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Applications of Spreadsheets in Education The Amazing Power of a Simple Tool, 2011, 41–63 41 Optimal Control of Dynamical Systems Mark A. Lau 1,∗ and William E. Singhose 2 1 Department of Electrical and Computer Engineering Universidad del Turabo, Gurabo, Puerto Rico, USA 2 The George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology, Atlanta, Georgia, USA CHAPTER 3 ∗ Address correspondence to: Dr. Mark A. Lau, Department of Electrical and Computer Engineering, Universidad del Turabo, Road 189 Km 3.3, Gurabo, Puerto Rico 00778-3030, USA; Tel: (+1) 787-743-7979, Ext. 4174; E-mail: mlau@suagm.edu Abstract: Optimal control techniques have numerous applications in engineering, economics, finance, biology, medicine, and many other fields. In spite of the utility of these techniques, the presentation of the topic of optimal control is normally reserved for graduate studies within specialties (e.g., systems engineering). In this chapter we present some illustrative examples in optimal control whose numerical solutions are obtained using the built-in solver capabilities of spreadsheets. Our hope is that the important and interesting topic of optimal control can be introduced to undergraduate students in a less intimidating manner when spreadsheets are used. Keywords: optimal control, Pontryagin’s minimum principle, dynamical systems. 3.1 Introduction Optimal control theory provides the means to drive a machine as fast as possible. Or, it can provide a controller that moves a machine in the most efficient possible way. Optimal control can even provide the best method for balancing the needs for fast and efficient motion. It is the branch of mathematics that furnishes the conditions or techniques for deriving functions (control laws) so as to optimize a given criterion (cost functional). The foundations of the theory were established in the 1960s thanks to the work of Lev Pontryagin, his collaborators, and Richard Bellman. A typical control problem includes: 1. A cost functional that is a function of state and control variables. 2. A set of differential equations that models the interactions between control variables and state variables. Mark Lau and Stephen Sugden (Eds) All rights reserved – c○2011 Bentham Science Publishers Ltd.
42 Applications of Spreadsheets in Education The Amazing Power of a Simple Tool Lau and Singhose 3. In some optimal control problems, ancillary constraints may be included. For example, fixed end states, energy usage, control saturation, and so on. The solid mathematical foundations that underpin the theory of optimal control have provided the impetus for finding applications in engineering, economics, finance, biology, medicine, among other fields. Only a brief outline of the mathematical background will be provided in this chapter; for a more detailed study of the subject the reader is referred to some of the classic books in optimal control [1–3]. In spite of the elegance of its theory and the richness of applications, most undergraduate engineering students receive very limited or no exposure to optimal control. In fact, our experience as instructors reveals that the presentation of this topic is normally reserved for graduate studies within specialties (e.g., systems engineering) that make use of the theory of optimal control. We believe that optimal control theory is filled with profound ideas and that undergraduate engineering students will benefit from the study of this topic. It is our intention in this chapter to offer an accessible introduction to optimal control with the aid of spreadsheets. In particular, the Microsoft Excel Solver tool will be employed to solve optimal control problems. The spreadsheet approach has been used in the teaching of undergraduate economics and business courses [4–6]. In [4], detailed examples of continuous-time systems with first order dynamics are solved using the Solver tool; likewise, [5] uses the same tool to explain the optimization problem in a first order, discrete-time system, while [6] presents spreadsheet implementations for solving dynamic optimization problems. In this chapter we present some illustrative examples dealing with second order, continuous-time systems that might appeal to undergraduate engineering students. The reason for using spreadsheets is that most students are familiar with such tools. Additionally, spreadsheets are widely available and very straightforward to set up, thus making the programming requirement fairly straightforward. The rest of the chapter is organized as follows. Section 3.2 briefly presents the mathematical background for solving optimal control problems. Subsequently, Section 3.3 discusses a number of optimal control examples and solutions using spreadsheets. In Section 3.4, we offer our perspective on classroom use, and finally, some concluding remarks are given in Section 3.5. 3.2 Mathematical Background 3.2.1 Pontryagin’s Minimum Principle The most important component of optimal control is the minimization of a cost function, such as move time or fuel usage. Here we provide a statement of necessary conditions for the minimization of a cost functional. More details can be found in [1–3]. Consider a dynamical system modeled by the ordinary differential equation (with ˙ (·) denoting the derivative with respect to time) ˙x(t)=f t,x(t),u(t) (3.1) together with x(t0)=x0, u(t)∈U , t ∈[t0, t f] (3.2) where f : [t0,t f]×R n × U ↦→ R n is a continuously differentiable function, x(t) ∈ R n is the state vector, and u(t) is an input vector from the set of admissible controls U ⊂ R m . The system is
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