Linear Dynamical Systems Research Articles

A Deterministic Least Squares Approach for Simultaneous Input and State Estimation

This paper considers a deterministic estimation problem to find the input and state of a linear dynamical system which minimise a weighted integral squared error between the resulting output and the measured output. A completion of squares approach is used to find the unique optimum in terms of the solution of a Riccati differential equation. The optimal estimate is obtained from a two-stage procedure that is reminiscent of the Kalman filter. The first stage is an end-of-interval estimator for the finite horizon which may be solved in real time as the horizon length increases. The second stage computes the unique optimum over a fixed horizon by a backwards integration over the horizon. A related tracking problem is solved in an analogous manner. Making use of the solution to both the estimation and tracking problems a constrained estimation problem is solved which shows that the Riccati equation solution has a least squares interpretation that is analogous to the meaning of the covariance matrix in stochastic filtering. The paper shows that the estimation and tracking problems considered here include the Kalman filter and the linear quadratic regulator as special cases. The infinite horizon case is also considered for both the estimation and tracking problems. Stability and convergence conditions are provided and the optimal solutions are shown to take the form of left inverses of the original system.

Tight Bounds for Uncertain Time-Correlated Errors With Gauss–Markov Structure in Kalman Filtering

Safety-critical navigation applications require that estimation errors be reliably quantified and bounded. This can be challenging for linear dynamic systems if the process noise or measurement errors have uncertain time correlation. In many systems (e.g., in satellite-based or inertial navigation systems), there are sources of time-correlated sensor errors that can be well modeled using Gauss-Markov processes (GMP). However, uncertainty in the GMP parameters, particularly in the correlation time constant, can cause misleading error bounds. In this paper, we develop time-correlated models that ensure tight upper bounds on the estimation error variance, assuming that the actual error is a stationary first-order GMP with a variance and time constant that are only known to reside within an interval. We first use frequency-domain analysis to derive stationary GMP models in both the continuous and discrete time domains, which outperform models previously described in the literature. Then, we derive an even tighter estimation error bound using a non-stationary GMP model, for which we determine the minimum initial variance that guarantees bounding conditions. Both models can easily be implemented in a linear estimator like the Kalman filter.

On stable solution of the problem of disturbance reduction in a linear dynamical system

On stable solution of the problem of disturbance reduction in a linear dynamical system

Improving transfer entropy and partial transfer entropy for relative detection of effective connectivity strength between time series

Information theory-based measures have recently been considered in determining functional and effective connectivity. Transfer Entropy (TE) and Partial Transfer Entropy (Partial TE) are two such measures that investigate causal connectivity in bivariate and multivariate systems, respectively. The existence or absence of causal connectivity between variables is determined through statistical testing that compares TE or Partial TE values under different conditions or between the original TE or Partial TE values and surrogate data. However, using normal TE or Partial TE provides no information about coupling strength between variables. In this study, we present a repeating statistical test method for TE and Partial TE values in linear dynamic systems to determine the relative coupling strength between connectivity in the networks. In this method, segments with different time lengths (N) were considered for each variable in a system. Using the proposed method, the number of segments with statistically significant TE or partial TE values for each pair of time series was counted and analyzed as N increased. Our research results demonstrated that the proposed statistical test distinguishes stronger linear couplings with fewer segments and shorter time lengths (N) from weaker couplings. The proposed method also confirmed that Partial TE is more capable of determining direct causal connectivity than TE. Therefore, our proposed method can not only detect direct causal connections, but also determine the causal effect size of linear connectivity between time series relative to each other in a system, thus improving the performance of TE and Partial TE criteria.

Conditional Probability in Machine Learning

To help teaching of machine learning course, manipulation rules and application examples of conditional probabilities in machine learning are presented. The emphasis is to make a clear distinction between reasonable assumptions and logical deductions developed from assumptions and axioms. The formula for conditional probability of conditional probability is presented with examples in Bayesian coin tossing, Bayesian linear regression, and Gaussian processes for regression and classification. The signal + noise model is formulated in terms of a proposition and exemplified by linear-Gaussian models and linear dynamical systems.

Comparison of Bootstrap Methods for Estimating Causality in Linear Dynamic Systems: A Review.

In this study, we present a thorough comparison of the performance of four different bootstrap methods for assessing the significance of causal analysis in time series data. For this purpose, multivariate simulated data are generated by a linear feedback system. The methods investigated are uncorrelated Phase Randomization Bootstrap (uPRB), which generates surrogate data with no cross-correlation between variables by randomizing the phase in the frequency domain; Time Shift Bootstrap (TSB), which generates surrogate data by randomizing the phase in the time domain; Stationary Bootstrap (SB), which calculates standard errors and constructs confidence regions for weakly dependent stationary observations; and AR-Sieve Bootstrap (ARSB), a resampling method based on AutoRegressive (AR) models that approximates the underlying data-generating process. The uPRB method accurately identifies variable interactions but fails to detect self-feedback in some variables. The TSB method, despite performing worse than uPRB, is unable to detect feedback between certain variables. The SB method gives consistent causality results, although its ability to detect self-feedback decreases, as the mean block width increases. The ARSB method shows superior performance, accurately detecting both self-feedback and causality across all variables. Regarding the analysis of the Impulse Response Function (IRF), only the ARSB method succeeds in detecting both self-feedback and causality in all variables, aligning well with the connectivity diagram. Other methods, however, show considerable variations in detection performance, with some detecting false positives and others only detecting self-feedback.

