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1

Fast computational approach to the Levenberg-Marquardt algorithm for training feedforward neural networks

Bilski Jarosław, Smoląg Jacek, Kowalczyk Bartosz, Grzanek Konrad, Izonin Ivan

Journal of Artificial Intelligence and Soft Computing Research

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2023

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Vol. 13, No. 2

45--61

EN

This paper presents a parallel approach to the Levenberg-Marquardt algorithm (LM). The use of the Levenberg-Marquardt algorithm to train neural networks is associated with significant computational complexity, and thus computation time. As a result, when the neural network has a big number of weights, the algorithm becomes practically ineffective. This article presents a new parallel approach to the computations in Levenberg-Marquardt neural network learning algorithm. The proposed solution is based on vector instructions to effectively reduce the high computational time of this algorithm. The new approach was tested on several examples involving the problems of classification and function approximation, and next it was compared with a classical computational method. The article presents in detail the idea of parallel neural network computations and shows the obtained acceleration for different problems.

2

Smoothing Levenberg-Marquardt algorithm for solving non-Lipschitz absolute value equations

Yilmaz Nurullah, Kayacan Aysegul

Journal of Applied Analysis

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2023

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Vol. 29, nr 2

277--286

EN

In this study, we concentrate on solving the problem of non-Lipschitz absolute value equations (NAVE). A new Bezier curve based smoothing technique is introduced and a new Levenberg-Marquardt type algorithm is developed depending on the smoothing technique. The numerical performance of the algorithm is analysed by considering some well-known and randomly generated test problems. Finally, the comparison with other methods is illustrated to demonstrate the efficiency of the proposed algorithm.

3

Forward and inverse kinematics solution of a robotic manipulator using a multilayer feedforward neural network

Sharkawy Abdel-Nasser

Journal of Mechanical and Energy Engineering

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2022

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Vol. 6, No 2

1--17

EN

In this paper, a multilayer feedforward neural network (MLFFNN) is proposed for solving the problem of the forward and inverse kinematics of a robotic manipulator. For the forward kinematics solution, two cases are presented. The first case is that one MLFFNN is designed and trained to find solely the position of the robot end-effector. In the second case, another MLFFNN is designed and trained to find both the position and the orientation of the robot end-effector. Both MLFFNNs are designed considering the joints’ positions as the inputs. For the inverse kinematics solution, a MLFFNN is designed and trained to find the joints’ positions considering the position and the orientation of the robot end-effector as the inputs. For training any of the proposed MLFFNNs, data is generated in MATLAB using two different cases. The first case is that data is generated assuming an incremental motion of the robot’s joints, whereas the second case is that data is obtained with a real robot considering a sinusoidal joint motion. The MLFFNN training is executed using the Levenberg-Marquardt algorithm. This method is designed to be used and generalized to any DOF manipulator, particularly more complex robots such as 6-DOF and 7-DOF robots. However, for simplicity, this is applied in this paper using a 2-DOF planar robot. The results show that the approximation error between the desired output and the estimated one by the MLFFNN is very low and it is approximately equal to zero. In other words, the MLFFNN is efficient enough to solve the problem of the forward and inverse kinematics, regardless of the joint motion type.

4

An improved stress-dependent model for magnetomechanical effect simulation of Terfenol-D rods

Nait Ouslimane Ahmed, Mohellebi Hassane, Phan Quang Anh

Przegląd Elektrotechniczny

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2021

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R. 97, nr 1

115--120

EN

Terfenol-D is one of the smart materials widely used in the fabrication of magnetostriction based sensors and actuators due to its high material properties. However, using Terfenol-D in industrial applications rely on the ability of predicting its hysteresis by mathematical models. In this paper, we present an improved hysteresis model for reproducing hysteresis curves of Terfenol-D. Levenberg–Marquardt algorithm is used to estimate the optimal parameters of the improved model. The simulation and experimental results show the performances of the proposed model.

XX

Artykuł zajmuje się Terfenolem-D – dość powszechnie stosowanym materiałem magnetostrykcyjnym. Niestety dotychczas brakowało matematycznego modelu tego materiału uwzględniającego histerezę. Wykorzystano algorytm Levenberg–Marquardt do bardziej szczegółowego opisu parametrów Terfenolu.

