Discrete-time Stochastic Systems Research Articles

Sampled-Data H∞ Dynamic Output-Feedback Control for NCSs With Successive Packet Losses and Stochastic Sampling.

The problem of sampled-data H∞ dynamic output-feedback control for networked control systems with successive packet losses (SPLs) and stochastic sampling is investigated in this article. The aim of using sampled-data control techniques is to alleviate network congestion. SPLs that occur in the sensor-to-controller (S-C) and controller-to-actuator (C-A) channels are modeled using a packet loss model. Additionally, it is assumed that stochastic sampling follows a Bernoulli distribution. A model is established to capture the stochastic characteristics of both the SPL model and stochastic sampling. This model is crucial as it allows us to determine the probability distribution of the sampling interval between successive update instants, which is essential for stability analysis. An exponential mean-square stability condition for the constructed equivalent discrete-time stochastic system, which also guarantees the prescribed H∞ performance, is established by incorporating probability theory. The desired controller is designed using a step-by-step synthesis approach, which may offer lower design conservatism compared to some existing methods. Finally, our designed approach using a networked F-404 engine system model is validated and its merits relative to existing results are discussed. The proposed method is finally validated by employing a networked model of the F-404 engine system. Furthermore, the advantages of our method are presented in comparison to previous results.

Robust static output feedback stabilization of stochastic discrete-time systems using stochastic geometric dissipativity

This paper delves into the realm of geometrically mean square stable in probability (GMSSiP) for stochastic discrete-time systems (SDTS). To achieve this, we construct a stochastic geometric QSR-dissipativity (SG-QSR-D) inequalities and devise a linear static output feedback (SOF) controller. This framework is instrumental in establishing GMSSiP and asymptotical stability in probability. Additionally, we establish sufficient and necessity conditions for GMSSiP of the SDTS by using SG-QSR-D. Expanding on this foundation, a linear SOF controller is developed by establishing connections between GMSSiP and SG-QSR-D. Specifically, we demonstrate that the asymptotical stability in probability of the closed-loop linear time-invariant (LTI) SDTS is ensured through a linear matrix inequality (LMI) condition and a linear SOF controller. Two illustrative examples are given to show the effectiveness of the obtained results.

Partial Stability in Probability of Nonlinear Stochastic Discrete-Time Systems with Delay

A system of nonlinear stochastic functional-difference equations with finite delay is considered. By assumption, this system has a “partial” trivial equilibrium (with respect to part of the state variables). The problem under study is to analyze partial stability in probability of this equilibrium: stability is considered with respect to part of the variables determining it. The problem is solved using a discrete-stochastic modification of the method of Lyapunov–Krasovskii functionals. Conditions for partial stability in probability are established. An example is provided to illustrate the features of the proposed approach and the rationale for introducing a one-parameter family of functionals.

Event-triggered finite-time $$H_{2}/H_\infty $$ control for discrete-time stochastic mean-field systems

Event-triggered finite-time $$H_{2}/H_\infty $$ control for discrete-time stochastic mean-field systems

Almost sure exponential stability and stochastic stabilization of discrete-time stochastic systems with impulses

This paper considers the almost sure exponential stability and stochastic stabilization problems of discrete-time stochastic systems (DTSSs) with impulsive effects, where the average impulsive interval is taken into account. By using the average impulsive interval approach and the strong law of large numbers, we not only establish the criteria for almost sure exponential stability of general nonlinear discrete-time impulsive stochastic systems (DTISSs) but also design an exact method of a stochastic perturbation to stabilize a given unstable impulsive discrete-time systems. Adopting the average impulsive interval approach and the strong law of large numbers, we established a criterion for almost sure exponential stability of general nonlinear DTISSs. Furthermore, a method of stochastic perturbation has been developed to stabilize an unstable impulsive discrete-time system. Finally, two simulation examples demonstrate the effectiveness of the derived results.

SVD-Based Parameter Identification of Discrete-Time Stochastic Systems with Unknown Exogenous Inputs

This paper addresses the problem of parameter identification for discrete-time stochastic systems with unknown exogenous inputs. These systems form an important class of dynamic stochastic system models used to describe objects and processes under a high level of a priori uncertainty, when it is not possible to make any assumptions about the evolution of the unknown input signal or its statistical properties. The main purpose of this paper is to construct a new SVD-based modification of the existing Gillijns and De Moor filtering algorithm for linear discrete-time stochastic systems with unknown exogenous inputs. Using the theoretical results obtained, we demonstrate how this modified algorithm can be applied to solve the problem of parameter identification. The results of our numerical experiments conducted in MATLAB confirm the effectiveness of the SVD-based parameter identification method that was developed, under conditions of unknown exogenous inputs, compared to maximum likelihood parameter identification when exogenous inputs are known.

