Linear Discrete Dynamic Systems Research Articles

Decoding of MDP Convolutional Codes over the Erasure Channel under Linear Systems Point of View

This paper attempts to highlight the decoding capabilities of MDP convolutional codes over the erasure channel by defining them as discrete linear dynamical systems, with which the controllability property and the observability characteristics of linear system theory can be applied, in particular those of output observability, easily described using matrix language. Those are viewed against the decoding capabilities of MDS block codes over the same channel. Not only is the time complexity better but the decoding capabilities are also increased with this approach because convolutional codes are more flexible in handling variable-length data streams than block codes, where they are fixed-length and less adaptable to varying data lengths without padding or other adjustments.

Quantum simulation of discrete linear dynamical systems and simple iterative methods in linear algebra

Quantum simulation is capable of simulating certain dynamical systems in continuous time—Schrödinger’s equations being the most direct and well known—more efficiently than classical simulation. Any linear dynamical system can in fact be transformed into a system of Schrödinger’s equations via a method called Schrödingerisation (Jin et al. 2022. Quantum simulation of partial differential equations via Schrödingerisation. ( https://arxiv.org/abs/2212.13969 ) and Jin et al. 2023. Phys. Rev. A 108 , 032603. ( doi:10.1103/PhysRevA.108.032603 )). We show how Schrödingerisation allows quantum simulation to be directly used for the simulation of continuous-time versions of general (explicit) iterative schemes or discrete linear dynamical systems. In particular, we use this new method to solve linear systems of equations and for estimating the maximum eigenvector and eigenvalue of a matrix, respectively. This method is applicable using either discrete-variable quantum systems or on hybrid continuous-variable and discrete-variable quantum systems. This framework provides an interesting alternative to solve linear algebra problems using quantum simulation.

On the сontrollability, observability, and optimization of discrete nonstationary linear Volterra systems

In this article, we study discrete nonstationary linear dynamic systems of Volterra type. An essential feature of such kind of systems is that their current states depend on the previous states of this system. The formula Cauchy, which gives us the solution of linear Volterra systems with the control inputs, is obtained. The necessary and sufficient conditions of the pointwise controllability, pointwise output controllability, and observability are proven. Also the linear-quadratic optimization problem for the nonstationary Volterra control systems is studied.

Diagonal canonical form of interval matrices and applications on dynamical systems

Finding the simplest form of a set of quantities is an important aspect of any branch of Mathematics. Of course, the simplest form or the canonical form as we often call it in mathematics, must possess all the important characteristics of the set of quantities. A real square matrix satisfying certain conditions can be brought to diagonal form which is its simplest form such that the diagonal form retains the eigenvalues, determinants, trace, rank, nullity,.. of the original matrix. Many computations with matrices become easier if one can diagonalize the matrices. In this article, we suggest an approach for diagonalizing interval matrices employing a novel methodology called the pairing technique, which will make it simpler and more effective to classify and investigate interval matrices. This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. We also discuss two real world applications on planar systems and linear discrete dynamical systems.

NUMERICAL ESTIMATION OF SOIL DENSITY BY THE METHOD OF FINITE DIFFERENCE GRIDS IN MATHEMATICAL MODELING OF COMPACTION OF SUBSIBLE SOILS BY DEEP EXPLOSIONS

