Consider the reduced-order observer designed in Problem 9.4-2. In this problem, velocity [ d y / d t ] [ d y / d t

Consider the reduced-order observer designed in Problem 9.4-2. In this problem, velocity $\left[dy/dt\right]$ is estimated, using position $\left[y\right]$ plus other information. We could simply calculate velocity, using one of the numerical differentiators described in Section 8.8. This calculated velocity would then replace the estimate of velocity in the control system. (This is the approach taken in the PID controller.)

Problem 9.4-2

(a) Consider the plant observer as an open-loop system, as shown in Fig. 9-4. Calculate the transfer function $Q\left(z\right)/Y\left(z\right)$ with $U\left(z\right)=0$, where $q\left(kT\right)$ is the observer estimate of velocity.

Use (9-62) with $u\left(k\right)=0$.

(b) Plot the Bode diagram for the transfer function in part (a) versus

(c) One numerical differentiator given in Section 8.8 is

$$D\left(z\right)=\frac{z-1}{Tz}$$

Plot the Bode diagram for this transfer function on the same graph as in part (b).

(d) Comment on the similarities and differences in the two frequency responses.

Consider the satellite control system of Problem 9.2-4. (a) Design a reduced-order observer for this system, with the time constant equal to one-half the value of Problem 9.2-4(b). (b) To check the results of part (a), use (9-63) to show that these results yield the desired observer charac- teristic equation. (c) Find the control-observer transfer function Dee(z) in Fig. 9-8. Use the control gain matrix of Problem 9.2-4(b), K [0.3893 1.769]. (d) The characteristic equation of the closed-loop system of Fig. 9-8 is given by 1 + Dce(z)G(z) = 0 Use G(z) as given and Dce(z) in part (c) to show that this equation yields the same characteristic equation as a (z)a(z) = 0.