Robust Filtering for Linear Polytopic Discrete-Time Periodic Systems*

Periodic Control Systems — PSYCO 2010 Antalya, Turkey, August 26-28, 2010

Robust Filtering for Linear Polytopic Discrete-Time Periodic Systems ? Carlos E. de Souza ∗ Daniel F. Coutinho ∗∗ ∗ Department

of Systems and Control, Laborat´orio Nacional de Computac¸a˜ o Cient´ıfica – LNCC/MCT, Petr´opolis, RJ 25651-075, Brazil (e-mail: [email protected]). ∗∗ Group

of Automation and Control Systems, Pontif´ıcia Universidade Cat´olica do Rio Grande do Sul (PUCRS), Porto Alegre, RS 90619-900, Brazil (e-mail: [email protected]). Abstract: This paper is concerned with the problems of robust H¥ and guaranteed variance filtering for linear discrete-time periodic systems with polytopic-type parameter uncertainty in the matrices of the system state-space model. Filtering methods are derived for designing linear periodic asymptotically stable filters with either a prescribed upper-bound on the `2 -gain from the noise signals to the estimation error, in the H¥ case, or a guaranteed upper-bound on the average steady-state variance of the estimation error variance, for the guaranteed variance filtering, in spite of the parameter uncertainty. The proposed methods are based on parameter-dependent Lyapunov functions and are given in terms of linear matrix inequalities. The potentials of the proposed filtering methods are demonstrated by an example. Keywords: Robust H¥ filtering, guaranteed variance filtering, linear periodic systems, uncertain systems, discrete-time systems, parameter-dependent Lyapunov function. 1. INTRODUCTION Over the past three decades periodic systems have been attracting significant interest within the control community. One of the motivations for this interest is the fact that cyclic processes arise very often in nature and engineering and applications of periodic systems may be found in a large spectrum of different fields, such as economics, biology, aeronautics, control and filtering of linear systems subject to cyclostationary noise, and control of multirate plants; see, Bittanti [1986], Bittanti and Colaneri [2009], and the references therein. A considerable amount of attention has been paid to linear periodic systems and a number of important results on control and filtering problems have been reported in the literature. As for control design, techniques have been developed to solve a variety of problems, as for instance, pole placement (Colaneri [1991]), characterization of all stabilizing controllers (Bittanti and Colaneri [1999]), robust stabilization (de Souza and Trofino [2000], Farges et al. [2007]), robust tracking (Grasselli et al. [1996]), H¥ control (Colaneri and de Souza [1992]), H2 control (Wi´sniewski and Stoustrup [2001]), and robust H2 control (Farges et al. [2007]). In the context of filtering, the minimum mean-square state estimation (i.e. the celebrated periodic Kalman filter) and properties of the associated periodic Riccati equation, have been widely studied during the 80’s and early 90’s (see, e.g. Bittanti et al. [1988], de Souza [1991], Bittanti et al.. [1991], De Nicolao [1992], and the references therein). On the other hand, the H¥ filtering problem for continuous-time periodic systems has been solved in Xie and de Souza [1993] in terms of a periodic Riccati equation, whereas the discrete-time case has been treated in Bittanti and ? This work was supported by CNPq, Brazil, under grants 30.3440/2008-2/PQ and 30.1461/2008-2/PQ.

