Distance Measures for Uncertain Linear Systems: A General Theory

In this paper, we propose a generic notion of distance between systems that can be used to measure discrepancy between open-loop systems in a feedback sense under several uncertainty structures. When the uncertainty structure is chosen to be four-block (or equivalently, normalized coprime factor) uncertainty, then this generic distance measure reduces to the well-known nu-gap metric. Associated with this generic distance notion, we also define a generic stability margin notion that allows us to give the distance measure a feedback interpretation by deriving generic robust stability and robust performance results. The proposed distance notion and the corresponding results exploit a powerful generalization of the small-gain theorem which handles perturbations in RfrL _infin, rather than only in RfrH _infin. When the uncertainty structure is fixed to one of the standard structures (e.g., additive, multiplicative, inverse multiplicative, coprime factor, four-block or any mixtures of the above), we give a step-by-step procedure (based on model validation ideas) that shows how the generic notion of distance and the correspondingly generic winding number conditions can be reduced to simple formulae. This work provides a unified framework that captures and embeds previous results in this area and also completes the picture by showing how other results of a similar nature can be obtained from the same framework. The techniques used involve only basic linear algebra, so they also provide a simplification of previous advanced proofs. Furthermore, the various distance measures so created can be used for non-conservative model embedding into the smallest uncertain family. An illustrative example is also given that demonstrates the superior qualities, above the nu-gap metric, of a particular distance measure obtained from this work in situations where the plant is lightly-damped. All systems considered in this paper are linear time-invariant.