Synthesis of anisotropic suboptimal controllers by convex optimization

This paper considers a disturbance attenuation problem for a linear discrete time invariant system under random disturbances with imprecisely known probability distributions. The statistical uncertainty is measured in terms of relative entropy using the mean anisotropy functional. The disturbance attenuation capabilities of the system are quantified by the anisotropic norm which is a stochastic counterpart of the H-infinity norm. The designed anisotropic suboptimal controller generally is a dynamic fixed-order output-feedback compensator which is required to stabilize the closed-loop system and keep its anisotropic norm below a prescribed threshold value.

💡 Research Summary

This paper addresses the problem of attenuating stochastic disturbances whose probability distributions are only partially known, by employing the mean anisotropy functional as a measure of statistical uncertainty. The authors consider a linear discrete‑time invariant (LDTI) plant and assume that the disturbance sequence is a stationary Gaussian process whose covariance matrix is unknown but whose mean anisotropy does not exceed a prescribed non‑negative level a. Under this assumption the disturbance attenuation capability of a closed‑loop system is quantified by the a‑anisotropic norm ‖·‖ₐ, a stochastic counterpart of the H∞ norm that continuously interpolates between the H₂ norm (as a → 0) and the H∞ norm (as a → ∞).

The design goal is to find a fixed‑order dynamic output‑feedback controller K(z) that internally stabilizes the closed‑loop system and guarantees that its a‑anisotropic norm is bounded by a user‑specified threshold γ. To achieve this, the paper builds on the Anisotropic Norm Bounded Real Lemma (ANBRL), which provides a necessary and sufficient condition for ‖T_{ZW}‖ₐ ≤ γ in terms of a determinant inequality and two linear matrix inequalities (LMIs) involving reciprocal matrices. Earlier formulations of ANBRL were non‑convex; recent work (cited as