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

Authors: Van Van Huynh; Yao-Wen Tsai; Phan Van Duc
Journal: Mathematical Problems in Engineering
 (2015)

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## 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.


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## Body

## 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 H∞ 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. XTdenotes the transpose of matrix X. In and 0n×m are used to denote the n×n identity matrix and the n×m zero matrix, respectively. The subscripts n and n×m are omitted where the dimension is irrelevant or can be determined from the context. x stands for the Euclidean norm of vector x and A stands for the matrix induced norm of the matrix A. The expression A>0 means that A is a symmetric positive definite. Rn denotes the n-dimensional Euclidean space. For the sake of simplicity, sometimes function xi(t) is denoted by xi.


## 2. Problem Formulations and Preliminaries

We consider a class of interconnected time-delay systems that is decomposed intoL subsystems. The state space representation of each subsystem is described as follows:(1)x˙i=Ai+ΔAixi+Biui+Git,xi,xidi&22+∑j=1j≠iLHij+ΔHijt,xj,xjdjxjdj,yi=Cixi,where xi∈Rni, ui∈Rmi, and yi∈Rpi with mi<pi<ni are the state variables, inputs, and outputs of the ith subsystem, respectively. The triplet Ai,Bi,Ci and Hij represent known constant matrices of appropriate dimensions. The notations xidi∶=xi(t-di) and yidi∶=yi(t-di) represent delayed states and delayed outputs, respectively. The symbol di∶=di(t) is the time-varying delay, which is assumed to be known and is bounded by d-i for all di where d-i>0 is constant. The initial conditions are given by xit=χitt∈-d-i,0, where χi(t) are continuous in [-d-i,0] for i=1,2,3,…,L. The matrices ΔAixi and ΔHij(t,xj,xjdj) represent mismatched parameter uncertainties in the state matrix and mismatched uncertain interconnections with rank[Bi:ΔAi:ΔHij]>rank(Bi)=mi. The matrix BiGi(t,xi,xidi) is the disturbance input. In this paper, only output variables yi 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 (Ai,Bi) are completely controllable.Assumption 2.
The matricesBi and Ci are full rank and rank(CiBi)=mi.Assumption 3.
The exogenous disturbanceGi(t,xi,xidi) is assumed to be bounded and to satisfy the following condition:(2)Git,xi,xidi≤ci+bixi,where bi and ci 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 asΔAi=DiΔFi(xi(t),t)Ei, where ΔFi(xi(t),t) is unknown but bounded as ΔFi(xi(t),t)≤1 and Di, Ei are known matrices of appropriate dimensions.Assumption 5.
The mismatched uncertain interconnections are given asΔHij=DijΔFij(t,xj,xjdj)Eij, where ΔFij(t,xj,xjdj) is unknown but bounded as ΔFij(t,xj,xjdj)≤1 and Dij, Eij are any nonzero matrices of appropriate dimensions.Remark 1.
Assumptionrank(CiBi)=mi is a limitation on the triplet (Ai,Bi,Ci) 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 outputsyi. That is Gi(t,xi,xidi)≤gi(t,yi,yidi), where gi(t,yi,yidi) is known. This condition is quite restrictive.(ii)
The sliding matrixFi satisfies ΓiCi=FiCiAi to guarantee sliding condition Si(xi)=Fiyi=0. 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:(3)x˙i=Aiixi+Biui+∑j=1j≠iLAijxj,yi=Cixi,where xi∈Rni, ui∈Rmi, and yi∈Rpi are the state variables, inputs, and outputs of the ith subsystem, respectively. Under assumption rank(CiBi)=mi, it follows from [3, 26] that there exists a coordinate transformation xi→zi=Tixi such that the interconnected system (3) has the following regular form: (4)z˙i=Aii1Aii2Aii3Aii4zi+∑j=1j≠iLAij1Aij2Aij3Aij4zj+0Bi2ui,yi=0Ci2zi,where TiAiiTi-1=Aii1Aii2Aii3Aii4, TiAijTj-1=Aij1Aij2Aij3Aij4, and TiBi=0Bi2, CiTi-1=0Ci2. The matrices Bi2∈Rmi×mi and Ci2∈Rpi×pi are nonsingular and Aii1 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. Sincerank(CiBi=mi), it follows from Lemma 3 that there exists a coordinate transformation zi=Tixi such that the system (1) has the following regular form:(5)z˙i=Ai1Ai2Ai3Ai4+Di1Di2ΔFiEi1Ei2zi203+0Bi2ui+Git,Ti-1zi,Ti-1zidi203+∑j=1j≠iLHij1Hij2Hij3Hij4+Dij1Dij2ΔFijEij1Eij2zjdj,yi=0Ci2zi,where Ti=Ti1Ti2, Ti-1=Wi1Wi2, TiAiTi-1=Ai1Ai2Ai3Ai4, TiHijTj-1=Hij1Hij2Hij3Hij4, TiDiΔFiEiTi-1=Di1Di2ΔFiEi1Ei2, TiDijΔFijEijTj-1=Dij1Dij2ΔFijEij1Eij2, and TiBi=0Bi2, CiTi-1=0Ci2. The matrices Bi2∈Rmi×mi and Ci2∈Rpi×pi are non-singular and Ai1 is stable.Lettingzi=zi1zi2, where zi1∈Rni-mi and zi2∈Rmi, the first equation of (5) can be rewritten as(6)z˙i1=Ai1+Di1ΔFiEi1zi1+Ai2+Di1ΔFiEi2zi2+∑j=1j≠iLHij1+Dij1ΔFijEij1zj1dj11111111+Hij2+Dij1ΔFijEij2zj2dj,(7)z˙i2=Ai3+Di2ΔFiEi1zi1+Ai4+Di2ΔFiEi2zi2+Bi2ui+Git,Ti-1zi,Ti-1zidi+∑j=1j≠iLHij3+Dij2ΔFijEij1zj1dj11111111+Hij4+Dij2ΔFijEij2zj2dj.Obviously, the system (6) represents the sliding-motion dynamic of the system (5), and, hence, the corresponding sliding surface can be chosen as follows:(8)σixi=KiCi2-1yi=0,where Ki=Fi1Fi2=0mi×pi-miFi2, Fi2=ΞiPiΞiT, the matrix Pi∈R(ni-mi)×(ni-mi) is defined later, and the matrix Ξi∈Rmi×(ni-mi) is selected such that Fi2 is nonsingular. Then, by using the second equation of (5), we have(9)σixi=KiCi2-1yi=KiCi2-10Ci2zi=KiNi0pi-mi×mi0mi×ni-miImiz=Fi1NiFi2zi=Fi2zi2=0,where Ni=0(pi-mi)×(ni-pi)I(pi-mi). In addition, the Newton-Leibniz formula is defined as(10)zi2di=zi2t-di=zi2t-∫t-ditz˙i2sds.Therefore, in sliding modes σi(xi)=0 and σ˙i(xi)=0, we have zi2=0 and zj2dj=0. 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(11)z˙i1=Ai1+Di1ΔFiEi1zi1+∑j=1j≠iLHij1+Dij1ΔFijEij1zj1dj.


### 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: (12)ΨiPiDi1Ei1TDi1TPi-φiImi0Ei10-φi-1Imi<0,i=1,2,…,L,where Ψi=Ai1TPi+PiAi1+L-1/εiPi+∑j=1,j≠iL(qεjHji1TPjHji1+φ-i-1PiDij1Dij1TPi+q^φ-jEji1TEji1), Pi∈R(ni-mi)×(ni-mi) is any positive matrix, and L is the number of subsystems and the scalars q>1, q^>1, φi>0, εi>0, φ-i>0, i=1,2,…,L. Then, we can establish the following theorem.Theorem 4.
