## Quiz on Proportional Integral (PI) Control     1. Increasing the integral time constant \tau_I has what effect on a Proportional Integral (PI) controller?

$$u(t) = u_{bias} + K_c \, e(t) + \frac{K_c}{\tau_I}\int_0^t e(t)dt$$

A. The controller has a more aggressive response
Incorrect. A larger value in the denominator makes the overall value of the integral term smaller
B. The controller responds more quickly
Incorrect. A larger value in the denominator makes the overall value of the integral term smaller, leading to a slower response
C. The controller has a less aggressive response
Correct. A larger value in the denominator makes the overall value of the integral term smaller, leading to a slower response to set point error.

2. Given K_p=2, \theta_p=2.0, and \tau_p=200.0, find K_c for a PI controller using ITAE regulatory control tuning for a PI controller.

A. 38.663
Correct. $$K_c = \frac{0.859}{K_p}\left(\frac{\theta_p}{\tau_p}\right)^{-0.977} \quad \tau_I = \frac{\tau_p}{0.674}\left(\frac{\theta_p}{\tau_p}\right)^{0.680}\quad\mathrm{Disturbance\;rejection}$$
B. 19.901
Incorrect. Regulatory control is disturbance rejection, the equation is found under ITAE tuning correlations
C. 29.569
Incorrect. Regulatory control is disturbance rejection, the equation is found under ITAE tuning correlations
D. 5.182
Incorrect. Regulatory control is disturbance rejection, the equation is found under ITAE tuning correlations

3. A nuclear plant provides a steady amount of electricity that is either stored or sold to the grid depending on current demand. A proposed control system is using a PI controller to adjust power level for load following and is currently losing money due to a slow reaction time. What adjustment could be made to the controller parameters could best address the problem?

A. Double the value of K_c
Correct. This is a good way to tune a controller that is too slow. You may need a regulatory commission approval for changes.
B. Decrease \theta_p
Incorrect. The question asks for adjustments to the controller parameters, where dead-time is a parameter that describes the system response.
C. Double the value of \tau_c
Incorrect. This would slow the response.
D. Divide the value of K_c by two.
Incorrect. This would slow the response.

4. What is the purpose of anti-reset windup for the integral error ie[i]?

if op[i] < op_lo:  # check lower limit
op[i] = op_lo
ie[i] = ie[i] - e[i] * delta_t # anti-reset windup

This code section is from the example PID control code.

A. It provides a lower limit to the controller output
Incorrect. Anti-reset windup pauses the change in the integral ie[i] when the controller output is saturated.
B. It provides an upper limit to the controller output
Incorrect. Anti-reset windup pauses the change in the integral ie[i] when the controller output is saturated.
C. It resets the integral term to zero when the controller output reaches an upper or lower limit
Incorrect. Anti-reset windup does not reset the integral term but pauses the change in the integral ie[i] when the controller output is saturated.
D. Pauses the change in the integral ie[i] when the controller output is saturated
Correct. Anti-reset windup is needed to avoid a large integral term if the controller output saturates at a lower or upper limit and the process variable (PV) is unable to meet the set point (SP) for an extended period of time. This persistent offset would lead to a large or small integral term that would cause the controller output to continue at the saturation limit even if the controller set point were to be brought back to an attainable level.