# Observer-Based Decentralized Tracking Control with Preview Action for a Class of Nonlinear Interconnected Systems

Authors: Xiao Yu
Journal: Complexity
 (2020)

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## Abstract

In this paper, for the first time, the observer-based decentralized output tracking control problem with preview action for a class of interconnected nonlinear systems is converted into a regulation problem forN augmented error subsystems composed of the tracking error dynamics, the difference equation of the state observer, and the available future reference trajectory dynamics associated with each individual subsystem. The developed innovative formulation of an observer-based decentralized preview tracking control scheme consists of the integral control action, the observer-based state feedback control action, and the preview action of the desired trajectory. The controller design feasibility conditions are formulated in terms of a linear matrix inequality (LMI) by using the Lyapunov function approach to ensure the existence of the suggested observer-based decentralized control strategy. Furthermore, both decentralized observer gain matrices and decentralized tracking controller gain matrices can be efficiently and simultaneously computed through a one-step LMI procedure. Stability analysis of the closed-loop augmented subsystem is carried out to illustrate that all tracking errors asymptotically converge toward zero. Finally, a numerical example is provided to demonstrate the effectiveness of the suggested control approach.


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## Body

## 1. Introduction

In the past few years, the output tracking control for interconnected systems that are composed of interconnected lower-dimensional individual subsystems has attracted a great deal of interests from researchers. At present, such systems appear in many practical applications such as telecommunication and transportation networks [1], power systems [2], wind turbines [3], complex mobile robots [4], 3-DOF helicopters [5], and so on. It is well known that compared to the traditional centralized control strategy, the decentralized scheme is considered to be the opposite design method, resulting in satisfactory output tracking with a straightforward and more economical implementation [6, 7]. As a matter of fact, the main feature of decentralized control lies in designing the control law for each individual subsystem utilizing only locally available information. The collection of these control laws constitutes the control structure of the whole interconnected system. Some interesting results in this field are presented in [5, 8–14].It should be pointed out that, first, the previously mentioned decentralized control scheme involves the state feedback. In most practical problems, however, the state variable knowledge of each subsystem is not always fully available, mainly due to the technical and/or economic limitations. To deal with this issue, a feasible alternative is to reconstruct the inaccessible state variables via a dynamic state observer. Much effort has been devoted to this area, and we refer the readers to [15–18]. Among them, several studies concentrate on the state reconstruction problem of interconnected systems. The interconnected state observer, where the interconnection information between local observers is used to design the observer, is studied in [18, 19]. However, the state reconstruction in a decentralized framework, which depends only on the local control input and output of the subsystem, provides an efficient and low-cost observer design to replace the usual state feedback for interconnected systems [10, 11].Second, when the future information of the desired trajectory is known beforehand, i.e., it is previewable, the tracking performance of the closed-loop system can be effectively improved by a preview control technique [20–22]. The salient feature of the preview control design is that it inserts the preview compensation action that incorporates the available future knowledge of the desired trajectory into the implemented control structure. Up until now, there have been systematic research frameworks on the preview controller design for the linear, the uncertain, and the stochastic systems. We refer the readers to [21–25] for some recent work in this area. In addition to the theoretical developments, many experimental studies have demonstrated that the preview control has potential applications in the fields of active suspension systems [26, 27], intelligent electric vehicles [28, 29], robots [30, 31], water tank level control systems [32], brushless DC motors [33], etc.In this paper, the observer-based decentralized tracking controller design with preview action for interconnected nonlinear systems is studied for the first time. It should be noted that observer-based controller design is not an easy task in this situation as the popular separation principle may not be applicable [11, 34]. To address this problem, the decentralized tracking control and decentralized observer design are simultaneously taken into account to ensure that the output of each subsystem asymptotically tracks the desired trajectory. For this purpose, we combine the state observation equation, the tracking error dynamics, and the known future desired trajectory dynamics for every individual subsystem to derive a new augmented error subsystem. The integral control action is naturally included using the classical difference approach and the tracking error dynamics. Furthermore, the global augmented error dynamics of the overall interconnected systems can be obtained using the lifting method, which converts the decentralized tracking control problem into a decentralized stabilization problem. With some special mathematical derivations, the proposed controller design method is formulated as an LMI feasibility problem, which can be efficiently resolved through a one-step procedure for the simultaneous computation of the decentralized tracking control gain matrices and the decentralized observation gain matrices. Finally, two numerical examples are provided to verify the theoretical results. The main contributions of the present study are concluded from the following two folds. First, a novel observer-based decentralized tracking control scheme with preview action is proposed for a class of Lipschitz nonlinear interconnected systems that are widely employed in practical applications. Second, the tracking control synthesis procedure is simplified by solving the observer gain and the tracking controller gain simultaneously with only one LMI problem, which effectively reduces the design complexity and the computational load.The rest of the paper is organized as follows. Section2 gives the problem formulation and control objective. In Section 3, the construction of the augmented error system is developed. In Section 4, the observer-based decentralized preview tracking controller design method is proposed for a class of nonlinear interconnected systems. Simulation results are provided in Section 5. Section 6 concludes this paper.Notations. Throughout this paper, Rn denotes the n-dimensional Euclidean space, and Rn×m denotes the n×m matrix space. The notation P>0 (or P<0) is used to define a symmetric positive-definite (or negative-definite) matrix. For matrices P and Q, P>Q stands for P−Q>0. The notation diagA,B represents a block-diagonal matrix whose diagonal elements are matrices A and B. MT denotes the transpose of matrix M. symA is used to represent A+AT. ⋅ denotes the Euclidean norm. In large matrix expressions, the symbol “∗” replaces terms that are induced by symmetry.


