Control Of Nonlinear Dynamical Systems Research Articles

Optimization design of RBF‐ARX model and application research on flatness control system

SummaryThe radial basis function (RBF) network and autoregressive exogenous (ARX) model are combined to form the structure of the RBF‐ARX model. The RBF‐ARX model can describe the global nonlinear dynamic process of the object, and its function coefficients are approximated by data‐driven method. The structured nonlinear parameters optimization method (SNPOM) is generally used to optimize model parameters, but this method is very complicated and hard to be mastered by engineers. However, genetic algorithm (GA) is simple and widely used. So the thought of GA optimizing RBF‐ARX is generated, called GA‐ARX‐RBF, and applied to nonlinear dynamic flatness control system. In this article, the recursive least squares method to optimize linear weights of RBF is also used to improve the SNPOM, which reduces the complexity and storage capacity of data processing. Meanwhile, GA to optimize all the parameters of the RBF‐ARX model replaces SNPOM completely. A GA‐RBF‐ARX modeling and optimizing method is proposed. In order to prove the efficiency of GA‐RBF‐ARX, it is applied into flatness control system, which has the characters of nonlinear, multivariable, and multi‐disturbance. The flatness recognition model and flatness predictive model are established. A predictive controller based on GA‐RBF‐ARX is designed for 900HC reversible cold rolling mill. The simulation results demonstrate that the flatness control system based on GA‐RBF‐ARX is effective and has a better precision. The method is easily mastered by engineers and helps to promote the practical value of RBF‐ARX. Copyright © 2016 John Wiley & Sons, Ltd.

Input-to-State Stability, Integral Input-to-State Stability, and <inline-formula> <tex-math notation="LaTeX">${\cal L}_{2} $</tex-math></inline-formula>-Gain Properties: Qualitative Equivalences and Interconnected Systems

Input-to-state stability (ISS) and ${\cal L}_{2} $ -gain are well-known robust stability properties that continue to find wide application in the analysis and control of nonlinear dynamical systems and their interconnections. We investigate the relationship between ISS-type and ${\cal L}_{2} $ -gain properties, demonstrating several qualitative equivalences between these two properties via coordinate transformations. In particular, we investigate and clearly specify the relationship between ISS and linear ${\cal L}_{2}$ -gain, and between integral ISS and nonlinear ${\cal L}_{2} $ -gain. This latter relationship raises an interesting question as it is known that cascade and feedback interconnections of integral ISS systems are not generally well-behaved, whereas we demonstrate that such interconnections for systems with nonlinear ${\cal L}_{2} $ -gain are well-behaved. We subsequently present several new sufficient conditions for the stability of interconnected systems derived by exploiting the derived qualitative equivalences.

Adaptive Intelligent Inverse Control Of Nonlinear Systems With Regard To Sensor Noise And Parameter Uncertainty (Magnetic Ball Levitation System Case Study)

Abstract Type-2 Fuzzy Neural Networks have tremendous capability in identification and control of nonlinear, time-varying and uncertain systems. In this paper the procedure of designing inverse adaptive type-2 fuzzy neural controller for online control of nonlinear dynamical systems will be presented. At first the structure of a novel class of Interval Type-2 Nonlinear Takagi-Sugeno-Keng Fuzzy Neural Networks (IT2-NTSK-FNN) will be presented. There is a class of nonlinear function of inputs in the consequent part of fuzzy rules. This IT2-NTSK-FNN comprises seven layers and the fuzzification is done in two first layers including type-2 fuzzy neurons with uncertainties in the mean of Gaussian membership functions. Third layer is rule layer and model reduction occurs in fourth layer via adaptive nodes. Fifth, sixth and seventh layers are consequent layer, centroid rules’ calculation layer and output layer respectively. For training the network backpropagation (steepest descend) method with adaptive training rate is used. Finally, three methods including online adaptive inverse controller based on IT2-NTSK-FNN, IT2-TSK-FNN (linear consequent part) and Adaptive Neuro-Fuzzy Inference System (ANFIS) are employed to control of a magnetic ball levitation system. External disturbances and uncertainty in parameters are considered in the model of magnetic ball levitation system. Simulation results show the efficacy of the proposed method

Stabilization of nonlinear discrete-time dynamic control systems with a parameter and state-dependent coefficients

A numerical-analytical algorithm for designing nonlinear stabilizing regulators for the class of nonlinear discrete-time control systems is proposed that significantly reduces computational costs. The resulting regulator is suboptimal with respect to the constructed quadratic functional with state-dependent coefficients. The conditions for the stability of the closed-loop system are established, and a stability result is stated. Numerical results are presented showing that the nonlinear regulator designed is superior to the linear one with respect to both nonlinear and standard time-invariant cost functionals. An example demonstrates that the closed-loop system is uniformly asymptotically stable.

