# State-Estimator-Based Asynchronous Repetitive Control of Discrete-Time Markovian Switching Systems

Authors: Xinghua Liu; Guoqi Ma; Prabhakar R. Pagilla; Shuzhi Sam Ge
Journal: Complexity
 (2020)

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## Abstract

This paper investigates the problem of asynchronous repetitive control for a class of discrete-time Markovian switching systems. The control goal is to track a given periodic reference without steady-state error. To achieve this goal, an asynchronous repetitive controller that renders the overall closed-loop switched system mean square stable is proposed. To reflect realistic scenarios, the proposed approach does not assume that the system modes are available synchronously to the controller but instead designs a detector that provides estimated values of the system modes to the controller. Based on a detected-mode-dependent estimator, the plant and asynchronous repetitive controller are formulated as a closed-loop stochastic system. By utilizing tools from stochastic Lyapunov–Krasovskii stability theory, we develop sufficient conditions in terms of linear matrix inequalities (LMIs) such that the closed-loop system is mean square stable and also simultaneously establish a synthesis procedure for obtaining the gain matrices. We provide numerical simulations on an electrical circuit switched system to illustrate the approach.


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## Body

## 1. Introduction

As a special class of hybrid dynamic systems, Markovian switching systems are modeled by a set of linear or nonlinear governing equations with the switching between systems in the set determined by Markov chains [1]. Since Markovian switching systems can be used to describe abrupt variations caused by the random failures of components and sudden environmental changes, important and useful results have been reported in many practical applications, such as robot manipulators [2], power systems [3], economic systems [4], sensor networks [5], neural networks [6], multiagent systems [7], and networked control systems [8–10]. To date, a variety of results have been published to address the development of controllers for stabilization of Markovian switching systems, see [11–21] and references therein. These results on control and filtering of Markovian switching systems are based on the assumption that the mode information of the plant is fully available to the controller or estimator at every instant of time to ensure that the switching of controller or estimator is synchronous with that of the plant. Accordingly, the designed controller is typically referred to as mode-dependent or synchronous; this assumption restricts the application of the controller design approach to many practical systems because the mode information of the plant may not be accessible to the controller due to communication delays and/or missing measurements which may lead to asynchronous behavior between the controller and the system modes.Since an asynchronous controller has broader applicability than a synchronous controller, the asynchronous control design problem has received wider attention in recent years [22–27]. For general switched systems, a strategy to stabilize continuous-time switched systems with asynchronous switching was provided in [22]. Asynchronous H∞ filtering was studied for discrete-time switched Takagi–Sugeno (T-S) fuzzy systems in [23]. Based on the hidden Markov model, an H2 controller design was proposed to stochastically stabilize a class of Markovian switching systems with partial information in [24]. A passivity-based asynchronous control problem for Markovian switching systems was considered in [25]. An H2 filtering design for discrete-time hidden Markovian switching systems was discussed in [26]. In [27], an asynchronous sliding mode control design was proposed for a class of uncertain Markovian switching systems with time-varying delays and stochastic perturbation. Although recent published results are encouraging, there are still many relevant open problems of importance to practical applications, and we consider one such essential problem in this paper.Since the control tasks in many applications that can be modeled by such systems are often repetitive, increased use of repetitive control formulations can be found in many applications, disk drive systems [28], rotating machinery [29], micro-/nanomanipulation applications [30], and power electronics systems [31]; repetitive control strategies use error measurements from the previous period to reduce subsequent steady-state tracking errors for periodic exogenous input signals. There is a rich body of literature related to repetitive control design techniques [32–40]. Most repetitive control designs in the literature are developed for deterministic systems, whereas designs for switched stochastic dynamical systems are sparse. In particular, the problem of designing a mode-dependent repetitive controller for discrete-time Markovian switching systems was first studied in [40], where a set of sufficient conditions in terms of linear matrix inequalities is derived for stabilization by combining a 2D Lyapunov functional and singular value decomposition of the output matrix. However, the problem of asynchronous repetitive control for Markovian switching systems has not been investigated, mainly due to the complexity of addressing the asynchronous behavior between the controller modes and the Markovian switching system modes.In this paper, we design a state-estimator-based asynchronous repetitive controller for a class of discrete-time Markovian switching systems for asymptotic desired reference signal tracking of the output by considering the asynchronous behavior between the Markovian switching system modes and the controller modes. The proposed design provides the following contributions to the existing literature:(1)
The estimate of the system mode is obtained by a detector via a hidden Markov model with a mode detection probability matrix which is utilized in the state estimator and controller. This approach relaxes the assumption prevalent in the existing literature that the system mode is available to the controller, and the controller and system mode switching is synchronized.(2)
By employing the proposed state-estimator-based asynchronous repetitive control scheme, we develop the closed-loop system as a hidden Markovian jump system. We show that the closed-loop system can achieve mean square asymptotically stable tracking under a set of sufficient conditions which can be expressed in terms of solvable LMIs.(3)
We also provide a synthesis procedure to obtain the estimator and controller design matrices; further, to facilitate the solution of the LMIs, we pose a stochastic optimization problem for determining the key gain parameters.The rest of the paper is organized as follows. InSection 2, we describe the plant and the associated asynchronous repetitive control approach and formulate the closed-loop governing equations for the overall system. The main theoretical framework together with the synthesis procedure is provided in Section 3. Application of the approach to an RLC circuit system and numerical simulations are presented in Section 4. Section 5 provides a summary of this work and potential topics for the future.


