Reduced inversion methods for solving discrete periodic Riccati matrix equations

Here, a gradient‐based neural network (GNN) model is constructed for solving the discrete periodic Lyapunov matrix equation (DPLME) associated with discrete‐time linear periodic systems. In practical applications, the recurrent neural network model should not only converge rapidly, but also be able to tolerate noise. However, the influence of noise on GNN models was seldom considered in the past. In order to improve the convergence and robustness of the GNN model, a novel type of non‐linear activation function is applied to the GNN model. Compared with the traditional activation functions, the activation function used here makes the GNN model to achieve fixed‐time convergence. Besides, when disturbed by bounded noise, the unique positive definite solution of the DPLME can still be obtained by using the GNN model. Finally, simulation experiment is performed to verify the effectiveness and superiority of the proposed GNN model.

In this paper, two iterative algorithms are constructed to obtain the positive definite solutions of the discrete periodic algebraic Riccati matrix equations. In these two algorithms, the estimation of the unknown matrices are updated by using the available estimation information at the current iteration step. The convergence properties of the proposed algorithms are also given. Finally, numerical examples are employed to illustrate the effectiveness of the proposed algorithms.

There are important relationships between the discrete‐time linear periodic descriptor systems and the discrete‐time periodic matrix equations. In the present paper, we introduce the matrix form of the biconjugate residual (BCR) algorithm for solving the discrete‐time periodic Sylvester matrix equations AiXiBi+CiXi+1Di=Ei,i=1,2,.... It is shown that the introduced algorithm converges to the solutions within a finite number of iterations in the absence of round‐off errors. Finally, three numerical examples are given to demonstrate the efficiency and the performance of the algorithm.

A parametric poles assignment algorithm for second-order linear periodic systems

Neural Networks (NN) have a wide range of applications in analog and digital signal processing. Nonlinear activation function is one of the main building blocks of artificial neural networks. Hyperbolic tangent and sigmoid are the most used nonlinear neural activation functions of NN. This project proposes a knowledge-based neural network (KBNN) modeling approach with new hyperbolic tangent function using Hashing trick. The KBNN embeds the existing FPGA analytical models (AM) into an NN. For fast computation of neuron in NN, we use new approximation scheme with hashing algorithm for the hyperbolic tangent function calculation. Hashing trick algorithm eliminates the less weight and using average weighting function. The approximation is based on mathematical analysis considering the maximum allowable error as design parameter. The NN can complement the analytical models according to their needs to provide further increased model accuracy, while maintaining the meaningful trends successfully captured in the analytical models. The proposed KBNN coded using verilog HDL and simulated using Xilinx 12.1. Also the proposed KBNN with new activation function and hashing trick method results in reduction of number of multiplications, area, delay, computation cost and power in VLSI implementation of artificial neural networks with hyperbolic tangent activation function.

Optical nonlinear activation function is an indispensable part of the optical neural network. While linear matrix computation has thrived in an integrated optical neural network, there are many challenges for nonlinear activation function on a chip such as large latency, high power consumption and high threshold. Here, we demonstrate that Ge/Si hybrid structure would be a qualified candidate owing to its property of CMOS-compatibility, low nonlinear threshold and compact footprint. Thanks to the strong thermal-optic effect of germanium in conjunction with micro-ring resonator, we experimentally demonstrate three different types of nonlinear function (Radial basis, Relu and ELU functions) with a lowest threshold of 0.74 mW among our measured nonlinear functions and they can work well with a repetition rate below 100 kHz. Simultaneous size shrinkage of germanium and resonance constraint inside germanium is proposed to speed up response time. Furthermore, we apply our measured nonlinear activation function to the task of classification of MNIST handwritten digit image dataset and improve the test accuracy from 91.8% to 94.8% with feedforward full-connected neural network containing three hidden layers. It proves that our scheme has potential in the future optical neural network.

The main purpose of the present study is to develop some artificial neural network (ANN) models for the prediction of limit pressure (PL) and pressuremeter modulus (EM) for clayey soils. Moisture content, plasticity index, and SPT values are used as inputs in the ANN models. To get plausible results, the number of hidden layer neurons in all models is varied between 1 and 5. In addition, both linear and nonlinear activation functions are considered for the neurons in output layers while a nonlinear activation function is employed for the neurons in the hidden layers of all models. Logistic activation function is used as a nonlinear activation function. During the modeling studies, total eight different ANN models are constructed. The ANN models having two outputs produced the worst results, independent from activation function. However, for PL, the best results are obtained from the feed-forward neural network with five neurons in the hidden layer, and logistic activation function is employed in the output neuron. For EM, the best model producing the most acceptable results is Elman recurrent network model, which has 4 neurons in the neurons in the hidden layer, and linear activation function is used for the output neuron. Finally, the results show that the ANN models produce the more accurate results than the regression-based models. In the literature, when empirical equations based on regression analysis were used, the best root mean square error (RMSE) values obtained to date for PL and EM have been 0.43 and 5.65, respectively. In this study, RMSE values for PL and EM were found to be 0.20 and 2.99, respectively, by using ANN models. It was observed that using ANN approach drastically increases the prediction accuracy in terms of RMSE criterion.