ISBN 978-3-902661-86-9/11/$20.00 © 2010 IFAC

Cuzzola [2001] using a linear matrix inequality (LMI) approach. The aforementioned filtering methods are restricted to systems with an uncertainty-free state-space model and as such they do not provide a guaranteed performance in the presence of parameter uncertainty. In the case of uncertain systems, a robust H¥ filtering method for linear continuous-time periodic systems with norm-bounded parameter uncertainty has been developed in Xie et al. [1991] using a Riccati equation approach. In this paper we focus on robust filtering for linear discretetime periodic systems subject to polytopic-type parameter uncertainty in the all the matrices of the state-space model. Two robust filtering problems are considered. For the first one, the problem of robust H¥ filtering, we address the design of a linear periodic asymptotically stable filter that provides a prescribed upper-bound on the `2 -gain from the noise signals to the estimation error, in spite of the parameter uncertainty. On the other hand, for the second problem, the one of guaranteed variance filtering, the filter is required to ensure a guaranteed upperbound for the average steady-state variance of the estimation error variance. The proposed filter designs build on uncertaintydependent Lyapunov functions based on periodic matrices and are tailored via LMIs. An example is presented to illustrate the effectiveness of the developed robust filtering methods. Notation. Z is the set of integers, Rn is the n-dimensional Euclidean space, Rm×n is the set of m×n real matrices, In is the n×n identity matrix, diag{· · · } stands for a block-diagonal matrix, and a matrix Sk is denoted N-periodic, 0 < N ∈ Z, if Sk+N =Sk , ∀ k ∈Z. The transpose of a matrix S is denoted by ST , tr ( · ) is the matrix trace, and Sk > 0 (Sk ≥ 0) for a real periodic matrix Sk means that Sk = SkT and positive definite (positive semidefinite) for all k ∈ Z. `2 stands for the space of square summable vector sequences over [0, ¥) with norm k·k2 .

1

on the H¥ norm of the estimation error system (7) over the uncertainty domain W . For the second problem, the one of guaranteed variance filtering, the noise signal wk is assumed to be a zero-mean white sequence with an N-periodic covariance matrix, which is uncorrelated with x0 for all k ≥ 0, and the filter is required to ensure the asymptotic stability of system (7) and an upper-bound (optimized in a certain sense) on the average steady-state variance of the estimation error over one period N, for all system matrices in W . This paper develops LMI methodologies of robust H¥ and guaranteed variance filtering based on uncertainty-dependent Lyapunov functions.

2. PROBLEM STATEMENT Consider the following uncertain linear discrete-time periodic system:  xk+1 = Ak xk + Bk wk yk = Ck xk + Dk wk (1)  zk = Lk xk + Mk wk where xk ∈ Rn is the state vector, wk ∈ Rnw is the noise signal (including process and measurement noises), yk ∈ Rny is the measurement, zk ∈ Rnz is the signal to be estimated, and Ak , Bk , Ck , Dk , Lk and Mk are N-periodic real matrices with appropriate dimensions which are uncertain but assumed to belong to the following convex polytope: ) ( W where

k

=

P P

k

:P

k

=

u

l iP

i=1

(i) k ,

l i ≥ 0,

u

l i= 1

k

(i)

(i)

(i)

(2)

i=1

= [ Ak Bk Ck Dk Lk Mk ], h i (i) (i) (i) (i) (i) (i) P k = A(i) , B C D L M k k k k k k (i)

We end this section by recalling some definitions for linear periodic systems and for that consider system (1) without uncertainty. First, note that the transition matrix F k,l of Ak is N-periodic in k and l, i.e. F k+N,l+N = F k,l , ∀ k, l ∈ Z, and the eigenvalues of F k+N,k , known as characteristic multipliers of Ak , are independent of k. Next, we recall that the unforced uncertainty-free system of (1) is asymptotically stable if and only if all the characteristic multipliers of Ak are inside the open unit disc. Throughout this paper, the uncertain system (1) is said to be robustly stable if the unforced system of (1) is asymptotically stable for all matrices Ak ∈ W .

(i)

(3) (4)

(i)

and where Ak , Bk , Ck , Dk , Lk and Mk , i = 1, . . . , u , k = 1, . . . , N, are given real matrices. For notation simplicity, we shall denote © ª W := W k , k = 1, . . . , N (5)

Finally, we present a version of the periodic Lyapunov lemma. Lemma 1. (Bolzern and Colaneri [1988]) The unforced system of (1) without uncertainty is asymptotically stable if and only if there exists an N-periodic matrix Pk > 0 such that

as the uncertainty domain. Note that u = 1 corresponds to the case where system (1) is uncertainty free.

Ak Pk ATk − Pk+1 < 0, ∀ k ∈ Z.

Observe that in the case where the noise signals affecting the state, measurement and signal z equations are different, say B˘ k rk , D˘ k vk and M˘ k uk , respectively, where B˘ k , D˘ k and M˘ k are N-periodic, we simply set wk = [ rkT vTk uTk ]T , Bk = [ B˘ k 0 0 ], Dk = [ 0 D˘ k 0 ] and Mk = [ 0 0 M˘ k ] in the system model (1) .

