Govt. Exams
Entrance Exams
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.
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.
A limit cycle is a closed trajectory in phase plane representing self-sustained oscillations with constant amplitude and frequency, independent of initial conditions.
Kv = lim[s→0] s·Kp·G(s) = lim[s→0] s·10·1/[s(s+2)] = 10/2 = 5
Breakaway/break-in points satisfy dK/ds = 0, derived from the magnitude condition where multiple roots exist on the real axis.
Lyapunov's second method directly analyzes stability without solving differential equations. It's applicable to linear and nonlinear systems.
The lag section (lag compensator) increases low-frequency gain without significantly affecting transient response, thus improving steady-state error performance.
For complete state controllability, rank of [B AB A²B ... Aⁿ⁻¹B] must equal n, the order/number of states of the system.
