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Polytopic quasi-linear parameter-varying (quasi-LPV) models of nonlinear processes allow the usage linear matrix inequalities (LMIs) to guarantee some performance goal on them (in most cases, locally, over a so-called modeling region). In order to get a finite number of LMIs, nonlinearities are embedded on the convex hull of a finite set of linear models. However, for a given system, the quasi-LPV representations are not unique, yielding different performance bounds depending on the model choice. To avoid such drawback, earlier literature on the topic used annihilator-based approaches, which require gridding on the modeling region, and nonconvex BMI conditions for controller synthesis; optimal performance bounds are obtained, but with a huge computational burden. This paper proposes building a model by minimizing the projection of the nonlinearities onto directions, which are deleterious for performance.For a small modeling region, these directions are obtained from LMIs with the linearized model. Additionally, these directions will guide the selection of the polytopic embedding's vertices. The procedure allows gridding-free LMI controller synthesis, as in standard LPV setups, with a very reduced performance loss with respect to the aforementioned BMI+gridding approaches, at a fraction of the computational cost. KEYWORDS gain scheduling, linear matrix inequalities, linear-parameter-varying systems, quasi-LPV systems, robust control, Takagi-Sugeno systems 1230

Given a nonlinear system, the sector-nonlinearity methodology provides a systematic way of transforming it in an equivalent Takagi-Sugeno model. However, such transformation is not unique: conservatism of shape-independent performance conditions in the form of linear matrix inequalities results in some models yielding better results than others. This paper provides some guidelines on choosing a sector-nonlinearity Takagi-Sugeno model, with provable optimality (in a particular sense) in the case of quadratic nonlinearities. The approach is based on Hessian and restrictions of a function onto a subspace.

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This report generalises recent results on stability analysis and estimation of the domain of attraction of nonlinear systems via exact piecewise affine Takagi-Sugeno models. Algorithms in the form of linear matrix inequalities are proposed that produce progressively better estimates which are proved to asymptotically render the actual domain of attraction; regions already proven to belong to such domain of attraction can be removed and the estimate can contain significant portions of the modelling region boundary; in this way, level-set approaches in prior literature can be significantly improved. Illustrative examples and comparisons are provided.