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Showing papers on "Nonlinear system published in 2014"


BookDOI
10 Sep 2014
TL;DR: In this paper, the authors use a state-space approach and focus on stability analysis and the synthesis of stabilizing control laws in both local and global contexts, and propose methods and algorithms based on the use of linear programming and linear matrix inequalities for computing estimates of the basin of attraction.
Abstract: This monograph details basic concepts and tools fundamental for the analysis and synthesis of linear systems subject to actuator saturation and developments in recent research. The authors use a state-space approach and focus on stability analysis and the synthesis of stabilizing control laws in both local and global contexts. Different methods of modeling the saturation and behavior of the nonlinear closed-loop system are given special attention. Various kinds of Lyapunov functions are considered to present different stability conditions. Results arising from uncertain systems and treating performance in the presence of saturation are given. The text proposes methods and algorithms, based on the use of linear programming and linear matrix inequalities, for computing estimates of the basin of attraction and for designing control systems accounting for the control bounds and the possibility of saturation. They can be easily implemented with mathematical software packages.

639 citations


Journal ArticleDOI
TL;DR: The proposed controller theoretically guarantees a prescribed tracking transient performance and final tracking accuracy, while achieving asymptotic tracking performance in the absence of time-varying uncertainties, which is very important for high-accuracy tracking control of hydraulic servo systems.
Abstract: In this paper, an output feedback nonlinear control is proposed for a hydraulic system with mismatched modeling uncertainties in which an extended state observer (ESO) and a nonlinear robust controller are synthesized via the backstepping method. The ESO is designed to estimate not only the unmeasured system states but also the modeling uncertainties. The nonlinear robust controller is designed to stabilize the closed-loop system. The proposed controller accounts for not only the nonlinearities (e.g., nonlinear flow features of servovalve), but also the modeling uncertainties (e.g., parameter derivations and unmodeled dynamics). Furthermore, the controller theoretically guarantees a prescribed tracking transient performance and final tracking accuracy, while achieving asymptotic tracking performance in the absence of time-varying uncertainties, which is very important for high-accuracy tracking control of hydraulic servo systems. Extensive comparative experimental results are obtained to verify the high-performance nature of the proposed control strategy.

586 citations


Book
24 Sep 2014
TL;DR: In this article, the authors present a survey on model order reduction of coupled systems, including linear systems, eigenvalues, and projection, and propose a unified Krylov projection framework for structure-preserving model reduction via proper orthogonal decomposition.
Abstract: Basic Concepts.- to Model Order Reduction.- Linear Systems, Eigenvalues, and Projection.- Theory.- Structure-Preserving Model Order Reduction of RCL Circuit Equations.- A Unified Krylov Projection Framework for Structure-Preserving Model Reduction.- Model Reduction via Proper Orthogonal Decomposition.- PMTBR: A Family of Approximate Principal-components-like Reduction Algorithms.- A Survey on Model Reduction of Coupled Systems.- Space Mapping and Defect Correction.- Modal Approximation and Computation of Dominant Poles.- Some Preconditioning Techniques for Saddle Point Problems.- Time Variant Balancing and Nonlinear Balanced Realizations.- Singular Value Analysis and Balanced Realizations for Nonlinear Systems.- Research Aspects and Applications.- Matrix Functions.- Model Reduction of Interconnected Systems.- Quadratic Inverse Eigenvalue Problem and Its Applications to Model Updating - An Overview.- Data-Driven Model Order Reduction Using Orthonormal Vector Fitting.- Model-Order Reduction of High-Speed Interconnects Using Integrated Congruence Transform.- Model Order Reduction for MEMS: Methodology and Computational Environment for Electro-Thermal Models.- Model Order Reduction of Large RC Circuits.- Reduced Order Models of On-Chip Passive Components and Interconnects, Workbench and Test Structures.

