Asynchronous partially mode-dependent control for switched larger-scale nonlinear systems with bounded sojourn time

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Highlights

  • Both the bounded sojourn time and the sojourn probabilities are considered within a unified switching framework to better reflect the change of the interconnections among subsystems in large-scale systems.

  • A novel structure of mode storage rule is designed to adequately utilize the available modes.

  • A new stability criterion is obtained with available modes, the sum of these modes’ sojourn probabilities and the bounds of sojourn time.

  • Under the condition that the corresponding Lyapunov function can be applied to the partially mode-dependent case, an asynchronous controller is designed by using the stochastic information.

Abstract

In this paper, the asynchronous partially mode-dependent control problem is investigated for switched larger-scale systems (SLSSs) with inherent nonlinearities, known sojourn probabilities and bounded sojourn time. The interconnection among subsystems is nonlinear and subject to stochastic switching. The resultant part available mode information is that the modes of controllers and the plant are asynchronous. By introducing a mode storage rule and transforming the asynchronous system into a general switched system, the stability criteria is obtain with available modes, the sum of these modes’ sojourn probabilities and the bounds of sojourn time. Then, some sufficient conditions are derived to guarantee the mean-square stability (MSS) by employing Young inequality and the desired asynchronous controller is designed by using available softwares. At last, an example is provided to illustrate the main methods in this paper.

Introduction

Over the past several decades, large-scale systems have become an active topic in signal processing and control because the huge scale and the high complexity in modern industrial production [1], [2], [3]. A large-scale system is composed of some coupled and interactive subsystems. At the same time, control problem arises in large-scale systems like aerospace, telecommunication network, etc. [4], [5]. Among them, earlier works about the decentralized control for the proposed systems with nonlinear were mainly focused on the nonlinear under a linear growth condition [6]. Recently, corresponding results were extended to a general nonlinear system where the violates the linear growth condition, see [7] and [8].

In the practical system, the interconnection of subsystems may subject to abrupt changes because of random failures, repairs of the component, environmental changes, and so on [9]. For instance, for the smart grid system, the connectivity of its topology inevitably changes due to the plug and play operating [10]. For the nonlinear coupled networks, the variation of coupling mode can be governed by a set of switching signals including Markovian Jump Systems (MJSs) [11], Semi-MJSs (S-MJSs) [12], sojourn probability systems [13] and persistent dwell-time systems [14]. It is noteworthy that MJSs theories are available only when the sojourn time between two jumps is geometric distribution in the discrete-time domain. Although S-MJSs are capable of describing the much broader scope of stochastic switched systems by breaking away from the constraint condition of distribution, both of them need more probabilities in contrast with the switched systems with known sojourn probabilities. Meanwhile, in recent years, the persistent dwell-time switching rule, as a kind of time-dependent switching rule, has been used to manage the switchings among nonlinear singularly perturbed subsystems [15], [16]. However, the random characteristics were ignored in those works. Recently, in [17], the random case has been considered and piecewise-constant transition probabilities have been studied subject to the persistent dwell-time switching rule.

Based on the above discussion, a more general switched rule dependent on both unconstrained sojourn time and sojourn probabilities was proposed in [18]. Furthermore, considering that the sojourn times are usually finite, a recent attempt in [19] was given to make the sojourn time of each system mode be bounded. Inspired by these analyses, we aim to extend the sojourn time in [18] to a bounded one and apply the extended rule to large-scale systems. However, the decentralized control for switched large-scale systems with stochastic switching rule has not been adequately investigated, not to mention the case that both general nonlinear growing conditions and switched rules with bounded sojourn time are involved.

