mu-Synthesis

mu-synthesis designs controllers that are robust against structured uncertainty by minimizing the structured singular value of the closed loop.

Beyond unstructured robustness

H-infinity control guarantees robustness against a single full-block, norm-bounded perturbation. Real uncertainty is usually structured: several independent parameters, each bounded separately, entering at known locations. Treating them as one lumped block is conservative. mu-synthesis targets the structured case directly.

The structured singular value

The structured singular value mu of a matrix M with respect to an uncertainty structure is the reciprocal of the size of the smallest structured perturbation that makes I minus M-Delta singular. Robust stability against all admissible perturbations of size below one holds if and only if the peak of mu over frequency is below one. mu is therefore the exact robustness measure for structured uncertainty.

D-K iteration

mu is hard to compute exactly, so synthesis uses an upper bound built from scaling matrices D that commute with the uncertainty structure. The D-K iteration alternates two convex steps: with D fixed, solve an H-infinity problem for the controller K; with K fixed, fit frequency-dependent scalings D that minimize the bound. Repeating drives the scaled H-infinity norm, and hence mu, downward.

The iteration is not jointly convex, so it can stall at a local minimum, and the fitted D scalings inflate controller order, usually requiring model reduction afterward. Despite this, D-K iteration is the workhorse method for robust multivariable design.

• Handles independent, separately bounded uncertainties exactly
• Robust stability iff peak mu below one
• D-K iteration alternates H-infinity synthesis and scaling fits
• Not jointly convex; high-order controllers need reduction

For a multi-coil magnetic system where each power supply gain and each inductance is known only within a tolerance, mu-synthesis captures the parameter structure that unstructured methods blur. Such analyses for the Hyperion design remain simulation studies.

The complementary analysis question, given a fixed controller, is robustness analysis via mu; synthesis closes the loop by shaping the controller to reduce it.