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Guaranteed cost and H∞ filtering for discrete-time polytopic uncertain systems with time delay
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Journal of the Franklin Institute 342 (2005) 365–378 www.elsevier.com/locate/jfranklin
Guaranteed cost and H 1 filtering for discretetime polytopic uncertain systems with time delay Jong Hae Kim, Seong Joon Ahn, Seungjoon Ahn Division of Electronics, Information and Communication Engineering, Sunmoon University, Asan-si, Chungnam 336-708, Republic of Korea
Abstract The design methods of guaranteed cost filtering and H 1 filtering for discrete-time uncertain linear systems with time delay are investigated in this paper. The uncertain parameters are assumed to be unknown but belonging to known convex compact set of polytope type less conservative than norm bounded parameter uncertainty. The objective is to design stable guaranteed cost filter and H 1 filter guaranteeing asymptotic stability for filtering error dynamics and minimizing performance measures. The sufficient conditions for the existence of filters and filter design methods are established by linear matrix inequality (LMI) approach, which can be solved efficiently by so called interior point algorithm. The developed algorithms are illustrated by numerical simulation. r 2005 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Guaranteed cost filtering; H 1 filtering; Discrete-time case; Time delay systems; Linear matrix inequality; Convex bounded uncertainty
1. Introduction The problem of filtering (or estimation) has been studied for more than three decades in various branches of science and engineering [1]. Especially, since the filtering design that can handle model uncertainties has been one of the interesting Tel.: +82 41 530 2352; fax: +82 41 530 2910.
E-mail address: [email protected] (J.H. Kim). 0016-0032/$30.00 r 2005 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2005.01.002
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problems, much effort has been devoted to the development of filtering design method. Since the Kalman filtering theory has been introduced, many results have been developed in continuous-time case. Moreover, the problem of robust H 1 filtering for discrete-time uncertain linear systems has been addressed. Theodor et al. [2] considered a game theory approach to robust discrete-time H 1 estimation by algebraic Riccati equation (ARE) approach. Xie et al. [3] treated H 1 estimation for discrete-time linear uncertain systems. Li and Fu [4] considered the robust H 1 filtering problem for a general class of norm bounded uncertain systems described by the so-called integral quadratic constraints based on the ideas proposed by Gahinet [5] and Iwasaki et al. [6] in continuous-time case. Their main contribution was to show that the robust H 1 filtering problem could be solved using LMI techniques. However, it is not easy to find out a robust H 1 filter satisfying three LMIs which are mutually coupled. Recently, Geromel et al. [7,8] dealt with H 2 and H 1 robust filtering for convex bounded uncertain systems and for discrete-time linear systems by LMI techniques. Palhares and Peres [9] considered the problem of robust H 1 filtering design with pole constraints for discrete-time systems using LMI approach. However, the existing methods of robust H 1 filter design do not directly apply to the case where the system exhibits time delay in state variables. Since the delayed state is very often causes for instability and poor performance of systems, the filtering problem for discrete-time dynamic systems with time delay has been attracting attention over the fast few years. Wang and Mau [10] dealt with stabilization and state observer design method for perturbed discrete-time delay large scale systems. Robust Kalman filter synthesis for uncertain multiple time delay stochastic systems was developed by Hsiao and Pan [11]. Palhares et al. [12] addressed the problem of robust H 1 filtering for discrete-time convex bounded uncertain linear systems with multiple state delays. However, they [12] ignored the initial state value and time delay. It is very important to address filtering design problems for uncertain discrete time linear systems with time delay. The performance of robust H 1 filter is varied according to the initial value. Also, they [12] just discussed on robust H 1 filtering problem with general H 1 performance measure. This paper considers two filter design algorithms including robust H 1 filter and robust guaranteed cost filter for discrete time uncertain systems with time delay. Therefore, another new design method of robust H 1 filtering for convex bounded uncertain systems with time delay is addressed by LMI technique with modified H 1 performance measure considering initial state value. Of course, this approach can be directly applied to multiple time delay systems. More recently, Guan et al. [13] proposed guaranteed cost control methods for discrete-time uncertain system with time delay using ARE approach. However, their approach is somewhat conservative because some variables are pre-selected in order to find guaranteed cost controller. Also, their upper bound of guaranteed cost function is not optimal. Therefore, Kim [14] proposed a new robust guaranteed cost controller design method for discrete-time parameter uncertain systems with time delays both state and control input in order to eliminate those problems. Unfortunately, there are no papers considering both time delay and parameter
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uncertainties to get the upper bound of guaranteed cost function in discrete-time filtering problem. Hence, the aim of this paper is to present a systematic method of deriving sufficient LMIs for robust H 1 filter and robust guaranteed cost filter based on Lyapunov functional approach and LMI techniques. In this paper, the guaranteed cost filter and the H 1 filter design methods for discrete-time uncertain linear systems with time delay are proposed by LMI approach, respectively. The uncertain parameters are assumed to belong to a given convex bounded polytope. The sufficient conditions for the existence of filters and filter design methods are proposed by LMI conditions in terms of all finding variables. The obtained filters not only guarantee the asymptotic stability of filtering error dynamics but also minimize the guaranteed cost function or modified H 1 performance measure. The notations are fairly standard. I and 0 stands for the the zero P identity and 2 1=2 matrices with proper dimensions, respectively. kxðkÞk2 is ð 1 kxðkÞk Þ ; where k¼0 k k denotes the Euclidean vector. The symbol n represents the submatrices that lie below the main diagonal. X 40 (or X o0) means positive (or negative) definite symmetric matrix.
2. Problem formulation Consider the following linear convex bounded uncertain system with time delay: xðk þ 1Þ ¼ AxðkÞ þ Ad xðk dÞ þ BwðkÞ, yðkÞ ¼ CxðkÞ þ DwðkÞ, xðkÞ ¼ f1 ðkÞ; dpkp0,
ð1Þ
n
r
where xðkÞ 2 R is the state vector, yðkÞ 2 R is the measurement output vector, wðkÞ 2 Rm is the noise signal vector, and f1 ðkÞ is an initial value function. Time delay d is positive integer number. Here, system matrices are assumed to be unknown but belonging to a known convex compact set of polytope type, i.e. ðA; Ad ; B; C; DÞ 2 C,
(2)
where ( C:¼ ðA; Ad ; B; C; DÞjðA; Ad ; B; C; DÞ ¼
k X i¼1
ti ðAi ; Adi ; Bi ; C i ; Di Þ;
k X
) ti ¼ 1
ð3Þ
i¼1
and LðÞ denotes the set of i; i ¼ 1; 2; . . . ; k vertices of the above convex polytope. The kind of convex bounded parameter uncertainty has been widely used. Note that in many practical cases, very often, only a few entries of the matrices of systems dynamics contain uncertain parameters. And we assume that system (1) is asymptotically stable. This assumption guarantees that the boundedness of the
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filtering error holds, since the asymptotic stability of the filtering error dynamics depends on the states of system (1). Our aim is to design a stable linear guaranteed cost and H 1 filter described by ^ þ 1Þ ¼ G xðkÞ ^ xðk þ KyðkÞ,
(4)
where, G and K are filter variables. If we take the error state vector as follows: ^ eðkÞ ¼ xðkÞ xðkÞ,
(5)
then the error dynamics is obtained eðk þ 1Þ ¼ GeðkÞ þ ðA KC GÞxðkÞ þ Ad xðk dÞ þ ðB KDÞwðkÞ, zðkÞ ¼ LeðkÞ
ð6Þ
by defining the error state output as zðkÞ ¼ LeðkÞ: Define the following augmented state vector: " # xðkÞ xf ðkÞ ¼ (7) eðkÞ such that the filtering error dynamics is given by xf ðk þ 1Þ ¼ Af xf ðkÞ þ Adf xf ðk dÞ þ Df wðkÞ, zðkÞ ¼ C f xf ðkÞ, xf ðkÞ ¼ ff ðkÞ ¼
"
f1 ðkÞ f2 ðkÞ
# ;
dpkp0,
ð8Þ
where f2 ðkÞ is an initial error value function and some notations are denoted by " # " # Ad 0 A 0 ; Adf ¼ Af ¼ , Ad 0 A KC G G " # B ð9Þ ; C f ¼ ½0 L . Df ¼ B KD Also, we introduce guaranteed cost function J1 ¼
1 X
zðkÞT zðkÞ
(10)
k¼0
and modified H 1 performance measure considering initial states values J2 ¼
1 X k¼0
T
zðkÞ zðkÞ g
2
1 X k¼0
T
T
wðkÞ wðkÞ þ xf ð0Þ R0 xf ð0Þ þ
1 X
! T
xf ðiÞ Rd xf ðiÞ
i¼d
(11) with positive definite symmetric matrices R0 and Rd : Here, R0 and Rd are weighting factors which can be selected according to the initial states value.
