Observer-based quantized control for discrete-time switched systems with infinitely distributed delay

https://doi.org/10.1016/j.jfranklin.2022.03.012Get rights and content

Abstract

This paper studies the globally almost surely exponential stabilization of discrete-time switched systems (DSSs) with infinitely distributed delay. On account of the limitation of communication resources in the actual environment, a novel class of observer-based quantized control scheme is designed that incorporates the quantization of three kinds of signals: the measurement output, the state of observer, and the measurement output of observer. By employing S-procedure and some matrix inequality techniques, an algorithm is given to design the controller parameters. To reduce the conservativeness of the obtained results, new multiple Lyapunov–Krasovskii functionals (LKFs) with negative terms are proposed to deal with the infinitely distributed delay and mode-dependent average dwell time (MDADT) switching based on transition probability (TP) is introduced to study the stabilization of DSSs with both stable and unstable modes. It is worth highlighting that the improved stabilization conditions for DSSs can release the restriction on the length of dwell time (DT) of stable and unstable subsystems. Finally, a simulation example is presented to demonstrate the validity of the proposed method.

Introduction

As an important and special class of hybrid systems, switched systems, which consist of a limited number of subsystems and evolve according to a certain switching law, have been widely applied in such a series of engineering fields as power electronics [1], intelligent traffic control [2], aircraft navigation control system [3], etc. Since the switching law determines the active state of the subsystem at each moment, it has a great impact on the dynamic behavior of the whole system and has aroused the research interest of many scholars. There are a number of different switching laws that have been put forward, such as slow switching [4], arbitrary switching [5], average dwell time (ADT) switching [6], [7], [8], [9], [10], and so on. Compared with the other switching laws mentioned above, ADT switching has received more attention due to its flexible and effective characteristics. However, it should be noted that, the meaning behind the ADT switching is that all modes have to enjoy common ADT. Apparently, it is limited and impractical in application since each mode has its own dynamic evolution, so the results of ADT switching that do not take into account differences between modes may be conservative. Whereupon, improved switching laws have been proposed from two different perspectives: one is time-dependent and the other is mode-dependent. As a typical time-dependent switching rule, persistent dwell time (PDT) switching has been wildly used in the research of switched systems because it has great advantages in describing the system with the large uncertainty by focusing on fast switching and slow switching that take place in turn, such as singularly perturbed switched systems [11], [12]. Moreover, Wang et al. [13] have also applied the PDT switching idea to investigate the nonfragile H synchronization issue for a class of discrete-time Takagi–Sugeno fuzzy Markov jump systems. For the mode-dependent switching, Zhao et al. [14] integrated this idea into the ADT switching, and proposed a more flexible and applicable MDADT switching, in which the ADT associated with each mode can be different and the obtained stability criteria for switched systems are mode-dependent. To further reduce the conservativeness of the results, Yang et al. [15] introduced TP that describes the possibility of each mode being activated into MDADT switching and called it TP-based MDADT switching. Based on this switching law, the DT of unstable subsystems can also be free from the limitation of upper bound by playing the role of the TP matrix. Therefore, Yang et al. [16] further researched the globally almost surely exponential synchronization of complex networks with TP-based MDADT switching by designing controllers via actuator fault and impulsive effects. Considering the potential application value of discrete-time systems in the fields of system identification, time series analysis and image processing, ADT switching and MDADT switching have been extended to the stability research of DSSs in recent years [17], [18], [19]. Unfortunately, very few papers have been studied the stability of the DSSs with TP-based MDADT switching, which inspires the research of this article.

Time delay is an inherent phenomenon in the engineering field, involving communication, chemical processes, nuclear reactors, biological systems, etc. The appearance of it often has a great influence on the performance of the system, such as instability, oscillation and chaos. Therefore, the stability analysis and control design of time-delayed systems have attracted extensive attention from researchers [20], [21], [22], [23]. Especially, the infinitely distributed delay has been commonly considered since signal transmission exists in a certain period of time, which often makes the current state of the system affected by the previous state for a long time. For instance, Wan et al. [24] investigated exponential synchronization of semi-Markovian coupled neural networks with time-varying delay and infinitely distributed delay. In [25], the consensus of linear multiagent systems with distributed infinite transmission delays was researched by designing a low gain controller. Although infinitely distributed delay has been widely studied, most of the existing results focus on continuous-time systems. To fill the gap, Liu et al. [26] first defined the infinitely distributed delay in discrete time domain, and discussed the synchronization and state estimation of a class of discrete-time complex networks with discrete and distributed time delays. Subsequently, some literature further extended the study of infinitely distributed delay to switched systems [27], [28]. It is remarkable that the common method to deal with the delay in existing papers is to design such Lyapunov–Krasovskii functional d=1βdv=kdk1x(v)Pσ(k)x(v), where d=1βdx(kd) denotes the infinitely distributed delay and Pσ(k) is a symmetric positive definite matrix with switching signal σ(k). Although it is effective, there is room for new design techniques of LKFs aiming at less conservative criteria. Moreover, there is no article that introduces infinitely distributed delay into stability analysis for DSSs under the framework of TP-based MDADT switching, which is our second motivation in the present paper.

