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This paper considers large-scale linear stochastic systems representing, e.g., spatially discretized stochastic partial differential equations. Since asymptotic stability can often not be ensured in such a stochastic setting (e.g., due to larger noise), the main focus is on establishing model order reduction (MOR) schemes applicable to unstable systems. MOR is vital to reduce the dimension of the problem in order to lower the enormous computational complexity of for instance sampling methods in high dimensions. In particular, a new type of Gramian-based MOR approach is proposed in this paper that can be used in very general settings. The considered Gramians are constructed to identify dominant subspaces of the stochastic system as pointed out in this work. Moreover, they can be computed via Lyapunov equations. However, covariance information of the underlying systems enters these equations which is not directly available. Therefore, efficient sampling-based methods relying on variance reduction techniques are established to derive the required covariances and hence the Gramians. Alternatively, an ansatz to compute the Gramians by deterministic approximations of covariance functions is investigated. An error bound for the studied MOR methods is proved yielding an a priori criterion for the choice of the reduced system dimension. This bound is new and beneficial even in the deterministic case. The paper is concluded by numerical experiments showing the efficiency of the proposed MOR schemes. [ABSTRACT FROM AUTHOR]

The fundamental lemma by Jan C. Willems and co-workers is deeply rooted in behavioral systems theory and it has become one of the supporting pillars of the recent progress on data-driven control and system analysis. This tutorial-style paper combines recent insights into stochastic and descriptor-system formulations of the lemma to further extend and broaden the formal basis for behavioral theory of stochastic linear systems. We show that series expansions – in particular Polynomial Chaos Expansions (PCE) of L 2 -random variables, which date back to Norbert Wiener's seminal work – enable equivalent behavioral characterizations of linear stochastic systems. Specifically, we prove that under mild assumptions the behavior of the dynamics of the L 2 -random variables is equivalent to the behavior of the dynamics of the series expansion coefficients and that it entails the behavior composed of sampled realization trajectories. We also illustrate the short-comings of the behavior associated to the time-evolution of the statistical moments. The paper culminates in the formulation of the stochastic fundamental lemma for linear time-invariant systems, which in turn enables numerically tractable formulations of data-driven stochastic optimal control combining Hankel matrices in realization data (i.e. in measurements) with PCE concepts. [ABSTRACT FROM AUTHOR]

In this paper, the joint input and state estimation problem is considered for linear discrete-time stochastic systems. An event-based transmission scheme is proposed with which the current measurement is released to the estimator only when the difference from the previously transmitted one is greater than a prescribed threshold. The purpose of this paper is to design an event-based recursive input and state estimator such that the estimation error covariances have guaranteed upper bounds at all times. The estimator gains are calculated by solving two constrained optimisation problems and the upper bounds of the estimation error covariances are obtained in form of the solution to Riccati-like difference equations. Special efforts are made on the choices of appropriate scalar parameter sequences in order to reduce the upper bounds. In the special case of linear time-invariant system, sufficient conditions are acquired under which the upper bound of the error covariance of the state estimation is asymptomatically bounded. Numerical simulations are conducted to illustrate the effectiveness of the proposed estimation algorithm. [ABSTRACT FROM AUTHOR]

This paper is written to show a new detection method for integrity attack on the measurements. It is proved that the normality of the residual error from Kalman filter is equivalent to that of the measurement. Our method utilizes this property and it could not only detect the distribution changes at current time, but also detect the its time correlations. Overall, it is an improved detection method for a wider range of attacks. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a network realization theory for a class of mixed quantum–classical linear stochastic systems. Two forms, the standard form and the general form, of this class of linear mixed quantum–classical systems are proposed. Necessary and sufficient conditions for their physical realizability are derived. Based on these physical realizability conditions, a network synthesis theory for this class of linear mixed quantum–classical systems is developed, which clearly exhibits the quantum component, the classical component, and their interface. An example is used to illustrate the theory presented in this paper. [ABSTRACT FROM AUTHOR]

