Consider the system of Problem 11.4-5.

(a) Find the feedback gain required to minimize the given cost function for the infinite-time problem.

(b) Find the closed-loop system characteristic equation.

(c) Find the closed-loop time constant τ as a function of the sample period T.

(d) Why is the settling time in part (c) less than that found in Problem 11.4-6(c), where τ = 0.410T.

Problem 11.4-5

Given the plant of Problem 11.4-4, with the cost function

(a) Calculate the feedback gains required to minimize the cost function, using the difference-equation approach of Section 11.4.

(b) Find the maximum magnitude of u(k) as a function of x(0).

(c) Compare the maximum magnitude of u(k) in part

(b) with the value in Problem 11.4-4(c). Explain the difference.

Problem 11.4-4

Given the first-order plant described by

(a) Calculate the feedback gains required to minimize the cost function, using the partial-differentiation procedure of Section 11.3.

(b) Repeat part (a) using the difference-equation approach of Section 11.4.

(c) Find the maximum magnitude of u(k) as a function of x(0).

3 J, = 2x2(k) k = 0