Relative Stability Analysis

Motivation

  • 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 \(-1\).

Example

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

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  • \(G(s)H(s)\) 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).

Gain Margin

  • 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.

Gain Margin

  • 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 \[K_g = \frac{1}{|G(j\omega_\phi)H(\omega_\phi)|}.\]
  • In terms of decibles,\[K_g(dB) = -20\log|G(j\omega_\phi)H(\omega_\phi)|.\]

Phase Margin

  • 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 \[\phi_m = 180^\circ+\angle G(j\omega_g)H(\omega_g).\]

Gain and Phase Margin on Bode Plot

  • 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 Nyquist Plot)

Stable System

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Unstable System

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Unstable System

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Stability Margins and Closed-Loop Stability

  • 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.

Stability Margins and System Performance

  • 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.

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  • 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.

Example: System with Good Stability Margins but Poor 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 \(-0.01\pm 0.7j\).

Example: System with Good Stability Margins but Poor Performance

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Example: System with Good Stability Margins but Poor Performance

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Example

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 \(K = 100\).

\(K=10\)

\(\phi_m = 25^\circ\) and \(K_g=9.5 dB\). The system is stable.

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\(K=100\)

\(\phi_m = -23^\circ\) and \(K_g=-10.5 dB\). The system is unstable.

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