Adaptive Output Feedback Sliding Mode Control for Complex Interconnected Time-Delay Systems

Abstract

We extend the decentralized output feedback sliding mode control (SMC) scheme to stabilize a class of complex interconnected time-delay systems. First, sufficient conditions in terms of linear matrix inequalities are derived such that the equivalent reduced-order system in the sliding mode is asymptotically stable. Second, based on a new lemma, a decentralized adaptive sliding mode controller is designed to guarantee the finite time reachability of the system states by using output feedback only. The advantage of the proposed method is that two major assumptions, which are required in most existing SMC approaches, are both released. These assumptions are (1) disturbances are bounded by a known function of outputs and (2) the sliding matrix satisfies a matrix equation that guarantees the sliding mode. Finally, a numerical example is used to demonstrate the efficacy of the method.

Academic Editor:Hak-Keung Lam

Department of Mechanical and Automation Engineering, Da-Yeh University, No. 168 University Road, Changhua 51591, Taiwan

Received 3 June 2014; Revised 3 September 2014; Accepted 8 September 2014; 5 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

Advancement in the field of engineering has led to increasingly complex large-scale systems [1]. In addition, time-delay systems often feature in real-world problems, for example, chemical processes, biological systems, economic systems, and hydraulic/pneumatic systems. Time delay commonly leads to a degradation and/or instability in system performance (e.g., [2, 3]). The stability of interconnected time-delay systems has therefore been the focus of much research, which has achieved useful results [4-8]. However, the solutions proposed by previous studies necessarily require that all state variables are available for measurements.

In many practical systems, the state variables are not accessible for direct measurement or the number of measuring devices is limited. Recently, various control approaches have been employed to overcome the above obstacles. In [9-11], based on the assumption that each isolated subsystem is of triangular form and includes internal dynamics, a class of decentralized stabilizing dynamic output feedback controller was proposed for interconnected time-delay systems. In [12], based on two adaptive neural networks, a class of decentralized stabilizing output feedback controllers was proposed for a class of uncertain nonlinear interconnected time-delay systems with immeasurable states and triangular structures. In [13], based on adaptive fuzzy control theory, a decentralized robust output feedback controller was proposed for a class of strict-feedback nonlinear interconnected time-delay systems. In [14], a new adaptive robust state observer was designed for a class of uncertain interconnected systems with multiple time-varying delays. By including fuzzy logic systems and fuzzy state observer, the authors of [15] presented an adaptive decentralized fuzzy output feedback control for interconnected systems when system states cannot be measured. The work in [16] investigated the issue of robust and reliable decentralized [figure omitted; refer to PDF] tracking control for fuzzy interconnected time-delay systems. In [1], based on Lyapunov stability theory and the corresponding linear matrix inequalities (LMI), the design of a dynamic output feedback controller was proposed for uncertain interconnected systems of neutral type. The authors of [17] proposed two new stability criteria of the synchronization state for interconnected time-delay systems. The above work obtained important results related to decentralized control using only output variables. However, it should be noted that most of the existing results for interconnected time-delay systems can only be obtained when the systems conform to a special structure [9-13]. The approaches proposed by [14-17] cannot be applied for interconnected time-delay systems with mismatched parameter uncertainties in the state matrix of each isolated subsystem. Therefore, it is important to develop a decentralized adaptive output feedback sliding mode control (SMC) law to stabilize interconnected time-delay systems with a more general structure.

