Govt. Exams
Entrance Exams
Lead-lag compensator combines lead (improves transient) and lag (improves steady-state) characteristics for overall system improvement.
Peak overshoot Mp = e^(-πζ/√(1-ζ²)). It depends only on damping ratio ζ, not on ωₙ.
Standard form: ωₙ² = 100, so ωₙ = 10 rad/s. 2ζωₙ = 10, so ζ = 10/(2×10) = 0.5
Lead compensator (a > 1) provides phase lead in mid-frequency range, improving transient response and system speed, hence used for transient improvement.
For a system to be completely observable, the observability matrix [C; CA; CA²; ...] must have rank equal to n (number of states).
For 2% criteria, settling time ts ≈ 4/(ζωₙ). This is the standard formula for second-order systems.
Gain margin in dB = 20log₁₀(GM). Therefore, 6 = 20log₁₀(GM), GM = 10^(6/20) ≈ 2
The Nyquist criterion checks encirclements of the critical point (-1, 0) in the complex plane. No encirclement indicates stability for minimum phase systems.
For ramp input, ess = 1/Kv. Given ess = 0.2, therefore Kv = 1/0.2 = 5 sec⁻¹
Kv = lim(s→0) s·G(s) = lim(s→0) s·10/[s(s+5)] = 10/5 = 2 sec⁻¹