Pseudo-inverse Matrix Research Articles

Higher-Order Optimality Conditions for Degenerate Unconstrained Optimization Problems

In this paper necessary and sufficient conditions of a minimum for the unconstrained degenerate optimization problem are presented. These conditions generalize the well-known optimality conditions. The new optimality conditions are presented in terms of polylinear forms and Hesse’s pseudoinverse matrix. The results are illustrated by examples.The formulation and appearance of these conditions differ from high-order optimality conditions by other authors. The suggested representation of high-order optimality conditions makes them convenient for the evaluation of the convergence rate for unconstrained optimization methods in the case of a singular minimum point, for example, for the analysis of Newton’s and quasi-Newton’s methods.

Development of a Solvability Map.

From time to time, it is necessary to determine whether there are sufficient measurements for the image reconstruction task especially when a non-standard scanning geometry is used. When the imaging system can be approximately modeled as a system of linear equations, the condition number of the system matrix indicates whether the entire system can be stably solved as a whole. When the system as a whole cannot be stably solved, the Moore-Penrose pseudo inverse matrix can be evaluated through the singular value decomposition (SVD) and then a generalized solution can be obtained. However, these methods are not practical because they require the computer memory to store the whole system matrix, which is often too large to store. Also, we do not know if the generalized solution is good enough for the application in mind. This paper proposes a practical image solvability map, which can be obtained for any practical image reconstruction algorithm. This image solvability map measures the reconstruction errors for each location using a large number of computer-simulated random phantoms. In other words, the map is generated by a Monte Carlo approach.

Nonequilibrium molecular dynamics method based on coarse-graining formalism: Application to a nonuniform temperature field system.

A nonequilibrium molecular dynamics method is proposed to produce nonequilibrium states flexibly. In this method, virtual points are set in a simulation box, and coarse-grained physical quantities at these points are constrained using Gauss's principle of least constraint. The coarse-grained physical quantities are evaluated by averaging microscopic quantities with an appropriate weight. To obtain the weight to evaluate the coarse-grained physical quantities, a shape function matrix is initially constructed from the particle configuration. This matrix expresses an interpolation of the physical quantities at particle positions from the coarse-grained quantities at the virtual points. Then, a matrix form of the weight is calculated as the Moore-Penrose pseudoinverse matrix for the shape function matrix. This method is applied to constrain the coarse-grained kinetic energy and produce a nonuniform temperature field in the system. The temperature profile at a nonequilibrium steady state depends on the method for constructing the shape function matrix. In particular, a local temperature coincides with the coarse-grained temperature when the shape function matrix is constructed based on a higher-order interpolation.

A new approach for displacement and stress monitoring of tunnel based on iFEM methodology

Structural monitoring plays a key role for underground structures such as tunnels. Strain readings are expected to report structural conditions during construction and at the final delivery of the works. Furthermore, it is increasingly requested an extension to long-term monitoring from contractors with possible use of the same system in service during construction. A robust and efficient monitoring methodology from discrete strain measurements is the inverse finite element method (iFEM), which allows to reconstruct the structural response without input data on the load pattern applied to the structure as well as material and inertial properties of the elements and therefore it is interesting for structural configurations affected by uncertain loading conditions, such as the tunnel. The formulation presented in this paper, based on the iFEM theory, is improved from the previous work available in literature for both the shape functions used and the computational procedure. Indeed, the approach allows to overcome inconsistencies related to structural loading conditions and a pseudo-inverse matrix preserve all the rigid body modes without imposing specific constraints which is typical for tunnels. Numerical validation of the iFEM procedure is performed by simulating the input data coming from a tunnel working in a heterogeneous soil under different loading conditions with direct FEM analysis.