Intermittent faults (IFs) have the properties of unpredictability, non-determinacy, inconsistency and repeatability, switching systems between faulty and healthy status. In this paper, the fault detection and isolation (FDI) problem of IFs in a class of linear stochastic systems is investigated. For the detection and isolation of IFs, it includes: (1) to detect all the appearing time and the disappearing time of an IF; (2) to detect each appearing (disappearing) time of the IF before the subsequent disappearing (appearing) time; (3) to determine where the IFs happen. Based on the outputs of the observers we designed, a novel set of residuals is constructed by using the sliding-time window technique, and two hypothesis tests are proposed to detect all the appearing time and disappearing time of IFs. The isolation problem of IFs is also considered. Furthermore, within a statistical framework, the definition of the diagnosability of IFs is proposed, and a sufficient condition is brought forward for the diagnosability of IFs. Quantitative performance analysis results for the false alarm rate and missing detection rate are discussed, and the influences of some key parameters of the proposed scheme on performance indices such as the false alarm rate and missing detection rate are analysed rigorously. The effectiveness of the proposed scheme is illustrated via a simulation example of an unmanned helicopter longitudinal control system. [ABSTRACT FROM PUBLISHER]

Risk-sensitive identification of AR-processes was first considered in [12]. The purpose of this paper is to extend this original approach to ARMA-processes and even to multi-variable linear stochastic systems. We provide a new definition of a risk-sensitive identification criterion. For this we first consider a recursive identification procedure which is parameterized by a weight-matrix K acting on the stochastic gradient. Using the asymptotic theory of recursive estimation a suitably scaled version of the error process will be approximated by a stationary Gaussian process, see Chapter 4.5, Part II of [1]. The new risk sensitive criterion will be defined in terms of this associated stationary Gaussian process in a familiar manner via an exponential-quadratic cost. The main result of the paper is the minimization of the proposed new criterion with respect to the weight-matrix K over a feasible set EK, see (22), where the cost function is known to be finite, Theorem 6.1. This results will then be extended to the case when minimization over a feasible set E°K is considered, see (26), on the complement of which the cost function is known to be infinite, Theorem 6.1. The starting point of our analysis is an expression of the cost function given in LEQG-theory, in particular a result of [10]. A new expression for the cost function will be also given, using stochastic realization theory, as the mutual information rate between two stochastic processes. [ABSTRACT FROM AUTHOR]

This paper presents a novel method for simultaneous unknown input and state estimation (UISE) with multi-step delay for linear stochastic systems with rank-deficient direct feed-through matrix. The augmented Kalman filter and smoothing are established with an input model, where the input signal follows a specific probability distribution with finite covariance. Then, in light of the established properties of matrix limiting, the infinity filtering is deduced as the covariance matrix of the input signal tends to infinity. Moreover, the infinity smoothing algorithm is also derived to obtain the multi-step delayed unknown input and state estimation. Furthermore, the infinity filtering is strictly proved to coincide with the existing five-step recursive filter (FSRF). It is shown that some previous UISE algorithms, such as the unbiased minimum-variance (UMV) filter and recursive three-step filter (RTSF), are special cases of the proposed infinity filtering algorithm. This work reveals that the existing UMV, RTSF, and FSRF algorithms are essentially the particular limit of augmented Kalman filter and smoothing, which gives a new insight into the area of UISE. [ABSTRACT FROM AUTHOR]

This paper concentrates on the linear least mean square (LLMS) filtered and smoothed estimators for networked linear stochastic systems. Multiple packet losses, Markovian communication constraints, and superposed process noise are considered simultaneously. In order to reduce the channel load during communication, at every step, just one transmission node is permitted to send data packets. Hence, a Markovian communication protocol is utilized to arrange the packets of these transmission nodes. Moreover, multiple data packet dropouts occur during transmission due to an imperfect communication channel. Therefore, the global observation information cannot be obtained by the state estimator. The real state of Markov chain is assumed to be unknown to the estimator except the transition probability matrix. By means of the innovation analysis approach and orthogonal projection principle, we design Kalman-like estimators in a recursive form. Finally, through simulation experiments, we verify the effectiveness and superiority of the designed algorithm. [ABSTRACT FROM AUTHOR]