Suppose that LMI (12) has solution Pi>0 and the scalars q>1, q^>1, φi>0, εi>0, φ-i>0, i=1,2,…,L. Suppose also that the SMC law is (13)uit=-Fi2Bi2-1κiηit+κ-iyi+κ^iyidi11111111111111+ζit+αiyidiσiσi,i=1,2,…,L,where κi=Fi2(Ai3+Di2Ei1)+∑j=1,j≠iLβiFj2(Hji3+Dji2Eji1),  κ-i=Fi2Ai4+Di2Ei2Fi2-1KiCi2-1, κ^i=∑j=1,j≠iLFj2(Hji4+Dji2Eji2)Fi2-1KiCi2-1, and the scalars αi>0, βi>1, and the time functions ζi(t) and ηi(t) will be designed later. The sliding surface is given by (8). Then, the dynamics of system (11) restricted to the sliding surface σi(xi)=0 is asymptotically stable.Before proofing Theorem4, we recall the following lemmas.Lemma 5 (see [27]).
LetX, Y, and F be real matrices of suitable dimension with FTF≤I; then, for any scalar φ>0, the following matrix inequality holds:(14)XFY+YTFTXT≤φ-1XXT+φYTY.Lemma 6 (see [28]).
The linear matrix inequality:(15)QxΠxΠxTRx>0,where Q(x)=QxT, R(x)=RxT, and Π(x) depend affinely on x, is equivalent to R(x)>0, Q(x)-Π(x)Rx-1ΠxT>0.Lemma 7.
Assume thatx∈Rn, y∈Rn, N∈Rn×n, and N is a positive definite matrix. Then, the inequality(16)xTNy+yTNx≤1εxTNx+εyTNyholds for all ε>0.Proof of Lemma7.
For anyn×n matrix N>0, N1/2 is well defined and N1/2>0. Let vector (17)ϑ=1εN1/2x-εN1/2y.Then, we have (18)ϑTϑ=1εN1/2x-εN1/2yT1εN1/2x-εN1/2y=1εxTNx-xTNy-yTNx+εyTNy.Since ϑTϑ≥0, it is obvious that (19)xTNy+yTNx≤1εxTNx+εyTNy.The proof is completed.Proof of Theorem4.
Now we are going to prove that the system (11) is asymptotically stable. Let us first consider the following positive definition function:(20)V=∑i=1Lzi1TPizi1,where the matrix Pi∈R(ni-mi)×(ni-mi) is defined in (12). Then, the time derivative of V along the state trajectories of system (11) is given by(21)V˙=∑i=1Lzi1TAi1TPi+PiAi1+PiDi1ΔFiEi1111111111+Ei1TΔFiTDi1TPizi1+∑i=1L∑j=1j≠iLzj1djTHij1TPizi1+zi1TPiHij1zj1dj11111111111+zi1TPiDij1ΔFijEij1zj1dj11111111111+zj1djTEij1TΔFijTDij1TPizi1.Applying Lemma 5 to (21) yields(22)V˙≤∑i=1Lzi1TAi1TPi+PiAi1+φi-1PiDi1Di1TPi111111111+φiEi1TEi1zi1+∑i=1L∑j=1j≠iLzj1djTHij1TPizi1+zi1TPiHij1zj1dj111111111+φ-i-1zi1TPiDij1Dij1TPizi1111111111+φ-izj1djTEij1TEij1zj1dj,where the scalars φi>0 and φ-i>0. By Lemma 7, it follows that for any εi>0(23)∑i=1L∑j=1j≠iLzi1TPiHij1zj1dj+zj1djTHij1TPizi1≤∑i=1L∑j=1j≠iL1εizi1TPizi1+εizj1djTHij1TPiHij1zj1dj.From (22) and (23), it is obvious that(24)V˙≤∑i=1Lzi1TAi1TPi+PiAi1+φi-1PiDi1Di1TPi11111111+φiEi1TEi1zi1+∑i=1L∑j=1j≠iLεizj1djTHij1TPiHij1zj1dj+1εizi1TPizi11111111111+φ-izj1djTEij1TEij1zj1dj1111111111+φ-i-1zi1TPiDij1Dij1TPizi11εi.Then, by using (24) and properties(25)∑i=1L∑j=1j≠iLεizj1djTHij1TPiHij1zj1djlllllllllllllll=∑i=1L∑j=1j≠iLεjzi1diTHji1TPjHji1zi1di,∑i=1L∑j=1j≠iLφ-izj1djTEij1TEij1zj1djllllllllllll=∑i=1L∑j=1j≠iLφ-jzi1diTEji1TEji1zi1di,it generates(26)V˙≤∑i=1Lzi1TAi1TPi+PiAi1+φi-1PiDi1Di1TPi11111111+φiEi1TEi1zi1+∑i=1L∑j=1j≠iL1εiεjzi1diTHji1TPjHji1zi1di11111111111+1εizi1TPizi1+φ-jzi1diTEji1TEji1zi1di11111111111+φ-i-1zi1TPiDij1Dij1TPizi11εi.According to Assumption 5, Eij is a free-choice matrix. Therefore, we can easily select matrix Eij such that the matrix Eji1TEji1 is semipositive definite. Since the zi1 for i=1,2,…,L are independent of each other, then, from equation (31) of paper [3], the following is true:(27)Vz11d1,z21d2,z31d3,…,zn1dn≤qVz11,z21,z31,…,zn1for q>1, and is equivalent to(28)∑i=1L∑j=1j≠iLεjzi1diTHji1TPjHji1zi1di≤q∑i=1L∑j=1j≠iLεjzi1THji1TPjHji1zi1which implies that(29)∑i=1L∑j=1j≠iLφ-jzi1diTEji1TEji1zi1di≤q^∑i=1L∑j=1j≠iLφ-jzi1TEji1TEji1zi1,where the scalar q^>1. Thus, from (26), (28), and (29), we achieve(30)V˙≤∑i=1Lzi1T∑j=1j≠iLqεjHji1TPjHji1+φ-i-1PiDij1Dij1TPiAi1TPi+PiAi1+φi-1PiDi1Di1TPi1111ll111+φiEi1TEi1+L-1εiPi1111ll111+∑j=1j≠iLqεjHji1TPjHji1+φ-i-1PiDij1Dij1TPi1111111l111111+q^φ-jEji1TEji1∑j=1j≠iLqεjHji1TPjHji1+φ-i-1PiDij1Dij1TPiDi1Tzi1.In addition, by applying Lemma 6, LMI (12) is equivalent to the following inequality:(31)Ai1TPi+PiAi1+φiEi1TEi1+L-1εiPi+φi-1PiDi1Di1TPi+∑j=1j≠iLqεjHji1TPjHji1111111+φ-i-1PiDij1Dij1TPi+q^φ-jEji1TEji1<0.According to (30) and (31), it is easy to get(32)V˙<0.The inequality (32) shows that LMI (12) holds, which further implies that the sliding motion (11) is asymptotically stable.Remark 8.
