Symbolic Math Toolbox Research Articles

Introspection of UBNIN and Modified-UBNIN Algorithms for Structural MRI. Reply to Kelly et al. A Comparison of Brain-State Representations of Binary Neuroimaging Connectivity Data. Comment on "Samantaray et al. Unique Brain Network Identification Number for Parkinson's and Healthy Individuals Using Structural MRI. Brain Sci. 2023, 13, 1297".

The purpose of this reply is to address the comments given by Kelly et al. on our original paper "Unique Brain Network Identification Number for Parkinson's and Healthy Individuals using Structural MRI". We agree to the inadvertent rounding pitfall in our original paper due to the non-inclusion of symbolic math toolbox (MATLAB). We now provide the actual ranges (with decimal values) of the UBNIN values of healthy individuals and those with Parkinson's disease and further observations. Upon further introspection, we propose another variant, called Modified-UBNIN (UBNIN-MT,MN) which is highly weighted on the node with the highest network degree (i.e., connections). The italicized sentences within inverted commas are statements from Kelly et al.'s comment paper.

Паралельний алгоритм розв’язування систем інтегральних рівнянь Вольтерри II роду

The problem of increasing the effectiveness of the study of integral models of dynamic systems is considered. A parallel algorithm for solving a system of Volterra linear integral equations of the second kind based on the quadrature method of numerical integration is proposed. The algo-rithm was implemented in the MATLAB computer mathematics system in the form of an m-function. The program uses the MATLAB Distributed Computing Toolbox infrastructure to manage workflows and distribute computations between them on multi-core processors. Com-putational experiments were conducted on a model example using the Symbolic Math Toolbox package for symbolic calculations and a comparison of the execution of parallel calculations with the execution time of the implementation of a sequential algorithm. The results showed a significant increase in the speed of research of integral models on multi-core processors when using the proposed algorithm and its computer implementation.

Modeling and Trajectory Tracking Control for a Multi-Section Continuum Manipulator

This paper presents a dynamic modeling and trajectory tracking control design for a hyper-redundant continuum manipulator (HRM), which is able to continuously bend along its length. The HRM has the advantages of high flexibility, large workspace, and low inertia, but also poses challenges of complex kinematics, and dynamic coupling. To address these challenges, the kinematics model was developed based on the piecewise constant curvature (PCC) assumption that captures the relationship between the cable lengths, joint angles, and end-effector pose. Inverse kinematics is presented geometrically by solving trigonometric equations. The dynamic model for the continuum robot (CR) is derived using the Euler–Lagrange representation, which incorporates the effects of gravity and elasticity. Accordingly, three control strategies are developed and applied to a two-section continuum robot, which are the inverse dynamic proportional integral derivative controller (PID), fuzzy logic controller (FLC), and sliding mode controller (SMC). We validate our proposed methods through simulations on various 3D trajectories, utilized MATLAB symbolic math toolbox in conjunction with Simulink simulation; demonstrate and evaluate different close-loop dynamic control responses for different scenarios. This simulation used in the creation of a unique animated simulation for a whole 3D continuum robot while tracking its desired trajectories. Finally, the simulation results demonstrate that our proposed methods can effectively control the HRM with high tracking performance, exhibiting enhanced responses in terms of settling time, rising time, and steady-state error. While the developed SMC outperforms Poth's PID and FLC in terms of robustness and settling time (51% and 10.3%), respectively.

Predictive Coupled Control of the Road Train Longitudinal and Lateral Motion on a Curved Path

