Modal occupation measures and LMI relaxations for nonlinear switched systems control

Abstract

This paper presents a linear programming approach for the optimal control of nonlinear switched systems where the control is the switching sequence. This is done by introducing modal occupation measures, which allow to relax the problem as a primal linear programming (LP) problem. Its dual linear program of Hamilton-Jacobi-Bellman inequalities is also characterized. The LPs are then solved numerically with a converging hierarchy of primal-dual moment-sum-of-squares (SOS) linear matrix inequalities (LMI). Because of the special structure of switched systems, we obtain a much more efficient method than could be achieved by applying standard moment/SOS LMI hierarchies for general optimal control problems.through an adequate switching strategy, in order to impose global stability and/or performance. Interested readers may refer to the survey papers [17,31,44,33] and the books [32,46] and the references therein.In this context, these systems are generally controlled by intrinsically discontinuous control signals, whose switching rule must be carefully designed to guarantee stability and performance properties. As far as optimality is concerned, several results are available in two main different contexts.The first category of methods exploits necessary optimality conditions, in the form of Pontryagin's maximum principle (the so-called indirect approaches), or through a large nonlinear discretization of the problem (the so-called direct approaches). The first contributions can be found in [8,22,36] where the problem has been formulated and partial solutions provided through generalized Hamilton-Jacobi-Bellman (HJB) equations or inequalities and convex optimization. In [38,47,43], the maximum principle is generalized to the case of general hybrid systems with nonlinear dynamics. The case of switched systems is discussed in [4,42,2]. For general nonlinear problems and hence for switched systems, only local optimality can be guaranteed , even when discretization can be properly controlled [51]. The subject is still largely open and we are far from a complete and numericaly tractable solution to the switched optimal control problem.The second category collects extensions of the performance indices H 2 and H ∞ originally developed for linear time invariant systems without switching, and use the flexibility of Lyapunov's approach, see for instance [20,16] and references therein. Even for linear switched systems, the proposed results are based on nonconvex optimization problems (e.g., bilinear matrix inequality conditions) difficult to solve directly. Sufficient convex Linear Matrix Inequality (LMI) design conditions may be obtained, yielding computed solutions which are suboptimal, but at the price of introducing a conservatism (pessimism) and a gap to optimality which is hard, if not impossible, to evaluate. Since the computation of this optimal strategy is a difficult task, a suboptimal solution is of interest only when it is proved to be consistent, meaning that it imposes to the switched system a performance not worse th...

A semidefinite programming approach for polynomial switched optimal control problems

“…Note that both measure LP (8) and its conic dual (9) have the switching control variable, s(t), in their formulations. In contrast, the work of Claeys et al 29 defines modal occupation measures to control the switching, and the resulting dual problem includes a system of HJB inequalities without any variable for controlling the switchings. Indeed, their approach uses several modal occupation measures to eliminate a need for an extra input to control the switches but, by contrast, ours applies one state-action occupation measure with an additional switching control variable.…”

“…Lately, the works of Henrion et al 28 and Claeys et al 29 employed the approach introduced by Lasserre et al 20 for solving optimal control of nonlinear switched systems using a variant of embedding transformation via probability measures. The work of Claeys et al 29 presented a much more efficient method with less computational complexity in contrast to standard LMI hierarchies of general polynomial optimal control problems by defining modal occupation measures.…”

“…In our formulation, we do not make any preassumption about the number of switches and the switching signal. Inspired by work done by Mojica-Nava et al, 30 we first introduce a new variation of the presented formulation in the work of Claeys et al 29 by applying a polynomial representation to simulate the behavior of switched systems using a new continuous control variable. Then, based on the method of moments, the embedded optimal control problem is formulated as an infinite-dimensional LP over the space of occupation measures and also, the corresponding dual problem that is the primary concern of this contribution is introduced.…”

“…Although we present and use a new version of the formulation in the work of Claeys et al, 29 through a different framework, we investigate the problem of this article, and our main contributions can be listed as follows:…”

“…Although this type of modeling for primal-dual moment-S.O.S LMI may be regarded as a limitation since modal occupation measures introduced in the work of Claeys et al 29 serve to reduce the number of variables in the state-action occupation measure and allow for better scalability of moment relaxations, as numerical experiments show (see Sections 6.1 and 6.2) on some problems that have chattering phenomenon, our proposed dual problem performs superiorly over that of Claeys et al 29 in controlling the switched system on chattering arcs. Finally, in this paper, all subsystems are combined through the continuous control variable of switchings into one system, and the discussion of determining other possible choices to find the optimal control policy with less computational efforts in general switched optimal control problems with polynomial data is beyond the scope of this article and will be investigated separately.…”