Theorem4 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 intoL subsystems(33)v˙i=Aii+ΔAiivi+∑j=1j≠iLAijvjdj,where vi=vi1vi2 are the state variables of the ith subsystem with vi1∈Rni-mi and vi2∈Rmi. The matrix Aii=Aii1Aii2Aii3Aii4 is known matrices of appropriate dimensions. The matrices ΔAii=ΔAii1ΔAii2ΔAii3ΔAii4 and Aij=Aij1Aij2Aij3Aij4 are unknown matrices of appropriate dimensions. The notation vidi:=vi(t-di) represents delayed states. The symbol di:=di(t) is the time-varying delay, which is assumed to be known and is bounded by d-i for all di. The initial conditions are given by vit=χit(t∈[-d-i,0]), where χi(t) are continuous in [-d-i,0] for i=1,2,3,…,L. If the matrix Aii1 is stable then ∑i=1Lvi1(t) is bounded by ∑i=1Lϕi(t) for all time, where ϕi(t) is the solution of(34)ϕ˙it=k^iϕit+ki∑j=1j≠iLAji2vi2diAii2+ΔAii2vi211111111111111+∑j=1j≠iLAji2vi2di,i=1,2,…,Lin which k^i=ki(ΔAii1+∑j=1,j≠iLβiAji1)+λi<0, ki>0. λi is the maximum eigenvalue of the matrix Aii1 and the scalar βi>1.Proof of Lemma10.
We are now in the position to prove Lemma10. From (33), it is obvious that(35)v˙i1t=Aii1+ΔAii1vi1+Aii2+ΔAii2vi2+∑j=1j≠iLAij1vj1dj+Aij2vj2dj.From system (35), we have(36)vi1t=exp⁡Aii1vi1011+∫0texpAii1t-τ111111×∑j=1j≠iLAij1vj1dj+Aij2vj2djΔAii1vi1+Aii2+ΔAii2vi2111111111+∑j=1j≠iLAij1vj1dj+Aij2vj2djdτ.According to (36), we obtain(37)vi1t≤expAii1tvi1011111+∫0texp⁡Aii1t-τAii2+ΔAii21111111111×vi2dτ11111+∫0texpAii1t-τ∑j=1j≠iLAij1vj1djΔAii1vi111111111111111111111111111+∑j=1j≠iLAij1vj1dj11111111111111111111111111+∑j=1j≠iLAij2vj2djdτ.The stable matrix Aii1 implies that exp(Aii1t)≤kiexp(λit) for some scalars ki>0,i=1,2,…,L. Therefore, the above inequality can be rewritten as(38)vi1texp⁡-λit≤kivi10111111111lll1111+∫0tkiexp⁡-λiτAii2+ΔAii211111111111l11111×vi2dτ111111111lll1111+∫0tkiexp⁡-λiτ11111111111l111111×+∑j=1j≠iLAij2vj2djΔAii1vi1t11111111111111111111111+∑j=1j≠iLAij1vj1dj1111111111111111111111+∑j=1j≠iLAij2vj2djdτ.Let si(t) be the right side term of the inequality (38)(39)sit=kivi10+∫0tkiexp-λiτAii2+ΔAii2vi2dτ+∫0tkiexp-λiτ+∑j=1j≠iLAij1vj1djΔAii1vi1lllllllllllllllllllllllllllllllllllll+∑j=1j≠iLAij1vj1djllllllllllllllllllllllllllllllllllll+∑j=1j≠iLAij2vj2djdτ.Then, by taking the time derivative of si(t), we can get that (40)ddtsit=kiexp-λitAii2+ΔAii2vi211+kiexp-λit+∑j=1j≠iLAij1vj1dj+∑j=1j≠iLAij2vj2djΔAii1vi11111111111111111+∑j=1j≠iLAij1vj1dj1111111111111111+∑j=1j≠iLAij2vj2dj.For the above equation, we multiply the term 1/kiexp(λit) on both sides; then (41)1kiexpλitddtsit=Aii2+ΔAii2vi211+ΔAii1vi1+∑j=1j≠iLAij1vj1dj11+∑j=1j≠iLAij2vj2dj.Then, by taking the summation of both sides of the above equation, we have(42)∑i=1L1kiexpλitddtsit=∑i=1LAii2+ΔAii2vi2+∑i=1LΔAii1vi111+∑i=1L∑j=1j≠iLAji2vi2di+∑i=1L∑j=1j≠iLAji1vi1di.Since the vi1 for i=1,2,…,L are independent of each other, then, from equation (32) of paper [3], it is clear that (43)vi1di≤βivi1,i=1,2,…,Lfor some scalars βi>1,i=1,2,…,L. Then, by substituting (43) into (42), we achieve (44)∑i=1L1kiexpλitddtsit=∑i=1LAii2+ΔAii2vi2+∑i=1LΔAii1vi111+∑i=1L∑j=1j≠iLAji2vi2di+∑i=1L∑j=1j≠iLβiAji1vi1.For the above equation, we multiply the term kiexp(-λit) to both sides. Since vi1exp(-λit)≤si(t), one can get that(45)∑i=1Lddtsit≤∑i=1Lkiexp-λit11×Aii2+ΔAii2vi2+∑j=1j≠iLAji2vi2di11+∑i=1Lk-isit,where k-i=kiΔAii1+∑j=1,j≠iLβiAji1. For the above inequality, we multiply the term exp(-k-it) to both sides, then(46)∑i=1Lddtsitexp-k-it≤∑i=1Lkiexp-λit11×Aii2+ΔAii2vi2+∑j=1j≠iLAji2vi2di×exp-k-it.Since vi1exp(-λit)≤si(t), integrating the above inequality on both sides, we obtain(47)∑i=1Lvi1≤∑i=1Lkivi10expk-i+λit11+∑i=1L∫0tkiexp-λiτ∑j=1j≠iLAji2vi2diAii2+ΔAii2vi211111111111111111111111+∑j=1j≠iLAji2vi2di1111111111∫0tkiexp-λiτ∑j=1j≠iLAji2vi2diAii2+ΔAii2vi2×exp-k-iτdτexpk-itexpλit=∑i=1L∑j=1j≠iLAji2vi2diϕi0expk-i+λit+∫0tkiexpk-i+λit-τ111111×Aii2+ΔAii2vi2+∑j=1j≠iLAji2vi2didτ=∑i=1Lϕitifϕi0≥kivi10,where the time function ϕi(t) satisfies (34). Hence, we can see that ∑i=1Lϕi(t)≥∑i=1Lvi1 for all time, if ϕi(0) is sufficiently large.Remark 11.
It is obvious that the time functionϕi(t) is dependent on only state variable vi2. Therefore, we can replace state variable vi1 by a function of state variable vi2 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(48)uit=-Fi2Bi2-1κiηit+κ-iyi+κ^iyidi1111111111111+ζit+αiκ^iyidiσiσi,i=1,2,…,L,where κi=Fi2Ai3+Di2Ei1+∑j=1,j≠iLβiFj2Hji3+Dji2Eji1, κ-i=Fi2Ai4+Di2Ei2Fi2-1KiCi2-1, κ^i=∑j=1,j≠iLFj2Hji4+Dji2Eji2Fi2-1KiCi2-1, and the scalars αi>0 and βi>1. The adaptive law is defined as(49)ζit≥b^iFi2Bi2111×Wi1ηit+Wi2Fi2-1KiCi2-1yi111+Fi2Bi2c^i+q˘ici24,where b^i and c^i are the solution of the following equations:(50)b^˙i=q-iFi2Bi2111×Wi1ηit+Wi2Fi2-1KiCi2-1yi,c^˙i=q~i-q˘ic^i+Fi2Bi2in which Wi1Wi2=Ti-1 and the scalars q-i>0, q~i>0, and q˘i>0. The time function ηi(t) 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 Pi>0 and the scalars q>1, q^>1, φi>0, εi>0, φ-i>0, i=1,2,…,L. 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 Theorem12.
