Variance-constrained H∞ state estimation for time-varying multi-rate systems with redundant channels: The finite-horizon case

Information Sciences 501 (2019) 222–235

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Information Sciences journal homepage: www.elsevier.com/locate/ins

Variance-constrained H∞ state estimation for time-varying multi-rate systems with redundant channels: The finite-horizon case Licheng Wang a, Zidong Wang b,c,∗, Guoliang Wei d, Fuad E. Alsaadi e a

Shanghai Key Lab of Modern Optical System, Department of Control Science and Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China b College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China c Department of Computer Science, Brunel University London, Uxbridge, Middlesex, UB8 3PH, United Kingdom d College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China e Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia

a r t i c l e

i n f o

Article history: Received 29 January 2019 Revised 11 May 2019 Accepted 26 May 2019 Available online 28 May 2019 Keywords: State estimation Multi-rate systems Redundant channels Finite-horizon H∞ performance Error variance constraint

a b s t r a c t This paper deals with the H∞ state estimation problem for a class of networked multi-rate time-varying systems with estimation error variance constraint. The redundant channel transmission scheme is employed to reduce the packet dropout rate and improve the quality of the data delivery. By utilizing the lifting technique, an augmented estimation error system is established with a uniform sampling rate. The objective of this paper is to design a time-varying state estimator such that, in the simultaneous presence of the asynchronous sampling, probabilistic packet dropouts as well as stochastic noises, the error dynamics of the state estimation satisfies both the prescribed H∞ performance requirement and the prescribed estimation error variance constraints. Through intensive stochastic analysis, sufficient conditions are established to ensure the existence of the desired estimator whose parameters are determined by solving a set of recursive linear matrix inequalities. A numerical example is presented to show the validity of the proposed estimation strategy. © 2019 Elsevier Inc. All rights reserved.

1. Introduction The state estimation (or filtering) problem, which has been well regarded as one of the fundamental research topics in the areas of control, communications and signal processing, has been attracting considerable research interests [7,9,18,19,22,27,32,34]. During the past few decades, many different kinds of estimator design approaches have been proposed with respect to different estimation performance specifications, see, e.g. [13,14,21,33,35,43–45]. Among others, the robust filtering algorithms have aroused particular research attention for their clear advantages in coping with parameter uncertainties and exogenous disturbances without the need to know the noise statistics. As a typical class of robust filtering schemes, the H∞ filter algorithm aims to achieve the prescribed disturbance attenuation level on the estimation error against the external disturbances [7,8]. Furthermore, since almost all real-world systems contain time-varying parameters,

∗

Corresponding author at: College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China. E-mail address: [email protected] (Z. Wang).

https://doi.org/10.1016/j.ins.2019.05.073 0020-0255/© 2019 Elsevier Inc. All rights reserved.

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one would be naturally interested in the transient behaviors of system dynamics over a finite horizon. As such, much research effort has recently been devoted to the so-called finite-horizon H∞ filtering problem for time-varying systems. For certain filtering tasks such as tracking maneuvering targets, the filtering error variance is often required to stay below a prescribed bound that is determined a priori according to practical requirements, and this gives rise to the varianceconstrained filtering problems [1,23]. In comparison with the conventional optimal filtering approaches (e.g. Kalman filtering), the variance-constrained filtering strategy no longer pursues the optimality (i.e., the minimum variance) and therefore offers much design freedom for considering other desired performance indices such as regional stability, performance robustness and disturbance rejection/attenuation levels. Consequently, the variance-constrained multi-objective filter design problem has stirred a great deal of research interest in the past decade with many important results available in the literature, see e.g. [30,41] on the H∞ filtering problem with variance constraints. In nowadays popular networked control systems (NCSs), signals are typically transmitted in a digital manner through networks by executing the A/D and D/A conversions, and this has motivated the rapid development of the sampled-data systems, see [6,16,17,38–40]. Due primarily to the physical limitations, it is quite difficult to sample signals from different system components (e.g., plant, sensor, controller and actuator) at a unified sampling-rate. On the other hand, a larger sampling rate means faster data updates that facilitate better system performance but at the cost of consuming more resources. As such, a carefully chosen sampling rate would alleviate the implementation cost in the interest of maintaining satisfactory system performance. From both the perspectives of physical constraints and resource savings, the multi-rate sampling scheme has recently become an attractive focus of research focus and the corresponding control/filtering problems have recently received an increasing research interest, see e.g. [10,36,41,42]. As one of the most commonly encountered network-induced phenomena, the packet dropout is known to be a source for instability, oscillation and poor performance of the system [26]. Consequently, in practical applications, it is desirable that the data exchange between different components is conducted with a high reliability, and this is particularly true for some real-time critical tasks such as the tracking problem for highly maneuvering targets [31] as well as the high-speed and high-accuracy positioning problem [5]. So far, most existing literature has been based on the assumption that only one channel is available for data transmissions. In order to improve the reliability of the data transmission, a seemingly natural way is to employ redundant channels to compensate for possible communication failures (most often packet dropouts) that are likely to occur in a single channel. To date, some initial effort has been made on the analysis/design issues for NCSs with redundant channels, see [20,26,46] and the references therein. Nevertheless, despite its obvious engineering insight, the variance-constrained H∞ state estimation (VCHSE) problem has not fully been investigated yet for stochastic time-varying systems with redundant channels, not to mention the case when the multi-rate sampling is involved as well. In view of such a situation, we are motivated to launch an investigation in this paper on the VCHSE problem and shorten the identified gap. Concluding the above discussions, in this paper, our aim is to look into the variance-constrained H∞ state estimation problem for a class of stochastic time-varying systems. The multi-rate sampling scheme is taken into consideration where the sampling period for the measurement outputs is integer multiples of the one for the plant. Moreover, the redundant channels are employed from the sensors to the estimators with hope to enhance the communication reliability. The main contributions of the paper can be outlined as follows: 1) a general measurement model with redundant channels is established so as to make the data transmission more reliable, where the number of the channel is extended to M (M ≥ 2); 2) by means of the lifting technique, the variance-constrained H∞ state estimation problem is, for the first time, investigated for time-varying multi-rate systems over a finite-horizon; and 3) the time-varying estimator parameters are obtained by solving a set of forward recursive matrix inequalities, thereby facilitating the online applications. The remainder of this paper is organized as follows. In Section 2, the discrete time-varying system model, the measurement output model, and the estimator model are, respectively, introduced and the problem under consideration is formulated. In Section 3, the main theorems are presented to address the analysis and design problems of the multi-rate timevarying H∞ state estimator under the error variance constraints. A simulation example is shown in Section 4 to illustrate the main results derived. Finally, we conclude the paper in Section 5. Notation. The notations are quite standard. Throughout this paper, Z+ , Rn and Rn×m denote, respectively, the positive integer space, the n-dimensional Euclidean space and the set of all n × m real matrices. AT represents the transpose of A. The notation X ≥ Y (respectively, X > Y), where X and Y are symmetric matrices, means that X − Y is positive semi-definite (respectively, positive definite). diagN {Ai } stands for the block-diagonal matrix diag{A1 , A2 · · · , AN }, and colN {xi } denotes the column vector [xT1 xT2 · · · xTN ]T . I is the identity matrix with appropriate dimension. E{x} stands for the expectation of stochastic variable x. x describes the Euclidean norm of a vector x. The symbol  represents the Kronecker product. 2. Problem formulation Consider a class of linear discrete time-varying stochastic systems described by the following state-space model:



x(Ts+1 ) = A(Ts )x(Ts ) + B(Ts )w(Ts ) z(Ts ) = M (Ts )x(Ts )

(a ) (b )

(1)

where x(Ts ) ∈ Rn represents the state vector, z(Ts ) ∈ R p is the output to be estimated, and w(Ts ) ∈ R denotes a zero-mean Gaussian white noise sequence with E{w2 (Ts )} = 1. h  Ts+1 − Ts (∀s = 1, 2, . . .) is the sampling period of system (1). A(Ts ), B(Ts ) and M(Ts ) are the known time-varying matrices with appropriate dimensions.

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Fig. 1. Diagram of the multi-rate sampling systems with the redundant channel transmission scheme.

To mitigate the negative impact from the network-induced phenomenon of packet losses during the data transmission, the redundant-channel-based transmission mechanism is taken into consideration in this paper. More specifically, the measured output received by the estimator via M redundant channels is characterized by the following model:



y(tk ) =

α1 (tk )(C1 (tk )x(tk ) + D1 (tk )v1 (tk ) ) +

M s−1   s=2

 (1 − αl (tk ))αs (tk )(Cs (tk )x(tk ) + Ds (tk )vs (tk ) )

(2)

l=1

where y(tk ) ∈ Rm is the measurement output at sampling instant tk (k = 0, 1, 2, . . . ) and tk+1 − tk = bh with b ≥ 2 being a known positive integer. Also, tk satisfies 0 = t0 < t1 < · · · < tk < tk+1 < · · · and limk→+∞ tk = +∞. For i ∈ {1, 2, . . . , M}, vi (tk ) ∈ R is the measurement noise of the ith channel, which is a zero-mean Gaussian white noise process with E{v2i (tk )} = 1 and is uncorrelated with w(tk ). Ci (tk ) and Di (tk ) are known real time-varying matrices with appropriate dimensions. The random variables α i (tk ), which govern the randomly occurring packet loss phenomenon for the ith channel, obey the following Bernoulli distribution:

Prob{αi (tk ) = 1} = α¯ i , Prob{αi (tk ) = 0} = 1 − α¯ i

(3)

where α¯ i ∈ [0, 1] (i = 1, 2, . . . , M ) are known constants. It is assumed that the random variables α i (tk ) and α j (tk ) are mutually independent for i = j (i, j ∈ {1, 2, . . . , M} ). Remark 1. According to the measurement model (2), the data is transmitted via M redundant channels with the arrival rate

α¯ i (i = 1, 2, · · · , M ). In particular, if the first channel is available at time tk (i.e., α1 (tk ) = 1), it will be assigned to transmit the measurement data no matter what the conditions of the rest channels are. If the first channel fails to work and the second channel is available, then the data will be transmitted via the second one. Similarly, the data would be delivered via the sth channel if and only if all the previous s − 1 channels suffer from packet dropouts and the sth channel does not. Obviously, compared with the traditional single channel case, such a transmission strategy typically increases the arrival rate of the transmitted data as the more redundant channels the higher arrival rate. By applying the evolution relationship (1) iteratively, one derives the following dynamic equations:

⎧ x(tk+1 ) = Ab (tk )x(tk ) + B1b (tk )wb (tk ) ⎪ ⎪ 2 ⎪ ⎪ ⎨x(tk+1 − h ) = Ab−1 (tk )x(tk ) + Bb−1 (tk )wb−1 (tk ) ..

