Non-stationary Dynamical Systems Research Articles

Rational dimension of a basis of a regression model for adaptive short-term forecasting the state of a discrete nonstationary dynamic system

Relevance. Today, there are many methodologies for predicting power consumption of various objects. However, there is no a general methodology that is suitable for all types of energy systems, including the sectoral characteristics of small northern settlements and other objects with the stochastic nature of electricity consumption schedules. At the same time, during the development of problem-oriented forecasting methods, it is necessary to take into account computational and statistical features of forecasted time series to the maximum and apply them adequately. The mentioned circumstance prompts the creation of criteria-indicators that allow evaluating the quality of the applied model for solving the forecasting problem, correctness of its construction and correctness of applying a priori information about the object and its physical properties.Aim. Develop and apply the criteria-indicators, which allow evaluating the quality of the forecast regression model and the influence of the dimensionality of such model base on a forecasting error. Methods. The choice of rational dimensionality of the regression model basis for the adaptive forecasting problem is based on the known and developed criteria-indicators. The main provisions of such criteria-indicators were formulated, which provide an assessment of the quality of conditioning of an equivalent square matrix, the presence of uninformative elements of the matrix, and linear dependence of the columns. Results. Based on the analysis of criteria-indicators, the authors selected a rational dimension of the regression model basis for the problem of adaptive short-term forecasting of the state of discrete non-stationary dynamic systems. Conclusions. The authors have previously selected the most promising criteria-indicators and developed a normalized difference factor of diagonal predominance. This allows us to evaluate the influence of the basis size change on the regression model quality when building an approach of adaptive short-term forecasting of electricity consumption by autonomous power systems of small northern settlements on the basis of regression analysis methods. Based on the analysis of criteria-indicators the authors obtained information about the influence of the regression model basis dimension on the forecasting problem solution error. The authors stated the further stages of research to reduce this error. The paper introduces and describes one of the ways to improve the forecasting model quality. The dependence of the forecasting error on the size of the regression model basis were revealed; the criteria-indicators considered in the article were successfully applied. It is confirmed that the pre-selected and developed criteria-indicators make it possible, at the stage of compiling an equivalent square matrix and performing preliminary actions on it, to track changes within the matrix. The changes will lead to improvement in the solution of the problem of adaptive short-term forecasting.

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.

Достаточные условия существования Н∝ -наблюдателя состояния линейных непрерывных динамических систем

<p>The article deals with the problem of finding the observer of the state vector of linear continuous non-stationary dynamical systems with uncertainty of the initial conditions, limited external influences and measurement errors over a finite time interval. Sufficient conditions for the existence of an observer are formulated and proved on the basis of the expansion principle. Relationships are obtained for finding the parameters of the observer and the worst laws of change in external influences and measurement errors. As a limiting case, the problem of observer synthesis for stationary linear dynamical systems on a semi-infinite time interval is considered. Two applied problems of estimating the aircraft state vector based on the results of incomplete and inaccurate measurements are solved.</p>

Convergence rates in the functional CLT for [formula omitted]-mixing triangular arrays

We obtain convergence rates (in the Levi–Prokhorove metric) in the functional central limit theorem (CLT) for partial sums Sn=∑j=1nξj,n of triangular arrays {ξ1,n,ξ2,n,…,ξn,n} satisfying some mixing and moment conditions (which are not necessarily uniform in n). For certain classes of additive functionals of triangular arrays of contracting Markov chains (in the sense of Dobrushin) we obtain rates which are close to the best rates obtained for independent random variables. In addition, we obtain close to optimal rates in the usual CLT and a moderate deviations principle and some Rosenthal type inequalities. We will also discuss applications to some classes of local statistics (e.g. covariance estimators), as well as expanding non-stationary dynamical systems, which can be reduced to non-uniformly mixing triangular arrays by an approximation argument. The main novelty here is that our results are obtained without any assumptions about the growth rate of the variance of Sn. The result are obtained using a certain type of block decomposition, which, in a sense, reduces the problem to the case when the variance of Sn is “not negligible” in comparison with the (new) number of summands.

A review on the numerical and experimental modeling of the floatover installations

Floatover is a large deck installation method that the integrated deck is slowly transferred from the floatover vessel to the pre-installed substructure by falling tide or ballasting vessel. Compared with other installation methods, the floatover method has the advantages of high efficiency, economy and large lifting capacity, which has been widely used in both fixed and floating platforms. The inherently complex problems of floatover systems involve complex hydrodynamic interactions and nonlinear multi-body coupled dynamics. Establishing an accurate numerical or experimental model to investigate the dynamic responses of the vessel, topside, and substructure, the tension of the mooring system, and the impact loads of load buffer devices induced by environmental loads are essential to the design and field operation of floatover installations. In this review, we introduce the state-of-the-art progress of numerical and experimental modeling and analysis of floatover systems made by industrial and academic researchers, including the modeling and analysis of the load buffer devices, hydrodynamic analysis, and the analysis of the nonstationary nonlinear dynamical system. Finally, the major challenges and future research prospects of floatover are provided.

