Control Systems
Control Systems tests transfer functions, block-diagram reduction, time response, Routh-Hurwitz stability, root locus, Bode plots and state-space models. ESE numericals often ask for steady-state error, stability range or a gain margin.
Core rules
- For unity negative feedback, T(s)=G(s)/[1+G(s)]; the characteristic equation is 1+G(s)=0.
- Type number is the number of poles at s=0 in the open loop. Step error is zero for Type 1; ramp error is 1/Kv, where Kv=lim sG(s).
- Routh array: necessary condition is all coefficients positive; sufficient condition is all first-column entries positive. Treat zero first-column entries and entire zero rows with the auxiliary polynomial.
- Bode: gain crossover gives phase margin; phase crossover gives gain margin. Minimum-phase systems have no right-half-plane zeros or poles.
- Root locus lies on the real axis to the left of an odd number of poles and zeros; branches end at zeros or infinity.
Worked example
G(s)=10/[s(s+2)] with unity feedback. Kv=lim(s to 0) sG(s)=10/2=5. A unit ramp input gives ess=1/Kv=0.2. The reasoning uses the final-value theorem on a stable closed loop; the closed-loop poles are roots of s^2+2s+10, which are stable.
Traps
Steady-state error formulas assume closed-loop stability. Do not cancel a pole at the origin with a zero and then call the system Type 0. In Routh, a zero row requires differentiation of the auxiliary equation. In Bode, a negative gain margin in dB indicates instability, not a large margin.