Sliding mode control is a robust fast-response control strategy that has been successfully applied to a wide variety of practical engineering systems [2, 3, 18]. Generally speaking, SMC is attained by applying a discontinuous control law to drive state trajectories onto a sliding surface and force them to remain on it thereafter (this process is called reaching phase), and then to keep the state trajectories moving along the surface towards the origin with the desired performance (such motion is called sliding mode) [2, 3, 18]. Earlier work on decentralized adaptive SMC mainly focused on interconnected systems or nonlinear systems that satisfy the matching condition [19-22]. If the matching condition is not satisfied, then the mismatched uncertainty will affect the dynamics of the system in sliding mode. Thus, system behavior in sliding mode is not invariant to mismatched uncertainty. Many techniques, such as [23-25], have been applied to deal with mismatched uncertainties in sliding mode. The authors of [23] proposed a decentralized SMC law for a class of mismatched uncertain interconnected systems by using two sets of switching surfaces. In [24], a decentralized dynamic output feedback based on a linear controller was proposed for the same systems. In [25], by using a multiple-sliding surface, a new control scheme was presented for a class of decentralized multi-input perturbed systems. However, time delays are not included in the above approaches [23-25]. The existence of delay usually leads to a degradation and/or instability in system performance [2, 3]. In the limited available literature, results on applying sliding mode techniques to interconnected time-delay systems are very few [2, 3, 18]. A decentralized model reference adaptive control scheme was proposed for interconnected time-delay systems in [18]. An interconnected time-delayed system with dead-zone input via SMC in which all system state variables are available for feedback was considered in [2]. The authors of [3] investigated the global decentralized stabilization of a class of interconnected time-delay systems with known and uncertain interconnections. Their proposed approach uses only output variables. Based on Lyapunov stability theory, they designed a composite sliding surface and analyzed the stability of the associated sliding motion. As a result, the stability of interconnected time-delay systems is assured under certain conditions, the most important of which are that the disturbances must be bounded by a known function of outputs and that the sliding matrix must satisfy a matrix equation in order to guarantee sliding mode. However, in practical cases, these assumptions are difficult to achieve. Therefore, it would be worthwhile to design a decentralized adaptive output feedback SMC scheme for complex interconnected time-delay systems with a more general structure in which two of the above limitations are eliminated. To the best of our knowledge, no decentralized adaptive output feedback SMC scheme has so far been proposed for interconnected time-delay systems with unknown disturbance, mismatched parameter uncertainties in the state matrix, and mismatched interconnections and without the measurements of the states.

In this technical note, we extend the concept of decentralized output feedback sliding mode controller, introduced by Yan et al. in [3], for the aim of stabilizing complex interconnected time-delay systems. The main contributions of this paper are as follows.

(i) The interconnected time-delay systems investigated in this study include mismatched parameter uncertainties in the state matrix, mismatched interconnections, and unknown disturbance. Therefore, we consider a more general structure than the one considered in [2, 3, 18-25].

(ii) This approach uses the output information completely in the sliding surface and controller design. Therefore, conservatism is reduced and robustness is enhanced.

(iii): The two major limitations in [3] are both eliminated (disturbances must be bounded by a known function of outputs and the sliding matrix must satisfy a matrix equation in order to guarantee sliding mode). Hence, the proposed method can be applied to a wider class of interconnected time-delay systems.

Notation. The notation used throughout this paper is fairly standard. [figure omitted; refer to PDF] denotes the transpose of matrix [figure omitted; refer to PDF] . [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are used to denote the [figure omitted; refer to PDF] identity matrix and the [figure omitted; refer to PDF] zero matrix, respectively. The subscripts [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are omitted where the dimension is irrelevant or can be determined from the context. [figure omitted; refer to PDF] stands for the Euclidean norm of vector [figure omitted; refer to PDF] and [figure omitted; refer to PDF] stands for the matrix induced norm of the matrix [figure omitted; refer to PDF] . The expression [figure omitted; refer to PDF] means that [figure omitted; refer to PDF] is a symmetric positive definite. [figure omitted; refer to PDF] denotes the [figure omitted; refer to PDF] -dimensional Euclidean space. For the sake of simplicity, sometimes function [figure omitted; refer to PDF] is denoted by [figure omitted; refer to PDF] .

2. Problem Formulations and Preliminaries

We consider a class of interconnected time-delay systems that is decomposed into [figure omitted; refer to PDF] subsystems. The state space representation of each subsystem is described as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] with [figure omitted; refer to PDF] are the state variables, inputs, and outputs of the [figure omitted; refer to PDF] th subsystem, respectively. The triplet [figure omitted; refer to PDF] and [figure omitted; refer to PDF] represent known constant matrices of appropriate dimensions. The notations [figure omitted; refer to PDF] and [figure omitted; refer to PDF] represent delayed states and delayed outputs, respectively. The symbol [figure omitted; refer to PDF] is the time-varying delay, which is assumed to be known and is bounded by [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is constant. The initial conditions are given by [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] are continuous in [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . The matrices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] represent mismatched parameter uncertainties in the state matrix and mismatched uncertain interconnections with [figure omitted; refer to PDF] . The matrix [figure omitted; refer to PDF] is the disturbance input. In this paper, only output variables [figure omitted; refer to PDF] are assumed to be available for measurements.