## 2. Problem Formulation

Consider the following interconnected system described by the connection ofN subsystems as follows:(1)xik+1=Aixik+Biuik+fixk,yik=Cixik,i=1,2,…,N,where Ai∈Rni×ni, Bi∈Rni×mi, and Ci∈Rpi×ni denote the state matrix, the control matrix, and the output matrix of each subsystem, respectively; xik=xi1kxi2k⋯xinikT∈Rni is the state vector of the ith subsystem; uik∈Rmi is the control input of the ith subsystem; yik∈Rpi is the output vector of the ith subsystem; xk=x1Tkx2Tk⋯xNTkT∈Rn is the state vector of the overall interconnected system; and fixk∈Rni stands for the vector of nonlinear interconnection terms of the ith subsystem and other subsystems.Assumption 1.
The nonlinear interconnection termfix (i=1,2,…,N) is globally Lipschitz with respect to x, i.e., there exists a constant γi>0 such that for all x,x¯∈Rn the following holds:(2)fix−fix¯≤γix−x¯.Remark 1.
Assumption1 is referred to as a globally Lipschitz condition. The structure of the nonlinearity satisfying this assumption extensively represents the nonlinear interconnection terms of a broad class of industrial processes; among them, a typical example is the multimachine power system [2, 7], and it has been widely used in the design of the decentralized control for complex interconnected systems [7, 8, 10, 11]. In particular, if an interconnected term fix is a linear function denoted by fix=Hix and Hi∈Rni×n, the restriction (2) is naturally satisfied with γi=Hi.The global system is characterized by the following state representation:(3)xk+1=Axk+Buk+fxk,yk=Cxk,where(4)xk=x1kx2k⋮xNk,uk=u1ku2k⋮uNk,yk=y1ky2k⋮yNk,fxk=f1xkf2xk⋮fNxk,A=diagAi,B=diagBi and C=diagCi with A∈RΣi=1Nni×Σi=1Nni, B∈RΣi=1Nni×Σi=1Nmi, and C∈RΣi=1Npi×Σi=1Nni, respectively.To consider the output tracking problem of an interconnected system (1), let the desired trajectory of the ith subsystem be rik∈Rpi (i=1,2,…,N). The following assumption is made concerning the preview ability of rik.Assumption 2.
There exists a constant vectorri such that limk⟶∞rik=ri. Furthermore, rik is previewable and the preview length is Mri; that is, at each time k, the current and Mri future values rik,rik+1,…,rik+Mri of the desired trajectory are available. Additionally, the future values of the desired trajectory beyond k+Mri are assumed to be unchanged, namely,(5)rik+j=rik+Mri,j=Mri+1,Mri+2,…Remark 2.
Assumption2 describes the previewable characteristics of the desired trajectory of the ith subsystem, and it is a rather standard hypothesis in the field of preview control [20, 26, 31, 32]. As commented in [33], the desired reference trajectories of control systems can be previewed in many applications. For example, an industrial robot manipulator usually follows a preset trajectory [30]. A humanoid robot also follows a walking pattern that can be previewed [31]. In this situation, the information of future reference trajectory can be fully utilized for the controller design to improve the tracking performance. Additionally, it should be noted that, as to an interconnected system (1), the output of each individual subsystem can track a different reference trajectory, and the desired trajectories are allowed to have different preview lengths, which is rather reasonable for practical applications.
Let the output tracking error of theith subsystem be(6)eik=yik−rik.


### 2.1. Control Objective

For system (1) (Assumptions 1 and 2 hold), find a set of observer-based decentralized preview tracking control policies u1k,…,uik,…,uNk such that the output vector yik of each subsystem asymptotically tracks the desired trajectory rik while being subjected to nonlinear interconnected perturbations, i.e., limk⟶+∞eik=0 for i=1,2,…,N.The following lemmas play important roles in the theoretical developments.Lemma 1.
(Schur Complement lemma) (see [35]). A given matrix S=S11S12S12TS22<0 is equivalent to any one of the following conditions:(i)
S22<0,S11−S12S22−1S12T<0(ii)
S11<0,S22−S12TS11−1S12<0Lemma 2.
(see [36]). For matrices T,H,V, and W with appropriate dimensions and scalar ξ, the inequality(7)T+WTHT+HW<0is fulfilled if the following condition holds:(8)T∗ξHT+VW−ξV−ξVT<0.Lemma 3.
(see [37]). If there are matrices P>0 and G with appropriate dimensions, then(9)−GTP−1G≤P−G−GT.