. ⎪ ⎪ ⎪ x(tk + h ) = A1 (tk )x(tk ) + Bb1 (tk )w1 (tk ) ⎪ ⎩ z(tk − mh ) = M (tk − mh )x(tk − mh )

(4)

where m ∈ {0, 1, 2, . . . , b − 1}, and for r = 1, 2, . . . , b,

Ar (tk ) =

b 

A(tk+1 − ih ),

i=b−r+1

wr (tk ) = [wT (tk )



Bsr (tk ) =

B¯ sj (tk ) =

···

wT (tk + (r − 1 )h )]T ,

B(tk ), r = 1, s = b, [B¯ s1 (tk ) · · · B¯ sr−1 (tk ) B(tk + (r − 1 )h )], r ∈ {2, . . . , b}, s = b − r + 1,

b− j 

A(tk+1 − ih )B(tk + ( j − 1 )h ).

i=s

The state estimator design problem is illustrated in Fig. 1 for the multi-rate sampling system (1)–(2) under the redundant-channel-based communication scheme. By utilizing the real but non-ideal measurements collected from the

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redundant channels, we construct the following state estimators:

⎧ ¯ tk )xˆ(tk )) xˆ(tk+1 ) = Ab (tk )xˆ(tk ) + Kb (tk )(y(tk ) − ( ⎪ ⎪ ⎪ ⎪ ¯ tk )xˆ(tk )) xˆ(t − h ) = Ab−1 (tk )xˆ(tk ) + Kb−1 (tk )(y(tk ) − ( ⎪ ⎪ ⎨ k+1 .

(5)

.. ⎪ ⎪ ⎪ ⎪ ¯ tk )xˆ(tk )) ⎪ xˆ(tk + h ) = A1 (tk )xˆ(tk ) + K1 (tk )(y(tk ) − ( ⎪ ⎩ zˆ(tk − mh ) = M (tk − mh )xˆ(tk − mh )

 ¯ tk )  α¯ 1C1 (tk ) + M { s−1 (1 − α¯ l )α¯ sCs (tk )} and Ks (tk ) (s = 1, 2, · · · , b) are the estimafor m ∈ {0, 1, 2, . . . , b − 1}, where ( s=2 l=1 tor parameters to be designed. Next, by resorting to the statistical information (3) of the occurrence of packet losses, the state estimator (5) is further rewritten as follows:

⎧ ˜ tk )x(tk ) xˆ(tk+1 ) = Ab (tk )xˆ(tk ) + Kb (tk )(( ⎪ ⎪ ⎪ ⎪ ¯ tk )(x(tk ) − xˆ(tk )) + vD (tk )) ⎪ + ( ⎪ ⎪ ⎪ ⎪ ˜ tk )x(tk ) ⎪ ⎪ xˆ(tk+1 − h ) = Ab−1 (tk )xˆ(tk ) + Kb−1 (tk )(( ⎪ ⎪ ⎪ ¯ tk )(x(tk ) − xˆ(tk )) + vD (tk )) ⎨ + (

(6)

.. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ˜ tk )x(tk ) ˆ ˆ(tk ) + K1 (tk )(( x ( t + h ) = A ⎪ 1 (tk )x k ⎪ ⎪ ⎪ ⎪ ¯ tk )(x(tk ) − xˆ(tk )) + vD (tk )) + ( ⎪ ⎪ ⎩ zˆ(tk − mh ) = M (tk − mh )xˆ(tk − mh ) ˜ tk ) = (tk ) − ( ¯ tk ) and for m ∈ {0, 1, 2, . . . , b − 1}, where (



(tk ) = α1 (tk )C1 (tk ) +

M s−1   s=2

(1 − αl (tk ))αs (tk )Cs (tk ) ,

l=1

vD (tk ) = α1 (tk )D1 (tk )v1 (tk ) +





M s−1   s=2

 (1 − αl (tk ))αs (tk )Ds (tk )vs (tk ) .

l=1

Denoting e(tk )  x(tk ) − xˆ(tk ) and z˜(tk )  z(tk ) − zˆ(tk ), one obtains the following estimation error dynamics:

⎧ ˜ tk )x(tk ) e(tk+1 ) = Ab (tk )e(tk ) − Kb (tk )( ⎪ ⎪ ⎪ ⎪ ¯ ⎪ − Kb (tk )(tk )e(tk ) − Kb (tk )vD (tk ) + B1b (tk )wb (tk ) ⎪ ⎪ ⎪ ⎪ ˜ tk )x(tk ) ⎪ e(tk+1 − h ) = Ab−1 (tk )e(tk ) − Kb−1 (tk )( ⎪ ⎪ ⎪ ⎪ ¯ tk )e(tk ) − Kb−1 (tk )vD (tk ) + B2 (tk )wb−1 (tk ) ⎨ − Kb−1 (tk )( b−1 .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ˜ tk )x(tk ) − K1 (tk )( ¯ tk )e(tk ) e(tk + h ) = A1 (tk )e(tk ) − K1 (tk )( ⎪ ⎪ ⎪ ⎪ ⎪ − K1 (tk )vD (tk ) + Bb1 (tk )w1 (tk ) ⎪ ⎪ ⎩ z˜(tk − mh ) = M (tk − mh )e(tk − mh )

for m ∈ {0, 1, 2, . . . , b − 1}. For the purpose of notation simplicity, we introduce the following notations:

xη (tk ) = col{x(tk ), x(tk − h ), · · · , x(tk − (b − 1 )h )}, xˆη (tk ) = col{xˆ(tk ), xˆ(tk − h ), · · · , xˆ(tk − (b − 1 )h )}, eη (tk ) = col{e(tk ), e(tk − h ), · · · , e(tk − (b − 1 )h )}, z˜η (tk ) = col{z˜(tk ), z˜(tk − h ), · · · , z˜(tk − (b − 1 )h )}, Aη (tk ) = col{Ab (tk ), Ab−1 (tk ), · · · , A1 (tk )}, Bη (tk ) = diag{B1b (tk ), B2b−1 (tk ), · · · , Bb1 (tk )}, Kη (tk ) = col{Kb (tk ), Kb−1 (tk ), · · · , K1 (tk )},

(7)

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Wη (tk ) = col{wb (tk ), wb−1 (tk ), · · · , w1 (tk )}, Mη (tk ) = diag{M (tk ), · · · , M (tk − (b − 1 )h )},

