Observer-based H control of discrete time-delay systems with random communication packet losses and multiplicative noises

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Abstract

This paper investigates the observer-based H control problem for a class of discrete-time mixed delay systems with random communication packet losses and multiplicative noises, where the mixed delays comprise both discrete and distributed time-varying delays, the random packet losses are described by a Bernoulli distributed white sequence that obeys a conditional probability distribution, and the multiplicative disturbances are in the form of a scalar Gaussian white noise with unit variance. In the presence of mixed delays, random packet losses and multiplicative noises, sufficient conditions for the existence of an observer-based feedback controller are derived, such that the closed-loop control system is asymptotically mean-square stable and preserves a guaranteed H performance. Then a linear matrix inequality (LMI) approach for designing such an observer-based H controller is presented. Finally, a numerical example is provided to illustrate the effectiveness of the developed theoretical results.

Introduction

H control problem has attracted great attention for its theoretical and practical significance in systems control. The objective of H control is to design controllers such that the closed–loop system is internally stable and its H norm from the external input to the controlled output is less than a prescribed level. Since the theory of H control has proposed by Zames [1], much effort has been made in H controller design in order to guarantee desired stability [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. However, this control is often based on the assumption that the entire state is available, which may not hold in many systems. Therefore, it is necessary to design observers that produce an estimate of the system state [13], [14], [15], [16].

Recently, networked control systems (NCSs) have been widely used in many areas such as industrial automation, unmanned vehicles, remote surgery, robots and so on, as their advantages in practical applications, for instance, the lower cost of installation and implementation, simpler installation and maintenance, etc. [17], [18], [19], [20]. In the control of the closed-loop system by the networks, there are many new problems such as intermittent data packet losses, network-induced time delay, and communication constraints, etc.

First, due to the unreliability of the network links, a networked control system typically exhibits significant communication delays and data loss across the network, which give rise to considerable research attention on how to design control systems with the consideration of the data loss issue [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35]. Furthermore, the problem of the random sensor-to-controller packet losses and the random controller-to-actuator packet losses considered simultaneously has attracted great attention [36], [37], [38], [39], this increases the difficulties in the controller design, as the existence of the random packet losses in the communication channel from the controller to the actuator.

On the other hand, it is inevitable that there exist time delays in dynamic systems due to measurement, transmission and transport lags, computational delays or unexpected inertia of system components, which has been known as a main source to degrade the performance of the control system [40]. In the last decade, significant progress has been made on the analysis and synthesis issues for systems with various types of delays [41], [42], [43], [44], [45], [46], [47], [48], [49]. As occurring very often in reality, discrete time-varying delays and distributed delays have been widely recognized, however, most of available results have been focused on continuous-time systems. With the increasing application of digital control systems, it is meaningful to investigate the issue of how discrete time-varying delays and distributed delays influence the dynamical behavior of a discrete-time system. And Wang et al. have done some pioneering work about this field. However, to the best of the author’s knowledge, the research on discrete-time systems with random packet losses and mixed delays is still an open problem that deserves further investigation.

Meanwhile, the control and filtering problems for systems with multiplicative noises have received much attention since many plants may be modeled by systems with multiplicative noises, and some characteristics of nonlinear system can be approximated by models with multiplicative noises rather than by linearized models [50], and the H output-feedback control as well as passive control of discrete-time systems with state-multiplicative noise has been investigated in [51], [52]. Costa and Benites have studied the linear minimum mean square filter for discrete-time linear systems with Markov jumps and multiplicative noises in [53]. Furthermore, Liu et al. have considered the robust reliable control for discrete-time-delay systems with multiplicative noises. Therefore, it is necessary to investigate observer-based H control of the systems with multiplicative noises.

Motivated by the above discussion, we consider the observer-based H control for discrete-time-delay systems with random packet losses and multiplicative noises. The objective is to design a observer-based H controller such that, in the presence of mixed delays, random packet losses and multiplicative noises, the closed–loop control system is asymptotically mean–square stable and also satisfies a given H disturbance attenuation index.

Notation. The notation used through the paper is fairly standard. N is the set of natural numbers and N+ stands for the set of nonnegative integers; Rn and Rn×m denote, respectively, the n dimensional Euclidean space and the set of all n×m real matrices. The notation P>0(0) means that P is real symmetric and positive definite (semi-definite). In symmetric block matrices or complex matrix expressions, we use an asterisk () to represent a term that is induced by symmetry and diag{} stands for a block-diagonal matrix. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. Moreover, we may fix a probability space (Ω,F,P) where, P, the probability measure, has total mass 1. Prob{α} means the occurrence probability of the event α,E{x} stands for the expectation of stochastic variable x. L2[0,+) is the space of square integrable vectors. The notation ||.|| stands for the usual L2[0,+) norm while |.| refers to the Euclidean vector norm. If A is a symmetric matrix, λmax(A) (respectively λmin(A)) denotes the largest (respectively, smallest) Eigenvalue of A.

Section snippets

Problem formulation

Consider the following discrete-time stochastic system with mixed delays and multiplicative noises:x(k+1)=Ax(k)+Avv(k)x(k)+Adx(k-d(k))+Alm=1+μmx(k-m)+Bω(k)+Du(k),z(k)=E1x(k)+E2ω(k),x(k)=ϕ(k),-<k0,where x(k)Rn is the state vector; u(k)Rm is the control input; z(k)Rr is the output; ω(k)Rq is the disturbance input, which belongs to L2[0,);d(k) denotes the time-varying delay with lower and upper bounds d̲d(k)d¯,kN+ where d̲,d¯ are known positive integers; ϕ(k) is the initial state of

Main results

The following Lemmas are essential in establishing our main results.

Lemma 1

Schur Complement [56]

Given constant matrices S1,S2 and S3, where S1=S1T and S2=S2T. Then S1+S3TS2-1S3<0 if and only ifS1S3TS3-S2<0,or-S2S3S3TS1<0.

Lemma 2

[57]

Let MRn×n be a positive semi-definite matrix, xiRn and constant αi>0(i=1,2,). If the series concerned is convergent, then we havei=1+αixiTMi=1+αixii=1+i=1+αixiTMxi.

In the following theorem, we will derive a sufficient condition such that the closed–loop system (11) is asymptotically mean–square

Numerical example

In this section, a satellite system example is used to demonstrate the effectiveness of the proposed observer-based H control for a class of discrete-time mixed delay systems with random packet losses and multiplicative noises, as shown in [49], [58]. Consider system (1) with measurement (4) under parameters as follows:A=0.800.50.9,Av=0.10.10.10.2,Ad=0.20.10.10.2,Al=-0.10.200.1,B=0.51T,D=0.10.10.10.1T,C1=0.10.1,E1=0.10.1,C2=0.05,E2=0.01,d(k)=1+1+(-1)k2,μm=3-(3+m).Then the following condition

Conclusions

In this paper, the observer-based H control has been studied for a class of discrete-time mixed delay systems with random packet dropouts and multiplicative noises. A new Lyapunov–Krasovskii functional, which is introduced to account for distributed and time-varying discrete-time delays, has been used to design the observer and controller, such that the closed–loop system is asymptotically mean–square stable. And the controller parameters can be obtained by solving certain LMIs. An

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    The work is supported by the Natural Science Foundation of China under Grants 60974021, 61203286 and 61125303, the 973 Program of China under Grant 2011CB710606, the Fund for Distinguished Young Scholars of Hubei Province under Grant 2010CDA081, National Priority Research Project NPRP 4-451-2-168, funded by Qatar National Research Fund.

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