Statement of the problem of assessing instability of passive quantum frequency standards in the presence of an error from the interaction

Construction of modern measuring complexes of the coordinate-time support system of Ukraine is impossible without improving mathematical models of quantum standards of frequency (QSF) used in group standards. This work is devoted to the analysis of methods for the stable solution of direct and inverse problems (methods for solving ill-posed problems) in models of the interaction of passive QSF in the process of their comparisons. The priority task is to use these methods for the numerical solution of problems in the design of group QSF and parallel quantum generators of random numbers. Methods for solving such problems are in demand, since they make it possible to create mathematical models of group QSF interaction. These models will enable the design of efficient parallel quantum random number generation devices for high-tech areas of cybersecurity.
 Varieties of methods such as the method of least squares or the method of the Moore-Penrose pseudo-inverse matrix are used, as a rule when evaluating the metrological parameters of QSF.Robust methods of regularization or filtering, for example, the Kalman or Wiener filter method, are used in the algorithms of group standards, due to the instability of the solution.However, these methods do not work in the presence of an error from the interaction of QSF in the process of their functioning in a group standard or in comparisons.
 The aim of this work is to analyze and substantiate the formulation of the problem of assessing the potential accuracy characteristics of passive QSF in the presence of an error from the interaction. Regularization parameters when determining the state vector of the group standard are found using signals transmitted by global navigation satellite systems such as GPS\GLONASS in the local differential correction mode.

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ КОЛЕБАНИЙ НЕОДНОРОДНЫХ КОЛЬЦЕВЫХ УЛЬТРАЗВУКОВЫХ ВОЛНОВОДОВ

The article describes technique for modelling of ultrasonic vibrations amplifiers, which are implemented in the form of non-uniform ring-shaped waveguides, based on application of harmonic balance method. Bending vibrations of the waveguide are described by means of non-uniform integral and differential equations equivalent to Euler–Bernoulli equations in order to simplify calculation of amplitude-frequency characteristics of vibrations, particularly, to exclude the need of working with singular matrices. Using harmonic balance method, equations of vibrations are reduced to overdetermined non-uniform linear system of algebraic equations, which least-squares solution is determined by means of pseudo-inverse matrix. On the basis of analysis of numerical example possibility of existence of variable-sign and constant-sign vibration modes of the waveguide is shown and it is determined that for realization of amplifying function it is necessary to use waveguide at constant-sign vibration mode. The constant-sign vibration modes are combinations of bending defor-mation and extensional deformation of central line of the waveguide and they are detected due to accounting extensibility of the central line in equations of vibrations. Validity of the obtained results is confirmed by comparing them to the results of modelling by means of finite element method.

Plane-Wave Synthesis Employing Propagating Plane-Wave Expansion for 3-D and 2-D RCS Prediction Including the Multiple Scattering Effects

The plane-wave synthesis (PWS) methods are proposed to predict the 3-D and 2-D RCS accurately with multiple scattering effects. The 3-D method employs the propagating plane-wave expansion and can be implemented identically as the near-field far-field transformation (NFFFT). Either the single-level or multilevel implementation can be applied, and in the single-level implementation, a single pseudo-inverse matrix can be used to synthesize plane waves propagating various incident directions. In addition to the 3-D formulation, the single-cut variation of the method (SC-PWS) is also presented for the 2-D RCS prediction. This method is based on the 2-D propagating plane-wave expansion and formulated employing the approximations on the 3-D Green's function. The numerical and experimental results are presented for the PWS and the SC-PWS.

Решение систем линейных алгебраических уравнений с использованием выборочной регуляризации

The solution of systems of linear algebraic equations, which matrices can be poorly conditioned or singular is considered. As a solution method, the original matrix is decomposed into triangular components by Gauss or Chole-sky with an additional operation, which consists in increasing the small or zero diagonal terms of triangular matrices during the decomposition process. In the first case, the scalar products calculated during decomposition are divided into two positive numbers such that the first is greater than the second, and their sum is equal to the original one. In further operations, the first number replaces the scalar product, as a result of which the value of the diagonal term increases, and the second number is stored and used after the decomposition process is completed to correct the result of calculations. This operation increases the diagonal elements of triangular matrices and prevents the appearance of very small numbers in the Gauss method and a negative root expression in the Cholesky method. If the matrix is singular, then the calculated diagonal element is zero, and an arbitrary positive number is added to it. This allows you to complete the decomposition process and calculate the pseudo-inverse matrix using the Greville method. The results of computational experiments are presented.

