Complexity reduction of Nonlinear Systems

A common problem in nonlinear control is the need to consider systems of high complexity. Here we consider systems, which although may be low order, have high complexity due to a complex right hand side of a differential equation (e.g. a right hand side which has many terms – such systems arise from coordinate transformations in constructive nonlinear control designs). This contribution develops a systematic method for the reduction of this complexity, complete with error bounds. In the case when the underling nonlinear system input/output operator is stable and differentiable, the operator Taylor expansion, truncated after a finite number of terms, is taken as the approximation. If the nonlinear system i/o operator is not stable, but admits a coprime factorizations, the Taylor approximation is made to both coprime factors. By bounding the gap between the polynomial system and the original nominal plant, and applying gap robust stability approaches, it is proved that local stability of approximation implies the local stability of the underlining nonlinear systems, and explicit robust stability margins and performance bounds obtained. For systems specified by a finite dimensional first order differential equation, the first order approximant is the system linearisation and the higher order approximants have greater state dimension but with polynomial right hand sides.