Control Systems — Complete Revision Notes

Signal Flow Graph Reduction

$$T = \frac{\sum P_k \Delta_k}{\Delta}$$

• $\Delta = 1 - \sum L_1 + \sum L_2 - \sum L_3 + \dots$
• $L_1$ = individual loop gains; $L_2$ = gain products of two non-touching loops.
• $\Delta_k$ = value of $\Delta$ for the subgraph that does not touch forward path $P_k$.

Damping Ratio $\zeta$ & Specifications

Standard closed-loop characteristic equation: $s^2 + 2\zeta \omega_n s + \omega_n^2 = 0$.

• Underdamped ($0 < \zeta < 1$): Poles $s = -\zeta \omega_n \pm j\omega_d$, where $\omega_d = \omega_n \sqrt{1-\zeta^2}$.
• Peak Overshoot: $\%M_p = e^{-\frac{\pi \zeta}{\sqrt{1-\zeta^2}}} \times 100\%$ (Depends purely on $\zeta$!).
• Peak Time: $t_p = \frac{\pi}{\omega_d}$.
• Settling Time: $t_s = \frac{4}{\zeta \omega_n} = 4\tau$ (2% tolerance band); $t_s = \frac{3}{\zeta \omega_n} = 3\tau$ (5% tolerance band).

Gain Margin, Phase Margin & Encirclements

• Phase Crossover Frequency $\omega_{pc}$: Frequency where $\angle G(j\omega_{pc})H(j\omega_{pc}) = -180^\circ$.
• Gain Margin (GM): $\text{GM} = -20\log_{10}|G(j\omega_{pc})H(j\omega_{pc})|\text{ dB}$.
• Gain Crossover Frequency $\omega_{gc}$: Frequency where $|G(j\omega_{gc})H(j\omega_{gc})| = 1$.
• Phase Margin (PM): $\text{PM} = 180^\circ + \angle G(j\omega_{gc})H(j\omega_{gc})$.
• Stability Condition: System is stable if $\omega_{gc} < \omega_{pc} \iff \text{GM} > 0\text{ dB}, \text{PM} > 0^\circ$.
• Nyquist Criterion: $N = P - Z$ ($N$ anticlockwise encirclements of $-1+j0$, $P$ open-loop RHP poles, $Z$ closed-loop RHP poles). For stability, $Z = 0 \implies N = P$.