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__Short Desciption__:

Control System Theory in Maple
by
Justin M.J. Wozniak, What is control theory? Control Systems Theory deals with the basic principles underlying the analysis and design of control systems. To “control” an object means to influence its behavior so as to achieve a desired goal. In order to implement this influence, engineers build devices that incorporate various mathematical techniques. ( control systems) A branch of the theory of control systems concerned with the special problems that arise in the design of control algorithms (that is, control policies and strategies) for complex systems. A branch of control theory concerned with large-scale systems. Control theory is an interdisciplinary branch of engineering and mathematics, that deals with the behaviour of dynamical systems. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain reference over.

__Summary__:

Control system theory is abroad field that uses many different areas of mathematics. Differential equations, matrix manipulation, complex functions and integral transforms are just some of the mathematical tools commonly used inmost basic control problems. The Maple computer algebra system is an excellent tool to use when approaching these types of problems, and the engineer attempting to solve a control problem with Maple would have access to all the functionality necessary. However, Maple was not designed primarily for engineering applications, and as a result engineers often use other more specialized software to meet their needs. The typical modern control system engineer has probably used a system that emphasizes floating point computation, often in matrix-vector form. Older control software written in Fortran or C++would fit into this category, as would modern MATLAB control software. Such systems tend to emphasize the state-space models of control systems in a floating point domain. In this thesis we examine the use of the computer algebra system Maple for control system theory. Maple is an excellent system for the manipulation of expressions used by control system engineers, but so far it is not widely used by the public for this purpose. Many of the algorithms in Maple were implemented because their application in control theory is known, but this functionality has not been compiled in a meaningful way for engineers. Over the past two years we have investigated what functionality control system engineers need and have attempted to fill in the gaps in Maple’s functionality to make it more useful to engineers. I would like to thank my supervisor, Dr. George Labahn, for his assistance and guidance in preparing this thesis. I also extend thanks to the readers, Dr. Justin Wan and Dr. Keith Geddes, whom I met in the classrooms of this university. A special debt of gratitude is owed to Venus, who accompanied me on our extended honeymoon in Waterloo this summer. Thank you for your patience during this project. I would like to thank Shannon Puddister for his input regarding the user interface from an engineering perspective. Thanks also to the Symbolic Computation Group, Waterloo Maple and ORCCA, for financial and academic support. n general, Maple’s floating point functionality is slower than that of competing software, but its polynomial arithmetic and related functionality make up the well- established, mature parts of the system. This indicates that Maple is best suited to operate on polynomial models of control systems in exactarithmetic. In this thesis we have undertaken two projects. The first is to implement in Maple the basic control system functionality that would benefit the control system engineer. This includes convenient plots, graphics, and engineering operations commonly used in control theory. The second project examines some algorithms already in Maple that are useful to the engineer, and report on how they help or hinder control engineering. We have approached the first project by designing and implementing the CST package in Maple. This package has a library of software algorithms commonly used by control engineers. It also has a Maplet that provides easy access to the graphical routines in the library. We have approached the second project by investigating two specific computational problems: solving the homogenous 1stordermatrix differential equation and eigen value assignment. 1.1 Symbolic Computation and Control System Theory. Existing Software for Control System Theory We begin with an examination of popular software tools for the analysis of control systems. We have chosen to examine three general purpose software systems which offer potential benefit to the control system engineer. 1.2.1 MATLAB MATLAB is a popular mathematical programming language based on matrix computation in the floating point domain. MATLAB also offers certain Maple functionality through the”Symbolic Toolbox”. It offers an interactive command-line interface, and provides a variety of plotting and graphical functionality in pop-up windows. MATLAB consists of the main program, with a variety of Toolboxes associated with it that maybe used for different applications. It is probably the most common choice for engineers working with control systems. The MATLAB Control System Toolbox allows for the creation and manipulation of Linear Time Invariant (LTI) objects, a MATLAB datatype. These objects maybe instantiated using a transfer function format or a state space model. LTI objects may also be created by pole placement methods or even through graphical. Mathematica is a programming language designed for a variety of mathematical applications in a symbolic context. The kernel of the Mathematica system is designed for list-based functional programming, but has been expanded to support a variety of programming styles. Users often prefer the document-driven interface, which is beneficial for plotting and interactive development, but the kernel is also accessible from the command line, or from other software systems, such as C, World Wide Web interfaces, or even Microsoft Excel. Mathematica is distributed by Wolfram Inc. Wolfram also offers software called the Control System Professional that is distributed separately from Mathematica. This software provides a variety of control functionality and analysis, such as plots and simulation. This functionality is available inline with the document-driven interface, allowing for seamless development with other problems. It allows for symbolic analysis of systems such as solving state equations, numerical system simulation, and classical plotting capabilities like Bode and Nyquist. Other available plotting includes time domain response to impulse, step, and ramp input. Control System Professional also allows for more advanced mathematics, such as Ackermann’s formula for eigenvalue assignment. Traditional and robust pole assignment methods are also available. A variety of system realizations maybe computed, including Jordan and Kalmanforms.

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"Control System Theory in Maple"

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