Reduced-Complexity LMI Conditions for Admissibility Analysis and Control Design of Singular Nonlinear Systems
Amine Dehak, Anh‐Tu Nguyen, Antoine Dequidt, Laurent Vermeiren, Michel Dambrine
Abstract
We present a reduced-complexity control approach for a class of descriptor nonlinear systems with a nonlinear derivative matrix, possibly singular. To this end, a systematic approach is proposed to obtain an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">equivalent</i> polytopic representation of a given nonlinear system within a compact set of the state-space. This modeling approach has two particular features compared to the related Takagi–Sugeno (T–S) fuzzy model-based framework. First, the model complexity only grows <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">proportionally</i> , rather than exponentially, with the number of premise variables. Second, the vertices of the proposed polytopic models can admit an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infinite</i> number of representations for the same predefined set of premise variables. This <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonuniqueness</i> feature allows introducing some specific slack variables at the modeling step to reduce the control design conservatism. Based on the proposed polytopic representation and Lyapunov stability theory, we derive reduced-complexity admissibility analysis and design conditions, expressed in terms of linear matrix inequalities, for the considered class of descriptor systems. In particular, a new nonlinear control law is proposed for regular descriptor systems to avoid using the extended redundancy form, which may yield numerically complex and conservative results due to the imposed special control structure. Both numerical and physically motivated examples are given to demonstrate the interests of the new control approach with respect to existing T–S fuzzy model-based control results.