### 1.1. Notations

The following notations are employed in the paper:ℤ0+ for the set of non-negative integers; ℝn for the n dimensional Euclidean space; ℝm×n for the set of all m×n matrices; I for the identity matrix with dimensions derived from the context; x and A for the Euclidean norm of a vector x and a matrix A; superscripts ⊤ and −1 for matrix transposition and matrix inverse; ⋆ to denote the terms due to symmetry in a matrix. X≺Y(X≻Y), where X and Y are both symmetric matrices, to mean the matrix X−Y is negative (positive) definite. λmaxA and λminA denote the eigenvalues of matrix A with maximum and minimum real parts, respectively. E denotes the mathematical expectation operator.


## 1.1. Notations

The following notations are employed in the paper:ℤ0+ for the set of non-negative integers; ℝn for the n dimensional Euclidean space; ℝm×n for the set of all m×n matrices; I for the identity matrix with dimensions derived from the context; x and A for the Euclidean norm of a vector x and a matrix A; superscripts ⊤ and −1 for matrix transposition and matrix inverse; ⋆ to denote the terms due to symmetry in a matrix. X≺Y(X≻Y), where X and Y are both symmetric matrices, to mean the matrix X−Y is negative (positive) definite. λmaxA and λminA denote the eigenvalues of matrix A with maximum and minimum real parts, respectively. E denotes the mathematical expectation operator.


## 2. Problem Formulation

### 2.1. System Description

To model the switching process as a discrete-time homogeneous Markov chain, we consider the parameterθk,k∈ℤ0+ which takes values from a finite set ℳ=1,2,3,…,M with the following mode transition probabilities:(1)Probθk+1=jθk=i=πij,where πij≥0, ∀i,j∈ℳ, and ∑j=1Mπij=1. Based on this Markov chain, we define the M×M transition probability matrix as π whose i,j-th element is πij.Letxk∈ℝn, uk∈ℝl and yk∈ℝm, respectively, denote the state, control input, and system output vectors. Let Aθk, Bθk, Cθk, and Dθk denote the system matrices with appropriate dimensions. The discrete-time linear Markovian switching system that we consider in this paper is given by(2)xk+1=Aθkxk+Bθkuk,yk=Cθkxk+Dθkuk.In the following, for ease of readability, forθk=i∈ℳ, the system matrices Aθk, Bθk, Cθk, and Dθk are denoted by Ai, Bi, Ci, and Di which are assumed to be known. The relative degree of the plant is assumed to be zero, which implies that Di≠0.Since it is generally not possible to accurately measure the evolution of the system modeθk, we consider a probabilistic detector that provides the estimated values of θk with a certain probability. Let the estimated system mode signal be denoted by ρk which is utilized by the controller and need not be synchronized with the system mode θk. We consider a hidden Markov model θk,ρk,ℳ,N to characterize the asynchronous phenomenon as follows:(3)Probρk=pθk=i=μip,where i∈ℳ, p∈N, N=1,2,3,…,N and μip is the mode detection probability. For any i∈ℳ, ∑p=1Nμip=1. The mode detection probability matrix is denoted as μ=μipM×N. The augmented process (θk, ρk, ℳ, N) can be regarded as a hidden Markov model. We assume that the cardinality of the set N is not greater than the cardinality of ℳ, i.e., N≤ℳ. The asynchronous model (3) covers the mode-dependent and mode-independent cases: (i) when N=ℳ and μip=1 for p=i, i.e., μ=I, the model (3) reduces to the synchronous mode-dependent case and (ii) when N=1, i.e., the model detector has only one mode ρk=1, the asynchronous model (3) becomes a mode-independent case.


### 2.2. Asynchronous Repetitive Control

We consider the repetitive control structure provided in Figure1, where rk is the periodic reference input signal with period T that needs to be tracked, vk is the output of the repetitive controller, and ek=rk−yk is the tracking error. The dynamic model of the discrete-time repetitive controller CRz is given by(4)vk=ek,0≤k<T,ek+vk−T,k≥T.Figure 1
The overall closed-loop system configuration.For the controller and estimator design, only the estimateρk of the system mode θk is assumed to be known. The following detected-mode-dependent estimator is utilized to provide the estimate of the state of the linear Markovian switching system (2):(5)x^k+1=Aepx^k+Bepyk,y^k=Cepx^k,where x^k∈ℝn is the estimated state, y^k is the output of the asynchronous estimator, and Aep, Bep and Cep,p∈N are estimator parameter matrices which need to be designed. Let the estimation error be δk=xk−x^k. Then, the estimator error governing equation is given by(6)δk+1=Ai−BepCixk−Aepx^k+Bi−BepDiuk.We consider the following detected-mode-dependent repetitive control input for the plant:(7)uk=Kvpvk+y^k=Kvpvk+Cepx^k,where Kvp,p∈N is the repetitive controller gain.Let the exogenous reference input signal be zero, i.e.,rk=0. Considering k>T, rk=0, and combining (2), (4), and (7), we can write the control input in terms of the estimation error as(8)uk=Eipx^k−FivpCiδk+Fivpvk−T,where Eip and Fivp, ∀i∈ℳ,p∈N are related to the control gain parameters Kvp and Cep as follows:(9)Eip+KvpCi=Cep,I+KvpDiFivp=Kvp.Remark 1.
The effect of the stochastic jumps between detected modes is reflected in the asynchronous repetitive control law (7) via gain matrices Kvp and Cep,p∈N. The above formulation translates the design of these gain matrices into finding parametric matrices Eip and Fivp,i∈ℳ,p∈N, which can be obtained as the solutions of a set of coupled matrix inequalities that depend on the transition and mode-detection probabilities πij and μip, respectively (cf. Theorem 2).