Link for citation: Tarasenko E.O. Numerical estimation of soil density by the method of finite difference grids in mathematical modeling of compaction of subsible soils by deep explosions. Bulletin of the Tomsk Polytechnic University. Geo Аssets Engineering, 2023, vol. 334, no. 5, рр.103-108. In Rus. Relevance. In order to ensure the duration of operation and strength of buildings and their expected design and construction on subsiding loess, it is necessary to carry out a numerical grade of the density characteristics of soils. Subsiding loesses are widespread in the world. They can be observed in Russia, China, the countries of Central Asia, etc. On the territory of Russia, their occurrence is noted at least at 17 % of the entire territory of the country. Almost ubiquitous distribution of loess soils occurs on the territory of the North Caucasus region (80–85 % of area). There is a large capacity of them (up to 50 m). This work is devoted to numerical modeling of the grade of density characteristics of subsiding soils using the finite difference grid method in the framework of a mathematical model of loess compaction by deep explosions. The use of the instrument of computational mathematics made it possible to solve the problem of estimating the compacted subsidence soil density depending on the depth of the explosive charge. The main aim of the research is to conduct a numerical assessment of soil density within the framework of mathematical modeling of compaction by the method of deep explosions of subsiding loess soils. Methods: application of the instrument of computational mathematics, differential equations, methods and results of solid state physics, engineering geology, soil mechanics. Experimental technique. Numerical modeling of the assessment of the increase in the compacted soil by deep explosions based on the grid method was carried out. Results. Based on the grid method using an explicit finite-difference scheme, discrete linear dynamic systems were constructed for numerical calculation of compacted subsidence soil density by deep explosions. The paper describs the layer-by-layer approximation to the solution of a differential equation in partial derivatives with given initial and boundary conditions, which mathematically describes the technological process of compaction of subsiding soil for the cases of surface ejection and camouflage compaction. The stability condition for the obtained finite-difference schemes is indicated. A computational experiment was carried out. It demonstrated the adequacy of the proposed method for estimating the compacted soil density to experimental data. Findings. The constructed numerical solutions of problems within the framework of mathematical modeling of the compaction of subsiding soils by deep explosions indicate the possibility of their practical application as calculation methods to reduce the cost of carrying out production work on the compaction of loess. The methods proposed in the work for determining soil density can be implemented in a number of areas of applied research in soil mechanics.

From Boolean networks to linear dynamical systems: a simplified route

ABSTRACT Boolean Networks can be converted to discrete linear dynamical systems on finite spaces by a semi-tensor-product approach. This approach has been used by many to study the dynamics and control of Boolean systems. However, the process of getting the linear representation using the semi-tensor-product method is complicated even for a simple three-node network and requires the help of a computer program. In this work, we show that we can skip the semi-tensor process and obtain the same linear representation with a straightforward mapping. Moreover, our approach produces a large number of isomorphic representations which provides a flexible framework. Importantly, it could simplify the analytical study of networks with unspecified number of nodes that have some structure.

Application of pseudoinverse matrices in optimal terminal control problems of linear discrete systems

The problem of linear-quadratic optimal terminal control of a linear discrete dynamic system with a vector input is considered. Solutions of the problem based on the apparatus of the optimization form of pseudo-circulation of matrices are given. The solution of the optimal terminal control problem is obtained, which ensures the achievement of the terminal control goal and the passage of the trajectory of the system as close as possible to the specified intermediate points. A complete set of solutions to the problem of optimal terminal control of a bundle of trajectories is obtained.

Bidimensional Jordan form and linear discrete dynamical systems with application in red blood cell production

Jordan canonical form (JCF) is one of the most important, and useful, concepts in linear algebra. Mathematics, physics, biology, science and engineering undergraduates often find the first application of real JCF in the discipline of differential equations (continuous models) to solving systems of differential equations. In this work, we apply real JCF to understand, through analytical and numerical methods, the two-dimensional linear discrete dynamical systems. After that, we present a schematic model of red blood cell production. An advantage of discrete models over continuous models is that they readily yield numerical exploration, whether by calculator or computer (in our case we will use Mathematica software). This fact provides us with a valuable pedagogical resource because we can find a simple model to show the links between mathematics and other areas of knowledge without the need for sophisticated mathematical concepts beyond linear algebra. Following Paul Halmos's suggestion, ‘The only way to learn mathematics is to do mathematics’, some classroom exercises are given to better understand the subject, and a project for further student investigations is suggested.

