Continuous-time State-space Model Research Articles

Estimation of planned and unplanned missing individual scores in longitudinal designs using continuous-time state-space models.

Latent change score (LCS) models within a continuous-time state-space modeling framework provide a convenient statistical approach for analyzing developmental data. In this study, we evaluate the robustness of such an approach in the context of accelerated longitudinal designs (ALDs). ALDs are especially interesting because they imply a very high rate of planned data missingness. Additionally, most longitudinal studies present unexpected participant attrition leading to unplanned missing data. Therefore, in ALDs, both sources of data missingness are combined. Previous research has shown that ALDs for developmental research allow recovering the population generating process. However, it is unknown how participant attrition impacts the model estimates. We have three goals: (a) to evaluate the robustness of the group-level parameter estimates in scenarios with empirically plausible unplanned data missingness; (b) to evaluate the performance of Kalman scores (KS) imputations for individual data points that were expected but unobserved; and (c) to evaluate the performance of KS imputations for individual data points that were outside the age ranged observed for each case (i.e., to estimate the individual trajectories for the complete age range under study). In general, results showed lack of bias in the simulated conditions. The variability of the estimates increased with lower sample sizes and higher missingness severity. Similarly, we found very accurate estimates of individual scores for both planned and unplanned missing data points. These results are very important for applied practitioners in terms of forecasting and making individual-level decisions. R code is provided to facilitate its implementation by applied researchers. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Supplemental Material for Estimation of Planned and Unplanned Missing Individual Scores in Longitudinal Designs Using Continuous-Time State-Space Models

Supplemental Material for Estimation of Planned and Unplanned Missing Individual Scores in Longitudinal Designs Using Continuous-Time State-Space Models

Supercritical and subcritical aeroelastic behaviors of a three-dimensional wing coupled with a nonlinear energy sink

This paper studies the flutter control of a three-dimensional wing using a nonlinear energy sink (NES), which has a purely cubic stiffness and a linear damper. The structural model is derived by Lagrange's equations and the aerodynamics is modeled using the auto regressive with exogenous input method. Both models are solved in a tightly coupling manner to obtain time domain results. Then, the structural describing function and the aerodynamic continuous-time state space model are used to derive the equivalent linearized aeroelastic system, so that eigenvalue analyses can be used to obtain stable and unstable limit cycle oscillations (LCOs). The LCOs obtained by time domain simulations agree well with the equivalent linearization method, while differences are observed for aperiodic motions which are only predicted in time domain for small damping coefficients. Results show the square of aeroelastic amplitude is inversely proportional to the NES stiffness coefficient for both periodic and aperiodic motions. The NES can induce LCOs below flutter speed for small damping coefficients, indicating a subcritical bifurcation which can change to a supercritical one by increasing the damping coefficient. It is discovered that the upper bound of partial suppression region is quantified by a turning point speed. The regions of complete suppression, partial suppression and no suppression are clarified.

From Inertial Measurements to Smart Cities [About This Issue

Autonomous vehicle technology is advancing rapidly. Their control capabilities often rely on high-bandwidth state estimation incorporating inertial measurements. High-performance state estimation incorporates inertial measurement error models through the process of state augmentation to enable on-the-fly instrument calibration. The feature article “Inertial Measurement Unit Error Modeling Tutorial” by Jay A. Farrell, Felipe O. Silva, Farzana Rahman, and J. Wendel provides a tutorial describing the process and issues related to developing a state-space model for the stochastic errors affecting an inertial measurement unit (IMU). The starting point is the instrument error characterization data sheet provided by the manufacturer, which is typically either an Allan standard deviation graph or the Allan variance parameters extracted from that graph. The desired output of the modeling process is a linear, discrete-time, state-space model of the IMU stochastic errors suitable for augmentation to the INS error-state model. Along with this tutorial, supplementary open source software is available. One software component does the following: 1) Given a continuous-time state-space IMU stochastic error model selected by the designer to match the Allan variance, the software computes a discrete-time equivalent state-space model. 2) Given that discrete-time model, it produces a stochastic error sequence suitable for Allan variance computations. 3) Given a sequence of stochastic errors, it computes and plots the Allan variance. Given Allan variance data and a specific continuous-time state-space IMU stochastic error model structure, a second software component implements an optimization-based approach for selecting the model parameters to match the Allan variance data.

