Network-Based $ H_ {\infty} $ Filtering for Discrete-Time Systems

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Correspondence Network-Based

Filtering for Discrete-Time Systems

Xian-Ming Zhang and Qing-Long Han

Abstract—This correspondence is concerned with network-based filtering for discrete-time systems. The output signals of the system under consideration are transmitted to the filter through a constraint communication network, which usually leads to network-induced delays and packet dropouts. By introducing a logic data packet processor to choose the newest data signal from the network to actuate the filter, network-induced delays and packet dropouts are modeled as a Markov chain taking values in a finite set. As a result, the filter to be designed is modeled as a Markov jumping linear filter. By introducing some slack matrix variables in terms of probability identity, a less conservative bounded real lemma (BRL) is derived to ensure that the filtering error system is stochastically stable with a prelevel. Based on this BRL, suitable filters are designed by scribed employing a cone complementary approach. A practical example on the Leslie model about some certain pest’s structured population dynamics is given to show the effectiveness of the proposed approach. Index Terms—Discrete-time systems, chain, transition probability.

filtering, networks, Markov

I. INTRODUCTION

H1

The concept of filtering was first introduced by the pioneering work in [1]. The main aim is to design a suitable filter to estimate filtering is insensystem states. Compared with Kalman filtering, sitive to the exact knowledge of the statistics of noise signals. Therefiltering has received growing fore, during the past two decades, attention in the area of state estimation (see, for example, [2]–[9] and references therein). With the rapid development of communication technology, communication networks have been increasingly used to transmit data between control components of a control system. In fact, the use of a communication network can offer a flexible control architecture with several advantages such as low cost, simply installation and maintenance and filtering high reliability [10]. On account of those advantages, in setting, one may use a communication network to transmit data from a filtering. physical plant to a filter, which leads to network-based What are the differences between traditional filtering and netfiltering? On one hand, in network-based filtering work-based setting, it is clear that the physical plant and the filter may be located remotely, while in a traditional filtering setting, they are usually filtering has located locally. It is thus implied that network-based

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Manuscript received August 24, 2011; accepted October 27, 2011. Date of publication November 07, 2011; date of current version January 13, 2012. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Benoit Champagne. This work was supported in part by the Australian Research Council Discovery Projects under Grant DP1096780 and Grant DP0986376, and the Research Advancement Awards Scheme Program (January 2010–December 2012) and the RDI Merit Grant Scheme Project under Grant RDIM1109 (January 2011–December 2011) at Central Queensland University, Australia. The authors are with the Centre for Intelligent and Networked Systems and the School of Information and Communication Technology, Central Queensland University, Rockhampton QLD 4702, Australia (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this correspondence are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2011.2175224

wider potential application scopes than traditional one. On the other hand, traditional filtering is based on an implicit assumption that there is an ideal channel to transmit output signals of the plant to the filter instantaneously and losslessly. As a result, the input of the filter is exactly equal to the output of the plant at any time. Nevertheless, this ideal assumption will be broken in network-based filtering due to the constrained quality of service of the communication network. That is, the input of the filter may be no longer equal to the output of the filtering techniques may be plant, which means that traditional inapplicable to network-based filtering. Therefore, it is significant filtering. For continuous-time plants, to investigate network-based some approaches have been proposed to design suitable network-based filters (see, for example, [11] and [12]). For discrete-time plants, however, few results have been reported on this issue, which is the first motivation of this study. What aspects should be considered in network-based filtering? The first is network-induced delays and packet dropouts, which are usually regarded as main causes for the deterioration of system dynamic performances or even system instability. The second is packet disorder, which may occur when a packet sent earlier arrives at the filter later. Once a disordered packet arrives at the filter, we should discard it actively. Otherwise it is meant that the “outdated” output signals will be used to design a filter to estimate system states at the “current” time, which is unreasonable. The third is that the designed filter should be able to reflect the effects of both network-induced delays and packet dropouts. During the data transmission over a network, at different times, network-induced delays are different and so are packet dropouts. Thus, the filter parameters to be designed should reflect these changes. In a word, the above three aspects should be taken into account when filtering problem. However, only dealing with the network-based parts of them have been considered in the existing publications. For filtering is investigated example, in [13] and [14], network-based for a class of discrete-time systems by taking into account network-induced delays and packet dropouts, but packet disorder is not involved. Moreover, the designed filter parameters are constant matrices, which means that the filter gain is irrespective of both network-induced delays and packet dropouts. Therefore, it remains challenging to address filtering the above three aspects when dealing with network-based problem, which is the second motivation of the present study. filThis correspondence will be concerned with network-based tering for discrete-time systems. The main contributions can be summarized in the following. • A logic data packet processor (DPP) is introduced to model network-induced delays and packet dropouts as a Markov chain. This DPP also has a logical capability of coping with packet disorder. • The filter to be designed is modeled as a Markov jumping filter, whose gain is dependent on both network-induced delays and packet dropouts. • A slack matrix variable approach based on a probability identity is introduced to formulate a novel bounded real lemma (BRL) for the filtering error system. This BRL is of quite generality because the transition probability from mode to mode is not necessarily known exactly. • Based on the novel BRL, a sufficient condition on the existence of desired filters is derived by using a cone complementary linearization approach, whose effectiveness is demonstrated through a practical example. Notation: The notation throughout this correspondence is quite standard. f1g stands for the mathematical expectation; The symbol “ ” denotes the symmetric term in a symmetric matrix.

