Performance‐oriented quasi‐LPV modeling of nonlinear systems

Abstract

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

Affine linear parameter‐varying embedding of non‐linear models with improved accuracy and minimal overbounding

“…Next to the computationally intensive methods in [3], recent developments include approaches based on linear fractional representation with an NL feedback block converted to an LPV model depending affinely on the scheduling variables in [15]. Choosing an LPV embedding for NL systems is investigated in [16] by minimising the projection of the non‐linearities onto directions deleterious for performance. The later problem is cast as a computationally intensive linear matrix inequality (LMI) based optimisation.…”

Gain‐scheduled control for discrete‐time non‐linear parameter‐varying systems with time‐varying delays

“…Often, uncertain LTI systems can be seen as a particular case of LPV systems. Besides, to make an additional distinction concerning pure LPV systems, if the time-varying parameters depend on some systems states (similar to membership functions in TS models), the system is referred to as qLPV system [17][18][19]. Notwithstanding, NLPV systems can preserve a nonlinear structure instead of reducing it to a purely LPV system.…”

Set‐based gain‐scheduled control via quasi‐convex difference inclusions

“…2 Determining contractive sets for LPV systems is a well-studied topic, and it can be approached either from set-based computations 1,[3][4][5][6] or via convex optimization (linear matrix inequalities, LMI). [7][8][9][10][11] Smooth nonlinear systems can be easily embedded in a polytopic LDI, giving rise to quasi-LPV models; [12][13][14][15][16] thus, LPV results can be applied to prove stability in some nonlinear control problems, with, of course, a dose of conservatism; 9 in fact, the quasi-LPV model of a nonlinear system is not unique, so the best one might depend on the required performance objectives. 16 A broader class of models is that of nonlinear parameter varying models (NLPV), x + = f (x,d,h,u); however, as such, they are too general to be useful.…”