S4-3.2 Time Response Analysis

What this is and why it exists

Ask a system to move and it answers in a shape. This unit is about reading that shape: how fast it rises, how far it overshoots, when it settles, and how wrong it remains at the end. The second-order system is the unit's centrepiece because two numbers — damping ratio and natural frequency — generate the whole family of shapes, and because most real systems are deliberately approximated as second-order for exactly that economy. The unit closes with PID, the controller that still runs most of the world's industry.

The vocabulary

• Standard test inputs — step, ramp and parabola: sudden command, steadily moving command, accelerating command.
• Time constant — a first-order system's speed: time to reach about 63 percent of a step's final value.
• Damping ratio (ζ) — how strongly oscillation is resisted: underdamped below one, critically damped at one, overdamped above.
• Natural frequency (ωn) — the frequency the system would oscillate at with no damping; the timescale of everything.
• Peak overshoot, rise time, settling time — the standard specification numbers read off a step response.
• System type — the number of pure integrations in the open loop; it decides which inputs are followed with zero error.
• Static error coefficients — Kp, Kv, Ka: the numbers that turn system type into an exact steady-state error.
• PID — control action assembled from the error's present (P), accumulated past (I) and predicted future (D).

The mental model

A door with a closing spring and a damper. Weak damper: the door swings past the frame, oscillates, and bangs — underdamped, fast but overshooting. Damper too strong: the door creeps shut — overdamped, no overshoot and no hurry. The critical setting closes it fastest without a slam. Every second-order step response is this door; ζ says which regime, ωn says how fast the whole drama runs, and the specification numbers are functions of the pair — overshoot depends on ζ alone (about 16 percent at ζ of 0.5), settling time runs inversely with ζ times ωn. Designing to a specification is choosing where those two numbers must sit.

Steady-state error is a different question: not how the system moves, but where it stops. The answer is mechanical once you know the system type. Each integration in the loop can absorb one degree of input growth: type 0 follows a step with finite error and loses a ramp entirely; type 1 zeroes the step error and follows a ramp with finite error; type 2 zeroes both and follows a parabola with finite error. The static error coefficients turn each finite case into a number. The table is worth memorising — it answers exam questions and design reviews in one look.

PID is those insights turned into a controller. The proportional term pushes on the present error — strong but leaves a residue. The integral term accumulates the past — it grinds the residual error to zero, effectively raising system type, at the cost of sluggishness and overshoot. The derivative term leans against the predicted future — it damps the overshoot the other two cause. Present, past, future: most industrial loops on earth run on that triad.

What you should now be able to explain or do

Sketch the step response of a second-order system for a given ζ and ωn and mark rise time, peak overshoot and settling time. Reproduce the type-versus-input error table and compute errors via Kp, Kv, Ka. Say what each PID term contributes and what it costs. Verify a simulated response against the formula predictions.

Check yourself