Linear State Space Research Articles

Quantification of Mismatch Error in Randomly Switching Linear State-Space Models

Switching Kalman Filters (SKF) are well known for solving switching linear dynamic system (SLDS), i.e., piece-wise linear estimation problems. Practical SKFs are heuristic, approximate filters and require more computational resources than a single-mode Kalman filter (KF). On the other hand, applying a single-mode mismatched KF to an SLDS results in erroneous estimation. This paper quantifies the average error an SKF can eliminate compared to a mismatched, single-mode KF before collecting measurements. Derivations of the first and second moments of the estimators errors are provided and compared. One can use these derivations to quantify the average performance of filters beforehand and decide which filter to run in operation to have the best performance in terms of estimation error and computation complexity. We further provide simulation results that verify our mathematical derivations.

Robust Divided Differences Gaussian Mixture Non Linear Filtering Applied to INS/GNSS Integrated Navigation System with Impulsive Measurement Noise

In this paper, robust INS/GNSS is designed to solve specific problem of non linear time variant state space estimation with non Gaussian measurement noise based on parallel divided difference filtering. Non linear approximation techniques such as Extended Kalman filter EKF, Interpolation Non Linear Filters called DD1 and DD2 are computed. Great comparisons are conduced and criticized in order to confirm and compare some different robust techniques explored in literature. The most tested noise in robust control and estimation theory has been simulated and disturbed GPS+GLONASS signals. Three parallel modified algorithms are simulated and compared with robust and modified forms of the same algorithms based on Root Mean Square Error RMSE Criteria.

On the Potential of Reduced Order Models for Wind Farm Control: A Koopman Dynamic Mode Decomposition Approach

The high dimensions and governing non-linear dynamics in wind farm systems make the design of numerical optimal controllers computationally expensive. A possible pathway to circumvent this challenge lies in finding reduced order models which can accurately embed the existing non-linearities. The work presented here applies the ideas motivated by non-linear dynamical systems theory—the Koopman Operator—to an innovative algorithm in the context of wind farm systems—Input Output Dynamic Mode Decomposition (IODMD)—to improve on the ability to model the aerodynamic interaction between wind turbines in a wind farm and uncover insights into the existing dynamics. It is shown that a reduced order linear state space model can reproduce the downstream turbine generator power dynamics and reconstruct the upstream turbine wake. It is further shown that the fit can be improved by judiciously choosing the Koopman observables used in the IODMD algorithm without jeopardizing the models ability to rebuild the turbine wake. The extensions to the IODMD algorithm provide an important step towards the design of linear reduced order models which can accurately reproduce the dynamics in a wind farm.

APPLYING STATE SPACE MODELS TO STOCHASTIC CLAIMS RESERVING

AbstractThe paper solves the loss reserving problem using Kalman recursions in linear statespace models. In particular, if one orders claims data from run-off triangles to time series with missing observations, then state space formulation can be applied for projections or interpolations of IBNR (Incurred But Not Reported) reserves. Namely, outputs of the corresponding Kalman recursion algorithms for missing or future observations can be taken as the IBNR projections. In particular, by means of such recursive procedures one can perform effectively simulations in order to estimate numerically the distribution of IBNR claims which may be very useful in terms of setting and/or monitoring of prudency level of loss reserves. Moreover, one can generalize this approach to the multivariate case of several dependent run-off triangles for correlated business lines and the outliers in claims data can be also treated effectively in this way. Results of a numerical study for several sets of claims data (univariate and multivariate ones) are presented.

