An exact handling of the gradient for overcoming persistent problems in nonlinear observer design via convex optimization techniques

Abstract

This paper proposes a comprehensive solution to persistent problems faced by nonlinear observer design via convex optimization techniques, namely, the construction of an exact error model and the handling of unmeasurable premises. Instead of using the differential mean value theorem, it is shown that the error system is always exactly amenable, via algebraic rearrangements, to an explicit matrix polynomial form of the gradient, where the error signal appears factorized at the rightmost side, thus allowing Lyapunov-based derivation of conditions in the form of linear matrix inequalities or sum of squares. An effective exploitation of the nonlinear characteristics of the observer while naturally decoupling available and non-available signals, is achieved via exact Takagi-Sugeno tensor-product models or polynomial rewriting of the error system, not the plant; generalizations such as H∞ disturbance rejection or decay rate are straightforward. In contrast with recent approaches, the output is not required to be linear nor the systems involved to be affine-in-control. Illustrative examples are provided that overcome recent approaches on the subject.