Consider a continuoustime dynamic system described by the differential equation gym + aw) = not) (1) 1. Let u(t) = 6(t), compute y(t) 2. Let u(t) =...

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Consider a continuous—time dynamic system described by the differential equation gym + aw) = not) (1) 1. Let u(t) = 6(t), compute y(t)2. Let u(t) = fimfi) — 4m(t) and let a:(t) = 6(t), now compute y(t). 3. Using (1), write down the transfer function G(s) for y(3) = G(s)u(s) and the condition on the valueof a so that the system is stable. 4. Let the value of a = 0 in (1), sketch the shape of the amplitude Bode plot of logmw versus10s“J IGUW) |- [MS] Consider the discrete—time system described by the difference equationy[n] — 0.64y[n — 2] = u['n,] + u[n — 1] (2)5. Show that the discrete—time system is stable.6. Let u[n] = 0.2” for n 2 0, compute y[n].7. Let u[n] = cos(21m) then y[n] = B cos(27rn + 9b) as n —> oo. Compute B. [2pts] 8. Consider a continuous—time dynamic system described by the differential equation gym + zjtym + 4:105) = no compute the resonance frequency and damping ratio of this resonant system.