Control Of Nonlinear Dynamical Systems Research Articles

The controllability of non-holonomic mechanical systems with constrained controls

A cycle of investigations, related to the problem of the controllability of non-linear dynamical systems, is developed. Systems which have a mechanical nature (wheeled means of transportation, transport and machining systems, and manipulators) are considered. Controllability criteria are established for the general class of mechanical systems which may contain non-holonomic constraints; similar criteria were established previously for holonomic systems in [1–4]. The controllability conditions obtained have a clear physical meaning. For example, for the controllability of a manipulation robot it is necessary that the control forces should predominate over the other generalized forces (the weight forces, and the resistance forces due to the external medium), Predominance is necessary as regards the amplitude of the forces. Additional conditions are related to the properties of constraints imposed on the system. This essentially requires that the constraint relations allow of the possibility of an appropriate change in the coordinates and velocities of the mechanical system in the region investigated.

Preface to the special issue on soft computing for modeling, simulation, and control of nonlinear dynamical systems

Preface to the special issue on soft computing for modeling, simulation, and control of nonlinear dynamical systems

Input uncertainty and conditioning in siliciclastic process modelling

Abstract Deterministic forward sedimentary process models enhance our quantitative understanding of sedimentary systems. They are also being used increasingly to assist in the reconstruction of the geological past and the inference of the present configuration of sedimentary deposits. Such usage presents the challenge of having to establish the initial and boundary conditions that will cause the model’s output to match present-day observations. This constitutes an inversion problem that can sometimes be handled by traditional optimization methods. Clastic sedimentation models, however, often incorporate complex non-linear relationships that preclude the use of these techniques. The problem must then be handled statistically by relaxing the requirement of honouring exact observations and matching only the spatial variability of the observed deposits. Recent advances in control of non-linear dynamic systems may also shed light on possible solutions to the inversion problem in siliciclastic models. This paper reviews known approaches to problems related to input uncertainty and conditioning, and presents original preliminary results on control of sedimentation models.

Covariance Control of Nonlinear Dynamic Systems via Exact Stationary Probability Density Function

This paper presents a method for designing covariance type controls of nonlinear stochastic systems. The method consists of two steps. The first step is to find a class of nonlinear feedback controls with undetermined gains such that the exact stationary PDF of the response is obtainable. The second step is to select the control gains in the context of the covariance control method by minimizing a performance index. The exact PDF makes the solution process of optimization very efficient, and the evaluation of expectations of nonlinear functions of the response very accurate. The theoretical results of various orders of response moments by the present method have been compared with Monte Carlo simulations. Special cases are studied when the approximate methods based on the maximum entropy principle or other closure schemes leads less accurate response estimates, while the present method still works fine.

