An iterative method for time optimal control of dynamic systems

Abstract

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.

Time-optimal control with direct collocation and variable discretization

“…Note that ∆t max might be determined by analyzing the system dynamics a-priori. To ensure that a solution to the optimal control problem exists, assume that there is an N > 0 for which (7) is feasible. In addition, necessary and sufficient optimality conditions for general nonlinear programs apply [22].…”

“…. , N − 1 to (7). Although the optimal solution is identical, the structure of the optimization problem differs slightly.…”

“…the switching times. An iterative indirect method is proposed in [7] with a dedicated initialization strategy to account for the difficulties in initializing the problem, which is generally a known problem for indirect methods. An approximate time-optimal control in the arc time space for single-input nonlinear systems is presented in [8].…”

Solving nonlinear optimal path tracking problem using a new closed loop direct–indirect optimization method: application on mechanical manipulators

“…A dynamic system with nonlinear state equation consider as (1). Also, the performance index function and its nonlinear term are defined as (2) and (3).…”

“…In ref. [2], an iterative method for time optimal control of a general type of dynamic systems is proposed. The method uses the indirect solution of the open-loop optimal control problem.…”