### 2.3. Closed-Loop Governing Equations and Preliminaries

Combining (2), (5), and (8) yields(10)x^k+1=Aep+BepCi+BepDiEipx^k+BepCi−BepDiFivpCiδk+BepDiFivpvk−T.Substituting (8) into (6) yields(11)δk+1=Ai+BiEip−Aep−BepCi−BepDiEipx^k+Ai+BepDiFivpCi−BepCi−BiFivpCiδk+BiFivp−BepDiFivpvk−T.Fork>T, we know that(12)vk=−yk+vk−T=−Cixk−Diuk+vk−T.Substituting (8) and simplifying, we can derive that(13)vk=−Ci+DiEipx^k+DiFivpCi−Ciδk+I−DiFivpvk−T.The closed-loop governing equations follow from (11), (10), (13) and can be expressed in compact form as follows:(14)ξk+1=Aipξk+ℬipvk−T,vk=Cipξk+Dipvk−T,where i∈ℳ,p∈N, ξk=x^⊤kδ⊤k⊤, Aip=A11ipA12ipA21ipA22ip, A11ip=Aep+BepCi+BepDiEip, A12ip=BepCi−BepDiFivpCi, A21ip=Ai+BiEip−Aep−BepCi−BepDiEip, A22ip=Ai+BepDiFivpCi−BepCi−BiFivpCi and(15)ℬip=BepDiFivpBiFivp−BepDiFivp,Dip=I−DiFivp,Cip=−Ci−DiEipDiFivpCi−Ci.Based on the above closed-loop governing equations, the problem considered in this paper can be stated as follows: for the discrete-time linear Markovian jump system (2), develop a design procedure to determine estimator matrices Aep, Bep, and Cep and controller gain matrices Kvp and Cep such that the closed-loop system (14) is mean square stable.To derive the main results, we will employ the following definition, assumption, and lemma.Definition 1 (see [41]).
The discrete-time linear Markovian switching system (2) with uk≡0 is said to be mean square stable if for every initial condition x0∈ℝn and θ0∈ℳ, the following holds:(16)limk⟶∞Exk2x0,θ0=0.Assumption 1.
The system output matrixCi∈ℝm×n, has full row rank, i.e., rankCi=m. The singular-value decomposition (SVD) of Ci can be obtained as follows:(17)Ci=GiΛi0Hi⊤,where Λi∈ℝm×m is a diagonal matrix with positive diagonal elements in descending order; 0∈ℝm×n−m is the matrix with zero as its elements; and Gi∈ℝm×m and Hi∈ℝn×n are orthogonal matrices.Lemma 1 (see [41]).
For a given matrixCi∈ℝm×n with rankCi=m, if X∈ℝn×n is a symmetric matrix, then there exists a matrix X¯∈ℝm×m such that CiX=X¯Ci if and only if(18)X=HdiagX11,X22H⊤,where H∈ℝn×n is the right orthogonal matrix of SVD, X11∈ℝm×m and X22∈ℝn−m×n−m.


## 2.1. System Description

To model the switching process as a discrete-time homogeneous Markov chain, we consider the parameterθk,k∈ℤ0+ which takes values from a finite set ℳ=1,2,3,…,M with the following mode transition probabilities:(1)Probθk+1=jθk=i=πij,where πij≥0, ∀i,j∈ℳ, and ∑j=1Mπij=1. Based on this Markov chain, we define the M×M transition probability matrix as π whose i,j-th element is πij.Letxk∈ℝn, uk∈ℝl and yk∈ℝm, respectively, denote the state, control input, and system output vectors. Let Aθk, Bθk, Cθk, and Dθk denote the system matrices with appropriate dimensions. The discrete-time linear Markovian switching system that we consider in this paper is given by(2)xk+1=Aθkxk+Bθkuk,yk=Cθkxk+Dθkuk.In the following, for ease of readability, forθk=i∈ℳ, the system matrices Aθk, Bθk, Cθk, and Dθk are denoted by Ai, Bi, Ci, and Di which are assumed to be known. The relative degree of the plant is assumed to be zero, which implies that Di≠0.Since it is generally not possible to accurately measure the evolution of the system modeθk, we consider a probabilistic detector that provides the estimated values of θk with a certain probability. Let the estimated system mode signal be denoted by ρk which is utilized by the controller and need not be synchronized with the system mode θk. We consider a hidden Markov model θk,ρk,ℳ,N to characterize the asynchronous phenomenon as follows:(3)Probρk=pθk=i=μip,where i∈ℳ, p∈N, N=1,2,3,…,N and μip is the mode detection probability. For any i∈ℳ, ∑p=1Nμip=1. The mode detection probability matrix is denoted as μ=μipM×N. The augmented process (θk, ρk, ℳ, N) can be regarded as a hidden Markov model. We assume that the cardinality of the set N is not greater than the cardinality of ℳ, i.e., N≤ℳ. The asynchronous model (3) covers the mode-dependent and mode-independent cases: (i) when N=ℳ and μip=1 for p=i, i.e., μ=I, the model (3) reduces to the synchronous mode-dependent case and (ii) when N=1, i.e., the model detector has only one mode ρk=1, the asynchronous model (3) becomes a mode-independent case.


