Question

Question 3 (10 pts): Consider the closed-loop system pictured below, with two inputs: the reference input z (ideally, to be tracked by output y) and a "disturbance" input d. (Note the minus sign at the bottom entry of the summing junction on the left.) Block H and G represent LTI systems; H has transfer function HL and G has transfer function GL. All blocks are causal (so that the closed-loop system is causal as well). Both z and d are assumed to be identically zero unti0ie, the inputs are zero for t<O but could potentially involve singularity functions (6,8,) occurring at t 0),. (a) (5 pts) Obtain an expression for the Laplace transform Y of y in terms of the Laplace transforms XL and Di of z and d. [Hint: As we did in class, first introduce an auxiliary quantity, say the Laplace transform Et, of the error signal ez-y, and write two equations; then eliminate Er

Answer 1

a) dce) Lt) La vit elt) t elt) t) Xit hohan Q& DレナyLXAL D t

fg /super position 나レ on Laplace Tr ラ レニー on LT ELEXL-Yレ @ 12 1 4 S(t P(s+2) Eレー 2 +25+l 23 t1

chara esrenstic eauationclosed Loo on clo 2 closed leop ertem is stable S 2 3 STABLE Aystem