• In practice it is not enough that a system is stable due to modelling uncertainties.
• There must also be some margins of stability that describe how stable the system is.
• We shall concentrate on minimum phase systems (\(P=0\)).
• In time-domain, the closer the dominant closed-loop poles to the imaginary axis, the poorer the system relative stability.
• Using Nyquist Stability Criterion, we know that the system is stable if and only if the Nyquist plot does not encircle \(-1\).

• Let us consider the following close-loop transfer function \[G(s)H(s) = \frac{K}{(s+1)\left[(s/10)^2+s/10+1\right]}\].

• \(G(s)H(s)\) is minimum phase
• To make the system closed-loop stable, the Nyquist plot should not encircle \(-1\).
• From the figure, the larger the value of \(K\) the closer the Nyquist plot is to the critical point \(–1\).
• The closeness of the plot to \(–1\) gives a measure on the relative stability of the system.
• In general, it is expected that the closer the Nyquist plot is to the critical point, the larger the maximum overshoot and the longer the settling time will be (worse performance).

• The gain margin \(K_g\) of a system is defined as the largest amount that the open loop gain can be increased before the closed loop system goes unstable.
• Define \(\omega_{\phi}\) to be the smallest frequency, where the phase of the open-loop transfer function is \(-180^\circ\).
• \(\omega_{\phi}\) is called the phase crossover frequency.
• If the open loop minimum-phase transfer function satisfies the following conditions:
  • The phase starts at 0 or -90.
  • The phase is monotonically decreasing
  • The magnitude is monotonically decreasing
• Then the gain margin for the system is given by \[K_g = \frac{1}{|G(j\omega_\phi)H(\omega_\phi)|}.\]
• In terms of decibles,\[K_g(dB) = -20\log|G(j\omega_\phi)H(\omega_\phi)|.\]

• The phase margin \(\phi_m\) is the amount of phase lag required to reach the stability limit.
• Define \(\omega_{g}\) to be the smallest frequency, where the magnitude of the open-loop transfer function is \(1\) or \(0dB\).
• \(\omega_{g}\) is called the gain crossover frequency.
• If the open loop minimum-phase transfer function satisfies the following conditions:
  • The phase starts at 0 or -90.
  • The phase is monotonically decreasing
  • The magnitude is monotonically decreasing
• Then the phase margin for the system is given by \[\phi_m = 180^\circ+\angle G(j\omega_g)H(\omega_g).\]

• Suppose the open loop minimum-phase transfer function satisfies the following conditions:
  • The phase starts at 0 or -90.
  • The phase is monotonically decreasing
  • The magnitude is monotonically decreasing
• We can find the gain and phase crossover frequency and get the stability margins from Bode plot.
• For most of the systems, we can find the stability margins from the Bode plot.
• However, the Bode plot interpretation of the gain and phase margins can be incorrect. (Need to look at Nyquist Plot)

• Unstable closed-loop system has negative gain and phase margin
• Stable closed-loop system has positive gain and phase margin
• Proper phase and gain margins ensure stability against modeling uncertainties and variations in system components.

• For minimum phase systems, to have satisfactory performance, the phase margin should be between \(30^\circ\) and \(70^\circ\) and gain margin should be greater than \(6dB\).
• For second-order system, the phase margin is directly related to the damping ratio of the closed loop system.

• The closed-loop transfer function is given by \[\frac{Y(s)}{R(s)} = \frac{1}{(s/\omega_n)^2+2\zeta s/\omega_n +1}.\]
• The phase margin is related to the damping ratio by \[\phi_m=\tan^{-1}\frac{2\zeta}{\sqrt{\sqrt{1+4\zeta^4}-2\zeta^2}}.\]
• The large the phase margin, the larger the damping ratio. In particular, for \(0\leq \zeta\leq 0.6\), \(\zeta \approx \phi_m/100\).
• For more complicated higher order systems, good stability margins may not be enough to guarantee good performance.

• Consider the following open-loop transfer function \[G(s)H(s) = \frac{0.38(s^2+0.1s+0.55)}{s(s+1)(s^2+0.06s+0.5)}.\]
• The system has infinite gain margin and \(70^\circ\) phase margin.
• However, the closed-loop system has two poles at \(-0.01\pm 0.7j\).

Consider an open-loop transfer function \[G(s)H(s) = \frac{K}{s(s+1)(s+5)}.\] Derive the stability margins of the system for \(K = 10\) and \(K = 100\).