Elsevier

Automatica

Volume 41, Issue 1, January 2005, Pages 11-27
Automatica

Optimal control of switching systems

https://doi.org/10.1016/j.automatica.2004.08.003Get rights and content

Abstract

This paper considers an optimal control problem for a switching system. For solving this problem we do not make any assumptions about the number of switches nor about the mode sequence, they are determined by the solution of the problem. The switching system is embedded into a larger family of systems and the optimization problem is formulated for the latter. It is shown that the set of trajectories of the switching system is dense in the set of trajectories of the embedded system. The relationship between the two sets of trajectories (1) motivates the shift of focus from the original problem to the more general one and (2) underlies the engineering relevance of the study of the second problem. Sufficient and necessary conditions for optimality are formulated for the second optimization problem. If they exist, bang–bang-type solutions of the embedded optimal control problem are solutions of the original problem. Otherwise, suboptimal solutions are obtained via the Chattering Lemma.

Introduction

This paper formulates sufficient and necessary conditions for solutions of an optimal control problem for a two-switched system. A control for such a system is a triplet (Sussmann, 1999) consisting of: (i) a sequence of switching instants; (ii) a sequence of modes; (iii) a sequence of control inputs (each control input function being associated with a mode). No constraints are imposed on the switching and the performance index contains no penalty on the switching. The development consists of four main steps: (i) the switched system is embedded into a larger family of systems and the optimal control problem is formulated for the larger family (Section 1); (ii) a relaxation of the embedded problem, and relationships between solutions of the switched optimal control problem and the embedded optimal control problem, are set forth (Section 3); (iii) sufficient conditions for the embedded optimal control problem are shown to be satisfied for a large class of nonlinear systems (namely, for systems linear in the control), and for other classes of systems it is shown that the relaxed problem always has a solution (Section 4); (iv) necessary conditions for optimality of a solution of the embedded optimal control problem are developed using the Maximum Principle (Section 5). When the necessary conditions indicate a bang–bang-type of solution, one obtains a solution to the original problem. In the cases when a bang–bang-type optimal switching solution does not exist, the solution to the embedded optimal control problem (assuming that it exists) can be approximated arbitrarily close by the trajectory of the switched system generated by an appropriate switching control. Section 6 illustrates the extension of the result to a three-switched system, and thus establishes a pattern for higher order extensions. Section 7 provides an example with three subcases showing the application of the theoretical development. The advantage of the formulation is that optimal and suboptimal solutions are obtained without imposing restrictions on the mode sequence or the number of mode switchings.

A wealth of literature is available for the modeling and control of hybrid systems that decompose into three main classes: (i) general/abstract models that allow for virtually all types of hybrid behavior as in Branicky, Borkar, and Mitter (1998), Cassandras, Pepyne, and Wardi (2001), Miller and Rubinovich (2003), Piccoli (1998) and Sussmann (1999); (ii) switched systems with vector fields restricted to Rn as in Bemporad and Morari (1999), Giua, Seatzu, and Van Der Mee (2001), Hedlund and Rantzer (1999), Riedinger, Kratz, Iung, and Zanne (1999b), Seidman (1987) and Xu and Antsaklis, 2000a, Xu and Antsaklis, 2003; and (iii) specific application-driven models such as those for hybrid electric vehicles and those for manufacturing as in Pepyne and Cassandras (2000). The formulation of this paper falls into class (ii).

In the general case, automata models are merged with continuous/discrete time models. The hybrid optimal control problems of Branicky et al. (1998), Cassandras et al. (2001), Miller and Rubinovich (2003), Piccoli (1998), and Sussmann (1999) have led to generalizations of the Maximum Principle.