We consider the following positive definite function:(51)V=∑i=1Lσi+0.5q-ib~i2+0.5q~ic~i2,where b~i(t)=bi-b^i(t) and c~i(t)=ci-c^i(t). Then, the time derivative of V along the trajectories of (9) is given by(52)V˙=∑i=1LσiTσiFi2z˙i2-1q-ib~ib^˙i-1q~ic~ic^˙i.
Substituting (7) into (52), we have(53)V˙=∑i=1LσiTσiFi2Ai3+Di2ΔFiEi1zi1hhhhhhhhhhh+Ai4+Di2ΔFiEi2zi2+∑i=1LσiTσiFi2Bi2ui+Git,Ti-1zi,Ti-1zidi-∑i=1L1q-ib~ib^˙i-∑i=1L1q~ic~ic^˙i+∑i=1L∑j=1j≠iLσiTσiFi2Hij3+Dij2ΔFijEij1zj1djhhhhhhhhhhhhhhhh+Hij4+Dij2ΔFijEij2zj2dj.From (53), properties AB≤AB and ΔFi≤1, ΔFij≤1 generate(54)V˙≤∑i=1LFi2Ai3+Di2Ei1zi1kkkkkkkkk+Ai4+Di2Ei2zi2+∑i=1L∑j=1j≠iLFi2Hij3+Dij2Eij1zj1djhhhhhhhhhhhhh+Hij4+Dij2Eij2zj2dj+∑i=1LFi2Bi2Gi+∑i=1LσiTσiFi2Bi2ui-∑i=1L1q-ib~ib^˙i-∑i=1L1q~ic~ic^˙i.Since Gi≤ci+bixi and xi=Wi1zi1+Wi2zi2, where Wi1Wi2=Ti-1, we obtain(55)V˙≤∑i=1LFi2Ai3+Di2Ei1zi1kkkkkkkkk+Ai4+Di2Ei2zi2+∑i=1LσiTσiFi2Bi2ui+∑i=1L∑j=1j≠iLFi2Hij3+Dij2Eij1zj1djhhhhhhhhhhhhhi+Hij4+Dij2Eij2zj2dj+∑i=1LbiFi2Bi2Wi1zi1+Wi2zi2+∑i=1LFi2Bi2ci-∑i=1L1q-ib~ib^˙i-∑i=1L1q~ic~ic^˙i.The facts ∑i=1L∑j=1,j≠iLFi2Hij3+Dij2Eij1zj1dj=∑i=1L∑j=1,j≠iLFj2Hji3+Dji2Eji1zi1di and ∑i=1L∑j=1,j≠iLFi2Hij4+Dij2Eij2zj2dj=∑i=1L∑j=1,j≠iLFj2Hji4+Dji2Eji2zi2di imply that(56)V˙≤∑i=1LFi2Ai3+Di2Ei1zi1kkkkkkkkk+Ai4+Di2Ei2zi2+∑i=1LσiTσiFi2Bi2ui+∑i=1L∑j=1j≠iLFj2Hji3+Dji2Eji1zi1dikkkkkkkkkkkkkk+Hji4+Dji2Eji2zi2di+∑i=1LbiFi2Bi2Wi1zi1+Wi2zi2+∑i=1LFi2Bi2ci-∑i=1L1q-ib~ib^˙i-∑i=1L1q~ic~ic^˙i.Equation (9) implies that(57)zi2=Fi2-1KiCi2-1yi,zi2di=Fi2-1KiCi2-1yidi.In addition, let vi1=zi1, vi2=zi2, vj1dj=zj1dj, vj2dj=zj2dj, Aii1=Ai1, ΔAii1=Di1ΔFiEi1, Aii2=Ai2, ΔAii2=Di1ΔFiEi2, Aij1=(Hij1+Dij1ΔFijEij1), Aij2=(Hij2+Dij1ΔFijEij2), and ϕi(t)=ηi(t). Then, by applying Lemma 10 to the system (6), we obtain(58)∑i=1Lzi1≤∑i=1Lηit,where ηi(t) is the solution of (59)η˙it=k^iηit+ki∑j=1j≠iLHji2+Dji1ΔFjiEji2zi2diAi2+Di1ΔFiEi2zi21111111111111+∑j=1j≠iLHji2+Dji1ΔFjiEji2zi2diin which k^i=(k-i+λi)<0 and k-i=kiDi1ΔFiEi1+∑j=1,j≠iLβiHji1+Dji1ΔFjiEji1. λi is the maximum eigenvalue of the matrix Ai1 and the scalars ki>0, βi>1.
From (57) and ΔFi≤1, ΔFji≤1, (59) can be rewritten as(60)η˙it=k^iηit111+ki∑j=1j≠iLHji2+Dji1Eji2Ai2+Di1Ei2Fi2-1KiCi2-1yi111111111+∑j=1j≠iLHji2+Dji1Eji2Fi2-11111111111111×KiCi2-1yidi∑j=1j≠iLHji2+Dji1Eji2,where k^i=(k-i+λi)<0 and k-i=ki(Di1Ei1+∑j=1,j≠iLβi(Hji1+Dji1Eji1)). By (56), (57), and (58), we have(61)V˙≤∑i=1LFi2Fi2-1Ai3+Di2Ei1ηi11111l1111+Ai4+Di2Ei2Fi2-1KiCi2-1yi11+∑i=1L∑j=1j≠iLFj2βiHji3+Dji2Eji1ηi11111111111111+Hji4+Dji2Eji211111111111111×Fi2-1KiCi2-1yidi11+∑i=1LbiFi2Bi2Wi1ηi+Wi2Fi2-1KiCi2-1yi11+∑i=1LFi2Bi2ci+∑i=1LσiTσiFi2Bi2ui11-∑i=1L1q-ib~ib^˙i-∑i=1L1q~ic~ic^˙i.By substituting the controller (48) into (61), it is clear that(62)V˙≤∑i=1LbiFi2Bi2Wi1ηi+Wi2Fi2-1KiCi2-1yi11-∑i=1Lζi-∑i=1Lαi+∑i=1LFi2Bi2ci11-∑i=1L1q-ib~ib^˙i-∑i=1L1q~ic~ic^˙i.Considering (50) and (62), the above inequality can be rewritten as(63)V˙≤∑i=1Lb^iFi2Bi2Wi1ηi+Wi2Fi2-1KiCi2-1yi11-∑i=1Lζi-∑i=1Lαi+∑i=1LFi2Bi2c^i11+∑i=1Lq˘i-c^i-ci22+ci24.By applying (49) to (63), we achieve(64)V˙≤-∑i=1Lαi-∑i=1Lq˘ic^i-ci22<0.The above inequality implies that the state trajectories of system (1) reach the sliding surface σi(xi)=0 in finite time and stay on it thereafter.Remark 13.
From sliding mode control theory, Theorems4 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 functionσi/σi by a continuous approximation such as σi/σi+μi, where μi 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 μi.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.


