Given a general first-order plant described by

(a) Show that for R = 0, the optimal gain K(k) is constant for all k ≥ 0.

(b) Give the input sequence u(k), k ≥ 0, for part (a), where u(k) is a function of x(0).

(c) Draw a flow graph of the closed-loop system for part (a).

(d) Give the closed-loop characteristic equation for part (a).

(e) How many sample periods are required for x(k), with initial condition x(0), to be driven to zero?

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with the cost function x(k+1) = Ax(k) + Bu(k) N JN = [Qx(k) + Ru(k)] k=0