Soft Computing for Control of Non-Linear Dynamical Systems

1 Introduction to Control of Non-Linear Dynamical Systems.- 2 Fuzzy Logic.- 2.1 Fuzzy Set Theory.- 2.2 Fuzzy Reasoning.- 2.3 Fuzzy Inference Systems.- 2.4 Type-2 Fuzzy Logic Systems.- 2.4.1 Type-2 Fuzzy Sets.- 2.4.2 Type-2 Fuzzy Systems.- 2.5 Fuzzy Modelling.- 2.6 Summary.- 3 Neural Networks for Control.- 3.1 Backpropagation for Feedforward Networks.- 3.1.1 The Backpropagation Learning Algorithm.- 3.1.2 Backpropagation Multilayer Perceptions.- 3.2 Adaptive Neuro-Fuzzy Inference Systems.- 3.2.1 ANFIS Architecture.- 3.2.2 Learning Algorithm.- 3.3 Neuro-Fuzzy Control.- 3.3.1 Inverse Learning.- 3.3.2 Specialized Learning.- 3.4 Adaptive Model-Based Neuro-Control.- 3.4.1 Indirect Neuro-Control.- 3.4.2 Direct Neuro-Control.- 3.4.3 Parameterized Neuro-Control.- 3.5 Summary.- 4 Genetic Algorithms and Simulated Annealing.- 4.1 Genetic Algorithms.- 4.2 Simulated Annealing.- 4.3 Applications of Genetic Algorithms.- 4.3.1 Evolving Neural Networks.- 4.3.1.1 Evolving Weights in a Fixed Network.- 4.3.1.2 Evolving Network Architectures.- 4.3.2 Evolving Fuzzy Systems.- 4.4 Summary.- 5 Dynamical Systems Theory.- 5.1 Basic Concepts of Dynamical Systems.- 5.2 Controlling Chaos.- 5.2.1 Controlling Chaos through Feedback.- 5.2.1.1 Ott-Grebogi-Yorke Method.- 5.2.1.2 Pyragas's Control Methods.- 5.2.1.3 Controlling Chaos by Chaos.- 5.2.2 Controlling Chaos without Feedback.- 5.2.2.1 Control through Operating Conditions.- 5.2.2.2 Control by System Design.- 5.2.2.3 Taming Chaos.- 5.2.3 Method Selection.- 5.3 Summary.- 6 Hybrid Intelligent Systems for Time Series Prediction.- 6.1 Problem of Time Series Prediction.- 6.2 Fractal Dimesion of an Object.- 6.3 Fuzzy Logic for Object Classification.- 6.4 Fuzzy Estimation of the Fractal Dimension.- 6.5 Fuzzy Fractal Approach for Time Series Analysis and Prediction.- 6.6 Neural Network Approach for Time Series Prediction.- 6.7 Fuzzy Fractal Approach for Pattern Recognition.- 6.8 Summary.- 7 Modelling Complex Dynamical Systems with a Fuzzy Inference System for Differential Equations.- 7.1 The Problem of Modelling Complex Dynamical Systems.- 7.2 Modelling Complex Dynamical Systems with the New Fuzzy Inference System.- 7.3 Modelling Robotic Dynamic Systems with the New Fuzzy Interence System.- 7.3.1 Mathematical Modelling of Robotic Systems.- 7.3.2 Fuzzy Modelling of Robotic Dynamic Systems.- 7.3.3 Experimental Results.- 7.4 Modelling Aircraft Dynamic Systems with the New Fuzzy Inference System.- 7.5 Summary.- 8 A New Theory of Fuzzy Chaos for Simulation of Non-Linear Dynamical Systems.- 8.1 Problem Description.- 8.2 Towards a New Theory of Fuzzy Chaos.- 8.3 Fuzzy Chaos for Behavior Identification in the Simulation of Dynamical Systems.- 8.4 Simulation of Dynamical Systems.- 8.5 Method for Automated Parameter Selection Using Genetic Algorithms.- 8.6 Method for Dynamic Behavior Identification Using Fuzzy Logic.- 8.6.1 Behavior Identification Based on the Analytical Properties of the Model.- 8.6.2 Behavior Identification Based on the Fractal Dimension and the Lyapunov Exponents.- 8.7 Simulation Results for Robotic Systems.- 8.8 Summary.- 9 Intelligent Control of Robotic Dynamic Systems.- 9.1 Problem Description.- 9.2 Mathematical Modelling of Robotic Dynamic Systems.- 9.3 Method for Adaptive Model-Based Control.- 9.3.1 Fuzzy Logic for Dynamic System Modelling.- 9.3.2 Neuro-Fuzzy-Fractal Adaptive Model-Based Control.- 9.4 Adaptive Control of Robotic Dynamic Systems.- 9.5 Simulation Results for Robotic Dynamic Systems.- 9.6 Summary.- 10 Controlling Biochemical Reactors.- 10.1 Introduction.- 10.2 Fuzzy Logic for Modelling.- 10.3 Neural Networks for Control.- 10.4 Adaptive Control of a Non-Linear Plant.- 10.5 Fractal Identification of Bacteria.- 10.6 Experimantal Results.- 10.7 Summary.- 11 Controlling Aircraft Dynamic Systems.- 11.1 Introduction.- 11.2 Fuzzy Modelling of Dynamical Systems.- 11.3 Neural Networks for Control.- 11.4 Adaptive Control of Aircraft Systems.- 11.5 Experimental Results.- 11.6 Summary.- 12 Controlling Electrochemical Processes.- 12.1 Introduction.- 12.2 Problem Description.- 12.3 Fuzzy Method for cControl.- 12.4 Neuro-Fuzzy Methof for Control.- 12.5 Neuro-Fuzzy-Genetic Method for Control.- 12.6 Experimental Results for the Three Hybrid Approaches.- 12.7 Summary.- 13 Controlling International Trade Dynamics.- 13.1 Introduction.- 13.2 Mathematical Modelling of International Trade.- 13.2.1 Oscillations in Autonomous Economies.- 13.2.2 International Trade as a Perturbation of Internal Oscillations.- 13.3 Fuzzy Logic for Model Selection.- 13.4 Adaptive Model-Based Control of International Trade.- 13.5 Simulation Results for Control of International Trade.- 13.6 Summary.- References.

