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 Выпуск 102
- Shchegolev A. On the central limit theorem for homogeneous discrete-time nonlinear Markov chains
The class of nonlinear Markov processes is characterized by the dependence of the current state of the process on its current distribution in addition to the dependence on the previous state. Due to this feature, these processes are characterized by complex limit behavior and ergodic properties, for which the usual criteria for Markov processes are not sufficient. Being a subclass of nonlinear Markov processes, nonlinear Markov chains have inherited these features. In this paper, conditions for the fulfillment of the central limit theorem for homogeneous nonlinear Markov chains in discrete time and with a discrete finite state space are studied. Also, a brief review of known results on the ergodic properties of nonlinear Markov chains is included. The obtained result complements the existing results in this area and may be useful for further applications in statistics.
- Zverkina G. Lorden's inequality and the rate of convergence of the distribution of one generalized Erlang -- Sevast'yanov queuing system
It is more important to estimate the rate of convergence to a stationary distribution rather than only to prove the existence one in many applied problems of reliability and queuing theory. This can be done via standard methods, but only under assumptions about an exponential distribution of service time, independent intervals between recovery times, etc. Results for such simplest cases are well-known. Rejection of these assumptions results to rather complex stochastic processes that cannot be studied using standard algorithms. A more sophisticated approach is needed for such processes. That requires generalizations and proofs of some classical results for a more general case. One of them is the generalized Lorden's inequality proved in this paper. We propose the generalized version of this inequality for the case of dependent and arbitrarily distributed intervals between recovery times. This generalization allows to find upper bounds for the rate of convergence for a wide class of complicated processes arising in the theory of reliability. The rate of convergence for a two-component process has been obtained via the generalized Lorden's inequality in this paper.
- Furtat I., Gushchin P., Nguyen B.H., Kolesnik N. Adaptive control with a guarantee of a given performance
The paper presents a modification of the classical output adaptive control algorithm in order to guarantee that the output signal belongs to a given set specified by the developer at any time. Unlike classical adaptive control schemes, where it is impossible to influence control performances, including transient and steady-state performance, it is proposed here to supplement the classical adaptive control procedure with a nonlinear control law to solve these problems. The nonlinear control law is based on the special coordinate transformation of the output variable so that the problem with constraints is reduced to the problem without constraints. For a transformed system without constraints, any existing adaptive control schemes can be applied to stabilize it. Moreover, in new coordinates, it is not required to guarantee the specified performance of transient processes at any time, and the value of the marginal error is not important. This is due to the fact that inverse transformations will always guarantee that the original signals are within the limits specified by the developer. The problem for plants with a relative degree one is solved in order to avoid cumbersome conclusions. However, all the results obtained can be directly extended to plants with an arbitrary relative degree. An example is given that illustrates the effectiveness of the proposed method and confirms the theoretical conclusions.
- Furtat I., Gushchin P., Kopisova E. Nonlinear control laws based on linear ones using odd functions
The paper investigates nonlinear control laws obtained from linear one by two types of coordinate change by using odd functions. The first coordinate change consists in passing each component of the state vector through a nonlinear function, the second coordinate change is in passing the entire linear control law through a nonlinear function. To study such systems, it is proposed to represent nonlinear functions as linear ones with a nonlinear slope. Such a representation will allow using the methods of linear matrix inequalities (LMI) to study the stability of the closed-loop systems. The stability domains and the domains for the initial conditions are found, obtained as a result of a multi-step solution search procedure using LMI. It is shown that the use of the proposed nonlinear control laws makes it possible to reduce the steady-state error compared to the linear one. The simulations illustrate the theoretical conclusions.
- Tyrsin A., Kashcheev S. Risk indicators of cascading failures in interconnected network structures
The behavior of real systems is often stochastic, and the connections between their elements can be adequately described as correlations. In recent years, there have been trends of increasing and complicating modern networks with the growth of their dependence on each other. We observe how several networks are combined into one interdependent network structure. This leads to an increase in the risks that the failure of nodes in one network may lead to the failure of dependent nodes in other networks. As a result of such failures, catastrophic cascade failures can occur in such interconnected network structures. Given the scale of such structures, which are often critical infrastructures, this problem becomes very relevant. The article introduces a scalar measure of the relationship between several arbitrarily distributed continuous random vectors. It allows you to assess the closeness of the relationship between different subsystems (networks) in network structures. Applied to Gaussian model network structures, the influence of the closeness of the relationship between subsystems on the risk of cascading failures has been studied. The probability of such failures was used as the risk value. As an indicator of the risk of cascading failures in the network structure, it is proposed to use the coefficient of correlation between its subsystems. And to reduce the risk of cascading failures in the network structure, it is necessary to reduce the tightness of correlation between the most interconnected elements of subsystems.
- Zadiran K., Shcherbakov M., Sai C. Forecasting of the remaining useful life in conditions of small data sample
In the article a method for forecasting the residual life of equipment using deep learning is proposed. The method is applicable in cases with a small amount of information about data failures, where existing classical methods may not provide the required accuracy. The process of maintaining the equipment in working condition is one of the most important processes in the operation of the equipment. At the same time, the maintenance process often suffers from inefficiency. Therefore, forecasting methods were developed, on the basis of which the concept of proactive maintenance process management was built, which allows optimizing the structure and costs of equipment management throughout the life cycle. However, these methods may show insufficient accuracy if there is not enough data to train them, for example, due to the rarity of equipment failures. To solve this problem, a new prediction method based on deep learning is proposed that can improve the prediction accuracy. In this method, the continuous prediction of the remaining service life over the entire interval is replaced by a model for generating signals containing the calculated prediction.
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