Order Nonlinear Systems Research Articles

Feedback stabilisation control design for fractional order non‐linear systems in the lower triangular form

Using the Lyapunov function method, this study investigates both state and output feedback stabilisation control design problems for fractional order non-linear systems in the lower triangular form, and presents a number of new results. First, some new properties for Caputo fractional derivative are presented. Second, by introducing appropriate transformations of coordinates, the feedback stabiliser design problem is converted into the determination of finding some parameters, which can be obtained by solving the Lyapunov equation and relevant matrix inequalities. Finally, based on the Lyapunov function method, both state and output feedback stabilisers are explicitly designed to make the closed-loop system asymptotically stable. The study of an illustrative example shows that the obtained results are effective in designing feedback stabilisers for fractional order non-linear systems in the lower triangular form.

Existence of positive solution for a fractional order nonlinear differential system involving a changing sign perturbation

Existence of positive solution for a fractional order nonlinear differential system involving a changing sign perturbation

Control problems for semilinear second order equations with cosine families

This paper aims to obtain the approximate controllability for the second order nonlinear control systems with a strongly cosine family and the associated with sine family.

Homoclinic orbits of first order nonlinear Hamiltonian systems with asymptotically linear nonlinearities at infinity

By using variational methods and critical point theory, in particular, a generalized weak linking theorem, we study a first order nonlinear Hamiltonian system with asymptotically linear nonlinearity at infinity. We obtain the existence of ground state homoclinic orbits for this nonlinear Hamiltonian system. In particular, we obtain a {\it necessary and sufficient condition} for the existence of ground state homoclinic orbits. To the best of our knowledge, there is no published result focusing on necessary and sufficient conditions of the existence of ground state homoclinic orbits for this system.

Stabilization conditions for fuzzy control of uncertain fractional order non‐linear systems with random disturbances

A novel fractional order Takagi–Sugeno (T–S) fuzzy control method for a class of fractional order non-linear systems is proposed in this study. On the basis of the fractional order interval theory, the generalised T–S fuzzy model for a class of uncertain fractional order non-linear systems is presented. By using fractional order Lyapunov stability theorem, the more relaxed and practical sufficient stability conditions which are presented as a new set of linear matrix inequalities are put forward, which are guaranteed by rigorous mathematical derivation. The proposed control scheme is developed for fractional order non-linear systems stabilisation in the presence of random disturbances. Typical instances including the fractional order three-dimensional (3D) non-linear system and the fractional order 4D non-linear Lorenz hyperchaos are implemented. Simulation results indicate the designed fractional order T–S fuzzy control scheme works well compared with the existing method.

Design a Methodology to Model-Reference Control of First Order Delays System

Design a nonlinear controller for second order nonlinear uncertain dynamical systems is one of the most important challenging works. This research focuses on the design, and analysis of a model-reference sliding mode controller for first order delay system, in presence of uncertainties. In order to provide high performance nonlinear methodology, model-reference sliding mode controller is selected. Pure sliding mode controller can be used to control of partly known nonlinear dynamic parameters. Conversely, pure sliding mode controller is used in many applications; it has an important drawback namely; chattering phenomenon. To attenuation the chattering, new filter based high speed control technique is introduced. In this technique, two type derivative techniques are used to improve the rate of delay as well stability, robustness and chattering attenuation. This technique cased to improve the rate of delay compare with conventional PID controller and conventional sliding mode controller.

Adaptive Mittag‐Leffler Stabilization of a Class of Fractional Order Uncertain Nonlinear Systems

AbstractThis paper deals with the stabilization of a class of commensurate fractional order uncertain nonlinear systems. The fractional order system concerned is of the strict‐feedback form with uncertain nonlinearity. An adaptive control scheme combined with fractional order update laws is proposed by extending classical backstepping control to fractional order backstepping scheme. The asymptotic stability of the closed‐loop system is guaranteed under the construction of fractional Lyapunov functions in the sense of generalized Mittag‐Leffler stability. The fractional order nonlinear system investigated can be stabilized asymptotically globally in presence of arbitrary uncertainty. Finally illustrative examples and numerical simulations are performed to verify the effectiveness of the proposed control scheme.

Multi-objective optimized fuzzy-PID controllers for fourth order nonlinear systems

In this paper, the Multi-objective Genetic Algorithm (MOGA) is used to obtain the Pareto frontiers of conflicting objective functions for the fuzzy-Proportional-Integral-Derivative (fuzzy-PID) controllers. The ball–beam and inverted pendulum fourth order nonlinear systems are regarded as nonlinear benchmarks. The considered objective functions for the ball–beam system are the distance error of the ball, the angle error of the beam, and the control effort. For the inverted pendulum system, the objective functions are the distance error of the cart, the angle error of the pendulum, and the control effort, which must be minimized simultaneously. The Pareto fronts are compared with those obtained by Multi-objective Particle Swarm Optimization (MOPSO). Four points are chosen from nondominated solutions of the obtained Pareto fronts based on the three conflicting objective functions and used for illustration of the state variables of the controlled systems. Obtained results elucidate the efficiency of the proposed controller in order to control nonlinear systems.