543 citations


Journal ArticleDOI
TL;DR: It is shown that the iterative performance index function is nonincreasingly convergent to the optimal solution of the Hamilton-Jacobi-Bellman equation and it is proven that any of the iteratives control laws can stabilize the nonlinear systems.
Abstract: This paper is concerned with a new discrete-time policy iteration adaptive dynamic programming (ADP) method for solving the infinite horizon optimal control problem of nonlinear systems. The idea is to use an iterative ADP technique to obtain the iterative control law, which optimizes the iterative performance index function. The main contribution of this paper is to analyze the convergence and stability properties of policy iteration method for discrete-time nonlinear systems for the first time. It shows that the iterative performance index function is nonincreasingly convergent to the optimal solution of the Hamilton-Jacobi-Bellman equation. It is also proven that any of the iterative control laws can stabilize the nonlinear systems. Neural networks are used to approximate the performance index function and compute the optimal control law, respectively, for facilitating the implementation of the iterative ADP algorithm, where the convergence of the weight matrices is analyzed. Finally, the numerical results and analysis are presented to illustrate the performance of the developed method.

535 citations


Journal ArticleDOI
TL;DR: It is proved that the proposed adaptive neural network (NN) consensus control method guarantees the convergence on the basis of Lyapunov stability theory.
Abstract: Because of the complicity of consensus control of nonlinear multiagent systems in state time-delay, most of previous works focused only on linear systems with input time-delay. An adaptive neural network (NN) consensus control method for a class of nonlinear multiagent systems with state time-delay is proposed in this paper. The approximation property of radial basis function neural networks (RBFNNs) is used to neutralize the uncertain nonlinear dynamics in agents. An appropriate Lyapunov–Krasovskii functional, which is obtained from the derivative of an appropriate Lyapunov function, is used to compensate the uncertainties of unknown time delays. It is proved that our proposed approach guarantees the convergence on the basis of Lyapunov stability theory. The simulation results of a nonlinear multiagent time-delay system and a multiple collaborative manipulators system show the effectiveness of the proposed consensus control algorithm.

528 citations


BookDOI
01 Jan 2014
TL;DR: In this paper, the authors give a short overview of the scopes of both the theory of rough paths and regularity structures and point out some analogies with other branches of mathematics.
Abstract: We give a short overview of the scopes of both the theory of rough paths and the theory of regularity structures. The main ideas are introduced and we point out some analogies with other branches of mathematics. 1.1 Controlled differential equations Differential equations are omnipresent in modern pure and applied mathematics; many “pure” disciplines in fact originate in attempts to analyse differential equations from various application areas. Classical ordinary differential equations (ODEs) are of the form Ẏt = f(Yt, t); an important sub-class is given by controlled ODEs of the form Ẏt = f0(Yt) + f(Yt)Ẋt , (1.1) where X models the input (taking values in R, say), and Y is the output (in R, say) of some system modelled by nonlinear functions f0 and f , and by the initial state Y0. The need for a non-smooth theory arises naturally when the system is subject to white noise, which can be understood as the scaling limit as h→ 0 of the discrete evolution equation Yi+1 = Yi + hf0(Yi) + √ hf(Yi)ξi+1 , (1.2) where the (ξi) are i.i.d. standard Gaussian random variables. Based on martingale theory, Ito’s stochastic differential equations (SDEs) have provided a rigorous and extremely useful mathematical framework for all this. And yet, stability is lost in the passage to continuous time: while it is trivial to solve (1.2) for a fixed realisation of ξi(ω), after all (ξ1, . . . ξT ;Y0) 7→ Yi is surely a continuous map, the continuity of the solution as a function of the driving noise is lost in the limit. Taking Ẋ = ξ to be white noise in time (which amounts to say that X is a Brownian motion, say B), the solution map S : B 7→ Y to (1.1), known as Ito map, is a measurable map which in general lacks continuity, whatever norm one uses to

