Problem $1.(20$pts$)$ Determine if the following statements are true of false,

$($Correct answer. $+2,$ no answer. $0,$ wrong answer. $-2).$

a$)$ For a stable linear system, the response to a sinusoidal input is asymptotically sinusoidal, with the same frequency and phase as the input.

b$)$ In general, by adding D$-$control action to P$-$control, we can improve the damping characteristic and steady$-$state error performance of the closed$-$loop system.

c$)$ In general, adding a LHP zero pulls the root locus to the left.

d$)$ When the input frequency is in the range of Bandwidth, the output magnitude follows at least $70\%$ of the input magnitude.

e$)$ Any unstable system can be stabilized with feedback control.

f$)$ A second$-$order system whose poles have negative real parts exhibits the impulse response with an exponentially decaying amplitude, and we called such a system stable.

g$)$ In terms of steady state error, higher order type system is always better than the lower one.

h$)$ Lag compensation acts mainly to speed up a response and to decrease the transient overshoot.

i$)$ If the relative degree of the open loop transfer function is larger than $2,$ the closed loop system becomes unstable for high gain.

j$)$ For a certain system, its state$-$space model is unique when the input and output are fixed.