Special issue on soft computing for control of non-linear dynamical systems

Special issue on soft computing for control of non-linear dynamical systems

Application of stochastic approximation techniques in neural modelling and control

Learning, i.e. estimation of weights and biases in neural networks, involves the minimization of an output error criterion, a problem which is usually solved using back-propagation algorithms. This paper aims to assess the potential of simultaneous perturbation stochastic approximation (SPSA) algorithms to handle this minimization problem. In particular, a variation of the first-order SPSA algorithm that makes use of several numerical artifices including adaptive gain sequences, gradient smoothing and a step rejection procedure is developed. For illustration purposes, several application examples in the identification and control of nonlinear dynamic systems are presented. This numerical evaluation includes the development of neural network models as well as the design of a model-based predictive neural PID controller.

Categories and Foundations of the Theory of Nonlinear Controlled Dynamical Systems: II

Categories and Foundations of the Theory of Nonlinear Controlled Dynamical Systems: II

Robust control for nonlinear dynamic systems

In this paper, a robust control scheme for a class of nonlinear dynamical system is proposed by using output-feedback linearization method. The presented control scheme is based on the VSS. We assume that the nonlinear dynamical system is minimum phase, i.e., the relative degree of the system is p = r1 + r2 + ···+ rm n and zero dynamics is stable. We also assume that the states of zero dynamics are not accessible. It is also shown that the global asymptotically stability is guaranted. And we verified that the proposed control scheme is then applied to the control of a two degrees of freedom (DOF) robotic manipulator with payload.

Adaptive intelligent control of aircraft systems with a hybrid approach combining neural networks, fuzzy logic and fractal theory

We describe in this paper a hybrid method for adaptive model-based control of nonlinear dynamic systems using neural networks, fuzzy logic and fractal theory. The new neuro-fuzzy-fractal method combines soft computing techniques with the concept of the fractal dimension for the domain of nonlinear dynamic system control. The new method for adaptive model-based control has been implemented as a computer program to show that the neuro-fuzzy-fractal approach is a good alternative for controlling nonlinear dynamic systems. It is well known that chaotic and unstable behavior may occur for nonlinear systems. Normally, we will need to control this type of behavior to avoid structural problems with the system. We illustrate in this paper our new methodology with the case of controlling aircraft dynamic systems. For this case, we use mathematical models for the simulation of aircraft dynamics during flight. The goal of constructing these models is to capture the dynamics of the aircraft, so as to have a way of controlling this dynamics to avoid dangerous behavior of the aircraft dynamic system.

Learning soft computing control strategies in a modular neural network architecture

Modelling and control of nonlinear dynamical systems is a challenging problem since the dynamics of such systems change over their parameter space. Conventional methodologies for designing nonlinear control laws, such as gain scheduling, are effective because the designer partitions the overall complex control into a number of simpler sub-tasks. This paper describes a new genetic algorithm based method for the design of a modular neural network (MNN) control architecture that learns such partitions of an overall complex control task. Here a chromosome represents both the structure and parameters of an individual neural network in the MNN controller and a hierarchical fuzzy approach is used to select the chromosomes required to accomplish a given control task. This new strategy is applied to the end-point tracking of a single-link flexible manipulator modelled from experimental data. Results show that the MNN controller is simple to design and produces superior performance compared to a single neural network (SNN) controller which is theoretically capable of achieving the desired trajectory.

Application of a New Theory of Fuzzy Chaos for the Simulation and Control of NonLinear Dynamical Systems

We describe in this article a new theory of chaos using fuzzy logic techniques. Chaotic behavior in nonlinear dynamical systems is very difficult to detect and control. Part of the problem is that mathematical results for chaos are difficult to use in many cases, and even if one could use them there is an underlying uncertainty in the accuracy of the numerical simulations of the dynamical systems. For this reason, we can model the uncertainty of detecting the range of values where chaos occurs, using fuzzy sets theory. Using fuzzy sets, we can build a new theory of Fuzzy Chaos, where we can use fuzzy set to describe the behaviors of a system. We illustrate our approach with two cases: Chua's circuit and Duffing's oscillator.

Adaptive extremum seeking control of nonlinear dynamic systems with parametric uncertainties

We pose and solve an extremum seeking control problem for a class of nonlinear systems with unknown parameters. Extremum seeking controllers are developed to drive system states to the desired set-points that extremize the value of an objective function. The proposed adaptive extremum seeking controller is “inverse optimal” in the sense that it minimizes a meaningful cost function that incorporates penalty on both the performance error and control action. Simulation studies are provided to verify the effectiveness of the proposed approach.

