Continuous-time Estimation Research Articles

Optimization based data enrichment using stochastic dynamical system models

We develop a general framework for state estimation in systems modeled with noise-polluted continuous time dynamics and discrete time noisy measurements. Our approach is based on maximum likelihood estimation and employs the calculus of variations to derive optimality conditions for continuous time functions. We make no prior assumptions on the form of the mapping from measurements to state-estimate or on the distributions of the noise terms, making the framework more general than Kalman filtering/smoothing where this mapping is assumed to be linear and the noises Gaussian. The optimal solution that arises is interpreted as a continuous time spline, the structure and temporal dependency of which is determined by the system dynamics and the distributions of the process and measurement noise. Similar to Kalman smoothing, the optimal spline yields increased data accuracy at instants when measurements are taken, in addition to providing continuous time estimates outside the measurement instances. We demonstrate the utility and generality of our approach via illustrative examples that render both linear and nonlinear data filters depending on the particular system. Application of the proposed approach to a Monte Carlo simulation exhibits significant performance improvement in comparison to a common existing method.

Regression estimation for continuous-time functional data processes with missing at random response

In this paper, we are interested in nonparametric kernel estimation of a generalised regression function based on an incomplete sample ( X t , Y t , ζ t ) t ∈ [ 0 , T ] copies of a continuous-time stationary and ergodic process ( X , Y , ζ ) . The predictor X is valued in some infinite-dimensional space, whereas the real-valued process Y is observed when the Bernoulli process ζ = 1 and missing whenever ζ = 0 . Uniform almost sure consistency rate as well as the evaluation of the conditional bias and asymptotic mean square error are established. The asymptotic distribution of the estimator is provided with a discussion on its use in building asymptotic confidence intervals. To illustrate the performance of the proposed estimator, a first simulation is performed to compare the efficiency of discrete-time and continuous-time estimators. A second simulation is conducted to discuss the selection of the optimal sampling mesh in the continuous-time case. Then, a third simulation is considered to build asymptotic confidence intervals. An application to financial time series is used to study the performance of the proposed estimator in terms of point and interval prediction of the IBM asset price log-returns. Finally, a second application is introduced to discuss the usage of the initial estimator to impute missing household-level peak electricity demand.

Exploring the Short-Term Memory of Heart Rate Variability through Model-Free Information Measures.

In this work, we perform a comparative analysis of discrete- and continuous-time estimators of information-theoretic measures quantifying the concept of memory utilization in short-term heart rate variability (HRV). Specifically, considering heartbeat intervals in discrete time we compute the measure of information storage (IS) and decompose it into immediate memory utilization (IMU) and longer memory utilization (MU) terms; considering the timings of heartbeats in continuous time we compute the measure of MU rate (MUR). All measures are computed through model-free approaches based on nearest neighbor entropy estimators applied to the HRV series of a group of 15 healthy subjects measured at rest and during postural stress. We find, moving from rest to stress, statistically significant increases of the IS and the IMU, as well as of the MUR. Our results suggest that both discrete-time and continuous-time approaches can detect the higher predictive capacity of HRV occurring with postural stress, and that such increased memory utilization is due to fast mechanisms likely related to sympathetic activation.

Rate of convergence of discretized drift parameters estimators in the Cox–Ingersoll–Ross model

The article is devoted to the drift parameters estimation in the Cox–Ingersoll–Ross model. We obtain the rate of convergence in probability of the maximum likelihood estimators based on the continuous-time estimators. Then we introduce the discrete versions of these estimators and investigate their asymptotic behavior. In particular, we establish the conditions for weak and strong consistency, asymptotic normality, and get the rate of convergence in probability.

Continuous-Time Trajectory Estimation for Differentially Flat Systems

Continuous-time estimation using splines on Lie groups has been gaining traction in the literature due to the ability to incorporate high-frequency sensor data without introducing new optimization parameters. However, evaluating time derivatives and Jacobians of Lie group splines is computationally expensive, limiting their use mainly to offline applications. Motivated by the trajectory planning literature, we develop a new estimation technique that leverages the differential flatness property of many dynamic systems to define the spline in the system's flat output space, which is often Euclidean. Doing so has the added benefit of providing a simple and effective way to include system inputs in the estimation process. We show an example of flatness-based estimation for the unicycle dynamic model. We then show that this new method can achieve similar performance as Lie group spline estimation with significantly less computation time, and validate its use in hardware using a differential-drive robot.

