Unknown Matrices Research Articles

A data-driven, physics-informed framework for forecasting the spatiotemporal evolution of chaotic dynamics with nonlinearities modeled as exogenous forcings

We introduce a data-driven, linear method for the spatiotemporal prediction of high-dimensional and chaotic dynamical systems. In this method, the observables are vector-valued and delay-embedded, and the nonlinearities are treated as external forcings. The proper representation of the nonlinear terms is found in a physics-informed way or a purely data-driven fashion, depending on whether any knowledge of governing dynamics is available or not. The unknown matrices appearing in the method are found using the dynamic mode decomposition with control technique. The distinctive features of the method enable it to accurately capture the system's dynamically important unstable modes while suppressing their unbounded growth via the included nonlinearities. The predictive capabilities of the method are demonstrated for well-known prototypes of chaotic dynamics such as the Kuramoto-Sivashinsky equation and Lorenz-96 system, for which the method predictions are accurate for several Lyapunov timescales. Similar performance is shown for two-dimensional lid-driven cavity flows at high Reynolds numbers.

Triangular Schlesinger systems and superelliptic curves

We study the Schlesinger system of partial differential equations in the case when the unknown matrices of arbitrary size (p×p) are triangular and the eigenvalues of each matrix form an arithmetic progression with a rational difference q, the same for all matrices. We show that such a system possesses a family of solutions expressed via periods of meromorphic differentials on the Riemann surfaces of superelliptic curves. We determine the values of the difference q, for which our solutions lead to explicit polynomial or rational solutions of the Schlesinger system. As an application of the (2×2)-case, we obtain explicit sequences of rational solutions and of one-parameter families of rational solutions of Painlevé VI equations. Using similar methods, we provide algebraic solutions of particular Garnier systems.

Guaranteed Synchronization Performance Control of Nonlinear Time-Delay MIMO Multiagent Systems With Actuator Faults.

This paper addresses the synchronization control problem of leader-follower multiagent systems with each follower described by a class of high-order nonlinear multiple-input-multiple-output (MIMO) dynamics in the presence of time delays and actuator faults. A distributed synchronization scheme with guaranteed synchronization performance based on the radial basis function neural network (RBF NN) is introduced. We propose an augmented quadratic Lyapunov function by incorporating the lower bounds of control gain matrices and the actuator healthy indicator, and the problems caused by the unknown time-varying control gain matrices, actuator faults, and coupling terms among agents are solved. Meanwhile, the output of followers can track that of the leader and the steady state, and the transient performance of synchronization can be guaranteed, while all the other signals in the closed-loop system are guaranteed to be bounded. Finally, numerical analysis has been carried out to verify the effectiveness of the proposed controller.

Robust Recursive Filtering for Stochastic Systems With Time-Correlated Fading Channels

This article is concerned with the robust recursive filtering (RF) problem for a class of stochastic uncertain systems subject to time-correlated fading channels. The measurement received by the sensor is transmitted to the remote filter through the time-correlated fading channel where the channel coefficient evolves according to a certain dynamics and hence exhibits a time-correlated nature. The parameter uncertainties of the system are described by norm-bounded unknown matrices. By introducing a class of auxiliary variables, an augmented system is constructed to reflect the dynamics of the fading coefficient and state simultaneously. Then, a recursive filter is designed which is capable of online computation. Furthermore, an upper bound is guaranteed for the filtering error covariance (FEC) for the possible parameter uncertainties as well as the time-correlated fading channels. With the help of the completing-the-squares technique, filter gains are parameterized by minimizing the obtained upper bound. Finally, two examples are employed to verify the effectiveness of the proposed robust RF method.

