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H∞ filtering for discrete-time systems with randomly varying sensor delays
Automatica 44 (2008) 1918–1922 www.elsevier.com/locate/automatica
Technical communique
H∞ filtering for discrete-time systems with randomly varying sensor delaysI Shaosheng Zhou a,∗ , Gang Feng b a Institute of Information and Control, School of Automation, Hangzhou Dianzi University, Hangzhou 310018, PR China b Department of Mechanical Engineering and Engineering Management, City University of Hong Kong, Hong Kong
Received 13 August 2006; received in revised form 23 May 2007; accepted 19 October 2007 Available online 10 March 2008
Abstract This paper investigates an H∞ filtering problem for discrete-time systems with randomly varying sensor delays. The stochastic variable involved is a Bernoulli distributed white sequence appearing in measured outputs. This measurement mode can be used to characterize the effect of communication delays and/or data-loss in information transmissions across limited bandwidth communication channels over a wide area. H∞ filtering of this class of systems is used to design a filter using the measurements with random delays to ensure the mean-square stochastic stability of the filtering error system and to guarantee a prescribed H∞ filtering performance. A sufficient condition for the existence of such a filter is presented in terms of the feasibility of a linear matrix inequality (LMI). Finally, a numerical example is given to illustrate the effectiveness of the proposed approach. c 2008 Elsevier Ltd. All rights reserved.
Keywords: Communication network; Data-loss; H∞ filtering; Random delays; Stochastic stability
1. Introduction Estimation of dynamic systems has found many practical applications and has attracted a lot of attention during the last decades. H∞ filtering is introduced as an alternative to classical Kalman filtering when the statistical property of noise sources is unknown or unavailable (Nagpal & Khargonekar, 1991). H∞ filtering is concerned with the design of estimators which ensure a bound on the L2 -induced gain from disturbance signals to estimation errors. Over the past decades, various approaches, such as the interpolation approach, the Riccati equation-based approach, and the LMI-based approach have been developed to deal with the H∞ filtering problem in various settings such as deterministic systems with uncertainties and/or delays as well as various stochastic systems (Xie, Liu, Zhang, & Zhang, 2004). Recently, the study of the H∞ filtering problem for I This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Guoxiang Gu under the direction of Editor Andr´e Tits. This work was supported in part by the National Natural Science Foundation of PR China under Grant 60574080 and 60434020. ∗ Corresponding author. Tel.: +86 571 86878584; fax: +86 571 86878584. E-mail addresses: [email protected] (S. Zhou), [email protected] (G. Feng).
c 2008 Elsevier Ltd. All rights reserved. 0005-1098/$ - see front matter doi:10.1016/j.automatica.2007.10.026
systems with delays has gained growing interest. A delayindependent H∞ filtering result for discrete-time systems with multiple time delays has been given in Palhares, de Souza, and Peres (2001) while delay-dependent results for this problem have been reported more recently in Gao, Lam, Xie, and Wang (2004) and Gao and Wang (2005). It is noted that the time delays are assumed to be deterministic in the literature mentioned above. However, they may occur in a randomly varying way in many practical applications as pointed out in Wang, Ho, and Liu (2004). Recently, there has been some attention to the research of systems with randomly varying delays. A randomly varying delayed sensor mode was first introduced in Ray (1994). Since then, some results for randomly delayed systems have been reported in Wang et al. (2004), Yaz and Ray (1996) and Yang, Wang, Hung, and Gani (2006). A variance-constrained filtering approach was proposed for systems with random sensor delays in Wang et al. (2004). And more recently, the authors in Yang et al. (2006) investigated the H∞ control problem for this class of systems. In the meantime, the H∞ filtering and control problems of stochastic systems have also attracted a lot of attention over the past few years (Hinrichsen & Pritchard, 1998). In Xu and Chen (2003), an LMI-based filter design approach was proposed for impulsive stochastic systems, and
1919
S. Zhou, G. Feng / Automatica 44 (2008) 1918–1922
based on the projection lemma, the reduced-order H∞ filtering problem for a class of stochastic systems was investigated in Xu and Chen (2002). The authors of Berman and Shaked (2006) presented an H∞ control scheme for a class of discrete-time nonlinear stochastic systems more recently. Motivated by the works in Wang et al. (2004) and Yang et al. (2006), this paper focuses on the H∞ filtering problem for systems with randomly varying sensor delays. We are interested in designing filters such that for all randomly varying sensor delays, the filtering error system is exponentially mean-square stable and a prescribed H∞ filtering performance is achieved. This paper is organized as follows. Section 2 formulates the H∞ filtering problem. In Section 3, a novel H∞ filtering approach is proposed. A numerical example is given to demonstrate the effectiveness of the proposed method in Section 4, which is followed by conclusions in Section 5. Notations: Throughout this paper, Z + denotes the set of positive integers; R n denotes the n-dimensional Euclidean space; R m×n denotes the set of all m × n real matrices. A real symmetric matrix P > 0(≥ 0) denotes P being a positive definite (or positive semi-definite) matrix, and A > (≥)B means A − B > (≥)0. I denotes an identity matrix of appropriate dimension. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations. The superscript ‘τ ’ represents the transpose. For a matrix N , N −τ stands for the transpose of matrix N −1 . A star ‘∗’ is used as an ellipsis for a corresponding transposed block matrix that is induced by symmetry. The notation l2 [0, ∞) represents the space of square summable infinite vector sequences with the usual norm k · k2 . Suppose n¯ that the vectors qP vk ∈ R , a sequence v = {vk } ∈ l2 [0, ∞) if ∞ τ kvk2 = k=1 vk vk < ∞. Prob{.} stands for the occurrence probability of an event; E{.} denotes the expectation operator with respect to some probability measure. 2. Problem formulation Consider a discrete-time system Σ : xk+1 = Axk + Aω ωk z k = L xk + L ω ωk
(1) (2)
where xk ∈ R n is the state; ωk ∈ R p¯ is the deterministic disturbance signal in l2 [0, ∞); z k ∈ R q is the signal to be estimated; A, Aω , L and L ω are known constant matrices with compatible dimensions. The measurement with random delays is given by yk = C xk yck = (1 − θk )yk + θk yk−1
(3) (4)
where yk ∈ R p is the output, yck ∈ R p is the measured output, C is a known matrix, and the stochastic variable θk is a Bernoulli distributed white sequence taking the values of 0 and 1 with Prob{θk = 1} = E{θk } = ρ Prob{θk = 0} = E{1 − θk } = 1 − ρ where ρ ∈ [0, 1] and is a known constant.
