On Weak Topology for Optimal Control of Switched Nonlinear Systems

Optimal control of switched systems is challenging due to the discrete nature of the switching control input. The embedding-based approach addresses this challenge by solving a corresponding relaxed optimal control problem with only continuous inputs, and then projecting the relaxed solution back to obtain the optimal switching solution of the original problem. This paper presents a novel idea that views the embedding-based approach as a change of topology over the optimization space, resulting in a general procedure to construct a switched optimal control algorithm with guaranteed convergence to a local optimizer. Our result provides a unified topology based framework for the analysis and design of various embedding-based algorithms in solving the switched optimal control problem and includes many existing methods as special cases

Strong Stationarity Conditions for Optimal Control of Hybrid Systems

We present necessary and sufficient optimality conditions for finite time optimal control problems for a class of hybrid systems described by linear complementarity models. Although these optimal control problems are difficult in general due to the presence of complementarity constraints, we provide a set of structural assumptions ensuring that the tangent cone of the constraints possesses geometric regularity properties. These imply that the classical Karush-Kuhn-Tucker conditions of nonlinear programming theory are both necessary and sufficient for local optimality, which is not the case for general mathematical programs with complementarity constraints. We also present sufficient conditions for global optimality. We proceed to show that the dynamics of every continuous piecewise affine system can be written as the optimizer of a mathematical program which results in a linear complementarity model satisfying our structural assumptions. Hence, our stationarity results apply to a large class of hybrid systems with piecewise affine dynamics. We present simulation results showing the substantial benefits possible from using a nonlinear programming approach to the optimal control problem with complementarity constraints instead of a more traditional mixed-integer formulation.Comment: 30 pages, 4 figure

Consistent Approximations for the Optimal Control of Constrained Switched Systems

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

Determination of the adjoint state evolution for the efficient operation of a hybrid electric vehicle

To minimize the fuel consumption in hybrid electric vehicles, it is necessary to define a strategy for the management of the power flows within the vehicle. Under the assumption that the velocity to be developed by the vehicle is known a priori, this problem may be posed as a nonlinear optimal control problem with control and state constraints. We find the solution to this problem using the optimality conditions given by the Pontryagin Maximum Principle. This leads to boundary value problems that we solve using a software tool named PASVA4. On real time operation, the velocity to be developed by the vehicle is not known in advance. We show how the adjoint state obtained from the former problem may be used as a weighing factor, called ‘‘equivalent consumption’’. This weighing factor may be used to design suboptimal real time algorithms for power management.Fil: Perez, Laura Virginia. Universidad Nacional de Rio Cuarto. Facultad de Ingeniería. Grupo de Electronica Aplicada; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: de Angelo, Cristian Hernan. Universidad Nacional de Rio Cuarto. Facultad de Ingeniería. Grupo de Electronica Aplicada; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Pereyra, Víctor. San Diego State University; Estados Unido

Discrete Mechanics and Optimal Control Applied to the Compass Gait Biped

This paper presents a methodology for generating locally optimal control policies for simple hybrid mechanical systems, and illustrates the method on the compass gait biped. Principles from discrete mechanics are utilized to generate optimal control policies as solutions of constrained nonlinear optimization problems. In the context of bipedal walking, this procedure provides a comparative measure of the suboptimality of existing control policies. Furthermore, our methodology can be used as a control design tool; to demonstrate this, we minimize the specific cost of transport of periodic orbits for the compass gait biped, both in the fully actuated and underactuated case

Charging Scheduling of Electric Vehicles with Local Renewable Energy under Uncertain Electric Vehicle Arrival and Grid Power Price

In the paper, we consider delay-optimal charging scheduling of the electric vehicles (EVs) at a charging station with multiple charge points. The charging station is equipped with renewable energy generation devices and can also buy energy from power grid. The uncertainty of the EV arrival, the intermittence of the renewable energy, and the variation of the grid power price are taken into account and described as independent Markov processes. Meanwhile, the charging energy for each EV is random. The goal is to minimize the mean waiting time of EVs under the long term constraint on the cost. We propose queue mapping to convert the EV queue to the charge demand queue and prove the equivalence between the minimization of the two queues' average length. Then we focus on the minimization for the average length of the charge demand queue under long term cost constraint. We propose a framework of Markov decision process (MDP) to investigate this scheduling problem. The system state includes the charge demand queue length, the charge demand arrival, the energy level in the storage battery of the renewable energy, the renewable energy arrival, and the grid power price. Additionally the number of charging demands and the allocated energy from the storage battery compose the two-dimensional policy. We derive two necessary conditions of the optimal policy. Moreover, we discuss the reduction of the two-dimensional policy to be the number of charging demands only. We give the sets of system states for which charging no demand and charging as many demands as possible are optimal, respectively. Finally we investigate the proposed radical policy and conservative policy numerically

An hybrid system approach to nonlinear optimal control problems

We consider a nonlinear ordinary differential equation and want to control its behavior so that it reaches a target by minimizing a cost function. Our approach is to use hybrid systems to solve this problem: the complex dynamic is replaced by piecewise affine approximations which allow an analytical resolution. The sequence of affine models then forms a sequence of states of a hybrid automaton. Given a sequence of states, we introduce an hybrid approximation of the nonlinear controllable domain and propose a new algorithm computing a controllable, piecewise convex approximation. The same way the nonlinear optimal control problem is replaced by an hybrid piecewise affine one. Stating a hybrid maximum principle suitable to our hybrid model, we deduce the global structure of the hybrid optimal control steering the system to the target