Stability analysis and stabilization of linear symmetric matrix-valued continuous, discrete, and impulsive dynamical systems — A unified approach for the stability analysis and the stabilization of linear systems

Matrix-valued dynamical systems are an important class of systems that can describe important processes such as covariance/second-order moment processes, or processes on manifolds and Lie Groups. We address here the case of processes that leave the cone of positive semidefinite matrices invariant, thereby including covariance and second-order moment processes. Both the continuous-time and the discrete-time cases are first considered. In the LTV case, the obtained stability and stabilization conditions are expressed as differential and difference Lyapunov conditions which are equivalent, in the LTI case, to some spectral conditions for the generators of the processes. Convex stabilization conditions are also obtained in both the continuous-time and the discrete-time setting. It is proven that systems with constant delays are stable provided that the systems with zero-delays are stable—which mirrors existing results for linear positive systems. The results are then extended and unified into an impulsive formulation for which similar results are obtained. The proposed framework is very general and can recover and/or extend many of the existing results in the literature on linear systems related to (mean-square) exponential (uniform) stability. Several examples are discussed to illustrate this claim by deriving stability conditions for stochastic systems driven by Brownian motion and Poissonian jumps, Markov jump systems, (stochastic) switched systems, (stochastic) impulsive systems, (stochastic) sampled-data systems, and all their possible combinations.

Convolutional Code Theory based Steganography Technique

Steganography, as it is known, is a technique to hide a secret message within a message or collection of data that is not secret, and a problem in mathematics is to decipher the secret included in the message, to solve this problem a good tool It is the theory of codes. Unlike the existing works that use block codes to hide information using the steganographic process, in this work, we propose the use of convolutional coding theory in steganography to encrypt and decrypt messages methods to decrypt messages. Here, we suggest a steganographic protocol based on convolutional codes in which they are defined as discrete linear dynamical systems with which the properties on controllability and observability characteristic of linear systems theory can be applied, in particular the properties of output observability character which can be easily described using matrix language. The proposed decoding algorithm used for dissimulation is a linear decoding method, which has decreased both the time and space complexity, compared to the Viterbi decoding algorithm, sometimes used in other cases; indeed, we go from 2h.n to 2h=2.n, in memory space (with h: constraint height, and n: length of cover object). Moreover, the time complexity is better, while we can also denote that with the convolutional approach, we intend to take advantage of the time-depending transaction.

Internal model control design based on approximation of linear discrete dynamical systems

In this paper, a new direct discrete approximation based internal model control design is proposed to the linear discrete dynamical systems. The approximation method is used to determine an accurate and stable reduced-order model for the considered original higher-order discrete-time system. The method involves an enhanced Differential Evolution algorithm to ascertain the stable denominator polynomial coefficients, and the preferable reduced numerator polynomial coefficients are evaluated by using the improved discrete multi-point Padé approximation approach. The method deploys on discrete step integral square error minimization between the original dynamical system and the approximated model, together with retaining their discrete impulse response energy values. The approximated model has been considered an internal (predictive) model and proceeds with an optimal internal model controller design to improve the discrete dynamical system behaviour according to the reference input/the set point. The controller's best performance is attained by tuning the single filter parameter ′λ′ by minimizing the integral square error between the reference input and the actual output of the dynamical system using the enhanced differential evolution algorithm. The acceptability and applicability of the proposed process reduction-based controller design have been validated on a single-input single-output supersonic jet engine inlet dynamical model. The controller robust study is conducted by inserting 10% disruption uncertainty in the system dynamical model poles and zeros. The method has also been extended to the discrete multi-input multi-output dynamical model of the single machine infinite bus power system to develop an optimal internal model control-based power system stabilizer. The simulation results showing better reference input tracking, comparison of performance indices, and also highlight the efficacy of the proposed controller design.

Algorithm for solving optimal open-loop terminal control problem for spacecraft rendezvous with account of constraints on the state