Using machine learning to anticipate tipping points and extrapolate to post-tipping dynamics of non-stationary dynamical systems.

The ability of machine learning (ML) models to "extrapolate" to situations outside of the range spanned by their training data is crucial for predicting the long-term behavior of non-stationary dynamical systems (e.g., prediction of terrestrial climate change), since the future trajectories of such systems may (perhaps after crossing a tipping point) explore regions of state space which were not explored in past time-series measurements used as training data. We investigate the extent to which ML methods can yield useful results by extrapolation of such training data in the task of forecasting non-stationary dynamics, as well as conditions under which such methods fail. In general, we find that ML can be surprisingly effective even in situations that might appear to be extremely challenging, but do (as one would expect) fail when "too much" extrapolation is required. For the latter case, we show that good results can potentially be obtained by combining the ML approach with an available inaccurate conventional model based on scientific knowledge.

Non-stationary dynamical systems; Shadowing theorem and some applications

In the present paper, we mean a sequence of maps along a sequence of spaces by a non-stationary dynamical system. We use an Anosov family as a generalization of an Anosov map, which is a sequence of diffeomorphisms along a sequence of compact Riemannian manifolds, so that the tangent bundles split into expanding and contracting subspaces, with uniform bounds for the contraction and the expansion. Also, we introduce the shadowing property on non-stationary dynamical systems. Then, we prepare the necessary conditions for the existence of the shadowing property to prove the shadowing theorem in nonstationary dynamical systems. The shadowing theorem is a known result in dynamical systems, which states that any dynamical system with a hyperbolic structure has the shadowing property. Here, we prove that the shadowing theorem is established on any invariant Anosov family in a non-stationary dynamical system. Then, as in some applications of the shadowing theorem, we check the stability of Anosov families, and also we peruse the stability of isolated invariant Anosov families in non-stationary dynamical systems.

О некотором классе уравнений Вольтерра I рода в задаче идентификации линейной нестационарной динамической системы

This paper proposes an approach to the identification of a nonstationary linear dynamic system. Its input-output mathematical model is presented as a Volterra equation of the first kind. The problem of nonparametric identification of Volterra kernels is solved on the basis of an active experiment using test piecewise linear signals (that have a rising front). The problem statement is based on the conditions for modeling the dynamics of technical devices in the energy and power industry. The choice of an admissible family of input signals is driven by the complexity of generating piecewise-constant type signals for real energy objects. The original problem is reduced to solving Volterra integral equations of the first kind with two variable integration limits. A formula for the inversion of the integral equations under study is constructed. Sufficient conditions are obtained for the solvability of the corresponding equations with respect to Volterra kernels in the class of continuous functions.

Robust estimation of state vector coordinates in the controlled helicopter motion problem

The problem of finding H∞ – a observer of the state vector of a linear continuous non-stationary dynamical system with a semi-infinite time of functioning is considered. It is assumed that a mathematical model of a closed-loop linear continuous deterministic dynamical system with an optimal linear regulator, found as a result of minimization of the quadratic quality criterion, is known. For solving the state observer synthesis problem the reduction of the problem to a min-max optimal control problem is used. In this problem, the minimum of the quality criterion is sought by the observer’s gain matrix, and the maximum – by the external influence, measurement noise, and initial conditions. To solve this problem, the extension principle is applied and sufficient optimality conditions are obtained that requires the choice of auxiliary functions of the Krotov–Bellman type. As a result of the implementation of the procedure for choosing an auxiliary function and using the rules of matrix differentiation, relations for the synthesis of the observer and formulas for finding the best matrix of observer gains, as well as the laws for choosing the worst external influences and noise, were obtained. We find a solution to the problem of state vector coordinates estimation in the presence of limited external influences and disturbances in a linear model of the measuring system. As an example, the equations of motion of the Raptor-type helicopter are used.

Dynamic Regret Minimization for Control of Non-stationary Linear Dynamical Systems

We consider the problem of controlling a Linear Quadratic Regulator (LQR) system over a finite horizon T with fixed and known cost matrices Q,R, but unknown and non-stationary dynamics At, Bt. The sequence of dynamics matrices can be arbitrary, but with a total variation, VT, assumed to be o(T) and unknown to the controller. Under the assumption that a sequence of stabilizing, but potentially sub-optimal controllers is available for all t, we present an algorithm that achieves the optimal dynamic regret of Õ(VT2/5 T3/5). With piecewise constant dynamics, our algorithm achieves the optimal regret of Õ(√ST) where S is the number of switches. The crux of our algorithm is an adaptive non-stationarity detection strategy, which builds on an approach recently developed for contextual Multi-armed Bandit problems. We also argue that non-adaptive forgetting (e.g., restarting or using sliding window learning with a static window size) may not be regret optimal for the LQR problem, even when the window size is optimally tuned with the knowledge of VT. The main technical challenge in the analysis of our algorithm is to prove that the ordinary least squares (OLS) estimator has a small bias when the parameter to be estimated is non-stationary. Our analysis also highlights that the key motif driving the regret is that the LQR problem is in spirit a bandit problem with linear feedback and locally quadratic cost. This motif is more universal than the LQR problem itself, and therefore we believe our results should find wider application.