On Biased Harmonic Signal Estimation: Application to Electric Power Grid Monitoring

Parametric estimation of a biased harmonic signal is a significant technical challenge for many engineering applications. Such a problem is particularly important for electric utility grid-connected power electronic converters. This article utilizes a linear regression model of the signal to solve this interesting practical problem. A continuous-time dynamic regressor extension and mixing (DREM) based approach is then applied for parameter estimation. For practical implementation, continuous-time estimators are discretized using implicit and explicit Euler methods. We then prove that the implicit discretization can achieve fixed-time convergence for the unknown frequencies estimation. Thanks to the estimated frequencies, another DREM-based linear regression problem is solved for the parameter estimation purpose. The overall order of the proposed technique is the same as the number of unknown parameters, making the estimator suitable for real-time implementation in embedded devices. Theoretical results are validated through extensive comparative experimental studies.

Staggered HLL: Near-continuous-time cardinality estimation with no overhead

Most of existing cardinality estimation algorithms do not support natively interval queries under a sliding window model and are thereby insensitive to data recency. We present Staggered-HyperLogLog (ST-HLL), a probabilistic data structure that takes inspiration from HyperLogLog (HLL) and provides nearly continuous-time estimation of cardinality rates, rather than absolute counts. Our solution has zero-bit overhead with respect to vanilla HLL and negligible additional computational complexity. It is based on a periodic staggered reset of HLL registers and a register equalization operation at query times to compensate for staggered counting. We tested ST-HLL over both synthetic and real Internet traffic traces, showing its ability to track variations of the flow cardinality, quickly adapting to variations under non-stationary flow arrival processes. We show that for the same amount of memory footprint, our algorithm improves the accuracy up to a factor 2x with respect to the state-of-the-art solution, Sliding HLL.

Distributed Kalman Consensus Filter for Estimation With Moving Targets.

Consensus-based distributed Kalman filters for estimation with targets have attracted considerable attention. Most of the existing Kalman filters use the average consensus approach, which tends to have a low convergence speed. They also rarely consider the impacts of limited sensing range and target mobility on the information flow topology. In this article, we address these issues by designing a novel distributed Kalman consensus filter (DKCF) with an information-weighted consensus structure for random mobile target estimation in continuous time. A new moving target information-flow topology for the measurement of targets is developed based on the sensors' sensing ranges, targets' random mobility, and local information-weighted neighbors. Novel necessary and sufficient conditions about the convergence of the proposed DKCF are developed. Under these conditions, the estimates of all sensors converge to the consensus values. Simulation and comparative studies show the effectiveness and the superiority of this new DKCF.

Active Disturbance Rejection Control for Uncertain Nonlinear Systems With Sporadic Measurements

This paper deals with the problem of active disturbance rejection control (ADRC) design for a class of uncertain nonlinear systems with sporadic measurements. A novel extended state observer (ESO) is designed in a cascade form consisting of a continuous time estimator, a continuous observation error predictor, and a reset compensator. The proposed ESO estimates not only the system state but also the total uncertainty, which may include the effects of the external perturbation, the parametric uncertainty, and the unknown nonlinear dynamics. Such a reset compensator, whose state is reset to zero whenever a new measurement arrives, is used to calibrate the predictor. Due to the cascade structure, the resulting error dynamics system is presented in a non-hybrid form, and accordingly, analyzed in a general sampled-data system framework. Based on the output of the ESO, a continuous ADRC law is then developed. The convergence of the resulting closed-loop system is proved under given conditions. Two numerical simulations demonstrate the effectiveness of the proposed control method.

ChevOpt: Continuous-Time State Estimation by Chebyshev Polynomial Optimization

In this paper, a new framework for continuous-time maximum a posteriori estimation based on the Chebyshev polynomial optimization (ChevOpt) is proposed, which transforms the nonlinear continuous-time state estimation into a problem of constant parameter optimization. Specifically, the time-varying system state is represented by a Chebyshev polynomial and the unknown Chebyshev coefficients are optimized by minimizing the weighted sum of the prior, dynamics and measurements. The proposed ChevOpt is an optimal continuous-time estimation in the least squares sense and needs a batch processing. A recursive sliding-window version is proposed as well to meet the requirement of real-time applications. Comparing with the well-known Gaussian filters, the ChevOpt better resolves the nonlinearities in both dynamics and measurements. Numerical results of demonstrative examples show that the proposed ChevOpt achieves remarkably improved accuracy over the extended/unscented Kalman filters and extended batch/fixed-lag smoother, closes to the Cramer-Rao lower bound.