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3. Guaranteed cost filtering In this guaranteed cost filtering part, we assume that B ¼ 0 and D ¼ 0 in order to give an analytical upper bound of guaranteed cost function. Associated with guaranteed cost filter, we introduce the following filtering design objective: Determine filter variables G and K that achieve minimization of guaranteed cost in filtering error dynamics. Therefore, our aim is to develop the guaranteed cost filtering design method satisfying the above objective irrespective of uncertainties and time delay. In the following, we present LMI optimization problems to get the guaranteed cost filtering satisfying the prescribed aim by LMI techniques. Theorem 1. If there exist positive definite symmetric matrices (or scalar) P1 ; P2 ; S 1 ; S 3 ; a; Q, and matrices S 2 ; M 1 ; M 2 ; satisfying the following optimization problem minimize fa þ trðQÞg subject to 2 6 6 6 6 6 6 ðiÞ 6 6 6 6 6 4
P1
0
P1 Ai
0
P1 Adi
n
P2
P2 Ai M 2 C i M 1
M1
P2 Adi
n
n
P1 þ S1
S2
0 T
n
n
n
P2 þ S 3 þ L L
0
n
n
n
n
S 1
n
n
n
n
n
0
3
7 0 7 7 7 0 7 7 7 0 7 7 7 S 2 7 5 S 3
o0, ðiiÞ a þ f1 ð0ÞT P1 f1 ð0Þ þ f2 ð0ÞT P2 f2 ð0Þo0, ðiiiÞ Q þ N T1 S 1 N 1 þ N T2 S 2 N 1 þ N T1 S 2 N 2 þ N T2 S 3 N 2 o0
(12)
for all ðAi ; Adi ; C i Þ 2 LðCÞ; then (4) is a discrete-time guaranteed cost filter and J n ¼ a þ trðQÞ is an upper bound of guaranteed cost function. Here, some notations are defined as M 1 ¼ P2 G, M 2 ¼ P2 K, 1 X
" T
T
ff ðiÞff ðiÞ ¼ NN ¼
i¼d
N1
#
N2
½N T1 N T2 .
ð13Þ
Proof. First, the asymptotic stability of the filtering error dynamics is established. If we define a Lyapunov functional as follows: V ðxf ðkÞÞ ¼ xf ðkÞT Pxf ðkÞ þ
k1 X i¼kd
xf ðiÞT Sxf ðiÞ
(14)
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with positive definite symmetric matrices to be determined " # " # S1 S2 P1 0 P¼ ; S¼ . S T2 S 3 0 P2
(15)
Then, there exist positive scalars d1 and d2 such that d1 kxf ðkÞk2 pV ðxf ðkÞÞp d2 kxf ðkÞk2 : The difference of (14) is given by DV ¼ V ðxf ðk þ 1ÞÞ V ðxf ðkÞÞ ¼ xf ðk þ 1ÞT Pxf ðk þ 1Þ xf ðkÞT Pxf ðkÞ þ xf ðkÞT Sxf ðkÞ xf ðk dÞT Sxf ðk dÞ.
ð16Þ
Defining ZðkÞ ¼ ½xf ðkÞT xf ðk dÞT T ; (16) can be rewritten as follows: DV ¼ ZðkÞT X ZðkÞ,
(17)
where 2 X ¼4
ATf PAf P þ S
ATf PAdf
n
S þ ATdf PAdf
3 5.
(18)
Since the matrix X is affine with respect to the system matrices, it follows that the LMI (i) of (12) ensures that X o0 over the entire uncertainty domain C and against any time delay. Therefore, it results that the filtering error dynamics is asymptotically stable over C from Lyapunov–Krasovski stability result [15]. To establish guaranteed cost for the filtering error dynamics, first note that since filtering error dynamics (8) is asymptotically stable over the convex compact set of polytope type C; xf ðkÞ tends to zero as k ! 1: The LMI (i) in (12) implies that DV o zðkÞT zðkÞo0.
(19)
Therefore, we have "
xf ðkÞ xf ðk dÞ
#T 2 T Af PAf P þ S þ C Tf C f 4 n
3"
# xf ðkÞ 5 o0. xf ðk dÞ S þ ATdf PAdf ATf PAdf
(20) Then, the following matrix inequality which ensures asymptotic stability of the filtering error dynamics 2 3 ATf PAdf ATf PAf P þ S þ C Tf C f 4 5o0 (21) n S þ ATdf PAdf
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is changed to 2 P1 6 6 n 6 6 n 6 6 6 n 6 6 n 4 n
0
P1 A
0
P1 Ad
P2 n
P2 A P2 KC P2 G P1 þ S 1
P2 G S2
P2 Ad 0
n
n
P2 þ S 3 þ LT L
0
n
n
n
S1
n
n
n
n
371
0
3
7 7 7 7 7 7o0. 0 7 7 S 2 7 5 S 3 0 0
(22) ^ M 2 ¼ P2 K; it can be easily verified that Using some changes of variables, M 1 ¼ P2 A; the LMI (i) of (12) ensures that (22) over the entire convex bounded uncertainty domain C: Furthermore, by the summation of both sides of inequality (19) from 0 to T f 1; we obtain
TX f 1
zðkÞT zðkÞ4xf ðT f ÞT Pxf ðT f Þ xf ð0ÞT Pxf ð0Þ
k¼0
þ
TX f 1
xf ðiÞT Sxf ðiÞ
i¼T f d
1 X
xf ðiÞT Sxf ðiÞ.
ð23Þ
i¼d
As the filtering error dynamics is asymptotically stable, when T f ! 1 (or T f 1 ! 1) some terms go to zero. Hence we get 1 X
zðkÞT zðkÞpff ð0ÞT Pff ð0Þ þ
k¼0
1 X
ff ðiÞT Sff ðiÞ.
(24)
i¼d
This is an upper bound of guaranteed cost function. The first term of right-hand side in (24) is changed to a þ ff ð0ÞT Pff ð0Þo0: This is equivalent to (ii) in (12). The second term of right-hand side in (24) has the following relations: 1 X i¼d
ff ðiÞT Sff ðiÞ ¼
1 X
trðff ðiÞT Sff ðiÞÞ
i¼d
¼ trðNN T SÞ ¼ trðN T SNÞotrðQÞ. Therefore, Q þ N T SNo0 is equal to (iii) in (12).
ð25Þ
&
Hence, we can get the optimal discrete-time guaranteed cost filter satisfying the filtering design objective. Also, all solutions including filter variables (G ¼ P1 2 M 1; n K ¼ P1 2 M 2 ) and the upper bound of guaranteed cost function (J ¼ a þ trðQÞ) can be calculated simultaneously because the proposed sufficient conditions are LMIs regarding all finding variables.
