Discrete-time Observer Research Articles

Approximation of stochastic differential equations driven by subfractional Brownian motion at discrete time observation

In this paper, we consider discrete time approximations for stochastic differential equations with the form: where is a continuous function with locally bounded variation, are measurable functions, and the integral with respect to is the pathwise Riemann-Stieltjes integral, SH is a subfractional Brownian motion with σ is a deterministic (possibly discontinuous) function.

A Kalman particle filter for online parameter estimation with applications to affine models

In this paper we address the problem of estimating the posterior distribution of the static parameters of a continuous-time state space model with discrete-time observations by an algorithm that combines the Kalman filter and a particle filter. The proposed algorithm is semi-recursive and has a two layer structure, in which the outer layer provides the estimation of the posterior distribution of the unknown parameters and the inner layer provides the estimation of the posterior distribution of the state variables. This algorithm has a similar structure as the so-called recursive nested particle filter, but unlike the latter filter, in which both layers use a particle filter, our algorithm introduces a dynamic kernel to sample the parameter particles in the outer layer to obtain a higher convergence speed. Moreover, this algorithm also implements the Kalman filter in the inner layer to reduce the computational time. This algorithm can also be used to estimate the parameters that suddenly change value. We prove that, for a state space model with a certain structure, the estimated posterior distribution of the unknown parameters and the state variables converge to the actual distribution in L^p with rate of order {mathcal {O}}(N^{-frac{1}{2}}+varDelta ^{frac{1}{2}}), where N is the number of particles for the parameters in the outer layer and varDelta is the maximum time step between two consecutive observations. We present numerical results of the implementation of this algorithm, in particularly we implement this algorithm for affine interest models, possibly with stochastic volatility, although the algorithm can be applied to a much broader class of models.

Improved Results on Stabilization of $G$-SDEs by Feedback Control Based on Discrete-Time Observations

We provide feasible criteria for stabilization of stochastic differential equations driven by $G$-Brownian motion based on discrete-time observations. We extend the results in Ren, Yin, and Sakthiv...

McKean--Vlasov SDEs in Nonlinear Filtering

Various particle filters have been proposed over the last couple of decades with the common feature that the update step is governed by a type of control law. This feature makes them an attractive alternative to traditional sequential Monte Carlo which scales poorly with the state dimension due to weight degeneracy. This article proposes a unifying framework that allows to systematically derive the McKean-Vlasov representations of these filters for the discrete time and continuous time observation case, taking inspiration from the smooth approximation of the data considered in Crisan & Xiong (2010) and Clark & Crisan (2005). We consider three filters that have been proposed in the literature and use this framework to derive It\^{o} representations of their limiting forms as the approximation parameter $\delta \rightarrow 0$. All filters require the solution of a Poisson equation defined on $\mathbb{R}^{d}$, for which existence and uniqueness of solutions can be a non-trivial issue. We additionally establish conditions on the signal-observation system that ensures well-posedness of the weighted Poisson equation arising in one of the filters.

Logarithmic Regret for Episodic Continuous-Time Linear-Quadratic Reinforcement Learning Over a Finite-Time Horizon

We study finite-time horizon continuous-time linear-quadratic reinforcement learning problems in an episodic setting, where both the state and control coefficients are unknown to the controller. We first propose a least-squares algorithm based on continuous-time observations and controls, and establish a logarithmic regret bound of order $O((\ln M)(\ln\ln M))$, with $M$ being the number of learning episodes. The analysis consists of two parts: perturbation analysis, which exploits the regularity and robustness of the associated Riccati differential equation; and parameter estimation error, which relies on sub-exponential properties of continuous-time least-squares estimators. We further propose a practically implementable least-squares algorithm based on discrete-time observations and piecewise constant controls, which achieves similar logarithmic regret with an additional term depending explicitly on the time stepsizes used in the algorithm.