## 2.1. Control Objective

For system (1) (Assumptions 1 and 2 hold), find a set of observer-based decentralized preview tracking control policies u1k,…,uik,…,uNk such that the output vector yik of each subsystem asymptotically tracks the desired trajectory rik while being subjected to nonlinear interconnected perturbations, i.e., limk⟶+∞eik=0 for i=1,2,…,N.The following lemmas play important roles in the theoretical developments.Lemma 1.
(Schur Complement lemma) (see [35]). A given matrix S=S11S12S12TS22<0 is equivalent to any one of the following conditions:(i)
S22<0,S11−S12S22−1S12T<0(ii)
S11<0,S22−S12TS11−1S12<0Lemma 2.
(see [36]). For matrices T,H,V, and W with appropriate dimensions and scalar ξ, the inequality(7)T+WTHT+HW<0is fulfilled if the following condition holds:(8)T∗ξHT+VW−ξV−ξVT<0.Lemma 3.
(see [37]). If there are matrices P>0 and G with appropriate dimensions, then(9)−GTP−1G≤P−G−GT.


## 3. Construction of the Augmented Error System

The state observer of theith subsystem, which depends only on the control input and the output of the considered subsystem, can be described by(10)x^ik+1=Aix^ik+Biuik+Liyik−y^ik,y^ik=Cix^ik,i=1,2,…,N,where x^ik∈Rni is the local observed state vector and Li∈Rni×pi is the observation gain matrix of the ith subsystem.The state observer of the global system, which is composed ofN-local observers, can be expressed as follows:(11)x^k+1=Ax^k+Buk+Lyk−y^k,y^k=Cx^k,where x^k=x^1Tkx^2Tk⋯x^NTkT and L=diagLi,i=1,2,…,N is the block-diagonal observation gain matrix.It should be mentioned that the state observation structure of the global interconnected nonlinear system is totally decentralized, since there is no information interchange between local state observers, thereby leading to a readily implementable and more economical scheme.The main feature consists in developing a suitable augmented error system to transform the tracking control problem into a regulation problem. For this objective, we define the observation error between the real state and its estimation as follows:(12)exik=xik−x^ik.The dynamics of the observation errorexik, combing equations (1) and (10), can be described by(13)exik+1=Ai−LiCiexik+fixk.In the following, the classical difference approach is used to construct the augmented subsystem of theith subsystem. We select the first-order forward difference operator:(14)Δηk=ηk+1−ηk,where ηk is a column vector.From equation (6), the output equation of (1) and the definition of (14), the tracking error dynamics of the ith subsystem are given by the following state representation:(15)eik+1=eik+Δeik=eik+Δyik−Δrik=eik+CiΔx^ik+Δexik−Δrik.Applying the operatorΔ to the local state observer (10) yields(16)Δx^ik+1=AiΔx^ik+BiΔuik+LiΔyik−CiΔx^ik=AiΔx^ik+BiΔuik+LiCiΔxik−Δx^ik=AiΔx^ik+BiΔuik+LiCiΔexik.To introduce the available future knowledge of the desired trajectory, we define a new vector:(17)xrik=ΔrikΔrik+1⋮Δrik+Mri∈RpiMri+1.Note that theMri-steps future knowledge that is available from the current time k with respect to the desired trajectory rik is summarized into vector xrik. The introduction of xrik plays a significant role in preview controller design.Under Assumption2, the dynamics of xrik can be written as(18)xrik+1=Arixrik,with(19)Ari=0Ipi0⋯000Ipi⋯0⋮⋮⋮⋮000⋯Ipi000⋯0.To consider the known future values of the desired trajectory for the controller design, we define the augmented state vectorx¯ik=eiTkΔx^iTkxriTkT with x¯ik∈Rpi+ni+piMri+1, and equations (15)–(18) are combined together to yield the following augmented error subsystem:(20)x¯ik+1=A¯ix¯ik+B¯iΔuik+L¯iΔexik,where(21)A¯i=ICi−Gri0Ai000Ari,Gri=I0⋯0︸pi×piMri,B¯i=0Bi0,L¯i=CiLiCi0.The global system composed of local systems (20) is characterized by the following representation:(22)x¯k+1=A¯x¯k+B¯Δuk+L¯Δexk,where A¯=diagA¯i, B¯=diagB¯i, and L¯=diagL¯i with A¯∈RΣi=1Npi+ni+piMri+1×Σi=1Npi+ni+piMri+1, B¯∈RΣi=1Npi+ni+piMri+1×Σi=1Nmi, and L¯∈RΣi=1Npi+ni+piMri+1×Σi=1Nni.To analyze the dynamic behavior of the observation error equation (13) and synthesize the observer-based tracking control law with preview action, the difference operator Δ is applied to (13) to derive(23)Δexik+1=Ai−LiCiΔexik+Δfixk,with Δfixk=fixk+1−fixk.The global system of (23) is described by(24)Δexk+1=A−LCΔexk+Δfxk,where A,L, and C are defined as before, and(25)Δfx=Δf1xΔf2x⋮ΔfNx.By combining equations (22) and (24) together, the required augmented error system is derived as follows:(26)x¯k+1=A¯x¯k+B¯Δuk+L¯Δexk,Δexk+1=A−LCΔexk+Δfxk.If we find a suitable controller to ensure that the closed-loop system of system (26) is asymptotically stable, then the expected output tracking is achieved, since the tracking error is a component of the state vector x¯k. Therefore, the output tracking problem under consideration is reduced to a stabilization problem of system (26).


