A ν-gap distance for uncertain and nonlinear systems

This paper extends results on the approximation of uncertain systems in the ν-gap metric (1999), and those on the calculation of ν-gap distances for nonlinear systems (1998). We wish to be able to approximate nonlinear systems with complex uncertainty descriptions by linear systems with simpler uncertainty descriptions which are more amenable to robust control system design, and to evaluate the approximation errors in some suitable way. To this end, given two sets of systems P₀ and P₁, each pathwise connected in the graph topology and with a non-empty intersection, we define a directed distance δ¯(P₀, P₁). This distance provides a bound on the worst case performance of any controller over the set P₁ in terms of its worst case performance over the set P₀. So, if both δ¯(P₀, P₁) and δ¯(P₁,P₀) are small, then the worst case performance of any feedback controller over each set is similar. If we regard P₀ as a (simpler) approximation of P₁, and if δ¯(P ₀, P₁) is small, then any controller designed to function well over the set P₀ is also guaranteed to work well over the set P₁ If, in addition, δ¯(P₁,P₀) is small then any controller that fails to work well with some member of P₀ will also fail to work well with some member of P₁. That is, δ¯(P₁, P₀) is a measure of how conservative the approximation is. The first main result is an appropriate generalization to the nonlinear case of the ν-gap metric, which reduces to the ν-gap metric when applied to LTI systems. We then give two different definitions of S for uncertain systems. The first is based on the nonlinear ν-gap, and hence is applicable to nonlinear systems, the second is defined as the supremum over frequency of a pointwise distance and so is only applicable to LTI systems. Both definitions have the desired properties, and both are equivalent to the ν-gap when applied to the distance between single systems. The second main result of this paper is that, for sets of LTI systems, the second distance is less than the first. So, it would appear that there is no general definition of the distance between sets of systems which specializes to the right (i.e., least conservative) notion in the LTI case