A satellite control system is modeled as shown in Fig. P9.2-4. This system is described in Problem

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A satellite control system is modeled as shown in Fig. P9.2-4. This system is described in Problem 1.4-1. For this problem, let D(z)=1D(z)=1K=1,T=1 s,J=4. In addition, K=1,T=1 s,J=4Hk=1, and Hk=1z. From the zz1zz[14s3]=0.125(z+1)(z1)2-transform tables,

z1zz[14s3]=0.125(z+1)(z1)2x(k+1)=[1101]x(k)+[0.1250.25]u(k)y(k)=[10]x(k)

A state model for this system is given by

x(k+1)=[1101]x(k)+[0.1250.25]u(k)y(k)=[10]x(k)x1(k)

where x1(k)x2(k) is angular position and x2(k)K is angular velocity.R(s) + A/D Digital controller D(z) D/A Sensor Hk M(s) Amplifier and thrustors K T(S) Torque Satellite 1 Js

(a) Show that the closed-loop system is unstable.
(b) Using pole-placement design, find the gain matrix Kζ=0.707 that yields the closed-loop damping ratio ζ=0.707τ=4 s and the time constant τ=4 sK.

(c) Show that the gain matrix KDce(z) in part (b) yields the desired closed-loop characteristic equation, using (9-15).

Equation 9-15a(z) |zI A + BK = (z A)(z- A) (Z - A) - - = (9-15)

Problem 1.4-1

Given that (11-21) and (11-22) are valid, derive (11-23).

Equation 11-21- S = Si + x(N-1)Q(N-1)x(N-1) + u(N-1)R(N  1)u(N  1) (11-21)

Equation 11-22Si = [Ax(N1) + Bu(N 1) + Bu(N-1)] Q(N)[Ax(N  1) + Bu(N  1)]]u(N1) - (11-22)

Equation 11-23u(N - 1) = -K(N-1)x(N-1)  (11-23)

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Digital Control System Analysis And Design

ISBN: 9780132938310

4th Edition

Authors: Charles Phillips, H. Nagle, Aranya Chakrabortty

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