e = bdiag(µ1,eI_r, . . . , µ_m,eI_r) > 0 and S_w = bdiag(µ1,wI_q, . . . , µ_m,wI_q) > 0 satisfy

∆e(ε)Se = Se∆e(ε) and ∆w(ε)Sw = Sw∆w(ε) in order to take the structure of ∆e(ε) and ∆w(ε) into account (Boyd et al., 1994) (µ_i,e andµ_i,w,i= 1, . . . , m, are positive scalars).

LetY =Z^TP, then by using the bounded real lemma (Boydet al., 1994), system (37)-(38) is exponentially convergent and has a L2 gain from w to e less than or equal to γ if there exist P = P^T > 0, Se > 0, S_w >0,Y and a scalarρ >0 such that matrices in system (40) satisfy the following inequality













Υ • • • • 0 •

S_e^−1/2(A5^TP−C5^TY)−I 0 0 0 0 0 S_w^−1/2(B5^TP−G5^TY) 0 −I 0 0 0 0 γ⁻¹(B^TP−G^TY) 0 0 −I 0 • 0

S_e^1/2He 0 0 0 −I 0 0

0 0 0 γ⁻¹S_w^1/2Hw 0 −I 0

I 0 0 0 0 0 −I













<0.

Pre- and post-multiplying this inequality by

bdiag(In, S_e^1/2, S_w^1/2, γI_q, S_e^1/2, S_w^1/2, I_r) and using the Schur lemma lead to the LMI (39).

Under condition (28), by usingZ =P⁻¹Y^T and (27), the matrices of filter (3) are given by (13)-(14).

3 Robust reduced order unbiased H

filtering

In this section, the following uncertain bilinear system is considered

˙

x=(A0+∆A0(t))x+

"m i=1

(Ai+∆Ai(t))uix+(B+∆B(t))w (41a)

y= (C+ ∆_C(t))x+ (D+ ∆_D(t))w (41b)

z=Lx (41c)

where x(t), y(t), z(t), w(t) and u(t) have been defined in section 2. The uncertain matrices ∆A0(t),

∆_B(t), ∆_C(t), ∆_D(t) and ∆_Ai(t) are given by '∆A0(t) ∆B(t)

∆_C(t) ∆_D(t) (

= 'M_0,x

M_0,y (

∆0(t)%

E0,x E0,w

&

(42a)

%∆A1(t) . . . ∆Am(t)&

=%

M_1,x . . . M_m,x&

bdiag(∆₁(t), . . . ,∆_m(t))

7 89 :

∆(t)

%

E_1,x^T . . . E_m,x^T&T

(42b)

where M_i,x∈IR^n×!i,M_0,y ∈IR^p×!0, E_i,x∈IR^!i×n and E_0,w ∈IR^!0×q (i= 0, . . . , m) are known constant matrices which specify how the elements of the matrices of the nominal system are affected by the uncertain parameters in ∆i(t), i = 0, . . . , m. The continuous time-varying uncertainties in (42) are assumed to satisfy :

Assumption 2. There exist ∆i,j(t) ∈ IR^!i,j×!i,j, j= 1, . . . , s_i and i= 0, . . . , m, such that

; ∆i(t) = bdiag(∆i,1(t), . . . ,∆i,si(t)), i= 0, . . . , m

"∆_i(t)"!I_!

i,∀t"0, i= 0, . . . , m (43) where )_i =6si

j=1)_i,j.

The constraint ED= 0 in section 2.3 is replaced by E%

M_0,y D&

= 0 (44)

in the sequel of section 3. Then equations (25), (26), (27) and (28) must be replaced by KΣ =# %

0 0 LA

&

, (45a)

Σ =#

'M_0,y D CA

0 0 bdiag(C, . . . , C) (

, (45b)

K=%

0 0 LA

&

Σ#^†+Z(I_(m+2)p−Σ#Σ#^†), (45c)

rank









0 0 LA

M0,y D CA

0 0 bdiag(C, . . . , C) 0 0 bdiag(L, . . . , L)









= rank







M_0,y D CA

0 0 bdiag(C, . . . , C) 0 0 bdiag(L, . . . , L)





. (45d)

Introducing the following augmented state vectorξ =3

x^T e^T 4T, using (29) and the change of variable

in (30)-(32) enable one to express the system obtained by the concatenation of (3) and (41)-(42) as ξ(t) =˙