(10)

3. ROBUST H¥ FILTERING Firstly, we recall the notion of H¥ norm for linear discrete-time periodic systems and related results. To this end, let the system ½ xk+1 = Fk xk + Gk wk (11) x k = Hk xk + Jk wk

This paper is aimed at designing robust filters for the uncertain system of (1)-(4) which provide an estimate zˆ of the signal z with a guaranteed performance (to be specified in the sequel) for all system matrices in the uncertainty domain W . Attention is focused on the design of linear discrete-time periodic, asymptotically stable, filters of order n with state-space realization ( xˆk+1 = Aˆ k xˆk + Bˆ k yk , xˆ0 = 0 F: (6) zˆk = Cˆk xˆk + Dˆ k yk

where xk ∈ Rn is the state, wk ∈ Rnw is the input, x k ∈ Rnx is output, and Fk , Gk , Hk and Jk are known N-periodic real matrices with appropriate dimensions. Assuming that system (11) is asymptotically stable, the H¥ norm of (11), namely its `2 -induced gain from w to x , is defined by ½ ¾ ||x ||2 kH x w k¥ = sup : w 6≡ 0, x0 = 0 . ||w||2 w ∈ `2

where Aˆ k , Bˆ k , Cˆk and Dˆ k are N-periodic real matrices to be determined. Note that since the underlying signal model (1) is N-periodic, it is somewhat natural to focus on N-periodic filters.

Characterizations of the H¥ norm of system (11) in terms of LMIs are given in the next lemma. The proof is a simple variation of known versions of the bounded real lemma for linear time-varying discrete-time systems (see, e.g. de Souza et al. [2006]) and thus is omitted.

Denoting x =[ xT xˆT ]T and considering (1) and (6), the dynamics of the estimation error e := z− zˆ can be described by the following state-space model: x k+1 = A˜ k x k + B˜ k wk (7) ek = C˜k x k + D˜ k wk where · ¸ · ¸ Ak 0 Bk ˜ ˜ Ak = , Bk = , (8) Bˆ k Dk Bˆ k Ck Aˆ k £ ¤ C˜k = Lk − Dˆ k Ck −Cˆk , D˜ k = Mk − Dˆ k Dk . (9)

Lemma 2. Given a scalar g > 0, system (11) is asymptotically stable and kH x w k¥ < g if and only if any of the following equivalent condition holds: (a) There exists an N-periodic n× n matrix Xk >0 satisfying the following inequalities (Bittanti and Cuzzola [2001]):   −Xk+1 Fk Xk Gk 0    Xk FkT −Xk 0 Xk HkT   < 0, ∀ k ∈ Z.  (12)   T 0 −g I JkT   Gk 0 Hk Xk Jk −g I

Two robust filtering problems will be addressed in this paper. In the first, namely robust H¥ filtering, the filter F is required to provide asymptotic stability and a prescribed upper-bound

2

(

(b) There exist N-periodic n× n matrices Pk > 0 and Kk satisfying the following inequalities:   −Pk+1 Fk Kk Gk 0  T T   Kk Fk Pk − Kk − KkT 0 KkT HkT    < 0, ∀ k ∈ Z. (13)   T 0 −g I JkT   Gk 0 Hk Kk Jk −g I

u

(i)

i=1

(i)

l i Pk , Pk > 0, l i ≥ 0

Proof. First, note that (16) implies that ¡ k > 0, ∀ k ∈ Z, which ensures the nonsingularity of the matrices Rk and Zk for all k ∈ Z, and thus the filter of (21) is well defined. Moreover, by convexity, (16) is equivalent to the feasibility of the inequalities   −Xk+1 Ak Bk 0  AT Xk − ¡ k 0 CTk    k  < 0, k = 1, . . . , N (22)   BTk 0 −g I DTk  0 Ck Dk −g I

(14)

for all the matrices of system (1) in W , where · ¸ Rk+1 Ak 0 Ak = , Sk+1 Ak + Bk Ck + Ak Ak

where l i are the scalars as in (2), that ensure the feasibility of the following conditions for all the matrices of system (1) in W :   −Pk+1 A˜ k Kk B˜ k 0  T ˜T   Kk Ak Pk − Kk − KkT 0 KkT C˜kT   < 0, ∀ k ∈ Z . (15)   ˜T  0 −g I D˜ Tk   Bk 0 C˜k Kk D˜ k −g I