The paper proposes an algorithm for solving the optimal open-loop terminal control problem of two spacecraft rendezvous with constraints on their states. A system of nonlinear differential equations that describes the dynamics of the active (maneuvering) spacecraft relative to the passive spacecraft (station) in the central gravitational field of the Earth in the orbital coordinate system of coordinates related to the passive spacecraft center-of-mass is considered as an initial model. The obtained nonlinear model of the active spacecraft dynamics is linearized relative to the specified reference state trajectory of the passive spacecraft, and then it is discretized and reduced to linear recurrence relations. Mathematical formalization of the spacecraft rendezvous problem under consideration is carried out at a specified final moment of time for the obtained discrete-time controlled dynamical system. The quality of solving the problem is estimated by a convex functional taking into account the geometric constraints on the active spacecraft states and the associated control actions in the form of convex polyhedral-compacts in the appropriate finite dimensional vector space. We propose a solution of the problem of optimal terminal control over the approach of the active spacecraft relative to the passive spacecraft in the form of a constructive algorithm on the basis of the general recursive algebraic method for constructing the availability domains of linear discrete controlled dynamic systems, taking into account specified conditions and constraints, as well as using the methods of direct and inverse constructions. In the final part of the paper, the computer modeling results are presented and conclusions about the effectiveness of the proposed algorithm are made.

THE CRITERION OF TENDING TO ZERO THE POWERES OF A MATRIX AND ITS APPLICATIONS

In this article the criterion for convergence of a sequence of powers to zero matrix and its practical applications are presented. The linear discrete dynamic system with constant coefficients is explored and the sufficient and necessary condition for the all eigenvalues of a given matrix have a negative real part in the complex plane is obtained. It is proved that if the maximum of modules of the all eigenvalues of the Jacobi matrix at zero of a complex valued holomorphic vector function with several variables is less than one then the zero fixed point of the function is attractive.

New Sensitivity Analysis Methods for Complex Modal Parameters in Asymmetrical Damped System

Most of the sensitivity algorithms with respect to the design parameters adopted the normalization relationship with system property matrices weighted when analyzing an asymmetrical damping linear discrete dynamic system. In this paper, two new methods (an algebraic method and a direct method) based on the Euclidean norm as the normalization conditions are proposed. Both proposed methods are developed to calculate the first- and second-order sensitivities of the complex modal parameters for an asymmetric damped system. The precisions of the Taylor approximations of the second degree are also discussed in the proposed methods. Like other algebraic methods, this paper outlines the proof of numerical stability. Different from Nelson’s direct method (“Simplified Calculation of Eigenvector Derivatives,” AIAA Journal, Vol. 14, No. 9, 1976, pp. 1201–1205), this paper considers complexity of the solution space in complex vector space, and the partitioning scheme of the proposed direct algorithm is more practical. The usefulness and correctness are demonstrated by a one-parameter numerical experiment and a two-parameter numerical experiment.

A Dynamic Variance-Based Triggering Scheme for Distributed Cooperative State Estimation over Wireless Sensor Networks

Wireless sensor networks (WSNs) have been spawning many new applications where cooperative state estimation is essential. In this paper, the problem of performing cooperative state estimation for a discrete linear stochastic dynamical system over wireless sensor networks with a limitation on the sampling and communication rate is considered, where distributed sensors cooperatively sense a linear dynamical process and transmit observations each other via a common wireless channel. Firstly, a novel dynamic variance-based triggering scheme (DVTS) is designed to schedule the sampling of each sensor and the transmission of its local measurement. In contrast to the existing static variance-based triggering scheme (SVTS), the newly proposed DVTS can lead to the larger average intertrigger time interval and thus fewer total triggering number with almost approximate estimation accuracy. Second, a new Riccati equation of the prediction variance iteration for each estimator is obtained, which switches dynamically among the modes related to the variance of the previous step and the recently received measurements from other sensors. Furthermore, the stability issue is also mainly investigated. Finally, simulation results show the effectiveness and advantage of the proposed strategy.

Stability index of linear random dynamical systems

Given a homogeneous linear discrete or continuous dynamical system, its stability index is given by the dimension of the stable manifold of the zero solution. In particular, for the $n$ dimensional case, the zero solution is globally asymptotically stable if and only if this stability index is $n.$ Fixed $n,$ let $X$ be the random variable that assigns to each linear random dynamical system its stability index, and let $p_k$ with $k=0,1,\ldots,n,$ denote the probabilities that $P(X=k)$. In this paper we obtain either the exact values $p_k,$ or their estimations by combining the Monte Carlo method with a least square approach that uses some affine relations among the values $p_k,k=0,1,\ldots,n.$ The particular case of $n$-order homogeneous linear random differential or difference equations is also studied in detail.