## 4. Observer-Based Decentralized Preview Tracking Controller Design

As for theith augmented subsystem in (20) and (23), we introduce the state feedback controller(27)Δuik=Kix¯ik,where Ki∈Rmi×pi+ni+piMri+1 is the controller gain matrix of the ith augmented subsystem to be determined.The control law of the global system (26) is given in the following form:(28)uk=Kx¯k,where K=diagKi,i=1,2,…,N is the block-diagonal controller gain matrix.The closed-loop system of the augmented system (26), applying the control law (28), is given by(29)x¯k+1=A¯+B¯Kx¯k+L¯Δexk,Δexk+1=A−LCΔexk+Δfxk.To effectively compute the decentralized control gain matrixK=diagKi and the observation gain matrix L=diagLi, in the following, an innovative design approach for the asymptotic stability of the overall closed-loop system (29) is presented.Theorem 1.
Suppose that Assumptions1 and 2 hold. Given scalar ξ>0, if there are matrices P>0,S>0, G1,G2,X,U, and V such that(30)ϒ11∗∗∗∗∗∗∗0ϒ22∗∗∗∗∗∗00−I∗∗∗∗∗ϒ41ϒ420−P∗∗∗∗0ϒ52I0−S∗∗∗FxG10000−ηI∗∗0G20000−ηI∗0ϒ820ϒ84−ξUT00ϒ88<0,where(31)ϒ11=P−G1−G1T,ϒ41=A¯G1+B¯X,ϒ22=S−G2−G2T,ϒ42=TeCG2+TsUC,ϒ52=AG2−UC,ϒ82=CG2−VC,ϒ84=ξUTTsT,ϒ88=−ξV−ξVT,η=1∑i=1Nγi2,and Fx,Te, and Ts are given by (36), (47), and (48), respectively, then the closed-loop system (29) is globally asymptotically stable. Furthermore, the decentralized controller gain K=diagKi and the decentralized observer gain L=diagLi are derived by K=XG1−1 and L=UV−1, respectively.Proof: Consider the Lyapunov function(32)Vx¯,Δex=x¯TΔexTP−100S−1x¯Δex=x¯TP−1x¯+ΔexTS−1Δex,where P=diagPi>0 and S=diagSi>0 with Pi∈Rpi+ni+piMri+1×pi+ni+piMri+1 and Si∈Rni×ni, respectively. Thus, the function V is positive-definite. Taking its difference along the trajectory of (29) leads to(33)ΔVx¯,Δex=x¯Tk+1P−1x¯k+1+ΔexTk+1S−1Δexk+1−x¯TkP−1x¯k−ΔexTkS−1Δexk=A¯+B¯Kx¯k+L¯ΔexkTP−1A¯+B¯Kx¯k+L¯Δexk+A−LCΔexk+ΔfxkTS−1A−LCΔexk+Δfxk−x¯TkP−1x¯k−ΔexTkS−1Δexk=x¯TkA¯+B¯KTP−1A¯+B¯K−P−1x¯k+2x¯TkA¯+B¯KTP−1L¯Δexk+ΔexTkL¯TP−1L¯Δexk+ΔexTkA−LCTS−1A−LC−S−1Δexk+2ΔexTkA−LCTS−1Δfx+ΔfTxkS−1Δfxk.Notice that, by Assumption1, it follows that(34)γiΔxTkΔxk−ΔfiTxΔfix≥0.Furthermore, from equation (12), Δxk can be computed by(35)Δxk=Δx^k+Δexk≜Fxx¯k+Δexk,where(36)Thus, equation (34) can be rewritten as(37)γi2Fxx¯k+ΔexkTFxx¯k+Δexk−ΔfiTxΔfix≥0,that is,(38)γi2x¯TkFxTFxx¯k+2x¯TkFxTΔexk+ΔexTkΔexk−ΔfiTxΔfix≥0.Summing both sides of the above inequality fromi=1 to N results in(39)∑i=1Nγi2x¯TkFxTFxx¯k+2x¯TkFxTΔexk+ΔexTkΔexk−∑i=1NΔfiTxΔfix=∑i=1Nγi2x¯TkFxTFxx¯k+2x¯TkFxTΔexk+ΔexTkΔexk−ΔfTxΔfx≥0.Thus, from (39), ΔV is further bounded by(40)ΔVx¯,Δex≤x¯TkA¯+B¯KTP−1A¯+B¯K−P−1x¯k+2x¯TkA¯+B¯KTP−1L¯Δexk+ΔexTkL¯TP−1L¯Δexk+ΔexTkA−LCTS−1A−LC−S−1Δexk+2ΔexTkA−LCTS−1Δfx+ΔfTxkS−1Δfxk+∑i=1Nγi2x¯TkFxTFxx¯k+2x¯TkFxTΔexk+ΔexTkΔexk−ΔfTxΔfx=x¯TkΔexTkΔfTxkΞx¯TkΔexTkΔfTxkT,where(41)Ξ=Ξ11∗∗Ξ21Ξ22∗0Ξ32Ξ33,with(42)Ξ11=A¯+B¯KTP−1A¯+B¯K−P−1+η−1FxTFx,Ξ21=L¯TP−1A¯+B¯K+η−1Fx,Ξ22=L¯TP−1L¯+A−LCTS−1A−LC−S−1+η−1I,η=1∑i=1Nγi2,Ξ32=S−1A−LC,Ξ33=S−1−I.Therefore, if the inequalityΞ<0 is satisfied, then ΔV<0 holds for all x¯TkΔexTkT≠0. Based on the Lyapunov stability theorem, the closed-loop system (29) is globally asymptotically stable at the origin.Applying the Schur complement lemma (i.e., Lemma1), Ξ<0 in equation (40) is equivalent to(43)−P−1∗∗∗∗∗∗0−S−1∗∗∗∗∗00−I∗∗∗∗A¯+B¯KL¯0−P∗∗∗0A−LCI0−S∗∗Fx0000−ηI∗0I0000−ηI<0.By pre- and postmultiplying (43) by the invertible matrix diagG1T,G2T,I,I,I,I,I and its transpose, respectively, we can obtain that(44)−G1TP−1G1∗∗∗∗∗∗0−G2TS−1G2∗∗∗∗∗00−I∗∗∗∗A¯+B¯KG1L¯G20−P∗∗∗0A−LCG2I0−S∗∗FxG10000−ηI∗0G20000−ηI<0.Here,G1=diagG1i, G1i∈Rpi+ni+piMri+1×pi+ni+piMri+1 