499 citations


Journal ArticleDOI
TL;DR: It is proved that the proposed control approach can guarantee that all the signals of the resulting closed-loop system are bounded, and the tracking errors between the system outputs and the reference signals converge to a small neighborhood of zero by appropriate choice of the design parameters.
Abstract: This paper investigates the adaptive fuzzy decentralized fault-tolerant control (FTC) problem for a class of nonlinear large-scale systems in strict-feedback form. The considered nonlinear system contains the unknown nonlinear functions, i.e., unmeasured states and actuator faults, which are modeled as both loss of effectiveness and lock-in-place. With the help of fuzzy logic systems to approximate the unknown nonlinear functions, a fuzzy adaptive observer is designed to estimate the unmeasured states. By combining the backstepping technique with the nonlinear FTC theory, a novel adaptive fuzzy decentralized FTC scheme 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 tracking errors between the system outputs and the reference signals converge to a small neighborhood of zero by appropriate choice of the design parameters. Simulation results are provided to show the effectiveness of the control approach.

493 citations


Journal ArticleDOI
TL;DR: In this article, a broadband piezoelectric based vibration energy harvester with a triple-well potential induced by a magnetic field was proposed and the parameters of the linear energy harvesting system without magnetic force actuation were obtained through intelligent optimization of the minimum error between numerical simulations and experimental responses.

483 citations


Proceedings Article
01 Jan 2014
TL;DR: In this article, the authors show that deep linear networks exhibit nonlinear learning phenomena similar to those seen in simulations of nonlinear networks, including long plateaus followed by rapid transitions to lower error solutions, and faster convergence from greedy unsupervised pretraining initial conditions than from random initial conditions.
Abstract: Despite the widespread practical success of deep learning methods, our theoretical understanding of the dynamics of learning in deep neural networks remains quite sparse. We attempt to bridge the gap between the theory and practice of deep learning by systematically analyzing learning dynamics for the restricted case of deep linear neural networks. Despite the linearity of their input-output map, such networks have nonlinear gradient descent dynamics on weights that change with the addition of each new hidden layer. We show that deep linear networks exhibit nonlinear learning phenomena similar to those seen in simulations of nonlinear networks, including long plateaus followed by rapid transitions to lower error solutions, and faster convergence from greedy unsupervised pretraining initial conditions than from random initial conditions. We provide an analytical description of these phenomena by finding new exact solutions to the nonlinear dynamics of deep learning. Our theoretical analysis also reveals the surprising finding that as the depth of a network approaches infinity, learning speed can nevertheless remain finite: for a special class of initial conditions on the weights, very deep networks incur only a finite, depth independent, delay in learning speed relative to shallow networks. We show that, under certain conditions on the training data, unsupervised pretraining can find this special class of initial conditions, while scaled random Gaussian initializations cannot. We further exhibit a new class of random orthogonal initial conditions on weights that, like unsupervised pre-training, enjoys depth independent learning times. We further show that these initial conditions also lead to faithful propagation of gradients even in deep nonlinear networks, as long as they operate in a special regime known as the edge of chaos.

453 citations


Journal ArticleDOI
TL;DR: This formulation extends the integral reinforcement learning (IRL) technique, a method for solving optimal regulation problems, to learn the solution to the OTCP, and it also takes into account the input constraints a priori.

440 citations


Journal ArticleDOI
TL;DR: A stabilization problem for nonlinear uncertain systems via adaptive backstepping approach is considered, and a designed controller together with the quantizer ensures the stability of the closed-loop system in the sense of signal boundedness.
Abstract: In this paper, we study a general class of strict feedback nonlinear systems, where the input signal takes quantized values. We consider a stabilization problem for nonlinear uncertain systems via adaptive backstepping approach. The control design is achieved by introducing a hysteretic quantizer to avoid chattering and using backstepping technique. A guideline is derived to select the parameters of the quantizer. The designed controller together with the quantizer ensures the stability of the closed-loop system in the sense of signal boundedness.

Journal ArticleDOI
TL;DR: It is shown that the feasibility of the event-triggered MPC algorithm can be guaranteed if, the prediction horizon is designed properly and the disturbances are small enough and that the state trajectory converges to a robust invariant set under the proposed conditions.

Journal ArticleDOI
TL;DR: The numerical results show that the proposed continuous genetic algorithm is a robust and accurate procedure for solving systems of second-order boundary value problems and the obtained accuracy for the solutions using CGA is much better than the results obtained using some modern methods.