An Information-Theoretic Framework to Measure the Dynamic Interaction Between Neural Spike Trains

While understanding the interaction patterns among simultaneous recordings of spike trains from multiple neuronal units is a key topic in neuroscience, existing methods either do not consider the inherent point-process nature of spike trains or are based on parametric assumptions. This work presents an information-theoretic framework for the model-free, continuous-time estimation of both undirected (symmetric) and directed (Granger-causal) interactions between spike trains. The framework computes the mutual information rate (MIR) and the transfer entropy rate (TER) for two point processes X and Y, showing that the MIR between X and Y can be decomposed as the sum of the TER along the directions X → Y and Y → X. We present theoretical expressions and introduce strategies to estimate efficiently the two measures through nearest neighbor statistics. Using simulations of independent and coupled point processes, we show the accuracy of MIR and TER to assess interactions even for weakly coupled and short realizations, and demonstrate the superiority of continuous-time estimation over the standard discrete-time approach. We also apply the MIR and TER to real-world data, specifically, recordings from in-vitro preparations of spontaneously-growing cultures of cortical neurons. Using this dataset, we demonstrate the ability of MIR and TER to describe how the functional networks between recording units emerge over the course of the maturation of the neuronal cultures. the proposed framework provides principled measures to assess undirected and directed spike train interactions with more efficiency and flexibility than previous discrete-time or parametric approaches, opening new perspectives for the analysis of point-process data in neuroscience and many other fields.

Identification of Continuous-time Linear Time-varying Systems with Abrupt Changes in Parameters

Abstract In this paper, we propose an adaptive identification method for continuous-time systems with slowly time-varying parameters subject to infrequent abrupt changes using sampled data in the presence of measurement noise. The proposed estimator consists of two steps. First, the time-derivatives of the input and output signals are approximated through a combination of a Kalman filter and fixed-interval smoother. The parameters of the time-varying system are then estimated using the approximate time-derivatives by two Kalman filters running forward and backward in time to track the abrupt changes. The performance of the proposed estimator is verified through simulation studies where it is shown to perform better than an existing direct continuous-time estimator.

An Air-to-Fuel ratio estimation strategy for turbocharged spark-ignition engines based on sparse binary HEGO sensor measures and hybrid linear observers

An effective Air-to-Fuel Ratio (AFR) control is paramount to ensure a good combustion and high catalyst efficiency. This work addresses the problem of determining continuous-time estimates of AFR in turbocharged Spark Ignition (SI) engines on the basis of binary sparse measurements of the exhaust gas Oxygen. The latter are provided by a HEGO (Heated Exhaust Gas Oxygen) sensor installed at the catalytic converter input in place of a more expensive linear UEGO (Universal Exhaust Gas Oxygen) sensor, as nowadays common in commercial cars. The HEGO sensor outputs two voltage values only, corresponding respectively to low or high concentration of the residual Oxygen in the exhaust gas (on/off behavior). In view of this, it can be classified as a binary sensor generating irregular and sparse measurements in that the useful information is only present at the instants of the on/off and off/on transitions. An estimation scheme based on the use of a recursive least-squares algorithm has been designed by resorting to the theory of linear hybrid observers with quantized inputs. A detailed convergence analysis of the state reconstruction error is also provided. The proposed hybrid observer scheme is employed in a PI control-loop designed to maintain the AFR close to a desired value. The effectiveness of the proposed method is demonstrated by several numerical simulations based on both synthetic and real data.

Leader-following consensus for multi-agent systems with nonlinear dynamics subject to additive bounded disturbances and asynchronously sampled outputs

This paper is concerned with the leader-following consensus problem for a class of Lipschitz nonlinear multi-agent systems with uncertain dynamics, where each agent only transmits its noisy output, at discrete instants and independently from its neighbors. The proposed consensus protocol is based on a continuous-discrete time observer, which provides a continuous time estimation of the state of the neighbors from their discrete-time output measurements, together with a continuous control law. The stability of the multi-agent system is analyzed with a Lyapunov approach and the exponential practical convergence is ensured provided that the tuning parameters and the maximum allowable sampling period satisfy some inequalities. The proposed protocol is simulated on a multi-agent system whose dynamics are ruled by a Chua’s oscillator.