The family of hybrid systems in category (ii) have models that are less general, but, for certain optimization problems, there exist numerically sound algorithms for obtaining suboptimal solutions. For example, one can use a hybrid version of the Bellman inequality, as in Hedlund and Rantzer (1999), or a special type of quadratic programming, as in Bemporad and Morari (1999), which involves both continuous and integer variables. Industrial experience and simulations show that the performance of the solutions obtained by means of these methods are satisfactory. An algorithm based on a dynamic programming approach is proposed in Xu and Antsaklis, 2000b, Xu and Antsaklis, 2003. The algorithm is based on a two-stage optimization problem. In the first stage, suboptimal solutions are obtained by considering fixed switching sequences. For a pre-assigned switching sequence (fixed number of switching and fixed mode sequence), a method for solving an optimization problem for linear autonomous systems and quadratic costs is studied in Giua et al. (2001). Using an analytical derivation of the cost functional, the optimal control is obtained in state feedback form. A general setup uses viscosity solution techniques: the value functions associated with each mode of operation are shown to be solutions of a system of quasivariational inequalities in a weak sense; the optimal controls are then computed using PDE methods. Such an approach is used in Capuzzo Dolcetta and Evans (1984) for solving an optimization problem associated with a switched system with time-invariant autonomous subsystems. A general optimal hybrid control problem which enables a direct Maximum Principle approach was presented in Riedinger et al. (1999b). However, this work discusses neither sufficient conditions for optimality nor the singular cases. In our paper, we discuss these issues for a two-switched system.

In this work, using the embedding, we: (i) provide a rigorous development of sufficient and necessary conditions; (ii) formulate a rigorous investigation of both regular and singular solutions, capturing a number of results in the literature, with no explicit assumptions on the number of mode switches or the mode sequence; (iii) provide a constructive method for suboptimal trajectories.

Section snippets

The two-switched system case

The two-switched system model adopted in this paper has system state x(t)Rn at time t with dynamicsx˙(t)=fv(t)(t,x(t),u(t)),x(t0)=x0,where at each tt0, v(t){0,1} is the switching control, u(t)ΩRm is the usual control input constrained to the convex and compact set Ω, and f0 and f1 are real vector-valued functions, f0,f1:R×Rn×RmRn, of class C1. The control inputs, v and u, are both measurable functions. The state of the system described by Eq. (1) does not undergo jump discontinuities.

Relationships between trajectories of the switched and embedded systems

In this section, it is shown that the set of trajectories of the switched system (1) is dense in the set of trajectories of the embedded system (3). If a trajectory of the embedded system is obtained for some control inputs v(·) (with values in the interval [0,1]), u0(·) and u1(·) in Eq. (3), then the goal is to show that this trajectory can be approximated within any desired accuracy, ε, by a trajectory of the switched system (1) corresponding to a proper choice of the switching input vε(·),

Sufficient conditions for existence of a solution of the EOCP

In this section, our goal is to apply Theorem 5.1, p. 61, in Berkovitz (1974), to obtain sufficient conditions for the existence of an optimal solution to the embedded control problem. (Different perspectives in a more general context can be found in Branicky et al. (1998).) Among the conditions required by the previously mentioned theorem, condition (v) requires that for all (t,x) a certain set Q+ (defined below) be convex; this is difficult to meet. Nevertheless, there is class of embedded

Necessary conditions for system (1)

In deriving necessary conditions that a solution of the EOCP must satisfy, the following lemma is useful.

Lemma 10

IfXRnis a compact set andf,g:XRare continuous functions then the following hold:

  • (L1)

    Let(μ,ρ)[0,1]×Xsatisfyμf(ρ)+g(ρ)=max(μ,ρ)[0,1]×Xμf(ρ)+g(ρ). Thenf(ρ)0μ{0,1}.

  • (L2)

    max(μ,ρ)[0,1]×Xμf(ρ)+g(ρ)=max{maxρX[f(ρ)+g(ρ)],maxρXg(ρ)}.

  • (L3)

    Letρ1Xsatisfyf(ρ1)+g(ρ1)=maxρX[f(ρ)+g(ρ)]>maxρXg(ρ). Thenf(ρ1)>0 andf(ρ1)+g(ρ1)>μf(ρ)+g(ρ)for allμ[0,1)and for allρX.

