Characterizing control system by speed, accuracy, stability

A control system is judged by three basic requirements: how fast it responds, how accurately it follows the input, and whether it is stable.

(i) Speed

Speed is the quickness of the transient response. It is measured by:

• time constant $\tau$ (first order), settling time $t_s \approx 4\tau$;
• rise time $t_r$, peak time $t_p = \dfrac{\pi}{\omega_n\sqrt{1-\zeta^2}}$, settling time $t_s = \dfrac{4}{\zeta\omega_n}$ (2% band);
• bandwidth in frequency response (larger bandwidth = faster).

(ii) Accuracy

Accuracy is how closely the output follows the input in steady state, measured by steady-state error

$$e_{ss} = \lim_{s\to0} sE(s)$$

with error constants $K_p$, $K_v$, $K_a$: step $e_{ss} = 1/(1+K_p)$, ramp $1/K_v$, parabola $1/K_a$. Higher type number or gain gives better accuracy.

(iii) Stability

A system is stable if every bounded input gives a bounded output (BIBO); for an LTI system all closed-loop poles must lie in the left half of the s-plane. Tests: Routh–Hurwitz, root locus, Bode, Nyquist. Relative stability (how far from instability) is measured by damping ratio, peak overshoot, gain margin and phase margin.

These three conflict: higher gain improves speed and accuracy but reduces damping and can cause instability, so design is a compromise.