An Exact, Finite Dimensional Representation for Full-Block, Circle Criterion Multipliers

TL;DR

This paper introduces a finite-dimensional, computationally tractable characterization of the complete set of full-block, circle criterion multipliers for stability analysis of feedback systems with sector-bounded nonlinearities, reducing conservatism.

Contribution

It provides the first finite-dimensional, exact description of all full-block, circle criterion multipliers, enabling practical stability analysis with reduced conservatism.

Findings

Finite-dimensional characterization for full-block multipliers.

Exact conditions involve a finite number of matrix copositivity constraints.

Implementation is feasible for nonlinearities with input/output dimensions up to 4.

Abstract

This paper provides the first finite-dimensional characterization for the complete set of full-block, circle criterion multipliers. We consider the interconnection of a discrete-time, linear time-invariant system in feedback with a non-repeated, sector-bounded nonlinearity. Sufficient conditions for stability and performance can be derived using: (i) dissipation inequalities, and (ii) Quadratic Constraints (QCs) that bound the input/output pairs of the nonlinearity. Larger classes of QCs (or multipliers) reduce the conservatism of the conditions. Full-block, circle criterion multipliers define the complete set of all possible QCs for non-repeated, sector-bounded nonlinearities. These provide the least conservative conditions. However, full-block multipliers are defined by an uncountably infinite number of constraints and hence do not lead to computationally tractable solutions if left in this raw form. This paper provides a new finite-dimensional characterization for the set of full-block, circle criterion multipliers. The key theoretical insight is: the set of all input/output pairs of non-repeated sector-bounded nonlinearities is equal to the set of all incremental pairs for an appropriately constructed piecewise linear function. Our new description for the complete set of multipliers only requires a finite number of matrix copositivity constraints. These conditions have an exact, computationally tractable implementation for problems where the nonlinearity has small input/output dimensions $(\le 4)$. We illustrate the use of our new characterization via a simple example.