It is well known that large-scale systems are complex and the identification of the interconnections among subsystems is difficult. Accordingly, the decentralized controller is usually asynchronous as the mode information cannot be completely accessed in practical applications. In recent years, asynchronous problems have stirred a lot of research interests, and many relevant research results have been reported, see [20], [21] and [22]. The common reasons for appearing asynchronous information include: (1) time-delays, under which the received mode information is later than the system mode information [23]; (2) communication protocol, under which the mode information update till some conditions are satisfied, for example, event-triggered protocol [24], [25], Round-Robin protocol [26], [27], random access protocol [28]; (3) unobservable modes, which are difficult to identify [29]; (4) measurement outliers, which should be detected and eliminated the influence fleetly [30]. For the asynchronous problems, the authors in [31], [32] and [33] introduced Hidden Markov model to represent the asynchronous between the designed controller and the original system/quantizer. In [34], a partially mode-dependent filter was constructed, where the transmission of the mode was randomly and is modeled by a Bernoulli distributed sequence. In [35], [36] and [37], the authors transformed the systems with asynchronous information to switched models by integrating system modes and observed modes into an new switching sequence with a larger state space. However, as is known to us, the asynchronous controller based on new stochastic switching characteristics, which are dependent on both sojourn probabilities and bounded sojourn times, has not been addressed yet, which still remains as a challenging research issue and constitutes the main motivations for the present research.

This paper investigates the asynchronous partially mode-dependent control for switched large-scale systems with interconnected nonlinear and a more general stochastic switching rule. The major difficulties are given as follows. 1) How to depict the switched interconnections among subsystems in a proper way. 2) The resultant part available mode information is that the modes of controller and the plant is asynchronous, which gives rise to the second difficulties on the controller design. 3) The available modes are obtained randomly. Hence, how to use available modes by quantifying their impact on the controller synthesis in terms of the sojourn probabilities is another difficulty.

The main contributions of this paper can be highlighted as follows:

  • 1.

    both the bounded sojourn time and the sojourn probabilities are considered within a unified switching framework to better reflect the change of the interconnections among subsystems in large-scale systems;

  • 2.

    a novel structure of mode storage rule is designed to adequately utilize the available modes;

  • 3.

    a new stability criteria is obtained with available modes, the sum of these modes’ sojourn probabilities and the bounds of sojourn time;

  • 4.

    under the condition that the corresponding Lyapunov function can be applied to the partially mode-dependent case, an asynchronous controller is designed by using the stochastic information.

The rest of this paper is organized as follows. Section 2 states preliminaries and problem formulation, followed by main results in Section 3. In Section 4, an example is presented to demonstrate the effectiveness of the established design scheme. Finally, conclusions are drawn in Section 5.

Notation: In this paper, λmax(A) and λmin(A) denote the maximum and minimum eigenvalue of A, respectively. Rn denotes, respectively, the n dimensional Euclidean space and the set of all n×m real matrices. The superscript “T” stands for the transposition of considered matrices. The notation P>0 for PRn×n shows that P is positive definite. I denotes the identity matrix of compatible dimension. “col{·}” stands for a column vector and diag{·} means a diagonal matrix. card{I} denotes the number of set I.

Section snippets

System description

Consider a switched larger-scale system (SLSS) composed of m interconnected nonlinear subsystem, in which the dynamics of ith subsystems is described byxij(k+1)=vi(rk)φij,v(rk,k,xv(k))+xi,j+1(k),xin(k+1)=vi(rk)φin,v(rk,k,xv(k))+ui(k),j=1,2,,n1,i=1,2,,mwhere xi(k)=[xi1(k),xi2(k),,xin(k)]TRn and x(k)=[x1(k),x2(k),,xm(k)]TRmn are the state for the ith subsystem and the whole system, respectively. ui(k)R represents the control action. The stochastic process {rk}kZ+ denotes the

Main results

In this section, we devote to the investigations of stability and control problems for SLSS (4) with inherent nonlinearities and bounded sojourn time as well as partially available mode information.

Numerical simulation

Let the large-scale system be composed of 5 interconnected nonlinear subsystems, in which the dynamics are given as: Mode 1:x11(k+1)=|x1,2(k)|+x2,13(k)+x4,22(k)+x1,2(k),x12(k+1)=sin(x1,2(k))+u1(k),x21(k+1)=|x2,2(k)|+(1/2)x1,21/3(k)+x2,2(k),x22(k+1)=sin(x2,2(k))+u2(k),x31(k+1)=|x3,1(k)|+x2,1(k)x2,2(k)+x3,2(k),x32(k+1)=|x3,1(k)|+x4,24/3(k)+u3(k),x41(k+1)=|x1,1(k)|1/2(k)+|x5,1|3/2(k)+x4,2,x42(k+1)=x3,13(k)+u4(k),x51(k+1)=x4,13(k)+x5,2(k),x52(k+1)=|x2,1(k)|+|x5,1(k)|+u5(k).