Higher order nonlinear dynamic systems on time scales

Abstract In this paper, we consider a nonlinear system of higher order three-point boundary value problems on time scales. The Schauder fixed point theorem is used to investigate the existence of solutions of nonlinear dynamic systems on time scales. Furthermore, we establish the criteria for the existence of at least one, two and three positive solutions for higher order nonlinear dynamic systems on time scales by using the four functionals fixed point theorem, the Avery–Henderson fixed point theorem and the five functionals fixed point theorem, respectively.

Finite time tracking control of higher order nonlinear multi agent systems with actuator saturation

Abstract: In this paper a distributed leader follower finite time tracking control of higher order nonlinear multi agent systems (MAS) is presented subject to actuator saturation. A saturated continuous homogeneous consensus control is developed to obtain finite time convergence. For stability of the saturated actuators the geometric homogeneity theory is used and it is proven that all the states of the followers can converge to that of the leader in finite time. Switching control is designed based on super twisting algorithm to nullify the effect of uncertainties and external disturbances. It is also ensures that the sliding surface reaches the equilibrium in finite time. The control law can be effectively used for more general higher order nonlinear agent dynamics. Simulation results verify the effectiveness of the proposed scheme.

Second-order P-type iterative learning control for fractional order non-linear time-delay systems

This paper presents a second-order P-type iterative learning control (ILC) scheme for a class of fractional-order non-linear time-delay systems with fractional order α (0 &le; α < 1). First, by analysing the control and learning processes, a discrete system for P-type ILC is established and the ILC design problem is then converted to a stabilisation problem for such a discrete system. Next, by introducing the (Λ, ξ)-norm and using a generalised Gronwall-Bellman lemma, the sufficient condition for the convergence of the control input and the tracking errors is obtained. Finally, the validity of the method is verified by a numerical example.

Second-order P-type iterative learning control for fractional order non-linear time-delay systems

This paper presents a second-order P-type iterative learning control ILC scheme for a class of fractional-order non-linear time-delay systems with fractional order α 0 ≤ α < 1. First, by analysing the control and learning processes, a discrete system for P-type ILC is established and the ILC design problem is then converted to a stabilisation problem for such a discrete system. Next, by introducing the Λ, ξ-norm and using a generalised Gronwall-Bellman lemma, the sufficient condition for the convergence of the control input and the tracking errors is obtained. Finally, the validity of the method is verified by a numerical example.

Finite Time Control for Fractional Order Nonlinear Hydroturbine Governing System via Frequency Distributed Model

This paper studies the application of frequency distributed model for finite time control of a fractional order nonlinear hydroturbine governing system (HGS). Firstly, the mathematical model of HGS with external random disturbances is introduced. Secondly, a novel terminal sliding surface is proposed and its stability to origin is proved based on the frequency distributed model and Lyapunov stability theory. Furthermore, based on finite time stability and sliding mode control theory, a robust control law to ensure the occurrence of the sliding motion in a finite time is designed for stabilization of the fractional order HGS. Finally, simulation results show the effectiveness and robustness of the proposed scheme.

Logic-based reset adaptation design for improving transient performance of nonlinear systems

In this paper, logic-based switching and resetting principles are presented to devise adaptive control laws for a class of uncertain nonlinear systems in order to ensure both the transient bound and the asymptotical convergence of the state. A novel supervisor is constructed to decide when to reset the estimation parameter with the pre-given estimation value. A benchmark example is presented to demonstrate the effectiveness of the approach.

Adaptive Fuzzy Control of Strict-Feedback Nonlinear Time-Delay Systems With Unmodeled Dynamics.

In this paper, an approximated-based adaptive fuzzy control approach with only one adaptive parameter is presented for a class of single input single output strict-feedback nonlinear systems in order to deal with phenomena like nonlinear uncertainties, unmodeled dynamics, dynamic disturbances, and unknown time delays. Lyapunov-Krasovskii function approach is employed to compensate the unknown time delays in the design procedure. By combining the advances of the hyperbolic tangent function with adaptive fuzzy backstepping technique, the proposed controller guarantees the semi-globally uniformly ultimately boundedness of all the signals in the closed-loop system from the mean square point of view. Two simulation examples are finally provided to show the superior effectiveness of the proposed scheme.

