Poles and Stability
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The steady-state solution of stable systems is due to simple pole in the j-Omega axis of the s-plane coming from the input. Suppose the transfer function of the system is
H(s) = Y(s)/X(s) = 1 / [(s+1)^2 + 4]
(a) Find the poles and zeros of H(s) and plot them in the s-plane. Find then the corresponding impulse response h(t). Determine if the impulse response of this system is absolutely integrable so that the system is BIBO stable.
(b) Let the input x(t) = u(t). Find y(t) and from it determine the steady-state solution.
(c) Let the input x(t) = tu(t). Find y(t) and from it determine the steady-state response. What is the difference between this case and the previous one?
(d) To explain the behavior in the case above consider the following: Is the input x(t) = tu(t) bounded? That is, is there some finite value M such that |x(t)| < M for all times? So what would you expect the output to be knowing that the system is stable?
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