¸ Rk+1 Bk Bk = , Sk+1 Bk + Bk Dk Ck = [ Lk − Ck − Dk Ck Dk = Mk − Dk Dk ,

(i)

where

" (i)

Ak =

Dk

" (i) Bk

(i)

=

(i)

Sk+1 Bk + Bk Dk

h

Ck = Lk(i) − Ck − Dk Ck(i) " (i)

(i)

(i)

Dk = Mk − Dk Dk , ¡ k =

#

Ak

Cˆk = (Ck Zk−1 )Uk ,

i −Ck ,

Rk + RTk

SkT + ZkT

Sk + Zk

Zk + ZkT

(19)

(29)

Dˆ k = Dk .

(30)

Considering (8), (9), (20), (23)-(26) and (28)-(30), it can be readily shown that ( T T ˜ Yk+1 A˜ k Kk Yk = Ak , Yk+1 Bk = Bk , (31) T C˜k Kk Yk = Ck , Yk (Kk + KkT )Yk = ¡ k .

# .

¸ I 0 . Vk Vk

Next, consider the following matrices for the filter of (6): ( −1 −1 Aˆ k = Uk+1 (Ak Zk−1 )Uk , Bˆ k = Uk+1 Bk ,

(18)

,

(26)

Since the matrices Rk and Zk are nonsingular for all k ∈ Z, the matrices Uk , Vk , Yk and Kk are also nonsingular for all k ∈ Z.

(17)

,

· Kk Yk =

#

(i)

Rk+1 Bk (i)

0

i=1

(i)

l i Xk

where Uk and Vk are n×n nonsingular N-periodic matrices such that Uk Vk = Zk . Moreover, define the N-periodic matrix " # RTk SkT Yk = . (28) 0 UkT

(16)

(i) Rk+1 Ak (i) (i) Sk+1 Ak + Bk Ck + Ak

u

Let the N-periodic matrix Kk in (15) be parameterized as follows: #T " T R−1 R−1 k k Vk Kk = (27) T −Uk−1 Sk R−1 Uk−1VkT −Uk−1 Sk R−1 k k Vk

Observe that

−g I

(25)

−Ck ],

It will be shown in the sequel that if the inequalities of (22) hold, then the filter (21) ensures that (15) is satisfied with an uncertainty-dependent matrix Pk as in (14).

Theorem 1. Consider the uncertain system of (1)-(4). Given a scalar g > 0, suppose that there exist N-periodic matrices Ak ∈ Rn×n , Bk ∈ Rn×ny , Ck ∈ Rnz ×n , Dk ∈ Rnz ×ny , Rk ∈ Rn×n , (i) Sk ∈ Rn×n , Zk ∈ Rn×n , and Xk ∈ R2n×2n , i=1, . . . , u , satisfying the following LMIs:   (i) (i) (i) −Xk+1 Ak Bk 0  ¡ (i) ¢T ¡ (i) ¢T  ½ (i)   A Xk − ¡ k 0 Ck k = 1, . . . , N,   k < 0,  ¡ (i) ¡ (i) ¢T  i = 1, . . . , u   B )T 0 −g I Dk   k (i)

Xk =

(24)

with l i ≥ 0 being the scalars as in (2).

The robust H¥ filter design method is presented in the next theorem, which is derived from (15) with Pk as in (14) and applying appropriate congruence transformation and parameterizations of the filter matrices (Aˆ k , Cˆk ) and Kk .

Ck

(23)

·

By Lemma 2 (b), the above conditions imply that the estimation error system (7) is robustly stable and kHew k¥ < g over the uncertainty domain W .

0

(21)

ensures that the estimation error system (7) is robustly stable and kHew k¥ < g over the uncertainty domain W .