Минимаксные среднеквадратические оценки матричных параметров в задачах линейной регрессии в условиях неопределенности

Исследована проблема оценки параметров в задачах линейной регрессии со случайными матричными коэффициентами. При условии, что наблюдаются случайные линейные функции от неизвестных матриц со случайными погрешностями, имеющими неизвестные корреляционные матрицы, исследованы задачи гарантированного среднеквадратичного оценивания линейных функций от матриц. Получены оценки сверху и снизу гарантированных среднеквадратических погрешностей линейных оценок по наблюдениям линейных функций от матриц в том случае, когда известны множества, которым принадлежат неизвестные матрицы и корреляционные матрицы погрешностей наблюдений. Установлено, что в некотором частном случае такие оценки являются точными. При предположении, что множества ограничены, выпуклы и замкнуты, получены более точные двусторонние оценки для гарантированных погрешностей. Найдены условия, когда гарантированные среднеквадратические погрешности приближаются к нулю при увеличении количества наблюдений. Приведены необходимые и достаточные условия несовмещенности линейных оценок линейных функций от матриц. Введено понятие квазиоптимальных оценок для линейных функций от матриц и доказано, что в классе несмещенных оценок квазиоптимальные оценки существуют и единые. Для таких оценок получены условия сходимости к нулю гарантированных среднеквадратических погрешностей. Также для линейных оценок неизвестных матриц введено понятие квазимимимаксных оценок и доказано, что они не смещены. Для специальных множеств, которым принадлежат неизвестная матрица и корреляционные матрицы погрешностей наблюдений, такие оценки выражены из-за решения линейных операторных уравнений в конечномерном пространстве. Для квазимимимаксных оценок при определенных предположениях определен вид гарантированной среднеквадратичной погрешности оценки неизвестной матрицы. Показано, что такие погрешности ограничиваются сверху суммой следов известных матриц. Приведен пример нахождения минимаксной несмещенной линейной оценки для специального вида случайных матриц, входящих в уравнение наблюдения.

Analyzing Raman spectral data without separabiliy assumption

Raman spectroscopy is a well established tool for the analysis of vibration spectra, which then allow for the determination of individual substances in a chemical sample, or for their phase transitions. In the time-resolved-Raman-sprectroscopy the vibration spectra of a chemical sample are recorded sequentially over a time interval, such that conclusions for intermediate products (transients) can be drawn within a chemical process. The observed data-matrix M from a Raman spectroscopy can be regarded as a matrix product of two unknown matrices W and H, where the first is representing the contribution of the spectra and the latter represents the chemical spectra. One approach for obtaining W and H is the non-negative matrix factorization. We propose a novel approach, which does not need the commonly used separability assumption. The performance of this approach is shown on a real world chemical example.

Reduced-dimensional reinforcement learning control using singular perturbation approximations

We present a set of model-free, reduced-dimensional reinforcement learning (RL) based optimal control designs for linear time-invariant singularly perturbed (SP) systems. We first present a state feedback and an output feedback based RL control design for a generic SP system with unknown state and input matrices. We take advantage of the underlying time-scale separation property of the plant to learn a linear quadratic regulator (LQR) for only its slow dynamics, thereby saving significant amount of learning time compared to the conventional full-dimensional RL controller. We analyze the sub-optimality of the designs using SP approximation theorems, and provide sufficient conditions for closed-loop stability. Thereafter, we extend both designs to clustered multi-agent consensus networks, where the SP property reflects through clustering. We develop both centralized and cluster-wise block-decentralized RL controllers for such networks, in reduced dimensions. We demonstrate the details of the implementation of these controllers using simulations of relevant numerical examples, and compare them with conventional RL designs to show the computational benefits of our approach.

An integral architecture for identification of continuous-time state-space LPV models

Abstract This paper presents an integral architecture for direct identification of continuous-time linear parameter-varying (LPV) state-space models. The main building block of the proposed architecture consist of an LPV model followed by an integral block, which is used to approximate the continuous-time state map of an LPV representation. The unknown LPV model matrices are estimated along with the state sequence by minimizing a properly constructed dual-objective criterion. A coordinate descent algorithm is employed to optimize the desired objective, which alternates between computing the unknown LPV matrices and estimating the state sequence. Thanks to the linear parametric structure induced by the LPV models, the unknown parameters within each coordinate descent step can be computed analytically via ordinary least squares. The effectiveness of the proposed methodology is assessed via a numerical example.