## 2.2. Asynchronous Repetitive Control

We consider the repetitive control structure provided in Figure1, where rk is the periodic reference input signal with period T that needs to be tracked, vk is the output of the repetitive controller, and ek=rk−yk is the tracking error. The dynamic model of the discrete-time repetitive controller CRz is given by(4)vk=ek,0≤k<T,ek+vk−T,k≥T.Figure 1
The overall closed-loop system configuration.For the controller and estimator design, only the estimateρk of the system mode θk is assumed to be known. The following detected-mode-dependent estimator is utilized to provide the estimate of the state of the linear Markovian switching system (2):(5)x^k+1=Aepx^k+Bepyk,y^k=Cepx^k,where x^k∈ℝn is the estimated state, y^k is the output of the asynchronous estimator, and Aep, Bep and Cep,p∈N are estimator parameter matrices which need to be designed. Let the estimation error be δk=xk−x^k. Then, the estimator error governing equation is given by(6)δk+1=Ai−BepCixk−Aepx^k+Bi−BepDiuk.We consider the following detected-mode-dependent repetitive control input for the plant:(7)uk=Kvpvk+y^k=Kvpvk+Cepx^k,where Kvp,p∈N is the repetitive controller gain.Let the exogenous reference input signal be zero, i.e.,rk=0. Considering k>T, rk=0, and combining (2), (4), and (7), we can write the control input in terms of the estimation error as(8)uk=Eipx^k−FivpCiδk+Fivpvk−T,where Eip and Fivp, ∀i∈ℳ,p∈N are related to the control gain parameters Kvp and Cep as follows:(9)Eip+KvpCi=Cep,I+KvpDiFivp=Kvp.Remark 1.
The effect of the stochastic jumps between detected modes is reflected in the asynchronous repetitive control law (7) via gain matrices Kvp and Cep,p∈N. The above formulation translates the design of these gain matrices into finding parametric matrices Eip and Fivp,i∈ℳ,p∈N, which can be obtained as the solutions of a set of coupled matrix inequalities that depend on the transition and mode-detection probabilities πij and μip, respectively (cf. Theorem 2).


## 2.3. Closed-Loop Governing Equations and Preliminaries

Combining (2), (5), and (8) yields(10)x^k+1=Aep+BepCi+BepDiEipx^k+BepCi−BepDiFivpCiδk+BepDiFivpvk−T.Substituting (8) into (6) yields(11)δk+1=Ai+BiEip−Aep−BepCi−BepDiEipx^k+Ai+BepDiFivpCi−BepCi−BiFivpCiδk+BiFivp−BepDiFivpvk−T.Fork>T, we know that(12)vk=−yk+vk−T=−Cixk−Diuk+vk−T.Substituting (8) and simplifying, we can derive that(13)vk=−Ci+DiEipx^k+DiFivpCi−Ciδk+I−DiFivpvk−T.The closed-loop governing equations follow from (11), (10), (13) and can be expressed in compact form as follows:(14)ξk+1=Aipξk+ℬipvk−T,vk=Cipξk+Dipvk−T,where i∈ℳ,p∈N, ξk=x^⊤kδ⊤k⊤, Aip=A11ipA12ipA21ipA22ip, A11ip=Aep+BepCi+BepDiEip, A12ip=BepCi−BepDiFivpCi, A21ip=Ai+BiEip−Aep−BepCi−BepDiEip, A22ip=Ai+BepDiFivpCi−BepCi−BiFivpCi and(15)ℬip=BepDiFivpBiFivp−BepDiFivp,Dip=I−DiFivp,Cip=−Ci−DiEipDiFivpCi−Ci.Based on the above closed-loop governing equations, the problem considered in this paper can be stated as follows: for the discrete-time linear Markovian jump system (2), develop a design procedure to determine estimator matrices Aep, Bep, and Cep and controller gain matrices Kvp and Cep such that the closed-loop system (14) is mean square stable.To derive the main results, we will employ the following definition, assumption, and lemma.Definition 1 (see [41]).
The discrete-time linear Markovian switching system (2) with uk≡0 is said to be mean square stable if for every initial condition x0∈ℝn and θ0∈ℳ, the following holds:(16)limk⟶∞Exk2x0,θ0=0.Assumption 1.
The system output matrixCi∈ℝm×n, has full row rank, i.e., rankCi=m. The singular-value decomposition (SVD) of Ci can be obtained as follows:(17)Ci=GiΛi0Hi⊤,where Λi∈ℝm×m is a diagonal matrix with positive diagonal elements in descending order; 0∈ℝm×n−m is the matrix with zero as its elements; and Gi∈ℝm×m and Hi∈ℝn×n are orthogonal matrices.Lemma 1 (see [41]).
For a given matrixCi∈ℝm×n with rankCi=m, if X∈ℝn×n is a symmetric matrix, then there exists a matrix X¯∈ℝm×m such that CiX=X¯Ci if and only if(18)X=HdiagX11,X22H⊤,where H∈ℝn×n is the right orthogonal matrix of SVD, X11∈ℝm×m and X22∈ℝn−m×n−m.


