Lecture 6 - EE 3011 Modelling and Control

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 \[G(s)H(s) = \frac{K}{(s+1)\left[(s/10)^2+s/10+1\right]}.\]

\(G(s)H(s)\) 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

Unstable System

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.

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

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.

\(K=100\)

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