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This paper considers the characterization and computation of invariant sets for discrete-time, time-invariant, linear systems with disturbance inputs whose values are confined to a specified compact set but are otherwise unknown. The emphasis is on determining maximal disturbance-invariant sets X that belong to a specified subset Γ of the state space. Such d-invariant sets have important applications in control problems where there are pointwise-in-time state constraints of the form χ ( t ) ∈ Γ . One purpose of the paper is to unite and extend in a rigorous way disparate results from the prior literature. In addition there are entirely new results. Specific contributions include: exploitation of the Pontryagin set difference to clarify conceptual matters and simplify mathematical developments, special properties of maximal invariant sets and conditions for their finite determination, algorithms for generating concrete representations of maximal invariant sets, practical computational questions, extension of the main results to general Lyapunov stable systems, applications of the computational techniques to the bounding of state and output response. Results on Lyapunov stable systems are applied to the implementation of a logic-based, nonlinear multimode regulator. For plants with disturbance inputs and state-control constraints it enlarges the constraint-admissible domain of attraction. Numerical examples illustrate the various theoretical and computational results.

This paper introduces a new four-dimensional hyperchaotic financial system on the basis of an established three-dimensional nonlinear financial system and a dynamic model by adding a controller term to consider the effect of control on the system. In terms of the proposed financial system, the sufficient conditions for nonexistence of chaotic and hyperchaotic behaviors are derived theoretically. Then, the solutions of equilibria are obtained. For each equilibrium, its stability and existence of Hopf bifurcation are validated. Based on corresponding first Lyapunov coefficient of each equilibrium, the analytical proof of the existence of periodic solutions is given. The ultimate bound and positively invariant set for the financial system are obtained and estimated. There exists a stable periodic solution obtained near the unstable equilibrium point. Finally, the dynamic behaviors of the new system are explored from theoretical analysis by using the bifurcation diagrams and phase portraits. Moreover, the hyperchaotic financial system has been simulated using a specially designed electronic circuit and viewed on an oscilloscope, thereby confirming the results of the numerical integrations and its real contribution to engineering.

[EN] The main contribution of this study is to present a new optimal eighth-order scheme for locating zeros with multiplicity m > 1. An extensive convergence analysis is presented with the main theorem in order to demonstrate the optimal eighth-order convergence of the proposed scheme. Moreover, a local convergence study for the optimal fourth-order method defined by the first two steps of the new method is presented, allowing us to obtain the radius of the local convergence ball. Finally, numerical tests on some real-life problems, such as a Van der Waals equation of state, a conversion Chemical engineering problem and two standard academic test problems are presented, which confirm the theoretical results established in this paper and the efficiency of this proposed iterative method. We observed from the numerical experiments that our proposed iterative methods have good values for convergence radii. Further, they have not only faster convergence towards the desired zero of the involved function but they also have both smaller residual error and a smaller difference between two consecutive iterations than current existing techniques., This research was partially supported by Ministerio de Economia y Competitividad under grant MTM2014-52016-C2-2-P and by the project of Generalitat Valenciana Prometeo/2016/089.

This paper presents a high-performance nonsingular terminal sliding mode control method for uncertain second-order nonlinear systems. First, a nonsingular terminal sliding mode surface is introduced to eliminate the singularity problem that exists in conventional terminal sliding mode control. By using this method, the system not only can guarantee that the tracking errors reach the reference value in a finite time with high-precision tracking performance but also can overcome the complex-value and the restrictions of the exponent (the exponent should be fractional number with an odd numerator and an odd denominator) in traditional terminal sliding mode. Then, in order to eliminate the chattering phenomenon, a super-twisting higher-order nonsingular terminal sliding mode control method is proposed. The stability of the closed-loop system is established using the Lyapunov theory. Finally, simulation results are presented to illustrate the effectiveness of the proposed method.

This paper investigates the problem of global stabilization for a class of switched nonlinear systems using multiple Lyapunov functions (MLFs). The restrictions on nonlinearities are neither linear growth condition nor Lipschitz condition with respect to system states. Based on adding a power integrator technique, we design homogeneous state feedback controllers of all subsystems and a switching law to guarantee that the closed-loop system is globally asymptotically stable. Finally, an example is given to illustrate the validity of the proposed control scheme.

