Linear Stochastic Systems Research Articles

Covariance-analytic performance criteria, Hardy-Schatten norms and Wick-like ordering of cascaded systems

This paper is concerned with robust performance analysis of linear stochastic systems acting as an integral operator on a standard Wiener process at the input and producing a stationary Gaussian random process at the output. We propose a performance criterion which uses the trace of an analytic function of the spectral density of the output process. This class of “covariance-analytic” cost functionals includes the usual mean square and risk-sensitive criteria as particular cases. Due to the “cost-shaping” analytic function, the covariance-analytic performance criterion depends on higher-order Hardy-Schatten norms of the system transfer function. We discuss the links of these norms with the asymptotic behaviour of cumulants of finite-horizon quadratic functionals of the system output and their variational properties pertaining to system robustness to statistically uncertain inputs which differ from the standard Wiener process. In the case of strictly proper finite-dimensional systems, governed in state space by linear stochastic differential equations, we develop a method for recursively computing the Hardy-Schatten norms through a recently proposed technique of rearranging cascaded linear systems, which resembles the Wick ordering of mixed products of noncommuting annihilation and creation operators in quantum mechanics. This computational procedure, which is one of the main results of the paper, involves a recurrence sequence of solutions to algebraic Lyapunov equations and represents the covariance-analytic cost as the squared H2-norm of an auxiliary cascaded system. These results are also compared with an alternative approach, which uses higher-order derivatives of stabilising solutions of parameter-dependent algebraic Riccati equations, and illustrated by a numerical example.

Novel parameter estimation method for the systems with colored noises by using the filtering identification idea

Compared with the systems with white noise disturbances, the parameter identification of the systems with colored noises (i.e., correlated noises) is more difficult. In this letter, we use the model transformation to study the identification problem for the systems with colored noises by using the filtering identification idea. The basic idea is to transform a system with colored noise into two identification models with white noises and then to propose a novel parameter estimation algorithms, which can generate more accurate parameter estimates than some related existing identification algorithm. The proposed method can be applied to other linear or nonlinear stochastic systems with different structures and colored disturbance noises. The provided simulation example tests the proposed algorithm.

Extensive numerical simulations on competitive growth between the Edwards–Wilkinson and Kardar–Parisi–Zhang universality classes

Extensive numerical simulations and scaling analysis are performed to investigate competitive growth between the linear and nonlinear stochastic dynamic growth systems, which belong to the Edwards–Wilkinson (EW) and Kardar–Parisi–Zhang (KPZ) universality classes, respectively. The linear growth systems include the EW equation and the model of random deposition with surface relaxation (RDSR), the nonlinear growth systems involve the KPZ equation and typical discrete models including ballistic deposition (BD), etching, and restricted solid on solid (RSOS). The scaling exponents are obtained in both the (1 + 1)- and (2 + 1)-dimensional competitive growth with the nonlinear growth probability p and the linear proportion 1 – p. Our results show that, when p changes from 0 to 1, there exist non-trivial crossover effects from EW to KPZ universality classes based on different competitive growth rules. Furthermore, the growth rate and the porosity are also estimated within various linear and nonlinear growths of cooperation and competition.

Analytical Nonstationary Response of Linear Stochastic MDOF Systems Endowed with Half-Order Fractional Derivative Elements

Analytical Nonstationary Response of Linear Stochastic MDOF Systems Endowed with Half-Order Fractional Derivative Elements

The causality measure of partial mutual information from mixed embedding (PMIME) revisited.

The measure of partial mutual information from mixed embedding (PMIME) is an information theory-based measure to accurately identify the direct and directional coupling, termed Granger causality or simply causality, between the observed variables or subsystems of a high-dimensional dynamical and complex system, without any a priori assumptions about the nature of the coupling relationship. In its core, it is a forward selection procedure that aims to iteratively identify the lag-dependence structure of a given observed variable (response) to all the other observed variables (candidate drivers). This model-free approach is capable of detecting nonlinear interactions, abundantly present in real-world complex systems, and it was shown to perform well on multivariate time series of moderately high dimension. However, the PMIME presents some inefficiencies in its performance mainly when applied on strongly stochastic (linear or nonlinear) systems as it may falsely detect non-existent relationships. Moreover, and by construction, the measure cannot extract purely synergetic relationships present in a system. In the current work, the issue of false detections is addressed by introducing an improved resampling significance test and a procedure of rechecking the identified drivers (backward revision). Regarding the inability to detect synergetic relationships, the PMIME is further enhanced by checking pairs as candidate drivers for the response variable after having considered all drivers individually. The effects of these modifications are investigated in a systematic simulation study on properly designed systems involving strong stochasticity, regressor terms with synergetic effects, and a system dimension ranging from 3 to 30. The overall results of the simulations indicate that these modifications indeed improve the performance of PMIME and alleviate to a significant degree the issues of the original algorithm. Guidelines for balancing between accuracy and computational efficiency are also given, particularly relevant for real-world applications. Finally, the measure performance is investigated in the study of futures of various government bonds and stock market indices in the period around COVID-19 pandemic.