Another important issue in the study of dynamic behavior of DSSs is the control scheme and a variety of control techniques have been proposed [29], [30], [31], [32], [33]. In practice, the state information of the system is often unable to be directly measured, only the output signal of the system can be measured. Therefore, it is a very effective approach to use the output of the system to construct state observers, and then design controllers to achieve the desired performance of the system by using the observed states, which has been widely applied in the literature concerning stability of DSSs [34], [35], [36]. On the other hand, quantitative technology is a useful tool in saving bandwidth and channel resources. The principle is to convert a real-valued signal into a piecewise constant one taking on a finite or infinite set of values by means of a quantizer device. And only these discrete quantization levels are required to transmit in the state space. Thus it has been generally utilized in the design of controllers. However, as far as we know, the earlier studies focus on the quantification of information transmission on either measurement output [37], [38], [39], or the state of observer [40], [41], or the two [42], [43], [44]. There is a lack of a stability theory for DSSs concerned with the three kinds of quantized signals including the measurement output, the state of observer, and the measurement output of observer. And it is easy to find that if these transmission signals are quantized simultaneously, the proposed methods will no longer be applicable.

Based on the above discussions, this paper is concerned with the stabilization of DSSs with infinitely distributed delay by virtue of a class of observer-based quantized control strategy, which aims to improve the existing corresponding results from the perspectives of reducing the conservativeness and saving communication resources. The main challenges are: (1) How to handle the influence caused by the error between the three types of quantized signals and the real signals? (2) How to make some subsystems stable when affected by infinitely distributed delay under the designed controller? To overcome these two difficulties, a set of new analytical techniques have to be developed in the follow-up paper. Then the main contributions are outlined as follows:

  • (1)

    Compared with classical ADT switching and MDADT switching, a more flexible and practical TP-based MDADT switching law is considered to study globally almost surely exponential stabilization of DSSs with both stable and unstable modes. And our results indicate that even if only one mode is controlled, the stabilization of the whole switched system can still be achieved, and the lower bound of the DT of the controlled subsystem can be very small. Furthermore, the globally almost surely exponential stabilization criteria of discrete-time systems with Markovian switching are also given as a special case.

  • (2)

    Without directly using system state information, measured output signals are utilized to design observer-based controllers. Moreover, the quantization scheme focusing on three types of signals: the measurement output, the state of observer, and the measurement output of observer is introduced into the observer-based controller to effectively save communication resources. Thus the controller proposed in this paper is more practical and realistic.

  • (3)

    By proposing some summation inequalities, new multiple LKFs with negative terms are developed to deal with infinitely distributed delay, which can obtain the stabilization criteria with less conservativeness.

The rest of this paper is organized as follows. The considered model, definition, assumptions and lemmas are presented in Section 2. In Section 3, stabilization conditions are obtained by strict mathematical proofs. In Section 4, a numerical simulation is shown to testify the theoretical analysis. Finally, conclusions are given in Section 5.

Notations: N, N+, Rn and Rm×n denote the sets of nonnegative integers, positive integers, n×1 real vectors and all m×n real matrices, respectively. In represents the n×n identity matrix. The superscript means matrix or vector transposition. Symbol * in a matrix is used to describe the symmetric part of the matrix. λmin(A) denotes the minimum eigenvalue of matrix A. Denote A>0 if A is a symmetric positive definite real matrix. · stands for the standard Euclidean norm of a vector. E{·} is defined as the mathematical expectation with the probability measure Pr. sym{A} means A+A.

Section snippets

Preliminaries and problem formulation

Suppose that {ks,sN} is a time sequence satisfying 0=k0<k1<k2< and limsks=. Define a switching signal σ(k) mapping from N to I={1,2,,r}, r>1. When k[ks1,ks),sN+, σ(k)=iI, it is considered that the ith mode is activated. Moreover, hi(s)=ksks1 is said to be the DT of the ith mode for the sth visiting of σ(k).

Consider a class of DSSs with infinitely distributed delay as follows:{x(k+1)=Cσ(k)x(k)+Aσ(k)d=1βdx(kd)+Bσ(k)uσ(k)(k),y(k)=Qσ(k)x(k),where x(k)=(x1(k),x2(k),,xn(k))Rn

Main results

This section gives GES a.s. conditions for the system Eq. (1) with infinitely distributed delay under the observer-based quantized controller. Both multiple LKFs with negative terms and TP-based MDADT switching are utilized to reduce the conservativeness of the obtained results. Note that the quantization technique is introduced into three types of signals at the same time, it is difficult to overcome the sector-bounded uncertainty caused by the logarithmic quantizers by directly using the

Numerical examples

For proving the availability of the obtained theoretical analysis, an example is given in this section.

Consider a discrete-time switched agent system with three modes, in which each subsystem is a double integrator model in the discrete-time setting [50]. And the system parameters are given byC1=C2=C3=(10.101),B1=B2=B3=(00.1),A1=(0.20.100.12),A2=(0.1800.20.1),A3=(0.200.10.2),Q1=(0.30.10.10.3),Q2=(0.20.10.20.2),Q3=(0.20.10.10.2).Choosing δ1=0.6, δ2=1.2, δ3=1.5, βd=37d, it is easy to obtain

Conclusions

The GES a.s. problem for a class of DSSs with infinitely distributed delay has been investigated in this work. First, an observer-based quantized controller with three kinds of quantized signals has been proposed, which can not only reduce the communication burden but also avoid using unmeasurable system state information. The design method of controller parameters is also given explicitly. Next, by designing multiple LKFs with negative terms and introducing more realistic TP-based MDADT

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported in part by the National Key Research and Development Program of China (Grant no. 2018AAA0101100), in part by the National Natural Science Foundation of China (Grant nos. 61973241 and 62176127), in part by the Natural Science Foundation of Hubei Province (Grant no. 2019CFA007), and in part by the Major Natural Science Foundation of Jiangsu Higher Education Institutions (Grant no. 20KJA120002)..

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