Adaptive Robust Control in Continuous-Time

We propose a continuous-time version of the adaptive robust methodology introduced in Bielecki et al. (2019). An agent solves a stochastic control problem where the underlying uncertainty follows a jump-diffusion process and the agent does not know the drift parameters of the process. The agent considers a set of alternative measures to make the control problem robust to model misspecification and employs a continuous-time estimator to learn the value of the unknown parameters to make the control problem adaptive to the arrival of new information. We use measurable selection theorems to prove the dynamic programming principle of the adaptive robust problem and show that the value function of the agent is characterised by a non-linear partial differential equation. As an example, we derive in closed-form the optimal adaptive robust strategy for an agent who acquires a large number of shares in an order-driven market and illustrates the financial performance of the execution strategy.

Semiparametric Estimation in Continuous-Time: Asymptotics for Integrated Volatility Functionals with Small and Large Bandwidths

This article studies the estimation of integrated volatility functionals, which is a semiparametric two-step estimation problem in the nonstationary continuous-time setting. We generalize the asymptotic normality results of Jacod and Rosenbaum to a wider range of bandwidths. Moreover, we employ matrix calculus to obtain a new analytical bias correction and variance estimation method. The proposed method gives more succinct expressions than the element-by-element analytical method of the above cited article. In addition, it has a computational advantage over the jackknife/simulation-based method proposed by Li, Liu, and Xiu. Comprehensive simulation studies demonstrate that our method has good finite sample performance for a variety of volatility functionals, including quadraticity, determinant, continuous beta, and eigenvalues.

Nuclear norm subspace identification of continuous time state–space models with missing outputs

Subspace identification methods using Generalized Poisson Moment Functionals (GPMF) have been proposed previously to tackle the problem of derivative estimation in continuous time (CT) systems. In this paper, a convergence result underpinning the GPMF methods for continuous time identification is detailed. Based on this, a CT-MOESP method is proposed to estimate the system matrices in state–space models. Since these results hold in the asymptotic case where the number of data points tend to infinity, Nuclear Norm Minimization (NNM) is used to integrate the low rank approximation step in subspace identification with a goodness of fit criterion. This paper extends these existing discrete time methods to continuous time by formulating the NNM optimization into the framework of the Alternating Direction Method of Multipliers (ADMM) algorithm. On the numerical front, the accuracy of the proposed method is demonstrated with the help of simulations on two systems frequently cited in literature. An industrial dryer application is considered in order to demonstrate the practical applicability of proposed method.

Bayesian approach for parameter estimation of continuous-time stochastic volatility models using Fourier transform methods

We propose a two stage procedure for the estimation of the parameters of a fairly general, continuous-time stochastic volatility model. An important ingredient of the proposed method is the Cuchiero–Teichmann volatility estimator, which is based on Fourier transforms and provides a continuous-time estimate of the latent process. This estimate is then used to construct an approximate likelihood for the parameters of interest, whose restrictions are taken into account through prior distributions. The procedure is shown to be highly successful for constructing the posterior distribution of the parameters of a Heston model, while limited success is achieved when applied to the highly parametrized exponential-Ornstein–Uhlenbeck.

Frequency Domain Estimation of Continuous Time Cointegrated Models with Mixed Frequency and Mixed Sample Data

Recent work by the author on mixed frequency data analysis has focused on the estimation of cointegrated systems in continuous time based on a fully specified dynamic system of equations, while the estimation of cointegrating vectors in a discrete time system has been approached using a semiparametric frequency domain estimator. We extend the latter approach to cover the continuous time case, establishing the asymptotic properties of the frequency domain estimator and explore, in a simulation study, the effects of misspecifying the continuous time dynamic model in discrete time compared to treating the dynamics non‐parametrically. An empirical illustration is also provided.

Convergence of discrete-time Kalman filter estimate to continuous-time estimate for systems with unbounded observation

In this article, we complement recent results on the convergence of the state estimate obtained by applying the discrete-time Kalman filter on a time-sampled continuous-time system. As the temporal discretization is refined, the estimate converges to the continuous-time estimate given by the Kalman–Bucy filter. We shall give bounds for the convergence rates for the variance of the discrepancy between these two estimates. The contribution of this article is to generalize the convergence results to systems with unbounded observation operators under different sets of assumptions, including systems with diagonaliz-able generators, systems with admissible observation operators, and systems with analytic semigroups. The proofs are based on applying the discrete-time Kalman filter on a dense, numerable subset on the time interval [0,T] and bounding the increments obtained. These bounds are obtained by studying the regularity of the underlying semigroup and the noise-free output.