Optimal Switching Synthesis for Jump Linear Systems with Gaussian initial state uncertainty

This paper provides a method to design an optimal switching sequence for jump linear systems with given Gaussian initial state uncertainty. In the practical perspective, the initial state contains some uncertainties that come from measurement errors or sensor inaccuracies and we assume that the type of this uncertainty has the form of Gaussian distribution. In order to cope with Gaussian initial state uncertainty and to measure the system performance, Wasserstein metric that defines the distance between probability density functions is used. Combining with the receding horizon framework, an optimal switching sequence for jump linear systems can be obtained by minimizing the objective function that is expressed in terms of Wasserstein distance. The proposed optimal switching synthesis also guarantees the mean square stability for jump linear systems. The validations of the proposed methods are verified by examples. NOMENCLATURE ‖ · ‖ Without subscription denotes `2-norm R+ The set of positive real Z+ The set of non-negative integer I The set of switching modes tr(·) Trace operator for a square matrix m.s. Convergence in the mean square sense X ∼ ς (x) random variable X with probability density function (PDF) ς (x) N (μ,Σ) Gaussian PDF with mean μ and covariance Σ ∗Address all correspondence to this author. INTRODUCTION A jump linear system is defined as a dynamical system consisting of a finite number of subsystems and a switching rule that governs a switching between the family of linear subsystems. Over decades, a variety of researches for jump linear systems have been investigated because of its practical implementation. For example, a jump linear system can be used for power systems, manufacturing systems, aerospace systems, networked control systems, etc( [1], [2], [3]). In general, problems for jump linear systems branch out into two different fields. The first one is the stability analysis under given switching laws. Since a certain switching law between individually stable subsystem can make the jump linear system unstable [4], it is very important to identify conditions under which system can be stable. Interestingly, the jump linear system also can be stable by switching between unstable subsystems. Fang et al. [5] showed sufficient conditions for stability of jump linear systems under arbitrary switching using linear matrix inequalities(LMIs). Lin et al. [6] showed necessary and sufficient conditions for asymptotic stability of jump linear systems using finite n-tuple switching sequences, satisfying a certain condition. In addition, broad analysis regarding stability for jump linear systems has been accomplished within few decades( [7, 8, 9, 10, 11, 12]). On the other hand, switching synthesis problem, which is another branch of jump linear systems, is relatively new and few investigations have been achieved. Since the main objective is to design switching sequences that establish both the stability and the performance, this case is much harder than stability analysis 1 Copyright c © 2014 by ASME ar X iv :1 40 8. 48 59 v1 [ cs .S Y ] 2 1 A ug 2 01 4 problem. For instance, Das and Mukherjee [13] solved the problem for an optimal switching of jump linear systems using Pontryagin’s minimum principle. In this method, two-point boundary value problem was solved via relaxation method, where ordinary differential equations are approximated by finite difference equations on mesh points. Therefore, the optimality and computational cost depend on mesh size. In addition, the time to find optimal solution varies according to guess solution. Egerstedt et al. [14] addressed a method to find derivative of the cost function with respect to switching time. However, in this paper, switching sequences are already given and the main focus is to find switching time. Although several other researches regarding optimal control problem together with optimal switching were studied for switched nonlinear systems( [15, 16, 17]), they may not fit to pure optimal switching problem for jump linear systems. Here we address optimal switching problem for jump linear systems with given multi-controllers. Multi-controller switching scheme is widely used, such as plant stabilization [18], system performance [6], adaptive control [19], and resource-constrained scheduling [20]. Under the assumption that more than two controllers are given to user, our objective is to find the optimal switching sequence which attains the best performance of the system by controller switching. We can also extend our method to multi-model switching problem by generalizing the multicontroller switching problem. Consequently, we aim to synthesize switching protocols that result in the optimality for the performance of jump linear systems. Moreover, we address the optimal switching problem with initial state uncertainties. In the practical perspective, initial state may contain uncertainties that usually come from measurement errors or sensor inaccuracies. Then, the system state is expressed as random variables represented by PDFs. We assume that the initial state PDF has a form of Gaussian distribution that is very common for real implementation. In order to measure the performance of the jump linear system with a given Gaussian PDF, we need to adopt a proper metric. In this paper, Wasserstein metric that assesses the distance between PDFs is used as a tool for both the stability and