Showing papers on "Nonlinear system published in 2015"

A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

Abstract: The Koopman operator is a linear but infinite-dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems In this manuscript, we present a data-driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a “black box” integrator We will show that this approach is, in effect, an extension of dynamic mode decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the “stochastic Koopman operator” (Mezic in Nonlinear Dynamics 41(1–3): 309–325, 2005) Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data and two that show potential applications of the Koopman eigenfunctions

Reduced Basis Methods for Partial Differential Equations: An Introduction

Abstract: 1 Introduction.- 2 Representative problems: analysis and (high-fidelity) approximation.- 3 Getting parameters into play.- 4 RB method: basic principle, basic properties.- 5 Construction of reduced basis spaces.- 6 Algebraic and geometrical structure.- 7 RB method in actions.- 8 Extension to nonaffine problems.- 9 Extension to nonlinear problems.- 10 Reduction and control: a natural interplay.- 11 Further extensions.- 12 Appendix A Elements of functional analysis.

Paracontrolled distributions and singular PDEs

Abstract: We introduce an approach to study certain singular partial differential equations (PDEs) which is based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths. We illustrate its applicability on some model problems such as differential equations driven by fractional Brownian motion, a fractional Burgers-type stochastic PDE (SPDE) driven by space-time white noise, and a nonlinear version of the parabolic Anderson model with a white noise potential.

Fuzzy Adaptive Output Feedback Control of MIMO Nonlinear Systems With Partial Tracking Errors Constrained

Abstract: In this paper, a partial tracking error constrained fuzzy output-feedback dynamic surface control (DSC) scheme is proposed for a class of uncertain multi-input and multi-output (MIMO) nonlinear systems. The considered MIMO nonlinear systems contain unknown functions and without the requirement of their states being available for the controller design. With the help of fuzzy logic systems identifying the MIMO unknown nonlinear systems, a fuzzy adaptive observer is established to estimate the unmeasured states. By transforming the tracking errors into new virtual error variables and based on the DSC backstepping recursive design technique, a new adaptive fuzzy output-feedback control method is developed. It is proved that the proposed control approach can guarantee that all the signals of the resulting closed-loop system are bounded and the partial state tracking errors are confined all times within the prescribed bounds. The simulation results and comparisons with the previous control approaches confirm the effectiveness and utility of the proposed scheme.

Observer-Based Adaptive Fuzzy Tracking Control of MIMO Stochastic Nonlinear Systems With Unknown Control Directions and Unknown Dead Zones

Abstract: In this paper, an adaptive fuzzy backstepping output-feedback tracking control approach is proposed for a class of multi-input and multi-output (MIMO) stochastic nonlinear systems. The MIMO stochastic nonlinear systems under study are assumed to possess unstructured uncertainties, unknown dead-zones, and unknown control directions. By using a linear state transformation, the unknown control coefficients and the unknown slopes characteristic of the dead-zones are lumped together, and the original system is transformed to a new system on which the control design becomes feasible. Fuzzy logic systems are used to approximate the unstructured uncertainties, and a fuzzy state observer is designed to estimate the unmeasured states. By introducing a special Nussbaum gain function into the backstepping control design, a stable adaptive fuzzy output-feedback tracking control scheme is developed. The main features of the proposed adaptive control approach are that it can guarantee the stability of the closed-loop system, and the tracking errors converge to a small neighborhood of zero. Moreover, it can solve the problems of unknown control direction, unknown dead-zone, and unmeasured states simultaneously. Two simulation examples are provided to show the effectiveness of the proposed approach.

Chaotic Dynamics of Nonlinear Systems

Abstract: One-Dimensional Maps Universality Theory Fractal Dimension Differential Dynamics Nonlinear Examples with Chaos Two-Dimensional Maps Conservative Dynamics Measures of Chaos Complexity and Chaos Reprise Glossary References Index.

Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis

Abstract: Isotropic and Anisotropic Function Spaces Lebesgue and Sobolev Spaces with Variable Exponent History of function spaces with variable exponent Lebesgue spaces with variable exponent Sobolev spaces with variable exponent Dirichlet energies and Euler-Lagrange equations Lavrentiev phenomenon Anisotropic function spaces Orlicz spaces Variational Analysis of Problems with Variable Exponents Nonlinear Degenerate Problems in Non-Newtonian Fluids Physical motivation A boundary value problem with nonhomogeneous differential operator Nonlinear eigenvalue problems with two variable exponents A sublinear perturbation of the eigenvalue problem associated to the Laplace operator Variable exponents versus Morse theory and local linking The Caffarelli-Kohn-Nirenberg inequality with variable exponent Spectral Theory for Differential Operators with Variable Exponent Continuous spectrum for differential operators with two variable exponents A nonlinear eigenvalue problem with three variable exponents and lack of compactness Concentration phenomena: the case of several variable exponents and indefinite potential Anisotropic problems with lack of compactness and nonlinear boundary condition Nonlinear Problems in Orlicz-Sobolev Spaces Existence and multiplicity of solutions A continuous spectrum for nonhomogeneous operators Nonlinear eigenvalue problems with indefinite potential Multiple solutions in Orlicz-Sobolev spaces Neumann problems in Orlicz-Sobolev spaces Anisotropic Problems: Continuous and Discrete Anisotropic Problems Eigenvalue problems for anisotropic elliptic equations Combined effects in anisotropic elliptic equations Anisotropic problems with no-flux boundary condition Bifurcation for a singular problem modelling the equilibrium of anisotropic continuous media Difference Equations with Variable Exponent Eigenvalue problems associated to anisotropic difference operators Homoclinic solutions of difference equations with variable exponents Low-energy solutions for discrete anisotropic equations Appendix A: Ekeland Variational Principle Appendix B: Mountain Pass Theorem Bibliography Index A Glossary is included at the end of each chapter.

Adaptive fuzzy output feedback dynamic surface control of interconnected nonlinear pure-feedback systems.

Abstract: In this paper, an adaptive fuzzy decentralized output feedback control design is presented for a class of interconnected nonlinear pure-feedback systems The considered nonlinear systems contain unknown nonlinear uncertainties and the states are not necessary to be measured directly Fuzzy logic systems are employed to approximate the unknown nonlinear functions, and then a fuzzy state observer is designed and the estimations of the immeasurable state variables are obtained Based on the adaptive backstepping dynamic surface control design technique, an adaptive fuzzy decentralized output feedback control scheme is developed It is proved that all the variables of the resulting closed-loop system are semi-globally uniformly ultimately bounded, and also that the observer and tracking errors are guaranteed to converge to a small neighborhood of the origin Some simulation results and comparisons with the existing results are provided to illustrate the effectiveness and merits of the proposed approach

Continuous control of the nonlinearity phase for harmonic generations

Abstract: The capability of locally engineering the nonlinear optical properties of media is crucial in nonlinear optics. Although poling is the most widely employed technique for achieving locally controlled nonlinearity, it leads only to a binary nonlinear state, which is equivalent to a discrete phase change of π in the nonlinear polarizability. Here, inspired by the concept of spin-rotation coupling, we experimentally demonstrate nonlinear metasurfaces with homogeneous linear optical properties but spatially varying effective nonlinear polarizability with continuously controllable phase. The continuous phase control over the local nonlinearity is demonstrated for second and third harmonic generation by using nonlinear metasurfaces consisting of nanoantennas of C3 and C4 rotational symmetries, respectively. The continuous phase engineering of the effective nonlinear polarizability enables complete control over the propagation of harmonic generation signals. Therefore, this method seamlessly combines the generation and manipulation of harmonic waves, paving the way for highly compact nonlinear nanophotonic devices.

