H [infinity] Output-Feedback Tracking Control for Networked Control Systems

Abstract

This paper investigates the observer-based [subscript]H∞[/subscript] 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.

Academic Editor:Yun-Bo Zhao

1, 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; 29 July 2015

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.

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 [figure omitted; refer to PDF] overcoming the spatial limits of the traditional control system, [figure omitted; refer to PDF] expanding system setups, [figure omitted; refer to PDF] increasing flexibility, [figure omitted; refer to PDF] multitasking, and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] consists of square-integrable functions on [figure omitted; refer to PDF] . Throughout this paper, standard notions will be adopted. The notations [figure omitted; refer to PDF] and [figure omitted; refer to PDF] mean that [figure omitted; refer to PDF] is positive semidefinite and positive definite, respectively. In symmetric block matrices, [figure omitted; refer to PDF] is used as an ellipsis for terms that are induced by symmetry. For a square matrix [figure omitted; refer to PDF] , the notation [figure omitted; refer to PDF] denotes [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the transpose of [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is a column vector with entries [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is a diagonal matrix with diagonal entries [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . 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: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] denote the state to be estimated, the control input, and the output, respectively, and [figure omitted; refer to PDF] denotes the disturbance input such that [figure omitted; refer to PDF] . Here, as a way to estimate the immeasurable state variables of (1), we employ the following usual state observer: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes the estimated state and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] : [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes the reference input such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] can approach the real state [figure omitted; refer to PDF] asymptotically;

(2) the estimated state [figure omitted; refer to PDF] can track a reference signal [figure omitted; refer to PDF] over a communication network; that is, the state [figure omitted; refer to PDF] can track [figure omitted; refer to PDF] by [figure omitted; refer to PDF] ;

(3) a guaranteed [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] , and the initial condition of (3) is given as [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] denotes the initial time.

Figure 1: Networked output-feedback control systems (NOCSs) with observer-based controller.

[figure omitted; refer to PDF]

Remark 1.

Here, it should be noted that, by the NCS architecture of [3], the communication constraints, such as data transmission delays and packet dropouts, can be represented in terms of piecewise continuous-time-varying delays with the lower and upper bounds.

Next, let us consider the following control law, inferred by [3]: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] corresponds to the piecewise continuous-time-varying delay that occurs from data transmission delays and packet dropouts. Then, by letting [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , the control law (4) can be rewritten as [figure omitted; refer to PDF] Further, by setting [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and by combining (1), (2), (3), and (5), the closed-loop system is described as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes the desired output, [figure omitted; refer to PDF]

Before ending this section, we present the following lemma that will be used in the proof of our main results.

Lemma 2 (see [21]).

For real matrices [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] with appropriate dimensions, it is satisfied that [figure omitted; refer to PDF] and thus the following inequality holds: [figure omitted; refer to PDF] . Further if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a scalar. On the other hand, if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

3. Main Results

Choose a Lyapunov-Krasovskii functional of the following form: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are positive definite matrices and [figure omitted; refer to PDF] . For later convenience, we define an augmented state [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and then establish some block entry matrices [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Then the closed-loop system (6) can be rewritten as [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] . As a result, the time derivative of [figure omitted; refer to PDF] along the trajectories of (6) is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] By (11), the time derivative of [figure omitted; refer to PDF] becomes [figure omitted; refer to PDF] where [figure omitted; refer to PDF] [figure omitted; refer to PDF] . To deal with [figure omitted; refer to PDF] , we apply the Jensen inequality [22] to [figure omitted; refer to PDF] , which results in [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ; that is, the set of [figure omitted; refer to PDF] is convex. Furthermore, by taking the convexity of [figure omitted; refer to PDF] into account, we can get the following equality: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] Hence, we can see that the time derivative of [figure omitted; refer to PDF] satisfies that [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] . As a result, based on this derivation, the following stability criteria can be established.

Lemma 3 (stability criterion).

For [figure omitted; refer to PDF] , the stability criterion is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

Proof.

If [figure omitted; refer to PDF] holds, then [figure omitted; refer to PDF] [figure omitted; refer to PDF] .

Lemma 4 (stability criterion in the [figure omitted; refer to PDF] sense).

The stability criterion in the [figure omitted; refer to PDF] sense is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Proof.

Let us consider the [figure omitted; refer to PDF] tracking performance such that [figure omitted; refer to PDF] . Then, as reported in [19], the [figure omitted; refer to PDF] stability criterion can be readily derived by [figure omitted; refer to PDF] , which is assured by (18).

