In this paper we propose a procedure for synthesizing piecewise linear optimal controllers for hybrid systems and investigate conditions for closed-loop stability. Hybrid systems are modeled in discrete-time within the mixed logical dynamical (MLD) framework[8], or, equivalently [7], as piecewise affine (PWA) systems. A stabilizing controller is obtained by designing a model predictive controller (MPC), which is based on the minimization of a weighted 1/∞-norm of the tracking error and the input trajectories over a finite horizon. The control law is obtained by solving a mixed-integer linear program (MILP) which depends on the current state. Although efficient branch and bound algorithms exist to solve MILPs, these are known to be NP-hard problems, which may prevent their on-line solution if the sampling-time is too small for the available computation power. Rather than solving the MILP on line, in this paper we propose a different approach where all the computation is moved off line, by solving a multiparametric MILP (mp-MILP). As the resulting control law is piecewise affine, on-line computation is drastically reduced to a simple linear function evaluation. An example of piecewise linear optimal control of the heat exchange system [16] shows the potential of the method.

In this paper, optimal control problems for switched autonomous systems are studied. In particular, we fo-cus on problems in which a prespecified sequence of ac-tive subsystems is given and propose an approach to finding the optimal switching instants. The approach derives the derivatives of the cost with respect to the switching instants and uses nonlinear optimization tech-niques to locate the optimal switching instants. The ap-proach is then applied to general quadratic problems for switched linear autonomous systems and to reachability problems. Examples illustrate the results. 1

The paper deals with the optimal control of switched piecewise linear autonomous systems, where the objective is that of minimizing a quadratic performance index over an infinite time horizon. We assume that the switching sequence and the corresponding jump matrices sequence is known, while the unknown switching times are the optimization parameters. The optimal control for this class of systems, assuming a switching sequence of finite length, takes the form of a homogeneous state feedback, i.e., it is possible to identify a homogeneous region of the state space such that an optimal switch should occur if and only if the present state belongs to this region; we show how such a region can be computed with a numerical procedure. As the number of allowed switches goes to infinity, we study the stability of the system and discuss some preliminary results related to the convergence of the state feedback law. 1

This paper presents an approach for solving optimal control problems of switched systems. In general, in

Abstract — This paper considers an optimal control problem for switched dynamical systems, where the objective is to minimize a cost functional de£ned on the state, and where the control variable consists of the switching times. The gradient of the cost functional is derived on an especially simple form, which lends itself to be directly used in gradient-descent algorithms. This special structure of the gradient furthermore allows for the number of switching points to become part of the control variable, instead of being a given constant. Numerical examples testify to the viability of the proposed approach. I.

This paper presents an approach for solving optimal control problems for switched systems

We consider the problem of finding optimal gaits for a quadruped robot. Paths are sought which minimize the actuation energy required for walking in an attempt to approximate natural motion. The number of possible gaits for a quadruped is quite large when one considers varied orders of leg motion, different liftoff times, and various ground contact combinations for the legs. The problem is treated as a fully nonlinear optimal hybrid path planning problem on a 22dimensional state space. Modeling aspects, our numerical approach, and experimental results are discussed in this paper.

In optimal control problems of switched systems, in general, one needs to nd both optimal continuous inputs and optimal switching sequences, since the system dynamics vary before and after every switching instant. In a previous paper, weproved that an optimal control problem can be posed as a two stage optimization problem under some additional assumptions. In general, the two stage optimization problem is still dicult to solve analytically. In this paper, we develop a search algorithm to explore the solution of the two stage optimization problem and nd useful suboptimal solutions. This algorithm is motivated by the idea of dynamic programming which studies the value functions. The algorithm is used to determine suboptimal solutions for general switched linear quadratic problems. 1 Introduction Aswitched system is a hybrid system that consists of several subsystems and a switching law indicating the active subsystem at each time instant. Examples of switched systems can be found in c...

Abstract. Nonlinear hybrid dynamical systems are the main focus of this paper. A modeling framework is proposed, feedback control strategies and numerical solution methods for optimal control problems in this setting are introduced, and their implementation with various illustrative applications are presented. Hybrid dynamical systems are characterized by discrete event and continuous dynamics which have an interconnected structure and can thus represent an extremely wide range of systems of practical interest. Consequently, many modeling and control methods have surfaced for these problems. This work is particularly focused on systems for which the degree of discrete/continuous interconnection is comparatively strong and the continuous portion of the dynamics may be highly nonlinear and of high dimension. The hybrid optimal control problem is defined and two solution techniques for obtaining suboptimal solutions are presented (both based on numerical direct collocation for continuous dynamic optimization): one fixes interior point constraints on a grid, another uses branch-and-bound. These are applied to a robotic multi-arm transport task, an underactuated robot arm, and a benchmark motorized traveling salesman problem. 1

We present a hierarchical framework for approximately optimal control of redundant manipulators. The plant is augmented with a low-level feedback controller, designed to yield input-output behavior that captures the task-relevant aspects of plant dynamics but has reduced dimensionality. This makes it possible to reformulate the optimal control problem in terms of the augmented dynamics, and optimize a high-level feedback controller without running into the curse of dimensionality. The resulting control hierarchy compares favorably to existing methods in robotics. Furthermore, we demonstrate a number of similarities to �nonhierarchical � optimal feedback control. Besides its engineering applications, the new framework addresses a key unresolved problem in the neural control of movement. It has long been hypothesized that coordination involves selective control of task parameters via muscle synergies, but the link between these parameters and the synergies capable of controlling them has remained elusive. Our framework provides this missing link. © 2005 Wiley Periodicals, Inc. *To whom all correspondence should be addressed.