Consider the pole-placement design of Problem 9.2-2.

Problem 9.2-2

Consider the plant of Problem 9.2-1, which has the transfer function, from Example 9.5,

Problem 9.2-1

The plant of Example 9.1 has the state equations given (9-1). Find the gain matrix K required to realize the
closed-loop characteristic equation with zeros which have a damping ratio 1 of 0.46 and a time constant v
of 0.5 s. Use pole-assignment design.

(a) Design a predictor observer for this system, with the time constant equal to one-half the value of Problem 9.2-2(a) and with the observer critically damped.

(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 control gain matrix of Problem 9.2-2(a).

(d) The characteristic equation of the closed-loop system of Fig. 9-8 is given by

$$1+D_{ce}\left(z\right)G\left(z\right)=0$$

Use $G\left(z\right)$ as given and $D_{ce}\left(z\right)$ in part (c) to show that this equation yields the same characteristic equation as ${\alpha }_c\left(z\right){\alpha }_e\left(z\right)=0$

${\mathrm{Consider\; the\; plant\; of\; Problem\; 9.2-1,\; which\; has\; the\; transfer\; function,\; from\; Example\; 9.5,}}_{}$

${\mathrm{Consider\; the\; plant\; of\; Problem\; 9.2-1,\; which\; has\; the\; transfer\; function,\; from\; Example\; 9.5,\alpha }}_c\left(z\right){\alpha }_e\left(z\right)=0$

$\mathrm{<strong>Problem\; 9.2-2Problem\; 9.2-2Problem\; 9.2-2Problem\; 9.2-2Problem\; 9.2-2Problem\; 9.2-2Problem\; 9.2-2</strong>}$

$\mathrm{Problem\; 9.2-2Problem\; 9.2-2}$

G(z.) N N 1 [s (s + 1)] = 0.00484Z+0.00468 z1.905z + 0.905