Process Modeling, Identification Methods, and Control Schemes for Nonlinear Physical Systems – A Comprehensive Review

An adaptive system for linear systems with unknown parameters is a nonlinear system. The analysis of such adaptive systems requires similar techniques to analyse nonlinear systems. Therefore it is natural to treat adaptive control as a part of nonlinear control systems. Nonlinear and Adaptive Control Systems treats nonlinear control and adaptive control in a unified framework, presenting the major results at a moderate mathematical level, suitable for MSc students and engineers with undergraduate degrees. Topics covered include introduction to nonlinear systems; state space models; describing functions for common nonlinear components; stability theory; feedback linearization; adaptive control; nonlinear observer design; backstepping design; disturbance rejection and output regulation; and control applications, including harmonic estimation and rejection in power distribution systems, observer and control design for circadian rhythms, and discrete-time implementation of continuous-time nonlinear control laws.

Theory of nonlinear control

The objective of this study is to enhance the responsiveness and precision of control systems through the acceleration of evolutionary programming in the optimization of nonlinear fuzzy control systems. Nonlinear systems are ubiquitous in engineering practice, presenting significant challenges due to their complexity and uncertainty for traditional control methodologies. Fuzzy control, an intelligent control approach rooted in empirical rules, possesses the capability to navigate uncertain and nonlinear issues. However, conventional fuzzy controllers struggle to achieve optimal control performance when confronted with highly complex nonlinear systems. Consequently, this paper introduces the application of accelerated evolutionary programming (RGA) for the optimization of fuzzy controllers, aiming to augment their performance in nonlinear systems. By merging genetic algorithms with a fuzzy adaptive PID controller, the optimized fuzzy controller is better equipped to adapt to the dynamic changes of nonlinear systems, offering robust and efficient control strategies. The results of simulation experiments demonstrate a marked improvement in response time, stability, and error reduction with the fuzzy control system based on RGA optimization. This study offers novel insights and methodologies for the further optimization of control within nonlinear systems, while providing theoretical support for the practical applications of complex systems.

International audience

The efficacy of the multimodel framework (MMF) in modeling and identification of complex, nonlinear, and uncertain systems has been widely recognized in the literature owing to its simplicity, transparency, and mathematical tractability, allowing the use of well-known modeling analysis and control design techniques. The approach proved to be effective in addressing some of the shortcomings of other modeling techniques such as those based on a single nonlinear autoregressive network with exogenous inputs model or neural networks. A great number of researchers have contributed to this active field. Due to the significant amount of contributions and the lack of a recent survey, the review of recent developments in this field is vital. In this two-part paper, we attempt to provide a comprehensive coverage of the multimodel approach for modeling and identification of complex systems. The study contains a classification of different methods, the challenges encountered, as well as recent applications of MMF in various fields. Part 1 of this paper presents an overview of MMF for modeling and identification of nonlinear systems as well as the review of recent developments in the partitioning strategies employed.

Multidimensional membership functions in T–S fuzzy models for modelling and identification of nonlinear multivariable systems using genetic algorithms

A nonlinear control law for the quad-rotor of a low-complexity, global approximation-free from system uncertainties and external disturbances are described in this paper. The control law guarantees convergence to a small bounded error using a prescribed performance function. The stability of the proposed nonlinear control system is also proven by the Lyapunov stability theorem. The advantage of this technique is that it has a simpler form than any other nonlinear compensators and is applicable to any nonlinear systems without precise knowledge of the systems. In this paper, the proposed approach is applied to attitude/altitude control of a quad-rotor. Numerical simulations are performed to investigate the proposed nonlinear attitude control law by applying it to an uncertain quadcopter system with external disturbances.

Identification of nonlinear systems using NARMAX model

The current theory of nonlinear systems is still not perfect. The modeling and control of nonlinear system problem has always been the difficulty. In a variety of methods of its study, fuzzy system theory because of having the language descriptive way similar to the human mind, can obtain and deal with the qualitative information intelligently. The theory itself also has non-linear characteristics. Therefore the use of fuzzy systems theory to establish the fuzzy model of nonlinear system can well describe the nonlinear characteristics. T-S fuzzy systems, due to the combination of the good performance of the fuzzy system to deal with nonlinear problems with the simple linear expressions, are not only suitable for modeling the nonlinear system, but also use T-S fuzzy model and the linear control theory method to design the controller. So it has been widely used in nonlinear system control problems, and has also greatly developed the T-S fuzzy system theory, appearing a lot of methods of structural and parameter identification. However, this study of T-S fuzzy rules makes us have to face the difference of different ways to select the number of rules as well as online self-adaptability of the number of rules which off-line method lacks when using T-S fuzzy model to deal with nonlinear system modeling and control problem. In view of this, this paper researches on modeling and controlling of complex nonlinear systems based on TS model from different perspectives.

