Noise-Tolerant ZNN-Based Data-Driven Iterative Learning Control for Discrete Nonaffine Nonlinear MIMO Repetitive Systems

Aiming at the tracking problem of a class of discrete nonaffine nonlinear multi-input multi-output(MIMO) repetitive systems subjected to separable and nonseparable disturbances, a novel data-driven iterative learning control(ILC) scheme based on the zeroing neural networks(ZNNs) is proposed. First, the equivalent dynamic linearization data model is obtained by means of dynamic linearization technology, which exists theoretically in the iteration domain. Then, the iterative extended state observer(IESO) is developed to estimate the disturbance and the coupling between systems, and the decoupled dynamic linearization model is obtained for the purpose of controller synthesis. To solve the zero-seeking tracking problem with inherent tolerance of noise,an ILC based on noise-tolerant modified ZNN is proposed. The strict assumptions imposed on the initialization conditions of each iteration in the existing ILC methods can be absolutely removed with our method. In addition, theoretical analysis indicates that the modified ZNN can converge to the exact solution of the zero-seeking tracking problem. Finally, a generalized example and an application-oriented example are presented to verify the effectiveness and superiority of the proposed process.

Policy Iteration for Optimal Control of Discrete-Time Time-Varying Nonlinear Systems

Aimed at infinite horizon optimal control problems of discrete time-varying nonlinear systems,in this paper,a new iterative adaptive dynamic programming algorithm,which is the discrete-time time-varying policy iteration(DTTV)algorithm,is developed.The iterative control law is designed to update the iterative value function which approximates the index function of optimal performance.The admissibility of the iterative control law is analyzed.The results show that the iterative value function is non-increasingly convergent to the Bellman-equation optimal solution.To implement the algorithm,neural networks are employed and a new implementation structure is established,which avoids solving the generalized Bellman equation in each iteration.Finally,the optimal control laws for torsional pendulum and inverted pendulum systems are obtained by using the DTTV policy iteration algorithm,where the mass and pendulum bar length are permitted to be time-varying parameters.The effectiveness of the developed method is illustrated by numerical results and comparisons.

A Fixed-Point Iterative Method for Discrete Tomography Reconstruction Based on Intelligent Optimization

Discrete Tomography(DT)is a technology that uses image projection to reconstruct images.Its reconstruction problem,especially the binary image(0–1matrix)has attracted strong attention.In this study,a fixed point iterative method of integer programming based on intelligent optimization is proposed to optimize the reconstructedmodel.The solution process can be divided into two procedures.First,the DT problem is reformulated into a polyhedron judgment problembased on lattice basis reduction.Second,the fixed-point iterativemethod of Dang and Ye is used to judge whether an integer point exists in the polyhedron of the previous program.All the programs involved in this study are written in MATLAB.The final experimental data show that this method is obviously better than the branch and bound method in terms of computational efficiency,especially in the case of high dimension.The branch and bound method requires more branch operations and takes a long time.It also needs to store a large number of leaf node boundaries and the corresponding consumptionmatrix,which occupies a largememory space.

Three-dimensional gravity inversion based on sparse recovery iteration using approximate zero norm

This research proposes a novel three-dimensional gravity inversion based on sparse recovery in compress sensing. Zero norm is selected as the objective function, which is then iteratively solved by the approximate zero norm solution. The inversion approach mainly employs forward modeling; a depth weight function is introduced into the objective function of the zero norms. Sparse inversion results are obtained by the corresponding optimal mathematical method. To achieve the practical geophysical and geological significance of the results, penalty function is applied to constrain the density values. Results obtained by proposed provide clear boundary depth and density contrast distribution information. The method's accuracy, validity, and reliability are verified by comparing its results with those of synthetic models. To further explain its reliability, a practical gravity data is obtained for a region in Texas, USA is applied. Inversion results for this region are compared with those of previous studies, including a research of logging data in the same area. The depth of salt dome obtained by the inversion method is 4.2 km, which is in good agreement with the 4.4 km value from the logging data. From this, the practicality of the inversion method is also validated.

A New Kind of Iteration Method for Finding Approximate Periodic Solutions to Ordinary Diferential Equations

In this paper, a new kind of iteration technique for solving nonlinear ordinary differential equations is described and used to give approximate periodic solutions for some well-known nonlinear problems. The most interesting features of the proposed methods are its extreme simplicity and concise forms of iteration formula for a wide range of nonlinear problems.

