Vol.:(0123456789) 1 3 Granular Computing https://doi.org/10.1007/s41066-018-0138-x ORIGINAL PAPER Robust functional observer for stabilising uncertain fuzzy systems with time‑delay Syed Imranul Islam 1  · Peng Shi 1  · Cheng‑Chew Lim 1 Received: 7 August 2018 / Accepted: 5 October 2018 © Springer Nature Switzerland AG 2018 Abstract This paper presents a new technique for stabilising a Takagi–Sugeno (T-S) fuzzy system with time-delay and uncertainty. A robust fuzzy functional observer is employed to design a controller when the system states are not measurable. The model uncertainty is norm bounded, and the time-delay is time-varying but bounded. The parallel distributed compensation method is applied for defning the fuzzy functional observer to design this controller. The proposed procedure reduces the observer order to the dimension of the control input. Improved stability conditions are provided for the observer compared with the existing results of functional observer-based stabilisation of T-S fuzzy models. Lyapunov–Krasovskii functionals are used to construct delay-dependent stability conditions as linear matrix inequalities. The solution of these inequalities is used for calculating the observer parameters. The sensitivity of the estimation error to the model uncertainty is reduced by minimis- ing the L 2 gain. The new design method developed is illustrated and verifed using two examples. Keywords Takagi–Sugeno fuzzy model · Functional observer · Time-delay · Robust controller design 1 Introduction A functional observer estimates the function of states directly. The design problem of the functional observer has been an active research feld for the last few decades for its ability to estimate the function of states in a single step rather than performing in two steps. It also reduces the observer order. The existence conditions, stability analysis and construction procedure of functional observers for linear systems are well established (Darouach 2000; Ha et al. 2003; Trinh and Fernando 2007; Mohajerpoor et al. 2016); the existence conditions are presented as rank equality condi- tions while the stability conditions are presented as linear matrix inequalities (LMIs). The efects of parametric uncer- tainty and time-delay on the functional observer for linear systems are studied in Darouach (2001), Teh and Trinh (2012), Tran et al. (2015) and Boukal et al. (2016). The design and application of functional observers for nonlin- ear systems represented by fuzzy models, however, received less attention. The concept of fuzzy sets proposed by Zadeh (1965) has started a new era in set theories. Fuzzy sets have been suc- cessfully applied in classifcation and system identifcation problems (Wang and Chen 2008; Chen and Chang 2011; Chen et al. 2012; Wang et al. 2017; Yordanova et al. 2017; Lai et al. 2018; Liu and Zhang 2018). Many modern systems have been modeled by fuzzy reasoning. The fuzzy reasoning comprises fuzzy inference rules described by “IF-THEN” statements. “IF” statements are called premises while “THEN” statements are called consequents. Takagi–Sug- eno (T-S) fuzzy modeling is an efcient way of representing a highly nonlinear system in a simple way by applying the fuzzy reasoning. The overall system dynamics is expressed as a fuzzy summation of the linear consequents of fuzzy rules of a T-S fuzzy model (Takagi and Sugeno 1985). The linear consequent models of a T-S fuzzy model are intercon- nected with each other by membership functions to represent a nonlinear system for any degree of accuracy (Feng 2006). As a consequence, this modeling technique enables the use of existing linear tools and techniques for analysing and synthesising diferent problems of nonlinear systems. The stability of these model-based systems has been a vibrant research area for a long time. Controller design problem for nonlinear systems using T-S fuzzy model has been an active research area (Sun et al. * Syed Imranul Islam syedimranul.islam@adelaide.edu.au 1 School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide, SA 5005, Australia