Book
14 Mar 2014
TL;DR: In this article, the main and Adjoint Equations in Mathematical Physics have been used to solve the problem of finding the solution of global problems, and the difference analogue of Nonstationary Heat Diffusion Equation in Atmosphere and Ocean.
Abstract: Author's Preface to the English Edition. Introduction. Part I: Adjoint Equations and Perturbation Theory. 1. Main and Adjoint Equations. Perturbation Theory. 2. Simple Main and Adjoint Equations in Mathematical Physics. 3. Nonlinear Equations. 4. Inverse Problems and Adjoint Equations. Part II: Problems of Environment and Optimization Methods on the Basis of Adjoint Equations. 5. Analysis of Mathematical Models in Environmental Problems. 6. Adjoint Equations, Optimization. 7. Adjoint Equations and Models of General Circulation of Atmosphere and Ocean. 8. Adjoint Equations in Data Processing Problems. Appendix I: Splitting Methods in the Solution of Global Problems. Appendix II: Difference Analogue of Nonstationary Heat Diffusion Equation in Atmosphere and Ocean. Bibliography. Index.

Journal ArticleDOI
TL;DR: In this article, the effects of material property distribution, spring constants and porosity volume fraction on linear and nonlinear frequencies of functionally graded materials (FGMs) beams with porosity phases were investigated.

Journal ArticleDOI
TL;DR: The proposed RADP methodology can be viewed as an extension of ADP to uncertain nonlinear systems and has been applied to the controller design problems for a jet engine and a one-machine power system.
Abstract: This paper studies the robust optimal control design for a class of uncertain nonlinear systems from a perspective of robust adaptive dynamic programming (RADP). The objective is to fill up a gap in the past literature of adaptive dynamic programming (ADP) where dynamic uncertainties or unmodeled dynamics are not addressed. A key strategy is to integrate tools from modern nonlinear control theory, such as the robust redesign and the backstepping techniques as well as the nonlinear small-gain theorem, with the theory of ADP. The proposed RADP methodology can be viewed as an extension of ADP to uncertain nonlinear systems. Practical learning algorithms are developed in this paper, and have been applied to the controller design problems for a jet engine and a one-machine power system.

Journal ArticleDOI
TL;DR: In this paper, an efficient and simple refined shear deformation theory is presented for the vibration and buckling of exponentially graded material sandwich plate resting on elastic foundations under various boundary conditions.
Abstract: In this paper, an efficient and simple refined shear deformation theory is presented for the vibration and buckling of exponentially graded material sandwich plate resting on elastic foundations under various boundary conditions. The displacement field of the present theory is chosen based on nonlinear variations in the in-plane displacements through the thickness of the plate. By dividing the transverse displacement into the bending and shear parts and making further assumptions, the number of unknowns and equations of motion of the present theory is reduced and hence makes them simple to use. Equations of motion are derived from Hamilton’s principle. Numerical results for the natural frequencies and critical buckling loads of several types of symmetric exponentially graded material sandwich plates are presented. The accuracy of the present theory is verified by comparing the obtained results with solutions available in the literature. Numerical results show that the present theory can archive accuracy c...

Journal ArticleDOI
TL;DR: This paper studies the composite adaptive tracking control for a class of uncertain nonlinear systems in strict-feedback form and achieves smoother parameter adaption, better accuracy, and improved performance.
Abstract: This paper studies the composite adaptive tracking control for a class of uncertain nonlinear systems in strict-feedback form. Dynamic surface control technique is incorporated into radial-basis-function neural networks (NNs)-based control framework to eliminate the problem of explosion of complexity. To avoid the analytic computation, the command filter is employed to produce the command signals and their derivatives. Different from directly toward the asymptotic tracking, the accuracy of the identified neural models is taken into consideration. The prediction error between system state and serial-parallel estimation model is combined with compensated tracking error to construct the composite laws for NN weights updating. The uniformly ultimate boundedness stability is established using Lyapunov method. Simulation results are presented to demonstrate that the proposed method achieves smoother parameter adaption, better accuracy, and improved performance.