and G2=diagG2i, G2i∈Rni×ni have been used.By using theL¯i defined in (20), the element L¯G2 that appears in (44) is given by(45)L¯G2=diagL¯i×diagG2i=diagCiLiCi0×diagG2i=diagCiG2iLiCiG2i0.In view of Lemma3, inequality (44) holds if(46)P−G1−G1T∗∗∗∗∗∗0S−G2−G2T∗∗∗∗∗00−I∗∗∗∗A¯+B¯KG1diagCiG2iLiCiG2i00−P∗∗∗0AG2−LCG2I0−S∗∗FxG10000−ηI∗0G20000−ηI<0.By some mathematical operations, after denotingp=∑i=1Npi,n=∑i=1Nni, we obtain(47)(48)Thus, from the above equations (47) and (48), it follows that(49)diagCiG2iLiCiG2i0=diagCiG2i00+diag0LiCiG2i0=TeCG2+TsLCG2.Hence, inequality (46) becomes(50)P−G1−G1T∗∗∗∗∗∗0S−G2−G2T∗∗∗∗∗00−I∗∗∗∗A¯+B¯KG1TeCG2+TsLCG20−P∗∗∗0AG2−LCG2I0−S∗∗FxG10000−ηI∗0G20000−ηI<0.The matrix in the left side of (50) is denoted by Φ. With the change of variable X=KG1, the controller gain matrix is determined by K=XG1−1. Since X=diagXi, the local controller gain is Ki=XiG1i−1 for i=1,2,…,N. To effectively compute the observation gain, let us introduce auxiliary invertible matrices Vii=1,2,…,N and define Li=UiVi−1. Then, the following equation holds:(51)LiCiG2i=UiVi−1CiG2i−ViCi+UiCi.Furthermore, letU=diagUi and V=diagVi. Then, the element LCG2 in (50) is given by(52)LCG2=diagLiCiG2i=diagUiVi−1CiG2i−ViCi+UiCi=UV−1CG2−VC+UC.Using the variable substitution with (52), the matrix Φ on the left side of inequality (50) can be expressed in the form of(53)Φ=Σ1+sym000TsU−U00︸HV−10CG2−VC00000︸W,where(54)Σ1=P−G1−G1T∗∗∗∗∗∗0S−G2−G2T∗∗∗∗∗00−I∗∗∗∗A¯G1+B¯XTeCG2+TsUC0−P∗∗∗0AG2−UCI0−S∗∗FxG10000−ηI∗0G20000−ηI.According to Lemma1, the inequality Φ<0 can be guaranteed by the following condition:(55)Σ1∗Σ2−ξV−ξVT<0,where(56)Σ2=ξHT+VW=0CG2−VC0ξUTTsT−ξUT00.The inequality (55) can be formulated in the form of (30). Therefore, if condition (30) is satisfied, then condition (50) holds. Furthermore, Ξ<0 is ensured.Remark 3.
It should be noted that Theorem1 provides a new criterion to ensure the asymptotic stability of the overall closed-loop system (29) in terms of LMI formulations. With the help of the LMI toolbox in MATLAB, both the decentralized observer gain L=diagLi and the decentralized controller gain K=diagKi can be computed simultaneously. It can be noticed that the matrix variables L and G2 in (50) appear in a nonlinear coupling form, which results in some computation troubles for the observer design. To overcome this obstacle, the auxiliary matrix variables U=diagUi and V=diagVi in the form of (52) are introduced. As a result, the decentralized observer-based decentralized preview tracking controller can be resolved through a one-step procedure. Moreover, with the aid of additional matrices Gii=1,2, the slack variable method [38, 39] is used to derive the LMI-based relaxation condition. Additionally, our adopted design method can generalize the results in [7, 11, 40, 41] and can be effectively applied to the H∞ model reference decentralized tracking control of the interconnected systems in discrete-time domain.Remark 4.
It should be mentioned that, in Assumption1, the specific values for the interconnection bounds γi are set according to the nonlinear interconnection term fix. As a matter of fact, our obtained results can also be applied to the case where the parameter γi is unknown and fix may be uncertain. Inspired by [7, 40], in this situation, the parameter bounds of Lipschitz interconnected terms γii=1,2,…,N can be maximized and the overall closed-loop system remains asymptotically stable, whereas the parameters ηii=1,2,…,N are minimized as much as possible. As a result, the decentralized observer-based decentralized tracking controller design can be characterized by the following optimization problem:(57)minimizeηsubject toP>0,S>0,G1,G2,X,U,Vand30.
