Generation of multi-wing chaotic attractor in fractional order system

In this paper, a novel approach is proposed for generating multi-wing chaotic attractors from the fractional linear differential system via nonlinear state feedback controller equipped with a duality-symmetric multi-segment quadratic function. The main idea is to design a proper nonlinear state feedback controller by using four construction criterions from a fundamental fractional differential nominal linear system, so that the controlled fractional differential system can generate multi-wing chaotic attractors. It is the first time in the literature to report the multi-wing chaotic attractors from an uncoupled fractional differential system. Furthermore, some basic dynamical analysis and numerical simulations are also given, confirming the effectiveness of the proposed method.     

Control of an aircraft landing in windshear

The problem of the feedback control of an aircraft landing in the presence of windshear is considered. The landing process is investigated up to the time when the runway threshold is reached. It is assumed that the bounds on the wind velocity deviations from some nominal values are known, while information about the windshear location and wind velocity distribution in the windshear zone is absent. The methods of differential game theory are employed for the control synthesis.The complete system of aircraft dynamic equations is linearized with respect to the nominal motion. The resulting linear system is decomposed into subsystems describing the vertical (longitudinal) motion and lateral motion. For each subsystem, an, auxiliary antagonistic differential game with fixed terminal time and convex payoff function depending on two components of the state vector is formulated. For the longitudinal motion, these components are the vertical deviation of the aircraft from the glide path and its time derivative; for the lateral motion, these components are the lateral deviation and its time derivative. The first player (pilot) chooses the control variables so as to minimize the payoff function; the interest of the second player (nature) in choosing the wind disturbance is just opposite.The linear differential games are solved on a digital computer with the help of corresponding numerical methods. In particular, the optimal (minimax) strategy is obtained for the first player. The optimal control is specified by means of switch surfaces having a simple structure. The minimax control designed via the auxiliary differential game problems is employed in connection with the complete nonlinear system of dynamical equations.The aircraft flight through the wind downburst zone is simulated, and three different downburst models are used. The aircraft trajectories obtained via the minimax control are essentially better than those obtained by traditional autopilot methods.     

Stability of perturbed switched nonlinear systems with delays

This paper addresses the stability problems of perturbed switched nonlinear systems with time-varying delays. It is assumed that the nominal switched nonlinear system (perturbation-free system) is uniformly exponentially stable and that the perturbations satisfy a linear growth bound condition. It is revealed that there exists an upper bound of perturbation guaranteeing that the perturbed system preserves the stability property of the nominal system, locally or globally, depending on both perturbations and the nominal system itself. An example is provided to illustrate the proposed theoretical results.     

Exact and analytical solutions for a nonlinear sigma model

We consider wave solutions to nonlinear sigma models in n dimensions. First, we reduce the system of governing PDEs into a system of ODEs through a traveling wave assumption. Under a new transform, we then reduce this system into a single nonlinear ODE. Making use of the method of homotopy analysis, we are able to construct approximate analytical solutions to this nonlinear ODE. We apply two distinct auxiliary linear operators and show that one of these permits solutions with lower residual error than the other. This demonstrates the effectiveness of properly selecting the auxiliary linear operator when performing homotopy analysis of a nonlinear problem. From here, we then obtain residual error‐minimizing values of the convergence control parameter. We find that properly selecting the convergence control parameter makes a drastic difference in the magnitude of the residual error. Together, appropriate selection of the auxiliary linear operator and of the convergence control parameter is shown to allow approximate solutions that quickly converge to the true solution, which means that few terms are needed in the construction of such solution. This, in turn, greatly improves computational efficiency. Copyright © 2013 John Wiley & Sons, Ltd.     