## 3.1. Sliding Surface Design

Let us first design a sliding surface, which depends on only output variables. Sincerank(CiBi=mi), it follows from Lemma 3 that there exists a coordinate transformation zi=Tixi such that the system (1) has the following regular form:(5)z˙i=Ai1Ai2Ai3Ai4+Di1Di2ΔFiEi1Ei2zi203+0Bi2ui+Git,Ti-1zi,Ti-1zidi203+∑j=1j≠iLHij1Hij2Hij3Hij4+Dij1Dij2ΔFijEij1Eij2zjdj,yi=0Ci2zi,where Ti=Ti1Ti2, Ti-1=Wi1Wi2, TiAiTi-1=Ai1Ai2Ai3Ai4, TiHijTj-1=Hij1Hij2Hij3Hij4, TiDiΔFiEiTi-1=Di1Di2ΔFiEi1Ei2, TiDijΔFijEijTj-1=Dij1Dij2ΔFijEij1Eij2, and TiBi=0Bi2, CiTi-1=0Ci2. The matrices Bi2∈Rmi×mi and Ci2∈Rpi×pi are non-singular and Ai1 is stable.Lettingzi=zi1zi2, where zi1∈Rni-mi and zi2∈Rmi, the first equation of (5) can be rewritten as(6)z˙i1=Ai1+Di1ΔFiEi1zi1+Ai2+Di1ΔFiEi2zi2+∑j=1j≠iLHij1+Dij1ΔFijEij1zj1dj11111111+Hij2+Dij1ΔFijEij2zj2dj,(7)z˙i2=Ai3+Di2ΔFiEi1zi1+Ai4+Di2ΔFiEi2zi2+Bi2ui+Git,Ti-1zi,Ti-1zidi+∑j=1j≠iLHij3+Dij2ΔFijEij1zj1dj11111111+Hij4+Dij2ΔFijEij2zj2dj.Obviously, the system (6) represents the sliding-motion dynamic of the system (5), and, hence, the corresponding sliding surface can be chosen as follows:(8)σixi=KiCi2-1yi=0,where Ki=Fi1Fi2=0mi×pi-miFi2, Fi2=ΞiPiΞiT, the matrix Pi∈R(ni-mi)×(ni-mi) is defined later, and the matrix Ξi∈Rmi×(ni-mi) is selected such that Fi2 is nonsingular. Then, by using the second equation of (5), we have(9)σixi=KiCi2-1yi=KiCi2-10Ci2zi=KiNi0pi-mi×mi0mi×ni-miImiz=Fi1NiFi2zi=Fi2zi2=0,where Ni=0(pi-mi)×(ni-pi)I(pi-mi). In addition, the Newton-Leibniz formula is defined as(10)zi2di=zi2t-di=zi2t-∫t-ditz˙i2sds.Therefore, in sliding modes σi(xi)=0 and σ˙i(xi)=0, we have zi2=0 and zj2dj=0. 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(11)z˙i1=Ai1+Di1ΔFiEi1zi1+∑j=1j≠iLHij1+Dij1ΔFijEij1zj1dj.


## 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: (12)ΨiPiDi1Ei1TDi1TPi-φiImi0Ei10-φi-1Imi<0,i=1,2,…,L,where Ψi=Ai1TPi+PiAi1+L-1/εiPi+∑j=1,j≠iL(qεjHji1TPjHji1+φ-i-1PiDij1Dij1TPi+q^φ-jEji1TEji1), Pi∈R(ni-mi)×(ni-mi) is any positive matrix, and L is the number of subsystems and the scalars q>1, q^>1, φi>0, εi>0, φ-i>0, i=1,2,…,L. Then, we can establish the following theorem.Theorem 4.
Suppose that LMI (12) has solution Pi>0 and the scalars q>1, q^>1, φi>0, εi>0, φ-i>0, i=1,2,…,L. Suppose also that the SMC law is (13)uit=-Fi2Bi2-1κiηit+κ-iyi+κ^iyidi11111111111111+ζit+αiyidiσiσi,i=1,2,…,L,where κi=Fi2(Ai3+Di2Ei1)+∑j=1,j≠iLβiFj2(Hji3+Dji2Eji1),  κ-i=Fi2Ai4+Di2Ei2Fi2-1KiCi2-1, κ^i=∑j=1,j≠iLFj2(Hji4+Dji2Eji2)Fi2-1KiCi2-1, and the scalars αi>0, βi>1, and the time functions ζi(t) and ηi(t) will be designed later. The sliding surface is given by (8). Then, the dynamics of system (11) restricted to the sliding surface σi(xi)=0 is asymptotically stable.Before proofing Theorem4, we recall the following lemmas.Lemma 5 (see [27]).
LetX, Y, and F be real matrices of suitable dimension with FTF≤I; then, for any scalar φ>0, the following matrix inequality holds:(14)XFY+YTFTXT≤φ-1XXT+φYTY.Lemma 6 (see [28]).
The linear matrix inequality:(15)QxΠxΠxTRx>0,where Q(x)=QxT, R(x)=RxT, and Π(x) depend affinely on x, is equivalent to R(x)>0, Q(x)-Π(x)Rx-1ΠxT>0.Lemma 7.
Assume thatx∈Rn, y∈Rn, N∈Rn×n, and N is a positive definite matrix. Then, the inequality(16)xTNy+yTNx≤1εxTNx+εyTNyholds for all ε>0.Proof of Lemma7.
For anyn×n matrix N>0, N1/2 is well defined and N1/2>0. Let vector (17)ϑ=1εN1/2x-εN1/2y.Then, we have (18)ϑTϑ=1εN1/2x-εN1/2yT1εN1/2x-εN1/2y=1εxTNx-xTNy-yTNx+εyTNy.Since ϑTϑ≥0, it is obvious that (19)xTNy+yTNx≤1εxTNx+εyTNy.The proof is completed.Proof of Theorem4.