By solving this minimization problem, the maximized values of the Lipschitz connection bounds can be found so that the control scheme may be applied to a larger nonlinear coverage.
When the LMI feasibility problem in Theorem1 is solvable, the controller gain in (28) for the global augmented error system (26) is determined by K=XG1−1. Correspondingly, the local controller gain is obtained by Ki=XiG1i−1 for each local subsystem. To simplify the expression and to clarify the control structure of the proposed observer-based decentralized tracking controller with preview action, the local controller gain Ki is partitioned as(58)
Hence, the local controller (27) can be expressed by(59)Δuik=Keieik+Kx˜iΔx^ik+Krixrik.
By solving the control inputuik from the above equation, we obtain(60)uik=∑j=−Mri−1k−1Δuij+u−Mri−1=Kei∑j=0k−1eij+Kx˜ix^ik+Krixrik.
Here, the assumption thatx^ij=0,uij=0 and rij=0 for j<0 has been used.Theorem 2.
Suppose that Assumptions1 and 2 hold. If the LMI condition (30) in Theorem 1 is feasible, then the observer-based decentralized preview controller for system (1) is(61)uk=u1Tku2Tk⋯uNTkT,uik=Kei∑j=0k−1eij+Kx^ix^ik+∑j=0Mrikrijrik+j,i=1,2,…,N,where x^ik is given by (10) with the observer gain Li=UiVi−1 (i=1,2,…,N), and the controller gains Kei Kx˜i, and krijj=0,1,…,Mri are determined by (58). Using this controller, the output vector yik can asymptotically track the desired trajectory rik for i=1,2,…,N.Remark 5.
The observer-based preview tracking control law proposed in this paper has a completely decentralized structure, as shown in (61), since there is no information transfer between local individual controllers. Furthermore, it can be seen that the control law of each subsystem consists of three terms. More specifically, the first term, Kei∑j=0k−1eij, stands for the integral control action that is capable of eliminating the static error of the system, the second term, Kx˜ix^k, is the observer-based state feedback control action, and the last one, ∑j=0Mrikrijrik+j, refers to the preview compensation action of the desired trajectory that is conducive to improving the tracking performance of the overall interconnected system.Remark 6.
So far, there have been many interesting results concerning decentralized tracking control in the literature, such as [5, 9, 12, 39, 42–47]. Note that most of them are developed via backstepping control, adaptive dynamic programming, iterative learning control, etc. However, in this paper, for the first time, we propose a novel and effective decentralized tracking controller design methodology based on the preview control technique, which guarantees that even in the presence of Lipschitz nonlinear interconnections, all tracking errors asymptotically converge to zero. In contrast to [6, 8, 23, 24, 26, 27, 29, 43, 44], where all the state variables are required to be measurable, in this paper, a more complex and challenging problem, i.e., the observer-based decentralized preview tracking control problem, is considered and the merits of the developed design method consist in two aspects: (i) an efficient preview compensation mechanism, which is obtained by integrating the known future information of the desired trajectory, is included in the proposed tracking control structure in order to improve the tracking performance; (ii) only the local information is utilized in the design procedure for each local subsystem, which is more economical and straightforward for practical applications.


