Optimal Control of Switched Systems: A Polynomial Approach (original) (raw)
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A generalization of a polynomial optimal control of switched systems
A generalization of a polynomial approach to solve the optimal control problem of nonlinear switched systems is presented. We present a technique to deal with the more general non-polynomial switched systems, using the polynomial approach previously presented, combined with a recasting process. It is shown that the representation of the original switched problem into a continuous polynomial system allows us to use the generalized Maximum Principle. With this method and from a theoretical point of view, we provide necessary and sufficient conditions for the existence of minimizer by using particular features of its relaxed, convex formulation using the theory of moments.
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Optimal control problems for switched nonlinear systems are investigated. We propose an alternative approach for solving the optimal control problem for a nonlinear switched system based on the theory of moments. The essence of this method is the transformation of a nonlinear, nonconvex optimal control problem, that is, the switched system, into an equivalent optimal control problem with linear and convex structure, which allows us to obtain an equivalent convex formulation more appropriate to be solved by high-performance numerical computing. Consequently, we propose to convexify the control variables by means of the method of moments obtaining semidefinite programs.
Optimal control of switched systems: new results and open problems
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In optimal control problems of switched systems, we may need to find both an optimal continuous input and an optimal switching sequence since the system dynamics vary before and after every switching instant. In this paper, optimal control problems for both continuoustime and discrete-time switched systems are formulated and investigated. In particular, we regard an optimal control problem as a two stage optimization problem and discuss its solution algorithm. The dynamic programming (DP) approach is also studied. Difficulties and open problems are discussed.
Optimal Control of Switched Systems Based on Parameterization of the Switching Instants
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This paper presents a new approach for solving optimal control problems for switched systems. We focus on problems in which a prespecified sequence of active subsystems is given. For such problems, we need to seek both the optimal switching instants and the optimal continuous inputs. In order to search for the optimal switching instants, the derivatives of the optimal cost with respect to the switching instants need to be known. The most important contribution of the paper is a method which first transcribes an optimal control problem into an equivalent problem parameterized by the switching instants and then obtains the values of the derivatives based on the solution of a two point boundary value differential algebraic equation formed by the state, costate, stationarity equations, the boundary and continuity conditions, along with their differentiations. This method is applied to general switched linear quadratic problems and an efficient method based on the solution of an initial value ordinary differential equation is developed. An extension of the method is also applied to problems with internally forced switching. Examples are shown to illustrate the results in the paper.
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In this paper we address the problem of optimal switching for switched linear systems. The uniqueness of our approach lies in describing the switching action by multiple control inputs. This allows us to embed the switched system in a larger family of systems and apply Pontryagin's Minimum Principle for solving the optimal control problem. This approach imposes no restriction on the switching sequence or the number of switchings. This is in contrast to search based algorithms where a fixed number of switchings is set a priori. In our approach, the optimal solution can be determined by solving the ensuing two-point boundary value problem. Results of numerical simulations are provided to support the proposed method.
A polynomial approach for stability analysis of switched systems
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A polynomial approach to deal with the stability analysis of switched non-linear systems under arbitrary switching using dissipation inequalities is presented. It is shown that a representation of the original switched problem into a continuous polynomial system allows us to use dissipation inequalities for the stability analysis of polynomial systems. With this method and from a theoretical point of view, we provide an alternative way to search for a common Lyapunov function for switched non-linear systems.
A numerical method for the optimal control of switched systems
49th IEEE Conference on Decision and Control (CDC), 2010
Switched dynamical systems have shown great utility in modeling a variety of systems. Unfortunately, the determination of a numerical solution for the optimal control of such systems has proven difficult, since it demands optimal mode scheduling. Recently, we constructed an optimization algorithm to calculate a numerical solution to the problem subject to a running and final cost. In this paper, we modify our original approach in three ways to make our algorithm's application more tenable. First, we transform our algorithm to allow it to begin at an infeasible point and still converge to a lower cost feasible point. Second, we incorporate multiple objectives into our cost function, which makes the development of an optimal control in the presence of multiple goals viable. Finally, we extend our approach to penalize the number of hybrid jumps. We also detail the utility of these extensions to our original approach by considering two examples.
SIAM Journal on Control and Optimization, 2013
Though switched dynamical systems have shown great utility in modeling a variety of physical phenomena, the construction of an optimal control of such systems has proven difficult since it demands some type of optimal mode scheduling. In this paper, we devise an algorithm for the computation of an optimal control of constrained nonlinear switched dynamical systems. The control parameter for such systems include a continuous-valued input and discrete-valued input, where the latter corresponds to the mode of the switched system that is active at a particular instance in time. Our approach, which we prove converges to local minimizers of the constrained optimal control problem, first relaxes the discrete-valued input, then performs traditional optimal control, and then projects the constructed relaxed discrete-valued input back to a pure discrete-valued input by employing an extension to the classical Chattering Lemma that we prove. We extend this algorithm by formulating a computationally implementable algorithm which works by discretizing the time interval over which the switched dynamical system is defined. Importantly, we prove that this implementable algorithm constructs a sequence of points by recursive application that converge to the local minimizers of the original constrained optimal control problem. Four simulation experiments are included to validate the theoretical developments.
Stabilization of switched systems via optimal control
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In this paper we consider switched systems composed of LTI non Hurwitz dynamics and we deal with the problem of computing an appropriate switching law such that the controlled system is globally asymptotically stable. We first present a method to design a feedback control law that minimizes a linear quadratic performance index when an infinite number of switches is allowed and at least one dynamics is Hurwitz. Then, we show that this approach can be applied to stabilize switched systems whose modes are all unstable, by simply applying the proposed procedure to a "dummy" system, augmented with a stable dynamics. If the system with unstable modes is globally exponentially stabilizable, then our method is guaranteed to provide the feedback control law that minimizes the chosen quadratic performance index, and that guarantees the closed loop system to be globally asymptotically stable.