On convergence of occupational measures sets of a discrete-time stochastic control system, with applications to averaging of hybrid systems

We establish that, under certain conditions, the set of random occupational measures generated by the state-control trajectories of a discrete-time stochastic system as well as the set of their mathematical expectations converge to a non-random, convex and compact set. We apply these results to the averaging a hybrid system with a slow continuous-time component and a fast discrete-time component. It is shown that the solutions of the hybrid system are approximated by the solutions of a differential inclusion. The novelty of our results is that we allow the state-control space of the fast component to be non-denumerable.

H∞ Controller Design for Networked Systems With Two-Channel Packet Dropouts and FDI Attacks.

In this article, the stochastic analysis and H∞ controller design problems of networked systems with packet dropouts and false data injection attacks are investigated. Different from the existing literature, we focus on the linear networked systems with external disturbances and both sensor-controller channel and controller-actuator channel are studied. First, we present a discrete-time modeling framework that leads to a stochastic closed-loop system with randomly varying parameters. To facilitate the analysis and H∞ control of resulting discrete-time stochastic closed-loop system, an equivalent yet analyzable stochastic augmented model is further constructed by matrix exponential computation. Based on this model, a stability condition is derived in the form of linear matrix inequality (LMI) with the aid of a reduced-order confluent Vandermonde matrix, Kronecker product operation, and law of total expectation. Specifically, the dimension of the LMI obtained in this article does not increase as the upper bound of consecutive packet dropouts does, which is also different from the existing literature. Subsequently, a desired H∞ controller is obtained such that the original discrete-time stochastic closed-loop system is exponentially mean-square stable with a prescribed H∞ performance. Finally, a numerical example and a direct current motor system are exploited to substantiate the effectiveness and practicability of the designed strategy.

Fixed time stability of discrete-time stochastic dynamical systems

In this paper, we address fixed time stability in probability of discrete-time stochastic dynamical systems. Unlike finite time stability in probability, wherein the finite time almost sure convergence behavior of the dynamical system depends on the system initial conditions, fixed time stability in probability involves finite time stability in probability for which the stochastic settling-time is guaranteed to be independent of the system initial conditions. More specifically, we develop Lyapunov theorems for fixed time stability in probability for Itô-type stationary nonlinear stochastic difference equations including a Lyapunov theorem that involves a Lyapunov difference satisfying an exponential inequality of the Lyapunov function that gives rise to a minimum bound on the average stochastic settling-time characterized by the primary and secondary branches of the Lambert W function.

Stability Analysis and Synthesis for Networked Systems With Noisy Sampling Intervals, Random Transmission Delays, and Packet Dropouts

In this article, we study the stability analysis and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> controller synthesis problems for networked systems with random transmission delays and packet dropouts that are subject to noisy sampling intervals. The sampling errors and transmission delays are described by an independent and identically distributed sequence of continuous random variables and the transmission delays are assumed to be smaller than the noisy sampling intervals. First, a closed-loop discrete-time stochastic framework for networked system that allows investigating noisy sampling intervals, random transmission delays and packet dropouts simultaneously is established. To render the discrete-time stochastic system suitable for stability analysis and synthesis, an equivalent yet tractable stochastic augmented model is then reformulated with the help of matrix exponential computation. Based on this augmented model, a stability condition involving the expectation of a nonlinear and random coupling matrix is derived. By recurring to the law of total expectation and Kronecker product operation, the expectation operation of the coupling matrix subject to high nonlinearity and multiple randomness is decoupled. Subsequently, a controller algorithm is designed such that the exponential mean-square stability of the original discrete-time stochastic system with a prescribed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> performance is guaranteed. Finally, an example is provided to illustrate the effectiveness of the controller design algorithm.