Now we are going to prove that the system (11) is asymptotically stable. Let us first consider the following positive definition function:(20)V=∑i=1Lzi1TPizi1,where the matrix Pi∈R(ni-mi)×(ni-mi) is defined in (12). Then, the time derivative of V along the state trajectories of system (11) is given by(21)V˙=∑i=1Lzi1TAi1TPi+PiAi1+PiDi1ΔFiEi1111111111+Ei1TΔFiTDi1TPizi1+∑i=1L∑j=1j≠iLzj1djTHij1TPizi1+zi1TPiHij1zj1dj11111111111+zi1TPiDij1ΔFijEij1zj1dj11111111111+zj1djTEij1TΔFijTDij1TPizi1.Applying Lemma 5 to (21) yields(22)V˙≤∑i=1Lzi1TAi1TPi+PiAi1+φi-1PiDi1Di1TPi111111111+φiEi1TEi1zi1+∑i=1L∑j=1j≠iLzj1djTHij1TPizi1+zi1TPiHij1zj1dj111111111+φ-i-1zi1TPiDij1Dij1TPizi1111111111+φ-izj1djTEij1TEij1zj1dj,where the scalars φi>0 and φ-i>0. By Lemma 7, it follows that for any εi>0(23)∑i=1L∑j=1j≠iLzi1TPiHij1zj1dj+zj1djTHij1TPizi1≤∑i=1L∑j=1j≠iL1εizi1TPizi1+εizj1djTHij1TPiHij1zj1dj.From (22) and (23), it is obvious that(24)V˙≤∑i=1Lzi1TAi1TPi+PiAi1+φi-1PiDi1Di1TPi11111111+φiEi1TEi1zi1+∑i=1L∑j=1j≠iLεizj1djTHij1TPiHij1zj1dj+1εizi1TPizi11111111111+φ-izj1djTEij1TEij1zj1dj1111111111+φ-i-1zi1TPiDij1Dij1TPizi11εi.Then, by using (24) and properties(25)∑i=1L∑j=1j≠iLεizj1djTHij1TPiHij1zj1djlllllllllllllll=∑i=1L∑j=1j≠iLεjzi1diTHji1TPjHji1zi1di,∑i=1L∑j=1j≠iLφ-izj1djTEij1TEij1zj1djllllllllllll=∑i=1L∑j=1j≠iLφ-jzi1diTEji1TEji1zi1di,it generates(26)V˙≤∑i=1Lzi1TAi1TPi+PiAi1+φi-1PiDi1Di1TPi11111111+φiEi1TEi1zi1+∑i=1L∑j=1j≠iL1εiεjzi1diTHji1TPjHji1zi1di11111111111+1εizi1TPizi1+φ-jzi1diTEji1TEji1zi1di11111111111+φ-i-1zi1TPiDij1Dij1TPizi11εi.According to Assumption 5, Eij is a free-choice matrix. Therefore, we can easily select matrix Eij such that the matrix Eji1TEji1 is semipositive definite. Since the zi1 for i=1,2,…,L are independent of each other, then, from equation (31) of paper [3], the following is true:(27)Vz11d1,z21d2,z31d3,…,zn1dn≤qVz11,z21,z31,…,zn1for q>1, and is equivalent to(28)∑i=1L∑j=1j≠iLεjzi1diTHji1TPjHji1zi1di≤q∑i=1L∑j=1j≠iLεjzi1THji1TPjHji1zi1which implies that(29)∑i=1L∑j=1j≠iLφ-jzi1diTEji1TEji1zi1di≤q^∑i=1L∑j=1j≠iLφ-jzi1TEji1TEji1zi1,where the scalar q^>1. Thus, from (26), (28), and (29), we achieve(30)V˙≤∑i=1Lzi1T∑j=1j≠iLqεjHji1TPjHji1+φ-i-1PiDij1Dij1TPiAi1TPi+PiAi1+φi-1PiDi1Di1TPi1111ll111+φiEi1TEi1+L-1εiPi1111ll111+∑j=1j≠iLqεjHji1TPjHji1+φ-i-1PiDij1Dij1TPi1111111l111111+q^φ-jEji1TEji1∑j=1j≠iLqεjHji1TPjHji1+φ-i-1PiDij1Dij1TPiDi1Tzi1.In addition, by applying Lemma 6, LMI (12) is equivalent to the following inequality:(31)Ai1TPi+PiAi1+φiEi1TEi1+L-1εiPi+φi-1PiDi1Di1TPi+∑j=1j≠iLqεjHji1TPjHji1111111+φ-i-1PiDij1Dij1TPi+q^φ-jEji1TEji1<0.According to (30) and (31), it is easy to get(32)V˙<0.The inequality (32) shows that LMI (12) holds, which further implies that the sliding motion (11) is asymptotically stable.Remark 8.
Theorem4 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 intoL subsystems(33)v˙i=Aii+ΔAiivi+∑j=1j≠iLAijvjdj,where vi=vi1vi2 are the state variables of the ith subsystem with vi1∈Rni-mi and vi2∈Rmi. The matrix Aii=Aii1Aii2Aii3Aii4 is known matrices of appropriate dimensions. The matrices ΔAii=ΔAii1ΔAii2ΔAii3ΔAii4 and Aij=Aij1Aij2Aij3Aij4 are unknown matrices of appropriate dimensions. The notation vidi:=vi(t-di) represents delayed states. The symbol di:=di(t) is the time-varying delay, which is assumed to be known and is bounded by d-i for all di. The initial conditions are given by vit=χit(t∈[-d-i,0]), where χi(t) are continuous in [-d-i,0] for i=1,2,3,…,L. If the matrix Aii1 is stable then ∑i=1Lvi1(t) is bounded by ∑i=1Lϕi(t) for all time, where ϕi(t) is the solution of(34)ϕ˙it=k^iϕit+ki∑j=1j≠iLAji2vi2diAii2+ΔAii2vi211111111111111+∑j=1j≠iLAji2vi2di,i=1,2,…,Lin which k^i=ki(ΔAii1+∑j=1,j≠iLβiAji1)+λi<0, ki>0. λi is the maximum eigenvalue of the matrix Aii1 and the scalar βi>1.Proof of Lemma10.
We are now in the position to prove Lemma10. From (33), it is obvious that(35)v˙i1t=Aii1+ΔAii1vi1+Aii2+ΔAii2vi2+∑j=1j≠iLAij1vj1dj+Aij2vj2dj.From system (35), we have(36)vi1t=exp⁡Aii1vi1011+∫0texpAii1t-τ111111×∑j=1j≠iLAij1vj1dj+Aij2vj2djΔAii1vi1+Aii2+ΔAii2vi2111111111+∑j=1j≠iLAij1vj1dj+Aij2vj2djdτ.According to (36), we obtain(37)vi1t≤expAii1tvi1011111+∫0texp⁡Aii1t-τAii2+ΔAii21111111111×vi2dτ11111+∫0texpAii1t-τ∑j=1j≠iLAij1vj1djΔAii1vi111111111111111111111111111+∑j=1j≠iLAij1vj1dj11111111111111111111111111+∑j=1j≠iLAij2vj2djdτ.The stable matrix Aii1 implies that exp(Aii1t)≤kiexp(λit) for some scalars ki>0,i=1,2,…,L. Therefore, the above inequality can be rewritten as(38)vi1texp⁡-λit≤kivi10111111111lll1111+∫0tkiexp⁡-λiτAii2+ΔAii211111111111l11111×vi2dτ111111111lll1111+∫0tkiexp⁡-λiτ11111111111l111111×+∑j=1j≠iLAij2vj2djΔAii1vi1t11111111111111111111111+∑j=1j≠iLAij1vj1dj1111111111111111111111+∑j=1j≠iLAij2vj2djdτ.Let si(t) be the right side term of the inequality (38)(39)sit=kivi10+∫0tkiexp-λiτAii2+ΔAii2vi2dτ+∫0tkiexp-λiτ+∑j=1j≠iLAij1vj1djΔAii1vi1lllllllllllllllllllllllllllllllllllll+∑j=1j≠iLAij1vj1djllllllllllllllllllllllllllllllllllll+∑j=1j≠iLAij2vj2djdτ.Then, by taking the time derivative of si(t), we can get that (40)ddtsit=kiexp-λitAii2+ΔAii2vi211+kiexp-λit+∑j=1j≠iLAij1vj1dj+∑j=1j≠iLAij2vj2djΔAii1vi11111111111111111+∑j=1j≠iLAij1vj1dj1111111111111111+∑j=1j≠iLAij2vj2dj.For the above equation, we multiply the term 1/kiexp(λit) on both sides; then (41)1kiexpλitddtsit=Aii2+ΔAii2vi211+ΔAii1vi1+∑j=1j≠iLAij1vj1dj11+∑j=1j≠iLAij2vj2dj.Then, by taking the summation of both sides of the above equation, we have(42)∑i=1L1kiexpλitddtsit=∑i=1LAii2+ΔAii2vi2+∑i=1LΔAii1vi111+∑i=1L∑j=1j≠iLAji2vi2di+∑i=1L∑j=1j≠iLAji1vi1di.Since the vi1 for i=1,2,…,L are independent of each other, then, from equation (32) of paper [3], it is clear that (43)vi1di≤βivi1,i=1,2,…,Lfor some scalars βi>1,i=1,2,…,L. Then, by substituting (43) into (42), we achieve (44)∑i=1L1kiexpλitddtsit=∑i=1LAii2+ΔAii2vi2+∑i=1LΔAii1vi111+∑i=1L∑j=1j≠iLAji2vi2di+∑i=1L∑j=1j≠iLβiAji1vi1.For the above equation, we multiply the term kiexp(-λit) to both sides. Since vi1exp(-λit)≤si(t), one can get that(45)∑i=1Lddtsit≤∑i=1Lkiexp-λit11×Aii2+ΔAii2vi2+∑j=1j≠iLAji2vi2di11+∑i=1Lk-isit,where k-i=kiΔAii1+∑j=1,j≠iLβiAji1. For the above inequality, we multiply the term exp(-k-it) to both sides, then(46)∑i=1Lddtsitexp-k-it≤∑i=1Lkiexp-λit11×Aii2+ΔAii2vi2+∑j=1j≠iLAji2vi2di×exp-k-it.Since vi1exp(-λit)≤si(t), integrating the above inequality on both sides, we obtain(47)∑i=1Lvi1≤∑i=1Lkivi10expk-i+λit11+∑i=1L∫0tkiexp-λiτ∑j=1j≠iLAji2vi2diAii2+ΔAii2vi211111111111111111111111+∑j=1j≠iLAji2vi2di1111111111∫0tkiexp-λiτ∑j=1j≠iLAji2vi2diAii2+ΔAii2vi2×exp-k-iτdτexpk-itexpλit=∑i=1L∑j=1j≠iLAji2vi2diϕi0expk-i+λit+∫0tkiexpk-i+λit-τ111111×Aii2+ΔAii2vi2+∑j=1j≠iLAji2vi2didτ=∑i=1Lϕitifϕi0≥kivi10,where the time function ϕi(t) satisfies (34). Hence, we can see that ∑i=1Lϕi(t)≥∑i=1Lvi1 for all time, if ϕi(0) is sufficiently large.Remark 11.