Dynamic Surface Control Using Neural Networks for a Class of Uncertain Nonlinear Systems With Input Saturation

Abstract: In this paper, a dynamic surface control (DSC) scheme is proposed for a class of uncertain strict-feedback nonlinear systems in the presence of input saturation and unknown external disturbance. The radial basis function neural network (RBFNN) is employed to approximate the unknown system function. To efficiently tackle the unknown external disturbance, a nonlinear disturbance observer (NDO) is developed. The developed NDO can relax the known boundary requirement of the unknown disturbance and can guarantee the disturbance estimation error converge to a bounded compact set. Using NDO and RBFNN, the DSC scheme is developed for uncertain nonlinear systems based on a backstepping method. Using a DSC technique, the problem of explosion of complexity inherent in the conventional backstepping method is avoided, which is specially important for designs using neural network approximations. Under the proposed DSC scheme, the ultimately bounded convergence of all closed-loop signals is guaranteed via Lyapunov analysis. Simulation results are given to show the effectiveness of the proposed DSC design using NDO and RBFNN.

Adaptive Neural Output Feedback Control of Uncertain Nonlinear Systems With Unknown Hysteresis Using Disturbance Observer

Abstract: In this paper, an adaptive neural output feedback control scheme is proposed for uncertain nonlinear systems that are subject to unknown hysteresis, external disturbances, and unmeasured states. To deal with the unknown nonlinear function term in the uncertain nonlinear system, the approximation capability of the radial basis function neural network (RBFNN) is employed. Using the approximation output of the RBFNN, the state observer and the nonlinear disturbance observer (NDO) are developed to estimate unmeasured states and unknown compounded disturbances, respectively. Based on the RBFNN, the developed NDO, and the state observer, the adaptive neural output feedback control is proposed for uncertain nonlinear systems using the backstepping technique. The first-order sliding-mode differentiator is employed to avoid the tedious analytic computation and the problem of “explosion of complexity” in the conventional backstepping method. The stability of the whole closed-loop system is rigorously proved via the Lyapunov analysis method, and the satisfactory tracking performance is guaranteed under the integrated effect of unknown hysteresis, unmeasured states, and unknown external disturbances. Simulation results of an example are presented to illustrate the effectiveness of the proposed adaptive neural output feedback control scheme for uncertain nonlinear systems.

Introduction to nonlinear finite element analysis

Abstract: Preliminary concepts.- Nonlinear Finite Element Analysis Procedure.- Finite Element Analysis for Nonlinear Elastic Systems.- Finite Element Analysis for Elastoplastic Problems.- Finite Element Analysis for Contact Problems.

Nonlinear Regression Models and Applications in Agricultural Research

Abstract: Nonlinear regression models are important tools because many crop and soil processes are better represented by nonlinear than linear models. Fitting nonlinear models is not a single-step procedure but an involved process that requires careful examination of each individual step. Depending on the objective and the application domain, different priorities are set when fitting nonlinear models; these include obtaining acceptable parameter estimates and a good model fit while meeting standard assumptions of statistical models. We propose steps in fitting nonlinear models as described by a flow diagram and discuss each step separately providing examples and updates on procedures used. The following steps are considered: (i) choose candidate models, (ii) set starting values, (iii) fit models, (iv) check convergence and parameter estimates, (v) find the "best" model among competing models, (vi) check model assumptions (residual analysis), and (vii) calculate statistical descriptors and confidence intervals. The associated feedback mechanisms are also addressed (i.e., model variance homogeneity). In particular, we emphasize the first step (choose candidate models) by providing an extensive library of nonlinear functions (77 equations with the associated parameter meanings) and examples of typical applications in agriculture. We hope that this contribution will clarify some of the difficulties and confusion with the task of using nonlinear models.