(5) (6)
Remark 1. The system measurements with varying sensor delay modeled in (3) and (4) was first introduced in Ray (1994) and has been used to characterize the effect of communication delays and/or data-loss in information transmissions across limited bandwidth communication channels over a wide area such as navigating a vehicle based on the estimations from a sensor web of its current position and velocity (Sinopoli et al., 2004). The output yk produced at a time k is sent to the observer through a communication channel. If no packet-loss occurs, the measurement output yck takes value yk ; otherwise, the measurement output yck takes value yk−1 . When the probability of event packet-loss occurring is assumed as ρ, the measurement output yck in (4) thus takes the value yk with probability 1 − ρ, and the value yk−1 with probability ρ. For the delayed sensor mode (4), we assume that x−1 = 0, which implies from (3) that y−1 = 0. We consider the following filter for the estimation of z k : xˆk+1 = A f xˆk + B f yck (7) zˆ k = C f xˆk + D f yck where xˆk ∈ R n and zˆ k ∈ R q . A f , B f , C f and D f are matrices to be determined. Combining (1)–(4) and (7) the filtering error dynamics can be represented as Σ˜ : x¯k+1 = A(θk )x¯k + A1 (θk )H x¯k−1 + Aω (θk )ωk (8) z¯ k = L(θk )x¯k + L1 (θk )H x¯k−1 + Lω (θk )ωk where x¯k = xkτ
xˆkτ
τ
, z¯ k = z k − zˆ k , H = I, 0 A 0 Aω A(θk ) = , Aω = 0 )B C A (1 − θ k f f 0 A1 (θk ) = , L1 (θk ) = −θk D f C θk B f C L(θk ) = L − (1 − θk )D f C, −C f , Lω = L ω .
(9)
(10)
It is noted that the filtering error dynamics Σ˜ is a system with stochastic parameters since some of the parametric matrices in (10) are associated with the stochastic variable θk . We adopt the notion of stochastic stability in the mean-square sense from Yang et al. (2006) to formulate our filtering problem. Definition 1. The filtering error dynamics Σ˜ is said to be exponentially mean-square stable if with ωk ≡ 0, there exist constants α > 0 and τ ∈ (0, 1) such that E{kx¯k k2 } ≤ ατ k E{kx¯0 k2 },
for all x¯0 ∈ R 2n , k ∈ Z + .
The H∞ filtering problem addressed in this paper is to design a filter in the form of (7) such that for a given scalar γ and all nonzero ωk , the filtering error system Σ˜ is exponentially mean-square stable and under the zero initial condition, the filtering error z¯ k satisfies ∞ X k=0
E{k¯z k k2 } ≤ γ 2
∞ X
kωk k2 .
(11)
k=0
In such a case, the filtering error system is said to be exponentially mean-square stable with H∞ filtering performance γ .
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S. Zhou, G. Feng / Automatica 44 (2008) 1918–1922
Remark 2. In the performance criterion (11), the disturbance ωk is assumed to be a deterministic signal belonging to l2 [0, ∞). It is worth pointing out that this criterion can be easily modified to accommodate the case where the disturbance signals may be random ones.
For any nonzero ωk ∈ l2 [0, +∞) and zero initial condition, we have JN =
where ∗ denotes the corresponding transposed block matrix due to symmetry and the ρ-dependent matrices are defined as in (10) with θk replaced by ρ. Proof. We first prove the mean-square stability of the system Σ˜ with ωk ≡ 0. In this case, the dynamic equation in (8) becomes x¯k+1 = A(θk )x¯k + A1 (θk )H x¯k−1 .
(13)
Define a Lyapunov functional candidate as τ Vk = x¯kτ P x¯k + x¯k−1 H τ Q H x¯k−1 .
