Fault-tolerant Control of Networked Control System with Packet. Dropout and Transmission delays. He Huang, Xiaodong Han, Xiaopeng Ji and Zhiquan Wang.
Fault-tolerant Control of Networked Control System with Packet Dropout and Transmission delays He Huang, Xiaodong Han, Xiaopeng Ji and Zhiquan Wang Abstract-This paper deals with the problem of fault-tolerant control of networked control systems (NCSs) with packet dropout and transmission delays which is introduced by communication channel. By employing delay-dependent approach, introducing partial failure model and utilizing integral-inequality method, the integrity design of NCSs with actuator failure is analyzed. The sufficient stabilizability condition of system with memoryless networked state feedback controller is obtained. Based on this condition, the networked fault-tolerate controller can be derived by solving a set of linear matrix inequalities. Numerical examples illustrate the effectiveness of the proposed approach. I. INTRODUCTION
N ETWORKED control systems (NCSs) are spatially distributed systems in which the communication between sensors, actuators, and controllers occurs through a shared communication network. The use of communication network has several advantages such as modular and flexible system design, simple and fast implementation, ease of system maintenance, and increased system agility [1], [2]. However, the insertion of communication network in the control loop introduces some shortcomings such as introduced delay and data packet dropout. It also makes the analysis and synthesis of NCSs complex [3]. Consequently, there are growing theoretical interests in the field of NCSs. Because data packet dropout and transmission delays might be potential sources to instability and poor performance of NCSs, much work have been done. In [1 1], an augmented state vector method has been proposed to control a linear system over a periodic delay network. Random delays have been discussed in [2] via an optimal stochastic control methodology. The problem of stability of NCSs with packet dropout was investigated in [5] based on asynchronous dynamical system (ADS) theory. In [4], [6], [12], the problem of state feedback controller design of NCSs with both delay and packet dropout taken into account was considered. In these works, stability and controller design method was proposed based on a delay-dependent approach. This method can be applied to compute maximum network-induced delay which guarantees that the NCS is stable and design the networked memoryless gain controller such that the Manuscript received September 14, 2007. The work with H. Huang and Z. Q. Wang is supported by the National Natural Science Foundation of China under Grant #60574082. H. Huang is with College of Automation, Nanjing University of Science and Technology, Nanjing 210094, China (email: hh.huanghe Ogmail.com) X. D. Han, X. P. Ji and Z. Q. Wang are with College of Automation Nanjing University of Science and Technology, Nanjing 210094, China.
controller can need less memory. Thus, this method will be used in this paper. On the other hand, with the complexity that introduced by network, the NCSs are more vulnerable to faults. Defects in NCSs can be amplified by communication networks. And this kind of control-loop may hide the faults from being observed. An effective way to increase availability and reliability of NCSs is to introduce fault-tolerate control (FTC). FTC of NCSs is now a new research field. The problem of fault-tolerate controller design of NCS was investigated in [9], [10] with transmission delay and communication constraints taken into consideration respectively. FTC of NCSs with medium access constraints has been discussed in [8] based on model predictive control theory. With both packet dropout and transmission delay taken into account, this paper considers fault-tolerate controller design for NCS. The NCS in continuous time is modeled as linear system with time-varying input delays. Without losing generality, it is assumed that the NCS considered in this paper consists of time-driven sensor, event-driven controller and event-driver actuator. A delay-dependent result for fault-tolerate controller design of NCSs is obtained by Lyapunov-Krasovskii functional approach. The proving process employs a new method called integral-inequality method [7] which can be used to study the delay-dependent stabilization issue oftime delay system. It was shown that the new criterion proved by the method does not need any assumption about system matrices. The partial failure model [ 13] that used to represent actuator fault in this paper is closer to reality than 0-1 failure model. And the simulation validates the method. II. PROBLEM FORMULATION
In a NCS, data dropout is unavoidable because of limited bandwidth. And during data transmission if there is packet collision, it's better to drop the old packet than retransmission it. In this paper, we take transmission delay introduced by network into account: the sensor-to-controller delay zC, and the controller-to-actuator delay rca, which can be lumped together for consideration. The physical plant is a continuous linear system, which can be described as (1) x(t) = Ax(t) + Bu(t), where x(t)eI=iR and u(t) e R are the state vector and control input vector, respectively. A , B are known real constant matrices with appropriate dimensions. Suppose that
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the data are transmitted with a single packet, and the real input u(t) in (1) realized through a zero-order hold (ZOH). Then, the continuous-time controller is modeled as u(t)= Kx(t Tca),n tE[tk tk+l ) k=11,2,',' (2) where K is state feedback gain matrix to be designed, x(t) is the output of the network and we denote the sampling instant of the sampler as tk , k=1,2, ,o. The sampling period h is a positive real constant scalar. Based on the architecture of the NCS, we can know that the output of the network is the delayed sampling value of the state. In this paper, we employ the model in [6] to represent the NCS with packet loss and delay. Packet dropout in the NCS is modeled as a switch. When the switch is closed, the data is transmitted, whereas the switch is open, the packet gets lost and the controller employs the old data. We use d(k) to denote that there are d(k) packets dropout at time instant tk. If no packet gets lost at time tk, then d(k) = 0 . Take d(k) packets dropout at time tk for instance, the output ofnetwork can be expressed as follow: x(t) = x(tk - Tc - d (k)h) Thus the NCS with transmission delay and packet dropout can be described as follow: x(t) = Ax(t) + BKx(t-rj) (3) Ax(t) + BKx(t - r(t)), t E [tk, tk+1 The delay part r(t) = t - (tk-- - d(k)h - rca) may vary with time t which introduces transmission delay and data dropout into the system. For consideration of the actual property of delay and packet dropout, it could be noted that O.< (t) =t-tk +Tc +ia + d(k)h < h+rc +rca + d(k)h
