Variable Control Parameterization for Time-Optimal Problems 1 (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...

Extended variable parameterization method for optimal control

Proceedings. IEEE International Symposium on Computer Aided Control System Design, 2002

An extension of the variable parameterization method is presented. Conditions for switching generation and reduction are analysed. The previously used quasi-Newton algorithm is completed with the Newton method. The transversality conditions are modified. The ideas are illustrated by a numerical example showing the performance of the algorithm also for singular cases. A comparison with direct and indirect collocation approaches is given.

Model Identification Dedicated to the Time-Optimal Control

Proceedings of the 17th IFAC World Congress, 2008, 2008

A model identification procedure is applied to the well known benchmark problem of the pendulum hinged to a cart. There is a dynamical model of the entire system. The PWM control signal and DC motor impact introduced electrically by EMF are included. A concatenation of trajectories collected during several control experiments is used to fit the parameters of the pendulum-cart mathematical model. The identification of model parameters is dedicated to the control goal. Several collected points of trajectories are neglected. The model matching corresponds to intervals.

An iterative method for time optimal control of dynamic systems

Archives of Control Sciences, 2000

An iterative method for time optimal control of a general type of dynamic systems is proposed, subject to limited control inputs. This method uses the indirect solution of open-loop optimal control problem. The necessary conditions for optimality are derived from Pontryagin's minimum principle and the obtained equations lead to a nonlinear two point boundary value problem (TPBVP). Since there are many difficulties in finding the switching points and in solving the resulted TPBVP, a simple iterative method based on solving the minimum energy solution is proposed. The method does not need finding the switching point so that the resulted TPBVP can be solved by usual algorithms such as shooting and collocation. Also, since the solution of TPBVPs is sensitive to initial guess, a short procedure for making the proper initial guess is introduced. To this end, the accuracy and efficiency of the proposed method is demonstrated using time optimal solution of some systems: harmonic oscillator, robotic arm, double spring-mass problem with coulomb friction and F-8 aircraft.

A computational method for a class of non-standard time optimal control problems involving multiple time horizons

Mathematical and Computer Modelling, 2009

In this paper, we consider a class of non-standard time optimal control problems involving a dynamical system consisting of multiple subsystems evolving over different time horizons. Different subsystems are required to reach their respective target sets at different termination times. The goal is to minimize the maximum of these termination times. By introducing a discrete variable to represent the system termination ordering, we reformulate this problem as a discrete optimization problem. A discrete filled function method is developed to solve this discrete optimization problem. For illustration, a numerical example is solved.

Sequential linear programming for design of time-optimal controllers

2007

This paper presents a sequential linear programming approach for the determination of time-optimal controller for nonlinear systems. The sequential linear programming solution is used to update the control profile so as to satisfy the terminal conditions for an assumed maneuver time. A univariant minimization approach which brackets the optimal value of the maneuver time, such as the bisection algorithm is used in an outer loop to converge to the minimum time. The proposed technique is illustrated on two benchmark problems: the attitude control of a spacecraft and the minimum time control of a robot.

The control parametrization enhancing transform for constrained time--delayed optimal control problems

Australian & New Zealand industrial and applied mathematics journal, 2002

The Control Parametrization Enhancing Technique (cpet), is extended to a general class of constrained time-delayed optimal control problems. A model transformation approach is used to convert the time-delayed problem to an optimal control problem involving mixed boundary conditions, but without time-delayed arguments. The cpet is then used to solve this non delayed problem. Two test examples have been solved to illustrate the efficiencies of the cpet for time delayed problems.

Time Optimal Swing-Up of the Planar Pendulum

IEEE Transactions on Automatic Control, 2000

This paper presents qualitative results on the global structure of the time optimal trajectories of the planar pendulum on a cart. This mechanical system is a benchmark to test nonlinear control methods and various papers addressed the problem of computing time optimal open-loop controls. Relying on the theory of optimal synthesis, we provide a discontinuous feedback giving optimal solutions for

Fuel/time optimal control of the benchmark problem

Journal of Guidance, Control, and Dynamics, 1995

Design of fuel/time optimal control of the benchmark two-mass/spring system is addressed in the frequency domain. The optimal control profile is represented as the output of a time-delay filter, where the amplitude of the time-delayed signals are constrainted to satisfy the control bounds. The time delays of the filter are determined by solving a parameter optimization problem that minimizes a weighted fuel/time cost function subject to the constraint that the tune-delay filter cancel all the poles of the system and the control profile satisfies the rigid-body boundary conditions. It is shown that three control structures exist: a three-switch profile corresponding to the time optimal control problem that changes to a six-switch profile corresponding to a cost function that includes a small weight on the fuel. As the weight on the fuel increases beyond a critical value, the control profile changes to a two-switch profile. The value of the critical weight that represents the transition of the control profile from a six-switch to a two-switch control profile is determined.

A Direct Approach for Time Optimal Control Problem with Linear Differential System

The purpose of this paper is to present an approach to solve the time-optimal control problem. While searching the control as a piecewise constant function the optimal control problem is reduced to a nonlinear programming problem. Two examples are presented, in which cases the computation is carried out with the Mathematica software.