The nonlinear parametric vibration of an axially moving string made by rubber-like materials is studied in the paper. The fractional viscoelastic model is used to describe the damping of the string. Then, a new nonlinear fractional mathematical model governing transverse motion of the string is derived based on Newton’s second law, the Euler beam theory, and the Lagrangian strain. Taking into consideration the fractional calculus law of Riemann-Liouville form, the principal parametric resonance is analytically investigated via applying the direct multiscale method. Numerical results are presented to show the influences of the fractional order, the stiffness constant, the viscosity coefficient, and the axial-speed fluctuation amplitude on steady-state responses. It is noticeable that the amplitudes and existing intervals of steady-state responses predicted by Kirchhoff’s fractional material model are much larger than those predicted by Mote’s fractional material model.

Cuttings transport problem has long been recognized as one of the key difficulties in drilling horizontal wells, and the models in cuttings transport research are usually formulated with highly nonlinear equations set. When using Newton methods to solve real engineering problems with nonlinear equations set, the problems of result dependence on initial values, Jacobian matrix singularity, and variable outflow of its definition domain in iterations are three of the often-encountered difficulties. In this paper, the ant colony algorithm is applied to solve the two-layer cuttings transport model with highly nonlinear equations set. The solution-searching process of solving nonlinear equations set is transformed into an optimization process of searching the minimum value of an objective function by applying ant colony algorithm. Analyzing the results of the example, it can be concluded that ant colony algorithm can be used to solve the highly nonlinear cuttings transport model with good solution accuracy; transforming the solution-searching process of solving nonlinear equations set into an optimization process of searching the minimum value of the objective function is necessary; the real engineering problem should be simplified as much as possible to decrease the number of unknown variables and facilitate the use of ant colony algorithm.

Based on a nonlinear consensus protocol, this paper considers the group synchronization of complex dynamical networks with sampled data. Using the Lyapunov method, the group synchronization of the nonlinear complex networks is analyzed. All the nodes in each group can converge to their own synchronous state asymptotically, if the sampled period satisfies some matrix inequality conditions. Furthermore, the theoretical results are verified by some simulations.

In this paper, the differential transformation method is applied to the system of Volterra integral and integrodifferential equations with proportional delays. The method is useful for both linear and nonlinear equations. By using this method, the solutions are obtained in series forms. If the solutions of the problem can be expanded to Taylor series, then the method gives opportunity to determine the coefficients of Taylor series. Hence, the exact solution can be obtained in Taylor series form. In illustrative examples, the method is applied to a few types of systems.

Multifingered robots play an important role in manipulation applications. They can grasp various shaped objects to perform point-to-point movement. It is important to plan the motion path of the object and appropriately control the grasping forces for multifingered robot manipulation. In this paper, we perform the optimal grasping control to find both optimal motion path of the object and minimum grasping forces in the manipulation. The rigid body dynamics of the object and the grasping forces subjected to the second-order cone (SOC) constraints are considered in optimal control problem. The minimum principle is applied to obtain the system equalities and the SOC complementarity problems. The SOC complementarity problems are further recast as the equations with the Fischer-Burmeister (FB) function. Since the FB function is semismooth, the semismooth Newton method with the generalized Jacobian of FB function is used to solve the nonlinear equations. The 2D and 3D simulations of grasping manipulation are performed to demonstrate the effectiveness of the proposed approach.

This paper focuses on theoretical analysis of second-order consensus in multiagent system. As an extension of the general linear protocol, a nonlinear protocol is designed for multiagent system with undirected communication topology. The nonlinear protocol is also applied to achieve reference velocity consensus. Through choosing the appropriate Lyapunov functions and using LaSalle’s invariance principle, some consensus conditions are derived. Simulation examples are provided to demonstrate the effectiveness of the proposed results.