Here, a new fault detection and isolation (FDI) method based on linear optimisation is proposed for stochastic linear systems with unknown inputs. The stochastic system is discrete and its state and output equations are dependent on unknown inputs. Both stochastic/deterministic unknown inputs (external disturbances) and measurement noise are considered in the dynamic model; therefore, a simple and complete formulation of the robust FDI scheme is presented for stochastic systems. According to the proposed method, first an observable and controllable subsystem is found such that it is independent of the effect of unknown inputs. Then an innovation filter is used for the FDI task. Sufficient conditions to guarantee the stability and observability of the new subsystem are presented based on linear matrix inequalities. Simulation results demonstrate the effectiveness and capability of the proposed method for FDI in stochastic systems with unknown inputs. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies linear systems under sustained random perturbations. The stochastic perturbation model is given by a bounded Markov diffusion process, as it appears, e.g., in the description of load or generation uncertainties of power systems. For such systems, the Lyapunov exponents describe necessary and sufficient conditions for almost sure asymptotic stability. The paper presents several numerical methods for the computation of Lyapunov exponents and applies this methodology to linear oscillators in dimension 2 and 3, and to a one machine – infinite bus electric power system. [Copyright &y& Elsevier]

Abstract In this paper, we examine the role of the switching period on the stochastic stability of jump linear systems. More specifically, we consider a jump linear system in which the state matrix switches every m time steps randomly within a finite set of realizations, without a memory of past switching instances. Through the computation of the Lyapunov exponents, we study δ -moment and almost sure stability of the system. For scalar systems, we demonstrate that almost sure stability is independent of m , while δ -moment stability can be modulated through the selection of the switching period. For higher-dimensional problems, we discover a richer influence of m on stochastic stability, quantified in an almost sure and a δ -moment sense. Through the detailed analysis of an archetypical two-dimensional problem, we illustrate the existence of disconnected windows of opportunity where the system is asymptotically stable. Outside of these windows, the system is unstable, even though it switches between two Schur-stable state matrices. [ABSTRACT FROM AUTHOR]

Abstract In this paper, two iterative algorithms are proposed to solve stochastic algebraic Riccati matrix equations arising in the linear quadratic optimal control problem of linear stochastic systems with state-dependent noise. In the first algorithm, a standard Riccati matrix equation needs to be solved at each iteration step, and in the second algorithm a standard Lyapunov matrix equation needs to be solved at each iteration step. In the proposed algorithms, a weighted average of the estimates in the last and the previous steps is used to update the estimate of the unknown variable at each iteration step. Some properties of the sequences generated by these algorithms under appropriate initial conditions are presented, and the convergence properties of the proposed algorithms are analyzed. Finally, two numerical examples are employed to show the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

A Kalman filtering‐based robust iterative learning control algorithm is proposed in this study for linear stochastic systems with uncertain dynamics and unknown noise statistics. Firstly, a learning gain matrix is designed for the nominal case by minimising the trace of the mean‐square matrix of the input tracking error. Theoretical results show that the proposed algorithm guarantees not only the asymptotic but also monotonic convergence of the input tracking error in the mean‐square error sense, especially when random noises are Gaussian distributed the proposed algorithm is further proved to be asymptotically efficient. In addition, a new mean‐square error constrained approach is presented in designing the robust learning gain matrix, taking into account model uncertainties. A sufficient condition is provided such that the mean‐square matrix of the input tracking error is constrained within a predesigned upper bound which can monotonically converge to zero. Finally, numerical examples considering both structured and unstructured model uncertainties are included to illustrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we characterize random eigenspaces with a non-intrusive method based on the decoupling of random eigenvalues from their corresponding random eigenvectors. This method allows us to estimate the first statistical moments of the random eigenvalues of the system with a reduced number of deterministic finite element computations. The originality of this work is to adapt the method used to estimate each random eigenvalue depending on a global accuracy requirement. This allows us to ensure a minimal computational cost. The stochastic model of the structure is thus reduced by exploiting specific properties of random eigenvectors associated with the random eigenfrequencies being sought. An indicator with no additional computation cost is proposed to identify when the method needs to be enhanced. Finally, a simple three-beam frame and an industrial structure illustrate the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper, Stackelberg games for linear stochastic systems governed by Itô differential equations with multiple followers are investigated. Stackelberg strategies are developed under two different settings, that is, the followers act either cooperatively to attain Pareto optimality or non-cooperatively to arrive at a Nash equilibrium. After establishing the necessary conditions to obtain Stackelberg strategies by using sets of cross-coupled algebraic nonlinear matrix equations (CANMEs), two numerical algorithms are developed based on Newton’s method and the semidefinite programming (SDP) problem to solve the CANMEs. Finally, a numerical example is solved to demonstrate the validity of the algorithms. [ABSTRACT FROM AUTHOR]