Recursive identification of a nonlinear state space model

SummaryThe convergence of a recursive prediction error method is analyzed. The algorithm identifies a nonlinear continuous time state space model, parameterized by one right‐hand side component of the differential equation and an output equation with a fixed differential gain, to avoid over‐parametrization. The method minimizes the criterion by simulation using an Euler discretization. A stability analysis of the associated differential equations results in conditions for (local) convergence to a minimum of the criterion function. Simulations verify the theoretical analysis and illustrate the performance in the presence of unmodeled dynamics, by identification of the nonlinear drum boiler dynamics of a power plant model.

Empirical spectral processes for stationary state space models

In this paper, we consider function-indexed normalized weighted integrated periodograms for equidistantly sampled multivariate continuous-time state space models which are multivariate continuous-time ARMA processes. Thereby, the sampling distance is fixed and the driving Lévy process has at least a finite fourth moment. Under different assumptions on the function space and the moments of the driving Lévy process we derive a central limit theorem for the function-indexed normalized weighted integrated periodogram. Either the assumption on the function space or the assumption on the existence of moments of the Lévy process is weaker. Furthermore, we show the weak convergence in both the space of continuous functions and in the dual space to a Gaussian process and give an explicit representation of the covariance function. The results can be used to derive the asymptotic behavior of the Whittle estimator and to construct goodness-of-fit test statistics as the Grenander–Rosenblatt statistic and the Cramér–von Mises statistic. We present the exact limit distributions of both statistics and show their performance through a simulation study.

Discrete-Time State-Space Construction Method for SSO Analysis of Renewable Power Generation Integrated AC/DC Hybrid System

Construction of the linearized state-space model is the key step of eigenvalue analysis for SSO study of power systems with large-scale renewable power generation (RPG) integrated. This paper proposes a novel method to construct the state matrix of the AC/DC hybrid power system with RPG within the discrete-time domain. The continuous-time state-space model of each AC/DC component as well as RPG units in the power system, is discretized together with their corresponding control. The discretized equation of each component is then represented by an equivalent circuit consisting of parallel historical current and conductance. According to the power system configuration, the discrete-time equivalent circuit of the whole AC/DC hybrid system is established, and the nodal analysis method is applied for the construction of the discrete-time state-space model of the power system. In this way, the state matrix can be acquired much more conveniently than that of continuous-time state-space method, which needs topology analysis to find independent state variables of the network from the proper tree. The proposed method has been applied in the eigenvalue analysis for the SSO study of a case system adapted from the AC/DC hybrid transmission system in northwest China. Eigenvalue results are in high accordance with time-domain simulation results in PSCAD/EMTDC.

A combined invariant-subspace and subspace identification method for continuous-time state–space models using slowly sampled multi-sine-wave data

The problem of identification of linear continuous-time (CT) state–space (SS) models directly from time-domain data is considered. Given multi-sine waves as excitation signals, this paper develops a new identification method called cISSIM (combined Invariant-Subspace and Subspace Identification Method), which is able to consistently identify CT SS models from slowly-sampled time-domain data in an error-in-variables framework. No prefiltering of the input/output signals is required which distinguishes cISSIM from other existing direct CT identification methods. Moreover, asymptotic unbiased estimates of the system state are obtained. A new persistence of excitation condition is also derived for cISSIM, which is shown to be related with the rank of a certain Khatri–Rao product.