Observer and Command-Filter-Based Adaptive Fuzzy Output Feedback Control of Uncertain Nonlinear Systems

Abstract: In this paper, observer and command-filter-based adaptive fuzzy output feedback control is proposed for a class of strict-feedback systems with parametric uncertainties and unmeasured states. First, fuzzy logic systems are used to approximate the unknown and nonlinear functions. Next, a fuzzy state observer is developed to estimate the immeasurable states. Then, command-filtered backstepping control is designed to avoid the explosion of complexity in the backstepping design, and compensating signals are introduced to remove the effect of the errors caused by command filters. The proposed method guarantees that all signals in the closed-loop systems are bounded. The main contributions of this paper are the proposed control method can overcome two problems of linear in the unknown system parameter and explosion of complexity in backstepping-design methods and it does not require that all of the states of the system are measured directly. Finally, two examples are provided to illustrate its effectiveness.

Reinforcement-Learning-Based Robust Controller Design for Continuous-Time Uncertain Nonlinear Systems Subject to Input Constraints

Abstract: The design of stabilizing controller for uncertain nonlinear systems with control constraints is a challenging problem The constrained-input coupled with the inability to identify accurately the uncertainties motivates the design of stabilizing controller based on reinforcement-learning (RL) methods In this paper, a novel RL-based robust adaptive control algorithm is developed for a class of continuous-time uncertain nonlinear systems subject to input constraints The robust control problem is converted to the constrained optimal control problem with appropriately selecting value functions for the nominal system Distinct from typical action-critic dual networks employed in RL, only one critic neural network (NN) is constructed to derive the approximate optimal control Meanwhile, unlike initial stabilizing control often indispensable in RL, there is no special requirement imposed on the initial control By utilizing Lyapunov’s direct method, the closed-loop optimal control system and the estimated weights of the critic NN are proved to be uniformly ultimately bounded In addition, the derived approximate optimal control is verified to guarantee the uncertain nonlinear system to be stable in the sense of uniform ultimate boundedness Two simulation examples are provided to illustrate the effectiveness and applicability of the present approach

Nonlinear time-series analysis revisited

Abstract: In 1980 and 1981, two pioneering papers laid the foundation for what became known as nonlinear time-series analysis: the analysis of observed data-typically univariate-via dynamical systems theory. Based on the concept of state-space reconstruction, this set of methods allows us to compute characteristic quantities such as Lyapunov exponents and fractal dimensions, to predict the future course of the time series, and even to reconstruct the equations of motion in some cases. In practice, however, there are a number of issues that restrict the power of this approach: whether the signal accurately and thoroughly samples the dynamics, for instance, and whether it contains noise. Moreover, the numerical algorithms that we use to instantiate these ideas are not perfect; they involve approximations, scale parameters, and finite-precision arithmetic, among other things. Even so, nonlinear time-series analysis has been used to great advantage on thousands of real and synthetic data sets from a wide variety of systems ranging from roulette wheels to lasers to the human heart. Even in cases where the data do not meet the mathematical or algorithmic requirements to assure full topological conjugacy, the results of nonlinear time-series analysis can be helpful in understanding, characterizing, and predicting dynamical systems.

Efficient and accurate approximations of nonlinear convolutional networks

Abstract: This paper aims to accelerate the test-time computation of deep convolutional neural networks (CNNs). Unlike existing methods that are designed for approximating linear filters or linear responses, our method takes the nonlinear units into account. We minimize the reconstruction error of the nonlinear responses, subject to a low-rank constraint which helps to reduce the complexity of filters. We develop an effective solution to this constrained nonlinear optimization problem. An algorithm is also presented for reducing the accumulated error when multiple layers are approximated. A whole-model speedup ratio of 4× is demonstrated on a large network trained for ImageNet, while the top-5 error rate is only increased by 0.9%. Our accelerated model has a comparably fast speed as the “AlexNet” [11], but is 4.7% more accurate.

