Output feedback L2-gain control of networked control systems subject to round-robin protocol

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Abstract

This paper considers the output feedback L2-gain control problem for continuous-time networked control systems subject to round-robin protocol. The sensors of the considered system are distributed over a network. Round-robin protocol is introduced to schedule the measurement information transmitted from sensors to the controller. Then the closed-loop system is modeled as a kind of time-delay system. According to Lyapunov–Krasovskii method, sufficient conditions are derived to guarantee the exponential stability and the prescribed L2-gain of the closed-loop system. The output feedback controller gains are determined efficiently by solving linear matrix inequalities. Finally, one example is reported to illustrate the advantages of the proposed design scheme.

Introduction

Recently, the control issues of networked control systems (NCSs) have attracted intensive attention thanks to the development of wireless communication technology [1]. The main feature of NCSs is that the system components (sensors, actuators, controllers) are connected via a shared communication network, which leads to several attractive advantages such as easier installation, higher flexibility and lower cost [1], [2], [3]. NCSs have been widely studied in many engineering applications including multi-area power systems [4], traffic management systems [5] and car suspension systems [6].

In the research of NCSs, two issues should be considered. The first one is the network-induced time-delay problems or packet dropout problems [7], [8], [9]. These phenomenons may cause poor system performance and destroy the system stability [9]. The other one is the nodes collision problems. However, most existing literatures only consider the first issue, but neglect the second one [10]. In order to deal with the node collision problems, round-robin protocol generally is introduced into NCSs to schedule the distributed sensors measurement information [11].

Some nice results have been derived in the control issues of NCSs subject to round-robin protocol [11], [12], [13], [14], [15]. For example, the authors of [11] analyzed the performance of the exponential stability and L2-gain for NCSs in the presence of round-robin protocol via a time-delay approach. However, in [11], the authors only considered the situation of two sensor nodes and the controller gains were assumed to be known a priori. Distributed state feedback controller design problem for large-scale interconnected systems with round-robin protocol was studied in [12]. Robust model predictive control issue for Markov jump systems subject to round-robin protocol was considered in [13]. A common feature of [12], [13] was that the studied systems’ state variables were required to be available, which may limit the application of the developed results. Recently, noncooperative event-triggered strategy and sliding mode control strategy were investigated for discrete-time NCSs subject to round-robin protocol in [14], [15], respectively. However, the results in [14], [15] were difficult to be extended into the continuous-time cases.

This paper aims to deal with the output feedback control problems for continuous-time NCSs under round-robin protocol. Based on the scheduling rule of round-robin protocol, at every instant only one selected sensor node gets into the network in order to avoid the node collision problem and relieve the transmission burden. The sensor selection is in a periodic manner. According to the developed Lyapunov–Krasovskii method, sufficient conditions are derived for ensuring the exponential stability and the prescribed L2-gain of the closed-loop NCS. Then these derived stability conditions are transformed into the form of linear matrix inequalities (LMIs). The obtained LMIs are used to be solved to determine the output feedback controller gains. Finally, the validity of our theoretical results are demonstrated through one example at the end of this paper. We now summarize the main contributions of this paper:

  • 1.

    Compared with [11] where only two sensor nodes are considered, this paper considers the situation of N (N2) sensor nodes which are scheduled by round-robin protocol. Moreover, the output feedback controller gains can be designed in this paper. The authors of [11] did not give the method to design the controller gains, they assumed that the controllers were given in advance.

  • 2.

    Compared with [12], [13] where the state variables of the controlled system were assumed to be available, this paper does not require this assumption and studies the output feedback control problems for NCSs under round-robin protocol.

  • 3.

    Compared with the literatures [13], [14], [15] where discrete-time NCSs were considered, this paper investigates the controller design problems for continuous-time NCSs under round-robin protocol.

This paper is organized as follows: The system description and the problem formulation are given in Section 2. Section 3 establishes sufficient conditions to guarantee the stability with the L2-gain of the closed-loop system and develops the method of the controller design. Numerical examples are presented in Section 4. The last part is the conclusions of this paper.

Notation: N+,Rn and L2[0,) stand for, the set of positive integers, the n-dimensional Euclidean space and the Lebesgue space of Rn-valued vector function which is defined on [0,), respectively. Matrix A<0 (A0) means that A is negative definite (negative semi-definite). The symbol “*” in symmetric matrices means the symmetric terms.

Section snippets

Problem formulation

Consider a continuous-time NCS with the architecture shown in Fig. 1. The dynamics of the studied NCS are{x˙(t)=Ax(t)+Bu(t)+Ew(t),z(t)=Fx(t)+Dw(t),where x(t)Rnx,u(t)Rnu,w(t)Rnw(w(t)L2[0,)),z(t)Rnz stand for the state variables, the controller input, the disturbance, the controlled output of the studied system, respectively. The other matrices A,B,E,F and D are known system matrices.

There are n distributed sensor nodes embedded in the NCS. Each sensor takes measurementsyi(t)=Cix(t),iN:={1,

Main results

Sufficient conditions are derived in this section to ensure the exponential stability and the L2-gain of NCS Eq. (5). Then, the output feedback controller gains are determined based on LMIs techniques.

Before proceeding further, we first define the matrix Y and vectors ξ(t),δ(t) as follows.Y:=[II000000II000I000II000000II000I0000II000000II00I],ξ(t)=[x(t)xc(t)x(tτ)xs(t)]:=[x(t)x(tk)x(tkn+1)x(tτ)2ttktktx(s)ds2tktk1tk1tkx(s)ds2tkn+1t+τtτtkn+1x(s)ds],δ(t)=[δ

Example

In this section, we compare our proposed method and the methods in [18], [24] through one numerical example.

Consider that there are four sensor nodes in the NCS in Fig. 1, i.e., iN:={1,2,3,4}. The dynamics of the considered system are of the form Eq. (1) withA=[1.5+a0.51.8b003+2×b00.2+0.5×a00400001.2],B=[0.10.2+b0.20.1],F=[001.8×b+a0.20],D=0.1,E=[1.6ab0.5+0.5×bb1.20.4×b],C1=[1101],C2=[011+b0],C3=[1011],C4=[0011],where “a” and “b” can be chosen different values for comparison purpose.

Conclusions

This paper studies the exponential stability and L2-gain analysis of continuous-time NCSs subject to round-robin protocol. In such a protocol, only one sensor is selected to occupy the network to transmit the measurements to the controller at every time instant. The sensor is selected in a cyclic manner. According to the developed Lyapunov–Krasovskii method, sufficient conditions are derived to ensure the exponential stability and the L2-gain of the studied NCS. Output feedback controller gains

Declaration of Competing Interest

The authors declare no conflict of interest.

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