Control Systems
Control Systems tests transfer functions, stability, time response, root locus, frequency response and state-space properties.
Core rules
- Closed-loop transfer function T = G/(1+GH) for negative feedback; characteristic equation 1+GH=0.
- Routh–Hurwitz gives the number of right-half-plane roots from sign changes in the first column; a zero row needs the auxiliary equation.
- Steady-state error depends on system type: step error 1/(1+Kp), ramp error 1/Kv, parabola error 1/Ka.
- First-order step response reaches 63.2% at τ and about 98% at 4τ. Second-order ζ and ωn set overshoot and settling time.
- Bode: gain margin is the negative of gain at phase crossover; phase margin is 180° plus phase at gain crossover. Positive margins mean stable.
- Root locus branches start at open-loop poles and end at zeros; real-axis segments lie to the left of an odd number of poles and zeros.
- Controllability matrix [B AB A²B…] must be full rank; observability matrix [C; CA; …] must be full rank.
Worked example
For s³+2s²+3s+K=0, Routh rows: s³: 1, 3; s²: 2, K; s¹: (6−K)/2, 0; s⁰: K. For stability all first-column entries must be positive: K>0 and 6−K>0, so 0<K<6. At K=6 the s¹ row vanishes and the auxiliary equation 2s²+6=0 gives a pair of jω-axis roots, so the system is marginally stable. At K>6 there are two right-half-plane roots.
Traps
- Forgetting the sign convention in 1+GH.
- Applying steady-state error formulas to unstable systems.
- Reading gain margin as positive when it is actually negative in dB.
- Missing the special zero-row case in Routh.
- Confusing controllability with observability.