Based on Lemma 3, the stabilization problem of (6) with [figure omitted; refer to PDF] will be addressed in Section 3.1, and further, based on Lemma 4, the [figure omitted; refer to PDF] stabilization problem of (6) with [figure omitted; refer to PDF] will be investigated in Section 3.2. Here, to derive a set of linear matrix inequalities (LMIs), we first set [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Then, from (7), it follows that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Accordingly, the term [figure omitted; refer to PDF] becomes [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Remark 5.

Inspired by the work of [18], this paper also applied the reciprocally convex approach to reduce the computational complexity and the conservatism of the delay-dependent stability criteria that will be used to derive our main results.

3.1. Control Design for [figure omitted; refer to PDF]

Lemma 6.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] be prescribed. Suppose that there exist matrices [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] and symmetric matrices [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Then the closed-loop system (6) is asymptotically stable in the absence of [figure omitted; refer to PDF] for any time-varying delay [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF] . Moreover, the control and observer gain matrices can be reconstructed as follows: [figure omitted; refer to PDF]

Proof.

From Lemma 3, the stabilization condition is given as follows: (i) [figure omitted; refer to PDF] and (ii) [figure omitted; refer to PDF] + [figure omitted; refer to PDF] - [figure omitted; refer to PDF] + [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and thus [figure omitted; refer to PDF] . Let us consider the condition given in (ii). Then, by letting [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we can rewrite the condition, [figure omitted; refer to PDF] , as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . Further, since [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ), pre- and postmultiplying both sides of (26) by [figure omitted; refer to PDF] and its transpose yield [figure omitted; refer to PDF] where [figure omitted; refer to PDF] + [figure omitted; refer to PDF] in which [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . That is, by applying the Schur complement to (27), we can get [figure omitted; refer to PDF] Here, since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , it follows from Lemma 2 that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . In this sense, it is clear that (28) holds if [figure omitted; refer to PDF] where [figure omitted; refer to PDF] However, as shown in (29), there exist some nonconvex terms in [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] as follows: [figure omitted; refer to PDF] Here, note that all terms associated with [figure omitted; refer to PDF] in [figure omitted; refer to PDF] can be separated as follows: [figure omitted; refer to PDF] Furthermore, from Lemma 2, it follows that [figure omitted; refer to PDF] which allows that (22) implies (29), based on the Schur complement. Next, we need to convert the given condition in (i), that is, [figure omitted; refer to PDF] , into an LMI. To this end, let us pre- and postmultiply both sides of [figure omitted; refer to PDF] by [figure omitted; refer to PDF] and its transpose. Then we can get [figure omitted; refer to PDF] which becomes (23) due to [figure omitted; refer to PDF] [figure omitted; refer to PDF] .

3.2. Control Design for [figure omitted; refer to PDF]

Theorem 7.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] be prescribed. Suppose that there exist scalars [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; matrices [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; and symmetric matrices [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Then the closed-loop system (6) is asymptotically stable and satisfies [figure omitted; refer to PDF] for all nonzero [figure omitted; refer to PDF] and for any time-varying delay [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF] . Moreover, the control and observer gain matrices can be reconstructed as follows: [figure omitted; refer to PDF]

Proof.

From Lemma 4, the [figure omitted; refer to PDF] stabilization condition is given as follows: (i) [figure omitted; refer to PDF] and (ii) [figure omitted; refer to PDF] + [figure omitted; refer to PDF] - [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] . As in the proof of Lemma 6, we first consider the condition given in (ii) by letting [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then the condition, [figure omitted; refer to PDF] , can be converted by (20) into [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . Further, since [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , pre- and postmultiplying both sides of (40) by [figure omitted; refer to PDF] and its transpose yield [figure omitted; refer to PDF] where [figure omitted; refer to PDF] + [figure omitted; refer to PDF] + [figure omitted; refer to PDF] , in which [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . That is, by applying the Schur complement to (41), we can get [figure omitted; refer to PDF] Here, since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , it follows from Lemma 2 that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] [figure omitted; refer to PDF] . In this sense, it is clear that (42) holds if [figure omitted; refer to PDF] where [figure omitted; refer to PDF] However, as shown in (43), there exist some nonconvex terms in [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] as follows: [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the same as those defined in the proof of Lemma 6, and [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes the identity matrix. As in the proof of Lemma 6, to deal with the nonconvex terms, we apply Lemma 2 to (43), which boils down to [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . As a result, by applying the Schur complement to [figure omitted; refer to PDF] in [figure omitted; refer to PDF] , we can obtain (36). The next step is to convert the given condition in (i), that is, [figure omitted; refer to PDF] , into an LMI. To this end, let us pre- and postmultiply both sides of [figure omitted; refer to PDF] by [figure omitted; refer to PDF] and its transpose. Then we can get [figure omitted; refer to PDF] which becomes (23) due to [figure omitted; refer to PDF] .