Fault prognosis for linear stochastic systems with intermittent fault and strong noise via health indicator extraction approach

AbstractIn this paper, the problem of fault prognosis is investigated for linear stochastic systems with intermittent faults (IFs) and strong noise. Different from permanent faults, the degradation state of IFs are related to the fault duration time, leading to difficulties in the first step of fault prognosis, that is, the health indicator (HI) extraction. Generally, HI extraction of IFs is achieved by fault detection techniques. Therefore, the proposed fault prognosis approach mainly covers the following steps. First, the HI and degradation model are introduced for linear stochastic systems with IFs. Then, considering dynamic characteristic of systems and influence of strong noise, an IF detection algorithm under strong noise is proposed using weighted moving average methods to extract the HI. Moreover, an alternative HI extraction approach is given by an HI estimation way, which avoids using fault detection techniques. Subsequently, the degradation model is fitted by virtue of HIs and the least square method, and the remaining useful life is predicted with the aid of the obtained model. Finally, an experiment concerning the rotary steerable drilling tool system is provided to verify the effectiveness of the derived results.

Simultaneous input and state estimation with multi-step delay for linear stochastic systems based on infinity filtering and smoothing

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.

On the Optimization Landscape of Dynamic Output Feedback Linear Quadratic Control

The convergence of policy gradient algorithms hinges on the optimization landscape of the underlying optimal control problem. Theoretical insights into these algorithms can often be acquired from analyzing those of linear quadratic control. However, most of the existing literature only considers the optimization landscape for static full-state or output feedback policies (controllers). We investigate the more challenging case of dynamic output-feedback policies for linear quadratic regulation (abbreviated as dLQR), which is prevalent in practice but has a rather complicated optimization landscape. We first show how the dLQR cost varies with the coordinate transformation of the dynamic controller and then derive the optimal transformation for a given observable stabilizing controller. One of our core results is the uniqueness of the stationary point of dLQR when it is observable, which provides an optimality certificate for solving dynamic controllers using policy gradient methods. Moreover, we establish conditions under which dLQR and linear quadratic Gaussian control are equivalent, thus providing a unified viewpoint of optimal control of both deterministic and stochastic linear systems. These results further shed light on designing policy gradient algorithms for more general decision-making problems with partially observed information.

Relatively exact controllability of fractional stochastic neutral system with two incommensurate constant delays

This paper is devoted to analyzing a kind of fractional stochastic neutral system (FSNS). Firstly, by introducing the notion of newly defined two‐parameter Mittag–Leffler matrix function, we derive the solution of the corresponding linear stochastic system. Subsequently, for the linear case, by virtue of the Grammian matrix, we give a suffcient and necessary condition to guarantee the relatively exact controllability for the addressed case. Furthermore, for the nonlinear one, the relatively exact controllability is obtained by fixed point and explore it via Banach contraction principle. Finally, two examples are provided to intensify our theoretical conclusions.

Filtered generalized iterative parameter identification for equation‐error autoregressive models based on the filtering identification idea

SummaryBy using the collected batch data and the iterative search, and based on the filtering identification idea, this article investigates and proposes a filtered multi‐innovation generalized projection‐based iterative identification method, a filtered generalized gradient‐based iterative identification method, a filtered generalized least squares‐based iterative identification method, a filtered multi‐innovation generalized gradient‐based iterative identification method and a filtered multi‐innovation generalized least squares‐based iterative identification method for equation‐error autoregressive systems described by the equation‐error autoregressive models. These filtered generalized iterative identification methods can be extended to other linear and nonlinear scalar and multivariable stochastic systems with colored noises.

Unveiling the principles of stochastic resonance and complex potential functions for bearing fault diagnosis

This paper introduces a novel and complex Unsaturated Piecewise Linear Quad-Stable Stochastic Resonance System (UPLQSR) to address the issue of output saturation in the Classical Quad-Stable Stochastic Resonance (CQSR) system. By linearizing the structure of the potential function, the constraints imposed by high-order terms are effectively eliminated, allowing Brownian particles to move more freely. Numerical simulations demonstrate that UPLQSR achieves significantly higher output signal amplitudes compared to CQSR, highlighting its remarkable signal amplification capability. By analyzing the structure of the potential function, the relationship between the height of the potential barrier and the aggressiveness of the particles jumping is determined. Utilizing adiabatic approximation theory, the paper derives the Steady-state Probability Density (SPD), Mean First Passage Time (MFPT), and Spectral Amplification (SA), revealing the specific process of the particle jumping, as well as the influence of the parameters on the performance of the UPLQSR. After optimizing parameters using the Adaptive Genetic Algorithm (GA), UPLQSR is applied to the early fault diagnosis of various bearing models under Gaussian white noise, demonstrating superior fault detection capabilities. In summary, this study pioneered the theory of non-saturation of quad-stable systems, optimized the weak signal detection technique, and provided a more accurate means of signal identification and analysis, highlighting its great value of application in engineering.