For system (1), the following basic assumptions are made for each subsystem in this paper.

Assumption 1.

All the pairs ( [figure omitted; refer to PDF] ) are completely controllable.

Assumption 2.

The matrices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are full rank and [figure omitted; refer to PDF] .

Assumption 3.

The exogenous disturbance [figure omitted; refer to PDF] is assumed to be bounded and to satisfy the following condition: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are unknown bounds which are not easily obtained due to the complicated structure of the uncertainties in practical control systems.

Assumption 4.

The mismatched parameter uncertainties in the state matrix of each isolated subsystem are satisfied as [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is unknown but bounded as [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are known matrices of appropriate dimensions.

Assumption 5.

The mismatched uncertain interconnections are given as [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is unknown but bounded as [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are any nonzero matrices of appropriate dimensions.

Remark 1.

Assumption [figure omitted; refer to PDF] is a limitation on the triplet [figure omitted; refer to PDF] and has been utilized in most existing output feedback SMCs, for example [3, 26, 27]. This assumption guarantees the existence of the output sliding surface. Assumptions 4 and 5 were used in [6, 27].

Remark 2.

There are two major assumptions in [3].

(i) The exogenous disturbances are bounded by a known function of outputs [figure omitted; refer to PDF] . That is [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is known. This condition is quite restrictive.

(ii) The sliding matrix [figure omitted; refer to PDF] satisfies [figure omitted; refer to PDF] to guarantee sliding condition [figure omitted; refer to PDF] . This limitation is really quite strong.

In this paper, a decentralized adaptive output feedback SMC scheme is proposed for complex interconnected time-delay systems where the two above limitations are eliminated.

For later use, we will need the following lemma.

Lemma 3 (see [3, 26]).

Consider the following interconnected system: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the state variables, inputs, and outputs of the [figure omitted; refer to PDF] th subsystem, respectively. Under assumption [figure omitted; refer to PDF] , it follows from [3, 26] that there exists a coordinate transformation [figure omitted; refer to PDF] such that the interconnected system (3) has the following regular form: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . The matrices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are nonsingular and [figure omitted; refer to PDF] is stable.

3. Sliding Mode Control Design for Complex Interconnected Time-Delay Systems

In this section, we design a new decentralized adaptive output feedback SMC scheme for the system (1). There are three steps involved in the design of our decentralized adaptive output feedback SMC scheme. In the first step, a proper sliding function is constructed such that the sliding surface is designed to be dependent on output variables only. In the second step, we derive sufficient conditions in terms of LMI for the existence of a sliding surface guaranteeing asymptotic stability of the sliding mode dynamic. In the final step, based on a new Lemma, we design a decentralized adaptive output feedback sliding mode controller, which assures that the system states reach the sliding surface in finite time and stay on it thereafter.

3.1. Sliding Surface Design

Let us first design a sliding surface, which depends on only output variables. Since [figure omitted; refer to PDF] , it follows from Lemma 3 that there exists a coordinate transformation [figure omitted; refer to PDF] such that the system (1) has the following regular 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] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . The matrices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are non-singular and [figure omitted; refer to PDF] is stable.

Letting [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , the first equation of (5) can be rewritten as [figure omitted; refer to PDF] Obviously, the system (6) represents the sliding-motion dynamic of the system (5), and, hence, the corresponding sliding surface can be chosen as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , the matrix [figure omitted; refer to PDF] is defined later, and the matrix [figure omitted; refer to PDF] is selected such that [figure omitted; refer to PDF] is nonsingular. Then, by using the second equation of (5), we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . In addition, the Newton-Leibniz formula is defined as [figure omitted; refer to PDF] Therefore, in sliding modes [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then, from the structure of systems (6)-(7), the sliding mode dynamics of the system (1) associated with the sliding surface (8) is described by [figure omitted; refer to PDF]

3.2. Asymptotically Stable Conditions by LMI Theory

Now we are in position to derive sufficient conditions in terms of linear matrix inequalities (LMI) such that the dynamics of the system (11) in the sliding surface (8) is asymptotically stable. Let us begin with considering the following LMI: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is any positive matrix, and [figure omitted; refer to PDF] is the number of subsystems and the scalars [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] . Then, we can establish the following theorem.

Theorem 4.