In this paper, we consider the problem of a road train path-following on a curved path with an optimal velocity. To solve the problem, we propose a control algorithm based on the coupled model predictive control strategy. Model predictive control assumes the computation of a control sequence by solving an optimal control problem on a finite horizon for a current state of a nonlinear time-varying system. We use the truck steering angle and road train acceleration as control inputs. We describe the road train longitudinal and lateral dynamics using an implicit nonlinear model in continuous time. To derive a discrete linear time-varying state-space prediction model describing the deviations of system dynamics from a reference path we use the Euler method to discretize the original system and compute analytical formulae for its Jacobian by MATLAB Symbolic Math Toolbox. We calculate the reference path and corresponding reference values of the state vector applying the well-known geometric techniques, which utilize the path coordinates and its curvature information. We take the reference values of a truck and a semitrailer yaw angles to be equal. Thus, the reference value of the jackknifing angle is zero. The calculations of reference velocity take into account its skid and rollover limits. To validate the proposed path-following algorithm on the road train we design a simulation model in Simulink. The paper presents the simulation results of testing the movement of a road train along a given path for various values of the reference speed. We show that the algorithm provides high enough reference path-following accuracy, vehicle reference speed tracking, and low values of the jackknifing angle on the speed values up to 18 m/s and curvature radii down to 250 m. The proposed algorithm can be used in ADAS-systems and autonomous vehicles development.

Evaluating the Challenges in Pressure – Volume – Temperature (PVT) Analysis of Gas Condensate Reservoirs

Aims: The variations in production performances of the Black oil and compositional simulation models can be evaluated by simulating oil formation volume factor (Bo), gas formation volume factor (Bg), gas-oil ratio (Rs) and volatilized oil-gas ratio (Rv). The accuracy of these two models could be assessed.
 Methodology: To achieve this objective some basic parameters were keyed into matrix laboratory (MATLAB) using the symbolic mathematical toolbox to obtain accurate Pressure Volume Temperature (PVT) properties which were used in a production and systems analysis software to generate the production performance and hydrocarbon recovery estimation. Standard black oil PVT properties for a gas condensate reservoir was simulated by performing a series of flash calculations based on compositional modeling of the gas condensate fluid at the prescribed conditions through a constant volume depletion (CVD) path. These series of calculations will be carried out using the symbolic math toolbox. PVT property values obtained from both compositional modeling and black oil PVT prediction algorithm are incorporated to determine the production performance of each method for comparison.
 Results: The absolute open flow for the black oil PVT algorithm and the compositional model for the Rs value of 500 SCF/STB and Rs value of 720SCF/STB were 130,461 stb/d and 146,028 stb/d respectively showing a 10.66% incremental flow rate.
 Conclusion: In analyzing PVT properties for complex systems such as gas condensate reservoirs, the use of compositional modeling should be practiced. This will ensure accurate prediction of the reservoir fluid properties.

A history of MATLAB

The first MATLAB (the name is short for “Matrix Laboratory”) was not a programming language. Written in Fortran in the late 1970s, it was a simple interactive matrix calculator built on top of about a dozen subroutines from the LINPACK and EISPACK matrix software libraries. There were only 71 reserved words and built-in functions. It could be extended only by modifying the Fortran source code and recompiling it. The programming language appeared in 1984 when MATLAB became a commercial product. The calculator was reimplemented in C and significantly enhanced with the addition of user functions, toolboxes, and graphics. It was available initially on the IBM PC and clones; versions for Unix workstations and the Apple Macintosh soon followed. In addition to the matrix functions from the calculator, the 1984 MATLAB included fast Fourier transforms (FFT). The Control System Toolbox appeared in 1985 and the Signal Processing Toolbox in 1987. Built-in support for the numerical solution of ordinary differential equations also appeared in 1987. The first significant new data structure, the sparse matrix, was introduced in 1992. The Image Processing Toolbox and the Symbolic Math Toolbox were both introduced in 1993. Several new data types and data structures, including single precision floating point, various integer and logical types, cell arrays, structures, and objects were introduced in the late 1990s. Enhancements to the MATLAB computing environment have dominated development in recent years. Included are extensions to the desktop, major enhancements to the object and graphics systems, support for parallel computing and GPUs, and the “Live Editor”, which combines programs, descriptive text, output and graphics into a single interactive, formatted document. Today there are over 60 Toolboxes, many programmed in the MATLAB language, providing extended capabilities in specialized technical fields.

