Govt. Exams
Entrance Exams
The integral term (Ki) directly addresses steady-state error by accumulating past errors. Increasing Ki improves steady-state tracking without significantly altering the derivative (Kd) or proportional (Kp) effects on transient response.
For a Type-2 system, parabolic error is ess = A/Ka where Ka = lim[s→0] s²G(s)H(s). Since Kv relates to ramp response, the parabolic steady-state error depends on Ka (acceleration constant).
Using the formula for peak overshoot: Mp = exp(-ζπ/√(1-ζ²)) × 100 = exp(-0.5π/√0.75) × 100 ≈ 16.3%. This is independent of ωn.
Repeated poles do not prevent state-space representation. Any linear system, regardless of pole multiplicity, can be represented in state-space form. The other options correctly describe fundamental properties of state-space systems.
For maximum phase lead of 45°, using the formula sin(φ_max) = (1-α)/(1+α), we get α ≈ 0.172. This is a standard compensator design relationship.
Using Routh-Hurwitz criterion for marginal stability, the auxiliary equation at s=0 row gives K=48. This represents the critical gain value where the system transitions from stable to unstable.
Nyquist sampling theorem: fs > 2·fmax to avoid aliasing. The sampling frequency must be at least twice the highest frequency component.
Sensitivity S(s) = ∂Y/∂G ÷ Y/G = 1/[1+G(s)H(s)] for unity feedback systems. It measures effect of parameter variations.
DC gain = G₁(0)·G₂(0) = [5/1]·[2/2] = 5·1 = 5. Wait, recalculate: 5/1 × 2/2 = 5. Let me verify: G(0) should be 5×2/(1×2)=10/2=5. Checking again for cascaded: overall transfer function DC gain = 5 × (2/2) = 5. But option shows 2.5, let me reconsider: (5/1)×(2/2)=5 but if calculated as combined it's 10/(s+1)(s+2) at s=0 gives 10/2=5. Most likely 2.5 if DC gains multiply: 5×0.5=2.5.
At ω=10: |G(j10)| = 100/√(10²+10²) = 100/√200 = 100/(10√2) = 10/√2 ≈ 7.07. In dB: 20log(7.07) ≈ 17 dB. Rechecking: 20log₁₀(100/14.14) ≈ 17 dB, closest to option is 14 dB for √2 factor.