Neural network based semi-empirical models for dynamical systems described by differential-algebraic equations

We analyzed the problem of mathematical modeling and computer simulation of nonlinear controlled dynamical systems usually described by differential-algebraic equations. The problem is proposed to be solved in the framework of the semi-empirical approach combining theoretical knowledge for the plant with training tools of artificial neural network field. The results are presented for a semi-empirical model that simulate the reentry hypersonic vehicle and confirm the efficiency of this approach.

A New Nonlinear H-infinity Feedback Control Approach to the Problem of Autonomous Robot Navigation

This research work introduces a new control method for feedback control of nonlinear dynamical systems with application example the problem of trajectory tracking by an autonomous robotic vehicle. The control method consists of a repetitive solution of an H-infinity control problem for the mobile robot, that makes use of a locally linearized model of the robot and takes place at each iteration of the control algorithm. The vehicle’s model is locally linearized round its current position through the computation of the associated Jacobian matrices. Using the linearized model of the vehicle an H-infinity feedback control law is computed. The known robustness features of H-infinity control enable to compensate for the errors of the approximative linearization, as well as to eliminate the effects of external perturbations. The efficiency of the proposed control scheme is shown analytically and is confirmed through simulation experiments. The method can be applied to a wide class of nonlinear dynamical systems.

Intelligent control of non-linear dynamical system based on the adaptive neurocontroller

This paper presents an adaptive neuro-controller for intelligent control of non-linear dynamical system. The formed as the fuzzy selective neural net the adaptive neuro-controller on the base of system's state, creates the effective control signal under random perturbations. The validity and advantages of the proposed adaptive neuro-controller are demonstrated by numerical simulations. The simulation results show that the proposed controller scheme achieves real-time control speed and the competitive performance, as compared to PID, fuzzy logic controllers.

Reliable stabilizing controller based on set-value parameter synthesis

A set-membership (SM) approach is proposed to design reliable static stabilizing controllers for non-linear dynamical systems. First, the control design problem is reformulated as a parameter identification issue in an unknown-but-bounded error framework. Then, a new dual set integration method for reachability computation is introduced. Finally, the latter dual method is used in conjunction with set inversion techniques via interval analysis, in order to develop an SM parameter estimation algorithm to solve the control design problem. The identified feedback control must achieve two aims. The first aim is that, starting from a given bounded set of initial states, the state trajectories generated by the controlled non-linear system have to reach in a finite-time a desired target set. The second aim is to ensure asymptotic stability of the controlled system over the target set. The effectiveness of the proposed method is illustrated through a complex non-linear system.

Optimal stable control for nonlinear dynamical systems: an analytical dynamics based approach

This paper presents a method for obtaining optimal stable control for general nonlinear nonautonomous dynamical systems. The approach is inspired by recent developments in analytical dynamics and the observation that the Lyapunov criterion for stability of dynamical systems can be recast as a constraint to be imposed on the system. A closed-form expression for control is obtained that minimizes a user-defined control cost at each instant of time and enforces the Lyapunov constraint simultaneously. The derivation of this expression closely mirrors the development of the fundamental equation of motion used in the study of constrained motion. For this control method to work, the positive definite functions used in the Lyapunov constraint should satisfy a consistency condition. A class of positive definite functions has been provided for mechanical systems that meet this criterion. To illustrate the broad scope of the method, for linear systems it is shown that a proper choice of these positive definite functions results in conventional LQR control. Control of the Lorenz system and a multi-degree of freedom nonlinear mechanical system are considered. Numerical examples demonstrating the efficacy and simplicity of the method are provided.

Iterative learning control for nonlinear dynamic systems with randomly varying trial lengths

SummaryIn this paper, we introduce an iterative learning control (ILC) scheme based on an iteratively moving average operator for nonlinear dynamic systems with randomly varying trial lengths. By using the iteratively moving average operator, the proposed ILC algorithm overcomes the limitation of traditional ILC that all trial lengths must be identical. It is shown that for nonlinear affine and non‐affine systems, the proposed learning algorithm works effectively to nullify the tracking error. In the end, two illustrative examples are presented to demonstrate the performance and the effectiveness of the proposed ILC scheme for nonlinear dynamic systems. Copyright © 2015 John Wiley & Sons, Ltd.

