AP6.5 Consider the closed-loop system in Figure AP6.5.

Suppose that all gains are positive, that is,

K K 1 2 > > 0, 0, 0 K K 3 4 > > , 0, and K5 > 0.

(a) Determine the closed-loop transfer function

T s( ) = Y s( ) / . R s( )

(b) Obtain the conditions on selecting the gains

K K 1 2 , , K K 3 4 , , and K5, so that the closed

loop system is guaranteed to be stable.

(c) Using the results of part (b), select values of the five gains so that the closed-loop system is stable, and plot the unit step response.

R(s) -K3 R(s) K K (a) -KA 10 s-10 Y(s) 10 K 2x-4 Y(s) s-10 Ks (b) FIGURE AP6.5 Multiloop feedback control system. (a) Signal flow graph. (b) Block diagram.