Mode 2:x11(k+1)=x3,13(k)+

Conclusions

In this paper, the asynchronous control problem has been investigated for switched larger-scale systems with inherent nonlinearities and bounded sojourn time. Firstly, a nonlinear growth condition with switching interconnection structures has been used to depict the inherent nonlinearities. Then, utilizing the partial structure information and the elapsed time of observable instants, the asynchronous case has been equivalently reduced to a general switched system with a series of conditional

References (39)

  • G. Schweiger et al.

    Modeling and simulation of large-scale systems: a systematic comparison of modeling paradigms

    Appl. Math. Comput.

    (2020)
  • N. Sandell et al.

    Survey of decentralized control methods for large scale systems

    IEEE Trans. Automat. Control

    (1978)
  • D. Ding et al.

    Secure state estimation and control of cyber-physical systems: a survey

    IEEE Trans. Syst. Man Cybern.

    (2021)
  • J. Wang et al.

    Coding-decoding-based sliding mode control for networked persistent dwell-time switched systems

    Int. J. Robust Nonlinear Control

    (2021)
  • Y. Zhu et al.

    Predictor methods for decentralized control of large-scale systems with input delays

    Automatica

    (2020)
  • Y. Li et al.

    Adaptive fuzzy decentralized control for a class of large-scale nonlinear systems with actuator faults and unknown dead zones

    IEEE Trans. Syst. Man Cybern.

    (2017)
  • Z. Su et al.

    Global stabilization via sampled-data output feedback for large-scale systems interconnected by inherent nonlinearities

    Automatica

    (2018)
  • C. Zhang et al.

    Global dynamic nonrecursive realization of decentralized nonsmooth exact tracking for large-scale interconnected nonlinear systems

    IEEE Trans. Cybern.

    (2019)
  • D. Ding et al.

    A scalable algorithm for event-triggered state estimation with unknown parameters and switching topologies over sensor networks

    IEEE Trans. Cybern.

    (2020)
  • K.W. Hedman et al.

    A review of transmission switching and network topology optimization

    Power and Energy Society General Meeting

    (2011)
  • Y. Zhou et al.

    Non-fragile H finite-time sliding mode control for stochastic Markovian jump systems with time delay

    Appl. Math. Comput.

    (2021)
  • L. Zhang et al.

    Stability and stabilization of discrete-time semi-Markov jump linear systems via semi-Markov kernel approach

    IEEE Trans. Automat. Control

    (2016)
  • E. Tian et al.

    H filtering for discrete-time switched systems with known sojourn probabilities

    IEEE Trans. Automat. Control

    (2015)
  • H. Shen et al.

    l2l State estimation for persistent dwell-time switched coupled networks subject to round-robin protocol

    IEEE Trans. Neural Netw. Learn. Syst.

    (2021)
  • J. Wang et al.

    Extended dissipative control for singularly perturbed PDT switched systems and its application

    IEEE Trans. Circuits Syst. I

    (2020)
  • X.M. Liu et al.

    Interval type-2 fuzzy passive filtering for nonlinear singularly perturbed PDT-switched systems and its application

    J. Syst. Sci. Complexity

    (2021)
  • J. Wang et al.

    H synchronization for fuzzy Markov jump chaotic systems with piecewise-constant transition probabilities subject to PDT switching rule

    IEEE Trans. Fuzzy Syst.

    (2021)
  • J. Li et al.

    Quantized control for networked switched systems with a more general switching rule

    IEEE Trans. Syst. Man Cybern.Syst.

    (2020)
  • Z. Ning et al.

    Stability and stabilization of a class of stochastic switching systems with lower bound of sojourn time

    Automatica

    (2018)
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      In recent years, nonlinear systems have obtained extensive attention because of their wide practical applications, such as multi-agent systems [1,2], large-scale systems [3,4] and complex networks [5,6].

    This work was supported in part by the National Natural Science Foundation of China under Grants 61873169, 62103282 and Sponsored by Shanghai Sailing Program 20YF1433200.

    1

    This is the first author footnote. but is common to third author as well.

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