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Figure 2.32. Relation between the output sensitivity function and the modulus margin
Therefore, the reduction (or the minimization) of |S_yp (jω)|_max will lead to an increase (or maximization) of the modulus margin10.
Typical values for a good modulus margin are
∆ M ≥ 0.5 (- 6 dB) [min: 0.4 (-8 dB)]
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¹⁰|S_yp (jω)|_max corresponds to the H_∞ norm of the output sensitivity function
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Note that ∆M ≥ 0.5 implies a gain margin ∆G ≥ 2 (6 dB) and a phase margin ∆ > 29°. As a general rule, a good modulus margin guarantees satisfactory values for the gain and phase margins¹¹.
To summarize, typical values for the stability margins in a robust design are:
• gain margin: ∆ G ≥2 (6dB) [min.: 1.6 (4dB)]
• phase margin: 30° ≤ ∆φ ≤ 60°
• delay margin:∆τ = [min.: 0.75 T_s]
• modulus margin: ∆ M ≥ 0.5 (-6dB), [min.: 0.4 (-8dB)]
If the plant model is known with a very good precision for a certain region of operation, the imposed robustness margins can eventually be less restrictive.
The modulus margin is very important because:
• It defines the maximum admissible value for the modulus of the output sensitivity function and therefore the low limits of the performance in disturbance rejection;
• It defines the tolerance with respect to nonlinear or time varying elements that may belong to the system (the circle criterion - see below).

Tolerance with Respect to Nonlinear Elements

In control systems we frequently have components with static nonlinear or timevarying characteristics (often in the actuators).
The characteristics of these components, without being accurately known, generally lie inside the conic region defined by a minimum linear gain ( α ) and a maximum linear gain ( β ) – see Figure 2.33.

Figure 2.33. Nonlinear or time-varying characteristics, contained in the conic domain (α, β)
The closed-loop system looks, for example, like those in Figure 2.34a
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¹¹ The converse is not true. Systems having satisfactory gain and phase margins may have a very low modulus margin.
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