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4. H 1 filtering In this section, we present the H 1 filter design method of polytopic uncertain system with time delay. The objective of H 1 filter is summarized as follows: Determine filter variables G and K that achieve minimization of H 1 norm in filtering error dynamics. In other words, the objective is to determine filter variables (G and K) within the upper bound, i.e. kzðkÞk2
sup wðkÞ2l 2 ½0;1Þa0
T
kwðkÞk2 þ ff ð0Þ R0 ff ð0Þ þ
P1
i¼d ff ðiÞ
T
Rd ff ðiÞ
og.
(26)
In the case of filtering problems for the polytopic systems with no time delay, the g bound is replaced by sup wðkÞ2l 2 ½0;1Þa0
kzðkÞk2 og. kwðkÞk2 þ ff ð0ÞT R0 ff ð0Þ
(27)
Theorem 2. Consider system (6). For all ðAi ; Adi ; Bi ; C i ; Di Þ 2 LðCÞ and given initial weighted positive definite symmetric matrices R0 and Rd if the following optimization problem: minimize r subject to 2
P1
0
P1 Ai
0
P1 Adi
0
n
P2
P2 Ai M 2 C i M 1
M1
P2 Adi
0
n
n
P1 þ S1
S2
0
0
6 6 6 6 6 6 6 ðiÞ 6 6 6 6 6 6 6 4
T
n
n
n
P2 þ S3 þ L L
0
0
n
n
n
n
S 1
S2
n
n
n
n
n
S3
n
n
n
n
n
n
P1 B i
3
7 P2 Bi M 2 D i 7 7 7 7 0 7 7 7 0 7 7 7 0 7 7 7 0 5 rI
o0, " ðiiÞ
P1 0
" ðiiiÞ
S1 S T2
# 0 orR0 , P2 # S2 orRd S3
(28)
has a solution positive definite symmetric matrices (or scalar) P1 ; P2 ; S1 ; S 3 ; r and matrices S 2 ; M 1 ; M 2 ; then (4) is a discrete-time H 1 filter and r is an upper bound of discrete-time H 1 norm in filtering error dynamics. Here, some notations are
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defined as r ¼ g2 , M 1 ¼ P2 G, M 2 ¼ P2 K.
ð29Þ
Proof. First, the asymptotic stability can be proved by the same way in Theorem 1 with wðkÞ ¼ 0 and the same Lyapunov functional (14). To establish the H 1 performance for the filtering error dynamics, notice that the following relation: 1 X
½zðkÞT zðkÞ g2 wðkÞT wðkÞ
k¼0
¼
1 X
½zðkÞT zðkÞ g2 wðkÞT wðkÞ þ DV þ V ðxf ð0ÞÞ V ðxf ð1ÞÞ.
ð30Þ
k¼0
Note that V ðxf ðkÞÞ ! 0 as k ! 1 because filtering error dynamics is asymptotically stable. Therefore, we can get the following from the modified H 1 performance measure (11), Lyapunov functional (14), relation (30), and initial condition J2 ¼
1 X
½zðkÞT zðkÞ g2 wðkÞT wðkÞ þ DV
k¼0
þ V ðxf ð0ÞÞ g
2
T
ff ð0Þ R0 ff ð0Þ þ
1 X
! T
ff ðiÞ Rd ff ðiÞ
i¼d
¼ J a þ Jb. ð31Þ P1 T Here, zðkÞ g2 wðkÞT wðkÞ þ DV and J b ¼ ff ð0ÞT ðP g2 R0 Þff ð0Þ þ P1 J a ¼ T k¼0 ½zðkÞ 2 i¼d ff ðiÞ ðS g Rd Þff ðiÞ: Hence, J a o0 and J b o0 imply J a þ J b o0: If we define zðkÞ ¼ ½xf ðkÞT xf ðk dÞT wðkÞT T ; J a can be rewritten as follows: Ja ¼
1 X
zðkÞT Y zðkÞ,
(32)
k¼0
where 2 6 Y ¼6 4
ATf PAf P þ S þ C Tf C f
ATf PAdf
ATf PDf
n
S þ ATdf PAdf
ATdf PDf
n
n
2
g I þ
DTf PDf
3 7 7. 5
(33)
^ M 2 ¼ P2 K; it can be easily verified that Using some changes of variables, M 1 ¼ P2 A; the LMI (i) of (28) ensures that Y o0 over the entire convex bounded uncertainty domain C: Also, the conditions P rR0 o0 and S rRd o0 in (28) imply J b o0: &
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In the case of H 1 filtering design, all solutions can be obtained at the same time. 1 The filter variables (G ¼ P1 2 M 1 ; K ¼ P2 M 2 ) and the upper bound of H 1 norm pffiffiffi (g ¼ r) can be calculated simultaneously because the proposed sufficient conditions are LMIs in terms of all finding variables. Remark 1. The proposed algorithm can be directly applied to polytopic uncertain systems with no time delay. Also, guaranteed cost and H 1 filtering problems of the polytopic uncertain systems with multiple time delays can be solvable by slight modifications from the proposed methods.
5. Example In order to check the validities of the proposed filtering design algorithms, an example is given. Consider an uncertain linear discrete-time system with time delay " # " # " # 0:1 0 0:1 0 0 xðk þ 1Þ ¼ xðkÞ þ xðk dÞ þ wðkÞ, 0:1 0:5 þ D1 0:1 0:3 1 yðkÞ ¼ ½1 þ D2 0 xðkÞ þ wðkÞ, zðkÞ ¼ ½1 1 eðkÞ, ff ðkÞ ¼ ½ekþ1 0 0:1 1 T , d ¼ 2.
ð34Þ
Here, we treat 0:1pD1 p0:1 and 0pD2 p1 yielding an uncertain systems of k ¼ 4 vertices. In the case of guaranteed cost filtering problem (B ¼ 0 and D ¼ 0), all solutions are obtained simultaneously as follows: " # " # 0:2348 0:7156 1:3578 0:7590 P1 ¼ ; P2 ¼ , 0:7156 2:9904 0:7590 2:5710 " # " # 0:1619 0:4265 0:0015 0:0002 ; S2 ¼ , S1 ¼ 0:4265 1:3007 0:0006 0:0050 " # " # 0:0946 0:0077 0:1169 0:1556 S3 ¼ ; M1 ¼ , 0:0077 0:0062 0:2595 1:6019 " # 0:7674 3 ; a ¼ 1:7494, M 2 ¼ 10 0:2157 3 2 0:0011 0 0 0 7 6 6 0 0:5885 0:0008 0 7 7 6 ð35Þ Q¼6 7. 7 6 0 0:0008 0:0013 0 5 4 0 0 0 0:0011
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Therefore, the robust guaranteed cost filter and the upper bound of guaranteed cost are " ^ þ 1Þ ¼ xðk
0:1707
0:2799
0:1513
0:7057
#
" ^ xðkÞ þ 10
3
0:7331 0:3003
# yðkÞ,
J n ¼ 2:3414.
ð36Þ
Moreover, the obtained filter guarantees not only the asymptotic stability of filtering error dynamics but also minimization of guaranteed cost for all admissible uncertainties and time delay. The trajectories of error state vector and error state output are shown in Fig. 1. Therefore, the guaranteed cost filter guarantees the asymptotic stability of filtering error dynamics in (a) and (b) of Fig. 1 because the error states converge to zero as k ! 1: Also, the guaranteed cost filter ensures the minimization of guaranteed cost function. From (c) of Fig. 1, J can be calculated as J ¼ 1:5543ðoJ n Þ: Table 1 displays the actual upper bound of guaranteed cost function at the vertices of C: The effectiveness of the upper bound J n is apparent.