A Modified Discrete-Time Jacobi Waveform Relaxation Iteration

In this paper, we investigate an accelerated version of the discrete-time Jacobi waveform relaxation iteration method. Based on the well known Chebyshev polynomial theory, we show that significant speed up can be achieved by taking linear combinations of earlier iterates. The convergence and convergence speed of the new iterative method are presented and it is shown that the convergence speed of the new iterative method is sharper than that of the Jacobi method but blunter than that of the optimal SOR method. Moreover, at every iteration the new iterative method needs almost equal computation work and memory storage with the Jacobi method, and more important it can completely exploit the particular advantages of the Jacobi method in the sense of parallelism. We validate our theoretical conclusions with numerical experiments.

Iterative Learning Control for Discrete-time Stochastic Systems with Quantized Information

An iterative learning control (ILC) algorithm using quantized error information is given in this paper for both linear and nonlinear discrete-time systems with stochastic noises. A logarithmic quantizer is used to guarantee an adaptive improvement in tracking performance. A decreasing learning gain is introduced into the algorithm to suppress the effects of stochastic noises and quantization errors. The input sequence is proved to converge strictly to the optimal input under the given index. Illustrative simulations are given to verify the theoretical analysis. © 2014 Chinese Association of Automation.

Iterative learning based fault diagnosis for discrete linear uncertain systems

In order to detect and estimate faults in discrete lin-ear time-varying uncertain systems, the discrete iterative learning strategy is applied in fault diagnosis, and a novel fault detection and estimation algorithm is proposed. And the threshold limited technology is adopted in the proposed algorithm. Within the chosen optimal time region, residual signals are used in the proposed algorithm to correct the introduced virtual faults with iterative learning rules, making the virtual faults close to these occurred in practical systems. And the same method is repeated in the rest optimal time regions, thereby reaching the aim of fault diagnosis. The proposed algorithm not only completes fault detection and estimation for discrete linear time-varying uncertain systems, but also improves the reliability of fault detection and decreases the false alarm rate. The final simulation results verify the validity of the proposed algorithm.

Discrete-Time Nonlinear Stochastic Optimal Control Problem Based on Stochastic Approximation Approach

In this paper, a computational approach is proposed for solving the discrete-time nonlinear optimal control problem, which is disturbed by a sequence of random noises. Because of the exact solution of such optimal control problem is impossible to be obtained, estimating the state dynamics is currently required. Here, it is assumed that the output can be measured from the real plant process. In our approach, the state mean propagation is applied in order to construct a linear model-based optimal control problem, where the model output is measureable. On this basis, an output error, which takes into account the differences between the real output and the model output, is defined. Then, this output error is minimized by applying the stochastic approximation approach. During the computation procedure, the stochastic gradient is established, so as the optimal solution of the model used can be updated iteratively. Once the convergence is achieved, the iterative solution approximates to the true optimal solution of the original optimal control problem, in spite of model-reality differences. For illustration, an example on a continuous stirred-tank reactor problem is studied, and the result obtained shows the applicability of the approach proposed. Hence, the efficiency of the approach proposed is highly recommended.

Approximation law for discrete-time variable structure control systems

Two approximation laws of sliding mode for discrete-time variable structure control systems are proposed to overcome the limitations of the exponential approximation law and the variable rate approximation law. By applying the proposed approximation laws of sliding mode to discrete-time variable structure control systems, the stability of origin can be guaranteed, and the chattering along the switching surface caused by discrete-time variable structure control can be restrained effectively. In designing of approximation laws, the problem that the system control input is restricted is also considered, which is very important in practical systems. Finally a simulation example shows the effectiveness of the two approximation laws proposed.

Explicit Iterative Methods of Second Order and Approximate Inverse Preconditioners for Solving Complex Computational Problems

Explicit Exact and Approximate Inverse Preconditioners for solving complex linear systems are introduced. A class of general iterative methods of second order is presented and the selection of iterative parameters is discussed. The second order iterative methods behave quite similar to first order methods and the development of efficient preconditioners for solving the original linear system is a decisive factor for making the second order iterative methods superior to the first order iterative methods. Adaptive preconditioned Conjugate Gradient methods using explicit approximate preconditioners for solving efficiently large sparse systems of algebraic equations are also presented. The generalized Approximate Inverse Matrix techniques can be efficiently used in conjunction with explicit iterative schemes leading to effective composite semi-direct solution methods for solving large linear systems of algebraic equations.

