Mathematical Problems in Engineering

Volume 2015, Article ID 724389, 10 pages

http://dx.doi.org/10.1155/2015/724389

## Output-Feedback Tracking Control for Networked Control Systems

School of Electrical Engineering, University of Ulsan, Daehak-ro 93, Nam-Gu, Ulsan 680-749, Republic of Korea

Received 27 May 2014; Revised 8 August 2014; Accepted 12 August 2014

Academic Editor: Yun-Bo Zhao

Copyright © 2015 Sung Hyun Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper investigates the observer-based tracking problem of networked output-feedback control systems with consideration of data transmission delays, data-packet dropouts, and sampling effects. Different from other approaches, this paper offers a single-step procedure to handle nonconvex terms that appear in the process of designing observer-based output-feedback control, and then establishes a set of linear matrix inequality conditions for the solvability of the tracking problem. Finally, two numerical examples are given to illustrate the effectiveness of our result.

#### 1. Introduction

Recently, the research on networked control systems (NCSs) has been rapidly growing due to both the fast development of technology of communication networks and the benefits of NCSs that include overcoming the spatial limits of the traditional control system, expanding system setups, increasing flexibility, multitasking, and improving system diagnosis and maintenance (see [1–4]). In particular, more recently, the development of the embedded system that has various communication modules and digital signal processing (DPS) core has confirmed the necessity of further investigations on NCSs. However, it is worth noticing here that the signal transmission over communication channels inevitably gives rise to data transmission delay problem, data-packet dropout problem, and sampling problem (see [3, 5–8]), which may cause instability or serious deterioration in the performance of the resultant control systems. Thus, exploring such problems has been recognized as one of the most important issues in the application of control theory.

Over the past several years, numerous researchers have made considerable efforts to propose methods for solving the aforementioned problems, especially based on Lyapunov-Krasovskii functional approach (see [9–11] for stabilization of NCSs (S-NCSs); [12, 13] for stabilization of NOCSs (S-NOCSs); and [5, 14–16] for tracking control of NCSs (T-NCSs), where NOCSs is the abbreviation of networked output-feedback control systems). In addition, [17] investigated the problem of output tracking for NCSs on the basis of the Lyapunov function approach. However, it is worth pointing out here that, regardless of such abundant literature, little progress has been made toward solving the tracking problem of NOCSs (T-NOCSs) in light of the Lyapunov-Krasovskii functional approach. In fact, all states of the controlled plant are not fully measurable in many engineering applications, and thus the tracking problem has emerged as a topic of significant interest in parallel to the stabilization problem. Thus, it is quite meaningful to study the method of designing T-NOCSs, especially by establishing a set of linear matrix inequality (LMI) conditions for the solvability of the tracking problem.

Motivated by the above concern, we investigate the problem of designing an observer-based T-NOCS with consideration of data transmission delays, data-packet dropouts, and sampling effects. Specifically, the attention is focused on designing an observer-based NOCS in such a way that the plant state tracks the reference signal in the sense. The contributions of this paper are mainly threefold.(1)The problem of designing T-NOCSs is systematically covered with the help of the Lyapunov-Krasovskii functional approach, which helps our results to have more wide applications.(2)A single-step procedure is proposed to handle nonconvex terms that inherently appear in the process of designing observer-based output-feedback control, which allows the derived sufficient conditions for the solvability of the tracking problem to be established in terms of LMIs.(3)Through the control synthesis process, this paper shows that the stability criteria derived from the reciprocally convex approach [18] can be clearly applied to the problem of designing T-NOCSs, which offers the possibilities for the extension of the results [19, 20] on the stability analysis toward the design of T-NOCSs. Finally, two numerical examples are given to illustrate the effectiveness of our result.

*Notation*. The Lebesgue space consists of square-integrable functions on . Throughout this paper, standard notions will be adopted. The notations and mean that is positive semidefinite and positive definite, respectively. In symmetric block matrices, is used as an ellipsis for terms that are induced by symmetry. For a square matrix , the notation denotes , where is the transpose of . is a column vector with entries and and is a diagonal matrix with diagonal entries and . All matrices, if their dimensions are not explicitly stated, are assumed to be compatible with algebraic operation.

#### 2. System Description and Preliminaries

Consider a continuous-time plant of the following form:where , , and denote the state to be estimated, the control input, and the output, respectively, and denotes the disturbance input such that . Here, as a way to estimate the immeasurable state variables of (1), we employ the following usual state observer:where denotes the estimated state and is the observer gain to be designed. Further, in parallel to (1) and (2), we incorporate the following dynamic system that generates the reference signal :where denotes the reference input such that and is constructed to be an asymptotically stable matrix. In this paper, our interest is to design an observer-based networked output-feedback control system (NOCS), based on (1)–(3), such that(1)the estimated state can approach the real state asymptotically;(2)the estimated state can track a reference signal over a communication network; that is, the state can track by ;(3)a guaranteed tracking performance can be achieved.To this end, we first employ the networked control system (NCS) architecture proposed in [3], which contains an observer with time-driven sampler, an event-driven controller, and a packet analyzer with event-driven holder (see Figure 1). For brevity, this paper omits the sophisticated description for the NCS under consideration since it is analogue to that of [3]. However, different from [3], we assume that the initial condition of (2) is given as , for , and the initial condition of (3) is given as , for , where denotes the initial time.