Fractional ordering of activation functions for neural networks: A case study on Texas wind turbine

The purpose of this work is to stimulate a neural network with non-linear activation functions. The non-linear functions are simulated in Microsoft Visual Studio C++ 6.0 to observe the precision and to implement on the programmable logic devices. This network is realized to accept very small input values. The multiplication between input values and weight values is realized with the add-logarithm and exponential functions. One approximates all the non-linear functions with linear functions using shift-add blocks.

Parametric solutions to the discrete periodic regul ator equations

The convergence properties for the solution of the discrete time Riccati matrix equation are extended to Riccati operator equations such as arise in a gyroscope noise filtering problem. Stabilizability and detectability are shown to be necessary and sufficient conditions for the existence of a positive semidefinite solution to the algebraic Riccati equation which has the following properties (i) it is the unique positive semidefinite solution to the algebraic Riccati equation, (ii) it is converged to geometrically in the operator norm by the solution to the discrete Riccati equation from any positive semidefinite initial condition, (iii) the associated closed loop system converges uniformly geometrically to zero and solves the regulator problem, and (iv) the steady state Kalman–Bucy filter associated with the solution to the algebraic Riccati equation is uniformly asymptotically stable in the large. These stability results are then generalized to time-varying problems; also it is shown that even in infinite dimensions, controllability implies stabilizability.

Analysis and design of linear periodic control systems are closely related to the discrete-time periodic matrix equations. In this paper, we propose an iterative algorithm based on the conjugate gradient method on the normal equations (CGNE) for finding the solution group of the general coupled periodic matrix equations $$\begin{aligned} \left\{ \begin{array}{l} A_{1,i}X_iB_{1,i}+C_{1,i}X_{i+1}D_{1,i}=E_{1,i},\\ A_{2,i}X_iB_{2,i}+C_{2,i}X_{i+1}D_{2,i}=E_{2,i}, \end{array} \right. ~~~\mathrm {for}~~~i=1,2,3,\ldots . \end{aligned}$$ A 1 , i X i B 1 , i + C 1 , i X i + 1 D 1 , i = E 1 , i , A 2 , i X i B 2 , i + C 2 , i X i + 1 D 2 , i = E 2 , i , for i = 1 , 2 , 3 , ? . By proving some properties of the algorithm, we show that the solution group of the periodic matrix equations can be obtained within a finite number of iterations in the absence of roundoff errors. Numerical examples are given to illustrate the efficiency and accuracy of the proposed algorithm.

Landslide susceptibility mapping is well recognized as an essential element in supporting decision-making activities for preventing and mitigating landslide hazards as it provides information regarding locations where landslides are most likely to occur. The main purpose of this study is to produce a landslide susceptibility map of Mt. Umyeon in Korea using an artificial neural network (ANN) involving the factor selection method and various non-linear activation functions. A total of 151 historical landslide events and 20 predisposing factors consisting of Geographic Information System (GIS)-based morphological, hydrological, geological, and land cover datasets were constructed with a resolution of 5 x 5 m. The collected datasets were applied to information gain ratio analysis to confirm the predictive power and multicollinearity diagnosis to ensure the correlation of independence among the landslide predisposing factors. The best 11 predisposing factors that were selected in this study were randomly divided into a 70:30 ratio for training and validation datasets, which were used to produce ANN-based landslide susceptibility models. The ANN model used in this study had a multi-layer perceptron (MLP) structure consisting of an input layer, one hidden layer, and an output layer. In the output layer, the logistic sigmoid function was used to represent the result value within the range of 0 to 1, and six non-linear activation functions were used for the hidden layer. The performance of the landslide susceptibility models was evaluated using the receiver operating characteristic curve, Kappa index, and five statistical indices (sensitivity, specificity, accuracy, positive predictive value (PPV), negative predictive value (NPV)) with the training dataset. In addition, the landslide susceptibility models were validated using the aforementioned measures with the validation dataset and were compared using the Friedman test to check the significant differences among the six developed models. The optimal number of neurons was determined based on the aforementioned performance evaluation and validation results. Overall, the model with the best performance was the MLP model with the logistic sigmoid activation function in the output layer and the hyperbolic tangent sigmoid activation function with five neurons in the hidden layer. The validation results of the best model showed a sensitivity of 82.61%, specificity of 78.26%, accuracy of 80.43%, PPV of 79.17%, NPV of 81.82%, a Kappa index of 0.609, and AUC of 0.879. The results of this study highlight the effectiveness of selecting an optimal MLP model structure for shallow landslide susceptibility mapping using an appropriate predisposing factor section method.

Presents a novel neural network which works as a classifier. It uses Euclidean distance similarity measurement to form clusters which are represented by output units. Uniquely, output units in the proposed network have nonlinear hard-limiter activation functions. Through this nonlinear activation function, complex decision boundaries from input patterns can be approximated. Furthermore, it does not forget previously remembered training patterns as it remembers newly shown patterns. This is shown with illustrative proofs. Simulation results are presented and compared with those from the backpropagation neural network. They demonstrate that the network described, with its simple architecture and learning, it is able to capture continuous distributions of complex decision boundaries from discrete patterns. >

An iterative algorithm for discrete periodic Lyapunov matrix equations