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• W1489337560 abstract "This paper extends results on the approximation of uncertain systems in the /spl nu/-gap metric (1999), and those on the calculation of /spl nu/-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 P/sub 0/ and P/sub 1/, each pathwise connected in the graph topology and with a non-empty intersection, we define a directed distance /spl delta/~(P/sub 0/, P/sub 1/). This distance provides a bound on the worst case performance of any controller over the set P/sub 1/ in terms of its worst case performance over the set P/sub 0/. So, if both /spl delta/~(P/sub 0/, P/sub 1/) and /spl delta/~(P/sub 1/,P/sub 0/) are small, then the worst case performance of any feedback controller over each set is similar. If we regard P/sub 0/ as a (simpler) approximation of P/sub 1/, and if /spl delta/~(P/sub 0/, P/sub 1/) is small, then any controller designed to function well over the set P/sub 0/ is also guaranteed to work well over the set P/sub 1/ If, in addition, /spl delta/~(P/sub 1/,P/sub 0/) is small then any controller that fails to work well with some member of P/sub 0/ will also fail to work well with some member of P/sub 1/. That is, /spl delta/~(P/sub 1/,P/sub 0/) is a measure of how conservative the approximation is. The first main result is an appropriate generalization to the nonlinear case of the /spl nu/-gap metric, which reduces to the /spl nu/-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 /spl nu/-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 /spl nu/-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." @default.
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• W1489337560 date "2003-01-22" @default.
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• W1489337560 title "A ν-gap distance for uncertain and nonlinear systems" @default.
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• W1489337560 doi "https://doi.org/10.1109/cdc.1999.831313" @default.
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