Rapid Solution of Optimal Control Problems by a Functional Spreadsheet Paradigm: A Practical Method for the Non-Programmer

Chahid Ghaddar

doi:10.3390/mca23040054

Abstract

We devise a practical and systematic spreadsheet solution paradigm for general optimal control problems. The paradigm is based on an adaptation of a partial-parametrization direct solution method which preserves the original mathematical optimization statement, but transforms it into a simplified nonlinear programming problem (NLP) suitable for Excel NLP solver. A rapid solution strategy is implemented by a tiered arrangement of pure elementary calculus functions in conjunction with Excel NLP solver. With the aid of the calculus functions, a cost index and constraints are represented by equivalent formulas that fully encapsulate an underlining parametrized dynamical system. Excel NLP solver is then employed to minimize (or maximize) the cost index formula, by varying decision parameters, subject to the constraints formulas. The paradigm is demonstrated for several fixed and free-time nonlinear optimal control problems involving integral and implicit dynamic constraints with direct comparison to published results obtained by fundamentally different methods. Practically, applying the paradigm involves no more than defining a few formulas using basic Excel spreadsheet skills.

Highlights

Many researchers and academics often need to solve optimal control problems that are frequently postulated in various engineering, social, and life sciences [1,2,3]
Excel nonlinear programming problem (NLP) solver is invoked from the Data tab on Excel Ribbon and displays a dialog to enter the problem objective, variables and constraints
The strategy is based on an adaptation of the partial-parametrization direct solution solution method which preserves the structure of the original mathematical optimization statement, method which preserves the structure of the original mathematical optimization statement, but but transforms it into a simplified NLP problem suitable for Excel NLP solver

Summary

Introduction

Many researchers and academics often need to solve optimal control problems that are frequently postulated in various engineering, social, and life sciences [1,2,3]. An optimal control problem is concerned with finding control functions, (or policies), that achieve optimal trajectories for a set of controlled differential state variables. The optimal trajectories are determined by solving a constrained dynamical optimization problem, such that a cost index is minimized (or maximized), subject to constraints on state variables and control functions. An optimal control problem may be stated generally as follows (bold symbols indicate vector-valued functions): Find control functions u(t) = (u1 (t), u2 (t), . Xn (t)), t ∈ [t0 , t F ] which minimize (or maximize) the cost index.

Objectives

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Results

Conclusion