Lump solutions to a generalized Hietarinta-type equation via symbolic computation

Lump solutions are one of important solutions to partial differential equations, both linear and nonlinear. This paper aims to show that a Hietarinta-type fourth-order nonlinear term can create lump solutions with second-order linear dispersive terms. The key is a Hirota bilinear form. Lump solutions are constructed via symbolic computations with Maple, and specific reductions of the resulting lump solutions are made. Two illustrative examples of the generalized Hietarinta-type nonlinear equations and their lumps are presented, together with three-dimensional plots and density plots of the lump solutions.     

Decoupling problems around a trajectory: from linearity to nonlinearity

Given a nonlinear control system for which an admissible statetrajectory is specified, we solve approximately the input outputdecoupling problem around this nominal trajectory. An approximatesolution for this problem is obtained by dealing with the linearizedsystem along this trajectory. An exact solution to the inputoutput decoupling problem for the linearization is shown tobe an approximate solution to the input output decoupling problemaround the nominal trajectory for the original nonlinear system.In a similar way, we provide an approximate solution to thedisturbance decoupling problem around a specified trajectoryof the nonlinear system. The nonlinear model of a two link robotmanipulator is used to illustrate the results on input outputdecoupling.     

Schwarz problem for higher‐order complex partial differential equations in the upper half plane

Linear and nonlinear elliptic complex partial differential equations of higher‐order are considered under Schwarz conditions in the upper‐half plane. Firstly, using the integral representations for the solutions of the inhomogeneous polyanalytic equation with Schwarz conditions, a class of integral operators is introduced together with some of their properties. Then, these operators are used to transform the problem for linear equations into singular integral equations. In the case of nonlinear equations such a transformation yields a system of integro‐differential equations. Existence of the solutions of the relevant boundary value problems for linear and nonlinear equations are discussed via Fredholm theory and fixed point theorems, respectively.     

Robustness of minimax controllers to nonlinear perturbations

We study the robustness of minimax controllers, originally designed for nominal linear or nonlinear systems, to unknown static nonlinear perturbations in the state dynamics, measurement equation, and performance index. When the nominal system is linear, we consider both perfect state measurements and general imperfect state measurements; in the case of nominally nonlinear systems, we consider perfect state measurements only. Using a differential game theoretic approach, we show for the former class that, as the perturbation parameter (say, >0) approaches zero, the optimal disturbance attenuation level for the overall system converges to the optimal disturbance attenuation level for the nominal system if the nonlinear structural uncertainties satisfy certain prescribed growth conditions. We also show that anH -controller, designed based on a chosen performance level for the nominal linear system, achieves the same performance level when the parameter is smaller than a computable threshold, except for the finite-horizon imperfect state measurements case. For that case, we show that the design of the nominal controller must be based on a decreased confidence level of the initial data, and a controller thus designed again achieves a desired performance level in the face of nonlinear perturbations satisfying a computable norm bound. In the case of nominally nonlinear systems, and assuming that the nominal system is solvable, we obtain sufficient conditions such that the nominal controller achieves a desired performance in the face of perturbations satisfying computable norm bounds. In this way, we provide a characterization of the class of uncertainties that are tolerable for a controller designed based on the nominal system. The paper also presents two numerical examples; in one of these, the nominal system is linear; in the other one, it is nonlinear.This research was supported in part by the US Department of Energy, Grant DE-FG-02-88-ER-13939 and in part by the National Science Foundation, Grant ECS-91-13153.An abridged version was presented at the 32nd IEEE Conference on Decision and Control, San Antonio, Texas, December 15–17, 1993, and it appeared in the Conference Proceedings.     

