Question

3. (35 points) A plant is controlled by a proportional controller with a unity feedback as shown below: \(T(s)\) Controller Process \(Y(s)\) \(R(s)\) K \(G(s)\) The transfer function of the process is given by \(G(s) = \frac{s+2}{s(s+1)(s+b)}\). (a) Obtain the closed-loop transfer function \(T(s) = \frac{Y(s)}{R(s)}\). (b) Obtain the sensitivity, \(S_b\), of the closed-loop system to small change in the parameter b. (c) With \(R(s) = 0\), \(T_d(s) = \frac{A}{s}\) and the steady-state error \(E(s) = R(s) - Y(s)\), obtain the steady state error \(e_d\) due to the disturbance. (d) With \(R(s) = \frac{B}{s^2}\), \(T_d(s) = 0\) and the steady-state error \(E(s) = R(s) - Y(s)\), obtain the steady state error \(e_s\) due to the ramp input.

          3. (35 points) A plant is controlled by a proportional controller with a unity feedback as shown
below:
\(T(s)\)
Controller
Process
\(Y(s)\)
\(R(s)\)
K
\(G(s)\)
The transfer function of the process is given by \(G(s) = \frac{s+2}{s(s+1)(s+b)}\).
(a) Obtain the closed-loop transfer function \(T(s) = \frac{Y(s)}{R(s)}\).
(b) Obtain the sensitivity, \(S_b\), of the closed-loop system to small change in the parameter b.
(c) With \(R(s) = 0\), \(T_d(s) = \frac{A}{s}\) and the steady-state error \(E(s) = R(s) - Y(s)\), obtain the
steady state error \(e_d\) due to the disturbance.
(d) With \(R(s) = \frac{B}{s^2}\), \(T_d(s) = 0\) and the steady-state error \(E(s) = R(s) - Y(s)\), obtain the
steady state error \(e_s\) due to the ramp input.

3. (35 points) A plant is controlled by a proportional controller with a unity feedback as shown
below:
T(s)
Controller
Process
Y(s)
R(s)
K
G(s)
The transfer function of the process is given by G(s) = (s+2)/(s(s+1)(s+b)).
(a) Obtain the closed-loop transfer function T(s) = (Y(s))/(R(s)).
(b) Obtain the sensitivity, Sb, of the closed-loop system to small change in the parameter b.
(c) With R(s) = 0, Td(s) = (A)/(s) and the steady-state error E(s) = R(s) - Y(s), obtain the
steady state error ed due to the disturbance.
(d) With R(s) = (B)/(s^2), Td(s) = 0 and the steady-state error E(s) = R(s) - Y(s), obtain the
steady state error es due to the ramp input.

Added by Joseph S.

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