Unbiased estimation of the gradient of the log-likelihood for a class of continuous-time state-space models

Abstract In this paper, we consider static parameter estimation for a class of continuous-time state-space models. Our goal is to obtain an unbiased estimate of the gradient of the log-likelihood (score function), which is an estimate that is unbiased even if the stochastic processes involved in the model must be discretized in time. To achieve this goal, we apply a doubly randomized scheme, that involves a novel coupled conditional particle filter (CCPF) on the second level of randomization. Our novel estimate helps facilitate the application of gradient-based estimation algorithms, such as stochastic-gradient Langevin descent. We illustrate our methodology in the context of stochastic gradient descent (SGD) in several numerical examples and compare with the Rhee–Glynn estimator.

Maximum approximate likelihood estimation of general continuous-time state-space models

Continuous-time state-space models (SSMs) are flexible tools for analysing irregularly sampled sequential observations that are driven by an underlying state process. Corresponding applications typically involve restrictive assumptions concerning linearity and Gaussianity to facilitate inference on the model parameters via the Kalman filter. In this contribution, we provide a general continuous-time SSM framework, allowing both the observation and the state process to be non-linear and non-Gaussian. Statistical inference is carried out by maximum approximate likelihood estimation, where multiple numerical integration within the likelihood evaluation is performed via a fine discretization of the state process. The corresponding reframing of the SSM as a continuous-time hidden Markov model, with structured state transitions, enables us to apply the associated efficient algorithms for parameter estimation and state decoding. We illustrate the modelling approach in a case study using data from a longitudinal study on delinquent behaviour of adolescents in Germany, revealing temporal persistence in the deviation of an individual's delinquency level from the population mean.

Enhancing disturbance rejection performance for a class of networked cascade control systems: an H∞ approach

The emergence and integration of communication technologies in industrial control systems has created new requirements in terms of stability, performance and robustness that should be met to deal with the effects of network-induced delays. In this paper, the analytical design of a networked cascade control system exposed to communication delay and data dropout is investigated. First, an appropriate augmented continuous-time state-space model is proposed. Then, a mathematical approach is introduced to implement the augmented Lyapunov-Krasovskii functional and free-weighting matrix technique for the stability analysis purpose. The augmented representation simplifies the stability analysis of these systems significantly and makes it possible to correlate different variables in Lyapunov functional, which causes a remarkable improvement in H ∞ performance. Numerical simulations are presented to validate the feasibility and performance of the proposed approach. The obtained results indicate achieving much higher allowable delay bounds and robustness against disturbances.

Input-to-state stability for system identification with continuous-time Runge–Kutta neural networks

Runge–Kutta neural networks (RKNN) bridge the gap between continuous-time advantages and the discrete-time nature of any digital controller. RKNN defines a neural network in a continuous-time setting and explicitly discretises it by a variable sample time Runge–Kutta method. As a result, a-priori model knowledge such as the well-known continuous-time state space model can easily be incorporated directly into the neural network weights. Additionally, RKNN preserves a long-term prediction accuracy and a reduced parameter precision requirement. In this contribution, we enhance RKNN with global-asymptotic stability (GAS) and input-to-state stability (ISS) criteria. Based on a Lyapunov function, we develop constraints on the neural network weights for an arbitrary sample time. The constraints are independent of any measurements, have only a few hyperparameters, can be combined with any standard constraint optimisation algorithm and guarantee stability already during training. We apply the algorithm on two public real-world nonlinear identification tasks, an electro-mechanical positioning system and a cascaded water tank benchmark. Both benchmarks are solved with the same hyperparameters and the presented method is competitive to nonlinear identification methods beyond neural networks in the literature. In prediction configuration, all other black-box nonlinear identification approaches are outperformed regarding root mean squared error (RMSE) by an order of magnitude on the cascaded tank benchmark.

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.