Numerical Approximation of Real Finite Nonnegative Function by the Modulus of Discrete Fourier Transform

The numerical algorithms for finding the lines of branching and branching-off solutions of nonlinear problem on mean-square approximation of a real finite nonnegative function with respect to two variables by the modulus of double discrete Fourier transform dependent on two parameters, are constructed and justified.

Iterative approximate solutions of kinetic equations for reversible enzyme reactions

We study kinetic models of reversible enzyme reactions and compare two techniques for analytic approximate solutions of the model. Analytic approximate solutions of non-linear reaction equations for reversible enzyme reactions are calculated using the Homotopy Perturbation Method (HPM) and the Simple Iteration Method (SIM). The results of the approximations are similar. The Matlab programs are included in appendices.

CONTINUOUS AND DISCRETE N-CONVEX APPROXIMATIONS

We show that the best L_p-approximant to continuous functions by n-convex functions is the limit of discrete n-convex approximations.The techniques of the proof are then used to show the existence of near interpolants to discrete n-convex data by continuous n-convex functions if the data points are close.

Carlson iterating rational approximation and performance analysis of fractional operator with arbitrary order

The performance analysis of the generalized Carlson iterating process,which can realize the rational approximation of fractional operator with arbitrary order,is presented in this paper.The reasons why the generalized Carlson iterating function possesses more excellent properties such as self-similarity and exponential symmetry are also explained.K-index,P-index,O-index,and complexity index are introduced to contribute to performance analysis.Considering nine different operational orders and choosing an appropriate rational initial impedance for a certain operational order,these rational approximation impedance functions calculated by the iterating function meet computational rationality,positive reality,and operational validity.Then they are capable of having the operational performance of fractional operators and being physical realization.The approximation performance of the impedance function to the ideal fractional operator and the circuit network complexity are also exhibited.

Generalization of iterative perturbation theory and coherent potential approximation (ITP+CPA) to double exchange model with orbital degeneracy

A combination of the iterative perturbation theory （ITP） of the dynamical mean field theory （DMFT） and coherentpotential approximation （CPA） is generalized to the double exchange model with orbital degeneracy. The Hubbard interaction and the off-diagonal components for the hopping matrix tij^mn（m ≠ n） are considered in our calculation of spectrum and optical conductivity. The numerical results show that the effects of the non-diagonal hopping matrix elements are important.

Learning-based control for discrete-time constrained nonzero-sum games

A generalized policy-iteration-based solution to a class of discrete-time multi-player nonzero-sum games concerning the control constraints was proposed.Based on initial admissible control policies,the iterative value function of each player converges to the optimum approximately,which is structured by the iterative control policies satisfying the Nash equilibrium.Afterwards,the stability analysis is shown to illustrate that the iterative control policies can stabilize the system and minimize the performance index function of each player.Meanwhile,neural networks are implemented to approximate the iterative control policies and value functions with the impact of control constraints.Finally,two numerical simulations of the discrete-time two-player non-zero-sum games for linear and non-linear systems are shown to illustrate the effectiveness of the proposed scheme.

Solitary Solution of Discrete mKdV Equation by Homotopy Analysis Method

In this paper, we apply homotopy analysis method to solve discrete mKdV equation and successfully obtain the bell-shaped solitary solution to mKdV equation. Comparison between our solution and the exact solution shows that homotopy analysis method is effective and validity in solving hybrid nonlinear problems, including solitary solution of difference-differential equation.

Strong Convergence of an Implicit Iteration Process for a Finite Family of Asymptotically Ф-pseudocontractive Mappings

Strong convergence theorems for approximation of common fixed points of asymptotically Ф-quasi-pseudocontractive mappings and asymptotically C-strictly- pseudocontractive mappings are proved in real Banach spaces by using a new composite implicit iteration scheme with errors. The results presented in this paper extend and improve the main results of Sun, Gu and Osilike published on J. Math. Anal. Appl.

The Equivalence between Orthogonal Iterations and Alternating Least Squares

This note explores the relations between two different methods. The first one is the Alternating Least Squares (ALS) method for calculating a rank-k approximation of a real m×n matrix, A. This method has important applications in nonnegative matrix factorizations, in matrix completion problems, and in tensor approximations. The second method is called Orthogonal Iterations. Other names of this method are Subspace Iterations, Simultaneous Iterations, and block-Power method. Given a real symmetric matrix, G, this method computes k dominant eigenvectors of G. To see the relation between these methods we assume that G = AT A. It is shown that in this case the two methods generate the same sequence of subspaces, and the same sequence of low-rank approximations. This equivalence provides new insight into the convergence properties of both methods.