Multiphase Learning for an Interval-Based Hybrid Dynamical System

Linear dynamical systems over finite rings

The network oriented representation of the 3-D electromagnetic structures is based on two types of linear systems. The former one is a dynamic linear system which describes the interaction of electric and magnetic fields inside the structure while the latter one is a linear system consisting of delay lines meant for the reflection and propagation. The system identification of the linear dynamic system needs the estimation of the poles positions and the maximum duration of the delays of complementary linear system. The improved system identification scheme based on poles set stability criterion is introduced. This scheme offers to divide the restored impulse response into the finite duration entire part and the infinite pole part which pertain to the each linear system correspondingly. The simulation results of the model parameter estimation for the two-port coplanar resonator are demonstrated.

In this paper, we study the Poincare recurrence phenomenon in linear dynamical systems, that is, dynamical systems whose trajectories return infinitely often to neighborhoods of their initial condition. Specifically, we provide several equivalent notions of Poincare recurrence and review sufficient conditions for nonlinear dynamical systems that ensure that the system exhibits Poincare recurrence. Furthermore, we establish necessary and sufficient conditions for Poincare recurrence in linear dynamical systems. Finally, we show that in the case of linear systems the absence of volume-preservation is equivalent to the absence of Poincare recurrence implying irreversibility of a dynamical system

Optimal sensor scheduling for multiple linear dynamical systems

Identification of dynamical systems is, in essence, a problem of observability of system parameters from a related Bet of observations. In this article, a necessary and sufficient condition is obtained for the observability of parameters from a given record of inputs and states of a linear system with a class of parameter variations. Also obtained is n necessary and sufficient condition of observability of a Sub-class of such systems with polynomial system and constant output coefficient matrices when only the outputs instead of the states are accessible for measurement.

Introduction: This article is the result of the research “Analysis of the dynamic systems component of the engineering programs in control and automation and technology in industrial electronics,” developed at the Francisco José de Caldas District University in the years 2022-2023. Problem: The document addresses the lack of a clear and comprehensive structure in the analysis of linear dynamic systems, as well as the absence of an innovative approach to addressing these topics in traditional courses, which hinders their understanding and application in different disciplines. Objective: The objective is to present a novel approach to analyzing linear dynamic systems, providing a complete framework of concepts, relationships, and important tools in this analysis, as well as a literature review based on over seventeen years of experience in courses and studies of this kind. Methodology: A journey of concepts is proposed from signals and systems, through modeling with methods such as black box, white box, gray box, to linear analysis based on examples with calculation of differential equations, state representation, transfer function, and block diagrams. Results: The document aggregates and articulates all the concepts of dynamic systems, along with the relationships and tools used, offering a more practical and intuitive approach to understanding the material. Conclusion: The document provides a comprehensive and articulated view of key concepts in the analysis of linear dynamic systems, highlighting an innovative approach that facilitates their understanding. Its main contribution is to aggregate and articulate these concepts, along with the tools and relationships used, to offer a more practical and clear approach for their study. Originality: The originality lies in proposing a novel and structured approach to analyzing linear dynamic systems, addressing the lack of clarity and completeness in traditional approaches. Limitations: Although the document proposes a novel approach, it does not delve into specific aspects of some topics covered, which could limit the detailed understanding of certain concepts. Deepening: The document could delve into the practical application of the proposed concepts and tools in real cases of dynamic systems, as well as into the comparison with traditional approaches to highlight the differences and advantages of the proposed new approach.

Chapter 1: Introduction to dynamic modeling 1. Modeling drugs in the bloodstream 2. Terminology 3. Equilibrium values 4. Dynamic economic applications 5. Applications of dynamics using spreadsheets Chapter 2: First order dynamical systems 1. Solutions to linear dynamical systems with applications 2. Solutions to an affine dynamical system 3. An introduction to genetics 4. Solution to affine dynamical systems with applications 5. Applications to finance Chapter 3: Introduction to probability 1. The multiplication to probability 2. Introduction to probability 3. Multistage tasks 4. An introduction to Markov chains Chapter 4: Nonhomogeneous dynamical systems 1. Exponential terms 2. Exponential terms, a special case 3. Fractal geometry 4. Polynomial terms 5. Polynomial terms, a special case Chapter 5: Higher order linear dynamical systems 183 1. An introduction to second order linear equations 2. Multiple roots 3. The gambler's ruin 4. Sex-linked genes 5. Stability for second order affine equations 6. Modeling a vibrating string 7. Second order nonhomogeneous equations 8. Gambler's ruin revisited 9. A model of a national economy 10. Dynamical systems with order greater than two 11. Solutions involving trigonometric functions Chapter 6: Introduction to nonlinear dynamical systems 1. A model of population growth 2. Using linearization to study stability 3. Harvesting strategies 4. More linearization Chapter 7: Vectors and matrices 1. Introduction to vectors and matrices 2. Rules of linear algebra 3. Gauss-Jordan elimination 4. Determinants 5. Inverse matrices Chapter 8: Dynamical systems of several equations 1. Introduction to dynamical systems of several equations 2. Characteristic values 3. First order dynamical systems of several equations 4. Regular Markov chains 5. Absorbing Markov chains 6. Applications of absorbing Markov chains 7. Long term behavior of solutions 8. The heat equation 9. Bibliography

