Consider the temperature control system of Problem 9.2-3.

(a) Determine if this system is controllable.

(b) An observer is added to this system in Problem 9.3-2, with the equation [see (9-38)]

Construct a single set of state equations for the plant-observer system, with the state vector $\left[x\right(k\left)q\right(k){]}^T$ and the input $u\left(k\right)$ (i.e., there is no feedback).

(c) Show that this plant-observer system is uncontrollable.

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

q(k + 1) = (AGC)q(k) + Gy(k) + Bu(k) 0.3012q(k)+10.99y(k) + 1.0368u(k) =