Derive the SISO black-box identification algorithm described in Section 10.4 when the excitation input u(k) is a unit step signal.

SISO black-box

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10.4 BLACK-BOX IDENTIFICATION For simplicity of explanation, and without any loss of generality, we will assume (10-8) to be a single-input single-output (SISO) system, that is, m = p = 1. We will also assume the input u(k) to be of the form 1, k = 0 u(k) = lo, k = 1, 2,...,N (10-9) indicating that the input is a short-lived unit impulse. Since the system in (10-8) is linear, we also assume without any loss of generality that the initial value of the state x(k) is zero, that is, x(0) = 0. Before proceeding further, we need to make three important assumptions: Assumption 1: The LTI system (10-8) is both controllable and observable. Assumption 2: The input u(k) is persistently exciting, that is, the signal u(k) is rich enough to stir the contribution of all the dominant dynamic modes in the output y(k).