Takagi-Sugeno fuzzy models within orthonormal basis function framework and their application to process control

NOE TS fuzzy modelling of nonlinear dynamic systems with uncertainties using symbolic interval-valued data

Takagi-Sugeno (T-S) fuzzy model is a system described by a set of fuzzy if-then rules which give local linear representations of the underlying nonlinear system. Usually, a fuzzy control law is constructed by a set of controllers for each local linear representation to carry out the control of a T-S fuzzy system. While, simultaneous control addresses the stability and performance of multiple plants under a single feedback controller. In this paper, simultaneous linear control for nonlinear uncertain systems described by general T-S fuzzy model is considered. A procedure for simultaneous control of T-S fuzzy systems with parametric uncertainties is presented, by which robust reliability method is adopted to deal with uncertainties. Simultaneously linear controller design for an uncertain T-S fuzzy system is carried out by solving a problem of robust reliability based optimization, in which, both the robustness with respect to uncertainties and the control cost can be taken into account simultaneously. It is demonstrated by numerical simulation of the chaotic Lorenz system control that the presented method is effective and feasible.

Fuzzy models within the framework of orthonormal basis functions (OBF fuzzy models) were introduced in previous works and have shown to be a very promising approach to the areas of non-linear system identification and control since they exhibit several advantages over those dynamic model architectures usually adopted in the literature. In the present paper these models are reviewed and used as a basis for a predictive control scheme which is applied to the control of a process for ethyl alcohol (ethanol) production.

In this paper, the issue of human behavior learning (HBL) is addressed for a class of nonlinear human-in-the-loop (HiTL) systems where the human operator is viewed as a nonlinear optimal controller. Owing to its outstanding interpretability and strong nonlinear representation capability, the Takagi-Sugeno (T-S) fuzzy model is employed to represent the nonlinear HiTL control system and approximate the unknown human control law based on the parallel distributed compensation (PDC) scheme. A quadratic-like cost function with fuzzy weighting matrices is built to depict the human behavior, which conforms to human thinking and is unknown to the machine. The aim of the HBL is to retrieve the fuzzy weighting matrices such that the human control law will be optimal in the sense of minimizing the retrieved cost function. In the proposed HBL scheme, the state-dependent Riccati equation (SDRE) based nonlinear optimal control technique plays an important role, which has a similar structure to the linear quadratic regulator (LQR) theory and thus is of low computational complexity. With the help of the PDC based fuzzy approximator for the unknown human control law, a two-step procedure is proposed for the HBL. First, a filter-based adaptive law is developed to learn the gain matrices of the fuzzy approximator using the system state data only. The convergence analysis of the adaptive estimator is also given. Then, a semidefinite programming (SDP) problem with the quadratic objective function can be set up for determining the fuzzy weighting matrices of the cost function. The simulation study on a steering control system of the intelligent vehicle is given to show the effectiveness and applicability of the developed approach.

This paper proposes new algorithms based on the fuzzy c-regressing model algorithm for Takagi-Sugeno (T-S) fuzzy modeling of the complex nonlinear systems. A fuzzy c-regression state model (FCRSM) algorithm is a T-S fuzzy model in which the functional antecedent and the state-space-model-type consequent are considered with the available input-output data. The antecedent and consequent forms of the proposed FCRSM consists mainly of two advantages: one is that the FCRSM has low computation load due to only one input variable is considered in the antecedent part; another is that the unknown system can be modeled to not only the polynomial form but also the state-space form. Moreover, the FCRSM can be extended to FCRSM-ND and FCRSM-Free algorithms. An algorithm FCRSM-ND is presented to find the T-S fuzzy state-space model of the nonlinear system when the input-output data cannot be precollected and an assumed effective controller is available. In the practical applications, the mathematical model of controller may be hard to be obtained. In this case, an online tuning algorithm, FCRSM-FREE, is designed such that the parameters of a T-S fuzzy controller and the T-S fuzzy state model of an unknown system can be online tuned simultaneously. Four numerical simulations are given to demonstrate the effectiveness of the proposed approach.

