Switching TS fuzzy model-based dynamic sliding mode observer design for non-differentiable nonlinear systems

Abstract

This paper presents four sliding mode observers (SMOs) based on a novel approach in Takagi–Sugeno (TS) fuzzy modeling of multi-input multi-output (MIMO) non-linear systems that have non-differentiable operating points. A comprehensive approach is proposed to using the TS fuzzy model (TSFM) in the field of non-differentiable nonlinear systems, where the TSFM is an approximation with high accuracy and a negligible error (2$\epsilon$) of the nonlinear model. Furthermore, the considered system can be with measurable or unmeasurable premise variables. The observers are synthesized for the above two cases and dynamic observers for state estimations of MIMO non-linear Lipschitz systems. The dynamic gain of the observer is established from inspiring state-space representation of an LTI system with error as input, internal states, and the gain of the observer as output. The dynamics used in the gain of the observer will increase the degrees of freedom in the design procedure and a generalization to the one used in previous works. The proposed method is applicable for continuous-time, but not necessarily differentiable, nonlinear systems. Considering the inherent strongly nonlinear and coupling performance of the plant, the switching method driven by states is presented. This paper presents a comparison of four SMOs and multiple-model adaptive estimation (MMAE) for benchmark hydraulic wind power transfer (HWPT). Simulation results demonstrate improvement in the state observation convergence rate and simplicity and universality of the proposed approach.

Introduction

Dynamics of many industrial systems are described by nonlinear equations including non-differential or highly nonlinearity terms to enumerate a few: quadruple tank process (QTP) (Zare and Koofigar, 2014), renewable energy (Farbood et al., 2018), manipulator (Zhao et al., 2016), chemical process (Johansson, 2000), etc. Stability analysis and control design of such complicated nonlinear systems are one of the challenging research fields in control theory. The first step to analyze and design in the control is the system representation. Moreover, TSFM is known as a powerful tool to represent nonlinear systems. TSFM facilitates extending convex optimization techniques for such complicated systems. Representation of non-differential nonlinear systems in terms of TSFM is known as a hot and difficult research field in control theory. Up to now, there is no comprehensive study in this area (Zeng, 2014). One possible solution to describe highly nonlinear systems is switching TSFM (Lam et al., 2004, Tanaka et al., 2001).

A few articles working on TSFM of non-differentiable/highly nonlinearity terms in the nonlinear systems dynamics can be classified into continuous and discontinuous terms. Continuous and differentiable terms can be represented by linear TSFM. Continuous but non-differentiable terms cannot be described by linear TSFM and instead can be transformed to affine TSFM. For example, the square root function as a continuous and non-differentiable term cannot be described by linear TSFM (Zeng, 2014). The other category is about the discontinuous terms e.g. sign function which is stated that there does exist not only linear TSFM but also affine TSFM. Of course, there are different techniques to approximate such terms in TSFM as following: (a) approximating the non-smooth function by a smooth function, such as sigmoid function (Danca, 2015), hyperbolic tangent function (Na et al., 2014), and polynomial function (Bustamante et al., 2012), (b) representation by piecewise linear model (Li et al., 2009, Saifia et al., 2015), (c) considering as system disturbance (Vafamand et al., 2017). Therefore, the investigation of the research works concentrated on the non-differentiable terms shows that the TSFM representation is a necessity and an interesting topic that deserves more study.

TSFM representation of non-differentiable nonlinear systems provides a background to manipulate varieties of control issues including stability analysis, controller design, observer design, fault tolerant control, and so on. In some practical applications, the state variables of plants cannot be completely or partially measured (Kim et al., 2011) because of physical or sensors limitations. In these conditions, the state variables can be estimated from the available knowledge through the system’s inputs and outputs. The design of suitable observers/estimators has several applications in fault detection, observer-based control, process identification, and chaotic synchronization (Oveisi et al., 2018, Pierri et al., 2008), and so on. Several types of observers have been developed in the literature including Luenberger observers (Tanaka et al., 1998), unknown input observers (Blandeau et al., 2018), high gain observers (Khalil and Praly, 2014), proportional integral observer (Xie et al., 2017), the adaptive method (Kharrat et al., 2018), and $H_{\infty }$ observer (Xie et al., 2019). Moreover, the SMO design problem has attracted considerable interest of many researchers since it offers robustness properties concerning uncertainties and disturbances (Slotine et al., 1986, Utkin, 1977). Using the similar design principles of the sliding mode control (SMC), the trajectories of the estimation error of the SMO are constrained to reach estimation error sliding surface in a finite-time such that the error is less insensitive to the disturbances or uncertainties. However, TSFM representation of complicated non-differential nonlinear systems provides the possibility to establish the observer design based on its TSFM. Some excellent works about TSFM based observer design can be cited (Delrot et al., 2012, Ichalal et al., 2018).