In the sequel we shall develop an LMI method of robust H¥ filtering for system (1) based on Lemma 2 (b). Specifically, given a scalar g > 0, a filter F is designed such that there exist an N-periodic matrix Kk and an affine parameter-dependent N-periodic matrix Pk > 0 of the form Pk =

¡ ¢ xˆk+1 = Ak Zk−1 xˆk + Bk yk ¡ ¢ zˆk = Ck Zk−1 xˆk + Dk yk

(20)

Then, the filter as follows

3

Define the N-periodic positive definite matrix Pk = Yk−T Xk Yk−1 =

u i=1

l

−T (i) −1 i Yk Xk Yk

var {x } = (32)

k

N

¡ ¢ tr X k

(33)

k=1

where X k denotes the periodic equilibrium of the covariance ¤ matrix of x k .

and let the congruence transformation matrix Y

1 N

= diag{Yk+1 , Yk , Inw , Inz } .

Under the assumptions of Definition 1, the covariance matrix of the state vector of system (11) converges to an N-periodic equilibrium Qk , which is the unique positive semidefinite periodic solution to the periodic Lyapunov equation (de Souza [1991])

Now, consider the inequalities of (15) with the matrices Kk , (Aˆ k , Bˆ k , Cˆk , Dˆ k ) and Pk as in (27), (30) and (32), respectively. Pre- and post-multiplying (15) by Y Tk and Y k , respectively, and considering (31), we conclude that (22) ensures the feasibility of (15) for all matrices of system (1) in W . Hence, by Lemma 2 (b), the estimation error system (7) is robustly stable and kHew k¥ < g over the uncertainty domain W .

Fk Qk FkT − Qk+1 + Gk GTk = 0.

(34)

As a consequence, the covariance matrix of x k also converges to an N-periodic equilibrium X k given by

Finally, the filter matrices of (21) follow immediately from (30) by considering Uk as the matrix of a periodic similarity transformation. ¤

X

leading to

var {x } =

Theorem 1 provides an LMI method for designing an N-period filter of the form (6) for system (1) with a prescribed H¥ noise attenuation level g for all admissible modelling uncertainties. Note that, since the design conditions of (16) are affine in g , finding the optimal filter of Theorem 1, namely the filter that achieves the smallest possible g , is a convex optimization problem of minimizing g subject to the LMI constraints of (16).

k

= Hk Qk HkT + Jk JkT 1 N

N k=1

¢ ¡ tr Hk Qk HkT + Jk JkT .

(35) (36)

Notice that in the light of (34)-(36), var {x } coincides with the squared generalized H2 norm of system (11), which was considered in Wi´sniewski and Stoustrup [2001] and Farges et al. [2007]. The next lemma presents two characterizations of var {x } in terms of LMIs that are obtained from these works. Lemma 3. Consider system (11) subject to the assumptions of Definition 1. Then the following conditions holds:

Remark 1. Observe that the filter (21) of Theorem 1, in general, has a direct feedforward from the measured signal yk to the estimate zˆk . However, if desired, this feedforward can be readily suppressed but imposing the constraints Dk =0, k =1, . . . , N, to the LMIs of (16). As an abuse of terminology, we shall refer to these filters as strictly proper. ¤

(a) If there exist N-periodic matrices Xk > 0 and G k ≥ 0 satisfying the inequalities (Wi´sniewski and Stoustrup [2001])   −Xk+1 Fk Xk Gk   T −Xk 0  < 0, ∀ k ∈ Z , (37)  Xk Fk GTk

4. GUARANTEED VARIANCE FILTERING 

In parallel with stationary Kalman filtering, the guaranteed variance filter design is carried out under periodic steady-state regime of the estimation error system (7). To this end, it is assumed that the initial state x0 of system (1) is a zero-mean random variable and the noise signal wk is a zero-mean white sequence with an identity covariance matrix and uncorrelated with x0 for all k≥0, and the average steady-state variance of the estimation error over one period N (to be defined in the sequel) is chosen as the performance measure. The motivation for using the average steady-state variance is that, as the system and filter are N-periodic, it is well known (see, e.g. de Souza [1991]) that subject to the robust stability of the estimation error system (7), the covariance matrix of the state vector of (7) converges to an N-periodic equilibrium, and thus it is natural to consider the average steady-state error variance over one period. Note that in spite of the assumption of an identity covariance matrix for wk , the results of this section also apply to the more general situation where wk is a cyclostationary white noise with an N-periodic covariance matrix Wk . Indeed, this can be readily achieved by incorporating the square-root of Wk to the matrices Bk , Dk and Mk .