Inverse-Free Discrete ZNN Models Solving for Future Matrix Pseudoinverse via Combination of Extrapolation and ZeaD Formulas.

Time-varying matrix pseudoinverse (TVMP) problem has been investigated by many researchers in recent years, but a new class of matrix termed Zhang matrix has been found and not been handled by some conventional models, e.g., Getz-Marsden dynamic model. On the other way, future matrix pseudoinverse (FMP), as a more challenging and intractable discrete-time problem, deserves more attention due to its significant role-playing on some engineering applications, such as redundant manipulator. Based on the zeroing neural network (ZNN), this article concentrates on designing new discrete ZNN models appropriately for computing the FMPs of all matrices of full rank, including the Zhang matrix. First, an inverse-free continuous ZNN model for computing TVMP is derived. Subsequently, Zhang et al. discretization (ZeaD) formulas and equidistant extrapolation formulas are used to discretize the continuous ZNN model to two discrete ZNN models for computing FMPs with different truncation errors. The numerical experiments are conducted for the five conventional discrete models and two new discrete ZNN models. Distinct numerical results substantiate the effectiveness and choiceness of newly proposed models. Finally, one of the newly proposed models is implemented on simulating and physical instances of robot manipulators, respectively, to show its practicability.

Adaptive Reinforcement Learning-Enhanced Motion/Force Control Strategy for Multirobot Systems

This paper presents an adaptive reinforcement learning- (ARL-) based motion/force tracking control scheme consisting of the optimal motion dynamic control law and force control scheme for multimanipulator systems. Specifically, a new additional term and appropriate state vector are employed in designing the ARL technique for time-varying dynamical systems with online actor/critic algorithm to be established by minimizing the squared Bellman error. Additionally, the force control law is designed after obtaining the computation of constraint force coefficient by the Moore–Penrose pseudo-inverse matrix. The tracking effectiveness of the ARL-based optimal control is verified in the closed-loop system by theoretical analysis. Finally, simulation studies are conducted on a system of three manipulators to validate the physical realization of the proposed optimal tracking control design.

Nonlinear Control of Quadrotor for Fault Tolerance: A Total Failure of One Actuator

This paper deals with the problem of a quadrotor experiencing a total failure of one actuator. First, a nonlinear mathematical model for the faulty quadrotor is derived with three control inputs and six outputs, which includes the translational and rotational dynamics. Because of the limited inputs, the controllability of the yaw state is sacrificed, and the control inputs are reallocated to the other three healthy rotors. Second, a nonlinear controller is designed based on the proposed model. This presented controller includes two subcontrollers: 1) a roll angle, pitch angle, and altitude subcontroller and 2) a horizontal position subcontroller. In the controller design, the Moore–Penrose pseudoinverse of the coefficient matrix is applied to overcome the singularity of the roll angle. Third, the stability of both of the designed subcontrollers is verified by the Lyapunov stability theorem, and the convergence of the designed subcontrollers is analyzed. Finally, simulation results are provided to verify the effectiveness of the proposed model and the designed controller.

Particular properties of estimation of partial capacitances of insulation of three core power cables by applying aggregate measurements

This paper presents a description of specific properties of determining the values of partial capacitances of insulation gaps in power cables with paper insulation for various ways of forming and solving the system of linear algebraic equations. Possible ways of inspection the insulation of three core power cables for the estimation of values of partial capacitances by applying aggregate measurements which are based on various ways of connection of emittance meter to tested sample of power cable are given. Estimation of partial capacitances by the direct solution of a system of linear algebraic equations, by minimizing the root mean square error of solving an overdetermined system of equations by the least squares method, as well as by finding a normal solution of an indefinite system of equations by the pseudo-inverse matrix, is also considered. It is shown that minimization of the root mean square error by the least squares method and the direct solution of system of equations show quite similar results for the case of estimation of partial capacitances by means of aggregate measurements, at the same time the solution of an indefinite system of equations by the method of a pseudo-inverted matrix allows to reproduce rather accurately only 3 out of 6 values of partial capacitances. The uneven effect of frequency on the electrical capacitance of the insulation gaps between the cores of the power cable and between its cores and the sheath is shown. It was proposed to use the frequency dependence of the electrical capacitance of insulation gaps as an informative parameter about the technical state of insulating gaps between the cores of the power cable and between its cores and its sheath.
 