(14)
Let Fk be the minimal σ -algebra generated by {xˆi , 0 ≤ i ≤ k}. By (13) and some algebraic manipulations, we have τ E{Vk+1 |Fk } − Vk = x¯kτ xk−1 τ A (ρ)PA(ρ) + H τ Q H − P Aτ (ρ)PA1 (ρ) × ∗ Aτ1 (ρ)PA1 (ρ) − Q x¯k × . (15) xk−1 It can be checked by (12) and the Schur complement formula that τ A (ρ)PA(ρ) + H τ Q H − P Aτ (ρ)PA1 (ρ) < 0. τ ∗ A1 (ρ)PA1 (ρ) − Q (16) It then follows from the similar argument as in Yang et al. (2006) that system (13) is exponentially mean-square stable. In what follows, we shall show that the filtering error z¯ k satisfies (11). Let JN =
N −1 h X
E{k¯z k k2 } − γ 2 kωk k2
k=0
where N is a positive integer.
i
(17)
E{k¯z k k2 } − γ 2 kωk k2 + E{Vk+1 − Vk }
i
k=0
+ E V0 − E VN
At first, we establish a condition of mean-square stability and H∞ performance for the filtering error dynamics Σ˜ , which will be fundamental in the derivation of our H∞ filter design method. Lemma 3. Given a scalar γ > 0, the filtering error system Σ˜ is exponentially mean-square stable with a guaranteed H∞ filtering performance γ , if there exist matrices P and Q such that P 0 PA(ρ) PA1 (ρ) PAω ∗ I L(ρ) L1 (ρ) Lω ∗ ∗ P − Hτ QH 0 0 (12) >0 ∗ ∗ ∗ Q 0 ∗ ∗ ∗ ∗ γ 2I
N −1 h X
≤
N −1 X
E{ξkτ M(ρ)ξk }
(18)
k=0
where τ τ xk−1 , ωkτ τ A (ρ) M(ρ) = Aτ1 (ρ) P A(ρ) A1 (ρ) Aω Aτ ωτ L (ρ) + Lτ1 (ρ) L(ρ) L1 (ρ) Lω Lτ τω H QH − P 0 0 0 −Q 0 . + 0 0 −γ 2 I ξk = x¯kτ ,
(19)
(20)
It follows from (12) and by the Schur complement formula that M(ρ) < 0. This implies that for any N , J N < 0, which leads to that the filtering error z¯ k satisfies condition (11). Remark 4. Lemma 3 provides an H∞ filtering analysis result for systems with randomly varying sensor delays. This result can be readily extended to the case where the disturbance signals are random ones. 3. H∞ filter design Based on Lemma 3 we will give a sufficient condition for the existence of an H∞ filter in the form of (7) and present a method to construct the filter. Theorem 5. Consider the system of (1) and (2). Given a scalar γ > 0, there exists a filter in the form of (7) such that the filtering error system Σ˜ is exponentially mean-square stable with H∞ filtering performance γ , if there exist positive definite matrices X, Z , Q and matrices A¯ f , B¯ f , C¯ f , D¯ f such that
X
Z
0
X A + (1 − ρ) B¯ f C
∗
Z
0
ZA
∗
∗
I
L − (1 − ρ) D¯ f C
∗
∗
∗
X−Q
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
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S. Zhou, G. Feng / Automatica 44 (2008) 1918–1922
X A + (1 − ρ) B¯ f C + A¯ f
ρ B¯ f C
ZA
0
L − (1 − ρ) D¯ f C − C¯ f
−ρ D¯ f C
Z−Q
0
Z−Q
0
∗
Q
∗
∗
X Aω
It follows from (23) and (26) that
Z Aω Lω 0 > 0. 0 0
(21)
γ 2I
In this case, two nonsingular constant matrices M and N can always be obtained such that M N τ = I − X Z −1
X
I
0
X A + (1 − ρ)M B f C
∗
Y
0
A
∗
∗
I
L − (1 − ρ)D f C
∗
∗
∗
X−Q
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
X AY + (1 − ρ)M B f CY + M A f N τ
ρMB f C
AY
0
LY − (1 − ρ)D f CY − C f N τ
−ρ D f C
I − QY
0
Y − Y QY
0
∗
Q
∗
∗
(22)
and the parametric matrices of the H∞ filter in the form of (7) can be constructed by A f = M −1 A¯ f Z −1 N −τ B f = M −1 B¯ f (23) C f = C¯ f Z −1 N −τ D f = D¯ f . Proof. Suppose that inequality (21) holds. This inequality gives X Z >0 (24) Z Z which by the Schur complement formula gives X − Z > 0. From this, it is clear that there exist nonsingular matrices M and N such that (22) holds. Let X I I Y Π1 = , Π = , Y = Z −1 . (25) 2 Mτ 0 0 Nτ Pre- and post-multiplying the inequality of (21) by diag{I, Y, I, I, Y, I, I } lead to
X
I
0
X A + (1 − ρ) B¯ f C
∗
Y
0
A
∗
∗
I
L − (1 − ρ) D¯ f C
∗
∗
∗
X−Q
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
One can check via (10) and (25) that X I τ Π2 Π1 = I Y X A + (1 − ρ)M B f C Π1τ A(ρ)Π2 = A ρMBf C Π1τ A1 (ρ) = 0 X Aω Π1τ Aω = Aω
X Aω
Aω Lω 0 > 0. 0 0
(27)
γ2I
(28) Π12 AY
(29) (30) (31)
L(ρ)Π2 = [L − (1 − ρ)D f C LY − (1 − ρ)D f CY − C f N τ ] Q QY Π2τ H τ Q H Π2 = Y Q Y QY
(32) (33)
with Π12 = X AY + (1 − ρ)M B f CY + M A f N τ . Then, we have τ Π2 Π1 ∗ ∗ ∗ ∗
X AY + (1 − ρ) B¯ f CY + A¯ f Y
ρ B¯ f C
AY
0
LY − (1 − ρ) D¯ f CY − C¯ f Y
−ρ D¯ f C
I − QY
0
Y − Y QY
0
∗
Q
∗
∗
X Aω
Aω Lω 0 > 0. 0 0
γ2I
Π1τ A(ρ)Π2 L(ρ)Π2 Π2τ Π1 − Π2τ H τ Q H Π2 ∗ ∗
0 I ∗ ∗ ∗
Π1τ A1 (ρ) L1 (ρ) 0 Q ∗
Π1τ Aω Lω 0 > 0. (34) 0 γ2I
Pre-multiplying diag{Π2−τ , I, Π2−τ , I, I } and post-multiplying diag{Π2−1 , I, Π2−1 , I, I } to inequality (34) lead to (26)
Π1 Π −1 ∗2 ∗ ∗ ∗ > 0.