Robustness research of interval dynamic systems by algebraic method

The paper considers robust stability study of continuous and discrete interval dynamic systems by algebraic method. The original robustness results obtained for continuous and discrete linear interval dynamic systems within the algebraic direction of robustness stability are presented. The author formulated and proved the basic theorem on the robustness of linear continuous dynamic system with interval elements of the right-hand part matrix, which is determined through the separate angular coefficients of characteristic polynomial of the system. The basic theorem is proved on the basis of a lemma on the separative coefficients of the characteristic polynomial obtained by optimization methods of nonlinear programming on multiple interval elements of the system matrix. Their possible values can be the upper or lower limits of the corresponding interval or zero. A clarification note to the basic theorem for continuous systems is formulated. The idea lies in the need for a complete set of four angular polynomials for the robustness stability of the system, excluding multiple cases of the characteristic polynomial, when the set of Kharitonov polynomials degenerates and will consist of the less required four different polynomials. The theorem is obtained on the necessary and sufficient conditions of robustness stability for the polyhedron of interval matrices. A discrete analogue of the Kharitonov theorem is obtained for discrete systems. The algorithm of robustness stability determination for discrete interval dynamic systems is presented. Comparative characteristics of the results obtained in the works of well-known authors having studied the algebraic trend of robust stability problem are considered. They show the distinctive feature of this method, which consists in consideration of interval matrices of general type. The validity of the method is tested on the known counterexamples to Bialas’s theorem, as well as the other researchers studying robustness problems of interval dynamic systems.

Solving the Minimax Open-Loop Control Problem for Carrier Rocket Fuel Consumption

We propose a mathematical formalization and a method for solving the minimax open-loop terminal control problem for fuel consumption of a propulsion system in a carrier rocket. The initial nonlinear model of the control object is linearized along the reference trajectory and approximated by a linear discrete dynamic system. For the approximating system, we formulate the minimax open-loop terminal control problem, taking into account given geometric constraints on the control and disturbance vectors. We propose a new method and a numerical algorithm for solving the problem, which use a modification of the general recurrent algebraic method to construct generalized reachability sets for the linear discrete controlled system. We demonstrate the effectiveness of the proposed solution with computer simulation examples.

Dynamic identification of boundary conditions for convection-diffusion transport model in the case of noisy measurements

The paper addresses the problem of dynamic identification of mixed boundary conditions for one-dimensional convection-diffusion transport model based on noisy measurements of the function of interest. Using finite difference method the original model with the partial differential equation is replaced with the discrete linear dynamic system with noisy multisensor measurements in which boundary conditions are included as unknown input vector. To solve the problem, the algorithm of simultaneous estimation of the state and input vectors is used. The results of numerical experiments are presented which confirm the practical applicability of the proposed method.

Solving the problem of minimax closed-loop control of liquid-propellant launch vehicle fuel consumption control

The paper provides mathematical formalization and a method of solving the problem of minimax (guaranteed) closed-loop terminal control of fuel consumption of a liquid-propellant launch vehicle power plant. The initial discrete-continuous nonlinear model of the controlled object is linearized along the given reference phase path and is approximated by a linear discrete-time multistep dynamical system. The approximated system includes the state vector, the control vector and the disturbance vector that defines the error of formation of the approximated model. Taking into account the geometrical constrains of control and disturbance vectors in the approximated system, we formulate the main problem of minimax closed-loop terminal control of propellant consumption of the launch vehicles propulsion system. This problem consists in solving a number of auxiliary tasks of minimax open-loop terminal control. To solve each of these tasks we use an instrument of development and analysis of generalized attainability domains of the approximated linear discrete dynamical system. These techniques are implemented by modifying the general recurrent algebraic method. To solve the problems under consideration we propose an approach and an appropriate numerical algorithm that is reduced to the implementation of a finite sequence of only one-step algebraic and optimization operations. The efficiency of the proposed approach to solving the problem under consideration is demonstrated and verified by a computer simulation example. This simulation example consists in controlling the process of propellant consumption for Soyuz-2.1b launch vehicles third stage propulsion system.