G

k

0

 T  Xk Hk JkT

−I

Hk Xk Jk



Xk

 0  ≥ 0, ∀ k ∈ Z ,

0

I

then system (11) is asymptotically stable and 1 N ¡ ¢ var {x } ≤ tr G k . N k=1

(38)

(39)

(b) There exist N-periodic matrices Xk >0 and G k ≥0 satisfying (37) and (38) if and only if there exist N-periodic matrices Pk >0 and Kk satisfying the inequalities (Farges et al. [2007]):   −Pk+1 Fk Kk Gk   T T T 0  < 0, ∀ k ∈ Z , (40)  Kk Fk Pk − Kk − Kk 

GTk

0

−I

G

Hk Kk

Jk

k

 T T  Kk Hk JkT

Definition 1. Consider system (11) subject to the assumption that its initial state x0 is a zero-mean random variable and wk is a zero-mean white sequence with an identity covariance matrix and uncorrelated with x0 for all k ≥ 0. If the unforced system of (11) is asymptotically stable, the average steady-state variance of x k , denoted by var {x }, is defined by

Kk + KkT − Pk 0



 0  ≥ 0, ∀ k ∈ Z .

(41)

I

Remark 2. If no additional constraints are put on the periodic matrix Xk satisfying (37) and (38), the upper-bound on var{x } of (39) turns out to be non-conservative in the sense that the problem of minimizing the right-hand side of (39) with respect to Xk and G k , k = 1, . . . , N and subject to (37) and (38) leads to

4

N

1 var {x } = N

k=1



¡ ¢ tr Hk Qk HkT + Jk JkT

k  T C  k DTk

1 N ¡ ¢ tr G k + e N k=1

=



Gk  T ˜T  Kk Ck D˜ Tk

−I  D˜ k  0  ≥ 0, ∀ k ∈ Z .

C˜k Kk Kk + KkT

− Pk

(i)

 ¡ (i) ¢T  C  k ¡ (i) T Dk )

(i)

Ck

k

Dk (i)

¡ k − Xk 0

(i) Ak ,

½

k = 1, . . . , N, i = 1, . . . , u

minimize

(43)

N i=1

N

¡ ¢ tr G k , subject to (44) and (45)

5. AN EXAMPLE Consider the 3-periodic closed-loop uncertain system derived from Example 1 in de Souza and Trofino [2000], namely let system (1) with ¤ £ Ak = Ak +Bk Kk , Bk = Bk 0 , k = 1, 2, 3, · · · ¸ ¸ ¸ −3−a 2 −1−a 2 1−a 2 A1 = , A2 = , A3 = , −3 3 0.5 0 2.5 3 · ¸ · ¸ · ¸ 1 1 1 B1 = , B2 = , B3 = , 1 −0.5 1

(45)

K1 = [ 3 −2.2 ], K2 = [ 1 −2 ], K3 = [ −1.1 −2 ], Ck = [ 0 1 ], Dk = [ 0 1 ], k = 1, 2, 3,

(i) Ck ,

(i) Dk

Lk = [ 1 0 ], Mk = [ 0 0 ], k = 1, 2, 3, where a is an uncertain constant parameter satisfying |a | ≤ d . Theorems 1 and 2 have been applied to design strictly proper robust H¥ and guaranteed variance filters for the above system for different values of d . These designs will be denoted as (A).

ensures the robust stability the estimation error system (7) and the robust performance 1 N

= diag{ Inz , Yk , Inw } .

k=1

where the matrices and ¡ k are the same as in Theorem 1. Then the filter ( ¡ ¢ xˆk+1 = Ak Zk−1 xˆk + Bk yk (46) ¡ ¢ zˆk = Ck Zk−1 xˆk + Dk yk

var {e } <

k

where the optimization is carried out over the matrices Ak , Bk , (i) Ck , Dk , Sk , Zk , G k and Xk , i = 1, . . . , u , k = 1, . . . , N.