Speed of convergence to the Perron-Frobenius stationary distribution

In this paper, we study the second eigenvalue problem for non-negative matrices. Using the Hilbert projective metric, we give a new simple proof of the distance between the first and second eigenvalue whenever the spectral radius is a single peripheral eigenvalue. This article focuses on the convergence speed of (finite-dimensional) linear dynamical systems to the Perron-Frobenius stationary distribution. We extend these results to primitive matrices and cooperative ordinary differential equations, and our proof also extends to some non-autonomous discrete-time systems.

Robust virtual sensors design for linear and nonlinear systems under external disturbances

SummaryThe problem of robust virtual sensors design in linear and nonlinear dynamic systems is studied. To design such sensors, two different canonical forms are used: identification and Jordan canonical forms. The problems generally are solved in three steps: at the first step, the linear model invariant with respect to the disturbance is designed; at the second step, a possibility to take into account the nonlinear term and to estimate the given variable is checked; finally, stability of the observer is obtained. The relations allowing to design virtual sensor of minimal dimension estimating the prescribed component of the state vector of the system are obtained. The problem of virtual sensor location is studied and solved. The theoretical results are illustrated by the example of three tank system.

A distributed reduced-order unknown input observer

This paper presents a method of designing distributed observers for linear systems with unknown inputs. Under the requirement of distributed state estimation, outputs of the system are jointly and severally measured by a group of local observer nodes, for the sensing capability of each one is often limited. Therefore, at a local observer, an accurate full state estimation relies on communications with neighbors if the outputs available for use are restricted. In such distributed scenarios, this paper gives conditions for the unknown input decoupling state estimation, under which the designed distributed observer works and has a lower order. For a linear dynamical system of dimension n, the ith local observer is of dimension n−pi if pi-dimensional outputs are available. Each local observer can produce an n-dimensional state estimation with an exponential rate of convergence, free from unknown inputs and more efficient than its full-order counterparts. And beyond that, the design can be done in a robustly completely distributed way.

APPLICATION OF SPLINE FUNCTIONS AND WALSH FUNCTIONS IN PROBLEMS OF PARAMETRIC IDENTIFICATION OF LINEAR NONSTATIONARY SYSTEMS

Context. In this article, a generalized parametric identification procedure for linear nonstationary systems is proposed, which uses spline functions and orthogonal expansion in a series according to the Walsh function system, which makes it possible to find estimates of the desired parameters by minimizing the integral quadratic criterion of discrepancy based on solving a system of linear algebraic equations for a wide class of linear dynamical systems. The accuracy of parameter estimation is ensured by constructing a spline with a given accuracy and choosing the number of terms of the Walsh series expansion when solving systems of linear algebraic equations by the A. N. Tikhonov regularization method. To improve the accuracy of the assessment, an algorithm for adaptive partitioning of the observation interval is proposed. The partitioning criterion is the weighted square of the discrepancy between the state variables of the control object and the state variables of the model. The choice of the number of terms of the expansion into the Walsh series is carried out on the basis of adaptive approximation of non-stationary parameters in the observation interval, based on the specified accuracy of their estimates. The quality of the management of objects with variable parameters is largely determined by the accuracy of the evaluation of their parameters. Hence, obtaining reliable information about the actual nature of parameter changes is undoubtedly an urgent task.
 Objective. Improving the accuracy of parameter estimation of a wide class of linear dynamical systems through the joint use of spline functions and Walsh functions.
 Method. A generalized parametric identification procedure for a wide class of linear dynamical systems is proposed. The choice of the number of terms of the expansion into the Walsh series is made on the basis of the proposed algorithm for adaptive partitioning of the observation interval.
 Results. The results of modeling of specific linear non-stationary systems confirm the effectiveness of using the proposed approaches to estimating non-stationary parameters.
 Conclusions. The joint use of spline functions and Walsh functions makes it possible, based on the proposed generalized parametric identification procedure, to obtain analytically estimated parameters, which is very convenient for subsequent use in the synthesis of optimal controls of real technical objects. This procedure is applicable to a wide class of linear dynamical systems with concentrated and distributed parameters.

Averaging plus learning models and their asymptotics

We develop original models to study interacting agents in financial markets and in social networks. Within these models, randomness is vital as a form of shock or news that decays with time. Agents learn from their observations and learning ability to interpret news or private information in time-varying networks. Under general assumptions on the noise, a limit theorem is developed for the generalized averaging framework for certain type of conditions governing the learning. In this context, the agents’ beliefs (properly scaled) converge in distribution that is not necessarily normal. Fresh insights are gained not only from proposing a new setting for social learning models but also from using different techniques to study discrete-time random linear dynamical systems.