the performance measure. Hence, we introduce the optimal switching synthesis to achieve the optimality of the system performance with given Gaussian initial PDF by minimizing the objective function that is expressed in terms of Wasserstein distance. We also prove that the convergence of Wasserstein distance implies the mean square stability for the jump linear systems. Rest of this paper is organized as follows. Section II introduces the problem we want to solve. Brief explanations of Wasserstein distance are described in Section III. Section IV provides a way to solve optimal switching problems using receding horizon framework when Gaussian initial state uncertainty exists. Then, Section V demonstrates the validation of proposed methods by examples and Section VI concludes this paper. PROBLEM STATEMENT Consider a discrete-time linear system with multi-controller, given by x(k+1) = Ax(k)+Buσk(x), k ∈ Z,σk ∈I (1) uσk(x) = Kσk x (2) where the state vectors x ∈ Rn, control inputs uσ ∈ Rm, the system matrices A ∈ Rn×n, B ∈ Rn×m, and the set of modes I = {1,2, · · · ,m}. Note that the system matrix A is time-invariant and user can select one controller Kσk out of multiple choices. Without loss of generality, we can convert system (1)-(2) to the following jump linear systems by letting Aσk := A+BKσk . x(k+1) = Aσk x(k), k ∈ Z ,σk ∈I (3) where the system matrices Aσk ∈ Rn×n. The system in (3) represents not only the controller switching as depicted in (1)-(2), but also the system mode switching. Hence, we consider the jump linear system model (3) and we assume that individual subsystem dynamics Aσk are Schur stable. Our objective is to find the switching sequence, σ = {σ1,σ2, · · ·}, which guarantees the optimal performance of the switched system. For example, with multi-controller, we want to design a switching law which makes the system states reach the origin as fast as possible. Therefore, our aim is not to design controllers, but rather to synthesize the optimal switching sequence. For simplicity, we assume that there are two different controllers, which are good and poor in terms of system performance. The closed-loop dynamics are given by A1 and A2, respectively. In general, the reason to design multi-controller with respect to single system is to attain not only the system performance but also system stability, robustness, resource-optimal scheduling, etc. The convergence rate of system state is determined by spectral radius ρ(Aσ ) := max j |λ j σ |, where λσ = {λ 1 σ ,λ 2 σ , · · · ,λ n σ} is the set of eigenvalues for Aσ mode. According to characteristics of subsystem Aσ , there may exist a surge or an elevation in the state trajectory. In Fig. 1, we show one possibility where the switching is necessary for better performance of the system. Solid line represents the state trajectory of A1 while dashed line shows that of A2. In contrast to A2, which has slow convergence rate with no surge, A1 reaches the origin faster with a surge. Therefore, for better performance, it is clear that A2 mode has to be used from the beginning, and then system has to switch to A1 mode at time tk as described in arrows in Fig. 1. In this paper, motivated by the above example we address the following two questions. 2 Copyright c © 2014 by ASME FIG. 1. Schematic of Optimal Switching for Jump System 1. Is there a switching sequence for a jump linear system to get better performance compared to single mode? 2. If the above holds true, can we find the optimal switching sequence? In general, it is difficult to answer the first question directly. Instead, we want to show the case where the switching synthesis is not required because single mode attains the best performance. When ρ(A1) < ρ(A2), A1 mode has faster convergence to the origin than A2 mode. In addition, if ‖A1x(k)‖< ‖A2x(k)‖ for all k, then ‖x(k)‖ using A1 mode is always less than ‖x(k)‖ using A2 mode. As a result, A1 mode attains the best performance and jump is not necessary. For the second question, which is the main contribution of this paper, we introduce the optimal switching sequence using receding horizon framework and it is explained in section IV. Since, in most cases, initial condition of system state contains uncertainties, which come from measurement errors or sensor inaccuracies, we will use probability for initial state uncertainty of the system. Moreover, we assume that the type of initial state uncertainties is given by Gaussian distribution. The deterministic single initial state is a special case for Gaussian distribution with zero covariance. Therefore, in this paper we conceptually cover much broader one. Due to this Gaussian PDF, system states become a random number, and hence we cannot use `2-norm for the performance measure. As a consequence, we need to adopt a proper metric to quantify the distance between PDFs to measure the performance. For this reason, instead of using `2-norm ‖·‖`2 , Wasserstein distance is used as a tool for measuring the performance of jump linear systems. Brief explanations of Wasserstein distance are introduced in the next section. WASSERSTEIN DISTANCE Definition 1. (Wasserstein distance) Consider the metric space `2 (Rn) and let the vectors x1,x2 ∈Rn. Let P2(ς1,ς2) denote the collection of all probability measures ς supported on the product space R2n, having finite second moment, with first marginal ς1 and second marginal ς2. Then the L2 Wasserstein distance of order 2, denoted as 2W2, between two probability measures ς1,ς2, is defined as