Design Second Order Vibration Reduction

Design of a robust controller for multi input-multi output (MIMO) nonlinear uncertain dynamical system can be a challenging work. This paper focuses on the design and analysis of a high performance adaptive baseline sliding mode control for second order nonlinear uncertain system, in presence of uncertainties to reduce the vibration. In this research, sliding mode controller is a robust and stable nonlinear controller which selected to control of robot manipulator. The proposed approach effectively combines of design methods from switching sliding mode controller, adaptive model-free baseline controller and linear Proportional-Derivative (PD) control to improve the performance, stability and robustness of the sliding mode controller. Sliding mode controller has two important subparts, switching and equivalent. Switching part (discontinuous part) is very important in uncertain condition but it causes chattering phenomenon. To solve the chattering, the most common method used is linear boundary layer saturation method, but this method lost the stability. To reduce the chattering with respect to stability and robustness; linear controller is added to the switching part of the sliding mode controller. The linear controller is to reduce the role of sliding surface slope and switching (sign) function. The nonlinearity term of the sliding mode controller is used to eliminate the decoupling and nonlinear term of link’s dynamic parameters. However nonlinearity term of sliding mode controller is very essential to reliability but in uncertain condition or highly nonlinear dynamic systems it can cause some problems. To solve this challenge the baseline controller is used as online tune or adaptive controller. This controller improves the stability and robustness, reduces the chattering as well and reduces the level of energy due to the torque performance as well.

Nonlinear Alfvén wave dynamics in plasmas

Nonlinear Alfvén wave dynamics is presented using Lagrangian fluid approach in a compressible collisional magnetized plasma. In the framework of two fluid dynamics, finite electron inertia is shown to serve as a dispersive effect acting against the convective nonlinearity. In a moving frame, the Alfvén wave can, therefore, form an arbitrarily strong amplitude solitary wave structure due to the balance between nonlinearity and dispersion. Weak amplitude Alfvén waves are shown to be governed by a modified KdV equation, which extends for finite dissipation to a mKdV-Burgers equation. These equations have well known solutions. Next, we have analyzed the fourth order nonlinear Alfvén wave system of equations both numerically and by approximation method. The results indicate a collapse of the density and magnetic field irrespective of the presence of dispersion. The wave magnetic field, however, appears to be less singular showing collapse only when the dispersive effects are negligible. These results may contribute to our understanding of the generation of strongly localized magnetic fields (and currents) in plasmas and are expected to be of special importance in the astrophysical context of magnetic star formation.

Synchronization of Heterogeneous Multi‐Agent Systems by Adaptive Iterative Learning Control

AbstractIn this work, under a repeatable control environment, an adaptive iterative learning control method is applied to synchronize a group of uncertain heterogeneous agents. The agent dynamics are modeled by nonlinear equations, which contain both parametric and non‐parametric uncertainties. Furthermore, the uncertainties are assumed to be general nonlinear terms instead of the global Lipschitz functions. The communication among the followers is depicted by an undirected and connected graph, meanwhile, the virtual leader's trajectory is only accessible to a small portion of the followers. The proposed learning rules enable all the followers to learn and handle both parametric and non‐parametric uncertainties based on the local information such that the followers can synchronize their trajectories to the desired one. In comparison with the existing literature, most works assume first or second order nonlinear systems, and perfect initial conditions. In order to mitigate the identical initialization condition, the applicability of alignment condition and initial rectifying action are further explored. In addition, our developed algorithms can be applied to general high order nonlinear systems. Finally, synchronization examples of networked robotic manipulators are presented to demonstrate the effectiveness of the developed methods.

Distributed adaptive fuzzy iterative learning control of coordination problems for higher order multi-agent systems

In this paper, the adaptive fuzzy iterative learning control scheme is proposed for coordination problems of Mth order (M ≥ 2) distributed multi-agent systems. Every follower agent has a higher order integrator with unknown nonlinear dynamics and input disturbance. The dynamics of the leader are a higher order nonlinear systems and only available to a portion of the follower agents. With distributed initial state learning, the unified distributed protocols combined time-domain and iteration-domain adaptive laws guarantee that the follower agents track the leader uniformly on [0, T]. Then, the proposed algorithm extends to achieve the formation control. A numerical example and a multiple robotic system are provided to demonstrate the performance of the proposed approach.

An existence and uniqueness theorem for a second order nonlinear system with coupled integral boundary value conditions

The existence and uniqueness of positive solutions for a second order nonlinear system with coupled integral boundary value conditions are investigated. The main tools used in the proofs are a priori estimate method and a maximal principle.