I (i) Bk ,

(48)

Remark 3. Note that Remark 1 also applies to the guaranteed variance filtering method of Theorem 2. Moreover, the filter of Theorem 1 that minimizes the upper-bound on the average steady-state variance of the estimation error can be obtained via the following convex optimization problem:



 0   ≥ 0,

 0  ≥ 0, k = 1, . . . , N I

Finally, by Lemma 3, (42) and (43) imply the robust stability of the error system (7) and the robust performance of (47). ¤

Theorem 2. Consider the uncertain system of (1)-(4). Suppose there exist N-periodic matrices Ak ∈ Rn×n , Bk ∈ Rn×ny , Ck ∈ Rnz ×n , Dk ∈ Rnz ×ny , Rk ∈ Rn×n , Sk ∈ Rn×n , Zk ∈ Rn×n , G k ∈ (i) Rnz ×nz and Xk ∈ R2n×2n , i = 1, . . . , u , satisfying the LMIs  (i) (i) (i)  −Xk+1 Ak Bk ½  ¡ (i) ¢T  k = 1, . . . , N, (i)   A (44) Xk − ¡ k 0  < 0,  k i = 1, . . . , u ¡ (i) T Bk ) 0 −I G



T

The guaranteed variance filter design method is presented in the next theorem.



0

Dk

Pre- and post-multiplying (43) by Y k and Y k , respectively, and considering (31), it follows that (48) is equivalent to (43).

I

0

¡ k − Xk

Y

The guaranteed variance filtering method is based on Lemma 3. Specifically, the filter is designed such that there exist an N-period matrix Kk and an uncertainty-dependent N-periodic matrix Pk > 0 as in (14) that ensure the feasibility of the following inequalities for all the matrices of system (1) in W :   −Pk+1 A˜ k Kk B˜ k  T ˜T  (42)  Kk Ak Pk − Kk − KkT 0  < 0, ∀ k ∈ Z , 0

Ck

for all (Ck , Dk ) ∈ W , where the matrices Ck , Dk and Xk are as in (25) and (26). Next, as in the proof of Theorem 1, consider the matrix Yk given in (28) and let the matrix:

where e > 0 is an arbitrarily small scalar associated with the gap between Qk of (34) and Xk of (37) and (38). ¤

B˜ Tk

G

To illustrate the advantages of using parameter-dependent Lyapunov functions, we have also considered the strictly proper robust H¥ and guaranteed variance filter designs based on parameter-independent Lyapunov functions (referred to as (B) ). Note that these designs are readily obtained from The(i) orems 1 and 2 by setting Xk = Xk , i = 1, . . . , u , k = 1, . . . , N,

¡ ¢ tr G i , ∀ (Ak , Bk ,Ck , Dk , Lk , Mk ) ∈ W k , k = 1, . . . , N.

(47)

Proof. As (42) and (44) are similar to (15) and (16), respectively, using the same arguments as in the proof of Theorem 1, it can be readily shown that (44) ensures that (42) holds with the matrices Kk , (Aˆ k , Bˆ k ) and Pk as in (27), (30) and (32).

Fig. 1 displays the achieved minimum upper-bound g on kHew k¥ versus d obtained with designs (A) and (B). First, observe that design (B) did not provide a solution for d ≥0.56367, whereas design (A) can solve the problem for d ≤ 0.99279. These maximum values of d also apply to the solvability of the guaranteed variance filtering problem. Notice that designs (A) and (B) achieve the same g for d = 0, i.e. when the system

In the sequel, it will be shown that (45) ensures the feasibility of (43) with Kk and Pk as above and (Cˆk , Dˆ k ) given in (30). To this end, first note that, by convexity, (45) is equivalent to

5

is uncertainty-free, whereas for all the other values of d design (A) gives better performance than (B), in particular for large parameter uncertainty. For instance, for |a | ≤ 0.45 the minimum g achieved by the design (B) is 19.61% larger than that of (A).

error over one period have been developed. The proposed filter designs build on parameter-dependent Lyapunov functions in terms of periodic matrices and are tailored via linear matrix inequalities. REFERENCES