0 I ∗ ∗ ∗
Π2−τ Π1τ A(ρ) L(ρ) Π1 Π2−1 − H τ Q H ∗ ∗
Π2−τ Π1τ A1 (ρ) L1 (ρ) 0 Q ∗
Π2−τ Π1τ Aω Lω 0 0 2 γ I
(35)
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S. Zhou, G. Feng / Automatica 44 (2008) 1918–1922
Table 1 Optimal H∞ filtering performance ρ γ∗
1 0.0892
0.5 0.0835
0.3 0.0671
0.1 0.0367
0 9 × 10−6
−8.5349 Bf = −4.2744 D f = −0.7835.
5. Conclusions
Let Π1 Π2−1
−48.9874 98.9052 Af = , −24.7685 49.9976 C f = 15.5007 −31.3166 ,
(36)
= P.
It is easy to check via (25), (28) and (36) that P > 0. Substituting (36) into (35) leads to (12). The proof is thus completed by invoking Lemma 3. Remark 6. The main idea behind the derivation of Theorem 5 is the construction of the Lyapunov matrix P in (36), which is motivated by Palhares et al. (2001) and Zhou, Zhang, and Zheng (2006). It is noted that our method to construct Lyapunov matrix P is more general than that in Palhares et al. (2001). Remark 7. In view of Theorem 5, the H∞ filtering problem for systems with randomly varying sensor delays can be solved in terms of the feasibility of LMI in (21). Note that the inequality in (21) is not only linear with respect to matrix variables X, Z , Q, A¯ f , B¯ f , C¯ f , D¯ f but also linear with respect to scalar γ 2 . The H∞ filtering performance γ ∗ can thus be optimized by solving the following convex optimization problem: min
(X,Z ,Q, A¯ f , B¯ f ,C¯ f , D¯ f )
δ
subject to (21) with δ = γ 2 .
Then, the corresponding optimal H∞ filtering performance γ ∗ is given by γ ∗ = (min δ)1/2 . Remark 8. Following the similar arguments as in Palhares et al. (2001) and Zhou and Lam (2003), Theorem 5 can be readily extended to the cases in which the systems contain norm bounded parametric uncertainties or linear fractional parametric uncertainties. 4. A design example In this section, we present a numerical example to illustrate the theory developed in Section 3. Consider the system Σ with parametric matrices as follows: 0.2 0.05 0.1 A= , Aω = −0.02 0.3 −0.2 L = 0.5 −0.7 , C = 1 0.6 , L ω = 0.3. The objective is to design a filter in the form of (7) for this system such that the filtering error system Σ˜ is exponentially mean-square stable and an optimal H∞ filtering performance γ ∗ is achieved. By using the Matlab LMI Control Toolbox to solve the convex optimization problem as stated in Remark 7, we obtain the optimal H∞ filtering performance γ ∗ for different values of ρ, summarised in Table 1. It is shown that γ ∗ decreases as ρ decreases. In other words, a better H∞ filtering performance is achieved when more current and less delayed sensor information is used. In the case when ρ = 0.5, γ ∗ = 0.0835, M τ = 0.5 I2 , the parametric matrices of the desired filter are given as follows.