## 3. Main Results

In this section, we provide the asynchronous repetitive controller design approach for the system given by (2). First, we present the sufficient condition in the following lemma to ensure that the closed-loop system (14) is mean square stable.Theorem 1.
Letαi and βi, ∀i∈ℳ, denote positive scalars. Define the following matrices:(19)Pi≜diagαiP1i,P2i,Pi≜diagP1i,P2i,P1i≜∑j=1MπijαjP1j,P2i≜∑j=1MπijP2j,Qi≜∑j=1MπijβjQj,Ξ11ip≜∑p=1NμipAi⊤pPiAip+Ci⊤pQiCip,Ξ12ip≜∑p=1NμipAi⊤pPiℬip+Ci⊤pQiDip,Ξ22ip≜∑p=1Nμipℬi⊤pPiℬip+Di⊤pQiDip.The closed-loop asynchronous repetitive control system (14) is mean square stable if there exist matrices P1i≻0,P2i≻0 and Qi≻0,∀i∈ℳ with appropriate dimensions such that the following matrix inequality holds for all i∈ℳ, p∈N:(20)Θip≜−diagPi,βiQi+Ξ11ipΞ12ip⋆Ξ22ip≺0.Proof.
For any system modeθk=i∈ℳ, we construct a stochastic Lyapunov functional for the closed-loop system (14) as(21)Vξk,vk−T,i≜V1ξk,i+V2vk−T,i,where V1ξk,i and V2vk−T,i are given as follows:(22)V1ξk,i=ξ⊤kPiξk,V2vk−T,i=v⊤k−TβiQivk−T.
LettingEΔV≜EΔVξk,vk−T,i, we have EΔV=EΔV1ξk,i+EΔV2vk−T,i, where(23)EΔV1ξk,i=EV1ξk+1,θk+1=jξk,θk=i−EV1ξk,i,(24)EΔV2vk−T,i=EV2vk,θk=jvk−T,θk−1=i−EV2vk−T,i.
In the following developments, the expectation operator symbolE is omitted from the right-hand side of some expressions for presentation clarity. Based on the result of [41], utilizing (23) along the trajectories of (14), we simplify EΔV1ξk,i and EΔV2vk−T,i as follows:(25)EΔV1ξk,i=ξ⊤k+1Piξk+1−ξ⊤kPiξk=∑p=1NμipAipξk+ℬipvk−T⊤PiAipξk+ℬipvk−T−ξ⊤kPiξk=ξ⊤k∑p=1NμipAi⊤pPiAip−Piξk+2ξ⊤k∑p=1NμipAi⊤pPiℬipvk−T+v⊤k−T∑p=1Nμipℬi⊤pPiℬipvk−T.(26)EΔV2vk−T,i=v⊤kQivk−v⊤k−TβiQivk−T=∑p=1NμipCipξk+Dipvk−T⊤QiCipξk+Dipvk−T−v⊤k−TβiQivk−T=ξ⊤k∑p=1NμipCi⊤pQiCipξk+v⊤k−T∑p=1NμipDi⊤pQiDip−βiQivk−T+2ξ⊤k∑p=1NμipCi⊤pQiDipvk−T.
Combining (25) and (26), we obtain(27)EΔV=η⊤k−diagPi,βiQi+Ξipξk,where ηk≜ξkvk−T and Ξip≜Ξ11ipΞ12ip⋆Ξ22ip. Based on the matrix inequality (20), for any ηk≠0 we have(28)EΔV=η⊤kΘipηk≤−λmin−Θipη⊤kηk≤−γη⊤kηk,where λmin−Θip denotes the minimal eigenvalue of −Θip and γ=infλmin−Θip, for all i∈ℳ,p∈N. From (28), for any positive integer K>T≥1, we can obtain(29)∑k=0KEη⊤kηk≤1γEVη0,θ0−EVηK+1,θK+1≤1γEVη0,θ0.
AsK⟶+∞, we have the following:(30)∑k=0+∞Eη⊤kηk≤1γEVη0,θ0<+∞,which implies that(31)limk⟶∞Eηk2η0,θ0=0.
Therefore, the closed-loop asynchronous repetitive control system (14) is mean square stable in the sense of Definition 1.Remark 2.
In contrast to the repetitive control designs presented in the recent literature, Theorem1 presents a clear framework to ensure mean square stability for discrete-time Markovian switching systems with an estimator-based asynchronous repetitive controller. Further, the above approach employs multiple and totally mode-dependent Lyapunov functional (21) in arriving at the result in Theorem 1, which leads to a less conservative result compared with the approach given in [40] that includes the mode-independent term, such as vk−T⊤βiQivk−T vs vk−T⊤βQvk−T.
