Chapter 2

# 2.5 Relative Stability – Gain Margin

In the previous section we considered the system stability and introduced the Routh-Hurwitz Stability Criterion. The Criterion can give us the answer regarding the so-called Absolute Stability, i.e. is the system Stable or Unstable. Stability relates to the poles location – if any of the closed loop system poles are in the RHP, the system is unstable. The Criterion can also define a range of gains for a stable system operation. The upper limit of the stability range, i.e. the maximum gain computed from the Routh Array, is called a Critical Gain, $K_{crit}$. The controller gain at which the system operates is called the Operational Gain, $K_{op}$. The closer $K_{op}$ is to its critical value, the more oscillatory the response, the longer it takes to settle and the system is closer to becoming unstable. The measure of how “close” that is, is called Relative Stability.

Discussing Relative Stability is particularly relevant when system parameters are not well-identified. Let’s say based on the calculations for the system parameters the resulting poles are very close to the Imaginary Axis, yet still in the LHP. The answer to the Absolute Stability question is YES, the system is stable. Yet, due to uncertainties in the parameters, closed loop pole locations are not exactly known and it is possible that one or more poles are already in the RHP making the system unstable. If we have a measure of Relative Stability, we have a warning sign that the closed system poles are dangerously close to the Imaginary Axis and therefore the system is dangerously close to becoming unstable.

We will therefore define a measure of Relative Stability and will call it a Gain Margin:

 $G_{m}\frac{K_{crit}}{K_{op}}$ Equation 2-6

The larger the Gain Margin, the further away inside the LHP the system poles are. Note that excessively large Gain Margins mean that is very low, which may have a negative impact on the quality of the transient system response – recall Lab Projects – low gain is associated with a sluggish, slow and overdamped response (large settling and rise times), and with large errors. This dilemma describes limitations of Proportional Control – larger gains speed up the response and improve steady state errors (good), but also lead to oscillations, reduction in relative Stability, and eventually to Instability (bad).

Gain Margin $G_{m}$ becomes the additional system specification ensuring good relative Stability. We will hear more about it in a context of frequency response design, and we will define its measure using Bode plots. Because gains on frequency plots are traditionally described in decibels, it is customary to define Gain margin as either a non-dimensional ratio (referred to as volts per volts or V/V) or in logarithmic units, dB.

Based on Equation 2‑6 we have:

 If $K_{op} [latex]G_{m}>1$ in V/V units $G_{m}>0$ in dB units If $K_{op} = K_{crit}$, the system is marginally stable: $G_{m}=1$ in V/V units $G_{m}=0$ in dB units If $K_{op}>K_{crit}$, the system is unstable: $G_{m}<1$ in V/V units $G_{m}<0$ in dB units

Typically the requirement for Gain Margin is that it should be at least 6 dB or 2 V/V.