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Unformatted text preview: ________________________________________________________________________________ 1 TRANSFER FUNCTIONS AND BLOCK DIAGRAMS 1. Introduction 2. Transfer Function of Linear TimeInvariant (LTI) Systems 3. Block Diagrams 4. Multiple Inputs 5. Transfer Functions with MATLAB 6. Time Response Analysis with MATLAB 1. Introduction An important step in the analysis and design of control systems is the mathematical modelling of the controlled process. There are a number of mathematical representations to describe a controlled process: Differential equations : You have learned this before. Transfer function : It is defined as the ratio of the Laplace transform of the output variable to the Laplace transform of the input variable, with all zero initial conditions. Block diagram : It is used to represent all types of systems. It can be used, together with transfer functions, to describe the cause and effect relationships throughout the system. Statespacerepresentation : You will study it in the Control Systems Design course. 1.1. Linear TimeVariant and Linear TimeInvariant Systems Definition 1 : A timevariable differential equation is a differential equation in which one or more terms depend explicitly on the independent variable time, t. For example, the differential equation: ) t ( u ) t ( y dt ) t ( y d t 2 2 2 = + where u and y are dependent variables, is timevariable since the term t 2 d 2 y/dt 2 depends explicitly on t through the coefficient t 2 . Systems that are represented by differential equations whose coefficients are functions of time are called timevarying systems . An example of a timevarying control system is a spacecraft control system which the mass of spacecraft changes due to fuel consumption. Definition 2 : A timeinvariant differential equation is a differential equation in which none of the terms depends explicitly on the independent variable, t. For example, the differential equation: ) t ( u ) t ( y dt ) t ( dy b dt ) t ( y d m 2 2 = + + ________________________________________________________________________________ 2 where the coefficients m and b are constants, is timeinvariant since the equation depends only implicitly on t through the dependent variables y and u and their derivatives. Dynamic systems that are described by linear, constantcoefficient, differential equations are called linear timeinvariant (LTI) systems . 2. Transfer Function of Linear TimeInvariant (LTI) Systems The transfer function of a linear, timeinvariant system is defined as the ratio of the Laplace transform of the output (response function), Y(s) = ? {y(t)}, to the Laplace transform of the input (driving function) U(s) = ? {u(t)}, under the assumption that all initial conditions are zero....
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 Winter '08
 Gupta

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