PID Control

Selecting a gain based only on response speed can leave the system vulnerable to overshoot or model changes. Compare different performance metrics.

Starting conditions

Motor parameters: $J=.01$,$b=.1$,$K=.01$,$R=1$,$L=.5$. P=(500,0,0), PI=(500,300,0), PID=(500,300,30).

Procedure

1. Build three closed-loop models using negative unity feedback.
2. Plot the unit-step responses over $0\le t\le 2$ s in the same figure, and check the poles first.
3. Record overshoot, settling time, and steady-state error for stable models.

Results to retain

• A table with 3 controller rows, response plots, and a brief comparison of their roles.

Starting conditions

Starting from the PID baseline (500,300,30), change only Kp to 250,500,750. Keep Ki and Kd fixed.

Procedure

1. Use the PID result from Activity 1, $(500,300,30)$, as the baseline, and change only $K_p$ to 250 and 750 in turn. Keep $K_i=300$, $K_d=30$, the plant, and the time axis unchanged.
2. Plot the unit-step responses of both settings alongside the baseline curve in the same figure. Record overshoot, settling time, and steady-state error using the same ctl.step_infostepinfo settings and the same columns as Activity 1.
3. Choose one of the two additional settings. Use the table values to explain which metric you aim to improve over the baseline and which changes you are willing to accept.

Results to retain

• Response plots and a performance table for both settings. One sentence explaining your choice.

Starting conditions

1st-order plant $\dot y=-y+u$, reference 2→.5 (changes at 5 s), $u\in[0,1]$, dt=.01 s. Use $K_p=2$, $K_i=1$.

Procedure

1. In the supplied PI simulation, plot the actual input alongside the input before limiting.
2. Compare conditional integration enabled and disabled.
3. Explain why the error persists after lowering the reference, relating it to the integrator state.

Results to retain

• Three plots of output, input, and integrator state, and a comparison table.

In the comparison table, IAE is $\int_{5}^{t_{\mathrm{last}}}|r(t)-y(t)|\,dt$ from the reference-change time to the final recording time. Calculate input saturation duration by multiplying the number of samples where the limited $u$ is 0 or 1 over the entire record by dt.

Use the same interval and calculation method for both integration approaches.