This paper considers the H∞ filter design problem for a class of discrete-time systems with randomly varying sensor delays. Attention is focused on the design of a filter such that the filtering error system is mean-square stable and a prescribed level of H∞ filtering performance is guaranteed. An LMI-based filter design approach has been developed for this class of systems. It has been shown that the H∞ filtering problem can be solved in terms of the feasibility of an LMI. A numerical example has been provided to illustrate the effectiveness of the proposed approach. References Berman, N., & Shaked, U. (2006). H∞ control for discrete-time nonlinear stochastic systems. IEEE Transactions on Automatic Control, 51(6), 1041–1046. Gao, H., Lam, J., Xie, L., & Wang, C. (2004). New approach to mixed H2 /H∞ filtering for polytopic discrete-time systems. IEEE Transactions on Signal Process, 52(6), 1631–1640. Gao, H., & Wang, C. (2005). A delay-dependent approach to robust H∞ filtering for uncertain discrete-time state-delayed systems. IEEE Transactions on Signal Process, 53(8), 3183–3192. Hinrichsen, D., & Pritchard, A. J. (1998). Stochastic H∞ . SIAM Journal on Control Optimization, 36(5), 1504–1538. Nagpal, K. M., & Khargonekar, P. P. (1991). Filtering and smoothing in an H∞ setting. IEEE Transactions on Automatic Control, 36((2), 152–166. Palhares, R. M., de Souza, C. E., & Peres, P. L. D. (2001). Robust H∞ filtering for uncertain discrete-time state-delayed systems. IEEE Transactions on Signal Process, 49(8), 1696–1703. Ray, A. (1994). Output feedback control under randomly varying delays. Journal of Guidance Control and Dynamics, 17(4), 701–711. Sinopoli, B., Schenato, L., Franceschetti, M., Poola, K., Jordan, M. I., & Sastry, S. S. (2004). Kalman filtering with intermittent observations. IEEE Transactions on Automatic Control, 49(9), 1453–1464. Wang, Z., Ho, D. W. C., & Liu, X. (2004). Robust filtering under randomly varying sensor delay with variance constraints. IEEE Transactions on Circuits and System II, 51(6), 320–326. Xie, L., Liu, L., Zhang, D., & Zhang, H. (2004). Improved H2 and H∞ filtering for uncertain discrete-time systems. Automatica, 40(5), 873–880. Xu, S. Y., & Chen, T. (2002). Reduced-order H∞ filtering for stochastic systems. IEEE Transactions on Signal Process, 50(12), 2998–3007. Xu, S. Y., & Chen, T. (2003). Robust H∞ filtering for uncertain impulsive stochastic systems under sampled measurements. Automatica, 39(3), 509–516. Yang, F. W., Wang, Z., Hung, Y. S., & Gani, M. (2006). H∞ control for networked systems with random communication delays. IEEE Transactions on Automatic Control, 51(3), 511–518. Yaz, E., & Ray, A. (1996). Linear unbiased state estimation for random models with sensor delay. In Proc. conf. decision and control (pp. 47–52). Zhou, S. S., & Lam, J. (2003). Robust stabilization of delayed singular systems with linear fractional parametric uncertainties. Circuits Systems and Signal Processing, 22(6), 579–588. Zhou, S. S., Zhang, B. Y., & Zheng, W. X. (2006). Gain-scheduled H∞ filtering of parameter-varying systems. International Journal on Robust Nonlinear Control, 16(8), 397–411.
Technical communique
H∞ filtering for discrete-time systems with randomly varying sensor delaysI Shaosheng Zhou a,∗ , Gang Feng b a Institute of Information and Control, School of Automation, Hangzhou Dianzi University, Hangzhou 310018, PR China b Department of Mechanical Engineering and Engineering Management, City University of Hong Kong, Hong Kong
Received 13 August 2006; received in revised form 23 May 2007; accepted 19 October 2007 Available online 10 March 2008
Abstract This paper investigates an H∞ filtering problem for discrete-time systems with randomly varying sensor delays. The stochastic variable involved is a Bernoulli distributed white sequence appearing in measured outputs. This measurement mode can be used to characterize the effect of communication delays and/or data-loss in information transmissions across limited bandwidth communication channels over a wide area. H∞ filtering of this class of systems is used to design a filter using the measurements with random delays to ensure the mean-square stochastic stability of the filtering error system and to guarantee a prescribed H∞ filtering performance. A sufficient condition for the existence of such a filter is presented in terms of the feasibility of a linear matrix inequality (LMI). Finally, a numerical example is given to illustrate the effectiveness of the proposed approach. c 2008 Elsevier Ltd. All rights reserved.
Keywords: Communication network; Data-loss; H∞ filtering; Random delays; Stochastic stability
1. Introduction Estimation of dynamic systems has found many practical applications and has attracted a lot of attention during the last decades. H∞ filtering is introduced as an alternative to classical Kalman filtering when the statistical property of noise sources is unknown or unavailable (Nagpal & Khargonekar, 1991). H∞ filtering is concerned with the design of estimators which ensure a bound on the L2 -induced gain from disturbance signals to estimation errors. Over the past decades, various approaches, such as the interpolation approach, the Riccati equation-based approach, and the LMI-based approach have been developed to deal with the H∞ filtering problem in various settings such as deterministic systems with uncertainties and/or delays as well as various stochastic systems (Xie, Liu, Zhang, & Zhang, 2004). Recently, the study of the H∞ filtering problem for I This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Guoxiang Gu under the direction of Editor Andr´e Tits. This work was supported in part by the National Natural Science Foundation of PR China under Grant 60574080 and 60434020. ∗ Corresponding author. Tel.: +86 571 86878584; fax: +86 571 86878584. E-mail addresses: [email protected] (S. Zhou), [email protected] (G. Feng).
