How it works
Transfer function G(s) = C(s)/R(s) is defined under zero initial conditions. For the standard second-order system G(s) = ω_n² / (s² + 2ζω_n·s + ω_n²), the denominator is the characteristic polynomial, whose roots are the closed-loop poles. The order of the system equals the degree of the denominator polynomial. Type number equals the number of open-loop poles at the origin — a Type 1 system has one integrator in the forward path. DC gain is lim(s→0) G(s), provided the system has no poles at the origin.
Key points to remember
Poles are values of s where G(s)→∞; zeros are values where G(s)=0. Poles in the left half s-plane give stable transient responses that decay with time; poles on the jω axis give sustained oscillation; right-half-plane poles cause exponential growth. For G(s) = (s+2)/[(s+1)(s+3)], there is one zero at s = −2 and two poles at s = −1 and s = −3. The relative order (excess of poles over zeros) determines the high-frequency roll-off rate on a Bode plot: −20 dB/decade per excess pole. A transfer function with more zeros than poles is not physically realisable.
Exam tip
The examiner always asks you to find poles, zeros, order, and type number of a given G(s) and then comment on stability — write the factored form of numerator and denominator separately before listing roots.