Question

Problem 2: (25 pts) Suppose the open-loop unstable plant, $qquad P(s) = frac{s+1}{s^2-9}$ and the controller, $qquad C(s) = k(s+2)$ are structured in a unity feedback control scheme. (a) Find the closed-loop transfer function from $r$ to $y$ as a ratio of polynomials in $s$: $qquad G(s) = frac{Y(s)}{R(s)} = frac{b_ms^m + b_{m-1}s^{m-1} + dots + b_1s + b_0}{a_ns^n + a_{n-1}s^{n-1} + dots + a_1s + a_0}$ (b) Find the range of $k$ such that the closed-loop system $G(s)$ is stable. (c) Regarding the steady-state error for tracking, what is the System TYPE? (d) Assume $k = 5$. Can this system track a step input perfectly? If so, prove it mathematically. If not, determine the steady-state error $e(infty)$ when the reference $r(t)$ is a unit step function. (e) Suppose the above controller is replaced by a controller of the form: $qquad C_1(s) = frac{k}{s}$ Determine whether a controller of this form can produce a closed-loop system capable of track- ing a step reference with zero steady-state-error $e(infty)_{step} = 0.$

          Problem 2: (25 pts)
Suppose the open-loop unstable plant,
$qquad P(s) = frac{s+1}{s^2-9}$
and the controller,
$qquad C(s) = k(s+2)$
are structured in a unity feedback control scheme.
(a) Find the closed-loop transfer function from $r$ to $y$ as a ratio of polynomials in $s$:
$qquad G(s) = frac{Y(s)}{R(s)} = frac{b_ms^m + b_{m-1}s^{m-1} + dots + b_1s + b_0}{a_ns^n + a_{n-1}s^{n-1} + dots + a_1s + a_0}$
(b) Find the range of $k$ such that the closed-loop system $G(s)$ is stable.
(c) Regarding the steady-state error for tracking, what is the System TYPE?
(d) Assume $k = 5$. Can this system track a step input perfectly? If so, prove it mathematically. If
not, determine the steady-state error $e(infty)$ when the reference $r(t)$ is a unit step function.
(e) Suppose the above controller is replaced by a controller of the form:
$qquad C_1(s) = frac{k}{s}$
Determine whether a controller of this form can produce a closed-loop system capable of track-
ing a step reference with zero steady-state-error $e(infty)_{step} = 0.$

$Problem 2: (25 pts) Suppose the open-loop unstable plant, qquad P(s) = fracs+1s^2-9 and the controller, qquad C(s) = k(s+2) are structured in a unity feedback control scheme. (a) Find the closed-loop transfer function from r to y as a ratio of polynomials in s: qquad G(s) = fracY(s)R(s) = fracbms^m + bm-1s^m-1 + dots + b1s + b0ans^n + an-1s^n-1 + dots + a1s + a0 (b) Find the range of k such that the closed-loop system G(s) is stable. (c) Regarding the steady-state error for tracking, what is the System TYPE? (d) Assume k = 5. Can this system track a step input perfectly? If so, prove it mathematically. If not, determine the steady-state error e(infty) when the reference r(t) is a unit step function. (e) Suppose the above controller is replaced by a controller of the form: qquad C1(s) = fracks Determine whether a controller of this form can produce a closed-loop system capable of track- ing a step reference with zero steady-state-error e(infty)step = 0.$

Added by Julia W.

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