Dynamic Interval Systems Research Articles

Brief review of the development of theories of robustness, roughness and bifurcations of dynamic systems

The development issues of theories of robustness, roughness and bifurcations of dynamic systems are considered. In the modern theory of dynamic systems and automatic control systems, researches of the properties of roughness and robustness of systems are becoming more and more important. The work considers methods of research and ensuring robust stability of interval dynamic systems of both algebraic and frequency directions of robust stability. The main results of the original algebraic method of robust stability for continuous and discrete time are given. In the frequency direction of robust stability, the issues of a frequency-robust method to the analysis and synthesis of robust multidimensional control systems based on the use of the frequency condition number of the transfer matrix of the “input-output” ratio are considered. The main provisions of the theory and method of topological roughness of dynamic systems based on the concept of roughness according to Andronov-Pontryagin are presented with the introduction of a measure of roughness of systems in the form of a condition number of matrices of reduction to a diagonal (quasi-diagonal) basis at special points of phase space. Criteria for dynamic systems bifurcations are formulated. Applications of the topological roughness method to synergetic systems and chaos have been used to investigate many systems, such as Lorenz and Rössler attractors, Belousov-Jabotinsky, Chua systems, “predator-prey” and “predator-prey-food”, Hopf bifurcation, Schumpeter and Caldor economic systems, Henon mapping, and others. For research of weakly formalized and non-formalized systems, the use of the approach of analogies of theoretical-multiple topology and the abstract method to such systems is proposed. Further research suggests the development of roughness and bifurcation theories for complex nonlinear dynamical systems.

CLASSIFICATION OF ONE DIMENSIONAL DYNAMICAL SYSTEMS BY COUNTABLE STRUCTURES

AbstractWe study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to isomorphism equivalence relation of countable graphs. This solves a special case of Hjorth’s conjecture which states that every orbit equivalence relation induced by a continuous action of the group of all homeomorphisms of the closed unit interval is classifiable by countable structures. We also prove that conjugacy equivalence relation of Hilbert cube homeomorphisms is Borel bireducible to the universal orbit equivalence relation.

The modal sensitivity, robustness and roughness of dynamic systems (review article)

The review focuses on the development of theories of sensitivity and modal sensitivity, robustness and roughness of dynamic systems. In modern theory of dynamic systems and automatic control systems, there is a need to study the properties of sensitivity, robustness and roughness of systems in their interconnection. An import application of sensitivity theory is associated with the design and creation of high-precision and low-sensitivity systems. The most significant results were achieved in the development of differential methods for the analysis and synthesis of low-sensitivity systems. One approach to the analysis and synthesis of linear systems of low parametric sensitivity in the space of states was developed using the functions of modal sensitivity or, in other words, the method of modal sensitivity. A review of robustness theory considers methods of studying and ensuring the robust stability of interval dynamical systems, paying attention to both algebraic and frequency directions of robust stability. The work presents the main results of the original algebraic method of robust stability of interval dynamical systems for continuous and discrete time. The section “Basic development stages of the theory of roughness of systems” describes the main principles of the theory and method of topological roughness of dynamical systems, based on the concept of roughness according to Andronov–Pontryagin. Applications of the topological roughness method to synergetic systems and chaos have been used to investigate many systems, such as the Lorenz and Rössler attractors, the Belousov–Zhabotinsky, Chua, predator-prey, Hopf bifurcation, Schumpeter and Caldor economic systems, etc. The review proposes applying the approach of analogies of theoretical-multiple topology and the abstract method for studying weakly formalized and informal systems. Further research suggests the development of theories of sensitivity, robustness, roughness and bifurcation for complex nonlinear dynamic

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.

Slow Continued Fractions and Permutative Representations of

AbstractRepresentations of the Cuntz algebra ${\mathcal{O}}_{N}$ are constructed from interval dynamical systems associated with slow continued fraction algorithms introduced by Giovanni Panti. Their irreducible decomposition formulas are characterized by using the modular group action on real numbers, as a generalization of results by Kawamura, Hayashi, and Lascu. Furthermore, a certain symmetry of such an interval dynamical system is interpreted as a covariant representation of the $C^{\ast }$-dynamical system of the “flip-flop” automorphism of ${\mathcal{O}}_{2}$.