Реализация в среде MATLAB аналитических алгоритмов модального управления по состоянию и по выходу

The paper considers a problem of MATLAB-based programmed implementation (as an addition to expansion packs “Control System Toolbox” and “Symbolic Math Toolbox”, for extending possibilities of built-in functions ‘acker’ and ‘place’) of analytic algorithms of modal control with state-vector feedback synthesized by so called Full Pole Placement method, as well as with output-vector feedback (in case of incomplete information about state vector) using the Van der Woude idea. To calculate modal controllers a joined function with Boolean argument (0 for state, 1 for output) is proposed, which allows obtaining analytic solutions of the problem of modal control, that are actual for variable parameter systems, at any multiplicity of poles in original spectrum and in desirable spectrum. The developed algorithms are translated into MATLAB code.

Group delay equalisation of discrete Butterworth tan filters in the continuous domain

ABSTRACTA new optimisation method for equalising the group delay frequency response of the discrete transfer function of recursive digital filters, so-called tan filters, is presented in this paper. The method is based on an algorithm that is used for a stable all-pass filter approximation. Allpass filter acts as a group delay equaliser, with the purpose of equalising group delay of the discrete tan filter in a maximal flat sense. The algorithm relies on a set of simultaneous nonlinear equations. These set equations are derived directly from the flatness conditions of the equalised group delay response at the origin in the analogue -domain, in order to obtain the unknown coefficient values of the discrete allpass filter. The iterative method most widely used for solving set nonlinear equations and Symbolic Math Toolbox for Matlab features for finding numerical solutions are presented in detail.In the given examples, firstly a discrete Butterworth tan filter was designed by using transfer function poles in closed form. Secondly, the Butterworth tan filter was augmented with cascade connection of non-minimum phase all-pass discrete filter, in order for the group delay response of the whole filter to be equalised in a maximally flat sense.

Determination of natural frequencies and mode shapes of a wind turbine rotor blade using Timoshenko beam elements

Abstract. When simulating a wind turbine, the lowest eigenmodes of the rotor blades are usually used to describe their elastic deformation in the frame of a multi-body system. In this paper, a finite element beam model for the rotor blades is proposed which is based on the transfer matrix method. Both static and kinetic field matrices for the 3-D Timoshenko beam element are derived by the numerical integration of the differential equations of motion using a Runge–Kutta fourth-order procedure. In the general case, the beam reference axis is at an arbitrary location in the cross section. The inertia term in the motion differential equation is expressed using appropriate shape functions for the Timoshenko beam. The kinetic field matrix is built by numerical integration applied on the approximated inertia term. The beam element stiffness and mass matrices are calculated by simple matrix operations from both field matrices. The system stiffness and mass matrices of the rotor blade model are assembled in the usual finite element manner in a global coordinate system accounting for the structural twist angle and possible pre-bending. The program developed for the above-mentioned calculations and the final solution of the eigenvalue problem is accomplished using MuPAD, a symbolic math toolbox in MATLAB®. The natural frequencies calculated using generic rotor blade data are compared with the results proposed from the FAST and ADAMS software.

Multiprecision Algorithms for Computing the Matrix Logarithm

Two algorithms are developed for computing the matrix logarithm in floating point arithmetic of any specified precision. The backward error-based approach used in the state of the art inverse scaling and squaring algorithms does not conveniently extend to a multiprecision environment, so instead we choose algorithmic parameters based on a forward error bound. We derive a new forward error bound for Pad\'{e} approximants that for highly nonnormal matrices can be much smaller than the classical bound of Kenney and Laub. One of our algorithms exploits a Schur decomposition while the other is transformation-free and uses only the computational kernels of matrix multiplication and the solution of multiple right-hand side linear systems. For double precision computations the algorithms are competitive with the state of the art algorithm of Al-Mohy, Higham, and Relton implemented in \texttt{logm} in MATLAB\@. They are intended for computing environments providing multiprecision floating point arithmetic, such as Julia, MATLAB via the Symbolic Math Toolbox or the Multiprecision Computing Toolbox, or Python with the mpmath or SymPy packages. We show experimentally that the algorithms behave in a forward stable manner over a wide range of precisions, unlike existing alternatives.