0.4 0.3 0.2 e2 (k)
e1 (k)
0.1 0 -0.1 -0.2 -0.3 -0.4 0
10 20 30 40 50 60 70 80 90 100
0
10 20 30 40 50 60 70 80 90 100
(b)
k
k
1.2 1 0.8 0.6 z (k)
(a)
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
0.4 0.2 0 -0.2 -0.4 -0.6
(c)
0
10 20 30 40 50 60 70 80 90 100 k
Fig. 1. The trajectories of error state vector and error state output: (a) e1 ðkÞ; (b) e2 ðkÞ; (c) zðkÞ:
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Table 1 The actual upper bound of guaranteed cost function at the vertices of C Vertices of C
D1 ¼ 0:1 D2 ¼ 0
D1 ¼ 0:1 D2 ¼ 1
D1 ¼ 0:1 D2 ¼ 0
D1 ¼ 0:1 D2 ¼ 1
Upper bounds of guaranteed cost function
1.8621
1.8621
0.2594
0.2594
In the case of H 1 filtering method, if we set R0 2 3 2 10 0 0 0 10 0 6 0 10 0 7 6 0 7 6 6 0 10 R0 ¼ 6 7; Rd ¼ 6 4 0 4 0 0 10 0 5 0 0
0
0
10
0
0
and Rd as follows: 3 0 0 0 0 7 7 7 10 0 5 0
(37)
10
then, all solutions satisfying Theorem are obtained as follows: " # " # 208:7932 0:4059 200:0449 9:9198 P1 ¼ ; P2 ¼ , 0:4059 3:7533 9:9198 1:9909 " # " # 40:8781 0:0250 0:2493 0:0184 ; S2 ¼ , S1 ¼ 0:0250 1:4071 0:0065 0:0015 " # " # 63:3103 0:0600 54:7488 5:4723 ; M1 ¼ , S3 ¼ 0:0600 0:0035 9:0779 0:9924 " # 23:6141 ; r ¼ 21:0175. M2 ¼ 5:5193
ð38Þ
Therefore, the robust H 1 filter and the optimal H 1 norm are " # " # 0:0632 0:0035 0:0258 ^ þ 1Þ ¼ ^ yðkÞ, xðk xðkÞ þ 4:2447 0:4810 2:9008 gn ¼ 4:5845.
ð39Þ
Moreover, the obtained H 1 filter guarantees not only the asymptotic stability of filtering error dynamics but also the H 1 bound within an obtained g: Similarly, if we set noise signal as (a) in Fig. 2, the trajectories of error state vector and error state output are shown in (b–d) of Fig. 2. Therefore, the H 1 filter guarantees the asymptotic stability of filtering error dynamics in (b) and (c) of Fig. 2 against time delay and system matrices variations because the error states converge to zero as k ! 1: Also, the H 1 filter ensures the g bound. From the signal relationship (26) among noise signal (a) in Fig. 2, error state signal (d) in Fig. 2, initial condition, g can be calculated as 0.0852 which is less than the optimized gn ð¼ 4:5845Þ: On the other hand, Table 2 shows the actual H 1 norm of the transfer function from the noise
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0.6
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e1 (k)
w (k)
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0.4
377
-0.2
0.2
-0.4
0
-0.6 -0.8
-0.2 0
10 20 30 40 50 60 70 80 90 100 k
(a)
0
10 20 30 40 50 60 70 80 90 100 k
0
10 20 30 40 50 60 70 80 90 100 k
(b) 1.5
1
1 0.5
z (k)
e2 (k)
0.5 0 -0.5
0 -0.5
-1
-1
-1.5
-1.5
-2
-2 0
(c)
10 20 30 40 50 60 70 80 90 100 k
(d)
Fig. 2. The trajectories of noise signal, error state vector, and error state output: (a) wðkÞ; (b) e1 ðkÞ; (c) e2 ðkÞ; (d) zðkÞ:
Table 2 Actual H 1 norm at the vertices of C Vertices of C
D1 ¼ 0:1 D2 ¼ 0
D1 ¼ 0:1 D2 ¼ 1
D1 ¼ 0:1 D2 ¼ 0
D1 ¼ 0:1 D2 ¼ 1
H 1 norms
4.2915
4.3010
1.1129
1.2052
signal input to error state output at the vertices of C: The effectiveness of the gn guaranteed bound is apparent.
6. Conclusions This paper proposed guaranteed cost filter and H 1 filter design methods for convex polytopic uncertain systems with time delay in discrete-time case. From the guaranteed cost function and modified H 1 performance measure considering initial
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state value, the sufficient condition and design methods of such filters were developed in terms of LMI, which could be efficiently solved with global convergence assured. The guaranteed cost filter and H 1 filter assured asymptotic stability of filtering error dynamics irrespective of uncertain parameters and time delay. Moreover, guaranteed cost filter minimized the upper bound of guaranteed cost function and H 1 filter minimized g bound. The validity of presented filter design algorithms was checked through numerical example. It is well-known that delaydependent approach is less conservative than delay-independent one. Therefore, delay-dependent approach of H 1 filter and guaranteed cost filter design methods is a future research problem. And, another problem of the future researches is to apply the presented filter design algorithms to the real application systems. References [1] M.S. Mahmoud, L. Xie, Y.C. Soh, Robust Kalman filtering for discrete state-delay systems, IEE Proc. Control Theory Appl. 147 (2000) 613–618. [2] Y. Theodor, U. Shaked, C.E. de Souza, A game theory approach to robust discrete-time H 1 estimation, IEEE Trans. Signal Processing 42 (1994) 1486–1495. [3] L. Xie, C.E. de souza, M. Fu, H 1 estimation for discrete-time linear uncertain systems, Int. J. Robust Nonlinear Control 1 (1991) 111–123. [4] H. Li, M. Fu, A linear matrix inequality approach to robust H 1 filtering, IEEE Trans. Signal Processing 45 (1997) 2338–2350. [5] P. Gahinet, Explicit controllers formulas for LMI-based H 1 synthesis, Automatica 32 (1996) 1007–1014. [6] T. Iwasaki, R.E. Skelton, All controllers for the general H 1 control problem: LMI existence conditions and state space formulas, Automatica 30 (1994) 1307–1317. [7] J.C. Geromel, M.C. de Oliveira, H 2 and H 1 robust filtering for convex bounded uncertain systems, in: Proceedings of the Conference on Decision and Control, Tampa, FL, USA, 1998, pp. 146–151. [8] J.C. Geromel, J. Bernussou, M.C. de Oliveira, H 2 and H 1 robust filtering for discrete-time linear systems, in: Proceedings of the Conference on Decision and Control, Tampa, FL, USA, 1998, pp. 632–637. [9] R.M. Palhares, P.L.D. Peres, Robust H 1 filtering design with pole constraints for discrete-time systems: an LMI approach, in: Proceedings of the American Control Conference, San Diego, CA, USA, 1999, pp. 4418–4422. [10] W.J. Wang, L.G. Mau, Stabilization and estimation for perturbed discrete time-delay large-scale systems, IEEE Trans. Automat. Control 42 (1997) 1277–1282. [11] F.H. Hsiao, S.T. Pan, Robust Kalman filter synthesis for uncertain multiple time-delay stochastic systems, J. Dynamic Systems—Trans. ASME 118 (1996) 8803–8808. [12] R.M. Palhares, C.E. de Souza, P.L.D. Peres, Robust H 1 filter design for uncertain discrete-time state-delayed systems: an LMI approach, in: Proceedings of the Conference on Decision and Control, Phoenix, Arizona, USA, 1999, pp. 2347–2352. [13] X. Guan, Z. Lin, G. Duan, Robust guaranteed cost control for discrete-time uncertain systems with time delay, IEE Proc. Control Theory Appl. 146 (1999) 598–602. [14] J.H. Kim, Discrete time guaranteed cost control of uncertain time delay systems, in: Proceedings of IASTED International Conference on Modelling, Identification, and Control, Innsbruck, Austria, 2001, pp. 19–22. [15] J. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations, Springer, New York, 1997.