5

Local Levenberg-Marquardt algorithm for learning feedforwad neural networks

Bilski Jarosław, Kowalczyk Bartosz, Marchlewska Alina, Zurada Jacek M.

Journal of Artificial Intelligence and Soft Computing Research

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2020

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Vol. 10, No. 4

299--316

EN

This paper presents a local modification of the Levenberg-Marquardt algorithm (LM). First, the mathematical basics of the classic LM method are shown. The classic LM algorithm is very efficient for learning small neural networks. For bigger neural networks, whose computational complexity grows significantly, it makes this method practically inefficient. In order to overcome this limitation, local modification of the LM is introduced in this paper. The main goal of this paper is to develop a more complexity efficient modification of the LM method by using a local computation. The introduced modification has been tested on the following benchmarks: the function approximation and classification problems. The obtained results have been compared to the classic LM method performance. The paper shows that the local modification of the LM method significantly improves the algorithm’s performance for bigger networks. Several possible proposals for future works are suggested.

6

Modelling of back propagation neural network to predict the thermal performance of porous bed solar air heater

Ghritlahre Harish Kumar, Prasad Radha Krishna

Archives of Thermodynamics

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2019

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Vol. 40, no 4

103--128

EN

The objective of present work is to predict the thermal performance of wire screen porous bed solar air heater using artificial neural network (ANN) technique. This paper also describes the experimental study of porous bed solar air heaters (SAH). Analysis has been performed for two types of porous bed solar air heaters: unidirectional flow and cross flow. The actual experimental data for thermal efficiency of these solar air heaters have been used for developing ANN model and trained with Levenberg-Marquardt (LM) learning algorithm. For an optimal topology the number of neurons in hidden layer is found thirteen (LM-13).The actual experimental values of thermal efficiency of porous bed solar air heaters have been compared with the ANN predicted values. The value of coefficient of determination of proposed network is found as 0.9994 and 0.9964 for unidirectional flow and cross flow types of collector respectively at LM-13. For unidirectional flow SAH, the values of root mean square error, mean absolute error and mean relative percentage error are found to be 0.16359, 0.104235 and 0.24676 respectively, whereas, for cross flow SAH, these values are 0.27693, 0.03428, and 0.36213 respectively. It is concluded that the ANN can be used as an appropriate method for the prediction of thermal performance of porous bed solar air heaters.

7

Vehicle suspension control based on PID controller adapted by feed-forward neural network

Popardovský V., Popardovský E.

Zeszyty Naukowe Instytutu Pojazdów / Politechnika Warszawska

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2017

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z. 4/113

99--107

EN

Control of suspension systems in vehicles, brings the opportunity react to unknown road conditions (road inequalities) to preserve comfort of crew, driving safety and also to prevent excessive mechanical stress of useful load. The goal of this study is to design an adaptive PID controller with using neural network (used as „tuner“ for PID), for control of active vehicle suspension system. For this purpose, an ANN is designed to produce outputs for PID controller - Proportional (P), Integral (I) and Derivative (D) parameters. By this way was designed ANNAPID controller for vehicle suspension control.

PL

Sterowanie zawieszeniem w pojazdach daje możliwość reagowania na nieznane warunki drogowe (nierówności drogowe) w celu zachowania komfortu załogi, bezpieczeństwa jazdy, a także w celu uniknięcia nadmiernego obciążenia mechanicznego przy obciążeniu użytkowym. Celem tego badania jest opracowanie adaptacyjnego regulatora PID z wykorzystaniem sieci neuronowych (wykorzystywanej jako "tuner" dla PID), do sterowania aktywnym systemem zawieszenia pojazdu. W tym celu zaprojektowano sieć neuronową ANN, która na wyjściu zadaje sygnały sterujące dla regulatora PID, czyli odpowiednie parametry dla członów: proporcjonalnego (P), całkującego (I) i różniczkującego (D). W ten sposób zaprojektowano sterownik ANNAPID do sterowania zawieszeniem pojazdu.

8

Maximum likelihood estimation for identification of aircraft aerodynamic derivatives

Lichota P., Lasek M.