This paper is concerned with a state observer-based sliding mode control design methodology for a class of continuous-time state-delayed switched systems with unmeasurable states and nonlinear uncertainties. The advantages of the proposed scheme mainly lie in which it eliminates the need for state variables to be full accessible and parameter uncertainties to be satisfied with the matching condition. Firstly, a state observer is constructed, and a sliding surface is designed. By matrix transformation techniques, combined with Lyapunov function and sliding surface function, a sufficient condition is given to ensure asymptotic stability of the overall closed-loop systems composed of the observer dynamics and the estimation error dynamics. Then, reachability of sliding surface is investigated. At last, an illustrative numerical example is presented to prove feasibility of the proposed approaches.

By using the method of dynamical system, the exact travelling wave solutions of the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms are studied. Based on this method, all phase portraits of the system in the parametric space are given with the aid of the Maple software. All possible bounded travelling wave solutions, such as solitary wave solutions, kink and anti-kink wave solutions, and periodic travelling wave solutions, are obtained, respectively. The results presented in this paper improve the related previous conclusions.

Under the weaker assumption on nonlinear functions, the adaptive finite-time stabilization of more general high-order nonlinear systems with dynamic and parametric uncertainties is solved in this paper. To solve this problem, finite-time input-to-state stability (FTISS) is used to characterize the unmeasured dynamic uncertainty. By skillfully combining Lyapunov function, sign function, backstepping, and finite-time input-to-state stability approaches, an adaptive state feedback controller is designed to guarantee high-order nonlinear systems are globally finite-time stable.

This paper is devoted to the study of a nonlinear Carrier wave equation in an annular membrane associated with Robin-Dirichlet conditions. Existence and uniqueness of a weak solution are proved by using the linearization method for nonlinear terms combined with the Faedo-Galerkin method and the weak compact method. Furthermore, an asymptotic expansion of a weak solution of high order in a small parameter is established.

Under investigation in this paper are the coupled nonlinear Schrödinger (CNLS) equations with dissipation terms by the Hirota method, which are better than the formal Schrödinger equation in eliciting optical solitons. The bilinear form has been constructed, via which multisolitons and breathers are derived. In particular, the three-bright soliton solution and breathers are derived and simulated via some pictures. The propagation characters are analysed with the change of the parameters.

A new method to design a fuzzy bilinear observer (FBO) with unknown inputs is developed for a class of nonlinear systems. The nonlinear system is modeled as a fuzzy bilinear model (FBM). This kind of T-S fuzzy model is especially suitable for a nonlinear system with a bilinear term. The proposed fuzzy bilinear observer subject to unknown inputs is developed to ensure the asymptotic convergence of the error dynamic using the Lyapunov method. The proposed design conditions are given in linear matrix inequality (LMI) formulation. The paper studies also the problem of fault detection and isolation. An unknown input fuzzy bilinear fault diagnosis observer design is proposed. This work is given for both continuous and discrete cases of fuzzy bilinear models. Illustrative examples are chosen to provide the effectiveness of the given methodology.

This paper focuses on the principle for designing reduced-order fuzzy-observer-based actuator fault reconstruction for a class of nonlinear systems. The problem addressed can be indicated as an approach for a kind of reduced-order fuzzy observer design with special gain matrix structure that depends on a given matching condition specification. Using the Lyapunov theory, the stability conditions are obtained and expressed in terms of linear matrix inequalities, and the conditions for asymptotic estimation of actuator faults are derived. Simulation results illustrate the observer design procedure and demonstrate the actuator fault reconstruction effectiveness and performance.

The nonlinear model is crucial to prepare, supervise, and analyze mechanical system. In this paper, a new nonparametric and output-only identification procedure for nonlinear damping is studied. By introducing the concept of the stochastic state space, we formulate a stochastic inverse problem for a nonlinear damping. The solution of the stochastic inverse problem is designed as probabilistic expression via the hierarchical Bayesian formulation by considering various uncertainties such as the information insufficiency in parameter of interests or errors in measurement. The probability space is estimated using Markov chain Monte Carlo (MCMC). The applicability of the proposed method is demonstrated through numerical experiment and particular application to a realistic problem related to ship roll motion.

