On the impulse response of LTI systems

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^AtB ⩾ 0 for all t > 0 iff C(λI - A)^-k B ⩾ 0 for all k and for some λ > ρ(A), where ρ(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 - σ)² + w²) Hˆ (s), where σ > 0, and Hˆ (s) is of relative degree greater than 1, is less than or equal to the number of sign changes of Hˆ (s), whenever w is greater than a w*, which is a function of (Hˆ (s + σ)); in other words, the influence of the non-minimum phase zero is not felt by the transfer function ((s - σ)² + w²) Hˆ (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