Archive of Mechanical Engineering

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2013

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Vol. LX, nr 2

219--230

EN

This article investigates identification of aircraft aerodynamic derivatives. The identification is performed on the basis of the parameters stored by Flight Data Recorder. The problem is solved in time domain by Quad-M Method. Aircraft dynamics is described by a parametric model that is defined in Body-Fixed-Coordinate System. Identification of the aerodynamic derivatives is obtained by Maximum Likelihood Estimation. For finding cost function minimum, Lavenberg-Marquardt Algorithm is used. Additional effects due to process noise are included in the state-space representation. The impact of initial values on the solution is discussed. The presented method was implemented in Matlab R2009b environment.

PL

Artykuł zawiera informacje na temat identyfikacji pochodnych aerodynamicznych. Estymacja opiera się o parametry zapisywane przez Pokładowy Rejestrator Lotu. Zagadnienie jest rozważane w dziedzinie czasu przy użyciu podejścia Quad-M. Do opisu dynamiki samolotu wykorzystano model parametryczny zdefiniowany w układzie sztywno związanym z samolotem. Do identyfikacji wykorzystano Metodę Największej Wiarygodności. Do znalezienia minimum funkcji celu użyto algorytm Levenberga-Marquardta. W modelu uwzględniono wpływ dodatkowych czynników reprezentowany przez szum przetwarzania. Omówiono wpływ wartości początkowych na rozwiązanie. Prezentowane wyniki uzyskano w środowisku Matlab R2009b.

9

Stable gait synthesis and analysis of a 12 - degree of freedom eiped robot in sagittal and frontal planes

Sudheer A. P., Vijayakumar R., Mohandas K. P.

Journal of Automation Mobile Robotics and Intelligent Systems

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2012

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Vol. 6, No. 4

36-44

EN

Legged machines have not been offered biologically realistic movement patterns and behaviours due to the limitations in kinematic, dynamics and control technique. When the degrees of freedom (DOF) increases, the robot becomes complex and it affects the postural stability. A loss of postural stability of biped may have potentially serious consequences and this demands thorough analysis for the better prediction and elimination of the possibility of fall. This work presents the modelling and simulation of twelve degrees of freedom (DOF) biped robot, walking along a pre-defined trajectory after considering the stability in sagittal and frontal planes based upon zero moment point (ZMP) criterion. Kinematic modelling and dynamic modelling of the robot are done using Denavit-Hartenberg (DH) parameters and Newton-Euler algorithm respectively. This paper also proposes Levenberg- Marquardt method for finding inverse kinematic solutions and determines the size of the foot based on ZMP for the stable motion of biped. Biped robot locomotion is simulated, kinematic and dynamic parameters are plotted using MATLAB. Cycloidal gait trajectory is experimentally validated for a particular step length of the biped.

10

Analiza metody iteracyjnej minimalizacji w regularyzacji estymacji nieliniowej

Polak A.

Przegląd Elektrotechniczny

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2010

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R. 86, nr 9

27-30

PL

Pomiary pośrednie często polegają na estymacji parametrów modelu badanego obiektu, a proces estymacji może być źle uwarunkowana numerycznie. W celu poprawy uwarunkowania numerycznego stosowane są metody regularyzacji. Jednym z ostatnio zaproponowanych podejść do regularyzacji estymacji nieliniowej jest metoda iteracyjnej minimalizacji (IM), regulującej balans pomiędzy błędem systematycznym i losowym pomiaru pośredniego. Celem prezentowanych badań było porównanie doboru parametru regularyzacji za pomocą IM z powszechnie stosowną metodą Marquardta. W badaniach wykorzystano syntetyczne dane pomiarowe oraz metodę Monte Carlo. Z przeprowadzonych symulacji wynika, że algorytm IM ma lepsze właściwości metrologicznie niż algorytm Marquardta.

EN

Indirect measurements often amount to the estimation of parameters of a mathematical model that describes the object under investigation, and this process may numerically be ill conditioned. Various regularization techniques are used to solve the problem. The iterative minimisation (IM) is one of approaches proposed recently. It regulates the balance between systematic and random error of indirect measurement. The purpose of this study was to compare the selection of the regularisation parameter by IM with the commonly used Marquardt method. Synthetically generated measurement data and the Monte Carlo method were used to this end. From the performed simulations it stems that the IM algorithm has better metrological properties than Marquardt's one.

11

Error Analyses of Attitude and Heading Reference Systems

Reinstein M., Sipos M., Rohac J.

