A T–S Fuzzy Approach to the Local Stabilization of Nonlinear Discrete-Time Systems Subject to Energy-Bounded Disturbances

Abstract

This paper addresses the local stabilization problem of nonlinear discrete-time systems subject to energybounded disturbances by means of T-S fuzzy models. A fuzzy state feedback controller is designed such that the input-to-state stability in the 2 sense of the original nonlinear system is guaranteed in a bounded region of the state space. Such a region is related to the exactness of the T-S model and describes the domain around the origin where the convexity property remains valid. In addition, an estimate of the closed-loop reachable set is provided for a given class of 2 disturbances. Three (convex) optimization problems are proposed to either minimize the estimate of the reachable set, improve the disturbance tolerance or minimize the 2 -gain from the disturbance input to the regulated output. Numerical examples are considered to illustrate the approach demonstrating the effectiveness of the proposed technique for the control synthesis of nonlinear discrete-time systems.

Smoothing switched control laws for uncertain nonlinear systems subject to actuator saturation

“…Premultiplying and postmultiplying (16) and (20) by X D P 1 and performing the change of variables (7), if x.0/ 2 E.P; ı/ then x.t/ 2 E.P; ı/ for all t > 0. Considering Lemma 1 with ı D 1, LMIs (12c) ensure E.P; 1/ L.H k /, a sufficient condition such that (14) holds [31,32]; and also E.P; 1/ X , a sufficient condition for a state trajectory started with an initial condition x.0/ 2 E.P; 1/ to be in X for all t > 0 [34]. The proof is complete.…”

“…The problem of ensuring that the system state of (3), started with an initial condition in a suitable region within a given operating region, will remain in the operation region for all t > 0 is addressed, for example, in [34], considering an operation region X , ® x.t/ 2 < n W jN .h/ x.t/j 6 .h/ ; h 2 I p¯; (5) where N D h N The solution is achieved by determining an invariant region, say E X , such that, for all x.0/ 2 E, the state vector x.t/ of (3) remains in the operating region X for all t > 0. Consequently, (4) exactly represents the system (3) for t > 0 and all x.0/ 2 E.…”

Controle Chaveado de Sistemas Lineares Incertos Sujeitos à Saturação no Atuador Usando Realimentação Derivativa

“…com N e φ conhecidas. N (h) representa a linha h da matriz N e φ h a entrada h do vetor φ (Klug et al, 2015;Alves et al, 2016;de Oliveira et al, 2018). Teorema 1.…”

Estabilidad de sistemas Takagi-Sugeno bajo perturbaciones persistentes: estimación de conjuntos inescapables

“…La literatura sobre conjuntos inescapables en elámbito borroso es escasa. Básicamente, los autores sólo tienen constancia de los trabajos de Salcedo et al (2008); Wang y Liu (2013) y Klug et al (2015). El primero aborda el rechazo de perturbaciones persistentes con un pico máximo de amplitud (w T w ≤ γ), generalizando el trabajo de Abedor et al (1996) al caso borroso, mientras que los restantes dos abordan también el caso de perturbaciones de energía finita ( ∞ 0 w T w dt ≤ γ).…”

“…En el trabajo de Wang y Liu (2013) se plantea un análisis local para sistemas TS sujetos a perturbaciones con magnitud acotada o energía finita en donde, además de las condiciones H ∞ habituales, se incluyen condiciones para asegurar que ciertos conjuntos son invariantes. Recientemente, Klug et al (2015) han abordado también dicha problemática para el caso discreto, planteando problemas de optimización (convexa) para dar un estimado del mínimo conjunto alcanzable o una cota de la ganancia en norma-2 entre la entrada de perturbación y la salida controlada. Sin embargo, este trabajo no considera perturbaciones persistentes.…”