In Palm and Bergsten (2000), a SMO is designed with matched and unmatched uncertainties. It uses a transformation of the locally linearized model into a specific canonical form and assumes the nonlinear system to be linear dominant within a certain operating region. In Bergsten et al. (2002), a TS fuzzy system where the local affine dynamic models are off-equilibrium local linearization is investigated and two SMOs (the extension of Fuzzy-Thau Luenberger observer and dealing the affine TS fuzzy system as a dominant linear system subjected to known and unknown uncertainty to inspire and design SMO for linear system) are designed for such systems. Model/plant mismatches are represented as matched (the uncertainties appear in the output channels only) and unmatched uncertainties (the uncertainties do not appear in the output channels) (Bergsten et al., 2002). The analysis and design of a SMO are investigated on the basis of a TSFM subject both to unknown inputs and uncertainties in Akhenak et al. (2007). A SMO based SMC is designed for the TS fuzzy system to obtain both the state estimation and tracking control in Van et al. (2013). Special attention has already been paid to the application of SMO design for TS fuzzy system subject to fault diagnosis (Georg and Schulte, 2014). The authors in Li and Zhang (2019) developed the problem of SMO design for TS fuzzy descriptor fractional-order systems. Depending on the observer gains definition and design, the observers are categorized into two classes: static and dynamic observers. Static observer uses a constant gain, meanwhile, the dynamic observer makes use of dynamic gains which provides more degrees of freedom during design procedure and then brings more flexibility to achieve more accurate estimations and speedy convergence rate (Park, 1999). In recent years, dynamic observer has attracted more attention and the researchers further study on dynamic observer. Design of $H_{\infty }$ dynamic observer is studied in Pertew et al. (2005b) and its application in fault diagnosis and Lipschitz non-Linear systems is investigated. In Golabi et al. (2012), a robust controller is designed for uncertain TS fuzzy systems with $H_{\infty }$ performance through fuzzy dynamic observer with the cost of dimension increases. Design of dynamic observer for linear time-invariant systems is studied in Park et al. (2002) and shown the essential characteristics to be qualified as an observer such as the asymptotic behavior and the separation property hold. Special attention has already been paid to the application of $H_{\infty }$ dynamic observer design subject to fault diagnosis in Pertew et al. (2005a). In summary, TSFM based dynamic observer needs more attentions and research. Additionally, their extension to complicated non-differentiable nonlinear system seems interesting to be a challenging research topic.

In this paper, an approach is proposed for representing the non-differentiable nonlinear systems in TSFM and, moreover, a novel dynamic SMO (DSMO) is designed for the above-mentioned systems. The main contributions of this paper are enumerated as follows: (a) to represent the non-differentiable nonlinear system in terms of TSFMs, (b) to design not only static but also dynamic SMOs. In the literature, the SMOs have been designed using static observer gain. Based on the proposed TSFM for the non-differentiable nonlinear system, first, a static SMO is designed. Then, the SMO is extended to dynamic type. Two scenarios for both static and dynamic SMOs design are considered based on whether the premise variables are measurable or not. Finally, the proposed idea to transform the non-differentiable nonlinear system to TSFM is applied on HWPT, as a cross-coupled MIMO laboratory setup. HWPT systems are a new type of wind turbine drivetrains that offers several advantages over gearbox-based wind turbine drivetrains. Moreover, comparative results are presented to show the effectiveness of the proposed SMO method.

The rest of the paper is organized as follows: In Section 2, the model of a MIMO Lipschitz nonlinear system is introduced and the switching TSFM is established to represent the uncertain non-differentiable nonlinear systems. In Section 3, observer design procedures are discussed. Application of the proposed TSFM and observer design algorithms on HWPT and the simulation results are demonstrated in Section 4. Finally, the concluding remarks and future works are given in Section 5.