On the Relative Stability of Linear Time Invariant Systems

In this paper, state space representation based approach to relative stability of Linear Time Invariant (LTI) systems is discussed. Specifically, given a bound on the output, relative Bounded Input-Bounded Output (BIBO) stability of LTI systems is discussed. The bound on dominant pole is first determined using input-output description of LTI system. Using state space representation of homogeneous (no external input) LTI system, the components of state vector are upper bound. The results are applied to Continuous Time Markov Chains (i.e. homogeneous stochastic linear systems). Finally, using state space representation of arbitrary (with external input) Linear Time Invariant systems, the location of dominant pole is bounded. Using the upper bound, the location of line (parallel to imaginary-axis jω and to the left of it) is utilized for determining relative stability.

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.

Convergence Analysis of Ensemble Filters for Linear Stochastic Systems with Poisson-Sampled Observations

For continuous-time linear stochastic dynamical systems driven by Wiener processes, we consider the problem of designing ensemble filters when the observation process is randomly time-sampled. We propose a continuous-discrete McKean–Vlasov type diffusion process with additive Gaussian noise in observation model, which is used to describe the evolution of the individual particles in the ensemble. These particles are coupled through the empirical covariance and require less computations for implementation than the optimal ones based on solving Riccati differential equations. Using appropriate analysis tools, we show that the empirical mean and the sample covariance of the ensemble filter converges to the mean and covariance of the optimal filter if the mean sampling rate of the observation process satisfies certain bounds and as the number of particles tends to infinity.

New Approach to Feedback Stabilization of Linear Discrete Time-Varying Stochastic Systems

New Approach to Feedback Stabilization of Linear Discrete Time-Varying Stochastic Systems

Filtering of hidden Markov renewal processes by continuous and counting observations

<p>This paper introduces a subclass of Markov renewal processes (MRPs) and presents a solution to the optimal filtering problem in a stochastic observation system, where the state is modeled by an MRP and observed indirectly through noisy measurements. The MRPs considered here can be interpreted as continuous-time Markov chains (CTMCs) with a finite set of abstract states representing distributions of random vectors. The paper outlines the probabilistic properties of MRPs, emphasizing the ability to express any arbitrary function of the MRP as the solution to a linear stochastic differential system (SDS) with a martingale on the right-hand side (RHS). Using these properties, an optimal filtering problem is formulated in stochastic observation systems, where the hidden state belongs to the class of MRPs, and the observations consist of both diffusion and counting components. The drift terms in all observations depend on the system state. An optimal filtering estimate for a scalar function of the MRP is provided through the solution of an SDS with innovation processes on the RHS. Additionally, the paper presents a version of the Kushner-Stratonovich equation, describing the evolution of the conditional probability density function (PDF). To demonstrate the practical application of the estimation method, the paper presents a communications-related example, focusing on monitoring the qualitative state and numerical characteristics of a network channel using noisy observations of round-trip time (RTT) and packet loss flow. The paper also highlights the robustness of the filtering algorithm in scenarios where the MRP distribution is uncertain.</p>

ANALYSIS OF GLOBAL EXPONENTIAL STABILITY FOR NONLINEAR IMPULSIVE STOCHASTIC SYSTEMS WITH DELAY USING REGULARIZATION TECHNIQUES

The W-transform in the form of the regularization techniques, based on inverse-positive matrices, is used to obtain verifiable global stability conditions for linear and nonlinear stochastic delay systems with random coefficients and impulses at random times. The results are conveniently formulated in terms of coefficients of the systems. Several examples illustrating the main results are offered.

СТОХАСТИЧНА ДЕСКРИПТОРНА ГРА ПЕРЕСЛІДУВАННЯ

A differential pursuit game in a stochastic descriptor linear system is analyzed. The system dynamics is described by Ito’s stochastic differential algebraic equation. Solutions of the equation are presented by the formula of variation of constants in terms of the initial data and control unit. Constraints on the support functionals of two sets defined by the behaviors of the pursuer and evader are used to obtain the game completion conditions. The method of resolving functions is applied to construct a pursuer control bringing the dynamic vector of the system to a terminal set. The results are illustrated by an example of a stochastic descriptor system that describes transient states in a radio technical filter with random perturbations in the form of white noise. Keywords: stochastic differential algebraic equation, Wiener process, descriptor system, differential game, radio technical filter, white noise.

Output Regulation of Linear Stochastic Systems With Unstable Linear Exogenous Systems

Output Regulation of Linear Stochastic Systems With Unstable Linear Exogenous Systems

Optimal Covariance Steering for Continuous-Time Linear Stochastic Systems With Multiplicative Noise

Optimal Covariance Steering for Continuous-Time Linear Stochastic Systems With Multiplicative Noise