Journal of the Franklin Institute 342 (2005) 365–378 www.elsevier.com/locate/jfranklin
Guaranteed cost and H 1 filtering for discretetime polytopic uncertain systems with time delay Jong Hae Kim, Seong Joon Ahn, Seungjoon Ahn Division of Electronics, Information and Communication Engineering, Sunmoon University, Asan-si, Chungnam 336-708, Republic of Korea
Abstract The design methods of guaranteed cost filtering and H 1 filtering for discrete-time uncertain linear systems with time delay are investigated in this paper. The uncertain parameters are assumed to be unknown but belonging to known convex compact set of polytope type less conservative than norm bounded parameter uncertainty. The objective is to design stable guaranteed cost filter and H 1 filter guaranteeing asymptotic stability for filtering error dynamics and minimizing performance measures. The sufficient conditions for the existence of filters and filter design methods are established by linear matrix inequality (LMI) approach, which can be solved efficiently by so called interior point algorithm. The developed algorithms are illustrated by numerical simulation. r 2005 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Guaranteed cost filtering; H 1 filtering; Discrete-time case; Time delay systems; Linear matrix inequality; Convex bounded uncertainty
1. Introduction The problem of filtering (or estimation) has been studied for more than three decades in various branches of science and engineering [1]. Especially, since the filtering design that can handle model uncertainties has been one of the interesting Tel.: +82 41 530 2352; fax: +82 41 530 2910.
E-mail address: [email protected] (J.H. Kim). 0016-0032/$30.00 r 2005 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2005.01.002
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problems, much effort has been devoted to the development of filtering design method. Since the Kalman filtering theory has been introduced, many results have been developed in continuous-time case. Moreover, the problem of robust H 1 filtering for discrete-time uncertain linear systems has been addressed. Theodor et al. [2] considered a game theory approach to robust discrete-time H 1 estimation by algebraic Riccati equation (ARE) approach. Xie et al. [3] treated H 1 estimation for discrete-time linear uncertain systems. Li and Fu [4] considered the robust H 1 filtering problem for a general class of norm bounded uncertain systems described by the so-called integral quadratic constraints based on the ideas proposed by Gahinet [5] and Iwasaki et al. [6] in continuous-time case. Their main contribution was to show that the robust H 1 filtering problem could be solved using LMI techniques. However, it is not easy to find out a robust H 1 filter satisfying three LMIs which are mutually coupled. Recently, Geromel et al. [7,8] dealt with H 2 and H 1 robust filtering for convex bounded uncertain systems and for discrete-time linear systems by LMI techniques. Palhares and Peres [9] considered the problem of robust H 1 filtering design with pole constraints for discrete-time systems using LMI approach. However, the existing methods of robust H 1 filter design do not directly apply to the case where the system exhibits time delay in state variables. Since the delayed state is very often causes for instability and poor performance of systems, the filtering problem for discrete-time dynamic systems with time delay has been attracting attention over the fast few years. Wang and Mau [10] dealt with stabilization and state observer design method for perturbed discrete-time delay large scale systems. Robust Kalman filter synthesis for uncertain multiple time delay stochastic systems was developed by Hsiao and Pan [11]. Palhares et al. [12] addressed the problem of robust H 1 filtering for discrete-time convex bounded uncertain linear systems with multiple state delays. However, they [12] ignored the initial state value and time delay. It is very important to address filtering design problems for uncertain discrete time linear systems with time delay. The performance of robust H 1 filter is varied according to the initial value. Also, they [12] just discussed on robust H 1 filtering problem with general H 1 performance measure. This paper considers two filter design algorithms including robust H 1 filter and robust guaranteed cost filter for discrete time uncertain systems with time delay. Therefore, another new design method of robust H 1 filtering for convex bounded uncertain systems with time delay is addressed by LMI technique with modified H 1 performance measure considering initial state value. Of course, this approach can be directly applied to multiple time delay systems. More recently, Guan et al. [13] proposed guaranteed cost control methods for discrete-time uncertain system with time delay using ARE approach. However, their approach is somewhat conservative because some variables are pre-selected in order to find guaranteed cost controller. Also, their upper bound of guaranteed cost function is not optimal. Therefore, Kim [14] proposed a new robust guaranteed cost controller design method for discrete-time parameter uncertain systems with time delays both state and control input in order to eliminate those problems. Unfortunately, there are no papers considering both time delay and parameter
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uncertainties to get the upper bound of guaranteed cost function in discrete-time filtering problem. Hence, the aim of this paper is to present a systematic method of deriving sufficient LMIs for robust H 1 filter and robust guaranteed cost filter based on Lyapunov functional approach and LMI techniques. In this paper, the guaranteed cost filter and the H 1 filter design methods for discrete-time uncertain linear systems with time delay are proposed by LMI approach, respectively. The uncertain parameters are assumed to belong to a given convex bounded polytope. The sufficient conditions for the existence of filters and filter design methods are proposed by LMI conditions in terms of all finding variables. The obtained filters not only guarantee the asymptotic stability of filtering error dynamics but also minimize the guaranteed cost function or modified H 1 performance measure. The notations are fairly standard. I and 0 stands for the the zero P identity and 2 1=2 matrices with proper dimensions, respectively. kxðkÞk2 is ð 1 kxðkÞk Þ ; where k¼0 k k denotes the Euclidean vector. The symbol n represents the submatrices that lie below the main diagonal. X 40 (or X o0) means positive (or negative) definite symmetric matrix.
2. Problem formulation Consider the following linear convex bounded uncertain system with time delay: xðk þ 1Þ ¼ AxðkÞ þ Ad xðk dÞ þ BwðkÞ, yðkÞ ¼ CxðkÞ þ DwðkÞ, xðkÞ ¼ f1 ðkÞ; dpkp0,
ð1Þ
n
r
where xðkÞ 2 R is the state vector, yðkÞ 2 R is the measurement output vector, wðkÞ 2 Rm is the noise signal vector, and f1 ðkÞ is an initial value function. Time delay d is positive integer number. Here, system matrices are assumed to be unknown but belonging to a known convex compact set of polytope type, i.e. ðA; Ad ; B; C; DÞ 2 C,
(2)
where ( C:¼ ðA; Ad ; B; C; DÞjðA; Ad ; B; C; DÞ ¼
k X i¼1
ti ðAi ; Adi ; Bi ; C i ; Di Þ;
k X
) ti ¼ 1
ð3Þ
i¼1
and LðÞ denotes the set of i; i ¼ 1; 2; . . . ; k vertices of the above convex polytope. The kind of convex bounded parameter uncertainty has been widely used. Note that in many practical cases, very often, only a few entries of the matrices of systems dynamics contain uncertain parameters. And we assume that system (1) is asymptotically stable. This assumption guarantees that the boundedness of the
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filtering error holds, since the asymptotic stability of the filtering error dynamics depends on the states of system (1). Our aim is to design a stable linear guaranteed cost and H 1 filter described by ^ þ 1Þ ¼ G xðkÞ ^ xðk þ KyðkÞ,
(4)
where, G and K are filter variables. If we take the error state vector as follows: ^ eðkÞ ¼ xðkÞ xðkÞ,
(5)
then the error dynamics is obtained eðk þ 1Þ ¼ GeðkÞ þ ðA KC GÞxðkÞ þ Ad xðk dÞ þ ðB KDÞwðkÞ, zðkÞ ¼ LeðkÞ
ð6Þ
by defining the error state output as zðkÞ ¼ LeðkÞ: Define the following augmented state vector: " # xðkÞ xf ðkÞ ¼ (7) eðkÞ such that the filtering error dynamics is given by xf ðk þ 1Þ ¼ Af xf ðkÞ þ Adf xf ðk dÞ þ Df wðkÞ, zðkÞ ¼ C f xf ðkÞ, xf ðkÞ ¼ ff ðkÞ ¼
"
f1 ðkÞ f2 ðkÞ
# ;
dpkp0,
ð8Þ
where f2 ðkÞ is an initial error value function and some notations are denoted by " # " # Ad 0 A 0 ; Adf ¼ Af ¼ , Ad 0 A KC G G " # B ð9Þ ; C f ¼ ½0 L . Df ¼ B KD Also, we introduce guaranteed cost function J1 ¼
1 X
zðkÞT zðkÞ
(10)
k¼0
and modified H 1 performance measure considering initial states values J2 ¼
1 X k¼0
T
zðkÞ zðkÞ g
2
1 X k¼0
T
T
wðkÞ wðkÞ þ xf ð0Þ R0 xf ð0Þ þ
1 X
! T
xf ðiÞ Rd xf ðiÞ
i¼d
(11) with positive definite symmetric matrices R0 and Rd : Here, R0 and Rd are weighting factors which can be selected according to the initial states value.