 Keywords: root mean square error; least squares method; system of linear algebraic equations; dielectric losses; dielectric permittivity.

Multivariable Adaptive Control Method for Turbofan Engine With Dynamic and Input Uncertainties

Abstract The dynamic characteristics of the turbofan engine vary greatly in the full flight envelope, which makes the problem of dynamic uncertainty and input uncertainty very prominent. This brings different degrees of performance impact to the engine control system and even makes it lose stability. This paper proposes a multivariable adaptive control method for dealing with multivariable dynamic uncertainty and input uncertainty. First, for the case where the number of state variables is less than or equal to the number of input variables and the input matrix satisfies the full-rank condition of the row, the existence of the right pseudo-inverse matrix of the input matrix can be guaranteed. Then, the dynamic uncertainty and input uncertainty can be mathematically converted into standard matched uncertainty. Second, the Lyapunov quadratic function is constructed by the closed-loop tracking error vector and the adaptively adjustable control parameter estimation errors, and the Lyapunov stability constraint is considered. Then, under the premise of considering the dynamic characteristics of the actuator, an adaptive control algorithm for multivariable uncertainty model of turbofan engine is derived. Finally, ground and high-altitude simulations are carried out on the dual-loop control system based on the nonlinear dynamic model of the turbofan engine. The results show that the control system has robust stability and anti-interference performance for dynamic uncertainty and input uncertainty of turbofan engine in the full flight envelope. By introducing stronger parameter change rate information to the controller, its performance can be further improved, and the transient state control is more stable.

Fast OMP algorithm and its FPGA implementation for compressed sensing‐based sparse signal acquisition systems

Compressed sensing-based radio frequency signal acquisition systems call for higher reconstruction speed and low dynamic power. In this study, a novel low power fast orthogonal matching pursuit (LPF-OMP) algorithm is proposed for faster reconstruction of sparse signals from their compressively sensed samples and the reconstruction circuit consumes very low dynamic power. The searching time to find the best column is reduced by reducing the number of columns to be searched in successive iterations. A novel architecture of the proposed LPF-OMP algorithm is also presented here. The proposed architecture is implemented on field programmable gate array for demonstrating the performance enhancement. Computation of pseudoinverse in OMP is avoided to save time and storage requirement to store the pseudoinverse matrix. The proposed design incorporates a novel strategy to stop the algorithm without consuming any extra circuitry. A case study is carried out to reconstruct the RADAR test pulses. The design is implemented for K = 256, N = 1024 using XILINX Virtex6 device and supports maximum of K/4 iterations. The proposed design is faster, hardware efficient and consumes very less dynamic power than the previous implementations of OMP. In addition, the proposed implementation proves to be efficient in reconstructing low sparse signals.

A Vary-Parameter Convergence-Accelerated Recurrent Neural Network for Online Solving Dynamic Matrix Pseudoinverse and its Robot Application

Among this study, a vary-parameter convergence-accelerated neural network (VPCANN) model is generalized to solving dynamic matrix pseudoinverse, which can achieve super exponential convergence and noise-resistant, compared to the traditional Zhang neural network (ZNN) designed for dynamic problems. Simulative experiments reveal that the neural state solutions synthesized by the VPCANN can quickly approach to the theoretical pseudoinverse. Moreover, based on three types of noise disturbance including constant noise, random noise and dynamic noise, comparisons between the VPCANN and ZNN model are also investigated, verifying noise-resistant of the VPCANN model is better than the ZNN. In addition, to show the potential application of the VPCANN in practice, the kinematic motion planning of a six-links robot manipulator is considered, further substantiating the efficacy of the VPCANN in the dynamic matrix pseudoinverse.