β (tk ) = colM {βi (tk )}, β¯ = colM {β¯ i }, β¯ i = E{βi (tk )},  = [σi j ]M×M = E{(β (tk ) − β¯ )(β (tk ) − β¯ )T },  α (t ), i = 1 βi (tk ) = 1i−1k s=1 (1 − αs (tk ))αi (tk ), 2 ≤ i ≤ M. By setting ϑη (tk )  [WηT (tk ) in a compact form as follows:



vTD (tk )]T and combining (6) and (7), the estimation error system can be further rewritten

eη (tk+1 ) = Aη (tk )eη (tk ) − Fη (tk )xˆη (tk ) + Bη (tk )ϑη (tk )

(8)

z˜η (tk ) = M(tk )eη (tk ) where

Aη (tk ) = A¯ η (tk ) − Fη(2) (tk ), Fη (tk ) = Fη(2) (tk ) − Fη(1) (tk ), Bη (tk ) = [Bη (tk )

¯ tk ) 0 · · · 0], −Kη (tk )], Fη(1) (tk ) = [Kη (tk )(

  b−1

¯ Fη(2) (tk ) = [Kη (tk )(tk ) 0 ··0 ··0

· ] Aη (tk ) = [Aη (tk ) 0 · ]. b−1

b−1

According to (8), the state variance matrix is defined as

X (tk )  E{eη (tk )eTη (tk )}.

(9)

Based on the multi-rate sampling mechanism, the objective of this paper is to design a finite-horizon state estimator with form (5) for system (1) such that the following two performance requirements are met simultaneously: • (R1) For a prescribed disturbance attenuation level γ > 0 and a given weighted matrix W > 0, the estimation error output z˜η (k ) in (8) with respect to the energy bounded disturbance ϑη (tk ) satisfies the following H∞ performance criterion over a finite-horizon [0, tL ]:

E{z˜η (tk )2[0,L] } ≤ γ 2 E{ϑη (tk )2[0,L] } + γ 2 E{eTη (t0 )W eη (t0 )} L

(10) L

where E{z˜η (tk )2[0, L] } := k=0 E{z˜η (tk )2 } and E{ϑη (tk )2[0, L] } := k=0 E{ϑη (tk )2 }. • (R2) For a sequence of specified positive definite matrices { (tk )}1 ≤ k ≤ L , at each time instant tk , the estimation error covariance satisfies

E{eη (tk )eTη (tk )} ≤ (tk ),

∀k = 1, 2, . . . , L.

(11)

Remark 2. Compared with the existing multi-rate systems in [41,42], there are two prominent features with (8): 1) rather than the time-invariant systems considered in most existing literature, a class of time-varying systems is investigated and the variance-constrained H∞ performance is enforced on the state estimation problem over a finite-horizon; and 2) the redundant channel transmission strategy is employed so as to improve the reliability of the networked data communication. 3. Main results In this section, we shall address the state estimator analysis and synthesis issues for the time-varying system (1)-(2). First, sufficient conditions are given to ensure that the estimation error dynamics satisfies both the disturbance attenuation performance and the error variance constraints. Then, the estimator parameters are recursively derived by solving a set of recursive matrix inequalities. 3.1. The H∞ performance analysis Theorem 1. Consider the discrete time-varying stochastic multi-rate system (1)–(2). Let the estimator parameters {Ks (tk )}0≤k≤L (s = 1, 2, . . . , b), the initial positive definite matrix W > 0, the non-negative scalar μ0 and the prescribed disturbance attenuation level γ be given. If the following recursive matrix inequality

¯ tk ) + H  T (tk )P (tk+1 )H  (tk ) ≤ 0 (tk )  (

(12)

is feasible with the solution set {P (tk+1 )} for all 0 ≤ k ≤ L subject to the initial condition

E{eTη (t0 )P (t0 )eη (t0 )} + μ(t0 ) ≤ γ 2 E{eTη (t0 )W eη (t0 )}

(13)

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where

¯ tk ) = diag{ ¯ 1 (tk ), μ ¯ 2 (tk )}, μ ( ˜ (tk ),  ˜ (tk ) = μ(tk+1 ) − μ(tk ), ¯ 1 (tk ) = MTη (tk )Mη (tk ) + RT (tk )P (tk+1 )R(tk ) − P (tk ),  ¯ 2 (tk ) = − γ 2 I + BηT (tk )P (tk+1 )Bη (tk ),   (tk ) = [H¯ (tk ) H

H¯ (tk )xˆη (tk )

0], H¯ (tk ) = [H (tk )

0],

H (tk ) = (IM  Kη (tk ))C¯ (tk ), R(tk ) = A¯ η (tk ) − Fη(1) (tk ), C¯ (tk ) = [C1T (tk )

μ(tk ) = 

μ0

tk + 1

C2T (tk )

···

T CM (tk )]T ,

, P (tk+1 ) =   P (tk+1 ),

then the performance constraint defined in (10) is guaranteed for all nonzero ϑη (tk ). Proof. First, define the following real-valued function

V (tk )  eTη (tk )P (tk )eη (tk ) + μ(tk ). Then, calculating the difference of V(tk ) along the trajectory of (8) and taking the mathematical expectation, we have

E{ V (tk )}



= E eTη (tk+1 )P (tk+1 )eη (tk+1 ) − eTη (tk )P (tk )eη (tk ) + μ(tk+1 ) − μ(tk ) =E



Aη (tk )eη (tk ) − Fη (tk )xˆη (tk ) + Bη (tk )ϑ (tk )

T



× P (tk+1 ) Aη (tk )eη (tk ) − Fη (tk )xˆη (tk ) + Bη (tk )ϑ (tk ) − eT (tk )P (tk )e(tk ) + μ(tk+1 ) − μ(tk )