c 2008 Elsevier Ltd. All rights reserved. 0005-1098/$ - see front matter doi:10.1016/j.automatica.2007.10.026
systems with delays has gained growing interest. A delayindependent H∞ filtering result for discrete-time systems with multiple time delays has been given in Palhares, de Souza, and Peres (2001) while delay-dependent results for this problem have been reported more recently in Gao, Lam, Xie, and Wang (2004) and Gao and Wang (2005). It is noted that the time delays are assumed to be deterministic in the literature mentioned above. However, they may occur in a randomly varying way in many practical applications as pointed out in Wang, Ho, and Liu (2004). Recently, there has been some attention to the research of systems with randomly varying delays. A randomly varying delayed sensor mode was first introduced in Ray (1994). Since then, some results for randomly delayed systems have been reported in Wang et al. (2004), Yaz and Ray (1996) and Yang, Wang, Hung, and Gani (2006). A variance-constrained filtering approach was proposed for systems with random sensor delays in Wang et al. (2004). And more recently, the authors in Yang et al. (2006) investigated the H∞ control problem for this class of systems. In the meantime, the H∞ filtering and control problems of stochastic systems have also attracted a lot of attention over the past few years (Hinrichsen & Pritchard, 1998). In Xu and Chen (2003), an LMI-based filter design approach was proposed for impulsive stochastic systems, and
1919
S. Zhou, G. Feng / Automatica 44 (2008) 1918–1922
based on the projection lemma, the reduced-order H∞ filtering problem for a class of stochastic systems was investigated in Xu and Chen (2002). The authors of Berman and Shaked (2006) presented an H∞ control scheme for a class of discrete-time nonlinear stochastic systems more recently. Motivated by the works in Wang et al. (2004) and Yang et al. (2006), this paper focuses on the H∞ filtering problem for systems with randomly varying sensor delays. We are interested in designing filters such that for all randomly varying sensor delays, the filtering error system is exponentially mean-square stable and a prescribed H∞ filtering performance is achieved. This paper is organized as follows. Section 2 formulates the H∞ filtering problem. In Section 3, a novel H∞ filtering approach is proposed. A numerical example is given to demonstrate the effectiveness of the proposed method in Section 4, which is followed by conclusions in Section 5. Notations: Throughout this paper, Z + denotes the set of positive integers; R n denotes the n-dimensional Euclidean space; R m×n denotes the set of all m × n real matrices. A real symmetric matrix P > 0(≥ 0) denotes P being a positive definite (or positive semi-definite) matrix, and A > (≥)B means A − B > (≥)0. I denotes an identity matrix of appropriate dimension. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations. The superscript ‘τ ’ represents the transpose. For a matrix N , N −τ stands for the transpose of matrix N −1 . A star ‘∗’ is used as an ellipsis for a corresponding transposed block matrix that is induced by symmetry. The notation l2 [0, ∞) represents the space of square summable infinite vector sequences with the usual norm k · k2 . Suppose n¯ that the vectors qP vk ∈ R , a sequence v = {vk } ∈ l2 [0, ∞) if ∞ τ kvk2 = k=1 vk vk < ∞. Prob{.} stands for the occurrence probability of an event; E{.} denotes the expectation operator with respect to some probability measure. 2. Problem formulation Consider a discrete-time system Σ : xk+1 = Axk + Aω ωk z k = L xk + L ω ωk
(1) (2)
where xk ∈ R n is the state; ωk ∈ R p¯ is the deterministic disturbance signal in l2 [0, ∞); z k ∈ R q is the signal to be estimated; A, Aω , L and L ω are known constant matrices with compatible dimensions. The measurement with random delays is given by yk = C xk yck = (1 − θk )yk + θk yk−1
(3) (4)
where yk ∈ R p is the output, yck ∈ R p is the measured output, C is a known matrix, and the stochastic variable θk is a Bernoulli distributed white sequence taking the values of 0 and 1 with Prob{θk = 1} = E{θk } = ρ Prob{θk = 0} = E{1 − θk } = 1 − ρ where ρ ∈ [0, 1] and is a known constant.
(5) (6)
Remark 1. The system measurements with varying sensor delay modeled in (3) and (4) was first introduced in Ray (1994) and has been used to characterize the effect of communication delays and/or data-loss in information transmissions across limited bandwidth communication channels over a wide area such as navigating a vehicle based on the estimations from a sensor web of its current position and velocity (Sinopoli et al., 2004). The output yk produced at a time k is sent to the observer through a communication channel. If no packet-loss occurs, the measurement output yck takes value yk ; otherwise, the measurement output yck takes value yk−1 . When the probability of event packet-loss occurring is assumed as ρ, the measurement output yck in (4) thus takes the value yk with probability 1 − ρ, and the value yk−1 with probability ρ. For the delayed sensor mode (4), we assume that x−1 = 0, which implies from (3) that y−1 = 0. We consider the following filter for the estimation of z k : xˆk+1 = A f xˆk + B f yck (7) zˆ k = C f xˆk + D f yck where xˆk ∈ R n and zˆ k ∈ R q . A f , B f , C f and D f are matrices to be determined. Combining (1)–(4) and (7) the filtering error dynamics can be represented as Σ˜ : x¯k+1 = A(θk )x¯k + A1 (θk )H x¯k−1 + Aω (θk )ωk (8) z¯ k = L(θk )x¯k + L1 (θk )H x¯k−1 + Lω (θk )ωk where x¯k = xkτ
xˆkτ
τ
, z¯ k = z k − zˆ k , H = I, 0 A 0 Aω A(θk ) = , Aω = 0 )B C A (1 − θ k f f 0 A1 (θk ) = , L1 (θk ) = −θk D f C θk B f C L(θk ) = L − (1 − θk )D f C, −C f , Lω = L ω .
(9)
(10)
It is noted that the filtering error dynamics Σ˜ is a system with stochastic parameters since some of the parametric matrices in (10) are associated with the stochastic variable θk . We adopt the notion of stochastic stability in the mean-square sense from Yang et al. (2006) to formulate our filtering problem. Definition 1. The filtering error dynamics Σ˜ is said to be exponentially mean-square stable if with ωk ≡ 0, there exist constants α > 0 and τ ∈ (0, 1) such that E{kx¯k k2 } ≤ ατ k E{kx¯0 k2 },
for all x¯0 ∈ R 2n , k ∈ Z + .
The H∞ filtering problem addressed in this paper is to design a filter in the form of (7) such that for a given scalar γ and all nonzero ωk , the filtering error system Σ˜ is exponentially mean-square stable and under the zero initial condition, the filtering error z¯ k satisfies ∞ X k=0
E{k¯z k k2 } ≤ γ 2
∞ X
kωk k2 .
