Nonlinear Modeling by Interpolation Between Linear Dynamics and Its Application in Control

This paper proposes a finite series expansion to approximate general nonlinear dynamics models to arbitrary accuracy. The method produces an approximation of nonlinear dynamics in the form of an aggregate of linear models, weighted by unimodal basis functions, and results in a linear growth bound on the approximation error. Furthermore, this paper demonstrates that the proposed approximation satisfies the modeling assumptions for analysis based on linear matrix inequalities and hence widens the applicability of these techniques to the area of nonlinear control.

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Mixed Mathematical and Physical Modeling for Nonlinear Systems

Dynamic Systems and Control, Parts A and B ◽

10.1115/imece2005-81666 ◽

2005 ◽

Cited By ~ 2

Author(s):

Kiriakos Kiriakidis

Keyword(s):

Linear Growth ◽

Dynamic Models ◽

Linear Models ◽

Approximation Error ◽

Basis Functions ◽

Linear Matrix ◽

Matrix Inequalities ◽

Growth Bound ◽

Nonlinear Dynamic Models ◽

Mathematical And Physical Modeling

The paper proposes a finite series expansion to approximate general nonlinear dynamic models to arbitrary accuracy. The method produces an approximation of nonlinear dynamics in the form of an aggregation of linear models, weighted by unimodal basis functions, and results in a linear growth bound on the approximation error. Furthermore, the paper demonstrates that the proposed approximation satisfies the modeling assumptions for analysis based on linear matrix inequalities and hence widens the applicability of these techniques to the area of nonlinear control.

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On the Expansion of Nonlinear Models Using Bell-Shaped Basis Functions

Dynamic Systems and Control, Volumes 1 and 2 ◽

10.1115/imece2003-41256 ◽

2003 ◽

Cited By ~ 1

Author(s):

Kiriakos Kiriakidis

Keyword(s):

Nonlinear Dynamics ◽

Nonlinear Model ◽

Upper Bound ◽

Nonlinear Models ◽

Approximation Error ◽

Basis Functions ◽

Model Set

We propose a method that approximates any nonlinear model, without regard to complexity, by minimizing its distance from a rich model set. The method produces, potentially through an automated procedure, the approximation of the nonlinear dynamics in the form of a finite expansion associated with certain basis functions and provides an upper bound on the approximation error.

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Stable Fault Tolerant Controller Design for Takagi–Sugeno Fuzzy Model-Based Control Systems via Linear Matrix Inequalities: Three Conical Tank Case Study

Energies ◽

10.3390/en12112221 ◽

2019 ◽

Vol 12 (11) ◽

pp. 2221 ◽

Cited By ~ 8

Author(s):

Himanshukumar R. Patel ◽

Vipul A. Shah

Keyword(s):

Nonlinear System ◽

Nonlinear Systems ◽

Control Systems ◽

Linear Models ◽

Fault Tolerant ◽

Fuzzy Model ◽

Linear Matrix ◽

Matrix Inequalities ◽

Fuzzy Models ◽

Takagi Sugeno

This paper deals with a methodical design approach of fault-tolerant controller that gives assurance for the the stabilization and acceptable control performance of the nonlinear systems which can be described by Takagi–Sugeno (T–S) fuzzy models. Takagi–Sugeno fuzzy model gives a unique edge that allows us to apply the traditional linear system theory for the investigation and blend of nonlinear systems by linear models in a different state space region. The overall fuzzy model of the nonlinear system is obtained by fuzzy combination of the all linear models. After that, based on this linear model, we employ parallel distributed compensation for designing linear controllers for each linear model. Also this paper reports of the T–S fuzzy system with less conservative stabilization condition which gives decent performance. However, the controller synthesis for nonlinear systems described by the T–S fuzzy model is a complicated task, which can be reduced to convex problems linking with linear matrix inequalities (LMIs). Further sufficient conservative stabilization conditions are represented by a set of LMIs for the Takagi–Sugeno fuzzy control systems, which can be solved by using MATLAB software. Two-rule T–S fuzzy model is used to describe the nonlinear system and this system demonstrated with proposed fault-tolerant control scheme. The proposed fault-tolerant controller implemented and validated on three interconnected conical tank system with two constraints in terms of faults, one issed to build the actuator and sond is system component (leak) respectively. The MATLAB Simulink platform with linear fuzzy models and an LMI Toolbox was used to solve the LMIs and determine the controller gains subject to the proposed design approach.