Neural-Based Adaptive Output-Feedback Control for a Class of Nonstrict-Feedback Stochastic Nonlinear Systems

Abstract: In this paper, we consider the problem of observer-based adaptive neural output-feedback control for a class of stochastic nonlinear systems with nonstrict-feedback structure. To overcome the design difficulty from the nonstrict-feedback structure, a variable separation approach is introduced by using the monotonically increasing property of system bounding functions. On the basis of the state observer, and by combining the adaptive backstepping technique with radial basis function neural networks’ universal approximation capability, an adaptive neural output feedback control algorithm is presented. It is shown that the proposed controller can guarantee that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded in the sense of mean quartic value. Simulation results are provided to show the effectiveness of the proposed control scheme.

A Small-Gain Approach to Robust Event-Triggered Control of Nonlinear Systems

Abstract: This paper presents a new approach to event-triggered control for nonlinear uncertain systems by using the notion of input-to-state stability (ISS) and the nonlinear small-gain theorem. The contribution of this paper is threefold. First, it is proved that infinitely fast sampling can be avoided if the system is input-to-state stabilizable with the sampling error as the external input and the corresponding ISS gain is locally Lipschitz. No assumption on the existence of known ISS-Lyapunov functions is made in the discussions. Moreover, the forward completeness problem with event-triggered control is studied systematically by using ISS small-gain arguments. Second, the proposed approach gives rise to a new self-triggered sampling strategy for a class of nonlinear systems subject to external disturbances. If an upper bound of the external disturbance is known, then the closed-loop system can be designed to be robust to the external disturbance, and moreover, the system state globally asymptotically converges to the origin if the external disturbance decays to zero. Third, a new design method is developed for event-triggered control of nonlinear uncertain systems in the strict-feedback form. It is particularly shown that the ISS gain with the sampling error as the input can be designed to satisfy the proposed condition for event-triggered control and self-triggered control.

Off-Policy Reinforcement Learning for $ H_\infty $ Control Design

Abstract: The $H_\infty $ control design problem is considered for nonlinear systems with unknown internal system model. It is known that the nonlinear $ H_\infty $ control problem can be transformed into solving the so-called Hamilton–Jacobi–Isaacs (HJI) equation, which is a nonlinear partial differential equation that is generally impossible to be solved analytically. Even worse, model-based approaches cannot be used for approximately solving HJI equation, when the accurate system model is unavailable or costly to obtain in practice. To overcome these difficulties, an off-policy reinforcement leaning (RL) method is introduced to learn the solution of HJI equation from real system data instead of mathematical system model, and its convergence is proved. In the off-policy RL method, the system data can be generated with arbitrary policies rather than the evaluating policy, which is extremely important and promising for practical systems. For implementation purpose, a neural network (NN)-based actor-critic structure is employed and a least-square NN weight update algorithm is derived based on the method of weighted residuals. Finally, the developed NN-based off-policy RL method is tested on a linear F16 aircraft plant, and further applied to a rotational/translational actuator system.

Adaptive Fault-Tolerant Tracking Control for Linear and Lipschitz Nonlinear Multi-Agent Systems

Abstract: This paper considers the problem of fault-tolerant tracking control for linear and Lipschitz nonlinear multi-agent systems subject to actuator faults and the leader's bounded unknown input. The communication topology is the undirected subgraph with directed connections between the leader and the followers. Based on the relative states of neighbors and a general actuator fault model, an adaptive fault-tolerant control protocol is proposed to compensate for the failure effects on consensus tracking where the feedback matrices update the parameters by the online estimation of actuator faults. The criteria of reaching consensus tracking despite the actuator faults for both linear and Lipschitz nonlinear agents are derived, respectively. Finally, two examples are included to illustrate the theoretical results.