Teaching Aids for Modeling and Control of Hybrid Systems (CAMCHS)

Publication in the conference proceedings of EUSIPCO, Trieste, Italy, 1996

Designing minimum variance controllers (MVC) for nonlinear systems is confronted with many difficulties. The methods which are able to identify MIMO nonlinear systems are scarce, and linear models are not accurate in modeling nonlinear systems. In this paper, Vector ARX (VARX) models are proposed for designing MVC and generalized minimum variance controller (GMVC) for linear and nonlinear systems, and the accuracy of these models in approximating the nonlinear MIMO system is studied. However, the VARX is a linear model. It is shown that this model can identify some kinds of nonlinear systems with any desired accuracy. Therefore, the controller designed by the VARX is accurate, even for these nonlinear systems. The proposed controller is tested on a both linear system and a nonlinear four-tank benchmark process. In spite of the simplicity of designing GMVCs for the VARX models, the results show that the proposed method is accurate and implementable.

Fuzzy recurrent stochastic configuration networks (FRSCNs) are a class of randomized neurofuzzy models that have shown success in modeling nonlinear dynamic systems. However, the data generated from the real-world industry process often exhibits nonstationary characteristics, which could lead to the resulting model with poor generalization performance. This article presents an online self-learning FRSCN (OSL-FRSCN) for problem-resolving. OSL-FRSCNs integrate the self-organizing learning strategies into FRSCNs, maintaining the model's strong fuzzy inference capabilities while enhancing their continuous learning abilities for nonstationary data streams. The network parameters are updated online using the projection algorithm based on the newly arriving data streams. Moreover, the network structure can be dynamically adjusted in the light of the fuzzy recurrent stochastic configuration (FRSC) algorithm and an improved sensitivity analysis. Comprehensive comparisons over two nonlinear system identification tasks and two industrial applications are carried out. Numerical results clearly demonstrate that the proposed OSL-FRSCNs outperform other neurofuzzy and nonfuzzy models with sound generalization, verifying their effectiveness in modeling nonlinear systems with nonstationary dynamics.

This work extends previous work on nonlinear normal modes to include the case of forced response. This allows the nonlinear normal mode method to be applied to the feedback control problem providing a new method of controlling nonlinear multiple-degree-of-freedom systems. The proposed method uses a transformation proposed earlier for homogeneous systems written in state-space form. The coordinate transformation for the forcing vector is defined here in state space and related to the physical coordinate system. The result is a pseudomodal decoupling transformation of a nonlinear inhomogeneous system. Although interesting in its own statement, this transformation also provides a nonlinear modal control scheme. This result is applied to a known coupled two degree-of-freedom oscillator with a cubic stiffness term. The results illustrate the design of a nonlinear modal control law.

Identification of nonlinear systems is one of the important problems in engineering. Chaotic systems are among those nonlinear systems that have been in the center of attention of many researchers because of their complex and unpredictable behaviors. This paper proposes an algorithm for optimally estimating the parameters of system by minimizing the mean of squared errors (MSE) index. In this paper, Teaching Learning Based Optimization (TLBO) algorithm is used for solving both offline and online parameter estimation problems for chaotic systems. The validity of this algorithm in terms of convergence speed and parameter accuracy in comparison to other popular optimization algorithms such as Standard PSO (PSO), Differential Evolution (DE), Adaptive Particle Swarm Optimization (APSO) and Genetic Algorithm (GA) is shown through an illustrative example for the modeling of chaotic systems. Furthermore, in order to demonstrate the feasibility of this algorithm, it is applied to the problem of parameter identification of a well-known nonlinear Lorenz chaotic system. According to simulation results, the proposed algorithm is a very suitable algorithm for online parameter identification for a class of nonlinear chaotic systems.