Continuous-time system identification with neural networks: Model structures and fitting criteria

This paper presents tailor-made neural model structures and two custom fitting criteria for learning dynamical systems. The proposed framework is based on a representation of the system behavior in terms of continuous-time state-space models. The sequence of hidden states is optimized along with the neural network parameters in order to minimize the difference between measured and estimated outputs, and at the same time to guarantee that the optimized state sequence is consistent with the estimated system dynamics. The effectiveness of the approach is demonstrated through three case studies, including two public system identification benchmarks based on experimental data.

An integral architecture for identification of continuous-time state-space LPV models

Abstract This paper presents an integral architecture for direct identification of continuous-time linear parameter-varying (LPV) state-space models. The main building block of the proposed architecture consist of an LPV model followed by an integral block, which is used to approximate the continuous-time state map of an LPV representation. The unknown LPV model matrices are estimated along with the state sequence by minimizing a properly constructed dual-objective criterion. A coordinate descent algorithm is employed to optimize the desired objective, which alternates between computing the unknown LPV matrices and estimating the state sequence. Thanks to the linear parametric structure induced by the LPV models, the unknown parameters within each coordinate descent step can be computed analytically via ordinary least squares. The effectiveness of the proposed methodology is assessed via a numerical example.

Score-Based Parameter Estimation for a Class of Continuous-Time State Space Models

We consider the problem of parameter estimation for a class of continuous-time state space models (SSMs). In particular, we explore the case of a partially observed diffusion, with data also arriving according to a diffusion process. Based upon a standard identity of the score function, we consider two particle filter based methodologies to estimate the score function. Both methods rely on an online estimation algorithm for the score function, as described, e.g., in [P. Del Moral, A. Doucet, and S. S. Singh, M$2$AN Math. Model. Numer. Anal., 44 (2010), pp. 947--975], of $\mathcal{O}(N^2)$ cost, with $N\in\mathbb{N}$ the number of particles. The first approach employs a simple Euler discretization and standard particle smoothers and is of cost $\mathcal{O}(N^2 + N\Delta_l^{-1})$ per unit time, where $\Delta_l=2^{-l}$, $l\in\mathbb{N}_0$, is the time-discretization step. The second approach is new and based upon a novel diffusion bridge construction. It yields a new backward-type Feynman--Kac formula in continuous time for the score function and is presented along with a particle method for its approximation. Considering a time-discretization, the cost is $\mathcal{O}(N^2\Delta_l^{-1})$ per unit time. To improve computational costs, we then consider multilevel methodologies for the score function. We illustrate our parameter estimation method via stochastic gradient approaches in several numerical examples.

Sequential Inference Methods for Non-Homogeneous Poisson Processes With State-Space Prior

The non-homogeneous Poisson process provides a generalised framework for the modelling of random point data by allowing the intensity of point generation to vary across its domain of interest (time or space). The use of non-homogeneous Poisson processes have arisen in many areas of signal processing and machine learning, but application is still largely limited by its intractable likelihood function and the lack of computationally efficient inference schemes, although some methods do exist for the batch data case. In this paper, we propose for the first time a sequential framework for intensity inference which combines the non-homogeneous model of Poisson data with continuous-time state-space models for their time-varying intensity. This approach enables us to design efficient online inference schemes, for which we propose a novel sequential Markov chain Monte Carlo (SMCMC) algorithm, as is demanded by many applications where point data arrive sequentially and decisions need to be made with low latency. The proposed approach is compared with competing methods on synthetic datasets and tested with high-frequency financial order book data, showing in general improved performance and better computational efficiency than the main batch-based competitor algorithm, and better performance than a simple baseline kernel estimation scheme.