Vector-valued almost sure invariance principles for (non)stationary and random dynamical systems

We study the limit behavior of (non)stationary and random chaotic dynamical systems. Several (vector-valued) almost sure invariance principles for (non)stationary dynamical systems and quenched (vector-valued) almost sure invariance principles for random dynamical systems are proved. We also apply our results to stationary chaotic dynamical systems, which admit Young towers, and to (non)uniformly expanding non-stationary and random dynamical systems with intermittencies or uniform spectral gaps. It implies that the systems under study tend to a Brownian motion under various scalings.

Dynamic Regret Minimization for Control of Non-stationary Linear Dynamical Systems

We consider the problem of controlling a Linear Quadratic Regulator (LQR) system over a finite horizon T with fixed and known cost matrices Q,R, but unknown and non-stationary dynamics A_t, B_t. The sequence of dynamics matrices can be arbitrary, but with a total variation, V_T, assumed to be o(T) and unknown to the controller. Under the assumption that a sequence of stabilizing, but potentially sub-optimal controllers is available for all t, we present an algorithm that achieves the optimal dynamic regret of O(V_T^2/5 T^3/5 ). With piecewise constant dynamics, our algorithm achieves the optimal regret of O(sqrtST ) where S is the number of switches. The crux of our algorithm is an adaptive non-stationarity detection strategy, which builds on an approach recently developed for contextual Multi-armed Bandit problems. We also argue that non-adaptive forgetting (e.g., restarting or using sliding window learning with a static window size) may not be regret optimal for the LQR problem, even when the window size is optimally tuned with the knowledge of $V_T$. The main technical challenge in the analysis of our algorithm is to prove that the ordinary least squares (OLS) estimator has a small bias when the parameter to be estimated is non-stationary. Our analysis also highlights that the key motif driving the regret is that the LQR problem is in spirit a bandit problem with linear feedback and locally quadratic cost. This motif is more universal than the LQR problem itself, and therefore we believe our results should find wider application.

Industrial Forecasting with Exponentially Smoothed Recurrent Neural Networks

Time series modeling has entered an era of unprecedented growth in the size and complexity of data which require new modeling approaches. While many new general purpose machine learning approaches have emerged, they remain poorly understand and irreconcilable with more traditional statistical modeling approaches. We present a general class of exponential smoothed recurrent neural networks (RNNs) which are well suited to modeling nonstationary dynamical systems arising in industrial applications. In particular, we analyze their capacity to characterize the nonlinear partial autocorrelation structure of time series and directly capture dynamic effects such as seasonality and trends. Application of exponentially smoothed RNNs to forecasting electricity load, weather data, and stock prices highlight the efficacy of exponential smoothing of the hidden state for multistep time series forecasting. The results also suggest that popular, but more complicated neural network architectures originally designed for speech processing are likely over-engineered for industrial forecasting and light-weight exponentially smoothed architectures, trained in a fraction of the time, capture the salient features while being superior and more robust than simple RNNs and autoregressive models. Additionally, uncertainty quantification of Bayesian exponential smoothed RNNs is shown to provide improved coverage.

Methods for Control of Dynamical Systems with Delayed Feedback

We study the stabilization of linear nonstationary dynamical systems by introducing a linear nonstationary feedback with delay. We obtain sufficient stabilization conditions for the systems and propose a method for constructing stabilizing matrices.

Sufficient conditions for H-infinity control on the finite time interval

The article considers a problem of control of linear non-stationary continuous dynamical systems on a finite time interval. It is assumed that information about the linear output of the controlled system is available instead of the full state vector. The problem of the synthesis of H-infinity regulators is formulated. Based on the ideas of the extension principle, sufficient conditions for H-infinity control are formulated and proved, the best control law for an object and the worst control law for external disturbances with full state feedback are found. Since the exact implementation of the obtained control laws in the absence of complete information about the state is impossible, two approaches to the approximate solution of the problem are proposed. Both approaches are associated with finding an estimate of the state vector and using it in control. The first approach is based on the use of pseudoinverse matrices, which allow finding the estimate based on incoming measurements. The second approach utilizes the idea of using asymptotic observers that ensure that the estimation error tends to zero as well as the asymptotic stability of the closed-loop system. The solution of a model example is given, illustrating the application of the proposed approaches.