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With the logic DPP, network-induced delays and the number of packet dropouts can be readily calculated. If the k th data packet is successfully sent to the filter, then the network-induced delay of this 0, then k consecutive packet is 0 k . If k := k+1 0 k 0 1 packets time-stamped between k and k+1 have been lost during the transmission. Moreover, if one data packet is disordered, then the logic DPP drops it out actively. On the other hand, the above mechanism of the logic DPP forms an important time sequence f k gk=1;2;... . This sequence is non-decreasing, i.e., k  k+1 . At each time instant , the logic DPP uses the newest signal ( k ) to actuate the filter, which means that for each 0, there exists a unique k associated with . Moreover, integer . taking network-induced delays into account, we have k Let k := 0 k and := maxf k : = 1 2 . . .g 1. may be a small number because the logic DPP chooses the newest packet to drive the filter at each instant . To make it clear, in Fig. 2, the second data packet is dropped out by the logic DPP due to that it is disordered with the third and the fourth data packets when it arrives at the logic DPP. If this packet is not dropped out, then it will result in 8 = 2 and thus the value of 8 0 8 is 6 larger than the current value of 4. g. Since the random nature of Define a finite set S := f1 2 . . . network traffic, both network-induced delays and packet dropouts may 0, k takes values in the finite set S be random. Thus, for each randomly. In this correspondence, we suppose that f k :  1g is a discrete-time Markov chain taking values in S with the probability transition matrix 5 = ( ij )N 2N given by

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Fig. 1. A diagram for network-based

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Pfrk+1 = j jrk = ig = ij ; 8 i; j 2 S where ij  0 (i;j 2 S ) is the transition probability and N j =1 ij = 1; 8 i 2 S . The transition probability ij may be either known or unknown. For simplicity of presentation, denote that i := f : i Skn ij is knowng Sukn := f : ij is unknowng

j 

j 

;

: (2) Notice that at each instant k , the filter receives the signal y (k 0 rk ) to generalize the filtering signal. Since rk takes values in the finite set S randomly, in the following, we design an H1 filter of order n to be a discrete-time Markov jumping linear system described by

xf (k + 1) = Af (rk )xf (k) + Bf (rk )y(k 0 rk ) (3) zf (k) = Cf (rk )xf (k) + Df (rk )y(k 0 rk ) where Af (rk );Bf (rk );Cf (rk ), and Df (rk ) are filter parameters to be determined. Since rk takes values in S , the filter (3) contains N modes. For rk = i 2 S , the coefficient matrices of the ith mode are denoted by Afi ; Bfi ;Cfi and Dfi , which are real matrices of appro-

priate dimensions. Remark 1: By introducing a logic DPP, network-induced delays and packet dropouts are modeled as a Markov chain f k  1g. It is clear from the above analysis that network-induced delays and packet dropouts can be derived from the values of k . In contrast, in [14], a set of indicator functions are introduced to the output measures to describe network-induced delays and packet dropouts, that is, the output signal is described by q ( ) = f =0g 0 ( ) + f =d g j ( 0 j ) + ( ) j =1

r;k

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yk

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I p

Cxk

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I Pr  p p j ; ; ;q

Cxk d

Dw k

where f =0g and f =d g are the indicator functions with f f =0g g = f k = 0g = 0 and f f =d g g = f k = j g = j with j ( = 0 1 . . . ) being known positive scalars and k is a stochastic variable being used to determine network-induced delays and packet dropouts. During the transmission of the data packet ( ), it is quite difficult to capture the internal information j ( 0 j ) from the packet ( ), so it may be hard to work out the probabilities j ( = 0 1 . . . ). As a result, network-induced delays and packet dropouts may be not easily recognized from the stochastic variable k .