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3. Guaranteed cost filtering In this guaranteed cost filtering part, we assume that B ¼ 0 and D ¼ 0 in order to give an analytical upper bound of guaranteed cost function. Associated with guaranteed cost filter, we introduce the following filtering design objective: Determine filter variables G and K that achieve minimization of guaranteed cost in filtering error dynamics. Therefore, our aim is to develop the guaranteed cost filtering design method satisfying the above objective irrespective of uncertainties and time delay. In the following, we present LMI optimization problems to get the guaranteed cost filtering satisfying the prescribed aim by LMI techniques. Theorem 1. If there exist positive definite symmetric matrices (or scalar) P1 ; P2 ; S 1 ; S 3 ; a; Q, and matrices S 2 ; M 1 ; M 2 ; satisfying the following optimization problem minimize fa þ trðQÞg subject to 2 6 6 6 6 6 6 ðiÞ 6 6 6 6 6 4
P1
0
P1 Ai
0
P1 Adi
n
P2
P2 Ai M 2 C i M 1
M1
P2 Adi
n
n
P1 þ S1
S2
0 T
n
n
n
P2 þ S 3 þ L L
0
n
n
n
n
S 1
n
n
n
n
n
0
3
7 0 7 7 7 0 7 7 7 0 7 7 7 S 2 7 5 S 3
o0, ðiiÞ a þ f1 ð0ÞT P1 f1 ð0Þ þ f2 ð0ÞT P2 f2 ð0Þo0, ðiiiÞ Q þ N T1 S 1 N 1 þ N T2 S 2 N 1 þ N T1 S 2 N 2 þ N T2 S 3 N 2 o0
(12)
for all ðAi ; Adi ; C i Þ 2 LðCÞ; then (4) is a discrete-time guaranteed cost filter and J n ¼ a þ trðQÞ is an upper bound of guaranteed cost function. Here, some notations are defined as M 1 ¼ P2 G, M 2 ¼ P2 K, 1 X
" T
T
ff ðiÞff ðiÞ ¼ NN ¼
i¼d
N1
#
N2
½N T1 N T2 .
ð13Þ
Proof. First, the asymptotic stability of the filtering error dynamics is established. If we define a Lyapunov functional as follows: V ðxf ðkÞÞ ¼ xf ðkÞT Pxf ðkÞ þ
k1 X i¼kd
xf ðiÞT Sxf ðiÞ
(14)
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with positive definite symmetric matrices to be determined " # " # S1 S2 P1 0 P¼ ; S¼ . S T2 S 3 0 P2
(15)
Then, there exist positive scalars d1 and d2 such that d1 kxf ðkÞk2 pV ðxf ðkÞÞp d2 kxf ðkÞk2 : The difference of (14) is given by DV ¼ V ðxf ðk þ 1ÞÞ V ðxf ðkÞÞ ¼ xf ðk þ 1ÞT Pxf ðk þ 1Þ xf ðkÞT Pxf ðkÞ þ xf ðkÞT Sxf ðkÞ xf ðk dÞT Sxf ðk dÞ.
ð16Þ
Defining ZðkÞ ¼ ½xf ðkÞT xf ðk dÞT T ; (16) can be rewritten as follows: DV ¼ ZðkÞT X ZðkÞ,
(17)
where 2 X ¼4
ATf PAf P þ S
ATf PAdf
n
S þ ATdf PAdf
3 5.
(18)
Since the matrix X is affine with respect to the system matrices, it follows that the LMI (i) of (12) ensures that X o0 over the entire uncertainty domain C and against any time delay. Therefore, it results that the filtering error dynamics is asymptotically stable over C from Lyapunov–Krasovski stability result [15]. To establish guaranteed cost for the filtering error dynamics, first note that since filtering error dynamics (8) is asymptotically stable over the convex compact set of polytope type C; xf ðkÞ tends to zero as k ! 1: The LMI (i) in (12) implies that DV o zðkÞT zðkÞo0.
(19)
Therefore, we have "
xf ðkÞ xf ðk dÞ
#T 2 T Af PAf P þ S þ C Tf C f 4 n
3"
# xf ðkÞ 5 o0. xf ðk dÞ S þ ATdf PAdf ATf PAdf
(20) Then, the following matrix inequality which ensures asymptotic stability of the filtering error dynamics 2 3 ATf PAdf ATf PAf P þ S þ C Tf C f 4 5o0 (21) n S þ ATdf PAdf
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is changed to 2 P1 6 6 n 6 6 n 6 6 6 n 6 6 n 4 n
0
P1 A
0
P1 Ad
P2 n
P2 A P2 KC P2 G P1 þ S 1
P2 G S2
P2 Ad 0
n
n
P2 þ S 3 þ LT L
0
n
n
n
S1
n
n
n
n
371
0
3
7 7 7 7 7 7o0. 0 7 7 S 2 7 5 S 3 0 0
(22) ^ M 2 ¼ P2 K; it can be easily verified that Using some changes of variables, M 1 ¼ P2 A; the LMI (i) of (12) ensures that (22) over the entire convex bounded uncertainty domain C: Furthermore, by the summation of both sides of inequality (19) from 0 to T f 1; we obtain
TX f 1
zðkÞT zðkÞ4xf ðT f ÞT Pxf ðT f Þ xf ð0ÞT Pxf ð0Þ
k¼0
þ
TX f 1
xf ðiÞT Sxf ðiÞ
i¼T f d
1 X
xf ðiÞT Sxf ðiÞ.
ð23Þ
i¼d
As the filtering error dynamics is asymptotically stable, when T f ! 1 (or T f 1 ! 1) some terms go to zero. Hence we get 1 X
zðkÞT zðkÞpff ð0ÞT Pff ð0Þ þ
k¼0
1 X
ff ðiÞT Sff ðiÞ.
(24)
i¼d
This is an upper bound of guaranteed cost function. The first term of right-hand side in (24) is changed to a þ ff ð0ÞT Pff ð0Þo0: This is equivalent to (ii) in (12). The second term of right-hand side in (24) has the following relations: 1 X i¼d
ff ðiÞT Sff ðiÞ ¼
1 X
trðff ðiÞT Sff ðiÞÞ
i¼d
¼ trðNN T SÞ ¼ trðN T SNÞotrðQÞ. Therefore, Q þ N T SNo0 is equal to (iii) in (12).
ð25Þ
&
Hence, we can get the optimal discrete-time guaranteed cost filter satisfying the filtering design objective. Also, all solutions including filter variables (G ¼ P1 2 M 1; n K ¼ P1 2 M 2 ) and the upper bound of guaranteed cost function (J ¼ a þ trðQÞ) can be calculated simultaneously because the proposed sufficient conditions are LMIs regarding all finding variables.