ПРОСТОРОВО-РОЗНЕСЕНА ПЕРЕДАЧА СИГНАЛІВ У ЦИФРОВИХ ТРОПОСФЕРНИХ СТАНЦІЯХ

The article examines the possibility of increasing the normalized throughput of digital troposcatter stations on the way of using space-separated signal transmission. Shown is a block diagram of the transmitting path of a digital troposcatter station with space-separated signal transmission. Methods of separation at the reception of signals acting at the input of the space-time coding device of the transmission path of the microwave digital troposcatter station are analyzed. Three methods of signal addition are considered in detail: linear addition; auto selection; optimal (quasi-optimal) addition. Various variants of addition are analyzed: before the demodulator and after the demodulator. The calculation of the normalized channel capacity as a function of the signal-to-noise ratio for the case of two transmitting antennas with known and unknown channel matrices is carried out. It was found that with two transmitting antennas and an unknown channel matrix, the normalized bandwidth does not differ from the case of using one transmitting antenna. The effect of cross-polarization isolation on the normalized channel capacity is analyzed, when one transmitting antenna emits signals of horizontal polarization, and the second transmitting antenna emits signals of vertical polarization. The values of the error probability are obtained for spatially-separated signal transmission to two transmitting antennas for two laws of distribution of a random variable - Rayleigh and Rice. The Rice distribution law of a random variable, the error probability is expressed through the modified zero-order Bessel function. The calculated data are shown in the table. The calculation was carried out in two frequency ranges allowed for the use of troposcatter communication facilities and for a channel without fading and for a channel with intersymbol interference.

Approximate guaranteed estimates for matrices in linear regression problems with a small parameter

The problem of finding linear unbiased estimates of the linear operator of unknown matrices — components of the observations vector, is investigated. It is assumed that the observation vector additively depends on a random vector with zero expected value, and the unknown correlation matrix belongs to a known bounded set. For the introduced class of linear estimates, necessary and sufficient conditions for the existence of solutions of operator equations that determine the unknown parameters of the vector estimate, are proved. The form of the guaranteed mean square error of the estimate is introduced on the sets of constraints of the problem parameters. The influence on the linear unbiased estimate of small perturbations of known rectangular matrices, which are the composites of the observations vector components, is also investigated. The analytical form is given through the parameters of the perturbed set of singularities for the introduced special operators that depend on a small parameter, which determine the corresponding operator equations, as well as their approximate solutions, in the first approximation of the small parameter method. A test example of solving the problem of finding a linear unbiased estimate under the condition of perturbation of both linearly independent and linearly dependent known observation matrices is presented.

Simultaneous diagonalization of incomplete matrices and applications

We consider the problem of recovering the entries of diagonal matrices $\{U_a\}_a$ for $a = 1,\ldots,t$ from multiple incomplete samples $\{W_a\}_a$ of the form $W_a=PU_aQ$, where $P$ and $Q$ are unknown matrices of low rank. We devise practical algorithms for this problem depending on the ranks of $P$ and $Q$. This problem finds its motivation in cryptanalysis: we show how to significantly improve previous algorithms for solving the approximate common divisor problem and breaking CLT13 cryptographic multilinear maps.