A General Methodology to Measure Labour Market Dynamics

We propose a general methodology to measure labour market dynamics, inspired by the search and matching framework, based on the estimate of the transition rates between labour market states. We show how to estimate instantaneous transition rates starting from discrete time observations provided in longitudinal datasets, allowing for any number of states. We illustrate the potential of such methodology using Italian labour market data. First, we decompose the unemployment rate fluctuations into inflow and outflow driven components; then, we evaluate the impact of the implementation of a labour market reform, which substantially changed the regulations of temporary contracts.

Synchronization of multi-links systems with Lévy noise and application

ABSTRACT In previous work, the synchronization of multi-links systems with Lévy noise (MLSLN) has not been fully investigated. Therefore, the synchronization of MLSLN is concerned via feedback discrete-time observations control. Therein, we provide a novel Lyapunov functional for MLSLN. Moreover, the upper bound of the state observations interval duration is obtained. By making use of Lyapunov functional method, some techniques of inequality and graph theory, we derive some sufficient conditions to guarantee the mean-squared asymptotical synchronization of MLSLN. Then, the theoretical results are employed to study the synchronization of single-link robot arms. Meantime, numerical simulations are given to demonstrate the effectiveness of the developed results.

Feedback control based on discrete-time state observations for stabilization of coupled regime-switching jump diffusion with Markov switching topologies

AbstractIn this paper, the stabilization of coupled regime-switching jump diffusion with Markov switching topologies (CRJDM) is discussed. Particularly, we remove the restrictions that each of the switching subnetwork topologies is strongly connected or contains a directed spanning tree. Furthermore, a feedback control based on discrete-time state observations is proposed to make the CRJDM asymptotically stable. In most existing literature, feedback control only depends on discrete-time observations of state processes, while switching processes are observed continuously. Different from previous literature, feedback control depends on discrete-time observations of state processes as well as switching processes in this paper. Meanwhile, based on graph theory, stationary distribution of switching processes and Lyapunov method, some sufficient conditions are deduced to ensure the asymptotic stability of CRJDM. By applying the theoretical results to second-order oscillators with Markov switching topologies, a stability criterion is obtained. Finally, the effectiveness of the results is illustrated by a numerical example.

Current Control and Active Damping for Single Phase LCL-Filtered Grid Connected Inverter

LCL filter has been widely used in the grid connected inverter, since it is effective in attenuation of the switching frequency harmonics in the inverter. However, the resonance in this filter causes stability problems and must be damped effectively to achieve stability. There are some methods to damp the resonance; one method is passive damping of resonance by adding a series resistor with the filter capacitor, but passive element reduces the inverter efficiency. Other method uses active damping (AD) by adding a proportional control loop of filter capacitor current, but this method needs additional sensor to measure filter capacitor current; moreover, when the control loops are digitally implemented, the computation delay in AD control loop will lead to some difficulties in choosing control parameters and maintaining system stability. This paper presents current control scheme for the grid connected inverter with the LCL filter. The proposed scheme ensures the control of injected current into grid with AD of the resonance in the LCL filter while keeping system stability and eliminating the effect of computation delay of the AD loop. An estimation of filter capacitor current with one step ahead is performed using the discrete time observer based on measuring the injected current. This reduces the cost and increases the robustness of the system. Proportional Resonant (PR) controller is used to control the injected current. Design of control system and choosing its parameters are studied and justified in details to ensure suitable performance with adequate stability margins. Simulation and experimental results show the effectiveness and the robustness of the proposed control scheme.

Robust stochastic stability for multi-group coupled models with Markovian switching

As one of the most important topics in control theory, robust stability for the systems plays a key role in many areas of science and industry. This paper discusses the robust stochastic stability a class of multi-group coupled stochastic models with Markovian switching. Based on discrete-time observations in the diffusion part with time-delay, the robust stochastic exponential stability criterion in mean square sense is established in the forms of Lyapunov-type and coefficient-type. An application to the robust stochastic stability for a class of coupled oscillators is proposed.