Neural-network predictive control for nonlinear dynamic systems with time-delay

A new recurrent neural-network predictive feedback control structure for a class of uncertain nonlinear dynamic time-delay systems in canonical form is developed and analyzed. The dynamic system has constant input and feedback time delays due to a communications channel. The proposed control structure consists of a linearized subsystem local to the controlled plant and a remote predictive controller located at the master command station. In the local linearized subsystem, a recurrent neural network with on-line weight tuning algorithm is employed to approximate the dynamics of the time-delay-free nonlinear plant. No linearity in the unknown parameters is required. No preliminary off-line weight learning is needed. The remote controller is a modified Smith predictor that provides prediction and maintains the desired tracking performance; an extra robustifying term is needed to guarantee stability. Rigorous stability proofs are given using Lyapunov analysis. The result is an adaptive neural net compensation scheme for unknown nonlinear systems with time delays. A simulation example is provided to demonstrate the effectiveness of the proposed control strategy.

Stable adaptive fuzzy control of nonlinear systems preceded by unknown backlash-like hysteresis

This paper deals with adaptive control of nonlinear dynamic systems preceded by unknown backlash-like hysteresis nonlinearities, where the hysteresis is described by a dynamic equation. By utilizing this dynamic model and by combining a fuzzy universal function approximator with adaptive control techniques, a stable adaptive fuzzy control algorithm is developed without constructing a hysteresis inverse. The stability of the closed-loop system is shown using Lyapunov arguments. The effectiveness of the proposed method is demonstrated through simulations.

Neuro-fuzzy modelling and control of nonlinear dynamic systems

Neuro-fuzzy modelling and control of nonlinear dynamic systems

Neuro-fuzzy modelling and control of nonlinear dynamic systems

Neuro-fuzzy modelling and control of nonlinear dynamic systems

Adaptive control of nonlinear dynamical systems using a model reference approach

In this paper we consider using a model reference adaptive control approach to control nonlinear systems. We consider the controller design and stability analysis associated with these type of adaptive systems. Then we discuss the use of model reference adaptive control algorithms to control systems which exhibit nonlinear dynamical behaviour using the example of a Duffing oscillator being controlled to follow a linear reference model. For this system we show that if the nonlinearity is “small” then standard linear model reference control can be applied. A second example, which is often found in synchronization applications, is when the nonlinearities in the plant and reference model are identical. Again we show that linear model reference adaptive control is sufficient to control the system. Finally we consider controlling more general nonlinear systems using adaptive feedback linearization to control scalar nonlinear systems. As an example we use the Lorenz and Chua systems with parameter values such that they both have chaotic dynamics. The Lorenz system is used as a reference model and a single coordinate from the Chua system is controlled to follow one of the Lorenz system coordinates.

Orthogonal least squares algorithm for the approximation of a map and its derivatives with a RBF network

Radial basis function networks (RBFNs) are used primarily to solve curve-fitting problems and for non-linear system modeling. Several algorithms are known for the approximation of a non-linear curve from a sparse data set by means of RBFNs. Regularization techniques allow to define constraints on the smoothness of the curve by using the gradient of the function in the training. However, procedures that permit to arbitrarily set the value of the derivatives for the data are rarely found in the literature. In this paper, the orthogonal least squares (OLS) algorithm for the identification of RBFNs is modified to provide the approximation of a non-linear single-input single-output map along with its derivatives, given a set of training data. The interest in the derivatives of non-linear functions concerns many identification and control tasks where the study of system stability and robustness is addressed. The effectiveness of the proposed algorithm is demonstrated with examples in the field of data interpolation and control of non-linear dynamical systems.

Constrained exact controllability of semilinear systems

In the paper infinite-dimensional dynamical control systems described by semilinear abstract differential equations are considered. Using a generalized open-mapping theorem, sufficient conditions for constrained exact local controllability are formulated and proved. It is generally assumed, that the values of admissible controls are in a convex and closed cone with vertex at zero. Constrained exact local controllability of semilinear abstract second-order dynamical systems are also formulated and proved. As an illustrative example, constrained exact local controllability problem for semilinear hyperbolic type distributed parameters dynamical system is solved in details. Some remarks and comments on the existing results for controllability of nonlinear dynamical systems are also presented.