Appendix A Complex Variable TheoryTO ACCOMPANY AUTOMATIC CONTROL SYSTEMS EIGHTH EDITIONAUTOMATIC CONTROL SYSTEMS EIGHTH EDITIONAppendix CTO ACCOMPANY AUTOMATIC CONTROL SYSTEMS EIGHTH EDITIONElementary Matrix Theory and Algebra (original) (raw)
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Example: Suppose a linear system is represented by the differential equation d 2 y dt 2 + a 1 dy dt + a 0 y = u s 2 Y (s) + a 1 sY (s) + a 0 Y (s) = U (s). Taking Laplace Transforms with zero initial conditions, (s 2 + a 1 s + a 0)Y (s) = U (s) we see that the transfer function is G(s) = Y (s) U (s) = 1 s 2 + a 1 s + a 0. Remarks: – The transfer function is always computed with all initial conditions equal to zero. – The transfer function is the Laplace Transform of the impulse response function. [To see this, set U (s) = 1.] The transfer function of any linear system is a rational function G(s) = n(s) d(s) = b m s m + · · · + b 1 s + b 0 a n s n + · · · + a 1 s + a 0 where n(s), d(s) are the numerator and denominator polynomials of G(s), respectively – G(s) is proper if m = deg n(s) ≤ n = deg d(s) – G(s) is strictly proper if m < n. – For a proper or strictly proper rational function, the difference α = n − m is called the relative degree of the transfer function.
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The transfer function provides a basis for determining important system response characteristics without solving the complete differential equation. As defined, the transfer function is a rational function in the complex variable s = σ + jω, that is )
This book represents a substantial revision of the first edition which was published in 1971. Most of the topics of the original edition have been retained, but in a number of instances the material has been reworked so as to incorporate alternative approaches to these topics that have appeared in the mathematical literature in recent years.