/' M_0,x

ΨM0,x−6m

i=0α_iJ_iM_0,y (

∆₀(t)%

E_0,x 0&

+

"m i=0

α_i

/'A_i 0 0 H_i

( +

'M_i,x ΨMi,x

(

∆_i(t)%

E_i,x 0&0 +

"m i=1

σ_iε_i

/'A_i 0 0 H_i

( +

' M_i,x ΨMi,x

(

∆_i(t)%

E_i,x 0&

+

' 0

−J_iM_0,y (

∆₀(t)%

E_0,x 0&00 ξ

+









B ΨB−

"m i=0

α_iJ_iD



+

' M0,x

ΨM_0,x−6_m

i=0α_iJ_iM_0,y (

∆0(t)E0,w

+

"m i=1

σiεi

/' 0

−J_iD (

+

' 0

−J_iM_0,y (

∆0(t)E0,w

00 w

(46a)

e=% 0 I_r

&

ξ (46b)

where Ψ satisfies the unbiasedness relation (9). The uncertain system (46) is equivalent to the following one

ξ(t) =˙





"m i=0

α_iA_i 0

0 A−ZC



ξ+

' B

B−ZG (

w+

'A_σ 0 0

0 A−5 ZC5 B−5 ZG5 (



 p_x pe

p_w





 (47a)

+

' M_0,x M_α

BM−ZGM BMα−ZGMα

( 'p₀ p

( +

' 0 M_σ

B5Mσ−ZG5Mσ BMσ−ZGMσ

( 'p₀ p (

(47b)

#

q=Cqξ+Dqww (47c)

e=% 0 I_r

&

ξ (47d)

connected with

3p^T_x p^T_e p^T_w p^T₀ p^T p^T₀ p^T4T

7 89 :

b p

=

bdiag(∆x(ε),∆e(ε),∆w(ε),∆0(t),∆(t),∆0(ε, t),∆(ε, t))

7 89 :

∆(ε,t)b

3q^T_x q^T_e q^T_w q^T₀ q^T q^T₀ q^T 4T

7 89 :

b q

(48)

where A_σ =%

σ₁A₁ · · · σ_mA_m

&

, M_α =%

α₁M_1,x · · · α_mM_m,x

&

, M_σ =%

σ₁M_1,x · · · σ_mM_m,x

&

, M_α =%

α₀M_0,y^T · · · α_mM_0,y^T

&T

,BM =LM_0,x−%

0 0 LA

&

Σ#^†

'CM_0,x M_α,y

( , GM = (I_(m+2)p−Σ#Σ#^†)

'CM0,x

M_α,y (

,BMα =LM_α−%

0 0 LA

&

Σ#^†

'CMα

0 (

, GMα = (I_(m+2)p−Σ#Σ#^†)

'CMα

0 (

,BMσ =LMσ−%

0 0 LA

&

Σ#^† 'CMσ

0 (

, GMσ = (I_(m+2)p−Σ#Σ#^†)

'CM_σ 0

(

,B5Mσ =−%

0 0 LA

&

Σ#^†M,G5Mσ = (I_(m+2)p−Σ#Σ#^†)M,

M=

' 0

bdiag(σ₁M_0,y, ..., σ_mM_0,y) (

,C^q=













Hx 0 0 H_e

0 0

E0,x 0 E_x 0 E_0,x 0 E_x 0













, D^qw=











 00 H_w E0,w

0 E0,w

0











 ,

H_x=%

I_n . . . I_n

&T

∈IR^mn×n, E_0,x=%

E_0,x^T . . . E_0,x^T

&T

∈IR^m!0×n, E_x=%

E_1,x^T . . . E_m,x^T

&T

∈IR^{(!1+...+!m)×n}, E_0,w=%

E_0,w^T . . . E_0,w^T

&T

∈IR^m!0×q,

∆_x(ε) = bdiag(ε₁I_n, . . . , ε_mI_n),∆₀(ε, t) = bdiag(ε₁∆₀(t), . . . , ε_m∆₀(t)),

∆(ε, t) = bdiag(ε1∆1(t), . . . , εm∆m(t)).

The following inequalities hold from (32) and the definition of ∆i(t)

"∆_x(ε)"!1, ??∆₀(ε, t)??!1 and ??∆(ε, t)??!1. (49) Note that the unbiasedness condition (9) for the filter (3) is verified for the nominal case. In the system (47), the determination of gain matrixZ can be transformed into a robust static output feedback control problem by introducing the following auxiliary system









ξ˙=Aξ+Bww+Buu z=Czξ+Dzww y=Cyξ+Dyww

(50a)

u=−Z y (50b)

whereZ is the static output feedback controller to be designed in order to achieve stability and attenua- tion from the ‘augmented perturbation’ w^T(t) =3

#

p^T(t) w^T(t)4T to the ‘augmented controlled output’

z^T(t) =3

#

q^T(t) e^T(t)4

. u(t) andy(t) play the role of ‘control input’ and ‘measured output’, respectively.