Many problems with control theory have led to investigations into switched systems. One of the most urgent problems related to the analysis of the dynamics of switched systems is the stability problem. The stability of a switched system can be ensured by a common Lyapunov function for all switching modes under an arbitrary switching law. Finding a common Lyapunov function is still an interesting and challenging problem. The purpose of the present paper is to prove the stability of equilibrium in a certain class of nonlinear switched systems by introducing a common Lyapunov function; the Lyapunov function is based on generalized Kullback–Leibler divergence or Csiszár'sI-divergence between the state and equilibrium. The switched system is useful for finding positive solutions to linear algebraic equations, which minimize theI-divergence measure under arbitrary switching. One application of the stability of a given switched system is in developing a new approach to reconstructing tomographic images, but nonetheless, the presented results can be used in numerous other areas.

Existence and uniqueness theorem are the tool which makes it possible for us to conclude that there exists only one solution to a given problem which satisfies a constraint condition. How does it work? Why is it the case? We believe it, but it would be interesting to see the main ideas behind this. To this end, in this paper, we investigate existence, uniqueness, and other properties of solutions of a certain nonlinear fuzzy Volterra integrodifferential equation under strongly generalized differentiability. The main tools employed in the analysis are based on the applications of the Banach fixed point theorem and a certain integral inequality with explicit estimate. Also, some results for characterizing solution by an equivalent system of crisp Volterra integrodifferential equations are presented. In this way, a new direction for the methods of analytic and approximate solutions is proposed.

This paper witnesses the coupling of an analytical series expansion method which is called reduced differential transform with fractional complex transform. The proposed technique is applied on three mathematical models, namely, fractional Kaup-Kupershmidt equation, generalized fractional Drinfeld-Sokolov equations, and system of coupled fractional Sine-Gordon equations subject to the appropriate initial conditions which arise frequently in mathematical physics. The derivatives are defined in Jumarie’s sense. The accuracy, efficiency, and convergence of the proposed technique are demonstrated through the numerical examples. It is observed that the presented coupling is an alternative approach to overcome the demerit of complex calculation of fractional differential equations. The proposed technique is independent of complexities arising in the calculation of Lagrange multipliers, Adomian’s polynomials, linearization, discretization, perturbation, and unrealistic assumptions and hence gives the solution in the form of convergent power series with elegantly computed components. All the examples show that the proposed combination is a powerful mathematical tool to solve other nonlinear equations also.

The paper examines the problem of sliding mode manifold design for uncertain nonlinear system with discontinuous control. The original plant first is decomposed such that the problem is divided into a number of simpler sub-problems. Then the block control recursive procedure is presented in which nonlinear sliding manifold is derived. Finally combined high gain and Lyapunov functions techniques are applied to establish hierarchy of the control gains and to estimate the upper bounds of the sliding mode equation solutions.

Made available in DSpace on 2014-12-03T13:11:50Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-01-01Bitstream added on 2014-12-03T13:22:23Z : No. of bitstreams: 1 WOS000338053100001.pdf: 2350596 bytes, checksum: 489a95e3343a1f0d827a4eba58c11263 (MD5) Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) The paper proposes a new switched control design method for some classes of uncertain nonlinear plants described by Takagi-Sugeno fuzzy models. This method uses a quadratic Lyapunov function to design the feedback controller gains based on linear matrix inequalities (LMIs). The controller gain is chosen by a switching law that returns the smallest value of the time derivative of the Lyapunov function. The proposed methodology eliminates the need to find the membership function expressions to implement the control laws. The control designs of a ball-and-beam system and of a magnetic levitator illustrate the procedure. Fed Inst Educ, Dept Acad Areas Jata, BR-75804020 Jatai, GO, Brazil Univ Estadual Paulista, Dept Elect Engn, BR-15385000 Ilha Solteira, SP, Brazil Fed Inst Educ, Dept Acad Areas Janu, BR-39480000 Januaria, MG, Brazil Univ Estadual Paulista, Dept Elect Engn, BR-15385000 Ilha Solteira, SP, Brazil FAPESP: 11/17610-0

This paper presents a criterion for the existence and uniqueness of solutions to two and multipoint boundary value problems associated with annth order nonlinear Lyapunov system. A variation of parameters formula is developed and used as a tool to obtain existence and uniqueness. We discuss solution of the second order problem by the ADI method and develop a fixed point method to find the general solution of thenth order Lyapunov system. The results of Barnett (SIAM J. Appl. Anal.24(1), 1973) are a particular case.