This paper is concerned with the problem of designing fuzzy controllers for a class of nonlinear dynamic systems. The considered nonlinear systems are described by T-S fuzzy models with nonlinear local models, and the fuzzy models have fewer fuzzy rules than conventional T-S fuzzy models with local linear models. A new fuzzy control scheme with local nonlinear feedbacks is proposed, and the corresponding control synthesis conditions are given in terms of solutions to a set of linear matrix inequalities (LMIs). In contrast to the existing methods for fuzzy control synthesis, the new proposed control design method is based on fewer fuzzy rules and less computational burden. Moreover, the local nonlinear feedback laws in the new fuzzy controllers are also helpful in achieving good control effects. Numerical examples are given to illustrate the effectiveness of the proposed method.

Tins paper is concerned with a class of nonlinear networked control systems with network-induced delay shorter than one sample period. Based on the assumptions that the actuator node is time-driven and the controller node is event-driven, a Takagi-Sugeno (T-S) fuzzy model is used to approximate the nonlinear networked systems with delay. By means of Lyapunov functions, a method for designing a fuzzy state feedback controller to satisfy the prescribed H/sub /spl infin// performance index is proposed. The results are presented in terms of linear matrix inequalities (LMIs) which can be solved by LMI toolbox in Matlab 6.5. Finally, a numerical example is given to validate the proposed method.

Nonlinearities and system uncertainties are the most important difficulties in designing controllers that ensure stability and acceptable closed-loop performance. Many significant results on the stability and robust control of uncertain nonlinear systems using the TS fuzzy model have been reported over the past decades, and considerable advances have been made (Connor et al. 1992; Chen et al. 1996; Rocha et al. 2005; Boukhezzar et al. 2006; Kamal et al. 2011, 2012a, 2013; Khanl et al. 2011). However, as stated in Khan and Hossain (2011), many approaches for stability and robust control of uncertain systems are often characterized by conservatism when dealing with uncertainties. It has been well known that the TS fuzzy model is very effective representation of complex nonlinear systems. In the TS fuzzy model, the state space of a nonlinear system is divided into different fuzzy regions with a local linear model being used in each region. The overall model output is obtained by defuzzification using the center of gravity (COG) method. Once the fuzzy model is obtained, control design can be carried out via the so-called parallel and distributed computing (PDC) approach, which employs multiple linear controllers corresponding to the locally linear plant models (Boukhezzar et al. 2006). This class of systems is described as a weighted sum of some simple linear subsystems, and thus is easily analyzable.

This article proposes a fault diagnosis method for closed-loop satellite attitude control systems based on a fuzzy model and parity equation. The fault in a closed-loop system is propagated with the feedback loop, increasing the difficulty of fault diagnosis and isolation. The study uses a Takagi-Sugeno (T-S) fuzzy model and parity equation to diagnose and isolate a fault in a closed-loop satellite attitude control system. A fully decoupled parity equation is designed for the closed-loop satellite attitude control system to generate a residual that is sensitive only to a specific actuator and sensor. A T-S fuzzy model is used to describe the nonlinear closed-loop satellite attitude control system. With the combination of the T-S fuzzy model and fully decoupled parity equation, the fuzzy parity equation (FPE) of the nonlinear system can be obtained. Then this article uses a parameter estimator based on a Kalman filter to identify deviations and scale factor changes from information contained in the residuals generated by the FPE. The actuator and sensor fault detection and isolation simulation of the three-axis stable satellite attitude control system is provided for illustration.