Przegląd Elektrotechniczny

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2009

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R. 85, nr 8

114-118

EN

This paper describes the results of error analyses of two low-cost Attitude and Heading Reference Systems (AHRS). These error analyses concern both random sensor errors identified by Allan variance method and deterministic errors estimated during a calibration procedure. The calibration procedure is based on the Levenberg-Marquardt algorithm used to solve non-linear least squares estimation problem. The main contribution of this paper is to present data necessary for further inertial sensors signal processing by means of Kalman filtering.

PL

Artykuł opisuje rezultaty analizy błędu dwóch systemów nawigacji. Analiza koncentruje się na błędach czujników: przypadkowych zdefiniowanych przez wariancję Allana i systematycznych, określonych w procedurze kalibracji. Procedura kalibracji bazuje na algorytmie Levenberga-Marquardta stosowanym do rozwiązywania problemów estymacji metodą nieliniowych najmniejszych kwadratów. Głównym celem artykułu jest podanie danych niezbędnych do późniejszego przetwarzania sygnału czujników bezwładnościowych przy wykorzystaniu filtrów Kalmana.

12

Levenberg-Marquardt algorithm for optimization of mathematical models according to minimax objective function of measurement systems

Tomczyk K.

Metrology and Measurement Systems

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2009

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Vol. 16, nr 4

599-606

EN

The paper presents an application of Levenberg-Marquardt algorithm to parametric optimization of the minimax type of measurement systems. For the assumed objective function given by the integral square error, optimization of the third-order model is carried out and explained in detail. The optimization procedure is realized in three stages. The optimization method presented in the paper can find broad application in the process of determining optimum models of systems, especially for those that operate in dynamic states.

13

Regularyzacja identyfikacji obiektów złożonych opisanych modelami nieliniowymi

Polak A. G., Mroczka J.

Pomiary Automatyka Kontrola

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2007

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R. 53, nr 9 bis

190--193

PL

W referacie przedstawiono modyfikację regularyzacji Tichonowa-Phillipsa (TP) przystosowującą ją do estymacji parametrów modeli nieliniowych źle uwarunkowanych numerycznie. Zaproponowane podejście porównano z metodami Gaussa-Newtona (GN), Levenberga-Marquardta (LM) oraz metodą łączącą LM z TP (LMTP). Analizę właściwości czterech zaimplementowanych algorytmów przeprowadzono metodą Monte Carlo. Pokazała ona, że w przypadku identyfikacji modeli nieliniowych zawierających parametry słabo określone przez dane pomiarowe i jednocześnie charakteryzujące się "regularnym" rozkładem wartości w wektorze parametrów, najlepsze wyniki daje estymacja metodą Tichonowa-Phillipsa.

EN

In the paper a modification of the Tikhonov-Phillips regularisation enabling the estimation of parameters of numerically ill-conditioned nonlinear models is presented. This approach was compared with the Gauss-Newton (GN) and Levenberg-Marquardt (LM) methods, as well as with a method combining LM with TP one (LMTP). The analysis of the four computer-implemented algorithms was done by the Monte Carlo simulations. They have shown that the result of identification of a nonlinear model possessing weakly defined, however "regularly" distributed parameters, is the most accurate when using the Tikhonov-Phillips method.

14

Direct torque control for induction machines using neural networks

Messaif I., Berkouk E-M., Saadia N.

Archives of Control Sciences

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2007

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Vol. 17, no. 1

93-104

EN

In this work, a novel switching vector selector in Direct Torque Control of an induction machine using Artificial Neural Network is studied. In the first part, we describe design of a speed sensor-less Direct Torque Control (DTC) strategy of an induction motor supplied by a two-level voltage source. For this, a conventional look up table is applied which improves the performances. Due to the high computation load, this technique is not convenient for an one-line and real-time control. Thus, a simplified method of choosing the output vector for two-level voltage source inverter-fed induction machine is proposed in the second part, and a novel switching vector selector using Artificial Neural Network (ANN) is trained under the tutor of the method mentioned above. The ANN receives attention as controllers for many industrial applications. Although these networks eliminate the need for mathematical models, they require a lot of training to understand the model of plant or process. In fact, when the stator flux and electromagnetic torque are different from theirs respective references, the output vector can be expediently acquired. Simulation results showed that the ANN structure can replace successfully the conventional look up table of the DTC.