Control Systems
Control Systems tests transfer functions, stability, time response, and frequency response. Expect Routh-Hurwitz, steady-state error, and Bode or root-locus interpretation rather than block-diagram drawing alone.
Core rules
- Closed-loop transfer function T(s) = G(s)/(1 + G(s)H(s)); characteristic equation 1 + G(s)H(s) = 0.
- Routh-Hurwitz gives stability from coefficient signs and first-column sign changes; a necessary condition is all coefficients positive for a polynomial with positive leading coefficient.
- First-order step response reaches 63.2% at one time constant; second-order damping ratio zeta controls overshoot, and natural frequency omega_n controls speed.
- Steady-state error depends on system type: type 0 has finite position error; type 1 has zero position error but finite velocity error.
- Bode gain margin is read at phase crossover and phase margin at gain crossover. Negative feedback reduces sensitivity but can destabilize.
Worked example
For characteristic polynomial s^3 + 6s^2 + 11s + K = 0, form the Routh array: row s^3 has 1 and 11; row s^2 has 6 and K; row s^1 has (66 - K)/6 and 0; row s^0 has K. Stability requires K > 0 and 66 - K > 0, so 0 < K < 66. The reasoning is that every first-column term must be positive; at K = 66 a row of zeros gives marginal stability.
Traps
Applying steady-state error formulas to an unstable closed loop; confusing system order with type; mishandling Routh rows of zeros or sign changes; and reading phase margin without wrapping phase correctly.