(11)
k=0
In such a case, the filtering error system is said to be exponentially mean-square stable with H∞ filtering performance γ .
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S. Zhou, G. Feng / Automatica 44 (2008) 1918–1922
Remark 2. In the performance criterion (11), the disturbance ωk is assumed to be a deterministic signal belonging to l2 [0, ∞). It is worth pointing out that this criterion can be easily modified to accommodate the case where the disturbance signals may be random ones.
For any nonzero ωk ∈ l2 [0, +∞) and zero initial condition, we have JN =
where ∗ denotes the corresponding transposed block matrix due to symmetry and the ρ-dependent matrices are defined as in (10) with θk replaced by ρ. Proof. We first prove the mean-square stability of the system Σ˜ with ωk ≡ 0. In this case, the dynamic equation in (8) becomes x¯k+1 = A(θk )x¯k + A1 (θk )H x¯k−1 .
(13)
Define a Lyapunov functional candidate as τ Vk = x¯kτ P x¯k + x¯k−1 H τ Q H x¯k−1 .
(14)
Let Fk be the minimal σ -algebra generated by {xˆi , 0 ≤ i ≤ k}. By (13) and some algebraic manipulations, we have τ E{Vk+1 |Fk } − Vk = x¯kτ xk−1 τ A (ρ)PA(ρ) + H τ Q H − P Aτ (ρ)PA1 (ρ) × ∗ Aτ1 (ρ)PA1 (ρ) − Q x¯k × . (15) xk−1 It can be checked by (12) and the Schur complement formula that τ A (ρ)PA(ρ) + H τ Q H − P Aτ (ρ)PA1 (ρ) < 0. τ ∗ A1 (ρ)PA1 (ρ) − Q (16) It then follows from the similar argument as in Yang et al. (2006) that system (13) is exponentially mean-square stable. In what follows, we shall show that the filtering error z¯ k satisfies (11). Let JN =
N −1 h X
E{k¯z k k2 } − γ 2 kωk k2
k=0
where N is a positive integer.
i
(17)
E{k¯z k k2 } − γ 2 kωk k2 + E{Vk+1 − Vk }
i
k=0
+ E V0 − E VN
At first, we establish a condition of mean-square stability and H∞ performance for the filtering error dynamics Σ˜ , which will be fundamental in the derivation of our H∞ filter design method. Lemma 3. Given a scalar γ > 0, the filtering error system Σ˜ is exponentially mean-square stable with a guaranteed H∞ filtering performance γ , if there exist matrices P and Q such that P 0 PA(ρ) PA1 (ρ) PAω ∗ I L(ρ) L1 (ρ) Lω ∗ ∗ P − Hτ QH 0 0 (12) >0 ∗ ∗ ∗ Q 0 ∗ ∗ ∗ ∗ γ 2I
N −1 h X
≤
N −1 X
E{ξkτ M(ρ)ξk }
(18)
k=0
where τ τ xk−1 , ωkτ τ A (ρ) M(ρ) = Aτ1 (ρ) P A(ρ) A1 (ρ) Aω Aτ ωτ L (ρ) + Lτ1 (ρ) L(ρ) L1 (ρ) Lω Lτ τω H QH − P 0 0 0 −Q 0 . + 0 0 −γ 2 I ξk = x¯kτ ,
(19)
(20)
It follows from (12) and by the Schur complement formula that M(ρ) < 0. This implies that for any N , J N < 0, which leads to that the filtering error z¯ k satisfies condition (11). Remark 4. Lemma 3 provides an H∞ filtering analysis result for systems with randomly varying sensor delays. This result can be readily extended to the case where the disturbance signals are random ones. 3. H∞ filter design Based on Lemma 3 we will give a sufficient condition for the existence of an H∞ filter in the form of (7) and present a method to construct the filter. Theorem 5. Consider the system of (1) and (2). Given a scalar γ > 0, there exists a filter in the form of (7) such that the filtering error system Σ˜ is exponentially mean-square stable with H∞ filtering performance γ , if there exist positive definite matrices X, Z , Q and matrices A¯ f , B¯ f , C¯ f , D¯ f such that
X
Z
0
X A + (1 − ρ) B¯ f C
∗
Z
0
ZA
∗
∗
I
L − (1 − ρ) D¯ f C
∗
∗
∗
X−Q
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
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S. Zhou, G. Feng / Automatica 44 (2008) 1918–1922
X A + (1 − ρ) B¯ f C + A¯ f
ρ B¯ f C
ZA
0
L − (1 − ρ) D¯ f C − C¯ f
−ρ D¯ f C
Z−Q
0
Z−Q
0
∗
Q
∗
∗
X Aω
It follows from (23) and (26) that
Z Aω Lω 0 > 0. 0 0
(21)
γ 2I
In this case, two nonsingular constant matrices M and N can always be obtained such that M N τ = I − X Z −1
X
I
0
X A + (1 − ρ)M B f C
∗
Y
0
A
∗
∗
I
L − (1 − ρ)D f C
∗
∗
∗
X−Q
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
X AY + (1 − ρ)M B f CY + M A f N τ
ρMB f C
AY
0
LY − (1 − ρ)D f CY − C f N τ
−ρ D f C
I − QY
0
Y − Y QY
0
∗
Q
∗
∗
(22)