10

S. Bittanti. Deterministic and stochastic linear periodic systems. In S. Bittanti (Ed.), Time Series and Linear Systems. Springer-Verlag, Berlin, pp. 141–182, 1986. S. Bittanti and P. Colaneri. An LMI characterization of the class of stabilizing controllers for periodic discrete-time systems. In Proc. 14th IFAC World Congress, Beijing, China, 1999. S. Bittanti and P. Colaneri. Period Systems: Filtering and Control. Springer-Verlag, London, 2009. S. Bittanti, P. Colaneri, and G. De Nicolao. The difference periodic Riccati equation for the periodic prediction problem. IEEE Trans. Autom. Control, 33:706–712, 1988. S. Bittanti, P. Colaneri, and G. De Nicolao. The periodic Riccati equation. In S. Bittanti, A. Laub and J.C. Willems (Eds.), The Riccati Equation. Springer-Verlag, Berlin, pp. 127–162, 1991. S. Bittanti and F.A. Cuzzola. An LMI approach to periodic discrete-time unbiased filtering. Systems & Control Letts., 42(16):21–35, 2001. P. Bolzern and P. Colaneri. The periodic Lyapunov equation. SIAM J. Matrix Anal. Appl., 9:499–512, 1988. P. Colaneri. Output stabilization via pole-placement of discretetime linear periodic systems. IEEE Trans. Autom. Control, 36:739–742, 1991. P. Colaneri and C.E. de Souza. The H¥ control problem for continuous-time linear periodic systems. In Proc. 2nd IFAC Workshop System Structure and Control, Prague, Czech Republic, pp. 292–296, 1992. G. De Nicolao. Cyclomonotonicity and stabilizability properties of solutions of the difference periodic Riccati equation. IEEE Trans. Autom. Control, 37:1405–1410, 1992. C.E. de Souza. Periodic strong solution for the optimal filtering problem of linear discrete-time periodic systems. IEEE Trans. Autom. Control, 36:333–338, 1991. C.E. de Souza, K.A. Barbosa, and A. Trofino. Robust H¥ filtering for discrete-time linear systems with uncertain timevarying parameters. IEEE Trans. Signal Processing, 54(6): 2110–2118, 2006. C.E. de Souza and A. Trofino. An LMI approach to stabilization of linear periodic discrete-time systems. Int. J. Control, 73(8):696–703, 2000. C. Farges, D. Poucelle, D. Arzelier, and J. Daafouz. Robust H2 performance analysis and synthesis of linear polytopic discrete-time periodic systems via LMIs. Systems & Control Letts., 56:159–166, 2007. O.M. Grasselli, S. Longhi, A. Tornamb`e, and P. Valigi. Robust output regulation and tracking for linear periodic systems under structured uncertainties. Automatica, 32:1015-1019, 1996. R. Wi´sniewskiand and J. Stoustrup. Generalized H2 control synthesis for periodic systems. In Proc. 2001 American Control Conf., Arlington, VA, pp. 2600–2605, 2001. L. Xie and C.E. de Souza. H¥ state estimation for linear periodic systems. IEEE Trans. Autom. Control, 38:1704– 1707, 1993. L. Xie, C.E. de Souza, and M.D. Fragoso. H¥ filtering for linear periodic systems with parameter uncertainty. Systems & Control Letts., 17:343–350, 1991.

9 8 7

γ

6 5

(B)

4

(A) 3 2 1 0

0

0.1

0.2

0.3

0.4

0.5

δ

0.6

0.7

0.8

0.9

1

Fig. 1. Minimum g for the H¥ designs (A) and (B). The minimum upper-bound on var {e} of Theorem 2 (denoted by k ) obtained with the designs (A) and (B) are shown in Fig. 2. Similarly as for the H¥ filtering, design (A) achieves superior performance as compared to (B), in particular for large values of d . Notice that for d =0.45 the minimum k obtained with the design (B) is 71.03% larger than the value achieved with (A). 10 9 8 7

κ

6 5 4

(B)

(A)

3 2 1 0

0

0.1

0.2

0.3

0.4

0.5

δ

0.6

0.7

0.8

0.9

1

Fig. 2. Minimum k for the guaranteed variance designs (A) and (B). 6. CONCLUSION This paper has investigated the design of robust linear periodic filters for polytopic uncertain linear discrete-time periodic systems. Filters with either a prescribed robust H¥ performance, or a guaranteed average steady-state variance of the estimation

6