= E eTη (tk ) (A¯ η (tk ) − Fη(1) (tk ))T P (tk+1 )(A¯ η (tk ) − Fη(1) (tk ))



+ Fη(2)T (tk )P (tk+1 )Fη(2) (tk ) − Fη(1)T (tk )P (tk+1 )Fη(1) (tk ) − P (tk ) eη (tk )





+ 2eTη (tk ) Fη(2)T (tk )P (tk+1 )Fη(2) (tk ) − Fη(1)T (tk )P (tk+1 )Fη(1) (tk ) xˆη (tk )





+ xˆTη (tk ) Fη(2)T (tk )P (tk+1 )Fη(2) (tk ) − Fη(1)T (tk )P (tk+1 )Fη(1) (tk ) xˆη (tk )



+ ϑ T (tk )BηT (tk )P (tk+1 )Bη (tk )ϑ (tk ) + μ(tk+1 ) − μ(tk ) .

(14)

In order to facilitate our derivation, let us now deal with the term

(tk )  E{Fη(2)T (tk )P (tk+1 )Fη(2) (tk ) − Fη(1)T (tk )P (tk+1 )Fη(1) (tk )}. According to the definition of Fη(2 ) (tk ) and Fη(1 ) (tk ), one derives immediately that



(tk ) = E

˜ tk ) ˜ T (tk )KηT (tk )P (tk+1 )Kη (tk )( ∗



0 0

(15)

 ˜ tk )  (tk ) − ( ¯ tk ) = M β˜i (tk )Ci (tk ) with β˜i (tk )  βi (tk ) − β¯ i (tk ). where ( i=1 T ˜ T (tk )P (tk+1 )( ˜ tk )} as follows: Denoting P (tk+1 )  Kη (tk )P (tk+1 )Kη (tk ), we proceed to handle the term E{

˜ T (tk )P (tk+1 )( ˜ tk )} E{



T β˜1 (tk )C1 (tk ) + · · · + β˜M (tk )CM (tk ) P (tk+1 )  ×(β˜1 (tk )C1 (tk ) + · · · + β˜M (tk )CM (tk )     = E C¯ T (tk ) βˇ (tk )  P (tk+1 ) C¯ (tk ) =E

= C¯ T (tk )(  P (tk+1 ) )C¯ (tk ) = C¯ T (tk )(IM  Kη (tk ))T (  P (tk+1 ) )(IM  Kη (tk ))C¯ (tk )

(16)

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where β˜i j (tk )  β˜i (tk )β˜ j (tk ) (1 ≤ i, j ≤ M ), βˇ (tk ) = [β˜i j (tk )]M×M and  = [σi j ]M×M is the covariance matrix of the stochastic vector β (tk ). Therefore, (15) can be calculated as

(tk ) = diag{HT (tk )P (tk+1 )H (tk ), 0}

(17)

= H¯ T (tk )P (tk+1 )H¯ (tk ). It can be inferred from (14)–(17) that

E{ V (tk )}





≤ E eTη (tk ) (A¯ η (tk ) − Fη(1) (tk ))T P (tk+1 )(A¯ η (tk ) − Fη(1) (tk ))



+H¯ T (tk )P (tk+1 )H¯ (tk ) − P (tk ) eη (tk ) + eTη (tk )(H¯ T (tk )P (tk+1 )H¯ (tk ))xˆη (tk )

(18)

+ xˆTη (tk )(H¯ T (tk )P (tk+1 )H¯ (tk ))eη (tk ) + xˆTη (tk )(H¯ T (tk )P (tk+1 )H¯ (tk ))xˆη (tk )



+ ϑηT (tk )BηT (tk )P (tk+1 )Bη (tk )ϑη (tk ) + μ(tk+1 ) − μ(tk ) . Then, setting ξ (tk )  [eTη (tk ) 1 ϑηT (tk )]T and adding the zero term z˜η (tk )2 − γ 2 ϑη (tk )2 − (z˜η (tk )2 − 2 η (tk ) ) to the right-hand side of (18) lead to

γ 2 ϑ

E{ V (tk )} ≤ E{ξ T (tk )(tk )ξ (tk ) − (z˜η (tk )2 − γ 2 ϑη (tk )2 )}.

(19)

Subsequently, summing up (19) on both sides from 0 to L with respect to k yields

E{V (tL+1 ) − V (t0 )} ≤

L 

E



 ξ T (tk )(tk )ξ (tk ) − (z˜η (tk )2 − γ 2 ϑη (tk )2 ) .

(20)

k=0

By noting (12), the initial condition (13) and V (tL+1 ) ≥ 0, it can be directly derived from (20) that L 

E



z˜η (tk )

2





≤γ E 2

k=0

L 



ϑη (tk ) + eη (t0 )W eη (t0 ) , 2

T

(21)

k=0

which implies that the H∞ performance requirement is met and the proof is thus complete.



¯ tk )” is semi-negative definite Remark 3. It is observed from (14) that the term “−Fη(1 )T (tk )P (tk+1 )Fη(1 ) (tk )” denoted by “ (

and nonlinear with respect to the estimator parameters Ki (i = 1, 2, · · · , b), and this poses certain technical difficulties on the ¯ tk ) has been discarded estimator parameter design under the linear matrix inequality (LMI) framework. In [20], the term ( for the simplicity of mathematical analysis, but such a treatment would inevitably give rise to certain conservatism. In this paper, this term is properly handled in (15) and (16) by some equivalent operations, thereby avoiding unnecessary design conservatism. 3.2. Analysis on the estimation error variance After ensuring the H∞ performance index defined in (10), our next step is to establish easily verifiable conditions under which the estimation error dynamics (7) satisfies the variance performance requirement (11). Theorem 2. Consider the discrete multi-rate time-varying stochastic system (1) and (2). Let the estimator parameters {Ks (tk )}0 ≤ k ≤ L (s = 1, 2, . . . , b) in (6) and the positive scalar δ be given. Assume that there exists a family of positive definite matrices {Q (tk )}1≤k≤L+1 satisfying the following recursive matrix inequality:

  (1 + δ ) R(tk )Q (tk )RT (tk ) + S (tk )(  Q¯ (tk ))S T (tk )

¯ tk ))S T (tk ) + Bη (tk )W (tk )BηT (tk ) + (1 + δ −1 )S (tk )(  ( ≤ Q (tk+1 )

(22)

for all 0 ≤ k ≤ L subject to the initial condition X (t0 ) = Q (t0 ), where

S (tk ) = Kη (tk )Cˆ(tk ), Q¯ (tk ) = I1 Q (tk )I1T , Cˆ(tk ) = [C1 (tk )

C2 (tk )

···

CM (tk )],

¯ tk ) = I1 (tk )I T , ¯ (tk )}, ( W (tk ) = diag{I, W 1

(tk ) = xˆη (tk )xˆTη (tk ), I1 = [I

0], I = [Ii j ]b×b ,

L. Wang, Z. Wang and G. Wei et al. / Information Sciences 501 (2019) 222–235

¯ (tk ) = α¯ 1 D1 (tk )DT1 (tk ) + W

(1 − α¯ l )α¯ s Ds (tk )DTs (tk ),

s=2 l=1

 Ii j =

M  s−1 

229

T [Ib− j+1 I ji Ib−i+1

0T( j−i )×(b− j+1) ]T ,

i< j i> j i = j.

Then, we have

E{eη (tk )eTη (tk )} ≤ Q (tk ) (∀ 1 ≤ k ≤ L + 1 ). Proof. From the definition of state variance matrix X(tk ) in (9) and the augmented system dynamics (8) as well as the property of conditional expectation, the evolution of X(tk ) is governed by

X (tk+1 )

(23)

= E{eη (tk+1 )eTη (tk+1 )} =E

 

Aη (tk )eη (tk ) − Fη (tk )xˆη (tk ) + Bη (tk )ϑ (tk )

× Aη (tk )eη (tk ) − Fη (tk )xˆη (tk ) + Bη (tk )ϑ (tk )



  T



= E Aη (tk )X (tk )ATη (tk ) + Fη (tk )xˆη (tk )xˆTη (tk )FηT (tk ) − Aη (tk )eη (tk )xˆTη (tk )FηT (tk ) − Fη (tk )xˆη (tk )eTη (tk )ATη (tk )



+ Bη (tk )W (tk )BηT (tk ) .

(24)

With the help of the following elementary inequality

(δ 2 eη (tk ) + δ − 2 xˆη (tk ))(δ 2 eη (tk ) + δ − 2 xˆη (tk ))T ≥ 0 1

1

1

1

(25)

for any positive scalar δ > 0, it is easily derived that

− Aη (tk )eη (tk )xˆTη (tk )FηT (tk ) − Fη (tk )xˆη (tk )eTη (tk )ATη (tk ) ≤ δ Aη (tk )X (tk )ATη (tk ) + δ −1 Fη (tk )xˆη (tk )xˆTη (tk )FηT (tk ),

(26)

which implies that

X (tk+1 )



≤ E (1 + δ )Aη (tk )X (tk )ATη (tk ) + (1 + δ −1 )Fη (tk )xˆη (tk )xˆTη (tk )FηT (tk ) + Bη (tk )W (tk )BηT (tk )







= E (1 + δ ) (A¯ η (tk ) − Fη(1) (tk ))X (tk )(A¯ η (tk ) − Fη(1) (tk ))T + Fη2 (tk )X (tk )Fη(2)T (tk ) − Fη(1) (tk )X (tk )Fη(1)T (tk )



+ (1 + δ −1 )Fη (tk )xˆη (tk )xˆTη (tk )FηT (tk ) + Bη (tk )W (tk )BηT (tk )







= E (1 + δ ) (A¯ η (tk ) − Fη(1) (tk ))X (tk )(A¯ η (tk ) − Fη(1) (tk ))T



+ Fη (tk )X (tk )FηT (tk ) + (1 + δ −1 )Fη (tk )xˆη (tk )xˆTη (tk )FηT (tk ) + Bη (tk )W (tk )BηT (tk )



= X¯ (tk+1 ). In order to cope with the stochastic term Fη (tk ) in (27), one further has

E{Fη (tk )xˆ(tk )xˆT (tk )FηT (tk )} ˜ tk )( ¯ tk ) ˜ T (tk )KηT (tk )} = E{Kη (tk )(







¯ tk ) = E Kη (tk ) β˜1 (tk )C1 (tk ) + · · · + β˜M (tk )CM (tk ) (

(27)

230

L. Wang, Z. Wang and G. Wei et al. / Information Sciences 501 (2019) 222–235

×



β˜1 (tk )C1 (tk ) + · · · + β˜M (tk )CM (tk )η (tk )

T 

¯ tk ))CˆT (tk )KηT (tk ). = Kη (tk )Cˆ(tk )(  (

(28)

Along the same line, we can also obtain that

E{Fη (tk )X (tk )FηT (tk )} = Kη (tk )Cˆ(tk )(  X¯ (tk ))CˆT (tk )KηT (tk )

(29)

where X¯ (tk ) = I1 X (tk )I1T . In the following, let us finish the rest of the proof by induction. First, it follows from the initial condition that X (t0 ) = Q (t0 ). Then, assuming that X(tk ) ≤ Q(tk ), it remains to show that X (tk+1 ) ≤ Q (tk+1 ) is true. Before proceeding further, we introduce an auxiliary function as follows:

f (Q (tk ))  (1 + δ )(A¯ η (tk ) − Fη(1) (tk ))Q (tk )(A¯ η (tk ) − Fη(1) (tk ))T + (1 + δ )Kη (tk )Cˆ(tk )(  Q¯ (tk ))CˆT (tk )KηT (tk ) + (1 + δ