Neural-dynamics-enabled Jacobian inversion for model-based kinematic control of multi-section continuum manipulators

Continuum manipulators are a new generation of robotic systems that possess infinite number of degrees of freedom associated with inherent compliance, unlike traditional robotic manipulators which consist of a finite number of rigid links. Because of this characteristic, controlling continuum manipulators is more complicated and difficult based on only traditional control theory. Soft computing techniques are solid alternative for improving the control performance of such kinds of robots. In this paper, we employ two types of neural dynamic approaches, i.e., gradient neural dynamics and zeroing neural dynamics, to solve the real-time Jacobian matrix pseudo-inversion problem, thereby achieving model-based kinematic control of multi-section continuum manipulators. Different kinds of neural dynamic models are investigated and their performances in terms of tracking accuracy are shown with and without noise disturbances. Simulation validations with a two-section and a three-section continuum manipulator demonstrate the feasibility and robustness of the proposed models.

ПОСТРОЕНИЕ МОДЕЛИ ДВИЖЕНИЯ СУДНА ПО КУРСУ НА ОСНОВЕ ПСЕВДООБРАТНЫХ МАТРИЦ ИЗМЕРЕНИЙ ВЕКТОРА СОСТОЯНИЯ УПРАВЛЯЕМОГО ОБЪЕКТА

To configure the control systems of the ship movement along the trajectory, it is necessary to be concerned with the parameters of its controllability. There has been proposed building 
 a matrix model based on measurements of the state vector of a controlled object. The construction of the model is considered on the example of the problem of ship course control. An algorithm for determining the matrix coefficients of the selected model is proposed. The operation of the considered algorithm has been checked for square matrices by finding their inverse matrices, as well as for rectangular matrices for which the pseudo inverse matrix was found. The illustration of the proposed approach is carried out using the example of a simple 1-order linear Nomoto model. The considered approach is quite universal and can be applied to higher order models, including nonlinear ones.

Fault-tolerant control approach based on constraint control allocation for 4WIS/4WID vehicles

The four-wheel independently driven and steered electric vehicle is a promising vehicle model having a strong potential for handling stability, flexibility, and consumption reduction. However, failure of the actuators of 4WIS/4WID vehicles could lead to performance reduction and dangerous accidents owing to their complex system. A fault-tolerant control approach is adopted in the integrated chassis controller such that the autonomously driven vehicle maintains its safety and stability while actuator failures occur. A linear quadratic regulator is utilized to track the reference path by adjusting the total forces and moment. To resolve any actuator failures, a control allocation method based on the pseudo-inverse matrix is introduced for decoupling the forces and moment based on the current state of the tires with cycle and correction. In the actuator control layer, the desired forces of the tires are achieved by regulating the steering angles and driving torques based on the inverse tire models of normal and flat tires. Three sets of experiments are used to test the efficiency of the proposed method when applied to a 4WIS/4WID vehicle. The results demonstrate that the proposed fault-tolerant control method can greatly improve the tracking performance and stability of 4WIS/4WID vehicles under conditions of actuator failures.

Синтез H-inf регуляторов на конечном промежутке времени

The article considers a problem of stabilization of an aircraft, which can be solved by finding output control. Sufficient conditions for H-inf- output control for linear dynamic systems are obtained. Two approximate methods of closed-loop systems synthesis are proposed. The first one is based on finding a pseudo-inverse matrix of the linear measuring system model. The second one based on synthesizing an asymptotic observer, which allows to formulate an estimate of a state vector, giving a possibility of further use of the obtained estimate in the control law. The paper contains a step-by-step algorithm of finding an H-inf -control of a closed-loop system.

Regression assessment of the model based on the experimental planning matrix in composite materials’ analysis problems

An automated coefficients calculation of the regression model based on the experimental data in the form of a planning matrix is considered. The calculations are based on polynomial regression with possible consideration of the interaction effects between its factors. The application of the Moore – Penrose pseudoinverse matrix for determining the regression equation coefficients is shown. The choice of the calculated planning matrix is carried out taking into account the determination coefficient value and the factors’ calculated matrix rank. Calculation verification is carried out using the rank correlation between the experimental and calculated response functions. The experimental part of the work is given to determine the dependence of fungal resistance and fungicidal properties of filled cement composites on the type and grain size composition of the filler.