Suppose that LMI (12) has solution [figure omitted; refer to PDF] and the scalars [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] . Suppose also that the SMC law is [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 the scalars [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and the time functions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] will be designed later. The sliding surface is given by (8). Then, the dynamics of system (11) restricted to the sliding surface [figure omitted; refer to PDF] is asymptotically stable.

Before proofing Theorem 4, we recall the following lemmas.

Lemma 5 (see [27]).

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] be real matrices of suitable dimension with [figure omitted; refer to PDF] ; then, for any scalar [figure omitted; refer to PDF] , the following matrix inequality holds: [figure omitted; refer to PDF]

Lemma 6 (see [28]).

The linear matrix inequality: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] depend affinely on [figure omitted; refer to PDF] , is equivalent to [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Lemma 7.

Assume that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is a positive definite matrix. Then, the inequality [figure omitted; refer to PDF] holds for all [figure omitted; refer to PDF] .

Proof of Lemma 7.

For any [figure omitted; refer to PDF] matrix [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is well defined and [figure omitted; refer to PDF] . Let vector [figure omitted; refer to PDF] Then, we have [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , it is obvious that [figure omitted; refer to PDF] The proof is completed.

Proof of Theorem 4.

Now we are going to prove that the system (11) is asymptotically stable. Let us first consider the following positive definition function: [figure omitted; refer to PDF] where the matrix [figure omitted; refer to PDF] is defined in (12). Then, the time derivative of [figure omitted; refer to PDF] along the state trajectories of system (11) is given by [figure omitted; refer to PDF] Applying Lemma 5 to (21) yields [figure omitted; refer to PDF] where the scalars [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . By Lemma 7, it follows that for any [figure omitted; refer to PDF] [figure omitted; refer to PDF] From (22) and (23), it is obvious that [figure omitted; refer to PDF] Then, by using (24) and properties [figure omitted; refer to PDF] it generates [figure omitted; refer to PDF] According to Assumption 5, [figure omitted; refer to PDF] is a free-choice matrix. Therefore, we can easily select matrix [figure omitted; refer to PDF] such that the matrix [figure omitted; refer to PDF] is semipositive definite. Since the [figure omitted; refer to PDF] for [figure omitted; refer to PDF] are independent of each other, then, from equation (31) of paper [3], the following is true: [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , and is equivalent to [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] where the scalar [figure omitted; refer to PDF] . Thus, from (26), (28), and (29), we achieve [figure omitted; refer to PDF] In addition, by applying Lemma 6, LMI (12) is equivalent to the following inequality: [figure omitted; refer to PDF] According to (30) and (31), it is easy to get [figure omitted; refer to PDF] The inequality (32) shows that LMI (12) holds, which further implies that the sliding motion (11) is asymptotically stable.

Remark 8.

Theorem 4 provides a new existence condition of the sliding surface in terms of strict LMI, which can be easily worked out using the LMI toolbox in Matlab.

Remark 9.

Compared to recent LMI methods [1, 5-7], the proposed method offers less number of matrix variables in LMI equations making it easier to find a feasible solution.

In order to design a new decentralized adaptive output feedback sliding mode control scheme for complex interconnected time-delay system (1), we establish the following lemma.

Lemma 10.

Consider a class of interconnected time-delay systems that is decomposed into [figure omitted; refer to PDF] subsystems [figure omitted; refer to PDF] where [figure omitted; refer to PDF] are the state variables of the [figure omitted; refer to PDF] th subsystem with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The matrix [figure omitted; refer to PDF] is known matrices of appropriate dimensions. The matrices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are unknown matrices of appropriate dimensions. The notation [figure omitted; refer to PDF] represents delayed states. The symbol [figure omitted; refer to PDF] is the time-varying delay, which is assumed to be known and is bounded by [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . The initial conditions are given by [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] are continuous in [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . If the matrix [figure omitted; refer to PDF] is stable then [figure omitted; refer to PDF] is bounded by [figure omitted; refer to PDF] for all time, where [figure omitted; refer to PDF] is the solution of [figure omitted; refer to PDF] in which [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is the maximum eigenvalue of the matrix [figure omitted; refer to PDF] and the scalar [figure omitted; refer to PDF] .

Proof of Lemma 10.