## 5. Numerical Simulations

Example 1.
Consider an interconnected system described by(62)S1:x1k+1=00.352000.1450.4860.031−0.1420.565︸A1x1k+1.2510.550.302︸B1u1k+0.1sinx11k0.1cosx22k0︸f1x1k,x2k,y1k=110︸C1x1k,S2:x2k+1=0.713−0.9650.8120.32︸A2x2k+0.7011.52︸B2u2k+0.1arctanx11k0.25sinx22k︸f2x1k,x2k,y2k=1−1︸C2x2k,where x1k=x11kx12kx13kT∈R3 and x2k=x21kx22kT∈R2 are states, y1k,y2k∈R are outputs and u1k,u2k∈R are the control inputs. It is straightforward to compute that the nonlinear interconnections fix1,x2 (i=1,2) satisfy Assumption 1 with Lipschitz constants γ1=0.1 and γ2=0.25.
For the purpose of the simulation, the desired trajectories are set as(63)r1k=0,k<10,0.15k−10,10≤k≤30,3,k>30,(64)r2k=0,k<20,3,k≥20.
Here, we assume that the desired signalsr1k and r2k are both previewable, namely, Assumption 2 is satisfied. Furthermore, their preview lengths are assumed to be Mr1 and Mr2, respectively.
To compare the effect of the preview action of the desired trajectory on the output tracking performance, we consider three cases for the above interconnected nonlinear system, namely, (i)Mr1=0,Mr2=0, (ii) Mr1=2,Mr2=3, and (iii) Mr1=4,Mr2=6. Given ξ=1, the LMI feasibility problem (30) in Theorem 1 can be solved using the MATLAB LMI toolbox, and furthermore, the solutions with respect to the designed observer gain and controller gain are derived as follows:Case 1.
WhenMr1=0,Mr2=0, the decentralized observer gain and decentralized controller gain are(65)L1=0.17090.05010.0295,L2=0.65060.2055,Ke1=−0.0502,Ke2=0.0258,Kx^1=−0.0565−0.2834−0.2523,Kx^2=−0.5032−0.2594.Case 2.
WhenMr1=2,Mr2=3, the decentralized observer gain and decentralized controller gain are(66)L1=0.17360.05150.0303,L2=0.66340.1976,Ke1=−0.0585,Ke2=0.0311,Kx^1=−0.0647−0.2918−0.2536,Kx^2=−0.4823−0.3144,Kr1=0.05850.05830.0551,Kr2=−0.0312−0.0311−0.0341−0.0319.Case 3.
WhenMr1=4,Mr2=6, the decentralized observer gain and decentralized controller gain are(67)L1=0.17650.05040.0316,L2=0.67330.1987Ke1=−0.0721,Ke2=0.0383,Kx^1=−0.0783−0.3060−0.2553,Kx^2=−0.4672−0.3485,Kr1=0.07200.07160.06580.06240.0592,Kr2=−0.0383−0.0386−0.0417−0.0387−0.0354−0.0325−0.0295.
Based on Theorem2, for the predefined reference trajectories and the assumed preview lengths, the observer-based decentralized tracking control pair u1k,u2k can be derived in the form of (58), which can guarantee the asymptotic tracking of the desired trajectory for each subsystem while being subjected to nonlinear interconnected perturbations. In the following, the closed-loop output response, the tracking error, and the control input of subsystem S1 are shown in Figures 1–3, respectively. The corresponding simulation results for subsystem S2 are displayed in Figures 4–6, respectively. In the numerical simulation, the initial states of the subsystems are x1=000T and x2=00T, and the initial states of the observers are x^1=0.10.20T and x^2=−0.10T.
From these figures, we can see that the observer-based decentralized tracking controller proposed in this paper is very effective for the expected output tracking, no matter whether the preview action exists or not. Figures1 and 4 illustrate that the output vectors in all situations asymptotically track the desired trajectories. Compared to the conventional controller without preview action (i.e., Mr1=Mr2=0), the suggested decentralized tracking control law with preview action makes the system output produce faster responses and approach the desired value more quickly. The tracking error, as a good tool to evaluate the output tracking quality, is depicted in Figures 2 and 5, which indicates that the tracking error between the system output and the desired trajectory decreases dramatically as the preview length increases. Obviously, the developed preview control scheme improves the tracking performance of the overall system in spite of that the nonlinear interconnection perturbation affecting the subsystems. Furthermore, as Figures 3 and 6 show, the control signals remain bounded with a reasonable amplitude, thereby demonstrating the efficiency of the suggested control approach. Additionally, the integral error performance criteria, such as IAE, ISE, ITAE, and ITSE, are good tools to evaluate the tracking quality and are shown in Tables 1 and 2, where the effectiveness of the controller is evaluated and compared with classical tracking controller without preview action. It can be seen that all the performance indices values are reduced by using the preview controller. These merits mainly benefit from a novel preview compensation mechanism in the proposed tracking controller. The simulation results indicate that the suggested observer-based decentralized tracking controller with preview action has more remarkable and satisfactory control performance.Figure 1
The output response of subsystemS1.Figure 2
The tracking error response of subsystemS1.Figure 3
The control input response of subsystemS1.Figure 4
The output response of subsystemS2.Figure 5
The tracking error response of subsystemS2.Figure 6
The control input response of subsystemS2.Table 1
Performance index results for subsystemS1.
Performance index
Classical controller (Mr1=0,Mr2=0)
Preview controller (Mr1=2,Mr2=3)
Preview controller (Mr1=4,Mr2=6)
IAE
35.9558
22.8845
13.3241
ISE
33.0938
14.0228
4.9148
ITAE
1063.4
634.0
341.2
ITSE
917.8741
374.7498
126.2100Table 2
Performance index results for subsystemS2.
