Chapter 14

# 14.1 Why We Need Another Criterion of Stability

There are several ways of looking at the system stability in:

- S-domain
- Routh-Hurwitz Criterion
- Root Locus (still to come)

- Frequency domain
- Gain and Phase Margins

These approaches all have their limitations. The usefulness of s-domain based Routh-Hurwitz Criterion and the Root Locus method is limited by the fact that the system description has to be known accurately (i.e. exact system identification, no approximate models) in order to be able to establish the relative system stability. And, when the system description is known but the system order is high, hand-calculations based on Routh Array become very tedious. A computer-based approach is necessary, as in plotting the Root Locus using MATLAB, and then determining the critical gain [latex]K_{crit}[/latex] and critical frequency of oscillations [latex]\omega_{osc}[/latex] from the plot.

A simple alternative is to use open loop frequency response plots, which can be measured directly from the system, or sketched using linear approximations. Definitions of Gain Margin [latex]G_m[/latex] and Phase Margin [latex]\Phi_m[/latex] do not require that the system transfer function [latex]G(s)[/latex] be known, and this is their major advantage. As well, the procedure of determining Gain Margin [latex]G_m[/latex] and Phase Margin [latex]\Phi_m[/latex] does not increase in difficulty as the system order increases.

However, the limitation of determining the system stability through Gain Margin [latex]G_m[/latex] and Phase Margin [latex]\Phi_m[/latex] is that it only applies to systems that are open-loop stable and minimum-phase. While the vast majority of industrial control systems belong to this category, we need a more general stability criterion in frequency domain that would cover such special cases.

**Important Note:** If the system is open-loop unstable, no measurements of frequency response are possible, of course. However, the *theoretical *frequency response plot can be sketched, or computed, if the system transfer function [latex]G(s)[/latex] is known.

Note, that before Nyquist Criterion can be applied, we need to review a particular form of frequency response, called **Polar Plots**.