Order Nonlinear Systems Research Articles

A residue-type phenomenon and its applications to higher order nonlinear systems of Poisson type

In this paper, we establish a Residue-type phenomenon for the fundamental solution of the Laplacian. With the aid of the formula, we derive a higher order derivative formula for the Newtonian potential and its Hölder estimates with a gain of two derivatives. The estimates allow us to obtain the solvability of a type of higher order nonlinear Poisson system.

Robust stability analysis of uncertain fractional order neutral-type delay nonlinear systems with actuator saturation

In this paper, we study the robust stability of uncertain fractional order (FO) nonlinear systems having neutral-type delay and input saturation. From the Lyapunov–Krasovskii functional, sufficient criteria on asymptotic robust stability of such FO systems with the help of linear matrix inequalities are specified to compute the gain of state-feedback controller. An optimization is also derived using the cone complementarity linearization method for finding the controller gains subject to maximizing the domain of attraction. The main results are confirmed via numerical examples.

A STUDY ABOUT STABILITY OF TWO AND THREE SPECIES POPULATION MODELS

In this article, we have discussed the stability of second order linear and non-linear systems by characteristic roots. In the case of non-linear system, we linearize the nonlinear system under certain specified conditions and study the stability of critical points of the linearized systems. Necessary theories have been presented, applied, and illustrated with examples. A self-contained theory for a homogeneous linear system of third order is built by using the basic concept of the differential equation.

Existence and uniqueness of solutions for the system ofintegro-differential equations with three-point and nonlinear integral boundary conditions

The paper examines a system of nonlinear integro-differential equations with three-point and nonlinear integral boundary conditions. The original problem demonstrated to be equivalent to integral equations by using Green function. Theorems on the existence and uniqueness of a solution to the boundary value problems for the first order nonlinear system of integro- differential equations with three-point and nonlinear integral boundary conditions are proved. A proof of uniqueness theorem of the solution is obtained by Banach fixed point principle, and the existence theorem then follows from Schaefer’s theorem.

Asymptotic Solutions of Fifth Order Overdamped-Oscillatory Nonlinear Systems

Combo overdamp-oscillatory system plays an important role in natural phenomena in many engineering problems. In this paper, fifth order nonlinear damped-oscillatory differential system is studied to investigate an asymptotic analytical approximate solution in the fashion of overdamp-oscillations via an extension of the Krylov-Bogoliubov-Mitropolskii (KBM) method. The proposed method is demonstrated by its applications on a Duffing oscillators in the combined form of overdamp and oscillatory effects. The result obtained by the presented extended technique good agreement with the numerical solutions of the fourth order Runge-Kutta method.

Dynamic characteristics analysis of time-delay fractional order dynamic system

In this paper, a kind of time-delay fractional order dynamic system is studied. The classical integral sliding mode control method is used to design a single controller to control the time-delay fractional order Liu system from chaos to fixed point, and the single controller is further designed to control the time-delay fractional order Liu system from chaotic attractor to limit loop, so the chaos control of time-delay fractional order nonlinear dynamic system is achieved. On that basis, the dynamic characteristics of the time-varying delayed fractional-order Lorenz system are analyzed by the simulation, and the chaotic phenomena of the time-varying delayed fractional-order Lorenz system are verified under the provided time delay and order.

Event-triggered adaptive neural control of fractional-order nonlinear systems with full-state constraints

In this article, we present an event-triggered scheme for fractional order nonlinear systems under full-state constraints. The neural networks are employed to approximate the continuous nonlinear unknown functions in the system. Moreover, a new adaptive event-triggered strategy is designed under the unified framework of backstepping control method, which can largely reduce the amount of communications. Since the state constraints are frequently emerged in the control procedure, the barrier Lyapunov functions is used to avoid the violation of state constraints. The stability of the closed-loop system is ensured through fractional Lyapunov stability analysis. Finally, the effectiveness of the proposed scheme is verified by simulation examples.

Uniqueness Solution Of Abstract Fractional Order Nonlinear Dynamical Control Problems

The aim of this paper is to investigate the Uniqueness solution of Cauchy Problem represented for fractional order nonlinear dynamical control system involving certain control input and their approach of investigated depended on commutative composite semigroup and some certain conditions in certain space.

Multi-model estimation using neural network and fault detection in unknown time continuous fractional order nonlinear systems

In this paper, multi-model estimation and fault detection using neural network is proposed for an unknown time continuous fractional order nonlinear system. Fractional differentiation is considered based on Caputo concept and the fractional order is considered to be between 0 and 1. In order to estimate a time continuous fractional order nonlinear system with unknown term in its dynamic, single-layer and double-layer RBF neural network is used. First, a parallel-series neural network observer is designed for state estimation. Weights of the neural network are updated adaptively and updating laws are presented in fractional order form. Using Lyapunov method, it is proved that state estimation error and weight estimation error of the neural network are bounded. Parameters of the neural estimator converge to ideal parameters which satisfy excitation condition stability. Then, multi-model estimation structure of fractional order nonlinear systems is presented and its application in fault detection is investigated. Finally, simulation results are presented to show efficiency of the proposed method.

