Bibo Stabilisation of Continuous–Time Takagi–Sugeno Systems under Persistent Perturbations and Input Saturation

Abstract

This paper presents a novel approach to the design of fuzzy state feedback controllers for continuous-time non-linear systems with input saturation under persistent perturbations. It is assumed that all the states of the Takagi-Sugeno (TS) fuzzy model representing a non-linear system are measurable. Such controllers achieve bounded input bounded output (BIBO) stabilisation in closed loop based on the computation of inescapable ellipsoids. These ellipsoids are computed with linear matrix inequalities (LMIs) that guarantee stabilisation with input saturation and persistent perturbations. In particular, two kinds of inescapable ellipsoids are computed when solving a multiobjective optimization problem: the maximum volume inescapable ellipsoids contained inside the validity domain of the TS fuzzy model and the smallest inescapable ellipsoids which guarantee a minimum *-norm (upper bound of the 1-norm) of the perturbed system. For every initial point contained in the maximum volume ellipsoid, the closed loop will enter the minimum *-norm ellipsoid after a finite time, and it will remain inside afterwards. Consequently, the designed controllers have a large domain of validity and ensure a small value for the 1-norm of closed loop.

T–S fuzzy BIBO stabilisation of non-linear systems under persistent perturbations using fuzzy Lyapunov functions and non-PDC control laws

“…This work extends the results presented by Salcedo and Martinez (2008) or Salcedo et al (2018) when a non-PDC state-feedback controller is designed when minimising an upper bound for the 1-norm between φ(t) and y(t). Salcedo and Martinez (2008) or Salcedo et al (2018) considered only PDC controllers.…”

“…530 J.V. Salcedo et al 2018;Vafamand and Shasadeghi, 2017;Yoneyama, 2017;Vafamand, 2020a;2020b). In addition, some references have dealt with the case of persistent perturbations (Hu et al, 2019;Vafamand et al, 2016;2017b;2017a).…”

“…In this paper innovative approaches are developed to improve previous publications which deal with persistent perturbations (Salcedo and Martinez, 2008;Salcedo et al, 2018;Vafamand et al, 2017b;Hu et al, 2019). These innovative methodologies can be summarised as follows:…”

“…• Extension of the concepts of the -norm and inescapable ellipsoids corresponding to CQLFs (Salcedo and Martinez, 2008;Salcedo et al, 2018) when using FLFs, IDLFs and non-PDC controllers under persistent perturbations.…”

“…It is more complex to calculate the 1-norm than the 2-norm or the H ∞ -norm (Abedor et al, 1996;Sanchez Peña and Sznaier, 1998), although an upper bound can be estimated for it, called the star ( ) norm, based on LMIs only (Abedor et al, 1996;Salcedo et al, 2018). This bound allows us to tackle the design of fuzzy controllers using existing techniques (Tanaka and Wang, 2001;Liu and Zhang, 2003;Teixeira et al, 2003;Guerra et al, 2006).…”