Takagi-Sugeno (TS) fuzzy models have been the popular choice for multi-model based nonlinear system identification for controller design. However, the interpretation of the model from a dynamical systems point of view is still unclear. In this paper, we investigate the interpretation of the TS fuzzy models in terms of local dynamics of nonlinear systems in state space. The investigation is based on the comparison between the constant-affine recursive i/o equation and constant-affine state space representations. We show that the true local dynamics is not identifiable by TS fuzzy models by analysis and example. For the identification of constant-affine state space models, a subspace identification algorithm is proposed to identify the true local dynamics with zero initial conditions.

This paper addresses the issue of tracking the output of an activated sludge process using fuzzy integral control. First, the dynamics of the nonlinear process are modeled with a dynamic state space fuzzy model integrating the effect of external disturbances, and then an additional integral state of the output tracking error is introduced to obtain an augmented Takagi-Sugeno (TS) fuzzy model. The TS fuzzy model is able to describe the dynamics of complex nonlinear systems with an excellent degree of accuracy. It is formulated by fuzzy if-then rules which can give local linear representation of the overall nonlinear system. Second, the design of the fuzzy integral control is performed, in which the state feedback gains are obtained by solving linear matrix inequalities (LMI). The objective is to ensure trajectory tracking of an activated sludge process (ASP) by controlling two key variables: the substrate concentration and the level of dissolved oxygen. To assess the performance of the proposed control strategy, a comparative analysis is carried out with a gain scheduling PI (GS-PI) controller. Simulation results are provided to illustrate the effectiveness of the proposed approach. Where, the fuzzy integral control reduces the high energy consumption in water treatment plants.

ABSTRACTBased on Radial Basis Function neural networks, a stable adaptive control scheme is proposed for a class of unknown nonlinear dynamic systems. It is shown that the scheme is stable and convergent in the sense that all the signal in the loop remain bounded, and in the ideal case, output tracking error will converge to zero and in the non-ideal case, the error will converge to a residue which is proportional to the modelling error bound. Simulation results are also provided to demonstrate performance of the scheme.

On a class of nonlinear stochastic dynamical systems: Analysis of the transient behaviour

Electrochemical sensors have been broadly applied in various areas for diagnostics and controls. However, many electrochemical sensors are often cross-sensitive to one or more interfering compounds, which can make it rather challenging in attaining accurate concentration of the target compound. The sensor cross-sensitivity can cause instability issues in feedback control systems and inaccurate diagnostics. In many cases, a state estimation problem with cross-sensitive output is equivalent to a state estimation problem with limited number of state candidates that can lead to the same output. In this study, an innovative estimation error-based observer is proposed for state estimation for a general class of 1st-order nonlinear dynamic systems with cross-sensitive output. By constructing observers for each potential candidate and analyzing the respective estimation errors, the true state (or mode) of the system can be determined based on the cross-sensitive output measurement. The proposed algorithm was applied to estimate the ammonia coverage ratio of a Diesel selective catalytic reduction system with ammonia cross-sensitive <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\text{NO}_{\mathrm{x}}$</tex> sensor output. Simulation results show that the true ammonia coverage ratio can be accurately estimated with this method at both low and high ammonia coverage ratios. This method can be applied to a broad class of nonlinear dynamic systems with cross-sensitive outputs for state estimation.

The development of a mathematical model making it possible to represent ˝ as well as possible ˝ the dynamic behaviors of a complex real process represents a very important problem in the real world. Fuzzy logic and more, particularly the Takagi-Sugeno (TS) fuzzy model draws the attention of several researchers during these last decades. This is due to their capability to approximate the nonlinear system in several locally linear subsystems. Many clustering algorithms exist in literature allowing the identification of the parameters intervening in the TS fuzzy model. In this paper a new clustering algorithm noted NEPCM-PSO is proposed. The proposed algorithm represents a combination between New Extended Possibilistic C-Means algorithm (NEPCM) and Particle Swarm Optimization (PSO) algorithm. The effectiveness of this algorithm is tested on a nonlinear system and on an electro-hydraulic system. In this paper a comparative study between PCM algorithm, NEPCM algorithm and NEPCM-PSO algorithm are also presented.