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4. H 1 filtering In this section, we present the H 1 filter design method of polytopic uncertain system with time delay. The objective of H 1 filter is summarized as follows: Determine filter variables G and K that achieve minimization of H 1 norm in filtering error dynamics. In other words, the objective is to determine filter variables (G and K) within the upper bound, i.e. kzðkÞk2
sup wðkÞ2l 2 ½0;1Þa0
T
kwðkÞk2 þ ff ð0Þ R0 ff ð0Þ þ
P1
i¼d ff ðiÞ
T
Rd ff ðiÞ
og.
(26)
In the case of filtering problems for the polytopic systems with no time delay, the g bound is replaced by sup wðkÞ2l 2 ½0;1Þa0
kzðkÞk2 og. kwðkÞk2 þ ff ð0ÞT R0 ff ð0Þ
(27)
Theorem 2. Consider system (6). For all ðAi ; Adi ; Bi ; C i ; Di Þ 2 LðCÞ and given initial weighted positive definite symmetric matrices R0 and Rd if the following optimization problem: minimize r subject to 2
P1
0
P1 Ai
0
P1 Adi
0
n
P2
P2 Ai M 2 C i M 1
M1
P2 Adi
0
n
n
P1 þ S1
S2
0
0
6 6 6 6 6 6 6 ðiÞ 6 6 6 6 6 6 6 4
T
n
n
n
P2 þ S3 þ L L
0
0
n
n
n
n
S 1
S2
n
n
n
n
n
S3
n
n
n
n
n
n
P1 B i
3
7 P2 Bi M 2 D i 7 7 7 7 0 7 7 7 0 7 7 7 0 7 7 7 0 5 rI
o0, " ðiiÞ
P1 0
" ðiiiÞ
S1 S T2
# 0 orR0 , P2 # S2 orRd S3
(28)
has a solution positive definite symmetric matrices (or scalar) P1 ; P2 ; S1 ; S 3 ; r and matrices S 2 ; M 1 ; M 2 ; then (4) is a discrete-time H 1 filter and r is an upper bound of discrete-time H 1 norm in filtering error dynamics. Here, some notations are
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defined as r ¼ g2 , M 1 ¼ P2 G, M 2 ¼ P2 K.
ð29Þ
Proof. First, the asymptotic stability can be proved by the same way in Theorem 1 with wðkÞ ¼ 0 and the same Lyapunov functional (14). To establish the H 1 performance for the filtering error dynamics, notice that the following relation: 1 X
½zðkÞT zðkÞ g2 wðkÞT wðkÞ
k¼0
¼
1 X
½zðkÞT zðkÞ g2 wðkÞT wðkÞ þ DV þ V ðxf ð0ÞÞ V ðxf ð1ÞÞ.
ð30Þ
k¼0
Note that V ðxf ðkÞÞ ! 0 as k ! 1 because filtering error dynamics is asymptotically stable. Therefore, we can get the following from the modified H 1 performance measure (11), Lyapunov functional (14), relation (30), and initial condition J2 ¼
1 X
½zðkÞT zðkÞ g2 wðkÞT wðkÞ þ DV
k¼0
þ V ðxf ð0ÞÞ g
2
T
ff ð0Þ R0 ff ð0Þ þ
1 X
! T
ff ðiÞ Rd ff ðiÞ
i¼d
¼ J a þ Jb. ð31Þ P1 T Here, zðkÞ g2 wðkÞT wðkÞ þ DV and J b ¼ ff ð0ÞT ðP g2 R0 Þff ð0Þ þ P1 J a ¼ T k¼0 ½zðkÞ 2 i¼d ff ðiÞ ðS g Rd Þff ðiÞ: Hence, J a o0 and J b o0 imply J a þ J b o0: If we define zðkÞ ¼ ½xf ðkÞT xf ðk dÞT wðkÞT T ; J a can be rewritten as follows: Ja ¼
1 X
zðkÞT Y zðkÞ,
(32)
k¼0
where 2 6 Y ¼6 4
ATf PAf P þ S þ C Tf C f
ATf PAdf
ATf PDf
n
S þ ATdf PAdf
ATdf PDf
n
n
2
g I þ
DTf PDf
3 7 7. 5
(33)
^ M 2 ¼ P2 K; it can be easily verified that Using some changes of variables, M 1 ¼ P2 A; the LMI (i) of (28) ensures that Y o0 over the entire convex bounded uncertainty domain C: Also, the conditions P rR0 o0 and S rRd o0 in (28) imply J b o0: &
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In the case of H 1 filtering design, all solutions can be obtained at the same time. 1 The filter variables (G ¼ P1 2 M 1 ; K ¼ P2 M 2 ) and the upper bound of H 1 norm pffiffiffi (g ¼ r) can be calculated simultaneously because the proposed sufficient conditions are LMIs in terms of all finding variables. Remark 1. The proposed algorithm can be directly applied to polytopic uncertain systems with no time delay. Also, guaranteed cost and H 1 filtering problems of the polytopic uncertain systems with multiple time delays can be solvable by slight modifications from the proposed methods.
5. Example In order to check the validities of the proposed filtering design algorithms, an example is given. Consider an uncertain linear discrete-time system with time delay " # " # " # 0:1 0 0:1 0 0 xðk þ 1Þ ¼ xðkÞ þ xðk dÞ þ wðkÞ, 0:1 0:5 þ D1 0:1 0:3 1 yðkÞ ¼ ½1 þ D2 0 xðkÞ þ wðkÞ, zðkÞ ¼ ½1 1 eðkÞ, ff ðkÞ ¼ ½ekþ1 0 0:1 1 T , d ¼ 2.
ð34Þ
Here, we treat 0:1pD1 p0:1 and 0pD2 p1 yielding an uncertain systems of k ¼ 4 vertices. In the case of guaranteed cost filtering problem (B ¼ 0 and D ¼ 0), all solutions are obtained simultaneously as follows: " # " # 0:2348 0:7156 1:3578 0:7590 P1 ¼ ; P2 ¼ , 0:7156 2:9904 0:7590 2:5710 " # " # 0:1619 0:4265 0:0015 0:0002 ; S2 ¼ , S1 ¼ 0:4265 1:3007 0:0006 0:0050 " # " # 0:0946 0:0077 0:1169 0:1556 S3 ¼ ; M1 ¼ , 0:0077 0:0062 0:2595 1:6019 " # 0:7674 3 ; a ¼ 1:7494, M 2 ¼ 10 0:2157 3 2 0:0011 0 0 0 7 6 6 0 0:5885 0:0008 0 7 7 6 ð35Þ Q¼6 7. 7 6 0 0:0008 0:0013 0 5 4 0 0 0 0:0011
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Therefore, the robust guaranteed cost filter and the upper bound of guaranteed cost are " ^ þ 1Þ ¼ xðk
0:1707
0:2799
0:1513
0:7057
#
" ^ xðkÞ þ 10
3
0:7331 0:3003
# yðkÞ,
J n ¼ 2:3414.
ð36Þ
Moreover, the obtained filter guarantees not only the asymptotic stability of filtering error dynamics but also minimization of guaranteed cost for all admissible uncertainties and time delay. The trajectories of error state vector and error state output are shown in Fig. 1. Therefore, the guaranteed cost filter guarantees the asymptotic stability of filtering error dynamics in (a) and (b) of Fig. 1 because the error states converge to zero as k ! 1: Also, the guaranteed cost filter ensures the minimization of guaranteed cost function. From (c) of Fig. 1, J can be calculated as J ¼ 1:5543ðoJ n Þ: Table 1 displays the actual upper bound of guaranteed cost function at the vertices of C: The effectiveness of the upper bound J n is apparent.
