Included in at least one reviewed higher-education syllabus.
S4-3.2 Time Response Analysis
You can compute first/second-order responses to standard inputs, read time-domain specs, evaluate static error coefficients, and explain PD/PI/PID action.
The second-order response is the workhorse of control analysis: damping ratio and natural frequency between them set overshoot, rise time and settling time, and most real systems are deliberately approximated as second order to use exactly those results. This topic also covers static error coefficients — how wrong a system stays at steady state, organised by system type — and introduces PD, PI and PID action as three intuitions about the error's present, past and future. It sits early because these time-domain measures are the language every later design goal is stated in.
Work through these
Transfer function and impulse response; standard test inputs
The transfer function describes a system in the frequency domain, the impulse response describes it in time, and they hold the same information. Standard test inputs exist so that different systems can be compared on identical terms.
Transient response of first-order systems
A first-order system has one storage element and responds with a smooth exponential approach governed by a single time constant. It is the simplest useful model and the base case for everything after.
Transient response of second-order systems: damping cases
Second-order systems can overshoot, ring or crawl, and damping ratio and natural frequency between them decide which. Most real systems are deliberately approximated as second order so these results can be used.
Time-domain specifications: rise time, peak overshoot, settling time
Rise time, peak overshoot and settling time are how a customer states a requirement, and they translate directly into the damping ratio and natural frequency from the previous item. This is where analysis becomes specification.
System types, static error coefficients, error series
System type counts the integrators in the loop, and static error coefficients turn that count into a prediction of how wrong the output stays forever. It is a small table worth committing to memory.
PD, PI and PID controllers: what each term does
Proportional, integral and derivative action respond to the error now, the error accumulated, and the way the error is heading. These three intuitions still run most industrial control loops in the world.
Lab: simulate a second-order system and verify overshoot/settling against formulas
Simulate a second-order system and check the measured overshoot and settling time against the formulas. Small disagreements usually mean an assumption in the model, and finding which one is the exercise.
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