Control Systems
Control Systems
This domain tests transfer functions, block-diagram reduction, time response, stability and frequency response. Numerical questions often ask for damping ratio, overshoot or Routh array sign changes.
Core rules
- A transfer function is the Laplace ratio of output to input with zero initial conditions. Its poles are the roots of the denominator.
- First-order step response: time constant τ; about 63.2% of final value at one τ and about 98% at four τ.
- Second-order standard form: ωn²/(s² + 2ζωn s + ωn²). Peak overshoot Mp = exp(−πζ/√(1−ζ²)); peak time tp = π/(ωn√(1−ζ²)).
- Routh-Hurwitz: the number of sign changes in the first column equals the number of right-half-plane poles. A zero row is handled with the auxiliary polynomial.
- Steady-state error depends on system type and input. A type-0 system has finite step error; a type-1 system has finite ramp error.
- Gain margin is the negative of the gain at the phase crossover; phase margin is measured at the gain crossover. Both must be positive for stability.
- P action gives offset; I action removes offset; D action adds damping.
Worked example
For G(s) = 16/[s(s+4)] with unity feedback, the characteristic equation is s² + 4s + 16 = 0. Thus ωn = 4 rad/s and 2ζωn = 4, so ζ = 0.5. Overshoot = exp(−π×0.5/√0.75) = exp(−1.814) = 0.163, about 16.3%.
Traps
Do not use open-loop poles when the question asks for closed-loop stability. Check Routh special cases. Do not compute steady-state error for an unstable system. Watch the sign convention for margins.