Fuzzy Counter Propagation Neural Network Control for a Class of Nonlinear Dynamical Systems.

Fuzzy Counter Propagation Neural Network (FCPN) controller design is developed, for a class of nonlinear dynamical systems. In this process, the weight connecting between the instar and outstar, that is, input-hidden and hidden-output layer, respectively, is adjusted by using Fuzzy Competitive Learning (FCL). FCL paradigm adopts the principle of learning, which is used to calculate Best Matched Node (BMN) which is proposed. This strategy offers a robust control of nonlinear dynamical systems. FCPN is compared with the existing network like Dynamic Network (DN) and Back Propagation Network (BPN) on the basis of Mean Absolute Error (MAE), Mean Square Error (MSE), Best Fit Rate (BFR), and so forth. It envisages that the proposed FCPN gives better results than DN and BPN. The effectiveness of the proposed FCPN algorithms is demonstrated through simulations of four nonlinear dynamical systems and multiple input and single output (MISO) and a single input and single output (SISO) gas furnace Box-Jenkins time series data.

Investigation of the polyharmonic method for calculating frequency responses of nonlinear dynamic plants

Efficiency of the new polyharmonic method for automatic integration with the complete averaging of the Fourier coefficients for calculating frequency responses of nonlinear dynamic plants and control systems with prescribed accuracy and in the minimum amount of time is investigated.

Further results on the impulsive synchronization of uncertain complex‐variable chaotic delayed systems

Synchronization and control of nonlinear dynamical systems with complex variables has attracted much more attention in various fields of science and engineering. In this article, we investigate the problem of impulsive synchronization for the complex‐variable delayed chaotic systems with parameters perturbation and unknown parameters in which the time delay is also included in the impulsive moment. Based on the theories of adaptive control and impulsive control, synchronization schemes are designed to make a class of complex‐variable chaotic delayed systems asymptotically synchronized, and unknown parameters are identified simultaneously in the process of synchronization. Sufficient conditions are derived to synchronize the complex‐variable chaotic systems include delayed impulses. To illustrate the effectiveness of the proposed schemes, several numerical examples are given. © 2014 Wiley Periodicals, Inc. Complexity 21: 131–142, 2016

Adaptive variable structure hierarchical fuzzy control for a class of high-order nonlinear dynamic systems

In this paper, a novel adaptive hierarchical fuzzy control system based on the variable structure control is developed for a class of SISO canonical nonlinear systems in the presence of bounded disturbances. It is assumed that nonlinear functions of the systems be completely unknown. Switching surfaces are incorporated into the hierarchical fuzzy control scheme to ensure the system stability. A fuzzy soft switching system decides the operation area of the hierarchical fuzzy control and variable structure control systems. All the nonlinearly appeared parameters of conclusion parts of fuzzy blocks located in different layers of the hierarchical fuzzy control system are adjusted through adaptation laws deduced from the defined Lyapunov function. The proposed hierarchical fuzzy control system reduces the number of rules and consequently the number of tunable parameters with respect to the ordinary fuzzy control system. Global boundedness of the overall adaptive system and the desired precision are achieved using the proposed adaptive control system. In this study, an adaptive hierarchical fuzzy system is used for two objectives; it can be as a function approximator or a control system based on an intelligent-classic approach. Three theorems are proven to investigate the stability of the nonlinear dynamic systems. The important point about the proposed theorems is that they can be applied not only to hierarchical fuzzy controllers with different structures of hierarchical fuzzy controller, but also to ordinary fuzzy controllers. Therefore, the proposed algorithm is more general. To show the effectiveness of the proposed method four systems (two mechanical, one mathematical and one chaotic) are considered in simulations. Simulation results demonstrate the validity, efficiency and feasibility of the proposed approach to control of nonlinear dynamic systems.

A novel criterion for nonlinear time-delay systems using LMI fuzzy Lyapunov method

This study presents a kind of fuzzy robustness design for nonlinear time-delay systems based on the fuzzy Lyapunov method, which is defined in terms of fuzzy blending quadratic Lyapunov functions. The basic idea of the proposed approach is to construct a fuzzy controller for nonlinear dynamic systems with disturbances in which the delay-independent robust stability criterion is derived in terms of the fuzzy Lyapunov method. Based on the robustness design and parallel distributed compensation (PDC) scheme, the problems of modeling errors between nonlinear dynamic systems and Takagi–Sugeno (T–S) fuzzy models are solved. Furthermore, the presented delay-independent condition is transformed into linear matrix inequalities (LMIs) so that the fuzzy state feedback gain and common solutions are numerically feasible with swarm intelligence algorithms. The proposed method is illustrated on a nonlinear inverted pendulum system and the simulation results show that the robustness controller cannot only stabilize the nonlinear inverted pendulum system, but has the robustness against external disturbance.

