On the Impulse Response of LTI Systems
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The problem of inferring the oscillatory behavior of the impulse and step responses of a system from the location of poles and zeros of its transfer function has practical importance. The authors first show that all the induced norms of a LTI system equal the zero frequency gain of its transfer function, if its impulse response is nonnegative. An important step in tackling this problem is to characterize the oscillatory nature of the impulse response of a system, given its minimal representation. We use Bernstein's theorem to show that, for a minimal representation, (A, B, C), the impulse response, Ce/sup At/B /spl ges/ 0 for all t > 0 iff C(/spl lambda/I - A)/sup -k/ B /spl ges/ 0 for all k and for some /spl lambda/ > /spl rho/(A), where /spl rho/(A) is the spectral radius. Additionally, the authors show that the number of sign changes of the impulse response of a non-minimum phase system, ((s - /spl sigma/)/sup 2/ + w/sup 2/) H/spl circ/ (s), where /spl sigma/ > 0, and H/spl circ/ (s) is of relative degree greater than 1, is less than or equal to the number of sign changes of H/spl circ/ (s), whenever w is greater than a w*, which is a function of (H/spl circ/ (s + /spl sigma/)); in other words, the influence of the non-minimum phase zero is not felt by the transfer function ((s - /spl sigma/)/sup 2/ + w/sup 2/) H/spl circ/ (s). We use this result to refine the definition of a weakly non-minimum phase system and provide an example of designing a controller for a non-minimum phase system to achieve non-negative impulse and non-overshooting step responses.