Гарантованные среднеквадратические оценки линейных преобразований матриц в условиях статистической неопределенности

Линейная оценка наблюдений в условиях погрешностей разного вида с целью получения несмещаемых оценок является предметом исследования многочисленных научных публикаций. Задача линейного регрессионного анализа в условиях, когда элементами векторных наблюдений являются известные матрицы, допускающие малые отклонения от расчетных, исследовались в предыдущих публикациях авторов. С использованием технологии псевдообращенных операторов, а также метода возмущения задача была решена при условии, что мало возмущенными были линейно независимые матрицы наблюдений. Параметры линейных отметок были представлены в виде расписаний по малому параметру. Решения задач линейной оценки в условиях неопределенности в течение последних десятилетий осуществляются в рамках известного метода минимаксной оценки. Формально задачи, которые возникают в этом направлении решаются при наличии некоторых пространств для неизвестных параметров наблюдения, а также пространств, которым могут принадлежать погрешности наблюдений. Коэффициенты линейных оценок определяются в процессе оптимизации гарантированной среднеквадратичной погрешности искомой оценки. Таким образом, предметом научных исследований могут быть задачи линейного оценивания неизвестных прямоугольных матриц по наблюдениям с погрешностями с неизвестными корреляционными матрицами: неизвестные матрицы принадлежат какому-либо ограниченному пространству, корреляционные матрицы случайных возмущений вектора наблюдений неизвестны, но можно предположить случайно. ограниченном пространстве. Некоторые постановки задач линейной оценки наблюдений исследованы в предлагаемой публикации. Рассмотрена задача линейной оценки для вектора наблюдений специального вида, компоненты которого известны прямоугольные матрицы, которые подаются с малыми возмущениями. Предложены варианты постановки задачи, позволяющие получить в первом приближении малого параметра аналитическое решение. Приведен тестовый пример.

Distributed functional observers for fractional-order time-varying interconnected time-delay systems

Our main purpose in this paper is to design distributed functional observers for a class of fractional-order time-varying interconnected time-delay systems. The contributions of this paper conclude three aspects. First, we propose novel functional observers for subsystems of the fractional-order time-varying interconnected time-delay systems. The designed distributed functional observers in this paper work for a wider class of interconnected time-delay systems in the sense that when some existing design methods cannot be applied to estimate linear functions of the state vector, the distributed functional observers in this paper can still estimate linear functions of the original state vector. Then, we establish existence conditions for the proposed functional observers. Finally, we provide effective algorithms for computing unknown observer matrices.

Model-Free Reinforcement Learning of Minimal-Cost Variance Control

This letter proposes two reinforcement learning (RL) algorithms for solving a class of coupled algebraic Riccati equations (CARE) for linear stochastic dynamic systems with unknown state and input matrices. The CARE are formulated for a minimal-cost variance (MCV) control problem that aims to minimize the variance of a cost function while keeping its mean at an acceptable range using a noisy infinite-horizon full-state feedback linear quadratic regulator (LQR). We propose two RL algorithms where the input matrix can be estimated at the very first iteration. This, in turn, frees up significant amount of computational complexity in the intermediate steps of the learning phase by avoiding repeated matrix inversion of a high-dimensional data matrix. The overall complexity is shown to be less than RL for both stochastic and deterministic LQR. Additionally, the disturbance noise entering the model is not required to satisfy any condition for ensuring efficiency of either RL algorithms. Simulation examples are presented to illustrate the effectiveness of the two designs.

Fault Detection Finite-Time Filter Design for T–S Fuzzy Markovian Jump System with Missing Measurements

The primary purpose of this work was to address the problem of finite-time fault detection filtering for a class of discrete-time Takagi–Sugeno (T–S) fuzzy Markovian jump systems subject to randomly occurring uncertainties, time-varying delay, missing measurements and partly unknown transition probability matrices. Precisely the missing measurement phenomenon in the network environment satisfies the Bernoulli distributed white noise sequences. Firstly a fuzzy rule-dependent filter is constructed for estimating the unmeasured states of the system and the corresponding fault detection problem. Further, based on the filtering problem, the error between residual and fault is minimized with the prescribed strict $$(\mathsf {Q},\mathsf {S},\mathsf {R})$$ - $$\gamma $$ dissipativity performance. Secondly, the sufficient criteria are derived to ensure the filtering error system to be finite-time stochastic bounded. Finally, the applicability and usefulness of the proposed filter design through two numerical examples are verified.