Synthesis of robust control algorithms for linear interval dynamic systems in the electric power industry

When studying the objects of management in the field of electric power, we meet with various inaccuracies in determining their parameters. One of the methods of dealing with uncertainties is the use of various estimates of the parameters of the control object. modern science has developed various methods for assessing the uncertain parameters of the control object in the electric power industry. parameter uncertainty occurs when the set of parameters of the control object is more than one point. If this set is defined using probabilistic characteristics, then this is the so-called probabilistic uncertainty of the object parameters. If the boundaries of the intervals in which they are enclosed are known for the object parameters, then such uncertainty is called interval uncertainty. if the object parameters are set using the membership function, then the theory of fuzzy logic is used. Interval determination of parameters of the control object is used when working with values for which only the boundaries of the intervals within which their values are enclosed are known. The interval approach in the description of object parameters is used to account for rounding and errors that occur during calculations on a computer and is a strong method in the representation of objects with uncertain parameters, which are very common in the electric power industry. The reason why interval systems are used is the incompleteness of information about the control object, errors in measuring the parameters of the object, linearization errors, and so on. Various problems of the classical theory of automatic control allow us to replace the concentrated parameters with their interval analogues. Many interval problems are adequate for practical applications. The synthesis of linear quadratic regulators refers to the classical method, which allows us to obtain regulators that minimize the integral quality criterion with respect to the resource of regulated and regulatory quantities. In this paper, we study the possibility of synthesizing linear quadratic regulators for electric power facilities specified in intervals. This application of two well-known methods makes it possible to work with nonlinear objects using the classical linear control theory.

Robust of Interval Dynamic Systems

The frequency and algebraic directions of researches of robust stability are considered. Frequency or Tsypkin-Polyaka’s direction is considered briefly in a survey order. The algebraic or Kharitonov’s direction is considered more more widely, namely basic provisions and results of the Algebraic method of robust stability of interval dynamic systems developed within development algebraic or Kharitonov’s direction of robust stability are presented. Fundamental works of V. L. Kharitonov since issue have caused a huge flow of the publications connected with extreme relevance of the solution of problems of a robustness of systems. So far from a circle of problems of a robustness many issues of robust stability are resolved. Discrete analogs and versions of theorems of Kharitonov are received. Frequency conditions of robust stability are considered and solved in Ya. Z. Tsypkin, B. T. Polyak, Yu. I. Neymark works. However in a problem of robust stability not all issues are so far resolved, especially big contradictions have arisen in a continuous case. Also the tasks considered here for interval matrixes and polyhedrons of matrixes haven’t been solved. In work the theorem like the third theorem of Kharitonov which cancels counterexamples to former known results in this direction is formulated and proved and also on its basis the costal theorem for polyhedrons of matrixes is proved. The new costal theorem also cancels counterexamples for this case. To the main theorem of the considered algebraic method the specifying remark is formulated that in the absence of a full set of four angular polynoms of Kharitonov of a condition of this theorem are necessary, but can be insufficient for stability of system. Determination of angular separate coefficients of a characteristic polynom of system is generally carried out by means of use of methods of nonlinear programming. For a discrete case the discrete analog of the theorem of Kharitonov which is received on the basis of Schur’s theorem is presented. At the same time, the concepts of points and intervals of a variabless used for the theorem of an analog of a continuous case are entered. The algorithm of definition of a robustness of discrete systems is formulated. The main results, the Algebraic method of robust stability developed by the author are illustrated with cancellation of the counterexamples widely known from scientific literature.

The Study on D-stability of Dynamic Interval Systems Based on Robust State Observers

In this paper, we address the design problems of robust state observers for discrete-time dynamic interval systems with D-stability constrains. A method of designing the robust state observers is presented based on some matrix inequalities. It is not assumed that the system under consideration is necessarily regular and causal, the objective is to determine the observer gain matrix K such that the desired Robust Performance and Time-domain performance can be satisfied in the whole observe procedure and the closed-loop poles are placed in a specified disk while the regularity, causality and stability of the closed-loop system can be guaranteed for all admissible uncertainties, furthermore, the analytical expression of desired observer gain matrix K is also derived. Finally, a numerical example is given to demonstrate the application of the proposed method.