Loss of Memory and Moment Bounds for Nonstationary Intermittent Dynamical Systems

We study nonstationary intermittent dynamical systems, such as compositions of a (deterministic) sequence of Pomeau-Manneville maps. We prove two main results: sharp bounds on memory loss, including the "unexpected" faster rate for a large class of measures, and sharp moment bounds for Birkhoff sums and, more generally, "separately H\"older" observables.

Using machine learning to predict statistical properties of non-stationary dynamical processes: System climate,regime transitions, and the effect of stochasticity.

We develop and test machine learning techniques for successfully using past state time series data and knowledge of a time-dependent system parameter to predict the evolution of the "climate" associated with the long-term behavior of a non-stationary dynamical system, where the non-stationary dynamical system is itself unknown. By the term climate, we mean the statistical properties of orbits rather than their precise trajectories in time. By the term non-stationary, we refer to systems that are, themselves, varying with time. We show that our methods perform well on test systems predicting both continuous gradual climate evolution as well as relatively sudden climate changes (which we refer to as "regime transitions"). We consider not only noiseless (i.e., deterministic) non-stationary dynamical systems, but also climate prediction for non-stationary dynamical systems subject to stochastic forcing (i.e., dynamical noise), and we develop a method for handling this latter case. The main conclusion of this paper is that machine learning has great promise as a new and highly effective approach to accomplishing data driven prediction of non-stationary systems.

Control of a non-stationary dynamic system with estimating a strategy of human resources management by the integral indicators method

A general idea research is a lack of articles to estimate system indicator of the effectiveness of the strategy of human resources management (HR) at an economic object (enterprise). We are use the method of integral indicators for a comprehensive assessment of the activities of an economic object (enterprise). The economic object is formalized as a non-stationary dynamic system. The system has a dimension of 1.2 million parameters. The parameters of the object under research (business processes) are compared with staff responsibilities. The sanctions mode is set by blocking staff responsibilities in the interval in each time period t. The integral indicator is calculated according to the standard mode (Strategy 1) of the economic object (enterprise) and without blocking the staff responsibilities. Also, the integral indicator is calculated according to the non-standard operating mode of the enterprise (Strategy 2) with the blocking of staff responsibilities. The difference between the integral indicator of Strategy 2 and Strategy 1 is an estimation of the impact of the imposed sanctions. The resource consumption for the restoration of the normal operation of an economic object after the imposed sanctions is give (61.63 million rubles). Example 2 introduces equipment sanctions from America for a new project. An analysis of the indicators of a new project is carried out with the search and use of analog equipment

An algorithm for determining the state of a non-stationary dynamic system for assessing fire safety control in an enterprise by the method of integrated indicators

Analysis of the scientific literature showed that a lot of work is devoted to assessing the effectiveness of fire safety management in an enterprise. It is worth noting that today there is no universal method for the integrated assessment of fire safety management, taking into account the interconnectedness of all enterprise subsystems and the influence of environmental factors. One of the original approaches to assessing the effectiveness of the fire safety management system is the method of integral indicators. The method of integral indicators is used in the algorithm for analyzing the state of a dynamic non-stationary system for assessing fire safety management in an enterprise. The algorithm is implemented in the author’s complex of programs described in the text of the article. In the simulation, an analysis of 1.2 million values is performed on a well-studied economic object with the spaces identified at each time step: actual data, control and environmental parameters. In the experiment, the basic mode of operation of the enterprise does not contain the implementation of a fire safety management strategy. The research showed significant changes in the values of the integral indicator characterizing the state of the enterprise during the implementation of the fire safety management system at the enterprise.

Recovery of Characteristics of Non-Stationary Dynamic Systems from Three Test Signals

Non-stationary continuous and discrete dynamical systems are modeled, respectively, by Volterra integral equations of the first kind and their discrete analogues. Algorithms for the exact restoration of the impulse response (in analytical form) for continuous systems and the transient characteristic for discrete systems are constructed. We study an algorithm for the exact restoration of the impulse response of a non-stationary continuous dynamic system from three interconnected input signals. It is shown that the first signal can be arbitrary, and the second and third signals are connected by a first integral operator. The dynamic characteristic is chosen to be the exact Laplace transform formula of the impulse response, represented by an algebraic expression from the Laplace transform of the system output signals. A model example illustrating the performance of the presented algorithm is given. The practical application of the presented algorithm is discussed. An algorithm is also constructed for the exact restoration of the transient response of a non-stationary discrete dynamic system from three input signals that are interconnected. It is shown that the first signal can be arbitrary, and the second and third signals are associated with the first summation operator. We present the exact formula for the Z-transform of the transient response, which is represented by an algebraic expression from the Z-transform of the system output signals. A model example is given. The dynamic systems to which the proposed algorithms can be extended are described.