2. Compute the equation for the system error for the feedback system shown in the figure below. $W(s)$ $R(s)$ $\Sigma$ Controller $K_p$ $U(s)$ $\Sigma$ Plant $1/s(\tau s+1)$ $Y(s)$ $V(s)$ Sensor $1+K_t s$ $\Sigma$
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$E(s) = R(s) - (1+K_t s)Y(s)$ Show more…
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Problem 2: (20%) Consider the feedback control system with proportional controller shown in Fig. 2. (2a) Find the transfer functions Y(s)/R(s) and U(s) / R(s) in terms of the parameter Kp . (5%) (2b) Assume the reference input r(t) = us(t) is a unit step function. Find the steady-state response yss = lim y(t) = lim sY(s) , also in terms of Kp . What is the steady-state error? (2%) (2c) Is it possible to reduce the steady-state error to zero using the proportional feedback control approach? Explain. (2%) (2d) Choose Kp = 8 . Use the results of (2b) to compute the numerical values of yss and the steady-state error in percentage. (2%) (2e) Find the control input u(t) , and plot it as a function of time t. (5%) (2f) Assume u(t) is constrained to be between 0 and 12. Is the choice of Kp = 8 a good trade-off between the steady-state performance and the control-input constraint? Explain. (4%)
Adi S.
Consider the second-order plant with the transfer function G(s) = 1 / ((s + 1)(5s + 1)) and in a unity feedback structure. (a) Determine the system type and error constant with respect to tracking polynomial reference inputs of the system for P [Dc = kp], PD [Dc(s) = kp + kDs], and PID [Dc(s) = kp + kI/s + kDs] controllers. Let kp = 19, kI = 0.5, and kD = 4/19. (b) Determine the system type and error constant of the system with respect to disturbance inputs for each of the three regulators in part (a) with respect to rejecting polynomial disturbances w(t) at the input to the plant. (c) Is this system better at tracking references or rejecting disturbances? Explain your response briefly. (d) Verify your results for parts (a) and (b) using Matlab by plotting unit-step and -ramp responses for both tracking and disturbance rejection.
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