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# 107_notes03 - MAE107 Introduction of Modeling and Analysis...

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MAE107 Introduction of Modeling and Analysis of Dynamic Systems Lecture Notes #3 Prof. M’Closkey More on transfer functions, solutions to IVP’s with inputs of the form u ( t ) = e st Block diagram manipulation Textbook reading: 1. Section 8.9 introduces transfer functions as a means of describing the particular solutions of linear ODEs when the input is of exponential form, i.e. u ( t ) = e st for s C . 2. The introduction to Chapter 14, 14.1 and 14.2 give more detail on transfer func- tions.

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Transfer functions, poles, and zeros Recall the n th order linear, time-invariant ODE d n dt n ( x ( t ))+ a 1 d n - 1 dt n - 1 ( x ( t )) + · · · + a n - 1 d dt ( x ( t )) + a n x ( t ) = b 1 d n - 1 dt n - 1 ( u ( t )) + · · · + b n - 1 d dt ( u ( t )) + b n u ( t ) , (1) The transfer function was motivated by consider inputs of the form u ( t ) = e st , s C . (2) By assuming a particular solution of the form x p ( t ) = ce st , where c C is to be determined, we found c = b 1 s n - 1 + b 2 s n - 2 + · · · + b n - 1 s + b n s n + a 1 s n - 1 + a 2 s n - 2 + · · · + a n - 1 s + a n . (3) when s is not equal to a characteristic root. This is the system’s transfer function and it has greater significance beyond what we have demonstrated so far (more detail when we discuss Laplace transforms). Note that the condition that s not be equal to a characteristic root is only so that the assumed form of the particular solution actually satisfies the ODE, and in this case the particular solution is merely a scaled version of the input (the scaling factor being the transfer function value at s ). On the other hand, if s is equal to a characteristic root, then the particular solution is not of the assumed form, however, a unique solution to the IVP still exists of course. The transfer function can be denoted in several ways: 1. when it is useful to note the input function and the dependent variable explicitly, the transfer function can be denoted x/u or x u , where x and u represent the dependent variable and input function, respectively. 2. when a transfer function is shown in a block diagram it is useful to give it a generic symbol like “ H ”, “ G ”, “ P ”, etc. For example, 1
- u H - y If we treat the transfer function as a ratio of polynomials, then we see that the denomi- nator polynomial is actually the characteristic polynomial of the ODE whose roots (the characteristic roots) determine the nature of the homogeneous solutions of the ODE. There are two important definitions pertaining to the denominator and numerator poly- nomials: 1. Poles: The poles of a transfer function are the roots of s n + a 1 s n - 1 + a 2 s n - 2 + · · · + a n - 1 s + a n . In other words, the poles are the characteristic roots of the ODE. 2. Zeros: The zeros of a transfer function are the roots of b 1 s n - 1 + b 2 s n - 2 + · · · + b n - 1 s + b n . The zeros provide values of s C such that x p ( t ) = 0 .

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107_notes03 - MAE107 Introduction of Modeling and Analysis...

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