A Model Predictive Control Algorithm Applied To Non-Linear Processes

Model predictive control (MPC) is an optimization-based approach that has been successfully applied to a wide variety of control problems. In most of nonlinear strategies, the controllers are based on linear models with fixed parameters so that the vast body of linear control theory can be applied. Other solutions include the use of a nonlinear analytical model, combinations of linear empirical models, etc. This paper presents an MPC algorithm which uses on-line simulation and rule-based control. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Control of mechanical systems with uncertain parameters by means of small forces

An approach to the construction of a feedback control for non-linear Lagrange mechanical systems with uncertain parameters is developed. A Lagrange mechanical system with uncertain parameters, which is subject to the action of potential forces, control forces and unknown perturbations is considered is considered. It is assumed that the potential forces can be considerably greater than the control forces which, in their turn, are greater than the perturbations. An approach to the construction of a control, is proposed which enables one to bring a system from an arbitrary initial state to a specified final state in a finite time using a bounded control. A procedure, in which the specified nominal trajectory of the motion is tracked, is used. Initially, the trajectory, joining the specified initial and final states of the system, is constructed for a certain dynamical system which is close to the initial system but with completely known parameters. Then, using deviation equations, a control is constructed which brings the initial system onto this nominal trajectory in a finite time and subsequently forces the system to move along this nominal trajectory up to the final state. The control law used in tracking the nominal trajectory is based on a linear feedback, the gains of which depends on the discrepancy between the real trajectory and the nominal trajectory. The gain increase and tend to infinity as the discrepancies tend to zero but the control forces remain bounded and satisfy the conditions imposed on them. The results of numerical modelling of the controlled motions of a plane double pendulum are presented as an illustration.     

Monotone iterative methods for finite difference system of reaction-diffusion equations

Summary This paper presents an existence-comparison theorem and an iterative method for a nonlinear finite difference system which corresponds to a class of semilinear parabolic and elliptic boundary-value problems. The basic idea of the iterative method for the computation of numerical solutions is the monotone approach which involves the notion of upper and lower solutions and the construction of monotone sequences from a suitable linear discrete system. Using upper and lower solutions as two distinct initial iterations, two monotone sequences from a suitable linear system are constructed. It is shown that these two sequences converge monotonically from above and below, respectively, to a unique solution of the nonlinear discrete equations. This formulation leads to a well-posed problem for the nonlinear discrete system. Applications are given to several models arising from physical, chemical and biological systems. Numerical results are given to some of these models including a discussion on the rate of convergence of the monotone sequences.     

General soliton solutions to a reverse-time nonlocal nonlinear Schrödinger equation

General soliton solutions to a reverse-time nonlocal nonlinear Schrödinger (NLS) equation are discussed via a matrix version of binary Darboux transformation. With this technique, searching for solutions of the Lax pair is transferred to find vector solutions of the associated linear differential equation system. From vanishing and nonvanishing seed solutions, general vector solutions of such linear differential equation system in terms of the canonical forms of the spectral matrix can be constructed by means of triangular Toeplitz matrices. Several explicit one-soliton solutions and two-soliton solutions are provided corresponding to different forms of the spectral matrix. Furthermore, dynamics and interactions of bright solitons, degenerate solitons, breathers, rogue waves, and dark solitons are also explored graphically.     

Robust Output Tracking Control of an Uncertain Linear System via a Modified Optimal Linear-Quadratic Method

This study investigates the robust output tracking problem for a class of uncertain linear systems. The uncertainties are assumed to be time invariant and to satisfy the matching conditions. According to the selected nominal parameters, an optimal solution with a prescribed degree of stability is determined. Then, an auxiliary input via the use of an adapting factor, connected to the nominal optimal control, is introduced to guarantee the robustness and prescribed degree of stability for the output tracking control of the uncertain linear systems. This method is very simple and effective and can reject bounded uncertainties imposed on the states. A maglev vehicle model example is given to show its effectiveness.     

On the Riemann-Hilbert transformations for a Galilean invariant system

Summary As an application of the Riemann-Hilbert (RH) problem to mathematical physics, the RH transformations are considered for a Galilean invariant nonlinear system. Algebraic RH transformation gives rise to new solutions from the old via a calculation in linear algebra. It is proved that the infinitesimal RH transformations form an infinite-dimensional Lie algebra without using a hierarchy of potentials.     

