The objective of these exercises is to fit parameters to describe a second order underdamped system. A second order system differential equation has an output `y(t)`, input `u(t)` and four unknown parameters. The four parameters are the gain `K_p`, damping factor `\zeta`, second order time constant `\tau_s`, and dead time `\theta_p`.

$$\tau_s^2 \frac{d^2y}{dt^2} + 2 \zeta \tau_s \frac{dy}{dt} + y = K_p \, u\left(t-\theta_p \right)$$

The second order system can also be expressed in Laplace-domain form.

$$\frac{Y(s)}{U(s)} = \frac{K_p}{\tau_s^2 s^2 + 2 \zeta \tau_s s + 1}e^{-\theta_p s}$$

Where applicable, follow the steps to obtain a graphical approximation of a step response of an underdamped (oscillating) second order system. An underdamped system implies that `0 \ge \zeta > 1`.

1. Find `\Delta y` from step response.
2. Find `\Delta u` from step response.
3. Calculate `K_p = {\Delta y} / {\Delta u}`.
4. Calculate damping factor `\zeta` from overshoot `OS` or decay ratio `DR`.
5. Calculate `\tau_s` from equations for rise time `t_r`, peak time `t_p`, or period `P`.

See Second Order Graphical Methods for additional details on correlations for obtaining the unknown 2nd order system parameters. Assume a step input of 1.0 for each output graph for Exercises 1a-1c.

Exercise 1a

Exercise 1b

Exercise 1c

Use the following script to generate a step response for a 2nd order system. For each of the exercises (1a-1c), you can plug in the values for `K_p`, `\tau_s`, and zeta (`\zeta`) to generate a step response.

Exercise 2

A PI closed loop controller process variable (PV) rises to meet a new set point (SP) following a change from 50 to 70. The PV overshoots the new set point by 50% with two success peak values at 100 and 300 seconds after the set point change is initiated. The PV response eventually settles to the new SP with no offset. Determine the gain `K_p`, damping factor `\zeta`, and second order time constant `\tau_s` that describe the closed loop response as a second order underdamped system. Assume that dead time `\theta_p` is negligible.

What to Turn In

• Exercises 1a-1c: Graphical estimation of Kₚ, ζ, τₛ, and θₚ from the plotted step response. Show how each parameter was determined from the graph.
• Exercise 2: Determine Kₚ, ζ, and τₛ that describe the PI-controlled process. Show supporting calculations based on overshoot, decay ratio, and peak timing. Include a summary of how these parameters characterize the closed-loop system response.

Generative AI Learning

Use these prompts to test your understanding after completing the exercise. Direct the AI - the graphical readings are the skill being practiced.

Tip: The TCLab heater-to-sensor path is a live second-order system: fit it with the two-heater extension in the TCLab Simulation Studio and compare the fitted time constants with what your graphical zeta-tau readings imply.

What to Turn In

Submit a short report (PDF, 1-2 pages) that curates your results into a demonstration of what you learned. You may use Generative AI to help write the report, but you must supply the correct readings, justifications, and assumptions. Answer these questions:

1. For each assigned response: report Kp, tau, and zeta with the graphical readings (overshoot, period) that produced them, including one annotated plot.
2. Verify one fit by simulating the identified second-order model over the data. Include the overlay plot.
3. Give one physical example from chemical engineering for each regime: overdamped, critically damped, underdamped - and say what creates the oscillation in the underdamped one.
4. From the quiz prompt: one question you missed and the corrected answer.