−1

(30)

¯ tk ))C (tk )Kη (tk ) )Kη (tk )Cˆ(tk )(  ( ˆT

T

+ Bη (tk )W (tk )BηT (tk ). With the combination of (27)–(30) and through some straightforward manipulations, we have

X¯ (tk+1 ) − f (Q (tk )) = (1 + δ )(A¯ η (tk ) − Fη(1) (tk ))(X (tk ) − Q (tk ))(A¯ η (tk ) − Fη1 (tk ))T + (1 + δ )Kη (tk )Cˆ(tk )(  (X¯ (tk )

(31)

− Q¯ (tk )))CˆT (tk )KηT (tk ) ≤ 0. As a consequence, it is obvious that

X (tk+1 ) ≤ X¯ (tk+1 ) ≤ f (Q (tk )) ≤ Q (tk+1 ), which means that E{eη (tk

(32)

η k )} ≤ Q (tk ). Hence, the proof of Theorem 2 is complete.

)eT (t



3.3. Estimator design subject to mixed performance constraints The analysis results have been presented in Theorems 1 and 2 for the proposed multi-rate state estimation problem subject to mixed H∞ and variance performance requirements. Now, we are in a position to deal with the design problem for the estimator parameters, which can be obtained by solving a set of recursive matrix inequalities. Theorem 3. Consider the discrete time-varying stochastic multi-rate system (1) and (2) and state estimator (5). Let the initial positive definite matrix W > 0, the disturbance attenuation level γ , the positive scalar δ and a succession of prescribed estimation error variance bounds (tk )0≤k≤L+1 be given. Assume that there exist a set of matrices {Ks (tk )}0≤k≤L (s = 1, 2, . . . , b), two families of positive definite matrices {(tk )}1≤k≤L+1 and {Q (tk }1≤k≤L+1 such that the following recursive linear matrix inequalities

Q (tk ) − (tk ) ≤ 0,

⎡ 11 ϒ1 (tk ) ⎢ ∗ ϒ1 (tk ) = ⎣ ∗ ∗

and

⎡ϒ 11 (t ) k 2 ⎢ ∗ ϒ2 (tk ) =⎢ ⎣ ∗ ∗ ∗

(33)  T (tk ) H ϒ122 (tk ) ∗ ∗

ϒ113T (tk )

ϒ212 (tk ) −Q (tk )

ϒ213 (tk )

∗ ∗ ∗

0

ϒ133 (tk ) ∗

0 −Q¯  (tk ) ∗ ∗

⎤ ϒ114T (tk ) 0 ⎥ ⎦≤0 0 44 ϒ1 (tk ) ϒ214 (tk ) 0 0 −I ∗

ϒ215 (tk )⎤ 0 ⎥ 0 ⎥ ⎦≤0

(34)

(35)

0 ¯ (tk ) −W

are satisfied for all 0 ≤ k ≤ L under initial conditions (13) and E{eη (t0 )eTη (t0 )} = Q (t0 ) ≤ (t0 ). Then, the estimation error dynamics satisfies both the H∞ performance (R1) and the variance-constrained requirement (R2). In addition, the parameters are updated by

P (tk+1 ) = R−1 (tk+1 )

(36)

L. Wang, Z. Wang and G. Wei et al. / Information Sciences 501 (2019) 222–235

231

where

ϒ111 (tk ) = diag{ϒ¯ 111 (tk ), μ˜ (tk ), −γ 2 I}, ϒ113 (tk ) = [R(tk )

0

0],

ϒ122 (tk ) = − R (tk+1 ), ϒ133 (tk ) = ϒ144 (tk ) = −R(tk+1 ), Bη (tk )], R (tk+1 ) =  −1  R(tk+1 ), ϒ¯ 111 (tk ) = MT (tk )M(tk ) − P (tk ), ϒ213 (tk ) = δ¯ S (tk )Q¯  (tk ),

ϒ114 (tk ) = [0

0

(37)

¯ 1/2 (tk ), ϒ (tk ) = − Q (tk+1 ) + Bη (tk )IBη (tk ), ϒ (tk ) = δ˜ S (tk )  11 2

T

14 2

ϒ215 (tk ) = Kη (tk )W¯ (tk ), Q¯  (tk ) =   Q¯ (tk ),   ¯  (tk ) =   ( ¯ tk ), δ¯ = 1 + δ , δ˜ = 1 + δ −1  and other parameters are defined in Theorems 1 and 2. Proof. By using the Schur Complement Lemma, (12) holds if (34) can be guaranteed. Similarly, it is straightforward to see that (22) holds if (35) is satisfied. Furthermore, it is obvious from (33) that E{eη (tk )eTη (tk )} ≤ (tk ), and the rest of the proof follows immediately from that of Theorems 1 and 2. As such, the performance constraints (R1) and (R2) are simultaneously attained and the proof of this theorem is complete.  In Theorem 3, we have endeavored to settle the design problem of the time-varying estimator parameters Ks (tk ) (s = 1, 2, · · · , b). Note that the constraints (34) and (35) are linear with regard to all unknown variables, which can be expediently solved in terms of the semi-definite programming approach. In the following, an iterative algorithm (Algorithm 1) is given to show how to obtain the time-varying parameters recursively. Algorithm 1 : Multi-Objective-Based Recursive State Estimator Algorithm. Step 1. Initialization: given the disturbance attenuation level γ and positive definite matrix W . Set the initial values xη (t0 ), xˆη (t0 ), P (t0 ), Q (t0 ) and (t0 ) satisfying (14) and Q (t0 ) ≤ (t0 ). Set the time horizon L and t0 = 0. Step 2. With the known {P (tk ), Q (tk ), (tk )}, determine {Ks (tk ), Q (tk+1 ), R(tk+1 )} by solving a set of linear matrix inequalities (35)-(34). Step 3. Set k = k + 1 and obtain P (tk+1 ) according to the update eq. (37). Step 4. If k < L + 1, then repeat Step 2-Step 3, else stop.