This paper is devoted to the stability and convergence analysis of the two-step Runge-Kutta (TSRK) methods with the Lagrange interpolation of the numerical solution for nonlinear neutral delay differential equations. Nonlinear stability and D-convergence are introduced and proved. We discuss theGR(l)-stability,GAR(l)-stability, and the weakGAR(l)-stability on the basis of(k,l)-algebraically stable of the TSRK methods; we also discuss the D-convergence properties of TSRK methods with a restricted type of interpolation procedure.

This paper presents a design method of the optimal proportional-integral-derivative (PID) controller withɛ-Routh stability for different processes through Lyapunov approach. The optimal PID controller could be acquired by minimizing an augmented integral squared error (AISE) performance index which contains control error and at least first-order error derivative, or even may containnth-order error derivative. The optimal control problem could be transformed into a nonlinear constraint optimization (NLCO) problem via Lyapunov theorems. Therefore, optimal PID controller could be obtained by solving NLCO problem through interior method or other optimization methods. The proposed method can be applied for different processes, and optimal PID controllers under various control weight matrices andɛ-Routh stability are presented for different processes. Control weight matrix andɛ-Routh stability’s effects on system performances are studied, and different tuning methods’ system performances are also discussed.ɛ-Routh stability’s effects on disturbance rejection ability are investigated, and different tuning methods’ disturbances rejection ability is studied. To further illustrate the proposed method, experimental results of coupled water tank system (CWTS) under different set points are presented. Both simulation results and experiment results show the effectiveness and usefulness of the proposed method.

This paper investigates the linearization and stabilizing control design problems for a class of SISO Markovian jump nonlinear systems. According to the proposed relative degree set definition, the system can be transformed into the canonical form through the appropriate coordinate changes followed with the Markovian switchings; that is, the system can be full-state linearized in every jump mode with respect to the relative degree setn,…,n. Then, a stabilizing control is designed through applying the backstepping technique, which guarantees the asymptotic stability of Markovian jump nonlinear systems. A numerical example is presented to illustrate the effectiveness of our results.

Accurate wind speed forecasts are necessary for the safety and economy of the renewable energy utilization. The wind speed forecasts can be obtained by statistical model based on historical data. In this paper, a novel W-GP model (wavelet decomposition based Gaussian process learning paradigm) is proposed for short-term wind speed forecasting. The nonstationary and nonlinear original wind speed series is first decomposed into a set of better-behaved constitutive subseries by wavelet decomposition. Then these sub-series are forecasted respectively by GP method, and the forecast results are summed to formulate an ensemble forecast for original wind speed series. Therefore, the previous process which obtains wind speed forecast result is named W-GP model. Finally, the proposed model is applied to short-term forecasting of the mean hourly and daily wind speed for a wind farm located in southern China. The prediction results indicate that the proposed W-GP model, which achieves a mean 13.34% improvement in RMSE (Root Mean Square Error) compared to persistence method for mean hourly data and a mean 7.71% improvement for mean daily wind speed data, shows the best forecasting accuracy among several forecasting models.

Recently in the work of George, 2010, we considered a modified Gauss-Newton method for approximate solution of a nonlinear ill-posed operator equationF(x)=y, whereF:D(F)⊆X→Yis a nonlinear operator between the Hilbert spacesXandY. The analysis in George, 2010 was carried out using a majorizing sequence. In this paper, we consider also the modified Gauss-Newton method, but the convergence analysis and the error estimate are obtained by analyzing the odd and even terms of the sequence separately. We use the adaptive method in the work of Pereverzev and Schock, 2005 for choosing the regularization parameter. The optimality of this method is proved under a general source condition. A numerical example of nonlinear integral equation shows the performance of this procedure.

We consider the same class of systems in a previous paper, that is, a class of parametric strict-feedback nonlinear systems with additive disturbance. By using the backstepping technique, adaptive control algorithm is developed. Unlike the existing control schemes for systems with additive disturbance, no knowledge is assumed on two times continuous differentiability of the disturbance. Also, the developed backstepping controller does not require the uncertain parameters to be restricted within a known interval. Furthermore, overestimation of the system parameters is removed by employing the tuning functions method. It is shown that the proposed controller guarantees not only the stability in the sense of Lyapunov definition, but also the asymptotic tracking.