It is obvious that the time functionϕi(t) is dependent on only state variable vi2. Therefore, we can replace state variable vi1 by a function of state variable vi2 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(48)uit=-Fi2Bi2-1κiηit+κ-iyi+κ^iyidi1111111111111+ζit+αiκ^iyidiσiσi,i=1,2,…,L,where κi=Fi2Ai3+Di2Ei1+∑j=1,j≠iLβiFj2Hji3+Dji2Eji1, κ-i=Fi2Ai4+Di2Ei2Fi2-1KiCi2-1, κ^i=∑j=1,j≠iLFj2Hji4+Dji2Eji2Fi2-1KiCi2-1, and the scalars αi>0 and βi>1. The adaptive law is defined as(49)ζit≥b^iFi2Bi2111×Wi1ηit+Wi2Fi2-1KiCi2-1yi111+Fi2Bi2c^i+q˘ici24,where b^i and c^i are the solution of the following equations:(50)b^˙i=q-iFi2Bi2111×Wi1ηit+Wi2Fi2-1KiCi2-1yi,c^˙i=q~i-q˘ic^i+Fi2Bi2in which Wi1Wi2=Ti-1 and the scalars q-i>0, q~i>0, and q˘i>0. The time function ηi(t) 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 Pi>0 and the scalars q>1, q^>1, φi>0, εi>0, φ-i>0, i=1,2,…,L. 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 Theorem12.
We consider the following positive definite function:(51)V=∑i=1Lσi+0.5q-ib~i2+0.5q~ic~i2,where b~i(t)=bi-b^i(t) and c~i(t)=ci-c^i(t). Then, the time derivative of V along the trajectories of (9) is given by(52)V˙=∑i=1LσiTσiFi2z˙i2-1q-ib~ib^˙i-1q~ic~ic^˙i.
Substituting (7) into (52), we have(53)V˙=∑i=1LσiTσiFi2Ai3+Di2ΔFiEi1zi1hhhhhhhhhhh+Ai4+Di2ΔFiEi2zi2+∑i=1LσiTσiFi2Bi2ui+Git,Ti-1zi,Ti-1zidi-∑i=1L1q-ib~ib^˙i-∑i=1L1q~ic~ic^˙i+∑i=1L∑j=1j≠iLσiTσiFi2Hij3+Dij2ΔFijEij1zj1djhhhhhhhhhhhhhhhh+Hij4+Dij2ΔFijEij2zj2dj.From (53), properties AB≤AB and ΔFi≤1, ΔFij≤1 generate(54)V˙≤∑i=1LFi2Ai3+Di2Ei1zi1kkkkkkkkk+Ai4+Di2Ei2zi2+∑i=1L∑j=1j≠iLFi2Hij3+Dij2Eij1zj1djhhhhhhhhhhhhh+Hij4+Dij2Eij2zj2dj+∑i=1LFi2Bi2Gi+∑i=1LσiTσiFi2Bi2ui-∑i=1L1q-ib~ib^˙i-∑i=1L1q~ic~ic^˙i.Since Gi≤ci+bixi and xi=Wi1zi1+Wi2zi2, where Wi1Wi2=Ti-1, we obtain(55)V˙≤∑i=1LFi2Ai3+Di2Ei1zi1kkkkkkkkk+Ai4+Di2Ei2zi2+∑i=1LσiTσiFi2Bi2ui+∑i=1L∑j=1j≠iLFi2Hij3+Dij2Eij1zj1djhhhhhhhhhhhhhi+Hij4+Dij2Eij2zj2dj+∑i=1LbiFi2Bi2Wi1zi1+Wi2zi2+∑i=1LFi2Bi2ci-∑i=1L1q-ib~ib^˙i-∑i=1L1q~ic~ic^˙i.The facts ∑i=1L∑j=1,j≠iLFi2Hij3+Dij2Eij1zj1dj=∑i=1L∑j=1,j≠iLFj2Hji3+Dji2Eji1zi1di and ∑i=1L∑j=1,j≠iLFi2Hij4+Dij2Eij2zj2dj=∑i=1L∑j=1,j≠iLFj2Hji4+Dji2Eji2zi2di imply that(56)V˙≤∑i=1LFi2Ai3+Di2Ei1zi1kkkkkkkkk+Ai4+Di2Ei2zi2+∑i=1LσiTσiFi2Bi2ui+∑i=1L∑j=1j≠iLFj2Hji3+Dji2Eji1zi1dikkkkkkkkkkkkkk+Hji4+Dji2Eji2zi2di+∑i=1LbiFi2Bi2Wi1zi1+Wi2zi2+∑i=1LFi2Bi2ci-∑i=1L1q-ib~ib^˙i-∑i=1L1q~ic~ic^˙i.Equation (9) implies that(57)zi2=Fi2-1KiCi2-1yi,zi2di=Fi2-1KiCi2-1yidi.In addition, let vi1=zi1, vi2=zi2, vj1dj=zj1dj, vj2dj=zj2dj, Aii1=Ai1, ΔAii1=Di1ΔFiEi1, Aii2=Ai2, ΔAii2=Di1ΔFiEi2, Aij1=(Hij1+Dij1ΔFijEij1), Aij2=(Hij2+Dij1ΔFijEij2), and ϕi(t)=ηi(t). Then, by applying Lemma 10 to the system (6), we obtain(58)∑i=1Lzi1≤∑i=1Lηit,where ηi(t) is the solution of (59)η˙it=k^iηit+ki∑j=1j≠iLHji2+Dji1ΔFjiEji2zi2diAi2+Di1ΔFiEi2zi21111111111111+∑j=1j≠iLHji2+Dji1ΔFjiEji2zi2diin which k^i=(k-i+λi)<0 and k-i=kiDi1ΔFiEi1+∑j=1,j≠iLβiHji1+Dji1ΔFjiEji1. λi is the maximum eigenvalue of the matrix Ai1 and the scalars ki>0, βi>1.