Joint Bit Allocation and Hybrid Beamforming Optimization for Energy Efficient Millimeter Wave MIMO Systems

In this article, we aim to design highly energy efficient end-to-end communication for millimeter wave multiple-input multiple-output systems. This is done by jointly optimizing the digital-to-analog converter (DAC)/analog-to-digital converter (ADC) bit resolutions and hybrid beamforming matrices. The novel decomposition of the hybrid precoder and the hybrid combiner to three parts is introduced at the transmitter (TX) and the receiver (RX), respectively, representing the analog precoder/combiner matrix, the DAC/ADC bit resolution matrix and the baseband precoder/combiner matrix. The unknown matrices are computed as a solution to the matrix factorization problem where the optimal fully digital precoder or combiner is approximated by the product of these matrices. A novel and efficient solution based on the alternating direction method of multipliers is proposed to solve these problems at both the TX and the RX. The simulation results show that the proposed solution, where the DAC/ADC bit allocation is dynamic during operation, achieves higher energy efficiency when compared with existing benchmark techniques that use fixed DAC/ADC bit resolutions.

Uncovering heterogeneous social effects in binary choices

We identify and estimate heterogeneous social effects within groups of individuals that make binary choices. These heterogeneous social effects, which include peer and contextual effects, are modeled through unobserved influence matrices that summarize how the members within each group affect each other’s outcomes. We recover parameters in social effects as well as the unknown influence matrices by exploiting how these matrices are linked to the reduced-form effects of multiple characteristics. Monte Carlo experiments show that a nested fixed-point maximum-likelihood estimator for the social effects has good finite-sample performance. Using a new dataset, we analyze how college roommates influence each other’s decisions to participate in volunteering activities. Our estimates reveal substantial heterogeneity in the social effects among these students.

Secure synchronization control for a class of cyber-physical systems with unknown dynamics

This paper investigates the secure synchronization control problem for a class of cyber-physical systems &#x0028 CPSs &#x0029 with unknown system matrices and intermittent denial-of-service &#x0028 DoS &#x0029 attacks. For the attack free case, an optimal control law consisting of a feedback control and a compensated feedforward control is proposed to achieve the synchronization, and the feedback control gain matrix is learned by iteratively solving an algebraic Riccati equation &#x0028 ARE &#x0029 . For considering the attack cases, it is difficult to perform the stability analysis of the synchronization errors by using the existing Lyapunov function method due to the presence of unknown system matrices. In order to overcome this difficulty, a matrix polynomial replacement method is given and it is shown that, the proposed optimal control law can still guarantee the asymptotical convergence of synchronization errors if two inequality conditions related with the DoS attacks hold. Finally, two examples are given to illustrate the effectiveness of the proposed approaches.

Blind Kalman Filtering for Short-Term Load Forecasting

In this work we address the problem of short-term load forecasting. We propose a generalization of the linear state-space model where the evolution of the state and the observation matrices is unknown. The proposed blind Kalman filter algorithm proceeds via alternating the estimation of these unknown matrices and the inference of the state, within the framework of expectation-maximization. A mini-batch processing strategy is introduced to allow on-the-fly forecasting. The experimental results show that the proposed method outperforms the state-of-the-art techniques by a considerable margin, both on load profile estimation and peak load forecast problems.

Functional interval observer design for singular fractional-order systems with disturbances

In this study, we consider the problem of designing functional interval observers for a class of singular fractional-order systems. The goal of this work is to design not a full state interval observer for singular fractional-order systems, but to design an observer for state functions of this kind of systems. Conditions for the existence of such functional interval observers are given and an effective algorithm for computing unknown observer matrices is provided in this study. Two numerical examples and simulation results are provided to illustrate the effectiveness of the proposed design method.