Consider the satellite control system of Problem 11.4-7.

(a) By computer, find the feedback gains 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 constants as a function of the sample period T.

(d) Compare the time constants in part (c) with the value in Problem 11.6-5(c), which was τ = 3.24T. Explain the difference.

Problem 11.4-7

(a) Repeat Problem 11.4-6 for minimizing the cost function

Plot the gains on the graph from Problem 11.4-6(c)
(b) Explain the differences in the gains in part (a) and those in Problem 11.4-6.

Problem 11.4-6

A satellite control system is modeled as shown in Fig. P11.4-6. This system is described in Problem 1.4-1.
For this problem, ignore the sensor gain and let D(z) = 1. In addition, K = 1, T = 1 s, and J = 4. As stated in Problem 9.2-4, a state model for this system is given by

where x₁(k) is angular position and x₂(k) is angular velocity.

(a) Determine by hand the gains required to minimize the cost function

with N = 1. The value of N is chosen to be unity to limit the calculations.
(b) Use MATLAB to solve part (a) for N = 20. Sketch the calculated gains versus k.

Problem 1.4-1

Transcribed Image Text:

JN N = 2x00 [11]x(*) Ex() k=0 x(k) + 20u(k)