#### PROBLEMS

**8.1.** The control system illustrated in Figure P8.1(i) contains linear-state-variable-feedback elements *h*_{1} and *h*_{2}.

**(a)** Determine the gain *K* and the linear-state-variable-feedback constants *h*_{1} and *h*_{2} so that the resulting control system represents a zero steady-state step error system and its characteristic equation contains roots at −2 + *j*, −2 − *j*, and −8.

**Figure P8.1(i)**

**(b)** The same performance can be obtained as in part (a) if we implement a series controller, *G*_{c}(*s*), instead of using linear-state-variable-feedback as illustrated in the configuration shown in Figure P8.1(ii).

**Figure P8.1(ii)**

Determine the transfer function of *G*_{c}(*s*) in terms of *K*, *h*_{1}, and *h*_{2} obtained in part (a) and the other system parameters provided.

**8.2.** A control system containing a controller and a process are illustrated in the block diagram in Figure P8.2(i).

**Figure P8.2(i)**

**(a)** Determine the state equations of this control system.

**(b)** Determine the characteristic equation from knowledge of **P**.

**(c)** Determine the constant *h*_{1} and the gain *K* if the roots of the characteristic equation are at −6 and −8.

**(d)** Instead of using the controller configuration, the control-system engineer wishes to design ...