Computatıonal Algorıthm for the Numerıcal Solutıon of Systems of Volterra Integro-Dıfferentıal Equatıons

In this paper, we employ variational iterative method (VIM) to develop a suitable Algorithm for the numerical solution of systems of Volterra integro-differential equations. The formulated algorithm is used to solve first and second order linear and nonlinear system of Volterra integrodifferential equations which demonstrated a good numerical approach to overcome lengthen computational and integral simplification involves. Moreover, the comparison of the exact solution with the approximated solutions are made and approximate solutions p(x) q(t) proved to converge to the exact solutions p(x) q(t) respectively. The results reveal that the formulated algorithm are simple, effective and faster than analytical approach of solving Volterra integro-differential equations.

A Global Maximum Error Controller Based Nonlinearity Aware TPWL Method for Reducing Nonlinear Systems

Model order reduction is a common practice to reduce large order systems so that their simulation and control become easy. Nonlinearity aware trajectory piecewise linear is a variation of trajectory piecewise linearization technique of order reduction that is used to reduce nonlinear systems. With this scheme, the reduced approximation of the system is generated by weighted sum of the linearized and reduced sub-models obtained at certain linearization points on the system trajectory. This scheme uses dynamically inspired weight assignment that makes the approximation nonlinearity aware. Just as weight assignment, the process of linearization points selection is also important for generating faithful approximations. This article uses a global maximum error controller based linearization points selection scheme according to which a state is chosen as a linearization point if the error between a current reduced model and the full order nonlinear system reaches a maximum value. A combination that not only selects linearization points based on an error controller but also assigns dynamic inspired weights is shown in this article. The proposed scheme generates approximations with higher accuracies. This is demonstrated by applying the proposed method to some benchmark nonlinear circuits including RC ladder network and inverter chain circuit and comparing the results with the conventional schemes.

Analysis on control of a class of uncertain stochastic system by inequality technique

In this paper, some theoretical results of PID control of second order nonlinear uncertain stochastic system are given via inequalities. We extend the results of the corresponding deterministic systems to stochastic systems. Specifically, as long as we have a certain understanding of the upper bound of the derivative of the unknown nonlinear drift term and diffusion term, an analytic design method can be constructed for these three PID parameters to ensure the global stability and asymptotic stability of the closed-loop control systems. In addition, the numerical simulation results verify the theoretical analysis results.

Parameter estimation of a second order system via nonlinear identification algorithm

In this paper, a parameter estimation for a class of second order nonlinear system is presented. The considered system, can be represented in such way that it is linear respect its parameters. Since the system output is given, only, for the second integrator a state estimation of unmeasured state variables is reconstructed via nonlinear observer based on the terminal sliding mode observer. Therefore, the main contribution of this paper deals with a full order nonlinear observer algorithm design to enhance parameter identification of a nonlinear second order system. Finally, to illustrate the theoretical performance of the proposed identification algorithm, an experimental result of a mechanical system is presented.

Artificial Intelligence for Detection, Estimation, and Compensation of Malicious Attacks in Nonlinear Cyber-Physical Systems and Industrial IoT

This article proposes a hybrid intelligent-classic control approach for reconstruction and compensation of cyber attacks launched on inputs of nonlinear cyber-physical systems (CPS) and industrial Internet of Things systems, which work through shared communication networks. In this article, a class of n -order nonlinear systems is considered as a model of CPS while it is in presence of cyber attacks only in the forward channel. An intelligent-classic control system is developed to compensate cyber-attacks. Neural network (NN) is designed as an intelligent estimator for attack estimation and a classic nonlinear control system based on the variable structure control method is designed to compensate the effect of attacks and control the system performance in tracking applications. In the proposed strategy, nonlinear control theory is applied to guarantee the stability of the system when attacks happen. In this strategy, a Gaussian radial basis function NN is used for online estimation and reconstruction of cyber-attacks launched on the networked system. An adaptation law of the intelligent estimator is derived from a Lyapunov function. Simulation results demonstrate the validity and feasibility of the proposed strategy in car cruise control application as the testbed.

Fractional Sliding Mode Control for Nonlinear Aerospace Systems

This paper focuses on the study of fractional order terminal sliding mode control for nonlinear aerospace systems. Firstly, a novel fractional order integral terminal sliding mode control (FO-I-TSMC) method is proposed for the control of first order nonlinear system. FO-I-TSMC has three attractive advantages: i) Non-singular control law; ii) Elimination of the reaching phase; iii) Calculable finite convergence time. Furthermore, theory analysis is presented to reveal the potential advantages of the FO-I-TSMC method over its integer order counterparts. Secondly, a novel fractional order derivative integral-TSMC (FO-DI-TSMC) method is presented to deal with second order nonlinear system. Finally, FO-DI-TSMC is extended to deal with a general class of higher order control system. Simulation results are given to verify the effectiveness of the proposed methods.