and the parametric matrices of the H∞ filter in the form of (7) can be constructed by A f = M −1 A¯ f Z −1 N −τ B f = M −1 B¯ f (23) C f = C¯ f Z −1 N −τ D f = D¯ f . Proof. Suppose that inequality (21) holds. This inequality gives X Z >0 (24) Z Z which by the Schur complement formula gives X − Z > 0. From this, it is clear that there exist nonsingular matrices M and N such that (22) holds. Let X I I Y Π1 = , Π = , Y = Z −1 . (25) 2 Mτ 0 0 Nτ Pre- and post-multiplying the inequality of (21) by diag{I, Y, I, I, Y, I, I } lead to
X
I
0
X A + (1 − ρ) B¯ f C
∗
Y
0
A
∗
∗
I
L − (1 − ρ) D¯ f C
∗
∗
∗
X−Q
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
One can check via (10) and (25) that X I τ Π2 Π1 = I Y X A + (1 − ρ)M B f C Π1τ A(ρ)Π2 = A ρMBf C Π1τ A1 (ρ) = 0 X Aω Π1τ Aω = Aω
X Aω
Aω Lω 0 > 0. 0 0
(27)
γ2I
(28) Π12 AY
(29) (30) (31)
L(ρ)Π2 = [L − (1 − ρ)D f C LY − (1 − ρ)D f CY − C f N τ ] Q QY Π2τ H τ Q H Π2 = Y Q Y QY
(32) (33)
with Π12 = X AY + (1 − ρ)M B f CY + M A f N τ . Then, we have τ Π2 Π1 ∗ ∗ ∗ ∗
X AY + (1 − ρ) B¯ f CY + A¯ f Y
ρ B¯ f C
AY
0
LY − (1 − ρ) D¯ f CY − C¯ f Y
−ρ D¯ f C
I − QY
0
Y − Y QY
0
∗
Q
∗
∗
X Aω
Aω Lω 0 > 0. 0 0
γ2I
Π1τ A(ρ)Π2 L(ρ)Π2 Π2τ Π1 − Π2τ H τ Q H Π2 ∗ ∗
0 I ∗ ∗ ∗
Π1τ A1 (ρ) L1 (ρ) 0 Q ∗
Π1τ Aω Lω 0 > 0. (34) 0 γ2I
Pre-multiplying diag{Π2−τ , I, Π2−τ , I, I } and post-multiplying diag{Π2−1 , I, Π2−1 , I, I } to inequality (34) lead to (26)
Π1 Π −1 ∗2 ∗ ∗ ∗ > 0.
0 I ∗ ∗ ∗
Π2−τ Π1τ A(ρ) L(ρ) Π1 Π2−1 − H τ Q H ∗ ∗
Π2−τ Π1τ A1 (ρ) L1 (ρ) 0 Q ∗
Π2−τ Π1τ Aω Lω 0 0 2 γ I
(35)
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S. Zhou, G. Feng / Automatica 44 (2008) 1918–1922
Table 1 Optimal H∞ filtering performance ρ γ∗
1 0.0892
0.5 0.0835
0.3 0.0671
0.1 0.0367
0 9 × 10−6
−8.5349 Bf = −4.2744 D f = −0.7835.
5. Conclusions
Let Π1 Π2−1
−48.9874 98.9052 Af = , −24.7685 49.9976 C f = 15.5007 −31.3166 ,
(36)
= P.
It is easy to check via (25), (28) and (36) that P > 0. Substituting (36) into (35) leads to (12). The proof is thus completed by invoking Lemma 3. Remark 6. The main idea behind the derivation of Theorem 5 is the construction of the Lyapunov matrix P in (36), which is motivated by Palhares et al. (2001) and Zhou, Zhang, and Zheng (2006). It is noted that our method to construct Lyapunov matrix P is more general than that in Palhares et al. (2001). Remark 7. In view of Theorem 5, the H∞ filtering problem for systems with randomly varying sensor delays can be solved in terms of the feasibility of LMI in (21). Note that the inequality in (21) is not only linear with respect to matrix variables X, Z , Q, A¯ f , B¯ f , C¯ f , D¯ f but also linear with respect to scalar γ 2 . The H∞ filtering performance γ ∗ can thus be optimized by solving the following convex optimization problem: min
(X,Z ,Q, A¯ f , B¯ f ,C¯ f , D¯ f )
δ
subject to (21) with δ = γ 2 .
Then, the corresponding optimal H∞ filtering performance γ ∗ is given by γ ∗ = (min δ)1/2 . Remark 8. Following the similar arguments as in Palhares et al. (2001) and Zhou and Lam (2003), Theorem 5 can be readily extended to the cases in which the systems contain norm bounded parametric uncertainties or linear fractional parametric uncertainties. 4. A design example In this section, we present a numerical example to illustrate the theory developed in Section 3. Consider the system Σ with parametric matrices as follows: 0.2 0.05 0.1 A= , Aω = −0.02 0.3 −0.2 L = 0.5 −0.7 , C = 1 0.6 , L ω = 0.3. The objective is to design a filter in the form of (7) for this system such that the filtering error system Σ˜ is exponentially mean-square stable and an optimal H∞ filtering performance γ ∗ is achieved. By using the Matlab LMI Control Toolbox to solve the convex optimization problem as stated in Remark 7, we obtain the optimal H∞ filtering performance γ ∗ for different values of ρ, summarised in Table 1. It is shown that γ ∗ decreases as ρ decreases. In other words, a better H∞ filtering performance is achieved when more current and less delayed sensor information is used. In the case when ρ = 0.5, γ ∗ = 0.0835, M τ = 0.5 I2 , the parametric matrices of the desired filter are given as follows.
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