Modeling and Stability Analysis of Islanded DC Microgrids Under Droop Control

Abstract: The stability of dc microgrids (MG s ) depends on the control strategy adopted for each mode of operation. In an islanded operation mode, droop control is the basic method for bus voltage stabilization when there is no communication among the sources. In this paper, it is shown the consequences of droop implementation on the voltage stability of dc power systems, whose loads are active and nonlinear, e.g., constant power loads. The set of parallel sources and their corresponding transmission lines are modeled by an ideal voltage source in series with an equivalent resistance and inductance. This approximate model allows performing a nonlinear stability analysis to predict the system qualitative behavior due to the reduced number of differential equations. Additionally, nonlinear analysis provides analytical stability conditions as a function of the model parameters and it leads to a design guideline to build reliable (MG s ) based on safe operating regions.

Adaptive Control of Hydraulic Actuators With LuGre Model-Based Friction Compensation

Abstract: This paper concerns high-accuracy tracking control for hydraulic actuators with nonlinear friction compensation Typically, LuGre model-based friction compensation has been widely employed in sundry industrial servomechanisms However, due to the piecewise continuous property, it is difficult to be integrated with backstepping design, which needs the time derivation of the employed friction model Hence, nonlinear model-based hydraulic control rarely sets foot in friction compensation with nondifferentiable friction models, such as LuGre model, Stribeck effects, although they can give excellent friction description and prediction In this paper, a novel continuously differentiable nonlinear friction model is first derived by modifying the traditional piecewise continuous LuGre model, then an adaptive backstepping controller is proposed for precise tracking control of hydraulic systems to handle parametric uncertainties along with nonlinear friction compensation In the formulated nonlinear hydraulic system model, friction parameters, servovalve null shift, and orifice-type internal leakage are all uniformly considered in the proposed controller The controller theoretically guarantees asymptotic tracking performance in the presence of parametric uncertainties, and the robustness against unconsidered dynamics, as well as external disturbances, is also ensured via Lyapunov analysis The effectiveness of the proposed controller is demonstrated via comparative experimental results

Predicting nonlinear properties of metamaterials from the linear response.

Abstract: The discovery of optical second harmonic generation in 1961 started modern nonlinear optics. Soon after, R. C. Miller found empirically that the nonlinear susceptibility could be predicted from the linear susceptibilities. This important relation, known as Miller's Rule, allows a rapid determination of nonlinear susceptibilities from linear properties. In recent years, metamaterials, artificial materials that exhibit intriguing linear optical properties not found in natural materials, have shown novel nonlinear properties such as phase-mismatch-free nonlinear generation, new quasi-phase matching capabilities and large nonlinear susceptibilities. However, the understanding of nonlinear metamaterials is still in its infancy, with no general conclusion on the relationship between linear and nonlinear properties. The key question is then whether one can determine the nonlinear behaviour of these artificial materials from their exotic linear behaviour. Here, we show that the nonlinear oscillator model does not apply in general to nonlinear metamaterials. We show, instead, that it is possible to predict the relative nonlinear susceptibility of large classes of metamaterials using a more comprehensive nonlinear scattering theory, which allows efficient design of metamaterials with strong nonlinearity for important applications such as coherent Raman sensing, entangled photon generation and frequency conversion.

Observation of optical solitons in PT-symmetric lattices

Abstract: Controlling light transport in nonlinear active environments is a topic of considerable interest in the field of optics. In such complex arrangements, of particular importance is to devise strategies to subdue chaotic behaviour even in the presence of gain/loss and nonlinearity, which often assume adversarial roles. Quite recently, notions of parity-time (PT) symmetry have been suggested in photonic settings as a means to enforce stable energy flow in platforms that simultaneously employ both amplification and attenuation. Here we report the experimental observation of optical solitons in PT-symmetric lattices. Unlike other non-conservative nonlinear arrangements where self-trapped states appear as fixed points in the parameter space of the governing equations, discrete PT solitons form a continuous parametric family of solutions. The possibility of synthesizing PT-symmetric saturable absorbers, where a nonlinear wave finds a lossless path through an otherwise absorptive system is also demonstrated.