We are now in the position to prove Lemma 10. From (33), it is obvious that [figure omitted; refer to PDF] From system (35), we have [figure omitted; refer to PDF] According to (36), we obtain [figure omitted; refer to PDF] The stable matrix [figure omitted; refer to PDF] implies that [figure omitted; refer to PDF] for some scalars [figure omitted; refer to PDF] . Therefore, the above inequality can be rewritten as [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] be the right side term of the inequality (38) [figure omitted; refer to PDF] Then, by taking the time derivative of [figure omitted; refer to PDF] , we can get that [figure omitted; refer to PDF] For the above equation, we multiply the term [figure omitted; refer to PDF] on both sides; then [figure omitted; refer to PDF] Then, by taking the summation of both sides of the above equation, we have [figure omitted; refer to PDF] Since the [figure omitted; refer to PDF] for [figure omitted; refer to PDF] are independent of each other, then, from equation (32) of paper [3], it is clear that [figure omitted; refer to PDF] for some scalars [figure omitted; refer to PDF] . Then, by substituting (43) into (42), we achieve [figure omitted; refer to PDF] For the above equation, we multiply the term [figure omitted; refer to PDF] to both sides. Since [figure omitted; refer to PDF] , one can get that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . For the above inequality, we multiply the term [figure omitted; refer to PDF] to both sides, then [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , integrating the above inequality on both sides, we obtain [figure omitted; refer to PDF] where the time function [figure omitted; refer to PDF] satisfies (34). Hence, we can see that [figure omitted; refer to PDF] for all time, if [figure omitted; refer to PDF] is sufficiently large.

Remark 11.

It is obvious that the time function [figure omitted; refer to PDF] is dependent on only state variable [figure omitted; refer to PDF] . Therefore, we can replace state variable [figure omitted; refer to PDF] by a function of state variable [figure omitted; refer to PDF] in controller design. This feature is very useful in controller design using only output variables.

3.3. Decentralized Adaptive Output Feedback Sliding Mode Controller Design

Now, we are in the position to prove that the state trajectories of system (1) reach sliding surface (8) in finite time and stay on it thereafter. In order to satisfy the above aims, the modified decentralized adaptive output feedback sliding mode controller is selected to be [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and the scalars [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The adaptive law is defined as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the solution of the following equations: [figure omitted; refer to PDF] in which [figure omitted; refer to PDF] and the scalars [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The time function [figure omitted; refer to PDF] will be designed later. It should be pointed out that controller (48) uses only output variables.

Now let us discuss the reaching conditions in the following theorem.

Theorem 12.

Suppose that LMI (12) has solution [figure omitted; refer to PDF] and the scalars [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] . Consider the closed loop of system (1) with the above decentralized adaptive output feedback sliding mode controller (48) where the sliding surface is given by (8). Then, the state trajectories of system (1) reach the sliding surface in finite time and stay on it thereafter.

Proof of Theorem 12.

We consider the following positive definite function: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then, the time derivative of [figure omitted; refer to PDF] along the trajectories of (9) is given by [figure omitted; refer to PDF]

Substituting (7) into (52), we have [figure omitted; refer to PDF] From (53), properties [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] generate [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] , we obtain [figure omitted; refer to PDF] The facts [figure omitted; refer to PDF] and [figure omitted; refer to PDF] imply that [figure omitted; refer to PDF] Equation (9) implies that [figure omitted; refer to PDF] In addition, let [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] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Then, by applying Lemma 10 to the system (6), we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the solution of [figure omitted; refer to PDF] in which [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is the maximum eigenvalue of the matrix [figure omitted; refer to PDF] and the scalars [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

From (57) and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , (59) can be rewritten as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . By (56), (57), and (58), we have [figure omitted; refer to PDF] By substituting the controller (48) into (61), it is clear that [figure omitted; refer to PDF] Considering (50) and (62), the above inequality can be rewritten as [figure omitted; refer to PDF] By applying (49) to (63), we achieve [figure omitted; refer to PDF] The above inequality implies that the state trajectories of system (1) reach the sliding surface [figure omitted; refer to PDF] in finite time and stay on it thereafter.

Remark 13.

From sliding mode control theory, Theorems 4 and 12 together show that the sliding surface (8) with the decentralised adaptive output feedback SMC law (48) guarantee that (1) at any initial value the state trajectories will reach the sliding surface in finite time and stay on it thereafter; and (2) the system (1) in sliding mode is asymptotically stable.

Remark 14.

