Brief paper: Local energy minimization in optimal train control

The control parameterization method is a popular numerical technique for solving optimal control problems. The main idea of control parameterization is to discretize the control space by approximating the control function by a linear combination of basis functions. Under this approximation scheme, the optimal control problem is reduced to an approximate nonlinear optimization problem with a finite number of decision variables. This approximate problem can then be solved using nonlinear programming techniques. The aim of this paper is to introduce the fundamentals of the control parameterization method and survey its various applications to non-standard optimal control problems. Topics discussed include gradient computation, numerical convergence, variable switching times, and methods for handling state constraints. We conclude the paper with some suggestions for future research.

https://www.aimsciences.org/article/exportPdf?id=371849ed-608d-4e1a-9250-2ee3d01d59c1

The control parameterization method for nonlinear optimal control: a survey

Numerical methods for mixed-integer optimal control problems

SUMMARY It has been common practice to find controls satisfying only necessary conditions for optimality, and then to use these controls assuming that they are (locally) optimal. However, sufficient conditions need to be used to ascertain that the control rule is optimal. Second order sufficient conditions (SSC) which have recently been derived by Agrachev, Stefani, and Zezza, and by Maurer and Osmolovskii, are a special form of sufficient conditions which are particularly suited for numerical verification. In this paper we present optimization methods and describe a numerical scheme for finding optimal bang–bang controls and verifying SSC. A straightforward transformation of the bang–bang arc durations allows one to use standard optimal control software to find the optimal arc durations as well as to check SSC. The proposed computational verification technique is illustrated on three example applications. Copyright # 2005 John Wiley & Sons, Ltd.

Optimization methods for the verification of second order sufficient conditions for bang-bang controls

The Forward-Backward Sweep Method is a numerical technique for solving optimal control problems. The technique is one of the indirect methods in which the differential equations from the Maximum Principle are numerically solved. After the method is briefly reviewed, two convergence theorems are proved for a basic type of optimal control problem. The first shows that recursively solving the system of differential equations will produce a sequence of iterates converging to the solution of the system. The second theorem shows that a discretized implementation of the continuous system also converges as the iteration and number of subintervals increases. The hypotheses of the theorem are a combination of basic Lipschitz conditions and the length of the interval of integration. An example illustrates the performance of the method.

Convergence of the forward-backward sweep method in optimal control

An efficient algorithm, called the time-optimal switching (TOS) algorithm, is proposed for the time-optimal switching control of nonlinear systems with a single control input. The problem is formulated in the arc times space, arc times being the durations of the arcs. A feasible switching control, or as a special case bang-bang control, is found using the STC method previously developed by the authors to get from an initial point to a target point with a given number of switchings. Then, by means of constrained optimization techniques, the cost being considered as the summation of the arc times, a minimum-time switching control solution is obtained. Example applications of the TOS algorithm involving second-order and third-order systems are presented. Comparisons are made with a well-known general optimal control software package to demonstrate the efficiency of the algorithm.

Computational method for time-optimal switching control

We introduce and analyze a novel scalarization technique and an associated algorithm for generating an approximation of the Pareto front (i.e., the efficient set) of nonlinear multiobjective optimization problems. Our approach is applicable to nonconvex problems, in particular to those with disconnected Pareto fronts and disconnected domains (i.e., disconnected feasible sets). We establish the theoretical properties of our new scalarization technique and present an algorithm for its implementation. By means of test problems, we illustrate the strengths and advantages of our approach over existing scalarization techniques such as those derived from the Pascoletti---Serafini method, as well as the popular weighted-sum method.

A New Scalarization Technique to Approximate Pareto Fronts of Problems with Disconnected Feasible Sets

We propose a new scalarization technique for nonconvex multiobjective optimization problems and establish its theoretical properties. By combining our new scalarization approach with existing grid ...

A New Scalarization Technique and New Algorithms to Generate Pareto Fronts

We use a Fenchel duality scheme for solving control-constrained linear-quadratic optimal control problems We derive the dual of the optimal control problem explicitly, where the control constraints are embedded in the dual objective functional, which turns out to be continuously differentiable We specifically prove that strong duality and saddle point properties hold We carry out numerical experiments with the discretized primal and dual formulations of the problem, for which we implement powerful existing finite-dimensional optimization techniques and associated software We illustrate that by solving the dual of the optimal control problem, instead of the primal one, significant computational savings can be achieved Other numerical advantages are also discussed

A duality approach for solving control-constrained linear-quadratic optimal control problems ∗

We propose a new duality scheme based on a sequence of smooth minorants of the weighted-l1 penalty function, interpreted as a parametrized sequence of augmented Lagrangians, to solve non-convex con...

C. Y. Kaya

Papers

Computational method for time-optimal switching control

A New Scalarization Technique to Approximate Pareto Fronts of Problems with Disconnected Feasible Sets

A New Scalarization Technique and New Algorithms to Generate Pareto Fronts

A duality approach for solving control-constrained linear-quadratic optimal control problems ∗

A primal–dual penalty method via rounded weighted-ℓ1 Lagrangian duality