0.4 0.3 0.2 e2 (k)
e1 (k)
0.1 0 -0.1 -0.2 -0.3 -0.4 0
10 20 30 40 50 60 70 80 90 100
0
10 20 30 40 50 60 70 80 90 100
(b)
k
k
1.2 1 0.8 0.6 z (k)
(a)
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
0.4 0.2 0 -0.2 -0.4 -0.6
(c)
0
10 20 30 40 50 60 70 80 90 100 k
Fig. 1. The trajectories of error state vector and error state output: (a) e1 ðkÞ; (b) e2 ðkÞ; (c) zðkÞ:
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Table 1 The actual upper bound of guaranteed cost function at the vertices of C Vertices of C
D1 ¼ 0:1 D2 ¼ 0
D1 ¼ 0:1 D2 ¼ 1
D1 ¼ 0:1 D2 ¼ 0
D1 ¼ 0:1 D2 ¼ 1
Upper bounds of guaranteed cost function
1.8621
1.8621
0.2594
0.2594
In the case of H 1 filtering method, if we set R0 2 3 2 10 0 0 0 10 0 6 0 10 0 7 6 0 7 6 6 0 10 R0 ¼ 6 7; Rd ¼ 6 4 0 4 0 0 10 0 5 0 0
0
0
10
0
0
and Rd as follows: 3 0 0 0 0 7 7 7 10 0 5 0
(37)
10
then, all solutions satisfying Theorem are obtained as follows: " # " # 208:7932 0:4059 200:0449 9:9198 P1 ¼ ; P2 ¼ , 0:4059 3:7533 9:9198 1:9909 " # " # 40:8781 0:0250 0:2493 0:0184 ; S2 ¼ , S1 ¼ 0:0250 1:4071 0:0065 0:0015 " # " # 63:3103 0:0600 54:7488 5:4723 ; M1 ¼ , S3 ¼ 0:0600 0:0035 9:0779 0:9924 " # 23:6141 ; r ¼ 21:0175. M2 ¼ 5:5193
ð38Þ
Therefore, the robust H 1 filter and the optimal H 1 norm are " # " # 0:0632 0:0035 0:0258 ^ þ 1Þ ¼ ^ yðkÞ, xðk xðkÞ þ 4:2447 0:4810 2:9008 gn ¼ 4:5845.
ð39Þ
Moreover, the obtained H 1 filter guarantees not only the asymptotic stability of filtering error dynamics but also the H 1 bound within an obtained g: Similarly, if we set noise signal as (a) in Fig. 2, the trajectories of error state vector and error state output are shown in (b–d) of Fig. 2. Therefore, the H 1 filter guarantees the asymptotic stability of filtering error dynamics in (b) and (c) of Fig. 2 against time delay and system matrices variations because the error states converge to zero as k ! 1: Also, the H 1 filter ensures the g bound. From the signal relationship (26) among noise signal (a) in Fig. 2, error state signal (d) in Fig. 2, initial condition, g can be calculated as 0.0852 which is less than the optimized gn ð¼ 4:5845Þ: On the other hand, Table 2 shows the actual H 1 norm of the transfer function from the noise
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0.6
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0
e1 (k)
w (k)
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0.4
377
-0.2
0.2
-0.4
0
-0.6 -0.8
-0.2 0
10 20 30 40 50 60 70 80 90 100 k
(a)
0
10 20 30 40 50 60 70 80 90 100 k
0
10 20 30 40 50 60 70 80 90 100 k
(b) 1.5
1
1 0.5
z (k)
e2 (k)
0.5 0 -0.5
0 -0.5
-1
-1
-1.5
-1.5
-2
-2 0
(c)
10 20 30 40 50 60 70 80 90 100 k
(d)
Fig. 2. The trajectories of noise signal, error state vector, and error state output: (a) wðkÞ; (b) e1 ðkÞ; (c) e2 ðkÞ; (d) zðkÞ:
Table 2 Actual H 1 norm at the vertices of C Vertices of C
D1 ¼ 0:1 D2 ¼ 0
D1 ¼ 0:1 D2 ¼ 1
D1 ¼ 0:1 D2 ¼ 0
D1 ¼ 0:1 D2 ¼ 1
H 1 norms
4.2915
4.3010
1.1129
1.2052
signal input to error state output at the vertices of C: The effectiveness of the gn guaranteed bound is apparent.
6. Conclusions This paper proposed guaranteed cost filter and H 1 filter design methods for convex polytopic uncertain systems with time delay in discrete-time case. From the guaranteed cost function and modified H 1 performance measure considering initial
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state value, the sufficient condition and design methods of such filters were developed in terms of LMI, which could be efficiently solved with global convergence assured. The guaranteed cost filter and H 1 filter assured asymptotic stability of filtering error dynamics irrespective of uncertain parameters and time delay. Moreover, guaranteed cost filter minimized the upper bound of guaranteed cost function and H 1 filter minimized g bound. The validity of presented filter design algorithms was checked through numerical example. It is well-known that delaydependent approach is less conservative than delay-independent one. Therefore, delay-dependent approach of H 1 filter and guaranteed cost filter design methods is a future research problem. And, another problem of the future researches is to apply the presented filter design algorithms to the real application systems. References [1] M.S. Mahmoud, L. Xie, Y.C. Soh, Robust Kalman filtering for discrete state-delay systems, IEE Proc. Control Theory Appl. 147 (2000) 613–618. [2] Y. Theodor, U. Shaked, C.E. de Souza, A game theory approach to robust discrete-time H 1 estimation, IEEE Trans. Signal Processing 42 (1994) 1486–1495. [3] L. Xie, C.E. de souza, M. Fu, H 1 estimation for discrete-time linear uncertain systems, Int. J. Robust Nonlinear Control 1 (1991) 111–123. [4] H. Li, M. Fu, A linear matrix inequality approach to robust H 1 filtering, IEEE Trans. Signal Processing 45 (1997) 2338–2350. [5] P. Gahinet, Explicit controllers formulas for LMI-based H 1 synthesis, Automatica 32 (1996) 1007–1014. [6] T. Iwasaki, R.E. Skelton, All controllers for the general H 1 control problem: LMI existence conditions and state space formulas, Automatica 30 (1994) 1307–1317. [7] J.C. Geromel, M.C. de Oliveira, H 2 and H 1 robust filtering for convex bounded uncertain systems, in: Proceedings of the Conference on Decision and Control, Tampa, FL, USA, 1998, pp. 146–151. [8] J.C. Geromel, J. Bernussou, M.C. de Oliveira, H 2 and H 1 robust filtering for discrete-time linear systems, in: Proceedings of the Conference on Decision and Control, Tampa, FL, USA, 1998, pp. 632–637. [9] R.M. Palhares, P.L.D. Peres, Robust H 1 filtering design with pole constraints for discrete-time systems: an LMI approach, in: Proceedings of the American Control Conference, San Diego, CA, USA, 1999, pp. 4418–4422. [10] W.J. Wang, L.G. Mau, Stabilization and estimation for perturbed discrete time-delay large-scale systems, IEEE Trans. Automat. Control 42 (1997) 1277–1282. [11] F.H. Hsiao, S.T. Pan, Robust Kalman filter synthesis for uncertain multiple time-delay stochastic systems, J. Dynamic Systems—Trans. ASME 118 (1996) 8803–8808. [12] R.M. Palhares, C.E. de Souza, P.L.D. Peres, Robust H 1 filter design for uncertain discrete-time state-delayed systems: an LMI approach, in: Proceedings of the Conference on Decision and Control, Phoenix, Arizona, USA, 1999, pp. 2347–2352. [13] X. Guan, Z. Lin, G. Duan, Robust guaranteed cost control for discrete-time uncertain systems with time delay, IEE Proc. Control Theory Appl. 146 (1999) 598–602. [14] J.H. Kim, Discrete time guaranteed cost control of uncertain time delay systems, in: Proceedings of IASTED International Conference on Modelling, Identification, and Control, Innsbruck, Austria, 2001, pp. 19–22. [15] J. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations, Springer, New York, 1997.