The SMC scheme is often discontinuous which causes "chattering" in the sliding mode. This chattering is highly undesirable because it may excite high-frequency unmodelled plant dynamics. The most common approach to reduce the chattering is to replace the discontinuous function [figure omitted; refer to PDF] by a continuous approximation such as [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is a positive constant [29]. This approach guarantees not asymptotic stability but ultimate boundedness of system trajectories within a neighborhood of the origin depending on [figure omitted; refer to PDF] .

Remark 15.

The proposed controller and sliding surface use only output variables, while the bounds of disturbances are unknown. Therefore, this approach is very useful and more realistic, since it can be implemented in many practical systems.

4. Numerical Example

To verify the effectiveness of the proposed decentralized adaptive output feedback SMC law, our method has been applied to interconnected time-delay systems composed of two third-order subsystems, which is modified from [3].

The first subsystem's dynamics is given as [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] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The mismatched parameter uncertainties in the state matrix of the first subsystem are [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] [figure omitted; refer to PDF] . The mismatched uncertain interconnections with the second subsystem are [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The exogenous disturbance in the first subsystem is [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] can be selected by any positive value.

The second subsystem's dynamics is given as [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] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The mismatched parameter uncertainties in the state matrix of the second subsystem are [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The mismatched uncertain interconnection with the first subsystem is [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] [figure omitted; refer to PDF] . The exogenous disturbance in the second subsystem is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] can be selected by any positive value.

For this work, the following parameters are given as follows: [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] , [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] , [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] . According to the algorithm given in [3], the coordinate transformation matrices for the first subsystem and the second subsystem are [figure omitted; refer to PDF] . By solving LMI (12), it is easy to verify that conditions in Theorem 4 are satisfied with positive matrices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The matrices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are selected to be [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . From (8), the sliding surface for the first subsystem and the second subsystem are [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Theorem 4 showed that the sliding motion associated with the sliding surfaces [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is globally asymptotically stable. The time functions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the solution of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively. From Theorem 12, the decentralized adaptive output feedback sliding mode controller for the first subsystem and the second subsystem are [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] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Figures 5 and 6 imply that the chattering occurs in control input. In order to eliminate chattering phenomenon, the discontinuous controllers (67) and (68) are replaced by the following continuous approximations: [figure omitted; refer to PDF] From Figures 7 and 8, we can see that the chattering is eliminated.

The time-delays chosen for the first subsystem and the second subsystem are [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The initial conditions for two subsystems are selected to be [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively. By Figures 1, 2, 3, 4, 5, 6, 7, and 8 it is clearly seen that the proposed controller is effective in dealing with matched and mismatched uncertainties and the system has a good performance.

Figure 1: Time responses of states [figure omitted; refer to PDF] (solid), [figure omitted; refer to PDF] (dashed), and [figure omitted; refer to PDF] (dotted).

[figure omitted; refer to PDF]

Figure 2: Time responses of states [figure omitted; refer to PDF] (solid), [figure omitted; refer to PDF] (dashed), and [figure omitted; refer to PDF] (dotted).

[figure omitted; refer to PDF]

Figure 3: Time responses of sliding function [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 4: Time responses of sliding function [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 5: Time responses of discontinuous control input [figure omitted; refer to PDF] (67).

[figure omitted; refer to PDF]

Figure 6: Time responses of discontinuous control input [figure omitted; refer to PDF] (68).

[figure omitted; refer to PDF]

Figure 7: Time responses of continuous control input [figure omitted; refer to PDF] (69).

[figure omitted; refer to PDF]

Figure 8: Time responses of continuous control input [figure omitted; refer to PDF] (70).

[figure omitted; refer to PDF]

5. Conclusion

In this paper, a decentralized adaptive SMC law is proposed to stabilize complex interconnected time-delay systems with unknown disturbance, mismatched parameter uncertainties in the state matrix, and mismatched interconnections. Furthermore, in these systems, the system states are unavailable and no estimated states are required. This is a new problem in the application of SMC to interconnected time-delay systems. By establishing a new lemma, the two major limitations of SMC approaches for interconnected time-delay systems in [3] have been removed. We have shown that the new sliding mode controller guarantees the reachability of the system states in a finite time period, and moreover the dynamics of the reduced-order complex interconnected time-delay system in sliding mode is asymptotically stable under certain conditions.

Acknowledgment

The authors would like to acknowledge the financial support provided by the National Science Council in Taiwan (NSC 102-2632-E-212-001-MY3).

Conflict of Interests

The authors declare that they have no conflict of interests regarding to the publication of this paper.

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