The condition given in (20) contains nonlinear product terms of unknown gain matrices, and thus, it is a nonlinear matrix inequality. It is generally difficult to solve this inequality as there are no known established methods currently available in the literature to solve such a nonlinear matrix inequality. In this paper, the full row rank assumption on the system output matrix Ci, its SVD, and Lemma 1 are utilized to decompose the nonlinear product terms and solve the inequality.
The following theorem provides a sufficient condition to ensure mean square stability of the closed-loop asynchronous repetitive control system (20) in terms of an LMI and subsequently provides a method to synthesize the gain matrices via parametric matrices Eip, Fivp, ∀i∈ℳ,∀p∈N.Theorem 2.
For the given positive scalarsαi, βi, ∀i∈ℳ and under Assumption 1, if there exist matrices Xi≻0, Yi≻0, Z1i=HidiagZ11i,Z12iHi⊤≻0, Z2i=HidiagZ21i,Z22iHi⊤≻0,∀i∈ℳ with appropriate dimensions and arbitrary matrices ϒip, Φiep, Ψ1iep, Ψ2iep, Πivp, Uip, Vip, Wip, and Yip with appropriate dimensions such that the following LMI is satisfied for all i∈ℳ, p∈N:(32)Ωip=−Z1i00Ω14iΩ15ip⋆−Z2i0Ω24iΩ25ip⋆⋆−βi−1YiΩ34iΩ35ip⋆⋆⋆−Xi⟶0⋆⋆⋆⋆−Yi⟶≺0,where Xi=Pi−1, Xi⟶=diagXi,Xi,…,Xi︸N, Yi=Qi−1, Yi=Qi−1, Yi⟶=diagYi,Yi,…,Yi︸N, Z1i=αi−1P1i−1, Z2i=P2i−1 and for l=1,2,3:(33)Ωl4i≜ωl4i1ωl4i2⋯ωl4ip⋯ωl4iN,Ωl5i≜ωl5i1ωl5i2⋯ωl5ip⋯ωl5iN,and ∀p∈N,l=1,2,3 and ωl4ip are block matrices with one row and two columns, i.e.,(34)ω14ip=μipΦie⊤p+Ci⊤Ψ1ie⊤p+Ui⊤pZ1iAi⊤+ϒi⊤Bi⊤−Φie⊤p−Ci⊤Ψ1ie⊤p−Ui⊤p,ω24ip=μipCi⊤Ψ2ie⊤p−Wi⊤pZ2iAi⊤+Wi⊤p−Ci⊤Ψ2ie⊤p−CiTΠiv⊤pBi⊤,ω34ip=βi−1μipVi⊤pYi⊤pBi⊤−Vi⊤p,and ∀p∈N,l=1,2,3 and ωl5ip are denoted as(35)ω15ip=μip−Z1iCiT−ϒi⊤pDi⊤,ω25ip=μipCiTΠiv⊤pDi⊤−Z2iCi⊤,ω35ip=βi−1μipYi−Yi⊤pDi⊤,then the closed-loop asynchronous repetitive control system (14) is mean square stable. Furthermore, the estimator parametric matrices Aep, Bep and the related asynchronous repetitive control parametric matrices Eip and Fivp,∀i∈ℳ,∀p∈N are given by(36)Aep=ΦiepZ1i−1,Eip=ϒipZ1i−1,Bep=Ψ1iepGiΛiZ11i−1Λi−1Gi⊤=Ψ2iepGiΛiZ21i−1Λi−1Gi⊤,Fivp=ΠivpGiΛiZ21i−1Λi−1Gi⊤=YipYi−1,where Gi, Hi, and Λi,∀i∈ℳ are defined in Assumption 1.Proof.
The proof of this theorem follows from Theorem1. By Schur complement, (20) is equivalent to(37)−Pi0χ1iPi⟶χ3iQi⟶⋆−βiQiχ2iPi⟶χ4iQi⟶⋆⋆−Pi⟶0⋆⋆⋆−Qi⟶≺0,where Pi⟶=diagPi,Pi,…,Pi︸N, Qi⟶=diagQi,Qi,…,Qi︸N, and(38)χli=χli1,…,χlip,…,χliN,l=1,2,3,4,χ1ip=μipAi⊤p,χ2ip=μipℬi⊤p,χ3ip=μipCi⊤p,χ4ip=μipDi⊤p.
DefineZi=diagZ1i,Z2i=Pi−1, by pre-multiplying and post-multiplying matrix inequality (37) with the matrix(39)diagZi,βi−1Yi,Xi⟶,Yi⟶,we can obtain a matrix inequality as follows:(40)−Zi0Ziχ1iZiχ3i⋆−βi−1Yiβi−1Yiχ2iβi−1Yiχ4i⋆⋆−Xi⟶0⋆⋆⋆−Yi⟶≺0.
Choose anyp∈N, we notice that(41)Ziχ1ip=μipZ1iAi⊤p1Z2iAi⊤p2,Ziχ3ip=μipZ1iCi⊤p1Z2iCi⊤p2,where Ai⊤p1, Ai⊤p2 and Ci⊤p1, Ci⊤p2 denote the first and second rows of the block matrices Ai⊤p, Ci⊤p, respectively, i.e., Ai⊤p1=A11i⊤pA21i⊤p, Ai⊤p2=A12i⊤pA22i⊤p, and Ci⊤p1=−Ci⊤−Ei⊤pDi⊤, Ci⊤p2=Ci⊤Fiv⊤pDi⊤−Ci⊤.
Similarly, letχ1i1, χ3i1 denote the first row of the block matrix χ1i, χ3i, and χ1i2, χ3i2 for the second row of the block matrix χ1i, χ3i, respectively. Then, it can be found that the matrix inequality (40) is equivalent to(42)−Z1i00Z1iχ1i1Z1iχ3i1⋆−Z2i0Z2iχ1i2Z2iχ3i2⋆⋆−βi−1Yiβi−1Yiχ2iβi−1Yiχ4i⋆⋆⋆−Xi⟶0⋆⋆⋆⋆−Yi⟶≺0.
On the other hand, from Lemma1, there exist matrices Z1i¯ and Z2i¯ such that(43)CiZ1i=Z1i¯Ci,CiZ2i=Z2i¯Ci,∀i∈ℳ.