Journal ArticleDOI
TL;DR: The ROA can be computed by solving a convex linear programming (LP) problem over the space of measures and this problem can be solved approximately via a classical converging hierarchy of convex finite-dimensional linear matrix inequalities (LMIs).
Abstract: We address the long-standing problem of computing the region of attraction (ROA) of a target set (e.g., a neighborhood of an equilibrium point) of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints. We show that the ROA can be computed by solving an infinite-dimensional convex linear programming (LP) problem over the space of measures. In turn, this problem can be solved approximately via a classical converging hierarchy of convex finite-dimensional linear matrix inequalities (LMIs). Our approach is genuinely primal in the sense that convexity of the problem of computing the ROA is an outcome of optimizing directly over system trajectories. The dual infinite-dimensional LP on nonnegative continuous functions (approximated by polynomial sum-of-squares) allows us to generate a hierarchy of semialgebraic outer approximations of the ROA at the price of solving a sequence of LMI problems with asymptotically vanishing conservatism. This sharply contrasts with the existing literature which follows an exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix inequalities or conservative LMI conditions. The approach is simple and readily applicable as the outer approximations are the outcome of a single semidefinite program with no additional data required besides the problem description. The approach is demonstrated on several numerical examples.

Journal ArticleDOI
TL;DR: In this article, a nonlinear version of fluctuating hydrodynamics is developed, in which the Euler currents are kept to second order in the deviations from equilibrium and dissipation plus noise are added.
Abstract: With focus on anharmonic chains, we develop a nonlinear version of fluctuating hydrodynamics, in which the Euler currents are kept to second order in the deviations from equilibrium and dissipation plus noise are added. The required model-dependent parameters are written in such a way that they can be computed numerically within seconds, once the interaction potential, pressure, and temperature are given. In principle the theory is applicable to any one-dimensional system with local conservation laws. The resulting nonlinear stochastic field theory is handled in the one-loop approximation. Some of the large scale predictions can still be worked out analytically. For more details one has to rely on numerical simulations of the corresponding mode-coupling equations. In this way we arrive at detailed predictions for the equilibrium time correlations of the locally conserved fields of an anharmonic chain.

Journal ArticleDOI
TL;DR: It is proved that the proposed control approach can guarantee that all the signals of the closed-loop system are bounded in probability in the presence of the actuator failures and the unmodeled dynamics.
Abstract: This paper investigates fuzzy adaptive actuator failure compensation control for a class of uncertain stochastic nonlinear systems in strict-feedback form. These stochastic nonlinear systems contain the actuator faults of both loss of effectiveness and lock-in-place, unmodeled dynamics, and without direct measurements of state variables. With the help of fuzzy logic systems to approximate the unknown nonlinear functions, a fuzzy state observer is established to estimate the unmeasured states. By introducing the dynamical signal and the changing supply function technique design into the backstepping control design, a robust adaptive fuzzy fault-tolerant control scheme is developed. It is proved that the proposed control approach can guarantee that all the signals of the closed-loop system are bounded in probability in the presence of the actuator failures and the unmodeled dynamics. Simulation results are provided to show the effectiveness of the control approach.

Journal ArticleDOI
TL;DR: In this paper, a new alternating direction implicit Galerkin-Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed.
Abstract: In this paper, a new alternating direction implicit Galerkin-Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank-Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order 2 in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh-Nagumo model. Numerical results are provided to verify the theoretical analysis.

Journal ArticleDOI
TL;DR: It is shown that the proposed controller guarantees that all the signals in the closed-loop system are four-moment semiglobally uniformly ultimately bounded, and the tracking error eventually converges to a small neighborhood of the origin in the sense of mean quartic value.
Abstract: This paper considers the problem of adaptive neural control of stochastic nonlinear systems in nonstrict-feedback form with unknown backlash-like hysteresis nonlinearities. To overcome the design difficulty of nonstrict-feedback structure, variable separation technique is used to decompose the unknown functions of all state variables into a sum of smooth functions of each error dynamic. By combining radial basis function neural networks' universal approximation capability with an adaptive backstepping technique, an adaptive neural control algorithm is proposed. It is shown that the proposed controller guarantees that all the signals in the closed-loop system are four-moment semiglobally uniformly ultimately bounded, and the tracking error eventually converges to a small neighborhood of the origin in the sense of mean quartic value. Simulation results further show the effectiveness of the presented control scheme.