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Stabilization of Nonlinear Systems Using Quasi-Linear Models

10.1115/imece2000-2343 ◽

2000 ◽

Author(s):

Kiriakos Kiriakidis

Keyword(s):

Linear Matrix Inequalities ◽

Linear Models ◽

Nonlinear Models ◽

Linear Matrix ◽

Matrix Inequalities ◽

Analysis And Design ◽

Linear Differential Inclusions ◽

Linear Differential ◽

Generalized Lyapunov Functions ◽

Takagi Sugeno

Abstract Unconventional nonlinear models such as nonlinear ARMAX, Takagi-Sugeno fuzzy models, global linearizations, and linear hybrid systems are, at the highest level of abstraction, a sort of quasi-linear models, namely, Polytopic Linear Differential Inclusions (PLDIs). At present, quadratic stability has enabled, mainly via linear matrix inequalities, the analysis and design of a nonlinear system from the vertex matrices of its PLDI model. Proving stability by a globally quadratic Lyapunov function, however, entails conservatism. This paper proposes a less conservative framework by using piecewise-quadratic generalized Lyapunov functions. Further manipulation of the problem within such framework yields a set of bilinear rather than linear matrix inequalities.

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Stable and Optimal Controller Design for Takagi-Sugeno Fuzzy Model Based Control Systems via Linear Matrix Inequalities

Information Technologies and Control ◽

10.1515/itc-2017-0010 ◽

2016 ◽

Vol 14 (3) ◽

pp. 31-40 ◽

Cited By ~ 1

Author(s):

M. Namazov ◽

A. Alili

Keyword(s):

Nonlinear System ◽

Control Systems ◽

Linear Matrix Inequalities ◽

Linear Models ◽

Fuzzy Model ◽

Design Procedure ◽

Optimal Performance ◽

Linear Matrix ◽

Matrix Inequalities ◽

Takagi Sugeno

AbstractThis paper deals with a systematic design procedure that guarantees the stability and optimal performance of the nonlinear systems described by Takagi-Sugeno fuzzy models. Takagi-Sugeno fuzzy model allows us to represent a nonlinear system by linear models in different state space regions. The overall fuzzy model is obtained by fuzzy blending of these linear models. Then based on this model, linear controllers are designed for each linear model using parallel distributed compensation. Stability and optimal performance conditions for Takagi-Sugeno fuzzy control systems can be represented by a set of linear matrix inequalities which can be solved using software packages such as MATLAB’s LMI Toolbox. This design procedure is illustrated for a nonlinear system which is described by a two-rule Takagi-Sugeno fuzzy model. The fuzzy model was built in MATLAB Simulink and a code was written in LMI Toolbox to determine the controller gains subject to the proposed design approach.

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Improved over-approximation method for modelling and stability analysis of networked control systems

Transactions of the Institute of Measurement and Control ◽

10.1177/0142331215599837 ◽

2016 ◽

Vol 39 (2) ◽

pp. 131-140 ◽

Cited By ~ 2

Author(s):

Milad Farsi ◽

Reza Mahboobi Esfanjani

Keyword(s):

Control Systems ◽

Approximation Scheme ◽

Linear Models ◽

Networked Control Systems ◽

Networked Control ◽

Superior Performance ◽

Linear Matrix ◽

Matrix Inequalities ◽

Challenging Problem ◽

Analysis And Design

Networked control systems (NCSs) are commonly modelled by the switched systems involving exponential non-linearities. A challenging problem is to obtain a tight linear approximation of the mentioned non-linear models to derive analysis and design criteria in terms of linear matrix inequalities (LMIs), which can be easily handled using the well-established algorithms. The present paper introduces a novel procedure to achieve an improved polytopic over-approximation of the non-linearities emerged in the modelling of NCSs. Moreover, stability of the NCSs is analysed to verify the merits of the proposed method; to this end, a benchmark numerical example is presented to illustrate the superior performance of the suggested approximation scheme compared with the existing approaches in the literature.

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Model-Based State-of-Charge Estimation for a Valve-Regulated Lead-Acid Battery Using Linear Matrix Inequalities

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.4023766 ◽

2013 ◽

Vol 135 (4) ◽

Cited By ~ 1

Author(s):

Zheng Shen ◽

Christopher D. Rahn

Keyword(s):

Linear Matrix Inequalities ◽

Linear Models ◽

Observer Design ◽

State Of Charge ◽

Linear Matrix ◽

Matrix Inequalities ◽

Estimation Errors ◽

Luenberger Observer ◽

Averaged Models ◽

Lead Acid

State-of-charge (SOC) estimation for valve-regulated lead-acid (VRLA) batteries is complicated by the switched linear nature of the underlying dynamics. A first principles nonlinear model is simplified to provide two switched linear models and linearized to produce charge, discharge, and averaged models. Luenberger and switched SOC estimators are developed based on these models and propagated using experimental data. A design methodology based on linear matrix inequalities (LMIs) is used in the switched SOC estimator design to obtain a switched Luenberger observer with guaranteed exponential stability. The results show that estimation errors are halved by including switching in the observer design.

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