4. An illustrative example In this section, we verify the effectiveness of the estimation method proposed in this paper through a numerical example. Consider a discrete time-varying system with the following parameters:



0.78 + 0.2sin(2Ts ) A(Ts ) = 0.48 + 0.01



0.3 , B(Ts ) 0.6 + 0.1cos(Ts )





0.13 + 0.02sin(Ts ) = , M (Ts ) = [0.1 0.13 + 0.02sin(Ts )

0.1].

The transmission path of the sensor measurements consists of three redundant channels described by the following parameters:

C1 (tk ) = C2 (tk ) = C3 (tk )= [2.5 + 0.1sin(tk )

2.5],

D1 (tk ) = D2 (tk ) = D3 (tk ) = 0.15,

α¯ 1 = 0.6, α¯ 2 = 0.75, α¯ 3 = 0.95 where α¯ i (i = 1, 2, 3 ) are the probabilities of the successfully transmitted packets through the individual channels. In this simulation, suppose that the system under consideration possesses two sampling rates, namely, the update of the system state is at a fast rate with the period h while the sensor samples the available state information at a slow one with period 2h, i.e. b = 2. The H∞ performance level, the positive definite matrix W, (tk ) and the positive scalar are set as γ = 1.5, W = 5I, (tk ) = 1.8I and μ0 = 15, respectively. The initial state of the plant (1) and its estimation are, respectively, taken as x(t0 ) = [2.2 −2.3]T and xˆ(t0 ) = [1.5 −1.5]T . Moreover, the desired time-varying estimator parameters are listed in Table 1. The simulation results are shown in Figs. 2–5, where Figs. 2 and 3 depict the system state evolution of individual component xi (Ts ) and its estimation xˆi (Ts ) (i = 1, 2 ). Fig. 4 plots the variance upper bound and the actual variance of the estimation error for individual components, from which it can be clearly seen that the actual error variance of each state component is less than its upper bound. A comparison simulation regarding the filtering performance with different communication cases is conducted, and the comparison results are described in Fig. 5. From Fig. 5, we observe that the estimator performs very well with three channels and, however, the estimation error is divergent with two channels. In addition, when only one channel is used, the proposed

232

L. Wang, Z. Wang and G. Wei et al. / Information Sciences 501 (2019) 222–235 Table 1 The desired filter parameters. k K1 (tk )

0



0.20 0 0 0.2138

 K2 (tk )





0.2564 0.2358

1

 



0.1728 0.1903



0.1822 0.1783

2





0.2416 0.1984





0.2217 0.2406

3





0.1843 0.2210



 



0.2465 0.2377



30









0.2023 0.1948 0.1766 0.1990

Fig. 2. The actual state x1 (Ts ) and its estimation xˆ1 (Ts ).

Fig. 3. The actual state x2 (Ts ) and its estimation xˆ2 (Ts ).

Algorithm 1 is infeasible, which again shows the superiority of the adopted communication mechanism. As such, the numerical experiment confirms that the redundant channel transmission strategy indeed enhances both the communication reliability and the filtering performance. It is shown in the simulation results that the designed estimators achieve a satisfactory estimation performance even though the original target plant is unstable, thereby validating the effectiveness of the multi-objective state estimation scheme proposed in this paper.

L. Wang, Z. Wang and G. Wei et al. / Information Sciences 501 (2019) 222–235

233

Fig. 4. The actual estimation error variance and the corresponding upper bound.

Fig. 5. The absolute value of the estimation error under different communication cases.

5. Conclusions In this paper, the finite-horizon state estimation problem has been addressed for a class of time-varying stochastic multirate systems subject to both the estimation error variance and disturbance attenuation constraints. In order to enhance the data communication reliability, the measurement model with redundant channels has been established by introducing a set of Bernoulli-distributed white sequences with known probabilities. By resorting to the lifting technique, the original multi-rate system has been transformed into the tractable single-rate one. With the aid of stochastic analysis techniques, sufficient conditions have been presented to guarantee the expected performance requirements. Moreover, the estimator parameters have been explicitly expressed in terms of the solutions to certain recursive matrix inequalities. Finally, an illustrative example has been provided to demonstrate the effectiveness of the proposed estimator design scheme. Other possible research directions would be the further extension of the current results to the communication-protocol-based state estimation/filtering problems, see e.g. [2–4,24,37], and to some complicated distributed systems such as the complex networks [25,29], the sensor networks [15], the memristive neural networks [11,12], and the multi-agent systems [28].

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Declaration of Competing Interest The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (RG-15-135-40). The authors, therefore, acknowledge with thanks DSR for technical and financial support. This work was also supported in part by the National Natural Science Foundation of China under Grants 61873148 and 61873169, and the Alexander von Humboldt Foundation of Germany (https://www.humboldt-foundation.de). References [1] H. Dong, Z. Wang, B. Shen, D. Ding, Variance-constrained H∞ control for a class of nonlinear stochastic discrete time-varying systems: the event-triggered design, Automatica 72 (2016) 28–36. [2] X. Ge, Q.L. Han, Z. Wang, A dynamic event-triggered transmission scheme for distributed set-membership estimation over wireless sensor networks, IEEE Trans. Cybern. 49 (1) (2019) 171–183. [3] X. 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