Robust $H_{\infty }$ control for uncertain networked systems with communication constraints

Abstract

This paper considers robust $H_{\infty }$ control for networked control systems (NCSs) with communication constraints, network-induced delay and uncertainties in both system dynamic and access rates. Medium access constraints of both sensor–controller and controller–actuator channels are considered, that is, not all the sensors and actuators are granted access to the network simultaneously to transfer measurements, the access status of sensors or actuators are governed by some stochastic variables. Then new NCSs model is proposed by using these stochastic variables. The purpose of this paper is to design the robust $H_{\infty }$ controller to stabilize the NCSs with communication constraints and random network-induced delay. By using the Lyapunov functional method and the linear matrix inequality technology, sufficient conditions for the robust mean square stable (RMSS) of the NCSs with an $H_{\infty }$ norm bound $\gamma$ are obtained. A simulation example is given to demonstrate the effectiveness and application of the proposed method.

Introduction

Networked control systems (NCSs) are feedback control systems in which the communication between spatially distributed system components like sensors, actuators and controller occurs through band-limited digital communication networks, the structure of the NCSs can be seen in Fig. 1. The main difference between NCSs and traditional control systems is the insertion of the shared network with limited transmission capacity and non-ideal network conditions, such as network-induced delay, packets dropout, sensors/actuators fault and so on. Generally speaking, the researches on NCSs can be classified into two categories: researches focused on control [2], [5], [6], [8], [9], [16], [21], [23], [29] and researches focused on communication [4], [15], [17], [19], [20], [24]. The main task of the former is to build proper NCSs models and design networked controller to guarantee the quality of performance (QoP) of NCSs, such as robustness, $H_{\infty }$ performance and H₂ performance. The aim of the latter is to find proper network access scheduling to guarantee the quality of service (QoS) of NCSs.

Most of above mentioned researches assume that there is no restriction in the communication channels of sensor–controller or controller–actuator [4], [6], [15], [21], [23], that is, the communication medium can accommodate all sensors or actuators simultaneously. However, this is not always true in some practical systems [1], [7], [10], [11], [18], [26], [28], such as large-scale systems. In those systems, limitation on the number of sensors and actuators that are granted access to the network simultaneously to transfer measurements of sensors to the controller and control signals computed by the controller to the actuators. In the context of access constraints, (i) a communication sequence method is needed to decide which sensor/actuator has the permission to communicate with the controller/plant; (ii) a controller should be designed to stabilize the system in the presence of limited medium access constraints, random networked delay. In order to solve these problems, a control and communication co-design strategy is proposed in [27] for systems without time delay, wherein a periodic communication sequence is proposed. Later, improved co-design method can be seen in [13], [26], wherein the time delay is considered. However, the periodic communication policies require all the sensors and actuators to be strictly scheduled. Although periodic communication sequences are easy for implementation, they are not suitable for the dynamic, complex network conditions. The reason is that once the sensors/actuators fail, or packet losses during the transmission or large network-induced delay occurs, the designed communication sequences may fail to stabilize the NCSs. More recently, a stochastic access modeling method is proposed in [10], wherein the assignment of the active actuator subset is dependent on an event modeled by a Markov chain and the whole system is modeled as a Markovian jump systems. Among the pioneering papers, the work of [10] is closely related to this study, since stochastic assignment algorithm is proposed in both studies. However, in [10], (a) only access constraint of controller–actuator is considered; (b) the access or not of the communication is governed by a Markov chain, however, some of the elements in transition probability matrix are difficult or costly to be measured [25], which restricts their applications; and (c) robustness of access rates is not considered.

Generally speaking, when considering NCSs with limited communication capacity, most of the references mentioned above focusing on the communication approach except the work of [10], i.e., for a pre-designed controller feedback gain, design a communication sequence to stabilize the system. Different from these works, our investigation focuses on the control technology, the design process is as follows: firstly, a set of stochastic variables satisfying Bernoulli probability distribution are proposed to describe the access status of each sensor and actuator, “1” for access and “0” for not access. The access rates of sensors or actuators are described by the expectation of the corresponding stochastic variables, which is also related with the current working condition of the sensors and actuators. Including these variables in the system, a new NCSs model considering both limited access capacity and random networked delay is proposed. It should be noted that the considered uncertainties include both plant uncertainties and access rate uncertainties. Using Lyapunov functional method, sufficient conditions for the robust mean square stability (RMSS) of the proposed NCSs model are obtained. A corresponding algorithm is proposed to obtain the designed controller feedback gain. Finally, an example is given to illustrate the application and efficiency of the proposed method.

Compared with the existing researches, the main contributions of this paper include:

• The problem of medium access constraints of sensor–controller and controller–actuator are considered for NCSs, which has not been taken a full consideration in the existing researches.

• A stochastic access determine method is proposed for the communication scheduling, wherein the access status of each sensor or actuator is governed by a Bernoulli distributed white sequence, and the access rates in the proposed method are easier to be measured than the transition probabilities in [10].

• A novel co-design strategy of state feedback controller and random access assignment is proposed for NCSs, and a new Lyapunov functional method is proposed for the controller design.

• In the proposed NCSs model, random network-induced delays are taken into consideration and the uncertainties of the access rates of each sensor or actuator are considered, which has been omitted by most of existing researches.