Interval Quadratic Connected Control Systems in Industrial Robotics and Mechatronics

Industrial robotic and mechatronic systems in real machinery are quadratically connected dynamical systems with parametric uncertainties, whose mathematical models can be extremely complex. Program motion stabilization of such systems, especially those described in quasi-coordinates, is a topical scientific problem, which does not yet have an acceptable for engineering developments comprehensive solution. The article proposes a possible concept of such a solution, based on the Lyapunov’s direct method, extended to the interval dynamic systems domain with time derivatives of state variables written in the interval cubic forms. The authors conceptual approach is based on new concepts introduced in the article: covering form, form compensator, modular variables form. According to this approach, first of all, the transition to new complex system state variables formed with signatures is carried out. The mathematical apparatus is based on special interval matrix algebra transformations, which could not be included in the article because of the volume, and therefore are only stated. As a result, the authors obtain a new class of quasi-relay type controls with system state dependent modulated impact jumps, and therefore combining the properties of both soft smooth and hard forcing types.

Qualitative Analysis of Stable Reduced Order Models for Interval Systems Using Mixed Methods

ABSTRACT A large number of problems are available for complex higher order systems. These systems are difficult to analyze and synthesize. Therefore it is needful to approximate higher order models to lower order models. Interval systems have constant coefficients which are uncertain within a finite range. In this paper, the frequency domain reduction methods are applied for reducing the order of interval systems. The three mixed methods are considered for reducing the linear dynamic interval systems. In these mixed methods, denominators of the reduced models are obtained by using differentiation method while the numerators of the reduced models are obtained by using differentiation, factor division, and pade approximation methods. The novel feature of the proposed methods is that it guarantees the stability of reduced order models if the original interval systems are stable. In addition, the mixed methods are compared qualitatively in terms of integral square error and integral absolute error to know about the best method among three mixed methods. Numerical examples are given to show the effectiveness of the mixed methods.

Recurrence determinism and Li–Yorke chaos for interval maps

ABSTRACTRecurrence determinism, one of the fundamental characteristics of recurrence quantification analysis, measures predictability of a trajectory of a dynamical system. It is tightly connected with the conditional probability that, given a recurrence, following states of the trajectory will be recurrences. In this paper, we study recurrence determinism of interval dynamical systems. We show that recurrence determinism distinguishes three main types of ω-limit sets of zero entropy maps: finite, solenoidal without non-separable points, and solenoidal with non-separable points. As a corollary, we obtain characterizations of strongly non-chaotic and Li–Yorke (non-)chaotic interval maps via recurrence determinism. For strongly non-chaotic maps, recurrence determinism is always equal to one. Li–Yorke non-chaotic interval maps are those for which recurrence determinism is always positive. Finally, Li–Yorke chaos implies the existence of a Cantor set of points with zero determinism.

Generic points of shift-invariant measures in the countable symbolic space

AbstractWe are concerned with sets of generic points for shift-invariant measures in the countable symbolic space. We measure the sizes of the sets by the Billingsley-Hausdorff dimensions defined by Gibbs measures. It is shown that the dimension of such a set is given by a variational principle involving the convergence exponent of the Gibbs measure and the relative entropy dimension of the Gibbs measure with respect to the invariant measure. This variational principle is different from that of the case of finite symbols, where the convergent exponent is zero and is not involved. An application is given to a class of expanding interval dynamical systems.

Dynamics of a class of delayed reaction–diffusion systems with Neumann boundary condition

This paper considers a class of delayed reaction–diffusion systems under the Neumann boundary condition which arise in epidemiology and can describe the temporal and spatial evolutionary phenomena for the bacteria population and the human infective population. With the help of the iterative properties of interval mapping and dynamical system approaches, some positively invariant sets and attractive basins of the considered systems are analyzed detailedly. In addition, combining the global attractivity of interval mapping, we provide some sufficient conditions to ensure local or global attractivity of steady states of the systems. Finally, we apply these theoretical results to some models with different nonlinearity demonstrating “force of infection”, and then obtain some sufficient conditions about “force of infection” to describe the survival and extinction of bacteria and infective populations.