Tool for Analysing the Sensitivity and Tolerance of Mechatronic Systems in Matlab GUI

 Abstract—The article deals with the tool in Matlab GUI form that is designed to analyse a mechatronic system sensitivity and tolerance. In the analysed mechatronic system, a torque is transferred from the drive to the load through a coupling containing flexible elements. Different methods of control system design are used. The classic form of the feedback control is proposed using Naslin method, modulus optimum criterion and inverse dynamics method. The cascade form of the control is proposed based on combination of modulus optimum criterion and symmetric optimum criterion. The sensitivity is analysed on the basis of absolute and relative sensitivity of system function to the change of chosen parameter value of the mechatronic system, as well as the control subsystem. The tolerance is analysed in the form of determining the range of allowed relative changes of selected system parameters in the field of system stability. The tool allows to analyse an influence of torsion stiffness, torsion damping, inertia moments of the motor and the load and controller(s) parameters. The sensitivity and tolerance are monitored in terms of the impact of parameter change on the response in the form of system step response and system frequency-response logarithmic characteristics. The Symbolic Math Toolbox for expression of the final shape of analysed system functions was used. The sensitivity and tolerance are graphically represented as 2D graph of sensitivity or tolerance of the system function and 3D/2D static/interactive graph of step/frequency response.

Symbolic and numerical analysis of plates in bending using Matlab

The implementation of the meshless collocation method using radial basis functions for solving partial differential equations involves the symbolic manipulation of governing equations. In the present work, the meshless method is used to solve the bending of plates, using a third order shear deformation theory. To avoid manually writing all the terms, the Symbolic Math Toolbox in Matlab is used to symbolically manipulate expressions. The final terms are written as Matlab functions to be used by the numerical main program.

Application of Fuzzy Analytic Hierarchy Process to Bridge State Comprehensive Assessment and its Improvement Using Intelligent Control Technology

According to Ji-Shao Expressway ,based on the method of AHP, do the fuzzy evaluation and use Symbolic Math Toolbox based MATLAB established was grades assessment diagnostic model for the Intelligent Control Technology system, make rating of the damage status of each part with accumulative score method. Intelligent assessment method and the traditional AHP get Results are consistent，Show that soft computing based on Matlab modern engineering can be effectively applied to active duty bridges’ fuzzy evaluation. At the same time, it can provide reference and guidance for the general survey of other existing bridges.

Comparison of stochastic search optimization algorithms for the laminated composites under mechanical and hygrothermal loadings

The aim of the present study is to design the stacking sequence of the laminated composites that have low coefficient of thermal expansion and high elastic moduli. In design process, multi-objective genetic algorithm optimization of the carbon fiber laminated composite plates is verified by single objective optimization approach using three different stochastic optimization methods: genetic algorithm, generalized pattern search, and simulated annealing. However, both the multi- and single-objective approaches to laminate optimization have been used by considerably few authors. Simplified micromechanics equations, classical lamination theory, and MATLAB Symbolic Math toolbox are used to obtain the fitness functions of the optimization problems. Stress distributions of the optimized composites are presented through the thickness of the laminates subjected to mechanical, thermal, and hygral loadings.

Features of the interpretation of the dispersion Rayleigh-Lamb equation in the matlab environment

A new computer method of calculation of the Rayleigh-Lamb dispersion equation of longitudinal waves in elastic waveguides is сonsidered. A compact and effective program for building a normal waves spectrum and phase velocity dependence of elastic waves propagation from a waveguide thickness and frequency for traditional dispersion equation is created by means Symbolic Math Toolbox in MATLAB environment. With using MATLAB analytical approximation graphical dependencies in the form of polynomials of various degrees, which allows to estimate the level of dispersion distortions of transmitted pulse signals, are obtained

Symbolic equation for linear analog electrical circuits using Matlab

In this paper is presented a program which generates the modified nodal equation for electric analog circuits in a symbolic, partial symbolic and numerical mode. The program is an application, made in the environment of the program MATLAB version 7.1, which has a powerful symbolic math toolbox. MATLAB is a high-performance software package dedicated for numerical analysis and graphic representations in engineering applications. In this paper we try to explore the capabilities of this program in symbolic domain.