Plant-wide MPC control scheme for CO2 absorption/stripping system

Abstract The high contents of CO2 in natural gas processing industries cause various issues in the operation, and it is essential to reduce the amount of CO2 using amine-based absorption processes. An efficient and flexible control strategy is highly desirable for CO2 removal in CO2 absorption/stripping system. The excellent performance of Model Predictive Control (MPC) in setpoint tracking and disturbance rejection scenarios could make it a better option for the flexible controllability of a CO2 absorption/stripping plant. MPC performance depends significantly on the accuracy of the identified mathematical model of the plant. The state-space model is believed as the better option for MPC as it represents the plant model with true dynamics. Therefore, this study is focused on the design of the 2 × 2 MPC control strategy in MATLAB® MPC Designer Toolbox using 2nd order continuous-time state-space model. The main aim of this study is to develop a plant-wide control scheme based on MPC for the natural gas absorption/striping system. Step changes in the CO2 composition of sweet gas (±5%) and stripper temperature (±15%) have been introduced in the absorption/stripping simulation model. The results show that the MPC controller has achieved the new setpoint of CO2 composition within 0.5 sec in the setpoint tracking scenario. Similarly, the MPC controller has been able to reject the disturbances successfully introduced as ± 15% step change in stripper temperature within 7.5 sec. Hence, the performance of the MPC controller using the state-space model at higher step changes is adequate with no peak, closer to the setpoint, and no overshoot in the output.

Nonlinear subspace system identification based on output-only measurements

In this paper a new deterministic subspace approach is introduced for multi-dimensional nonlinear system identification based on output-only measurements. The strategy is to excite an arbitrary degree of freedom (DOF) and estimate linear and nonlinear parameters of other DOFs within continuous-time state space models. The nonlinearity is characterized in terms of the measured response and its derivatives in locations away from the excitation. Then, basis functions are assigned to predict the nonlinear restoring force and exploited instead of the input data in the identification process. It is demonstrated that projecting the output row space onto these functions, cancels the subspace corresponding to the nonlinear dynamics and the remaining space contains the dynamics of the underlying linear system as well as the contributions of unmeasured input and noise data to the system response. The recent concept and required assumptions are presented through a few mathematical statements and proofs. To ensure the validity of the proposed approach, several discrete and continuous systems with different structural nonlinearities are studied. The estimated parameters are compared with those obtained from traditional subspace identification algorithms and the direct parameter estimation method. The results highlight the robustness of the proposed approach in dealing with noise contaminated measurements.

A continuous-time state-space model for rapid quality control of argos locations from animal-borne tags

BackgroundState-space models are important tools for quality control and analysis of error-prone animal movement data. The near real-time (within 24 h) capability of the Argos satellite system can aid dynamic ocean management of human activities by informing when animals enter wind farms, shipping lanes, and other intensive use zones. This capability also facilitates the use of ocean observations from animal-borne sensors in operational ocean forecasting models. Such near real-time data provision requires rapid, reliable quality control to deal with error-prone Argos locations.MethodsWe formulate a continuous-time state-space model to filter the three types of Argos location data (Least-Squares, Kalman filter, and Kalman smoother), accounting for irregular timing of observations. Our model is deliberately simple to ensure speed and reliability for automated, near real-time quality control of Argos location data. We validate the model by fitting to Argos locations collected from 61 individuals across 7 marine vertebrates and compare model-estimated locations to contemporaneous GPS locations. We then test assumptions that Argos Kalman filter/smoother error ellipses are unbiased, and that Argos Kalman smoother location accuracy cannot be improved by subsequent state-space modelling.ResultsEstimation accuracy varied among species with Root Mean Squared Errors usually <5 km and these decreased with increasing data sampling rate and precision of Argos locations. Including a model parameter to inflate Argos error ellipse sizes in the north - south direction resulted in more accurate location estimates. Finally, in some cases the model appreciably improved the accuracy of the Argos Kalman smoother locations, which should not be possible if the smoother is using all available information.ConclusionsOur model provides quality-controlled locations from Argos Least-Squares or Kalman filter data with accuracy similar to or marginally better than Argos Kalman smoother data that are only available via fee-based reprocessing. Simplicity and ease of use make the model suitable both for automated quality control of near real-time Argos data and for manual use by researchers working with historical Argos data.