Consider a spring-mass system under the action of an external + ay + y =...

 

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Consider a spring-mass system under the action of an external ÿ + ay + y = u Here y is the horizontal displacement of the spring, u is the force, and a > 0 is a damping coefficient. (The spring's mass and stiffness coefficient have been normalized to 1.) We want Ke, where to make the output y track a reference input r by using a feedback controller u = e=r-y is the tracking error and K is a constant gain. force: 1) Draw a block diagram of the closed-loop system. Derive the transfer function of the closed-loop system from r to y. For what values of K is the closed-loop system stable? Justify your answer. 2) Suppose the reference is a constant signal (for example, a unit step, r(t) = 1(t)). Ex- plain whether the closed-loop system will achieve perfect tracking (e(t) → 0), imperfect tracking (e(t)→→ const 0), or no tracking (e(t) →∞). 3) Answer the same question as in part 2) but for the case when the reference is a ramp signal, r(t) = t. 1(t). 4) Let a = 1 and K = 1, and let r(t) = cos(t). Calculate the steady-state response of the closed-loop system to this input. (As in class, by "steady-state response" we mean the component of the output y(t) that persists after the transients have died down.) Consider a spring-mass system under the action of an external ÿ + ay + y = u Here y is the horizontal displacement of the spring, u is the force, and a > 0 is a damping coefficient. (The spring's mass and stiffness coefficient have been normalized to 1.) We want Ke, where to make the output y track a reference input r by using a feedback controller u = e=r-y is the tracking error and K is a constant gain. force: 1) Draw a block diagram of the closed-loop system. Derive the transfer function of the closed-loop system from r to y. For what values of K is the closed-loop system stable? Justify your answer. 2) Suppose the reference is a constant signal (for example, a unit step, r(t) = 1(t)). Ex- plain whether the closed-loop system will achieve perfect tracking (e(t) → 0), imperfect tracking (e(t)→→ const 0), or no tracking (e(t) →∞). 3) Answer the same question as in part 2) but for the case when the reference is a ramp signal, r(t) = t. 1(t). 4) Let a = 1 and K = 1, and let r(t) = cos(t). Calculate the steady-state response of the closed-loop system to this input. (As in class, by "steady-state response" we mean the component of the output y(t) that persists after the transients have died down.)