Abstract: This paper newly designs the recursive least-squares (RLS) Wiener fixed-lag smoother and filter using the covariance information in linear discrete-time stochastic systems. The estimators require the information of the observation matrix, the system matrix for the state variable, related with the signal, the variance of the state variable, the cross-variance function of the state variable with the observed value and the variance of the white observation noise. It is assumed that the signal is observed with additive white noise. The current fixed-lag smoothing algorithm has a characteristic, as shown in Theorem 1, that the fixed-lag smoothing estimate of the state vector is calculated in the reverse direction of time. [Copyright &y& Elsevier]

Abstract: This paper describes a design for a recursive least-squares Wiener fixed-interval smoother using the covariance information in linear discrete-time stochastic systems. The estimators require information from the observation matrix, the system matrix for the state variable, related to the signal, the variance of the state variable, the cross-variance function of the state variable with the observed value and the variance of the white observation noise. It is assumed that the signal is observed with additive white noise. [Copyright &y& Elsevier]

We consider a variant of the classical linear quadratic Gaussian regulator (LQG) in which penalties on the endpoint state are replaced by the specification of the terminal state distribution. The resulting theory considerably differs from LQG as well as from formulations that bound the probability of violating state constraints. We develop results for optimal state-feedback control in the two cases where i) steering of the state distribution is to take place over a finite window of time with minimum energy, and ii) the goal is to maintain the state at a stationary distribution over an infinite horizon with minimum power. For both problems the distribution of noise and state are Gaussian. In the first case, we show that provided the system is controllable, the state can be steered to any terminal Gaussian distribution over any specified finite time-interval. In the second case, we characterize explicitly the covariance of admissible stationary state distributions that can be maintained with constant state-feedback control. The conditions for optimality are expressed in terms of a system of dynamically coupled Riccati equations in the finite horizon case and in terms of algebraic conditions for the stationary case. In the case where the noise and control share identical input channels, the Riccati equations for finite-horizon steering become homogeneous and can be solved in closed form. The present paper is largely based on our recent work in [1], [2] and presents an overview of certain key results.

This paper newly designs the Wiener fixed-lag smoother and filter using covariance information in linear discrete-time stochastic systems. The estimators require the information of the observation matrix, the system matrix for the state variable, related with the signal, the cross-variance function of the state variable with the observed value and the variance of the white observation noise. It is assumed that the signal is observed with additive white noise. [Copyright &y& Elsevier]

The stochastic trapezoidal rule provides the only equidistant discretization scheme from the family of implicit Euler methods (see [12]) which possesses the same asymptotic (stationary) law as underlying continuous time, linear and autonomous stochastic systems with white or coloured noise. This identity holds even when integration time goes to infinity, independent of used integration step size! Especially, the asymptotic behaviour of first two moments of corresponding probability distributions is rigorously examined and compared in this paper. The coincidence of asymptotic moments is shown for autonomous systems with multiplicative (parametric) and additive noise using fixed point principles and the theory of positive operators. The key result turns out to be useful for adequate implementation of stochastic algorithms applied to numerical solution of autonomous stochastic differential equations. In particular, it has practical importance when accurate long time integration is required such as in the process of estimation of Lyapunov exponents or stationary measures for oscillators in mechanical engineering. [ABSTRACT FROM AUTHOR]

