Research Papers

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

[+] Author and Article Information
Brian C. Fabien

Department of Mechanical Engineering,
University of Washington,
Seattle, WA 98195
e-mail: fabien@uw.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received February 13, 2013; final manuscript received September 17, 2013; published online November 7, 2013. Assoc. Editor: Fu-Cheng Wang.

J. Dyn. Sys., Meas., Control 136(2), 021003 (Nov 07, 2013) (14 pages) Paper No: DS-13-1073; doi: 10.1115/1.4025596 History: Received February 13, 2013; Revised September 17, 2013

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.

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Fig. 1

Example 1, simplified trolley cable system

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Fig. 2

Example 1, continuation method convergence

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Fig. 3

Example 1, states, costates, control, and multipliers

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Fig. 4

Example 2, rigid body in a vertical plane

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Fig. 5

Example 2, continuation method convergence

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Fig. 6

Example 2, states and controls

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Fig. 7

Example 2, costates and multipliers

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Fig. 8

Example 3, R-P robot

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Example 3, continuation method convergence

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Fig. 10

Example 3, states and controls

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Fig. 11

Example 3, costates and multipliers




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