  • (L4)

    Letρ2XsatisfymaxρX1·f(ρ)+g(ρ)<max

Extension to three-mode switched systems

To extend the above development to three modes of operation suppose that the mode dynamics are: (i) x˙(t)=f00(t,x,u); (ii) x˙(t)=f01(t,x,u); (iii) x˙(t)=f1(t,x,u). Similar to Eq. (1), but with the switching controls v0,v1{0,1}, the switched system dynamic has the nested form:x˙(t)=[1-v1(t)]·{[1-v0(t)]·f00(t,x,u)+v0(t)·f01(t,x,u)}+v1(t)·f1(t,x,u).Similar to Eq. (3), the embedded system dynamic becomesx˙(t)=[1-v1(t)]·{[1-v0(t)]·f00(t,x,u00)+v0(t)·f01(t,x,u01)}+v1(t)·f1(t,x,u1)with (v0,v1)[0,1]×[

Examples

Solving the necessary conditions stated in Section 5, in conjunction with the employment of Propositions 3.1 and 3.2 in (and implicitly Theorem 1 and the Chattering Lemma) provides a method for obtaining at least suboptimal solutions of the SOCP. One general approach may consist of the following: (a) in the family of extremal trajectories for which the set T has measure zero, regular trajectories, determine (if any) the trajectories that have the smallest cost functional; (b) in the family of

Conclusions

This paper delineates a method for solving a hybrid optimal control problem formulated for a two-switched system. The embedding approach that is taken can be seen as a natural evolution of the framework set forth by Riedinger et al. (1999b). The relationship that exists between the set of trajectories of the switched and the set of trajectories of the embedded system (1) motivates the study of sufficient and necessary conditions for the embedded optimal control problem and (2) underlies the

Acknowledgements

The authors thank professor Leonard D. Berkovitz, Purdue University, for illuminating discussions that help clarify some aspects of this paper. The authors also thank the anonymous referees whose suggestions led to paper improvements.

Raymond DeCarlo, a native of Philadelphia, PA, received a B.S. and M.S. in Electrical Engineering from the University of Notre Dame in 1972 and 1974, respectively. In 1976, he received his Ph.D. under the direction of Dr. Richard Saeks from Texas Tech University. His doctoral research centered on Nyquist Stability Theory with applications to multidimensional digital filters. He taught at Texas Tech for one year before becoming an Assistant Professor of Electrical Engineering at Purdue

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    A characterization of the solutions of the embedded problem (16) can also be derived by applying the maximum principle. The analysis in this section is based on the results obtained by Bengea and DeCarlo (2005) for a more general class of switched systems. Bengea et al. (2011) provided a set of necessary and sufficient conditions that guarantee the existence of a solution to an embedded problem.

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Raymond DeCarlo, a native of Philadelphia, PA, received a B.S. and M.S. in Electrical Engineering from the University of Notre Dame in 1972 and 1974, respectively. In 1976, he received his Ph.D. under the direction of Dr. Richard Saeks from Texas Tech University. His doctoral research centered on Nyquist Stability Theory with applications to multidimensional digital filters. He taught at Texas Tech for one year before becoming an Assistant Professor of Electrical Engineering at Purdue University in the Fall of 1977 and an Associate Professor in 1982. He worked at the General Motors Research Laboratories during the summers of 1985 and 1986. He is a Fellow of the IEEE (1989), past Associate Editor for Technical Notes and Correspondence and past Associate Editor for Survey and Tutorial Papers, both for the IEEE Transactions on Automatic Control. He received the CSS distinguished member award in 1990 and the IEEE Third Millennium Medal in 2000. He has held various administrative positions within the CSS and has been a member of its the Board of Governors. He was Program Chairman for the 1990 IEEE Conference on Decision and Control, and was General Chairman in 1993. He has written three books and has over 100 publications in conferences and journals. His research interests include variable structure control, modeling the software development process, biomedical modeling and control, diesel engine control, hybrid electric vehicle control, hybrid and discrete event systems, decentralized control of large scale systems, analog and analog-digital fault diagnosis of circuits and systems, and digital and active filter design.

Sorin Bengea earned a B.S. and M.S. in Automatic Controls and Systems Engineering from Polytechnic University of Bucharest, Romania, in 1998 and 1999, respectively. He also earned a M.Sc. degree in Mathematics and a Ph.D. degree in Electrical and Computer Engineering from Purdue University, West Lafayette, IN, in 2003 and 2004, respectively. He is with the Innovation Center, Eaton Corporation, Eden Prairie, MN. His current research interests are in the broad area of hybrid system control.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Tongwen Chen under the direction of Editor I. Petersen.

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