Nonparametric Bayesian estimation of a Hölder continuous diffusion coefficient

We consider a nonparametric Bayesian approach to estimate the diffusion coefficient of a stochastic differential equation given discrete time observations over a fixed time interval. As a prior on the diffusion coefficient, we employ a histogram-type prior with piecewise constant realisations on bins forming a partition of the time interval. Specifically, these constants are realizations of independent inverse Gamma distributed randoma variables. We justify our approach by deriving the rate at which the corresponding posterior distribution asymptotically concentrates around the data-generating diffusion coefficient. This posterior contraction rate turns out to be optimal for estimation of a H\"older-continuous diffusion coefficient with smoothness parameter $0<\lambda\leq 1.$ Our approach is straightforward to implement, as the posterior distributions turn out to be inverse Gamma again, and leads to good practical results in a wide range of simulation examples. Finally, we apply our method on exchange rate data sets.

Parametric inference for hypoelliptic ergodic diffusions with full observations

Multidimensional hypoelliptic diffusions arise naturally in different fields, for example to model neuronal activity. Estimation in those models is complex because of the degenerate structure of the diffusion coefficient. In this paper we consider hypoelliptic diffusions, given as a solution of two-dimensional stochastic differential equations, with the discrete time observations of both coordinates being available on an interval $$T = n\varDelta _n$$ , with $$\varDelta _n$$ the time step between the observations. The estimation is studied in the asymptotic setting, with $$T\rightarrow \infty $$ as $$\varDelta _n\rightarrow 0$$ . We build a consistent estimator of the drift and variance parameters with the help of a discretized log-likelihood of the continuous process. We discuss the difficulties generated by the hypoellipticity and provide a proof of the consistency and the asymptotic normality of the estimator. We test our approach numerically on the hypoelliptic FitzHugh–Nagumo model, which describes the firing mechanism of a neuron.

Stabilisation of hybrid stochastic systems with Lévy noise by discrete-time feedback control

This paper is devoted to deal with the stabilisation problems of continuous-time hybrid stochastic systems (often described by stochastic differential equations with Markovian switching) with Lévy noise by feedback controls based on discrete-time state observations. There are so far few results on the stabilisation of continuous-time hybrid stochastic systems with Lévy noise based on discrete-time state observations, despite the stabilisation of stochastic systems by discrete-time feedback controls has been investigated in general. Precisely, it is discussed the stabilisation in the sense of -stability, asymptotic stability, mean-square exponential stability and almost sure exponential stability for such systems. It is also presented an upper bound on the duration τ between two consecutive state observations by means of the Lyapunov function. Two examples are given to illustrate the obtained results.

Periodically Intermittent Stabilization of Neural Networks Based on Discrete-Time Observations

In this brief, we design a periodically intermittent controller to stabilize a class of networks by using discrete-time observations on the states of white noise, which will cut costs by decreasing observation frequency and controlled time. The supremum of discrete-time observations is derived by a transcendental equation. Sufficient conditions are obtained to exponentially stabilize the underlying networks. A numerical example is provided to illustrate the effectiveness and advantages of the proposed new design technique.

Multilevel particle filters for the non-linear filtering problem in continuous time

In the following article we consider the numerical approximation of the non-linear filter in continuous-time, where the observations and signal follow diffusion processes. Given access to high-frequency, but discrete-time observations, we resort to a first order time discretization of the non-linear filter, followed by an Euler discretization of the signal dynamics. In order to approximate the associated discretized non-linear filter, one can use a particle filter (PF). Under assumptions, this can achieve a mean square error of $\mathcal{O}(\epsilon^2)$, for $\epsilon>0$ arbitrary, such that the associated cost is $\mathcal{O}(\epsilon^{-4})$. We prove, under assumptions, that the multilevel particle filter (MLPF) of Jasra et al (2017) can achieve a mean square error of $\mathcal{O}(\epsilon^2)$, for cost $\mathcal{O}(\epsilon^{-3})$. This is supported by numerical simulations in several examples.