Simulation and verification of Zhang neural network for online time-varying matrix inversion

Differing from gradient-based neural networks (GNN), a special kind of recurrent neural network has recently been proposed by Zhang et al. for real-time inversion of time-varying matrices. The design of such a recurrent neural network is based on a matrix-valued error function instead of a scalar-valued norm-based energy-function. In addition, it is depicted in an implicit dynamics instead of an explicit dynamics. This paper investigates the simulation and verification of such a Zhang neural network (ZNN). Four important simulation techniques are employed to simulate this system: (1) Kronecker product of matrices is introduced to transform a matrix-differential-equation (MDE) to a vector differential equation (VDE) [i.e., finally, there is a standard ordinary-differential-equation (ODE) formulation]. (2) MATLAB routine “ode45” with a mass-matrix property is introduced to simulate the transformed initial-value implicit ODE system. (3) Matrix derivatives are obtained using the routine “diff” and symbolic math toolbox. (4) Various implementation errors and different types of activation functions are investigated, further demonstrating the advantages of the ZNN model. Three illustrative computer-simulation examples substantiate the theoretical results and efficacy of the ZNN model for online time-varying matrix inversion.

(Teaching Mathematics for Economists Using MATLAB)

Arabic Abstract:يسعى هذا الكتاب لتقديم الرياضيات بوصفها مادة سهلة، تحتاج فقط إلى مهارات ذهنية بسيطة، ويركز على عرض المفاهيم الرياضية كأدوات جاهزة للاستعمال، ويكفي أن ندخل معطيات المشكلة التي ندرسها، لكي نحصل على الحل. وهو منهج مهم جدا لمن يستخدم الرياضيات في التطبيقات، دون الحاجة لأن تكون لديه قاعدة رياضية قوية، ومنهم المتخصصون في علوم الاقتصاد والإدارة. يبدأ الكتاب بمدخل عن أساسيات لغة ماتلاب، ثم يشرح كيفية استخدام ماتلاب في شرح المبادئ الرياضية التي يحتاجها المتخصصون في العلوم الاقتصادية والإدارية عند دراسة مواد التحليل الرياضي والإحصائي في المستويات المتقدمة، ابتداء بالأعداد، والأسس والجذور واللوغاريتمات، والمقادير الجبرية، وانتهاء بالدوال، ومبادئ التفاضل وتطبيقاته، والتكامل، وجبر المصفوفات، وحل معادلة أو نظم من المعادلات الخطية، والمجموعات. ويؤكد هذا الكتاب على أن جميع المفاهيم الرياضية التي يواجهها المتخصصون في علوم الاقتصاد والإدارة يمكن شرحها، وحل مسائلها باستخدام ماتلاب، وهو ما يفتح أمامهم آفاقا واسعة لاستخدام الحاسب الآلي في تطبيقات العلوم الاقتصادية والإدارية. English Abstract: Written for students, this book presents MATLAB as a tool to learn mathematical concepts as applied to economics and business courses. The book presents MATLAB to readers in a clear and easy-to-understand manner. Topics covered include numbers and sets, matrix mathematics, functions, calculus, and solving differential equations. An introduction to MATLAB is included, and many examples are presented using MATLAB and Symbolic Math Toolbox.

Challenges Concerning Symbolic Computations on Grids

Challenges Concerning Symbolic Computations on Grids

Tuning of Digital PID Controllers Based on Ziegler - Nichols Method

A unified algorithm for auto-tuning of digital РID controllers for n-order process model have been derived in this paper. Tuning procerdure uses the identification of ARX process model parameters using recursive least squares method (RLSM) with adaptive directional forgetting. The parameter estimates serve for computing of the ultimate (critical) proportional gain and period of oscillations on the basis of modified Ziegler-Nichols method for digital control loops. The relations for setting procedures in case of the 2nd and the 3rd order model are derived in the form of analytical formulas. In case of the higher order ARX models the MATLAB Symbolic Math Toolbox was used. These algorithms are suitable for automatic tuning of digital PID controllers or of the adaptive control of technological processes. The digital PID controller with weghting factor and noise filtering constant in the derivative component has been applied for temperature control of a laboratory fermentor. There has been developed the MATLABToolbox ATC PID for designing, testing, simulating and real time of auto-tuning digital PID controllers.