Additional necessary conditions for optimal control with state-variable inequality constraints

1982 ◽  
Vol 38 (2) ◽  
pp. 241-250 ◽  
Author(s):  
E. Kreindler
Keyword(s):  
Optimal Control ◽  
Necessary Conditions ◽  
Inequality Constraints ◽  
State Variable ◽  
Variable Inequality

A Simple Continuation Method for the Solution of Optimal Control Problems With State Variable Inequality Constraints

2013 ◽  
Author(s):  
Brian C. Fabien
Keyword(s):  
Optimal Control ◽  
Pure State ◽  
Original Problem ◽  
Optimal Control Problems ◽  
Necessary Conditions ◽  
Continuation Method ◽  
Inequality Constraints ◽  
Control Problems ◽  
State Variable ◽  
Variable Inequality

This paper develops a simple continuation method for the approximate solution of optimal control problems with pure state variable inequality constraints. The method is based on transforming the inequality constraints into equality constraints using nonnegative slack variables. The resultant equality constraints are satisfied approximately using a quadratic loss penalty function. The solution of the original problem is obtained by solving the transformed problem with a sequence of penalty weights that tends to zero. The penalty weight is treated as the continuation parameter. The necessary conditions for a minimum are written as a boundary value problem involving index-1 differential-algebraic equations (BVP-DAE). The BVP-DAE include the complementarity conditions associated with the inequality constraints. The paper shows that the necessary conditions for optimality of the original problem and the transformed problems are remarkably similar. In particular, the BVP-DAE for each problem differ by a linear term related to the Lagrange multipliers associated with the state variable inequality constraints. Numerical examples are presented to illustrate the efficacy of the proposed technique. Specifically, the paper presents results for; (1) the optimal control of a simplified model of a gantry crane system, (2) the optimal control of a rigid body moving in the vertical plane, and (3) the trajectory optimization of a planar two-link robot. All problems include pure state variable inequality constraints.

Indirect solution of optimal control problems with state variable inequality constraints: finite difference approximation

Robotica ◽  
2015 ◽  
Vol 35 (1) ◽  
pp. 50-72 ◽  
Author(s):  
A. Nikoobin ◽  
M. Moradi
Keyword(s):  
Optimal Control ◽  
Finite Difference ◽  
Inequality Constraints ◽  
Initial Guess ◽  
Control Signal ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Penalty Function Method ◽  
State Variable ◽  
Variable Inequality

SUMMARYIn this paper, a method for the indirect solution of the optimal control problem (OCP) in the presence of pure state variable inequality constraints (SVICs) and mixed state-control inequality constraints (SCIC), without a need for a close initial guess is presented. In the proposed method, using the finite difference approximation (FDA), the pure SVICs are converted to SCIC. Here, the distance of the constraint function to the feasibility bounds of the constraint is computed in every situation and the control signal is chosen appropriately to facilitate the constraint stays safe. In this method, prior knowledge of the numbers and sequences of activation times is not required. So, it can be simply implemented in continuous boundary value problem (BVP) solvers. The proposed method simply applies the SVICs and since the constraint is directly applied on the control signal, it improves the convergence. On the other hand, because of the convergence problem in the indirect solution of OCP, the simple homotopy continuation method (HCM) is used to overcome the initial guess problem by deploying a secondary OCP for which the initial guess can be zero. The proposed approach is applied on a few comprehensive problems in the presence of different constraints. Simulations are compared with the direct solution of the OCP to confirm the accuracy and with the penalty function method and the sequential constraint-free OCP to confirm the convergence. The results indicate that the FDA method for handling the constraints along with the HCM is easy to apply with acceptable accuracy and convergence, even for highly nonlinear problems in robotic systems such as the constrained time optimal control of a two-link manipulator (TLM) and a three-link common industrial robot.

Indirect Solution of Inequality Constrained and Singular Optimal Control Problems Via a Simple Continuation Method

2013 ◽  
Vol 136 (2) ◽  
Author(s):  
Brian C. Fabien
Keyword(s):  
Optimal Control ◽  
Original Problem ◽  
Optimal Control Problems ◽  
Necessary Conditions ◽  
Continuation Method ◽  
Singular Control ◽  
Inequality Constraints ◽  
Control Problems ◽  
Quadratic Loss ◽  
State Variable

This paper develops a simple continuation method for the approximate solution of optimal control problems. The class of optimal control problems considered include (i) problems with bounded controls, (ii) problems with state variable inequality constraints (SVIC), and (iii) singular control problems. The method used here is based on transforming the state variable inequality constraints into equality constraints using nonnegative slack variables. The resultant equality constraints are satisfied approximately using a quadratic loss penalty function. Similarly, singular control problems are made nonsingular using a quadratic loss penalty function based on the control. The solution of the original problem is obtained by solving the transformed problem with a sequence of penalty weights that tends to zero. The penalty weight is treated as the continuation parameter. The paper shows that the transformed problem yields necessary conditions for a minimum that can be written as a boundary value problem involving index-1 differential–algebraic equations (BVP-DAE). The BVP-DAE includes the complementarity conditions associated with the inequality constraints. It is also shown that the necessary conditions for optimality of the original problem and the transformed problem differ by a term that depends linearly on the algebraic variables in the DAE. Numerical examples are presented to illustrate the efficacy of the proposed technique.

Sign in / Sign up

Export Citation Format

Share Document