Stability of the Stochastic Model for Power Markets with Interval Parameters

Pertaining to the random nature of demand sides and the range of demand elasticity with suppliers and consumers, a stochastic model for power markets with interval parameters is described to illustrate uncertain external disturbances, which is a generalization of the Alvarado dynamic model, stochastic model, and interval model. The interval stochastic stability criteria of the provided model are investigated by the theory of economics, interval dynamical system, and the theory stability of stochastic differential equations. The conclusions indicate that the demand elasticity stable interval can be calculated and the random excitation intensity does not impact the system stability. Some numerical examples are given to show the applicability and validity of the obtained results from a statistical perspective.

Data-driven model for solar irradiation based on satellite observations

We construct a data-driven model for solar irradiation based on satellite observations. The model yields probabilistic estimates of the irradiation field every thirty minutes starting from two consecutive satellite measurements. The probabilistic nature of the model captures prediction uncertainties and can therefore be used by solar energy producers to quantify the operation risks. The model is simple to implement and can make predictions in realtime with minimal computational resources. To deal with the high-dimensionality of the satellite data, we construct a reduced representation using factor analysis. Then, we model the dynamics of the reduced representation as a discrete (30-min interval) dynamical system. In order to convey information about the movement of the irradiation field, the dynamical system has a two-step delay. The dynamics are represented in a nonlinear, nonparametric way by a recursive Gaussian process. The predictions of the model are compared with observed satellite data as well as with a similar model that uses only ground observations at the prediction site. We conclude that using satellite data in an area including the prediction site significantly improves the prediction compared with models using only ground observation site data.

Exponential Stability of Interval Nonlinear Systems

This article proposes an approach for investigating the exponential stability of a nonlinear interval dynamical system with nonlinearity of a quadratic type on the base of the Lyapunov’s direct method. It also constructs an inner estimate of the attraction domain to the origin for the system under consideration.

The Interval Stability of an Electricity Market Model

Combined with the electric power market dynamic model put forward by Alvarado, an interval model of electricity markets is established and investigated in this paper pertaining to the range of demand elasticity with suppliers and consumers. The stability of an electricity market framework with demand elasticity interval is analyzed. The conclusions characterizing the interval model provided are derived by constructing a suitable Lyapunov function and using the theory of interval dynamical system in differential equations and matrix inequality theory and so forth. Applying the corollary obtained can judge the system stability by available data about demand elasticity. The obtained results are validated and illustrated by a case example.

Bundled Software for the Desing of Interval Dynamic Systems

The article considers the system of an automatic control, that includes a linear controller and an interval control object. Using the coefficient estimations of quality factors the methodology of analysis of robust quality coefficients is developed that ensures maximum degree of stability and allowable system accuracy. The coefficient method of analysis of quality factors constitutes the basis of the method. On the basis of the obtained methods the bundled software RASILS is developed. A numerical illustration is presented in this paper.

Strong Invariance Principles with Rate for “Reverse” Martingale Differences and Applications

In this paper, we obtain almost sure invariance principles with rate of order $$n^{1/p}\log ^\beta n$$ , $$2< p\le 4$$ , for sums associated with a sequence of reverse martingale differences. Then, we apply those results to obtain similar conclusions in the context of some non-invertible dynamical systems. For instance, we treat several classes of maps of the interval (for possibly unbounded observables) or smooth dynamical systems under a very weak regularity condition on the observables.

An iterative algorithm for the stability analysis of dynamic interval systems

In this paper, a new iterative algorithm is proposed to analyze the stability of dynamic interval systems. Compared with existing researches, this algorithm takes much less computation time to obtain the superior of maximal eigenvalues and the inferior of minimal eigenvalues of a real interval matrix with real eigenvalues, under given precision. As a result, the stability of a dynamic interval system, which is determined by eigenvalues of its corresponding interval matrix, can be judged within a shorter time period. Furthermore, if the dynamic interval system is concluded to be stable, the output of our iterative algorithm also indicates the accurate maximal stability margin of this system. Finally, three numerical examples are given to demonstrate the applicability and effectiveness of this algorithm.