Consider the chamber temperature control system of Problem 9.2-3.

(a) Design a predictor observer for this system, with the time constant equal to one-half the value of Problem 9.2-3(b).

(b) To check the results of part (a), use (9-46) to show that these results yield the desired observer characteristic equation.

(c) Find the control-observer transfer function $D_{ce}\left(z\right)$ in Fig. 9-8. Use the value of control gain of Problem 9.2-3(b). Do not include the sensor gain in $D_{ce}\left(z\right)$.

(d) Draw a block diagram of the system in part (b), with $D\left(z\right)$ in Fig. P9.2-3 equal to $D_{ce}\left(z\right)$.

(e) Verify the system characteristic equation using the block diagram in part (d).

(f) What would be the effect of designing a reduced-order estimator for this system?

Problem 9.2-3

A chamber temperature control system is modeled as shown in Fig. P9.2-3. This system is described in
Problem 1.6-1. For this problem, ignore the disturbance input, T = 0.6 s, and let D(z) = 1. It was shown
in Problem 6.2-4 that

Note that the sensor gain is included in this transfer function.
(a) Draw a flow graph of the plant and sensor. Write the state equations with the state variable x(k) equal to
the system output and the output y(k) equal to the sensor output.
(b) Find the time constant v for this closed-loop system.
(c) Using pole-placement design, find the gain K that yield the closed-loop time constant . Notethat the sensor gain does not enter these calculations.(d) Show that the gain K in part (b) yields the desired closed-loop characteristic equation, using (9-15).
(e) Draw a block diagram for the system that includes the sensor. Let the digital computer realize a gain,
K₁, such that the closed-loop time constant is as given in part (b). The sensor in this system must have
the gain given.
(f) Using the characteristic equation for the block diagram of part (e),

Transcribed Image Text:

Z (0.04) N 1 2 Ls(s+ 0.5). 0.04147 z - 0.7408