Nonparametric Bayesian Estimation of a HHlder Continuous Diffusion Coefficient

We consider a nonparametric Bayesian approach to estimate the diffusion coefficient of a stochastic differential equation given discrete time observations over a fixed time interval. As a prior on the diffusion coefficient, we employ a histogram-type prior with piecewise constant realisations on bins forming a partition of the time interval. Specifically, these constants are realizations of independent inverse Gamma distributed randoma variables. We justify our approach by deriving the rate at which the corresponding posterior distribution asymptotically concentrates around the data-generating diffusion coefficient. This posterior contraction rate turns out to be optimal for estimation of a Holder-continuous diffusion coefficient with smoothness parameter 0<λ≤1. Our approach is straightforward to implement, as the posterior distributions turn out to be inverse Gamma again, and leads to good practical results in a wide range of simulation examples. Finally, we apply our method on exchange rate data sets.

Inner synchronisation of stochastic impulsive multi-links coupled systems via discrete-time state observations control

In this paper, the inner synchronisation of stochastic impulsive coupled systems with multi-links (SISM) is investigated. The feedback control based on discrete-time state observations, impulsive effects and multiple links are added to stochastic coupled systems simultaneously for the first time. Instead of the mean-square exponential synchronisation, the pth () moment exponential synchronisation is studied. A main theorem is proposed by average impulsive interval approach, graph theory and Lyapunov method to ensure the pth moment exponential synchronisation of SISM. Additionally, we apply the theoretical results to stochastic impulsive Rössler-like circuits with multi-links, which is the first attempt. Finally, a numerical example is provided to illustrate the effectiveness of the developed results.

Stabilization of nonlinear hybrid stochastic delay systems by feedback control based on discrete-time state and mode observations

ABSTRACT This paper is concerned with the stabilization problem for nonlinear stochastic delay systems with Markovian switching by feedback control based on discrete-time state and mode observations. By constructing an efficient Lyapunov functionals, we establish the sufficient stabilization criteria not only in the sense of exponential stability (both the mean-square stability and the almost sure stability) but also in other sense – that of stability and asymptotic stability. Meanwhile, the upper bound on the duration τ between two consecutive state and mode observations is obtained. Numerical examples are provided to demonstrate the effectiveness of our theoretical results.

Least-Squares Estimators of Drift Parameter for Discretely Observed Fractional Ornstein–Uhlenbeck Processes

We introduce three new estimators of the drift parameter of a fractional Ornstein–Uhlenbeck process. These estimators are based on modifications of the least-squares procedure utilizing the explicit formula for the process and covariance structure of a fractional Brownian motion. We demonstrate their advantageous properties in the setting of discrete-time observations with fixed mesh size, where they outperform the existing estimators. Numerical experiments by Monte Carlo simulations are conducted to confirm and illustrate theoretical findings. New estimation techniques can improve calibration of models in the form of linear stochastic differential equations driven by a fractional Brownian motion, which are used in diverse fields such as biology, neuroscience, finance and many others.

Interval Impulsive Observer for Linear Systems With Aperiodic Discrete Measurements

This article addresses the modeling and the design of an interval state observer for a linear time-invariant plant in the presence of sporadically available measurements corrupted by unknown-but-bounded errors and noise. The interval observer is modeled as an impulsive system where an impulsive correction is made whenever a measurement is available. The non-negativity of the observation error between two successive measurements is preserved by applying the internal positivity based on the Müller's existence theorem, while at measurement times a linear programming constraint is added. A new methodology for designing the discrete-time observer gain is proposed that guarantees both nonnegativity and stability of the estimation error. The synthesis is performed by solving a set of bilinear matrix inequalities (BMIs). The theoretical result is supported by numerical simulation.