Distributed Estimation and Smoothing for Linear Dynamic Systems Over Sensor Networks

This paper focuses on distributed estimation and smoothing for linear dynamic systems monitored by sensors that are organized as a network. To achieve low power consumption of sensors, the system is decomposed into several subsystems based on the observable canonical decomposition method. Each sensor in the network performs state estimation for one subsystem at most, while estimations for the majority of subsystems are obtained from the neighbors of each sensor. The optimal gains of the estimator and the smoother are pre-calculated offline, and a numerical solution is obtained by taking advantage of optimization algorithms. Furthermore, the boundedness of the prediction error covariance is guaranteed resulting from the bounded solution of a Riccati equation, and the performance of the smoother is analyzed. A numerical example is presented to validate the feasibility of the algorithm even if the sensor network suffers from network interruptions. The proposed algorithm is more robust against partial loss of predictions and recovers more rapidly, compared to some consensus-based distributed estimation algorithms.

A sliding mode technique for reconstruction of unknown control input in linear systems

The unknown input estimation problem for linear dynamic systems exposed to the disturbances is studied in the paper. A novel approach to designing sliding mode observer is suggested for systems not satisfying matching, minimum phase, and detectability conditions. To apply this approach, the reduced-order model of the original system is used. Such a model allows to reduce the dimension of the sliding mode observer, besides the restrictions placed on the original system are relaxed.

Generalize of some concepts on Linear Random Dynamical Systems

This work aims to study the concepts of proximal, distal, minimal, regionally proximal, and uniformly equicontinuous in the linear random dynamical systems with the study of some their properties.

Computing dispersion diagrams and forced responses of arbitrarily varying waveguides

Periodic structures can be used for vibration control thanks to their vibration attenuation stop bands. The dispersion diagram obtained for the structural unit cell can be used to characterize such wave attenuation bands. Several designs using spatially varying unit cells have recently been reported in the literature. Spatially varying properties can also be used to investigate the effect of variability on finite structures. Therefore, various methodologies to model spatially varying structures have been recently developed. This work presents new numerical methodologies to compute (i) dispersion diagrams and (ii) forced responses of one-dimensional spatially varying periodic waveguides. The first method extends previous work using a state-space formulation to compute dispersion relations using a linear time-varying system dynamics approach. The methodology is applied to elementary rods, Saint-Venant shafts, Euler–Bernoulli beams, and Timoshenko beams, and the results are validated with discretized spectral element models and the plane wave expansion method. The proposed approach is shown to be more efficient in cases where the plane wave expansion method presents a slow convergence. The second method is a recursive dynamic condensation to compute forced responses using the dynamic stiffness matrix of the unit cell. The results and computational times of the proposed recursive condensation are compared with the direct assembly of a global dynamic stiffness and its standard dynamic condensation. The two proposed methods are shown to be more efficient and accurate in the computation of dispersion diagrams and forced responses of one-dimensional structures for which the elastodynamic equations are known.

Multiplicity-induced-dominancy for delay systems: Comprehensive examples in the scalar neutral case

This article focuses on the characterization of a particular spectral property called Multiplicity-induced-dominancy applying for linear dynamical systems described by delay-differential equations. More precisely, we characterize the property in the scalar neutral case with respect to the system parameters. Particular attention is paid to the so-called over-order multiplicities corresponding to real double and triple characteristic roots.

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>

Recovering the Graph Underlying Networked Dynamical Systems under Partial Observability: A Deep Learning Approach

We study the problem of graph structure identification, i.e., of recovering the graph of dependencies among time series. We model these time series data as components of the state of linear stochastic networked dynamical systems. We assume partial observability, where the state evolution of only a subset of nodes comprising the network is observed. We propose a new feature-based paradigm: to each pair of nodes, we compute a feature vector from the observed time series. We prove that these features are linearly separable, i.e., there exists a hyperplane that separates the cluster of features associated with connected pairs of nodes from those of disconnected pairs. This renders the features amenable to train a variety of classifiers to perform causal inference. In particular, we use these features to train Convolutional Neural Networks (CNNs). The resulting causal inference mechanism outperforms state-of-the-art counterparts w.r.t. sample-complexity. The trained CNNs generalize well over structurally distinct networks (dense or sparse) and noise-level profiles. Remarkably, they also generalize well to real-world networks while trained over a synthetic network -- namely, a particular realization of a random graph.

Syndetic Sensitivity and Mean Sensitivity for Linear Operators

We study syndetic sensitivity and mean sensitivity for linear dynamical systems. For the syndetic sensitivity aspect, we obtain some properties of syndetic sensitivity for adjoint operators and left multiplication operators. We also show that there exists a linear dynamical system (X×Y,T×S) such that (X×Y,T×S) is cofinitely sensitive but (X,T) and (Y,S) are not syndetically sensitive. For the mean sensitivity aspect, we show that if (Y,S) is sensitive and not mean sensitive, where Y is a complex Banach space, the spectrum of T meets the unit circle. We also obtain some results regarding mean sensitive perturbations.

Stable approximation of SISO and MIMO linear dynamic systems

Stable approximation of SISO and MIMO linear dynamic systems