d 

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Remark 2: With the logic DPP, the filter is designed to be a Markov jumping filter, whose gain is dependent on network-induced delays and packet dropouts. However, in [13] and [14], the filter parameters are designed to be constant matrices, which are irrespective of network-induced delays and packet dropouts. Moreover, the transition probability of the Markov chain frk ; k  1g from mode i to mode j is not necessarily known exactly in this correspondence, while in [14], the probabilities pj (j = 0; 1; . . . ; q ) should be known in advance and so does in [13]. To proceed with, we use the lifting technique to formulate the filtering error system connecting with (1) and (3). Set X (k) := colfx(k); x(k 0 1); . . . ; x(k 0 N )g

kek < kwk2 is ensured for nonzero w(k) 2 l2 under zero initial T condition, where kek is defined by f 1 k=0 e (k)e(k)g . In this correspondence, we will develop a new method to solve the above network-based H1 filtering problem by introducing a new slack variable approach based on the probability identity N j =1 ij = 1 (i 2 S), which is inspired by the idea in [16].

1 FILTERING ANALYSIS

III. H

In this section, we will present a novel bounded real lemma (BRL) for the filtering error system (6) with partially known transition probabilities (2) using a slack variable approach. Proposition 1: For given > 0, the filtering error system (6) is stochastically stable with a prescribed H performance if there exist real matrices Pi > 0 and Mi = MiT , Ri , Qi = QiT (i 2 S) of appropriate dimensions such that for 8 i 2 S

1

W (k) := colfw(k); w(k 0 1); . . . ; w(k 0 N )g:

Then

07i01

y (k 0 rk ) = Cx(k 0 rk ) + B2 w(k 0 rk )

= CE (rk )X (k) + B2 H (rk )W (k)

(4)

where E (rk ) is a row-block vector with the (rk + 1)th block being an n 2 n identity matrix and others being n 2 n zero matrices; and H (rk ) is of the same form as E (rk ) with n 2 n block being replaced with q 2 q block. Consequently, the filter (3) can be rewritten as xf (k + 1) = Af (rk )xf (k) + Bf (rk )CE (rk )X (k) +Bf (rk )B2 H (rk )W (k) zf (k) = Cf (rk )xf (k) + Df (rk )CE (rk )X (k) +Df (rk )B2 H (rk )W (k):

(5)

0 0i I

? ? ? 0Pj01 ? ? ?

? ?

0 0I

?

where i = 1 0 i , ~ = i :=

pN +1

j 2S

Bi i Di i Ri i Qi 0 ~2 I

Bi Di 0Ri 0Qi

0 (i 2 S) such that

~ (k) + BW ~ (k) X (k + 1) = AX z (k) = LE0 X (k) + B3 H0 W (k)

where E0 := [In 0 1 1 1 0], H0 := [Iq 0 1 1 1 0] and

Ai Li

AE0 ~ := diagfB1 ; 0; . . . ; 0g: ; B (I 0)

Let  (k) = colfX (k); xf (k)g and e(k) = z (k) 0 zf (k). Then the filtering error system is given by  (k + 1) = A(rk ) (k) + B(rk )W (k) e(k) = L(rk ) (k) + D(rk )W (k)

Ai Li 0Mi

? ?

On the other hand, from (1), we have

~ := A

Ai i Li i Mi 0 Pi ?

Notice that

(6)

where

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j =1

ij Pj

0

0

I

Ai Li

Bi Pi 0 Di 0

N j =1 ij = 1. Then, we have N Mi Ri 10 ij ? Qi j =1

0

~2 I

= 0:

< 0: (10)

(11)

Adding (11) into (10) yields

~ A 0 Bf (rk )CE (rk ) Af (rk ) ~ B B(rk ) := Bf (rk )B2 H (rk ) L(rk ) := [LE0 0 Df (rk )CE (rk ) 0 Cf (rk )]

9i +

A(rk ) :=

D(rk ) := B3 H0 0 Df (rk )B2 H (rk ): It is clear that the filtering error system (6) is a Markov jumping system with N modes. For simplicity, for rk = i 2 S , the coefficient matrices of the ith mode of the system (6) are denoted by Ai ; Bi ; Li , and Di ; and E (rk ) and H (rk ) are denoted by Ei and Hi , respectively. The initial condition of (6) is supplemented as  () = colf0 ; 0; . . . ; 0g for   0 and r0 is defined by 0. The network-based H filtering problem to be addressed in this correspondence is stated as follows. Network-Based H Filtering Problem: Given a scalar > 0, design a full-order filter of the form (3) such that the filtering error system (6) has a prescribed H performance , i.e., 1) the system (6) with w(k)  0 is stochastically stable; and 2) the H performance