By utilizing the SVD of system output matrixCi given in Assumption 1, equation (43) can be expressed as(44)GiΛi0Z11i00Z12iHi⊤=Z1i¯GiΛi0Hi⊤,GiΛi0Z21i00Z22iHi⊤=Z2i¯GiΛi0Hi⊤,which yields(45)Z1i¯=GiΛiZ11iΛi−1Gi⊤,(46)Z2i¯=GiΛiZ21iΛi−1Gi⊤,i∈ℳ.
Moreover, for anyi∈ℳ and p∈N, we select the matrices ϒip, Φiep, Πivp, Ψ1iep, Ψ2iep and Yip as follows:(47)Φiep=AepZ1i,ϒip=EipZ1i,Πivp=FivpZ2i¯,Ψ1iep=BepZ1i¯,Ψ2iep=BepZ2i¯,Yip=FivpYi.
Define(48)Uip=BepDiEipZ1i,Vip=BepDiFivpYi,and Wip=BepDiFivpCiZ2i for any i∈ℳ and p∈N. Then, by substituting (43) and (47) into (42) and simplifying, we conclusively obtain the LMI (32). Concomitantly, the estimator parameter matrices Aep, Bep and the related asynchronous repetitive control parametric matrices Eip and Fivp,i∈ℳ,p∈N can be solved by (46) and (47).
Hence, if a solution to the LMI (32) exists, the closed-loop asynchronous repetitive control system (14) is mean square stable and the parametric matrices that need to be designed are obtained by using (36).Remark 3.
From (9) and (36) the design matrices Aep, Bep, Cep, Kvp can be obtained as follows:(49)Aep=ΦiepZ1i−1,Bep=Ψ1iepGiΛiZ11i−1Λi−1Gi⊤,Kvp=FivpI−DiFivp†,Cep=Eip+KvpCi,where I−DiFivp† denotes a possible generalized inverse of I−DiFivp. Although the design matrices associated with the estimator and controller appear to not only depend on the detection mode p∈N but also on the system mode i∈ℳ, we consider some additional constraints in the process of solving LMI (32) and enforce that all the system modes lead to common design matrices. For example, consider Φ1epZ11−1=Φ2epZ12−1=⋯=ΦMepZ1M−1 for any given p∈N, then Aep will be independent of the system mode i∈ℳ. Under such circumstances, the design matrices are able to coincide with the previous design approach as described in (5) and (7).Remark 4.
Notice that if the system matricesCi and Di are independent of the system mode i, i.e., Ci and Di are both constant matrices under all modes, then the additional constraints discussed in Remark 3 to render the gains Aep, Bep, Cep, and Kvp to be independent of the system mode will no longer be needed. However, such requirements on system matrices Ci and Di will reduce the applicability of the approach to a wider class of switched systems.Remark 5.
Theorem2 provides an LMI-based sufficient condition for mean square stability of the closed-loop asynchronous repetitive control system (14). One can use efficient interior-point methods with a polynomial-time complexity to determine whether the LMI is feasible; we have used MATLAB LMI toolbox to solve the LMI (32). The feasibility of the LMI (32) is facilitated by the tuning parameters αi, βi,i∈ℳ, which can be utilized to adjust the contribution of the control and learning parts, as described for the deterministic systems in [32].Remark 6.
By taking into consideration both the control input and tracking error, a new cost functional can be defined as follows:(50)J≜E∑k=0∞u2k+e2k.
Then, the optimal values ofαi, βi, i∈ℳ can be obtained by solving the following constrained stochastic optimization problem:(51)minimize:Jsubject to:24holds  for alli∈ℳ,p∈N.
The stochastic optimization problem (51) provides a framework to find optimal values for the adjustable parameters αi, βi, i∈ℳ.