On-Line Eigenvector Algorithms for the Identification of Dynamic Systems

This article presents a theoretical framework for the identification of nonlinear dynamic MISO systems with orthonormal base function models on fundamental basics of the Volterra theory. In the past the Volterra theory was used for the identification of nonlinear dynamic SISO systems (e.g. Hammerstein and Wiener models). In principle it is possible to extend these approaches to systems with more than one input (i.e. MISO systems). In former times this failed due to a lack of computational performance. In this paper an approach is presented which allows the identification of arbitrary coupled Hammerstein and Wiener models with multiple inputs. Some fundamental considerations for the identification of MISO systems based on arbitrary coupled Hammerstein models are made and extended to MISO systems based on arbitrary coupled Wiener models. This extended Volterra theory results in equations where the unknown parameters can be separated of the input values in a linear manner. In order to approximate the truncated impulse responses of the linear dynamic systems and to reduce the number of unknown parameters orthonormal base functions (OBFs) are introduced. As an example the proposed identification method is applied to a MISO system based on forward and backward coupled Hammerstein and Wiener models.

This paper presents uncented Kalman filter technique for identification of non linear dynamic systems. A novel unscented Kalman filter (UKF) has been proposed that has the advantage over EKF since it does not use linearization for state prediction and covariance's and costly calculations of derivatives. This leads to an accurate computation of Kalman gain and error covariance matrices which ultimately leads to an accurate identification of the system. The approach is shown to exhibit robustness characteristics and fast convergence property. A simulation example dealing with applications of the proposed algorithm is given.

In a presented research paper the problem of synthesis of the fault detection unit and failure occurrence locating in linear discrete dynamic time-invariant systems is considered. The result of synthesis is presented in the form of parallel type structure consisting of two independently functioning Kalman filter. The first of them calculates a system state vector estimation without taking note of faults, and the second - a degenerate type, creates a fault estimations. Linear combination of their exits forms the resulting state vector estimation. Both filters have dimensions smaller dimensions of the tested system and use the split procedure of an error differential signal. Splitting of the error signal is carried out before estimation process unlike Kitanidis filter. It allows to get a certain economy in computing costs, due to introduced restrictions and losses in accuracy. In general the obtained structure is suboptimal. Questions of stability and state vector estimation convergence of a dynamic system are briefly considered. Using of the computing resource of MatLab environment results of a functional methodcheck results are given.The article structure is constructed as follows. At first problem definition is executed and its resolvability from the mathematical point of view is analyzed. The following step is synthesis of the detection unit and localization of multiple faults then the convergence and stability of errors of estimation are analyzed. In final sections results of method operability check are given in the form of illustrative numerical example and the results of the performed research are summed up. In the conceptual plan research makes a generalization the known results for the continuous time systems.

Control problems in dynamic systems require optimal selection of input trajectories and the system parameters. In this paper, a novel procedure for optimization of linear dynamic system is proposed that solves simultaneously the parameter design problem and the optimal control problem using a specific system state transformation. Conventional optimization methods are also examined to compare with the proposed method. The limitations and advantages of both methods are discussed in terms of the number of states and inputs. Consequently, linear dynamic system examples are optimized under various constraints and the merits of the proposed method are examined.

This article presents a method of analyzing mixed logic/dynamic systems. The article describes the development of a computational tool for the analysis of piecewise linear (PL) dynamical systems. Unfortunately, many PL controllers are developed from ad hoc "intelligent systems" ideas which do not aim or allow the associated dynamic behavior to be predicted. An example of such a system is the anti-skid braking system in a car, where the controller is rule-based and designed using the engineer's knowledge of the system. The only current viable approach to testing such a system is by using extensive simulation and prototype testing, which must be repeated for each of the different car models on which it is installed. Anything that provides insight into the logic and dynamic interaction of such a system would be useful: hence the development of the work in this article. Similarly, systems with programmable logic controllers and gain schedulers also fall into the class of piecewise linear systems. The authors take ideas and known results from linear systems, convex set theory, and computational geometry and synthesize them to create an analysis tool for studying a class of systems that mix logic and dynamics. We choose to develop a computational analysis tool primarily because the traditional theoretical analysis of piecewise linear processes is intractable, except, that is, for certain local dynamic behavior.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

In the present paper finite-dimensional dynamical control systems described by linear higherorder ordinary differential state equations with multiple point delays in control are considered. Using algebraic methods, necessary and sufficient condition for relative controllability in a given time interval for linear dynamical system with multiple point delays in control is formulated and proved. This condition is generalization to relative controllability case some previous results concerning controllability of linear dynamical systems without multiple point delays in the control. Simple numerical example, which illustrates theoretical result is also given. Finally, some remarks and comments on the existing results for controllability of dynamical systems with delays in control are also presented.

The recursive stochastic algorithms for estimating the parameters of linear discrete-time dynamic systems in the presence of disturbance uncertainty has been considered in the paper. Problems related to the construction of min-max optimal recursive algorithms are demonstrated. In addition, the robustness of the proposed algorithms has been addressed. Since the min-max optimal solution cannot be achieved in practice, an approximate optimal solution based on a recursive stochastic Newton-Raphson type procedure is suggested. The convergence of the proposed practically applicable robustified recursive algorithm is established theoretically using the martingale theory. Both theoretical and experimental analysis related to the practical robustness of the proposed algorithm are also included. .