From (57) and ΔFi≤1, ΔFji≤1, (59) can be rewritten as(60)η˙it=k^iηit111+ki∑j=1j≠iLHji2+Dji1Eji2Ai2+Di1Ei2Fi2-1KiCi2-1yi111111111+∑j=1j≠iLHji2+Dji1Eji2Fi2-11111111111111×KiCi2-1yidi∑j=1j≠iLHji2+Dji1Eji2,where k^i=(k-i+λi)<0 and k-i=ki(Di1Ei1+∑j=1,j≠iLβi(Hji1+Dji1Eji1)). By (56), (57), and (58), we have(61)V˙≤∑i=1LFi2Fi2-1Ai3+Di2Ei1ηi11111l1111+Ai4+Di2Ei2Fi2-1KiCi2-1yi11+∑i=1L∑j=1j≠iLFj2βiHji3+Dji2Eji1ηi11111111111111+Hji4+Dji2Eji211111111111111×Fi2-1KiCi2-1yidi11+∑i=1LbiFi2Bi2Wi1ηi+Wi2Fi2-1KiCi2-1yi11+∑i=1LFi2Bi2ci+∑i=1LσiTσiFi2Bi2ui11-∑i=1L1q-ib~ib^˙i-∑i=1L1q~ic~ic^˙i.By substituting the controller (48) into (61), it is clear that(62)V˙≤∑i=1LbiFi2Bi2Wi1ηi+Wi2Fi2-1KiCi2-1yi11-∑i=1Lζi-∑i=1Lαi+∑i=1LFi2Bi2ci11-∑i=1L1q-ib~ib^˙i-∑i=1L1q~ic~ic^˙i.Considering (50) and (62), the above inequality can be rewritten as(63)V˙≤∑i=1Lb^iFi2Bi2Wi1ηi+Wi2Fi2-1KiCi2-1yi11-∑i=1Lζi-∑i=1Lαi+∑i=1LFi2Bi2c^i11+∑i=1Lq˘i-c^i-ci22+ci24.By applying (49) to (63), we achieve(64)V˙≤-∑i=1Lαi-∑i=1Lq˘ic^i-ci22<0.The above inequality implies that the state trajectories of system (1) reach the sliding surface σi(xi)=0 in finite time and stay on it thereafter.Remark 13.
From sliding mode control theory, Theorems4 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 functionσi/σi by a continuous approximation such as σi/σi+μi, where μi 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 μi.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(65)x˙1=A1+ΔA1x1+B1u1+G1x1,x1d1,t+H12+ΔH12x2d2,y1=C1x1,where x1=x11x12x13∈R3, u1∈R1, y1=y11y12∈R2, A1=-8010-81110, B1=001, C1=110001, and H12=0.100.10.0200.100.10.1. The mismatched parameter uncertainties in the state matrix of the first subsystem are ΔA1=D1ΔF1E1 with D1=-0.020.10.02T, E1=0.100.1, and ΔF1=0.4sin2(x112+t×x12+x13+t×x11x12). The mismatched uncertain interconnections with the second subsystem are ΔH12=D12ΔF12E12 with D12=0.10.020.1T, ΔF12=0.2sin2(x23d2+x23x22d2+t×x21x22), and E12=0.10.020.1. The exogenous disturbance in the first subsystem is G1(x1,x1d1,t)≤c1+b1x1, where b1 and c1 can be selected by any positive value.The second subsystem’s dynamics is given as(66)x˙2=A2+ΔA2x2+B2u2+G2x2,x2d2,t+H21+ΔH21x1d1,y2=C2x2,where x2=x21x22x23∈R3, u2∈R1, y2=y21y22∈R2, A2=-6010-61110, B2=001, C2=110001, and H21=0.10.020.100.10.10.020.10.02. The mismatched parameter uncertainties in the state matrix of the second subsystem are ΔA2=D2ΔF2E2 with D2=0.10.02-0.1T, E2=0.10.10.03, and ΔF2=0.45sin(x21+x232+t×x22+x21x22). The mismatched uncertain interconnection with the first subsystem is ΔH21=D21ΔF21E21 with D21=0.10.1-0.1T, E21=0.020.020.1, and ΔF21=0.5sin3(x11d1+t×x12d1+x11x12x13). The exogenous disturbance in the second subsystem is G2x2,x2d2,t≤c2+b2x2 where b2 and c2 can be selected by any positive value.For this work, the following parameters are given as follows:φ1=0.6, φ2=3.9, φ-1=0.08, φ-2=0.05, ε1=2, ε2=3, q=100, q^=100, q˘1=4, q˘2=3, q-1=q-2=q~1=q~2=1, β1=80, β2=150, k1=1.002, k2=1.01, b1=0.8, b2=0.1, c1=0.3, c2=0.5, α1=0.05, α2=0.06. According to the algorithm given in [3], the coordinate transformation matrices for the first subsystem and the second subsystem are T1=T2=0.7071-0.70710-1-1000-1. By solving LMI (12), it is easy to verify that conditions in Theorem 4 are satisfied with positive matrices P1=0.2104-0.0017-0.00170.2305 and P2=0.2669-0.0266-0.02660.2517. The matrices Ξ1 and Ξ2 are selected to be Ξ1=0.22360.6708 and Ξ2=0.8-0.4. From (8), the sliding surface for the first subsystem and the second subsystem are σ1=00-0.1137x11x12x13T=0 and σ2=00-0.2281x21x22x23T=0. Theorem 4 showed that the sliding motion associated with the sliding surfaces σ1 and σ2 is globally asymptotically stable. The time functions η1(t) and η2(t) are the solution of η˙1(t)=-7.954η1(t)+2.015y1+0.22y1d1 and η˙2t=-5.796η2(t)+2.024y2+0.215y2d2, respectively. From Theorem 12, the decentralized adaptive output feedback sliding mode controller for the first subsystem and the second subsystem are(67)u1t=y1d1ζ1+0.05+1.0862η1t+0.00028y1111+0.0068y1d1σ1σ1,(68)u2t=y1d1ζ2+0.06+1.1048η2t+0.000684y2111+0.0125y2d2σ2σ2,where ζ1≥0.113b^1(η1(t)+y1)+0.1137c^1+0.022, b^˙1=0.113(η1(t)+y1), c^˙1=-4c^1+0.113, ζ2≥0.228b^2(η2(t)+y2)+0.2281c^2+0.0625, b^˙2=0.228(η2(t)+y2), and c^˙2=-4c^2+0.2281. 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:(69)u1t=y1d1ζ1+0.05+1.0862η1t+0.00028y1111+0.0068y1d1σ1σ1+0.0001,(70)u2t=y1d1ζ2+0.06+1.1048η2t+0.000684y2111+0.0125y2d2σ2σ2+0.0001.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 ared1(t)=2-sin(t) and d2(t)=1-0.5cos(t). The initial conditions for two subsystems are selected to be χ1(t)=-10510T and χ2(t)=10-8-10T, 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 statesx11 (solid), x12 (dashed), and x13 (dotted).Figure 2
Time responses of statesx21 (solid), x22 (dashed), and x23 (dotted).Figure 3
Time responses of sliding functionσ1.Figure 4
Time responses of sliding functionσ2.Figure 5
Time responses of discontinuous control inputu1 (67).Figure 6
Time responses of discontinuous control inputu2 (68).Figure 7
Time responses of continuous control inputu1 (69).Figure 8
Time responses of continuous control inputu2 (70).


## 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.



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*Source: 239584-2015-07-05.xml*