A convex formulation of strict anisotropic norm bounded real lemma

This paper is aimed at extending the H-infinity Bounded Real Lemma to stochastic systems under random disturbances with imprecisely known probability distributions. The statistical uncertainty is measured in entropy theoretic terms 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. A state-space sufficient criterion for the anisotropic norm of a linear discrete time invariant system to be bounded by a given threshold value is derived. The resulting Strict Anisotropic Norm Bounded Real Lemma involves an inequality on the determinant of a positive definite matrix and a linear matrix inequality. It is shown that slight reformulation of these conditions allows the anisotropic norm of a system to be efficiently computed via convex optimization.

💡 Research Summary

The paper “A Convex Formulation of Strict Anisotropic Norm Bounded Real Lemma” extends the classical H‑∞ Bounded Real Lemma (BRL) to discrete‑time linear systems that are driven by stochastic disturbances with partially unknown probability distributions. The uncertainty of the disturbance is quantified by the mean anisotropy functional, an entropy‑based measure that captures how far a distribution deviates from a zero‑mean Gaussian with scalar covariance. For a given bound a > 0 on the mean anisotropy, the admissible disturbance set consists of all Gaussian‑white‑noise shaped by stable filters whose anisotropy does not exceed a.

The central performance metric is the a‑anisotropic norm ‖F‖ₐ of a system F, defined as the worst‑case ratio of output to input power over all admissible disturbances. This norm interpolates between the H₂ norm (a → 0) and the H∞ norm (a → ∞), thus providing a unified framework for robust stochastic control.

The main contribution is the Strict Anisotropic Norm Bounded Real Lemma (SANBRL). For a linear discrete‑time invariant (LDTI) system described by the state‑space equations xₖ₊₁ = Axₖ + Bwₖ, zₖ = Cxₖ + Dwₖ, the lemma states that ‖F‖ₐ < γ if and only if there exist a scalar q ∈ (0, min{γ⁻², ‖F‖_∞⁻²}) and a positive‑definite matrix R ≻ 0 such that two convex conditions hold:

1. A determinant inequality
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