Linear-quadratic Gaussian (LQG) optimal control systems subject to time-varying delay and nonlinear state perturbations are considered. Some robust stability conditions are derived which result in several bounds on the delayed state perturbations so that the uncertain linear-quadratic Gaussian optimal control systems with time-varying delay can remain stable in the sense of uniform ultimate boundedness. The modified Lyapunov equation and the improved Razumikhin-type theorem are employed to investigate such robust stability conditions. Finally, a numerical example is given to demonstrate the validity of the results. [ABSTRACT FROM AUTHOR]

Intensive research in the field of mathematical modeling of hydraulic servo systems has shown that their mathematical models have many important details which cannot be included in the model. Due to impossibility of direct measurement or calculation of dimensions of certain components, leakage coefficients or friction coefficients, it was supposed that parameters of the hydraulic servo system are random (stochastic nature). On the other side, it has been well known that the hydraulic servo cylinder can be approximated by a linear model with time-varying parameters. An estimation of states and time-varying parameters of linear state space models is of practical importance for fault diagnosis and fault tolerant control. Previous works on this topic consider estimation in Gaussian noise environment, but not in the presence of outliers. The known fact is that the measurements have inconsistent observations with the largest part of the observation population (outliers). They can significantly make worse the properties of linearly recursive algorithms which are designed to work in the presence of Gaussian noises. This paper proposes the strategy of parameter-state robust estimation of linear state space models in presence of non-Gaussian noises. The case of robust estimation of states and parameters of linear systems with parameter faults is considered. Because of its good features in robust filtering, the extended Masreliez-Martin filter represents a cornerstone for realization of the robust algorithm. The good features of the proposed robust algorithm to identification of the hydraulic servo cylinder are illustrated by intensive simulations.

The purpose of this work is to pose and solve the problem to guide a collection of weakly interacting dynamical systems (agents, particles, etc.) to a specified terminal distribution. This is formulated as a mean-field game problem, and is discussed in both non-cooperative games and cooperative games settings. In the non-cooperative games setting, a terminal cost is used to accomplish the task; we establish that the map between terminal costs and terminal probability distributions is onto. In the cooperative games setting, the goal is to find a common optimal control that would drive the distribution of the agents to a targeted one. We focus on the cases when the underlying dynamics is linear and the running cost is quadratic. Our approach relies on and extends the theory of optimal mass transport and its generalizations. [ABSTRACT FROM AUTHOR]

The fusion estimation problem for a class of multi‐sensor asynchronous sampling linear stochastic systems with missing measurements is considered, where the state is updated uniformly and each sensor non‐uniformly samples one measurement at most within a state update period. Based on the sampled measurement data of each sensor, the optimal local state estimators are designed at the state and measurement points by using the projection theory. The cross‐covariance matrices between estimation errors of any two local estimators are derived. Based on the obtained local estimators and cross‐covariance matrices, the distributed optimal fusion estimator is given by using matrix‐weighted fusion estimation algorithm in the linear minimum variance sense. A simulation example verifies the effectiveness of the algorithms. [ABSTRACT FROM AUTHOR]

In this brief, a methodology for detection and accommodation of actuator faults for a class of multi-input-multi-output (MIMO) stochastic systems is presented. First, a new real-time fault estimation module that estimates the actuator effectiveness is developed. The actuator fault diagnosis is based on the estimation of the state vector. Under some conditions, the stochastic system is transformed into two separate subsystems. One of them is not affected by actuator faults, so a reduced order Kalman filter can be used to estimate its states. The other, whose states are measurable, is affected by the faults. Then, the output of the nominal controller is reconfigured to compensate for the loss of actuator effectiveness in the system. Simulation results of a helicopter in vertical plane is presented to demonstrate the performance of the proposed fault-tolerant control scheme. [ABSTRACT FROM AUTHOR]

We apply Grenander's method of sieves to the problem of identification or estimation of the“drift” function for linear stochastic systems driven by a fractional Brownian motion (fBm). We use an increasing sequence of finite dimensional subspaces of the parameter space as the natural sieves on which we maximise the likelihood function. [ABSTRACT FROM AUTHOR]