The system considered here is the classical system of Doyle and Stein [9] to illustrate robustness problems. The system of Doyle and Stein is analog; the discrete model of the system is used here [11]. The sample period, T = 0.006 s, was chosen small so that the results approximate those of Doyle and Stein. The plant model is given by

(a) A pole-placement design is to yield closed-loop poles at s = -7 ± j2. Show that these s-plane poles
translate into the desired characteristic equation

(b) Find the gain matrix required to place the poles at the locations given in part (a).
(c) Find the steady-state Kalman filter gains for the case that R_w = R_v = 166.67.
(d) Find the plant transfer function G(z) and the control-estimator transfer function D_ce(z).
(e) Using the transfer functions of part (d), plot the Nyquist diagram for the system opened at the plant.
(f) Find the system phase and gain margins. Doyle and Stein found the phase margin of the analog system to be approximately 15°.

Transcribed Image Text:

x(k + 1) [ = 0.999946 -0.177854E-1 y(k) [21]x(k) + v(k) 0.592847E 2] 0.976233 2xCA x(k) + [0.178567E 4 0.592847E 2 u(k) 0.208907 -0.383511 w(k)