Bounded controllers for robust exponential convergence

We consider the problem of stabilizing an uncertain system when the norm of the control input is bounded by a prespecified constant. We treat continuous-time dynamical systems whose nominal part is linear and whose uncertain part is norm-bounded by a known affine function of the norm of the system state and the norm of the control input. Given a prespecified rate of convergence and a ball containing the origin of the state space, we present controllers which guarantee that, for all allowable uncertainties and nonlinearities, there is a region of attraction from which all solutions converge to the given ball with the prespecified convergence rate.This research was supported by the National Science Foundation under Grant MSS-90-57079.     

Semi-global finite-time output feedback stabilization for a class of large-scale uncertain nonlinear systems

This paper addresses the problem of semi-global finite-time decentralized output feedback control for large-scale systems with both higher-order and lower-order terms. A new design scheme is developed by coupling the finite-time output feedback stabilization method with the homogeneous domination approach. Specifically, we first design a homogeneous observer and an output feedback control law for each nominal subsystem without the nonlinearities. Then, based on the homogeneous domination approach, we relax the linear growth condition to a polynomial one and construct decentralized controllers to render the nonlinear system semi-globally finite-time stable.     

Sensitivity of solutions of linear DAE to perturbations of the system matrices

This paper studies the effect of perturbations in the system matrices of linear Differential Algebraic Equations (DAE) onto the solutions. It turns out that these may result in a more complicated perturbation pattern for higher index problems than in the case for (standard) additive perturbations. We give an analysis here for linear index-1 and index-2 problems, which, however, has clear ramifications in nonlinear problems (e.g., via the Newton process). This analysis is sustained by a number of examples. This revised version was published online in June 2006 with corrections to the Cover Date.     

Resonances, radiation damping and instabilitym in Hamiltonian nonlinear wave equations

We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations. The unperturbed dynamical system has a bound state, a spatially localized and time periodic solution. We show that, for generic nonlinear Hamiltonian perturbations, all small amplitude solutions decay to zero as time tends to infinity at an anomalously slow rate. In particular, spatially localized and time-periodic solutions of the linear problem are destroyed by generic nonlinear Hamiltonian perturbations via slow radiation of energy to infinity. These solutions can therefore be thought of as metastable states. The main mechanism is a nonlinear resonant interaction of bound states (eigenfunctions) and radiation (continuous spectral modes), leading to energy transfer from the discrete to continuum modes. This is in contrast to the KAM theory in which appropriate nonresonance conditions imply the persistence of invariant tori. A hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the Fermi golden rule, arising in the theory of resonances in quantum mechanics. The techniques used involve: (i) a time-dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues, (ii) a generalization of the Hamiltonian normal form appropriate for infinite dimensional dispersive systems and (iii) ideas from scattering theory. The arguments are quite general and we expect them to apply to a large class of systems which can be viewed as the interaction of finite dimensional and infinite dimensional dispersive dynamical systems, or as a system of particles coupled to a field. Oblatum: 6-XI-1998 & 12-VI-1998 / Published online: 14 January 1999     

Balanced POD for linear PDE robust control computations

A mathematical model of a physical system is never perfect; therefore, robust control laws are necessary for guaranteed stabilization of the nominal model and also ??nearby?? systems, including hopefully the actual physical system. We consider the computation of a robust control law for large-scale finite dimensional linear systems and a class of linear distributed parameter systems. The controller is robust with respect to left coprime factor perturbations of the nominal system. We present an algorithm based on balanced proper orthogonal decomposition to compute the nonstandard features of this robust control law. Convergence theory is given, and numerical results are presented for two partial differential equation systems.     

An Optimal Control Problem of a Coupled Nonlinear Parabolic Population System

An optimal control problem for a coupled nonlinear parabolic population system is considered. The existence and uniqueness of the positive solution for the system is shown by the method of upper and lower solutions. An explicit prior bound of solutions to the system is given by considering an auxiliary coupled linear system. The existence of the optimal control is proved and the characterization of the optimal control is established.