The matrices of system (50) are given by Bw =%

BpS^−1/2 γ⁻¹Bw

&

, Dyw=%

DypS^−1/2 γ⁻¹Dyw

&

,Cz =

'S^1/2Cq

Ce

( ,A=

'6m

i=0α_iA_i 0

0 A

( , Dzw=

'S^1/2DqpS^−1/2 γ⁻¹S^1/2Dqw

DepS^−1/2 γ⁻¹Dew

( ,Bu =

'0 I_r

(

,Cy =% 0 C&

,

%Bp Bw

&

=

'A_σ 0 0 M_0,x M_α 0 M_σ B 0 A5 5B BM B_Mα B5_Mσ B_Mσ B

( ,%

Dyp Dyw

&

=%

0 C5 G G5 ^M GMα G5Mσ GMσ G&

, 'Cq

Ce

(

=

' Cq

0 Ir

( ,

'Dqp Dqw

Dep Dew

(

=

D0 Dqw

0 0 E

.

where S is a block-diagonal matrix with the same ‘structure’ as the uncertain matrix ∆(ε, t) given in# (48), i.e.

S= bdiag(Sx, S_e, S_w, S_∆⁰, S_∆, S⁰_∆, S_∆)>0 (51)

with 





























S_x= bdiag(µ1,xI_n, . . . , µ_m,xI_n) Se= bdiag(µ1,eIr, . . . , µm,eIr) S_w = bdiag(µ_1,wI_q, . . . , µ_m,wI_q) S_∆⁰ = bdiag(µ0,1I_!0,1, . . . , µ_0,s0I_!0,s

0)

S_∆= bdiag(bdiag(µ_1,1I_!1,1, . . . , µ_1,s1I_!1,s1), . . . ,bdiag(µ_m,1I_!m,1, . . . , µ_m,smI_!m,sm)) S⁰_∆= bdiag(bdiag(µ5_1,1I_!0,1, . . . ,µ5_1,s0I_!0,s

0), . . . ,bdiag(µ5_m,1I_!0,1, . . . ,µ5_m,s0I_!0,s

0)) S_∆= bdiag(bdiag(µ_1,1I_!1,1, . . . , µ_1,s1I_!1,s1), . . . ,bdiag(µ_m,1I_!m,1, . . . , µ_m,smI_!m,sm))

where µ_i,x, µ_i,e, µ_i,w, µ_0,h, µ_i,j, µ5_i,h and µ_i,j are scalars to be chosen (i = 1, . . . , m, h = 1, . . . , s0 and j= 1, . . . , si).

Notice that the following relation holds

∆(ε, t)S# =S∆(ε, t).# (52)

Then, the gain Z is designed in the following theorem.

Theorem 2. Assume that assumptions 1 and 2 and relation (45d) hold, there exists a robust functional unbiased reduced order filter (3) for the uncertain system (41) if there exist matrices P = P^T > 0, Q=Q^T >0, S>0 and S>0 such that (with)=)1+. . .+)m, s=m(n+r+q) +)0(m+ 1) + 2))

DKy 0 0 Ir+s

ET









A^TP+PA PBw PBp C^T_e C^T_qS B^T_wP −γ²Iq 0 D^T_ewD^T_qwS B^T_pP 0 −S D^T_ep D^T_qpS

Ce Dew Dep −I_r 0 SCq SDqwSDqp 0 −S









DKy 0 0 Ir+s

E

<0, (53a)

DKu 0 0 I_q+s

ET











QA^T +AQ QC^T_e QC^T_q Bw BpS CeQ −Ir 0 Dew DepS CqQ 0 −S Dqw DqpS

B^T_w D^T_ew D^T_qw −γ²I_q 0 SB^T_p SD^T_epSD^T_qp 0 −S











DKu 0 0 I_q+s

E

<0, (53b)

In+r=PQ, (53c)

where S=S⁻¹ satisfy (51), and whereKy and Ku are two matrices whose columns span the null spaces of [Cy Dyp Dyw]and 3

B^T_u 04

, respectively. All gains Z are given by

Z =B^†_RKC^†_L+Z−B^†_RBRZCLC^†_L (54) with

K=−R⁻₁¹B^T_LV1C^T_R1

CRV1C^T_R2₋₁

+R⁻₁¹V^1/2₂ R2

1CRV1C^T_R2_−1/2 V1 =1

BLR⁻₁¹B^†_L−Q2₋1

>0 V2 =R1−B^T_L

F

V1−V1C^T_R1

CRV1C^T_R2₋1

CRV1

G BL

'B Q

• C (

=













−BuQA^T +AQ QC^T_e QC^T_q Bw BpS 0 CeQ −I_r 0 Dew DepS 0 CqQ 0 −S Dqw DqpS 0 B^T_w D^T_ew D^T_qw −γ²I_q 0 0 SB^T_p SD^T_epSD^T_qp 0 −S

• CyQ SDyp Dyw 0 0











