Control Systems by Hand: Open & Closed Loops, First- and Second-Order Responses, Steady-State Error, Dead Time, Stability Checks, Frequency Response in Decibels & Tuning Tables

What you'll build

What you will be able to do

Each lab ends with a concrete result that you build yourself and the checker confirms, cell by cell.

• Run a toy heater open loop and with proportional feedback, compute both outputs under a draught and an aged heater element, the error the closed loop still leaves and how much less the closed loop reacts to a gain change.
• Simulate the heater block step by step after a power step, compare it with the exact curve 1 − EXP(−t/τ), see 63 % at one time constant, compute the time to 98 % and chart both curves.
• Compute the natural frequency and damping ratio of a toy spring-mounted arm, tabulate its step response with the exact second-order formula, find the overshoot and build a table of overshoot over ζ.
• Compute the steady-state error and the closed-loop time constant of the heater under P control for four gains, simulate all four step responses side by side and chart them.
• Delay the heater's power signal by d steps with INDEX, simulate the late response, then compute the pole of a sampled P loop for six sample periods and label each loop smooth, oscillating or unstable.
• Sweep the controller gain from 0 to 8, compute the coefficients and the discriminant of s² + a·s + b, the real and imaginary parts of the roots in separate columns and flag each gain as stable or unstable.
• Tabulate magnitude in dB and phase of the heater loop over 18 frequencies, add a sensor lag and a dead time in series, and read the gain margin and the phase margin off the table with MATCH.
• Turn a fictional oscillation test into P and PI settings with XLOOKUP on a fictional tuning table, let the prepared simulator run both loops with dead time and compare overshoot and summed absolute error.