Journal ArticleDOI
TL;DR: This paper presents the energy‐conserving sampling and weighting (ECSW) hyper reduction method for discrete (or semi‐discrete), nonlinear, finite element structural dynamics models, which is natural for finite element computations and preserves an important energetic aspect of the high‐dimensional finite element model to be reduced.

Journal ArticleDOI
TL;DR: This paper analyzes distributed control protocols for first- and second-order networked dynamical systems and proposes a class of nonlinear consensus controllers where the input of each agent can be written as a product of a nonlinear gain, and a sum of non linear interaction functions.
Abstract: This paper analyzes distributed control protocols for first- and second-order networked dynamical systems. We propose a class of nonlinear consensus controllers where the input of each agent can be written as a product of a nonlinear gain, and a sum of nonlinear interaction functions. By using integral Lyapunov functions, we prove the stability of the proposed control protocols, and explicitly characterize the equilibrium set. We also propose a distributed proportional-integral (PI) controller for networked dynamical systems. The PI controllers successfully attenuate constant disturbances in the network. We prove that agents with single-integrator dynamics are stable for any integral gain, and give an explicit tight upper bound on the integral gain for when the system is stable for agents with double-integrator dynamics. Throughout the paper we highlight some possible applications of the proposed controllers by realistic simulations of autonomous satellites, power systems and building temperature control.

Journal ArticleDOI
TL;DR: In this paper, the effects of magnetic interaction number, slip factor and relative temperature difference on velocity and temperature profiles as well as entropy generation in magnetohydrodynamic (MHD) flow of a fluid with variable properties over a rotating disk are investigated using numerical methods.

Journal ArticleDOI
TL;DR: In this paper, a review of modeling approaches used for nonlinear crack-wave interactions is presented, including models of crack-induced elastic, thermo-elastic and dissipative nonlinearities.

Journal ArticleDOI
TL;DR: Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial-based spectral collocation element methods of arbitrary order for the compressible Navier--Stokes equations.
Abstract: Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial-based spectral collocation element methods of arbitrary order for the compressible Navier--Stokes equations. The new methods are similar to strong form, nodal discontinuous Galerkin spectral elements but conserve entropy for the Euler equations and are entropy stable for the Navier--Stokes equations. Shock capturing follows immediately by combining them with a dissipative companion operator via a comparison approach. Smooth and discontinuous test cases are presented that demonstrate their efficacy.

Journal ArticleDOI
TL;DR: This paper presents a new approach to construct more efficient reduced-order models for nonlinear partial differential equations with proper orthogonal decomposition and the discrete empirical interpolation method (DEIM).
Abstract: This paper presents a new approach to construct more efficient reduced-order models for nonlinear partial differential equations with proper orthogonal decomposition and the discrete empirical interpolation method (DEIM). Whereas DEIM projects the nonlinear term onto one global subspace, our localized discrete empirical interpolation method (LDEIM) computes several local subspaces, each tailored to a particular region of characteristic system behavior. Then, depending on the current state of the system, LDEIM selects an appropriate local subspace for the approximation of the nonlinear term. In this way, the dimensions of the local DEIM subspaces, and thus the computational costs, remain low even though the system might exhibit a wide range of behaviors as it passes through different regimes. LDEIM uses machine learning methods in the offline computational phase to discover these regions via clustering. Local DEIM approximations are then computed for each cluster. In the online computational phase, machine...

Journal ArticleDOI
TL;DR: In this paper, the authors revisited concurrent design of material and structure within FE2 nonlinear multiscale analysis framework for structural stiffness maximization at macroscopic scale, design variables are defined at the both scales.