Optimal performance of discrete-time control systems based on network-induced delay

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Abstract

The optimal tracking problem for single-input single-output (SISO) linear time-invariant discrete-time systems over communication channel with network-induced delay in the feedback path is studied in this paper. The tracking performance is measured by the energy of the error response between the output of a given plant and the reference signal. An H2 square error criterion is used as a measure for the tracking error. The result shows that the optimal tracking performance is constrained by nonminimum phase (NMP) zeros, unstable poles of a given plant and network-induced delay. It is also shown that, if the network-induced delay of the communication channel does not exist, the optimal tracking performance reduces to the existing tracking performance of the control system without communication channel. The result obtained in this work explicitly show how the optimal tracking performance is limited by the network-induced delay. A typical example is given to illustrate the obtained theoretical results.

Introduction

Networked control systems (NCSs) have been the focus of several research studies over the last few years [1], [6], [14], [15], [16], [17]. Compared with conventional point-to-point system connections, NCSs have advantages in installation, wiring, and maintenance cost and time. In an NCS, data travel through the communication channels from the sensors to the controller and from the controller to the actuators. A communication network in the feedback-control loop makes the analysis of NCS more complex, such as communication bandwidth, congestion, and contention for communication resources, quantization, packet dropout, delay, signal-to-noise ratio, which can degrade the performance of control systems, and can even destabilize the system. The technologies about modeling of the networked control systems and stabilization analysis are now fairly mature. Some other important issues are yet to be addressed, for example, optimal tracking design and attainable tracking performance of networked control systems constrained by the communication links in the feedback loop. Various information transmission constraints, such as data-rate limit, quantization precision, bandwidth constraints, time delays and data packet drop-out, are all likely to have a negative effect on tracking performance of networked control systems.

The performance limitations in control design have been an important issue of control science and engineering for many years [2], [4], [11], [13], [18]. In previous works, the paper [12] studied a tracking step signal performance problem for multi-input multi-output (MIMO), linear, time-invariant systems by using unity feedback control scheme. The results showed that the optimal performance depends critically upon the locations and directions of the unstable poles and nonminimum phase zeros of a given plant. In recent years, the topic of optimal tracking performance has been extended to networked control systems [5]. The paper [9] studied the tracking problem for linear time-invariant multi-input single-output (MISO) discrete-time systems with quantization effects. The results showed that the quantization limits the system tracking performance. The paper [7] studied the optimal tracking performance of MIMO LTI discrete-time systems with a power constrained additive white noise (AWN) channel in the feedback path. The paper [10] studied a necessary and sufficient condition on the channel signal-to-noise ratio (SNR) for closed-loop stabilizability using LTI feedback. The result gives a guideline in estimating the severity of the fundamental SNR limitation imposed by the plant unstable poles, NMP zeros, time-delay as well as the channel NMP zeros, bandwidth, and channel noise coloring. The paper [8] showed that time delay does worsen the performance, in terms of the mean square response of the system state to a Gaussian disturbance.

In networked control systems, the study of optimal tracking performance with network-induced delay constraints is seldom considered in the previous works. This paper studies the optimal tracking performance about SISO linear time-invariant discrete-time feedback system tracking a step signal over communication channel with network-induced delay. The result shows that the optimal tracking performance of system is determined by a given plant internal structure and network-induced parameters, no matter what compensator is adopted. The result will be guidance for the design of networked control systems. In general, network-induced delay is time varying. It is difficult in analyzing the performance of networked control systems due to the time-varying delay. When the relay devices such as gateways and routers do not exist in the communication network, the delay in the communication channel may be fixed, or can be transformed into constant delay by using some measures. For the convenience of the analysis, in this paper, we will study how the performance of control systems is affected by the constant delay in the communication channel. Without loss of generality, we consider the case where the system sensor is far away from the plant but the controller is close to the plant in this paper. So, the network-induced delay in the feedback channel is considered as Fig. 1. Our main contribution is to study a SISO linear time-invariant discrete-time system optimal tracking a step signal with network-induced delay. The performance is degraded by the nonminimum phase zeros, unstable poles of a given plant and network-induced delay.

The paper is organized as follows. Section 2 introduces some preliminaries and the problem formulation. In Section 3, we study optimal tracking performance of SISO linear time-invariant discrete-time feedback control system over communication channel with network-induced delay. A typical example is given to illustrate the obtained results in Section 4. The conclusions and future research directions are presented in Section 5.

Section snippets

Problem statement and preliminaries

We first describe the standard notation used throughout this paper. For any complex number z, we denote its complex conjugate by z¯. The open unit disc, the closed unit disc, the exterior of the closed unit disc and the unit circle are denoted by D{z:|z|<1}, D¯{z:|z|1}, D¯c{z:|z|>1} and D{z:|x|=1}, respectively. We define the Hilbert space L2 L2{f:f(z)measurableinD,f2212πππf(ejθ)2dθ<}.

The space L2 is the Hilbert space with inner product f,g12πππfH(ejθ)g(ejθ)dθ.which further

Optimal tracking with network-induced delay

Consider the system setup shown in Fig. 1. According to (2), (3), (6), (7), (8), the tracking performance in this case can be expressed as JJ=(1NYNMQ)1z122.

From (4), (11), we can rewrite JJ=infQRH(1NYNMQ)1z122

It is clear that in order to obtain J, Q must be appropriately selected.

Theorem 3.1

Let r be given by (5). If G(z) is factorized as in (6), (10), thenJ=i=1nseft|si|21|si1|2+i,j=1m(|pi|21)(|pj|21)b¯jbi(1p¯j)(1pi)γjHγjp¯jpi1,where bj=iNijmpjpi1pjp¯i,γj=1pjdLz1(pj).

Proof

From (9),

Illustrative example

In this section, an example is given to illustrate the theoretical results.

Consider the unstable plant model described by G=zk(z5)(z+0.5),where k[2,20].

This plant is nonminimum phase. The unstable pole is located at p1=5. For any k>1, it has a nonminimum phase zero at s1=k.

From Theorem 3.1, the optimal tracking performance is obtained J=k+1k1+1.5(15dLz1(5))2.

The optimal tracking performances SISO discrete-time control systems with different network-induced delays or NMP zeros is shown in

Conclusions

In this paper, the optimal tracking performance is given for SISO discrete-time of control systems over communication channel with network-induced delay. We have derived an explicit expression of the minimal tracking error for systems over communication channel in feedback path. Our main contribution is obtained by applying H2 square error criterion and the spectral factorization technique. The result shows that the optimal tracking performance only depends on the nonminimum phase zeros,

Acknowledgment

This work was partially supported by the National Natural Science Foundation of China under Grants 61100076, 61073065, 61170031, 61272069, 61272114. The authors are grateful to the associate editor and reviewers for their many constructive comments.

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