Anisotropy-Based Robust Performance Criteria for Linear Continuous Time Invariant Systems with Statistically Uncertain Noise

This paper is concerned with robust performance analysis for continuous time invariant systems governed by linear stochastic differential equations driven by statistically uncertain Ito processes. The uncertainty is understood as the deviation of imprecisely known probability distributions of the input disturbance from those of isotropic white-noise disturbances which, up to scaling, are organized as a standard Wiener process. Using Tustin’s transform with a one-parameter family of conformal maps of the unit disk in the complex plane onto the right half-plane for discrete and continuous time transfer functions, the deviation from the nominal isotropic white-noise model is quantified by the mean anisotropy for the input of a discrete-time counterpart of the original system. The parameter of this conformal correspondence specifies the time scale for filtered versions of the input and output of the system, in terms of which the worst-case root mean square gain is formulated subject to an upper constraint on the mean anisotropy. The resulting two-parameter counterpart of the anisotropy-constrained norm of the system for the continuous time case is computed in state space by using the methods of the discrete-time anisotropy-based theory of stochastic robust filtering and control, originated by the author in the mid-1990s. These results are illustrated by an application to an electric circuit with inductive coupling and statistically uncertain noise.