## 4. Numerical Example and Simulations

In this section, we consider a second-order RLC circuit system as shown in Figure2, which can be modeled as a discrete-time linear Markovian switching system (2).Figure 2
Configuration of the circuit system.By choosinguC and iL as state variables, u as input, and the load voltage uR2 as output, the state-space representation of this circuit system is given by(52)u˙Ci˙L=−1R1+R2C−R1R1+R2CR1LR1+R2−R1R2LR1+R2uCiL+1R1+R2CR2R1+R2u,uR2=−R2R1+R2−R1R2R1+R2uCiL+R2R1+R2u.Because of environmental changes and uncertainties, one can expect fluctuations in circuit parameters. We model the underlying system with these parameter fluctuations as a Markov jump system as follows:(53)x˙=Aθx+Bθu,y=Cθx+Dθu,where x=uC,iL⊤ and y=uR2.Upon discretization of these governing equations (zero-order hold approximation with sampling periodTs=0.01s), we obtain(54)xk+1=Adθxk+Bdθuk,yk=Cθxk+Dθuk,where Adθ=eAθTs and Bdθ=∫0TseAθtBθdt.We assume that the Markov chainsθk and ρk have three modes of operation, i.e., ℳ=1,2,3 and N=1,2,3. The transition probability matrices π and µ are correspondingly chosen as(55)π=0.30.50.20.70.10.20.60.30.1,μ=0.50.30.20.40.50.10.30.60.1.The evolution of the system and controller/estimator modes is illustrated in Figures3 and 4, respectively.Figure 3
Evolution of the system mode.Figure 4
Evolution of the controller mode.Forθ∈ℳ=1,2,3, the system parameters are chosen as(56)R11=1MΩ,R12=1.5MΩ,R13=0.5MΩ,R21=2MΩ,R22=1MΩ,R23=1.5MΩ,L1=0.5H,L2=0.6H,L3=0.4H,C1=3μF,C2=2.5μF,C3=4.5μF.The periodic reference signalrk is generated via sampling of the output of the following linear time-invariant (lti) exosystem:(57)ω˙t=ℱωt,rt=Gωt,where ℱ=01−π/220, G=12. The sampling period is Ts=0.01 seconds. The period of the continuous-time periodic reference signal rt is 4 seconds, i.e., T=4. Then, the number of samples for each period of the periodic reference signal rk is N=T/Ts=400.As stated in Remark5, we can choose a performance index,(58)J10=E∑k=010u2k+e2k,to seek the best overall control and learning performance. In this paper, we obtain α1=0.5, α2=1, α3=2, β1=1, β2=1.5, β3=2. By utilizing Theorem 2, we directly obtain that(59)Ae1=−0.150.210.35−0.56,Be1=−0.460.43,Ae2=−0.100.450.26−0.62,Be2=−0.670.31,Ae3=−0.290.150.23−0.36,Be3=−0.580.36,Ce1=0.130.26⊤,Ce2=0.320.29⊤,Ce3=0.560.23⊤,Kv1=−4.65,Kv2=−5.53,Kv3=−5.85.We choose the initial value ofωt as ω0=−2−1⊤. The initial state values are chosen as x0=2−1⊤, x^0=12⊤. Under the above parameters, evolution of the system state xk and estimation error δk are plotted in Figures 5 and 6, respectively. The trajectories of the reference signal rk and system output yk are shown in Figure 7. It can be observed that the overall closed-loop system achieves a satisfactory state estimation and reference tracking performance, which demonstrates the effectiveness of the proposed design scheme in Section 3.Figure 5
Trajectory of the system statexk.Figure 6
Evolution of the estimation errorδk=xk−x^k.Figure 7
Trajectories of the reference signalrk and system output yk.Remark 7.
Based on the numerical simulations results, the asynchronous repetitive control method proposed in this paper has the following advantages over the methods provided in [32-39]: (1) the proposed method does not require the controller or estimator to switch synchronously with the plant; (2) the proposed asynchronous repetitive control method is broader and inclusive in the sense that a variety of more restrictive cases can be obtained from it. For example, when N=ℳ and μip=1 for p=i, i.e., μ=I, the asynchronous repetitive control method reduces to the synchronous mode-dependent case. When N=1, i.e., the mode detector has only one mode ρk=1; it reduces to the asynchronous mode-independent case. When ℳ=N=1, it reduces to the synchronous mode-independent case, i.e., the general form of repetitive control.


## 5. Conclusion

In this work, we provided an asynchronous repetitive control strategy for discrete-time Markovian switching systems. A hidden Markov model was adopted to describe the asynchronous phenomenon appearing between the system modes and controller modes. With a state-estimator-based asynchronous repetitive controller, sufficient conditions for mean square stability of the closed-loop stochastic system were derived by using some basic results from stochastic analysis and matrix inequalities. With the proposed approach, all the design matrices can be obtained by solving a set of linear matrix inequalities. To illustrate the approach and its feasibility, numerical simulations on a circuit system were shown to demonstrate the effectiveness of the proposed design scheme. In this paper, we have illustrated the approach using a simple numerical example; one potential future topic is to apply the formulation to an engineering application and conduct numerical and experimental investigations; we plan to collaborate with practicing engineers to pursue this. Further, another possible future work is to develop both necessary and sufficient conditions for such systems, which will also open up more concrete opportunities for applying the method to more practical systems.



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