Stabilization of fuzzy systems with constrained controls by using positively invariant sets

Abstract

We deal with the extension of the positive invariance approach to nonlinear systems modeled by Takagi-Sugeno fuzzy systems. The saturations on the control are taken into account during the design phase. Sufficient conditions of asymptotic stability are given ensuring at the same time that the control is always admissible inside the corresponding polyhedral set. Both a common Lyapunov function and piecewise Lyapunov function are used.

Stabilization of saturated discrete-time fuzzy systems

“…Two main approaches have been developed in the literature: The first is the so-called positive invariance approach, which is based on the design of controllers that work inside a region of linear behavior where saturations do not occur (see [1,2,7] and the references therein). This approach has been extended to systems modelled by T-S systems [4,12]. The second approach, allows saturations to take effect, while guaranteeing asymptotic stability (see [5,15] and the references therein).…”

Stabilization of controlled positive discrete‐time T‐S fuzzy systems by state feedback control

“…The nonlinear system is represented by a set of linear models interpolated by membership functions and then model-based fuzzy controller is developed to stabilize the T-S fuzzy model by solving a set of linear matrix inequalities (LMIs) [4][5][6][7][8][9][10][11] and the references therein. Other approaches as positive invariance were also used [12].The most available methods use quadratic common Lyapunov function. However, it was recently proven for hybrid systems [13] and fuzzy systems [4,5,14] that the use of piecewise quadratic Lyapunov function leads to better results in the sense that a common quadratic Lyapunov function may not exist while a multiple one exists.…”

“…The nonlinear system is represented by a set of linear models interpolated by membership functions and then model-based fuzzy controller is developed to stabilize the T-S fuzzy model by solving a set of linear matrix inequalities (LMIs) [4][5][6][7][8][9][10][11] and the references therein. Other approaches as positive invariance were also used [12].…”

Robust stabilization and observation of positive Takagi-Sugeno systems

“…Our thesis work considers an additional problem that we frequently encounter in several dynamical systems: the nonnegativity of the states. (Benzaouia et al, 2006), (Benzaouia & Tadeo, 2010), , (Benzaouia et al, 2007), (Boukas & El Hajjaji, 2006), (El Hajjaji et al, 2006), (El Hajjaji & Chadli, 2008). The study of systems with nonnegative states is important in practice because many chemical, physical and biological processes involve quantities that have intrinsically constant and nonnegative signs: the concentration of substances, the levels of liquids, etc, are always nonnegative.…”