Optimal neuro-control strategy for nonlinear systems with asymmetric input constraints

In this paper, we present an optimal neuro-control scheme for continuous-time &#x0028 CT &#x0029 nonlinear systems with asymmetric input constraints. Initially, we introduce a discounted cost function for the CT nonlinear systems in order to handle the asymmetric input constraints. Then, we develop a Hamilton-Jacobi-Bellman equation &#x0028 HJBE &#x0029 , which arises in the discounted cost optimal control problem. To obtain the optimal neurocontroller, we utilize a critic neural network &#x0028 CNN &#x0029 to solve the HJBE under the framework of reinforcement learning. The CNN &#x02BC s weight vector is tuned via the gradient descent approach. Based on the Lyapunov method, we prove that uniform ultimate boundedness of the CNN&#x02BC s weight vector and the closed-loop system is guaranteed. Finally, we verify the effectiveness of the present optimal neuro-control strategy through performing simulations of two examples.

Oscillation and Asymptotic Behavior of Solution of nth Order System Nonlinear of Neutral Differential Equation

In this paper, authors obtained sufficient conditions to ensure that every bounded solution of nth order nonlinear neutral differential system oscillates or nonoscillatory converges to zero as t→∞, some examples were given to explain the results obtained.

Maximum power extraction from fractional order doubly fed induction generator based wind turbines using homotopy singular perturbation method

In this paper, a novel control mechanism is presented for the control of variable speed wind turbine (VSWT) with a fractional-order doubly fed induction generator (DFIG) to extract the maximum power in region 2. Firstly, the homotopy singular perturbation method (HSPM) is proposed to simplify the fractional order nonlinear model, to reduce the long memory attribute, and to decrease the mechanical stress. This method is a composition of fractional order singular perturbation method (SPM) and modification of the homotopy perturbation method (HPM). Using this method, the nonlinear fractional order DFIG based VSWT is separated into two lower order subsystems, including nonlinear integer order subsystem and linear fractional order subsystem. Integer order modeling is applied to have the maximum wind power extraction and to increase the trajectory tracking speed, while the fractional order modeling is exerted to reduce the mechanical loads and to ensure better tracking. Then, a sliding mode controller based on HSPM is designed and applied to the original fractional order nonlinear system. Stability of the original system is evaluated by investigating the Lyapunov stability theorem for each subsystem. The controller performance designed based on HSPM is compared to the designed controller performance for fractional order nonlinear system decomposed using fractional order SPM, integer order nonlinear system decomposed by SPM, and integer order nonlinear system without simplification. The simulation results affirm the effectiveness of the controller design based on the proposed method.

Identification of Wind Turbine using Fractional Order Dynamic Neural Network and Optimization Algorithm

In this paper, an efficient technique is presented to identify a 2500 KW wind turbine operating in Kahak wind farm, Qazvin province, Iran. This complicated system dealing with wind behavior is identified by using a proposed fractional order dynamic neural network (FODNN) optimized with evolutionary computation. In the proposed method, some parameters of FODNN are unknown during the process of identification, so a particle swarm optimization (PSO) algorithm is employed to determine the optimal values by which a fractional order nonlinear system can be completely identified with a high degree of accuracy. These parameters are very effective to achieve high performance of FODNN identifier and they include fractional order, initial values of states and weights of FODNN, and numerical algorithm step size for solving FODNN equation. Simulation results confirm the efficiency of the proposed scheme in term of accuracy. Furthermore, comparison of the results achieved by the proposed method and those of the integer order dynamic neural network (IODNN) depicts higher accuracy of the proposed FODNN.

Sub-optimal Control Design for Second Order Non-linear Systems using Krotov Sufficient Conditions

This article considers the problem of sub-optimal control design problem for second order non-linear systems. Traditionally, the computation of optimal control law(s) for non-linear systems is usually done using the iterative methods based on the standard tools of optimal control theory which viz. Calculus of Variations (CoV), Hamilton-Jacobi-Bellman equation, Pontryagin’s principle, etc. This work utilizes the rather less explored technique, namely the Krotov framework to obtain the non-iterative solutions. These conditions are derived by transforming the optimal control problem into another equivalent optimization problem. This translation is done via an ad-hoc selection of the so-called Krotov function. In this article, the Krotov function is chosen such that the equivalent optimization problem can be solved non-iteratively to obtain sub-optimal control laws for the original optimal control problem. The proposed methodology is demonstrated by a numerical example.