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Session 1566

Symbolic Algebra in Dynamic Systems and Controls Classes Joey K. Parker The University of Alabama

Abstract

Large sets of symbolic simultaneous linear equations occur frequently in the types of problems found in system dynamics and control courses. Students often have difficulty with algebraic manipulation of several symbolic equations. Three example problems (finding state variable equations for an electric circuit, developing transfer functions from sets of state variable equations, and block diagram reduction) show how symbolic algebra can be used to reduce tedious algebraic manipulation in system dynamics and control courses.

Introduction

Many mechanical engineering undergraduate programs include courses in dynamic system modeling and control, either separately or within a single course. Mechanical, electrical, and thermal systems with significant dynamic components are frequently modeled. Combinations of electrical and mechanical or electrical and thermal systems are also common. Modeling of these systems results in a small number of differential equations (typically linear, first-order, constant coefficients), with another larger set of related algebraic equations. These systems of equations must be reduced by algebraic manipulation to the proper form, typically state variable equations or transfer functions. Students often have difficulty with symbolic manipulation at this level of complexity - up to five or so "knowns" and five to ten "unknowns." The ability to manipulate these symbols algebraically is not indicative of student understanding of the fundamental concepts of dynamic systems and controls. In fact, many students spend most of their effort on such problems doing algebraic manipulations and are never able to move beyond this stage of solving a systems dynamics problem.

Modern computational systems, such as Matlab and to a lesser degree MathCad and Maple, are often used to solve problems in dynamic system/control system modeling and analysis. A number of textbook supplements 1,2,3,4,5 have been written that use Matlab for numerical solutions in control system design and analysis. An excellent set of tutorials with extensive examples using Matlab in control system design has been widely available for a few years 6. Matlab’s Simulink 7 and Visual Solutions/Mathsoft’s Micro-VisSim 8 have graphical interfaces that allow users to build control system block diagrams and perform numerical solutions on the modeled system. Gerber 9 reports the use of Maple to solve numerical problems in electric circuit analysis. A set of examples showing Maple used in both analog and linear control system analysis including inverse Laplace transforms is available on the Adept Scientific website 10. A set of Maple routines useful for classical frequency domain (Nyquist and Bode plots, root locus) analysis and linear regulator and stochastic optimal controller designs is also available 11.

Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition Copyright  2001, American Society for Engineering Education

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Parker, J. (2001, June), Symbolic Algebra In Dynamic Systems And Controls Classes Paper presented at 2001 Annual Conference, Albuquerque, New Mexico. 10.18260/1-2--9834