This paper considers the problem of assigning the dynamics of a nonlinear analytic system using nonlinear generalized sampled-data hold function (GSHF) control, in the neighborhood of an equilibrium point. On every sampling interval, the control input consists of a nonlinear time-periodic function applied to the sampled value of the state vector. The nonlinear monodromy map is the state transition map from one sample time to the next. It is shown that this map is arbitrarily assignable by GSHF feedback if and only if the linear part of the system is controllable. Two approaches are proposed to construct a GSHF controller that performs the assignment. The first approach matches the coefficients of the Taylor series expansion of the monodromy map around the equilibrium. The second approach interpolates an optimal control law at several points in the vicinity of the equilibrium. These approaches are illustrated on an example.

A new fault detection method using neural-networks-augmented state observer for nonlinear systems is presented in this paper. The novelty of the approach is that instead of approximating the entire nonlinear system with neural network, we only approximate the unmodeled part that is left over after linearization, in which a radial basis function (RBF) neural network is adopted. Compared with conventional linear observer, the proposed observer structure provides more accurate estimation of the system state. The state estimation error is proved to asymptotically approach zero by the Lyapunov method. An aircraft system demonstrates the efficiency of the proposed fault detection scheme, simulation results of which show that the proposed RBF neural network-based observer scheme is effective and has a potential application in fault detection and identification (FDI) for nonlinear systems.

This paper deals with a first-order differential equation with a polynomial nonlinear term. The integrability and existence of periodic solutions of the equation are obtained, and the stability of periodic solutions of the equation is derived.

The paper presents conditions suitable in design giving quadratic performances to stabilizing controllers for given class of continuous-time nonlinear systems, represented by Takagi-Sugeno models. Based on extended Lyapunov function and slack matrices, the design conditions are outlined in the terms of linear matrix inequalities to possess a stable structure closest to LQ performance, if premise variables are measurable. Simulation results illustrate the design procedure and demonstrate the performances of the proposed control design method.

This paper presents a study of linear control systems based on exact feedback linearization and approximate feedback linearization. As exact feedback linearization is applied, a linear controller can perform the control objectives. The approximate feedback linearization is required when a nonlinear system presents a noninvolutive property. It uses a Taylor series expansion in order to compute a nonlinear transformation of coordinates to satisfy the involutivity conditions.

The problems of radiative transfer give rise to interesting integral equations that must be faced with efficient numerical solver. Very often the integral equations are discretized to large-scale nonlinear equations and solved by Newton's-like methods. Generally, these kind of methods require the computation and storage of the Jacobian matrix or its approximation. In this paper, we present a new approach that was based on approximating the Jacobian inverse into a diagonal matrix by means of variational technique. Numerical results on well-known benchmarks integral equations involved in the radiative transfer authenticate the reliability and efficiency of the approach. The fact that the proposed method can solve the integral equations without function derivative and matrix storage can be considered as a clear advantage over some other variants of Newton's method.

This paper presents a high accuracy combination algorithm for solving the systems of nonlinear Volterra integral and integro-differential equations with weakly singular kernels of the second kind. Two quadrature algorithms for solving the systems are discussed, which possess high accuracy order and the asymptotic expansion of the errors. By means of combination algorithm, we may obtain a numerical solution with higher accuracy order than the original two quadrature algorithms. Moreover an a posteriori error estimation for the algorithm is derived. Both of the theory and the numerical examples show that the algorithm is effective and saves storage capacity and computational cost.

This paper presents a new semianalytical approach for geometrically nonlinear vibration analysis of Euler-Bernoulli beams with different boundary conditions. The method makes use of Linstedt-Poincaré perturbation technique to transform the nonlinear governing equations into a linear differential equation system, whose solutions are then sought through the use of differential quadrature approximation in space domain and an analytical series expansion in time domain. Validation of the present method is conducted in numerical examples through direct comparisons with existing solutions, showing that the proposed semianalytical method has excellent convergence and can give very accurate results at a long time interval.