Optimal control problems with switching points (original) (raw)
An efficient numerical solution for time switching optimal control problems
2020
In this paper, an efficient computational algorithm for the solution of Hamiltonian boundary value problems arising from bang-bang optimal control problems is presented. For this purpose, at first, based on the Pontryagin’s minimum principle, the first order necessary conditions of optimality are derived. Then, an indirect shooting method with control parameterization, in which the control function is replaced with piecewise constant function with values and switching points taken as unknown parameters, is presented. Thereby, the problem is converted to the solution of the shooting equation, in which the values of the control function and the switching points as well the initial values of the costate variables are unknown parameters. The important advantages of this method is that, the obtained solution satisfies the first order optimality conditions, further the switching points can be captured accurately which is led to an accurate solution of the bang-bang problem. However, solvi...
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.
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 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 Systems based on Bezier Control Points
International Journal of Intelligent Systems and Applications, 2012
This paper presents a new approach for solving optimal control problems for switched systems. We focus on problems in which a pre-specified sequence of active subsystems is given. For such problems, we need to seek both the optimal switching instants and the optimal continuous inputs. A Bezier control points method is applied for solving an optimal control problem which is supervised by a switched dynamic system. Two steps of approximation exist here. First, the time interval is divided into sub-intervals. Second, the trajectory and control functions are approximatedby Bezier curves in each subinterval. Bezier curves have been considered as piecewise polynomials of degree , then they will be determined by control points on any subinterval. The optimal control problem is there by converted into a nonlinear programming problem (NLP), which can be solved by known algorithms. However in this paper the MATLAB optimization routine FMINCON is used for solving resulting NLP.
2017
Optimal control problems involving hybrid binary-continuous control costs are challenging due to their lack of convexity and weak lower semicontinuity. Replacing such costs with their convex relaxation leads to a primal-dual optimality system that allows an explicit pointwise characterization and whose Moreau-Yosida regularization is amenable to a semismooth Newton method in function space. This approach is especially suited for computing switching controls for partial differential equations. In this case, the optimality gap between the original functional and its relaxation can be estimated and shown to be zero for controls with switching structure. Numerical examples illustrate the effectiveness of this approach.
International Journal of Computer Mathematics, 1995
Tn this paper, the shifted Chebyshev polynomial functions approximation IS extended to solve the linear ordinary differential equation of the two-point boundary-value problem. The linear ordinary differential equation of boundaryvalue problems are reduced to the linear functional differential equation of the initial-value problem. A new time-domain approach to the derivation of a Chebyshev transformation matrix is presented. Using the derived Chebyshev transformation matrix together with the Chebyshev integration matrix, the solution of the linear functional ordinary differential equation of initial-value problem can be obtained via shiffed Chebyshev series. Two examplesare given and the satisfactory computational results are compared with those of the exact solution. 2. Properties of shifted Chebyshev series Chebyshev polynomials T;(z) are defined as (Abramowitz and Stegun J964): T;(z)=cos(icos-Iz),-1~z~1 (1)
Optimal control of switching surfaces
2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601), 2004
This paper studies the problem of optimal switching surface design for hybrid systems. In particular, a formula is derived for computing the gradient of a given integral performance cost with respect to the switching surface parameters. The formula reflects the hybrid nature of the system in that it is based on a costate variable having a discrete element and a continuous element. A numerical example with a gradient descent algorithm suggests the potential viability of the formula in optimization.
Optimally switched linear systems
Automatica, 2008
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.