A numerical method for the optimal control of switched systems (original) (raw)
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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.
Optimal control of switched systems: new results and open problems
Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334), 2000
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 Hybrid Systems: A Brief Survey
2013
This paper surveys recent results in the field of optimal control of hybrid and switched systems. We summarize recent results that use different problem formulations and then explore the underlying relations among them. Based on the types of switching, we focus on two important classes of problems: internally forced switching (IFS) problems and externally forced switching (EFS) problems. For IFS problems, we focus on the optimal control techniques for piecewise affine systems. For EFS problems, the optimal control of autonomous and non-autonomous switched systems are investigated. Optimization methods found in the literature are discussed.
An Approximations Based Approach to Optimal Control of Switched Dynamic Systems
Mathematical Problems in Engineering, 2014
Our paper deals with a new computational approach to optimal control design for a class of switched systems. The control strategy we propose is based on the conventional proximal point method applied to a specific optimal control problem (OCP) with switched dynamics. The class of OCPs under consideration is widely applicable in optimization of real-world electronic systems. We create constructive approximations for the initially given sophisticated OCP, establish numerical stability (consistency) of the resulting algorithm, and develop an optimal control strategy. We finally discuss some implementability issues, study an illustrative example, and also point to possible generalizations of the elaborated control design in the context of nonlinear hybrid systems.
Optimal control of switched dynamical systems under dwell time constraints
53rd IEEE Conference on Decision and Control, 2014
This paper addresses the problem of optimally scheduling the mode sequence and mode duration for switched dynamical systems under dwell time constraints that describe how long a system has to stay in a mode before they can switch to another mode. The schedule should minimize a given cost functional defined on the state trajectory. The topology of the optimization space for switched dynamical systems with and without dwell time constraints is investigated and it is shown that the notion of local optimality must be replaced by stationarity with regards to a suitably chosen optimality function when dwell time constraints are present. Hence, an optimality function is proposed to characterize the solution to the dwell time problem as points that satisfy optimality condition defined in terms of optimality function. A conceptual algorithm is presented to solve the mode scheduling problem and its convergence to stationary points is proved. A numerical example is given to highlight the algorithm.
Optimal mode-switching for hybrid systems with varying initial states
Nonlinear Analysis: Hybrid Systems, 2008
This paper concerns a particular aspect of the optimal control problem for switched systems that change modes whenever the state intersects certain switching surfaces. These surfaces are assumed to be parameterized by a finite dimensional switching parameter, and the optimization problem we consider is that of minimizing a given cost-functional with respect to the switching parameter under the assumption that the initial state of the system is not a priori known. We approach this problem from two different vantage points by first minimizing the worst possible cost over the given set of initial states using results from min-max optimization. The second approach is based on a sensitivity analysis in which variational arguments give the derivative of the switching parameters with respect to the initial conditions. *
Hybrid Approach for Constrained Optimal Control of Nonlinear Switched Systems
Journal of Control, Automation and Electrical Systems
This paper examines the problem of the optimal control of nonlinear switched systems under mixed constraints. The solution to this problem consists mainly in determining two major factors; on the one hand, we need to determine the optimal control input, and on the other one, we have to find out the best switching instants that lead to minimizing a particular functional cost. In order to reach this objective, we use a hybrid approach that consists in dividing the problem in two stages. In the first one, we employ the method of Lagrange multipliers while considering the Karush-Kuhn-Tucker (KKT) conditions. This method must be accompanied by the bang-bang control. This combination of these optimization methods enables us to obtain the optimal control input while respecting the simultaneously imposed constraints on the state and the input. In the second stage, a metaheuristic method is used to define the optimum switching instants, and this is what we call the particle swarm optimization (PSO). In order to approve this approach, different examples are applied in this study.
Optimal Control of Switched Systems Based on Parameterization of the Switching Instants
IEEE Transactions on Automatic Control, 2004
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.
Optimal Control of Switching Surfaces in Hybrid Dynamical Systems
Discrete Event Dynamic Systems, 2005
This paper concerns an optimal control problem defined on a class of switched-mode hybrid dynamical systems. The system's mode is changed (switched) whenever the state variable crosses a certain surface in the state space, henceforth called a switching surface. These switching surfaces are parameterized by finite-dimensional vectors called the switching parameters. The optimal control problem is to minimize a cost functional, defined on the state trajectory, as a function of the switching parameters. The paper derives the gradient of the cost functional in a costate-based formula that reflects the special structure of hybrid systems. It then uses the formula in a gradient-descent algorithm for solving an obstacle-avoidance problem in robotics.
A provably convergent algorithm for transition-time optimization in switched systems
2005
This paper concerns a mode-sequencing and switching-time optimization problem defined on autonomous switched-mode hybrid dynamical systems. The design parameter consists of two elements: (i) the sequence of dynamicresponse functions associated with the modes, and (ii) the duration of each mode. The sequencing element is a discrete parameter which may render the problem of computing the optimal schedule exponentially complex. Therefore we are not seeking a global minimum, but rather a local solution in a suitable sense. To this end we endow the parameter space with a local continuous structure which allows us to apply gradientdescent techniques. With this structure, the problem is cast in the form of a nonlinear-programming problem defined on a sequence of nested Euclidean spaces with increasing dimensions. We characterize suboptimality in an appropriate sense, define a corresponding convergence criterion, and devise a provablyconvergent optimization algorithm.