On the Kolmogorov constant explicit form in the theory of discrete-time stochastic branching systems

AbstractWe consider a discrete-time population growth system called the Bienaymé–Galton–Watson stochastic branching system. We deal with a noncritical case, in which the per capita offspring mean $m\neq1$ . The famous Kolmogorov theorem asserts that the expectation of the population size in the subcritical case $m&lt;1$ on positive trajectories of the system asymptotically stabilizes and approaches ${1}/\mathcal{K}$ , where $\mathcal{K}$ is called the Kolmogorov constant. The paper is devoted to the search for an explicit expression of this constant depending on the structural parameters of the system. Our argumentation is essentially based on the basic lemma describing the asymptotic expansion of the probability-generating function of the number of individuals. We state this lemma for the noncritical case. Subsequently, we find an extended analogue of the Kolmogorov constant in the noncritical case. An important role in our discussion is also played by the asymptotic properties of transition probabilities of the Q-process and their convergence to invariant measures. Obtaining the explicit form of the extended Kolmogorov constant, we refine several limit theorems of the theory of noncritical branching systems, showing explicit leading terms in the asymptotic expansions.

Optimal consensus control of discrete-time stochastic multi-agent systems

Optimal consensus control of discrete-time stochastic multi-agent systems

Finite-Time Robust Distributed Estimate for Nonlinear Systems With Heterogeneous Sensors.

This article proposes a finite-time distributed state estimation (DSE) algorithm for discrete-time stochastic nonlinear systems with heterogeneous sensors. Considering the network with heterogeneous sensors, the distributed estimate framework is designed by three phases, namely, priori prediction, measurement update, and consensus fusion. To obtain the accurate priori prediction results, the interactive multiple model (IMM) method is adopted to calculate the priori state value in the priori prediction phase. By introducing the measurement probability matrix, a novel heterogeneous measurement information fusion algorithm is designed. Then the measurement information of each sensor is used to update the priori prediction estimates to calculate the estimate results in the measurement update phase. Based on the consensus method, the estimate results of each sensor are fused with consensus weight to calculate the distributed state estimates of nonlinear systems in the consensus fusion phase. Besides, with finite consensus fusion steps, the bounds of the proposed distributed estimate algorithm are proved to be existed. Finally, distributed state estimate simulation example for nonlinear system is set to validate the performance.

Verification of Parametric Properties of Linear Discrete-time Stochastic Systems

We study the problem of verifying parametric properties of linear discrete-time stochastic systems (linear DTSSs) over bounded and unbounded time horizons. A linear DTSS captures random processes, where the one-step transition relation between the current random vector X and the next-step random vector X’ is linear and is given by X’ = AX + W, where A is an n x n matrix and W is a random noise vector. We assume that the initial and noise random vectors are multivariate normal, each of which is expressed as a pair of two parameters, namely, mean and covariance. We are interested in verifying whether the system always satisfies a given parametric property specified as a subset of the parameters’ space over bounded or unbounded time horizons. For bounded time steps, we reduce the problem to the satisfiability of a semidefinite programming problem. For the unbounded time steps, we propose a novel abstraction procedure to reduce the verification problem to that of a finite graph, wherein, the nodes of the graph correspond to the regions of a partition of the random vector space, in contrast to existing works that partition the state-space. More precisely, we partition the parameter space of normal random vectors, namely, the space of means and covariance matrices, and apply semi-definite programming to compute the edges. We show that our abstraction procedure is sound. Our experiments with varying number of vehicle dynamics and different dimensions of random models show that our methods are reasonably scalable.

Active Diagnosis of Incipient Actuator Faults for Stochastic Systems

In this paper, the active diagnosis problem of incipient actuator faults is investigated for a class of discrete-time stochastic systems. The input is exclusively designed for diagnosis purposes to optimally discriminate among multiple mode hypotheses that correspond to the multiple incipient actuator fault scenarios. Due to infeasibility in evaluating the misdiagnosis probability in the closed form, its new upper bound is derived as an alternative. It is proven that this upper bound is a tighter bound on the misdiagnosis probability for incipient actuator faults, compared with the widely used Bhattacharyya upper bound. For computation complexity reduction and diagnosis performance improvement purposes, a novel input design criterion is constructed for active fault diagnosis (AFD) based on the derived upper bound, under which the input design problem can be solved to global optimality. By injecting the optimal input, to fully utilize the diagnostically relevant information in the output data so as to draw a reliable diagnosis conclusion, the resulting measurement output sequence is exploited to make the AFD decision on the basis of the minimum misdiagnosis probability rule. Finally, experimental results on a four-tank system illustrate the effectiveness and superiority of the proposed AFD approach.