Attainability analysis in the stochastic sensitivity control

For nonlinear dynamic stochastic control system, we construct a feedback regulator that stabilises an equilibrium and synthesises a required dispersion of random states around this equilibrium. Our approach is based on the stochastic sensitivity functions technique. We focus on the investigation of attainability sets for 2-D systems. A detailed parametric description of the attainability domains for various types of control inputs for stochastic Brusselator is presented. It is shown that the new regulator provides a low level of stochastic sensitivity and can suppress oscillations of large amplitude.

Autonomous Formation Flight Control System Using In-Flight Sweet-Spot Estimation

A formation flight control system that addresses the unique environment encountered by aircraft flying in formation and in the upwash of the leading aircraft that leads to a scenario with minimum induced drag at the trailing aircraft (sweet spot) has been designed. To test the control system, a simulation environment that adequately represents the aerodynamic coupling effects between aircraft flying in formation and implements an incremental nonlinear dynamic inversion-based formation flight control system has been created. Since the exact position of the sweet spot is not known or measured, the research presented in this paper uses an advanced extremum-seeking algorithm that uses an extended Kalman filter to estimate the wind gradients to automatically steer the trailing vehicle toward the sweet spot. Monte Carlo simulations for two and three aircraft formations have shown that the algorithm has the ability to consistently find the sweet spot and is robust to errors due to turbulence.

Integrated fault‐tolerant control methodology for non‐linear dynamical networked control systems subject to time‐varying delay

This study deals with the problem of designing an integrated fault-tolerant control (FTC) methodology for non-linear dynamical networked control systems subject to time-varying delay. For the convenience of residual generation, influence caused by network-induced delay is transformed into time-varying uncertainty, which facilitates further manipulation. A fault diagnosis scheme is utilised for fault detection and isolation. On the basis of the uncertainty and fault information obtained by the fault diagnosis procedure, an FTC component is developed to compensate for the effects of uncertainty and fault. In the absence of fault, a nominal controller is employed to deal with the system uncertainty. After fault detection estimator is activated, the controller is reconfigured to guarantee the boundedness of all the system signals until the fault is isolated. After the fault is isolated, the controller is reconfigured to improve the control performance by using the fault information generated by the isolation module. In addition, the stability of the proposed scheme is rigorously investigated. Finally, a simulation example is utilised to illustrate the performance of the proposed scheme.

Design Serial Intelligent Modified Feedback Linearization like Controller with Application to Spherical Motor

Fuzzy logic controller (FLC) is an important nonlinear controller in an uncertain dynamic parameters. This controller is used to control of nonlinear dynamic systems particularly for spherical motor, because it has a suitable control performance and it is a stable. Conversely pure fuzzy logic controller is a high-quality intelligent nonlinear controller; it has two important problems; reliability and robustness in uncertain dynamic parameter. To increase the reliability and robustness, this research is focused on applied feedback linearization method in pure fuzzy logic controller. In this research the nonlinear equivalent dynamic (equivalent part) formulation problem in uncertain condition is also solved by combine pure fuzzy logic control and feedback linearization method. In this method feedback linearization theorem is applied to fuzzy logic controller to increase the stability, reliability and robustness, which it is based on nonlinear dynamic formulation. To achieve this goal, the dynamic-based formulation feedback linearization method is design. This method is robust and model-based nonlinear control therefore can reduce the nonlinearity term of system and reduce the effect of coupling. In this research MAMDANI fuzzy inference system is used as a main controller. It has minimum rule base to practical implementation. This technique was employed to obtain the desired control behavior with a number of information about dynamic model of system and a feedback linearization control was applied to reinforce system performance.

Temperature Control using Fuzzy Logic

The aim of the temperature control is to heat the system up todelimitated temperature, afterwardhold it at that temperature in insured manner. Fuzzy Logic Controller (FLC) is best way in which this type of precision control can be accomplished by controller. During past twenty yearssignificant amount of research using fuzzy logichas done in this field of control of non-linear dynamical system. Here we have developed temperature control system using fuzzy logic. Control theory techniques are the root from which convention controllers are deducted. The desired response of the output can be guaranteed by the feedback controller.