4. Numerical Example

We provide two examples to verify the effectiveness of the proposed methods in Lemma 6 and Theorem 7. For the networked output-feedback control system (NOCS), we assume that the sampling period [figure omitted; refer to PDF] and the data transmission delay bounds are given by [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . As a result, from [3], it follows that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] denotes the maximum number of data-packet dropouts.

4.1. Example 1

Consider a continuous-time system of the following form: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a variable element. First of all, to show the applicability of the proposed method in Lemma 6, we search the maximum allowable upper bounds (MAUBs) for (48) with [figure omitted; refer to PDF] . To this end, let us set [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Then, from Lemma 6, we can obtain the MAUBs for [figure omitted; refer to PDF] , which are tabulated in Table 1. Now, let us analyze the behavior of the tracking response for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] of the NOCS in the case where [figure omitted; refer to PDF] by using the derived condition in Theorem 7. For this purpose, we set [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Then, from Theorem 7, we can obtain the following control and observer gain matrices: [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . In addition, the disturbance attenuation is given by [figure omitted; refer to PDF] . Here we assume that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] , where the initial time [figure omitted; refer to PDF] is set to zero. Figure 2(a) shows the [figure omitted; refer to PDF] - [figure omitted; refer to PDF] trajectories for four different initial conditions [figure omitted; refer to PDF] , which form a specific ellipse, made by the given reference input [figure omitted; refer to PDF] , as the time [figure omitted; refer to PDF] increases. Further, the behavior of the estimation error [figure omitted; refer to PDF] is depicted in Figure 2(d), from which we can see that the estimation error goes to zero as the time [figure omitted; refer to PDF] increases. Figures 2(b) and 2(c) show the behavior of the state [figure omitted; refer to PDF] of (49) for initial condition [figure omitted; refer to PDF] , where the network-induced delay [figure omitted; refer to PDF] is generated as shown in Figure 2(e) such that the data transmission delay [figure omitted; refer to PDF] and the data-packet dropouts [figure omitted; refer to PDF] . From Figures 2(b) and 2(c), we can see that the state [figure omitted; refer to PDF] tracks the reference signal [figure omitted; refer to PDF] well; that is, the tracking response of the NOCS with (2), (5), and (48) is in a good shape with respect to our control goal.

Table 1: Maximum allowable upper bounds (MAUBs) for [figure omitted; refer to PDF] of Example [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0.32 ( [figure omitted; refer to PDF] )	0.19 ( [figure omitted; refer to PDF] )	0.09 ( [figure omitted; refer to PDF] )

Figure 2: (a) The [figure omitted; refer to PDF] - [figure omitted; refer to PDF] trajectory for each different initial condition [figure omitted; refer to PDF] ; (b) the tracking response of [figure omitted; refer to PDF] for [figure omitted; refer to PDF] ; (c) the tracking response of [figure omitted; refer to PDF] for [figure omitted; refer to PDF] ; (d) the estimation error [figure omitted; refer to PDF] ; and (e) the network-induced delay [figure omitted; refer to PDF] . Here, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] denote the [figure omitted; refer to PDF] th element of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , respectively.

[figure omitted; refer to PDF]

4.2. Example 2

Consider the following satellite system, modified from [5]: [figure omitted; refer to PDF] , [figure omitted; refer to PDF] Through this example, we will achieve the [figure omitted; refer to PDF] performance for (49) based on Theorem 7 to design an observer-based NOCS in such a way that the state [figure omitted; refer to PDF] of (49) tracks the reference signal [figure omitted; refer to PDF] in the [figure omitted; refer to PDF] sense. The obtained [figure omitted; refer to PDF] performance for each upper bound [figure omitted; refer to PDF] is tabulated in Table 2, where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are assumed. From Table 2, we can see that the [figure omitted; refer to PDF] performance is improved as [figure omitted; refer to PDF] decreases from 0.1 to 0.02, which is reasonable.

Table 2: [figure omitted; refer to PDF] performance for each upper bound [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0.8265	0.9072	1.1337	2.3290

5. Concluding Remarks

This paper has addressed the observer-based [figure omitted; refer to PDF] tracking problem of NOCSs with network-induced delays. In the derivation, a single-step procedure is proposed to handle nonconvex terms that appear in the process of designing observer-based output-feedback control, and then a set of linear matrix inequality conditions are established for the solvability of the tracking problem.

Acknowledgment

This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2012R1A1A1013687).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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