On the practical stability of stochastic discrete-time systems

On the practical stability of stochastic discrete-time systems

Robust Estimators for Missing Observations in Linear Discrete-Time Stochastic Systems with Uncertainties

As a first approach to estimating the signal and the state, Theorem 1 proposes recursive least-squares (RLS) Wiener fixed-point smoothing and filtering algorithms that are robust to missing measurements in linear discrete-time stochastic systems with uncertainties. The degraded quantity is given by multiplying the Bernoulli random variable by the degraded signal caused by the uncertainties in the system and observation matrices. The degraded quantity is observed with additional white observation noise. The probability that the degraded signal is present in the observation equation is assumed to be known. The design feature of the proposed robust estimators is the fitting of the degraded signal to a finite-order autoregressive (AR) model. Theorem 1 is transformed into Corollary 1, which expresses the covariance information in a semi-degenerate kernel form. The autocovariance function of the degraded state and the cross-covariance function between the nominal state and the degraded state is expressed in semi-degenerate kernel forms. Theorem 2 shows the robust RLS Wiener fixed-point and filtering algorithms for estimating the signal and state from degraded observations in the second method. The robust estimation algorithm of Theorem 2 has the advantage that, unlike Theorem 1 and the usual studies, it does not use information on the existence probability of the degraded signal. This is a unique feature of Theorem 2.

Parameter Identification of the Discrete-Time Stochastic Systems with Multiplicative and Additive Noises Using the UD-Based State Sensitivity Evaluation

The paper proposes a new method for solving the parameter identification problem for a class of discrete-time linear stochastic systems with multiplicative and additive noises using a numerical gradient-based optimization. The constructed method is based on the application of a covariance UD filter for the above systems and an original method for evaluating state sensitivities within the numerically stable, matrix-orthogonal MWGS transformation. In addition to the numerical stability of the proposed algorithm to machine roundoff errors due to the application of the MWGS-UD orthogonalization procedure at each step, the main advantage of the obtained results is the possibility of analytical calculation of derivatives at a given value of the identified parameter without the need to use finite-difference methods. Numerical experiments demonstrate how the obtained results can be applied to solve the parameter identification problem for the considered stochastic system model.

Closed-Loop Continuous-Time Subspace Identification with Prior Information

This paper presents a closed-loop continuous-time subspace identification method using prior information. Based on a rational inner function, a generalized orthonormal basis can be constructed, and the transformed noises have ergodicity features. The continuous-time stochastic system is converted into a discrete-time stochastic system by using generalized orthogonal basis functions. As is known to all, incorporating prior information into identification strategies can increase the precision of the identified model. To enhance the precision of the identification method, prior information is integrated through the use of constrained least squares, and principal component analysis is adopted to achieve the reliable estimate of the system. Moreover, the identification of open-loop models is the primary intent of the continuous-time system identification approaches. For closed-loop systems, the open-loop subspace identification methods may produce biased results. Principal component analysis, which reliably estimates closed-loop systems, provides a solution to this problem. The restricted least-squares method with an equality constraint is used to incorporate prior information into the impulse response following the principal component analysis. The input–output algebraic equation yielded an optimal multi-step-ahead predictor, and the equality constraints describe the prior information. The effectiveness of the proposed method is provided by numerical simulations.

Moment propagation of polynomial systems through Carleman linearization for probabilistic safety analysis

We develop a method to approximate the moments of a discrete-time stochastic polynomial system. Our method is built upon Carleman linearization with truncation. Specifically, we take a stochastic polynomial system with finitely many states and transform it into an infinite-dimensional system with linear deterministic dynamics, which describe the exact evolution of the moments of the original polynomial system. We then truncate this deterministic system to obtain a finite-dimensional linear system, and use it for moment approximation by iteratively propagating the moments along the finite-dimensional linear dynamics across time. We provide efficient online computation methods for this propagation scheme with several error bounds for the approximation. Our results also show that precise values of certain moments at a given time step can be obtained when the truncated system is sufficiently large. Furthermore, we investigate techniques to reduce the offline computation load using reduced Kronecker power. Based on the obtained approximate moments and their errors, we also provide hyperellipsoidal regions that are safe for some given probability bound. Those bounds allow us to conduct probabilistic safety analysis online through convex optimization. We demonstrate our results on a logistic map with stochastic dynamics and a vehicle dynamics subject to stochastic disturbance.