Performance index
Classical controller (Mr1=0,Mr2=0)
Preview controller (Mr1=2,Mr2=3)
Preview controller (Mr1=4,Mr2=6)
IAE
35.5087
20.6670
14.2376
ISE
61.1314
22.0606
9.9347
ITAE
992.6933
548.3889
333.0283
ITSE
1479.1
511.4
211.6Example 2.
To illustrate the application of the proposed control scheme, we consider the two identical pendulums which are coupled by a spring and subject to two distinct inputs [47] as shown in Figure 7.
The mathematical model of such an interconnected system can be described by(68)x˙1t=A11x1t+B1u1t+A12x2t,y1t=C1x1t,x˙2t=A22x2t+B2u2t+A21x1t,y2t=C2x2t,with x1t=θ1tθ˙1tT, x2t=θ2tθ˙2tT, and(69)A11=A22=01gl−ka2ml2−μ,B1=B2=01ml2,A12=A21=00ka2ml20,C1=C2=11,where g represents the gravity, μ accounts for the friction, m1=m2=m are the masses of both pendulums, and k is the spring constant. The following parameters borrowed from [47] are used:(70)gl=1,1ml2=1,μ=1,km=2,al=0.5.
When the sampling period is 0.25 s, the approximate discrete-time model is of the following form:(71)S1:x1k+1=10.250.1251︸A1x1k+00.25︸B1u1k+00.125x21k︸f1x1k,x2k,y1k=11︸C1x1k.S2:x2k+1=10.250.1251︸A2x2k+00.25︸B2u2k+00.125x11k︸f2x1k,x2k,y2k=11︸C2x2k.
In this example, the nonlinear interconnectionsfix1,x2 (i=1,2) degenerate into linear functions, and Assumption 1 holds with γ1=γ2=0.125. For simulation purpose, the desired trajectories r1k and r2k are taken as (63) and (64). Furthermore, their preview lengths are assumed to be Mr1 and Mr2, respectively.
To compare the effect of the preview action of the desired trajectory on the tracking performance, the following three cases are considered, including (i)Mr1=0,Mr2=0, (ii) Mr1=4,Mr2=5, and (iii) Mr1=5,Mr2=9. Given ξ=1, by resorting to the MATLAB LMI toolbox, the feasible solutions to LMI (30) in Theorem 1 can be derived. Furthermore, the observer gain and controller gain are designed as follows:Figure 7
The coupled inverted pendulums.Case 1.
WhenMr1=0,Mr2=0, the decentralized observer gain and decentralized controller gain are(72)L1=0.28810.4844,L2=0.28810.4844,Ke1=−0.1014,Ke2=−0.1014,Kx^1=−8.4418−6.0674,Kx^2=−8.4418−6.0674.Case 2.
WhenMr1=4,Mr2=5, the decentralized observer gain and decentralized controller gain are(73)L1=0.28950.4861,L2=0.28380.4772,Ke1=−0.1581,Ke2=−0.1611,Kx^1=−8.5896−6.1464,Kx^2=−8.6504−6.1626,Kr1=0.15840.15770.14180.13260.1250,Kr2=0.16140.16200.14700.13790.13040.1239.Case 3.
WhenMr1=5,Mr2=9, the decentralized observer gain and decentralized controller gain are(74)L1=0.29460.4929,L2=0.28520.4784,Ke1=−0.1987,Ke2=−0.2066,Kx^1=−8.5990−6.1785,Kx^2=−8.7559−6.2206,Kr1=0.19900.19750.17430.16250.15340.1454,Kr2=0.20650.20960.18870.17760.16920.16310.15750.15210.14680.1408.
In numerical simulation, the initial states of the subsystems arex1=x2=00T, and the initial states of the observers are x^1=0.10T and x^2=−0.10T. The simulation results are displayed in Figures 8–13, from which we can see that the proposed preview control scheme makes the closed-loop system produce faster response and higher tracking precision compared to the traditional controller without preview. Furthermore, Tables 3 and 4 give the comparison of the proposed scenario with the conventional method. The use of the preview information in tracking control structure largely improves the performance indices values. The simulation results confirm the effectiveness and superiority of our proposed preview control method.Figure 8
The output response of subsystemS1.Figure 9
The tracking error response of subsystemS1.Figure 10
The control input response of subsystemS1.Figure 11
The output response of subsystemS2.Figure 12
The tracking error response of subsystemS2.Figure 13
The control input response of subsystemS2.Table 3
Performance index results for subsystemS1.
Performance index
Classical controller (Mr1=0,Mr2=0)
Preview controller (Mr1=4,Mr2=5)
Preview controller (Mr1=5,Mr2=9)
IAE
171.5137
99.7774
75.6379
ISE
227.8563
109.7278
73.8918
ITAE
13137
5757
3876
ITSE
11862
4631
2869Table 4
Performance index results for subsystemS1.
Performance index
Classical controller (Mr1=0,Mr2=0)
Preview controller (Mr1=4,Mr2=5)
Preview controller (Mr1=5,Mr2=9)
IAE
186.1303
105.5119
75.2243
ISE
287.7841
142.7642
90.1844
ITAE
14490
5950
3515
ITSE
14147
5482
3067


## 6. Conclusions

In this paper, the problem of asymptotic tracking control with preview action for a class of interconnected nonlinear systems is investigated. By means of the difference approach, the augmentation technique, and the partition method, we construct a novel augmented error system where the known future knowledge of the desired trajectory is fully taken into account. Thereby, the output tracking problem is transformed into a regulation problem. The proposed observer-based decentralized tracking controller design method is formulated by a feasibility problem involving an LMI, which can be efficiently computed through a one-step LMI procedure, to simultaneously determine the decentralized observation gain matrices and tracking controller gain matrices. Moreover, for each subsystem, the local tracking controller consists of the integral control action, the observer-based state feedback control action, and the preview action of the desired trajectory. Finally, the effectiveness of the suggested control approach is demonstrated by two numerical examples.



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