The Structured Singular Value

The structured singular value mu measures exactly how much structured uncertainty a system can tolerate before losing stability.

Definition

Given a complex matrix M and a set of allowed perturbation structures Delta, the structured singular value mu(M) is defined as the inverse of the smallest sigma-max of a structured Delta for which I minus M-Delta becomes singular. If no such Delta exists, mu is zero. It generalizes both the spectral radius and the largest singular value: for the full-block structure mu equals sigma-max, and for a scalar-times-identity structure it equals the spectral radius.

Robustness test

The main loop theorem states that a system with uncertainty pulled out into a Delta block connected to a nominal M(s) is robustly stable for all structured perturbations bounded by one if and only if the supremum over frequency of mu(M(jw)) is strictly below one. This turns robustness into a frequency-sweep computation of a single scalar.

Bounds

Computing mu exactly is NP-hard in general, so practice relies on bounds. A lower bound comes from a power iteration over destabilizing perturbations; an upper bound comes from scaling matrices D and G that exploit the structure. For structures with few blocks the gap is small, and the upper bound is what synthesis minimizes.

• mu equals sigma-max for one full block
• mu equals spectral radius for repeated scalar blocks
• Upper bound uses commuting D-scales; lower bound uses power iteration
• Exact computation is NP-hard, so bounds are used

The value of mu is the currency of robust design. A peak of 0.7 means the loop tolerates about 1.4 times the modeled uncertainty; a peak of 1.3 means a perturbation smaller than the model can destabilize it. Designers read the mu plot the way they read a gain margin.

mu underlies mu-synthesis and complements the small-gain theorem, which gives the conservative full-block special case. For structured parameter uncertainty in a design-stage magnet system, mu analysis quantifies margins the small-gain bound would understate.