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ij 9ij < 0

(12)

where

Bi T 7i Di 0 i Mi 0 Pi

Ai Li

9i := +

i RiT Ai Bi T 9ij := Li Di Mi Ri 0 RiT Qi

0 Ai Li i I i Ri i Qi 0 ~2 I

Pj

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0 I

Ai Li

Bi Di

Bi Di

with 7i and i being defined in (9) and i = 1 0 i . Therefore, the proof will be completed only if we can prove that the inequality (12) holds in case that (7) and (8) are true. In fact, by Schur complement, inequalities (7) and (8) are equivalent to 9i < 0 and 9ij < 0, respectively, which lead to (12) due to ij  0.

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Remark 3: The BRL for discrete-time Markovian jump linear systems with partially known transition probabilities was obtained in [18] (see Lemma 1 therein). Compared with [18, Lemma 1], a significant advantage of Proposition 1 in this correspondence is that some slack matrix variables i i and i in (11) are introduced based on the = 1. It is clear that, if we take i = probability identity N j =1 ij 2 , then Proposition 1 reduces to [18, Lemma i i = 0 and i = ~ 1] for the filtering error system (6). Therefore, Proposition 1 is less conservative than [18, Lemma 1]. Remark 4: As special cases of Proposition 1, the following is clear from Proposition 1: a) If the transition probability matrix 5 is completely known, then the filtering error system (6) is stochastically stable with a prescribed performance provided that the following matrix inequality is feasible for 8 2 S :

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P ? ? ?

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ij

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i

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Ai

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0

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< 0:

0 0 ~2 I

G G X 0G 701 G 0 GA GB L D ? 0 I ? ?  M 0 P  R 2 < 0 (16) ? ? ?  Q 0 ~ I 0G P 01 G 0 GA GB ? 0I L D ? ? 0M 0R < 0; 8j 2 Sukn : (17) ? ? ? 0Q Notice that G A = GG1 2A~A+~ +EX0 XBB CECE EX0 XAA G B = GG1 2B~B~++EX0 XBB BB2 H2H : If we set Y1 := X A and Y2 = X B , then G A and G B are linear on the matrix variables. Thus, all the terms in the matrix inequalities (16) and (17) are linear except for G 701 G and G P 01 G . To handle these nonlinear terms, we now introduce matrix variables S > 0 and Z > 0 such that G 701G  S ; G P 01G  Z where 1i , 2i , and i are real matrices of compatible dimensions. Then the matrix inequalities (7) and (8) are equivalent to, respectively, i

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b) If the transition probability matrix 5 is completely unknown, then the filtering error system (6) is stochastically stable with a prescribed 1 performance provided that the following matrix inequality is feasible for 8 2 S :

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which are equivalent to

G0011  0; P 01 G0011  0: ? S ? Z 0P 01 0 A B ~ := 701 , S~ := S 01 , P~ := P 01 , Z~ := Z 01 and G~ := Set 7 ? 0I L D < 0; j 2 S : 01 . Then, we have the following result. G (14) ? ? 0P 02 Proposition 2: For a given scalar > 0, an admissible network? ? ? 0 ~ I based H1 filter of form (3) exists for system (1) if there exist real ~ >0 matrices P > 0, S > 0, Z > 0, P~ > 0, S~ > 0, Z~ > 0, 7 ~ , M = M , R , Q = Q , Y1 and Y2 , Y3 and Y4 (i;j 2 S ) It is worth pointing out that, if the transition probability matrix 5 and G is completely known, both the conditions (13) and (14) for 8 i 2 S of appropriate dimensions and real matrices G (i 2 S ) of form (15) can be regarded as a BRL for the filtering error system (6). One natural such that for 8 i 2 S question arises: What is the relationship between them? In answering 0S 0 21 22 this question, we now establish the following fact. ? 0 I  23  24 Fact 1: Consider the filtering error system (6) with completely ? ?  M 0P  R 2 < 0 (18) known transition probability matrix 5. If the condition (14) holds, ? ? ?  Q 0 ~ I then so does the condition (13). 0 Z 0 2 2 1 2 Proof: By Schur complement, it is clear that (13) and (14) are ? 0I 23 24 < 0 equivalent, respectively, to (19) ? ? 0M 0R ? ? ? 0Q  F +0 P 0