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, $T=0.6s$, and let $D\left(z\right)=1$. 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 $x\left(k\right)$ equal to the system output and the output $y\left(k\right)$ equal to the sensor output.
(b) Find the time constant $\tau$ for this closed-loop system.
(c) Using pole-placement design, find the gain $K$ that yield the closed-loop time constant $\tau =1s$. Note that 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 (915).
(e) Draw a block diagram for the system that includes the sensor. Let the digital computer realize a gain, $K_1$, 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),
$$1+K_{1}G\left(z\right)H=0$$
verify that this block diagram yields the desired characteristic equation.

Problem 1.6-1

Problem 6.2-4

Shown in Fig. P6.2-4 is the block diagram of a temperature control system for a large test chamber. This
system is described in Problem 1.6-1. Ignore the disturbance input for this problem.

R(s) + Controller D(z) Volts Disturbance R, (s) 1-E S Ts M(s) Volts Chamber 2.5 s+0.5 2 s+0.5 Sensor 0.04 C(s)