Sequential sparse Bayesian learning with applications to system identification for damage assessment and recursive reconstruction of image sequences

Sparse Bayesian learning (SBL) has attracted substantial interest in recent years for reliable estimation of sparse parameter vectors of dimension much larger than the number of measurements. However, the theory of online sequential estimation of sparsely changing parameter vectors is much less studied. We present a sequential SBL framework for recursive learning of sparse vectors that also change sparsely between successive sampling time periods. Our method uses a hierarchical Bayesian model to recursively estimate the marginal posterior distribution of the parameter vector for each time period, incorporating the sparseness of both this vector and its temporal changes. Our Bayesian model is built around a linear Gaussian state space model and so many quantities of interest can be calculated by using the recursive Bayesian equations. The fast evidence maximization procedure for SBL is developed for recursive Bayesian analysis and the “noise” parameters are efficiently learned solely from the available data in an efficient manner. Numerical experiments verify that exploiting the sparseness of temporal changes of sparse vectors leads to better performance of sparse Bayesian learning. We also examine two applications of sequential SBL: structural system identification for estimating stiffness losses of sequential damage states and recursive reconstruction of image sequences. These illustrative applications validate the effectiveness and robustness of our method.

MPC-PI cascade control for the Kaibel dividing wall column integrated with data-driven soft sensor model

This paper proposes the MPC-PI cascade control for the Kaibel dividing wall column (Kaibel DWC) integrated with a data-driven soft sensor model. The PI control in the inner layer is able to stabilize the Kaibel DWC fast, while the model predictive control (MPC) in the outer layer can reduce the overshoot, and improve the control performances. The prediction model employs the nonlinear autoregressive with exogenous inputs (NARX) neural network model, which is suitable and effective for the nonlinear systems. The extended Kalman filter (EKF) is used for error corrections of the detected and the predicted compositions. The feasibility of the proposed scheme using the NARX model and the EKF is verified, and the results have been compared with those adopting the linear State Space model and the Kalman filter (KF). Results show that the steady-state deviations adopting the NARX model are far less than those employing the State Space model.

Integrated guidance and control framework for the waypoint navigation of a miniature aircraft with highly coupled longitudinal and lateral dynamics

A solution to the waypoint navigation problem for the fixed wing micro air vehicles (MAV) having a severe coupling between longitudinal and lateral dynamics, in the framework of integrated guidance and control (IGC) is addressed in this paper. IGC yields a single step solution to the waypoint navigation problem, unlike conventional multiple loop design. The pure proportional navigation (PPN) guidance law is integrated with the coupled MAV dynamics. A multivariable static output feedback (SOF) controller is designed for the linear state space model formulated in IGC framework. A waypoint navigation algorithm is proposed that handles the minimum turn radius constraint of the MAV and also evaluates the feasibility of reaching a waypoint. Non-linear simulations with and without wind disturbances are performed on a high fidelity 150 mm wingspan MAV model to demonstrate the proposed waypoint navigation algorithm.

Asymptotically Optimal Change Point Detection for Composite Hypothesis in State Space Models

This paper investigates change point detection in state space models, in which the pre-change distribution f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ0</sup> is given, while the post-distribution f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ</sup> after change is unknown. The problem is to raise an alarm as soon as possible after the distribution changes from f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ0</sup> to f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ</sup> , under a restriction on the false alarms. We investigate theoretical properties of a weighted Shiryayev-Roberts-Pollak (SRP) change point detection rule in state space models. By making use of a Markov chain representation for the likelihood function, exponential embedding of the induced Markovian transition operator, nonlinear Markov renewal theory, and sequential hypothesis testing theory for Markov random walks, we show that the weighted SRP procedure is second-order asymptotically optimal. To this end, we derive an asymptotic approximation for the expected stopping time of such a stopping scheme when the change time w = 1. To illustrate our method we apply the results to two types of state space models: general state Markov chains and linear state space models. Simulation study is given to illustrate the method.

State-space models for building control: how deep should you go?

Power consumption in buildings show nonlinear behaviours that linear models cannot capture, whereas recurrent neural networks (RNNs) can. This ability makes RNNs attractive alternatives for the model-predictive control (MPC) of buildings. However, RNNs are nonlinear and non-smooth functions which makes their use challenging in optimization problems. Therefore, this work systematically investigates whether using RNNs for building control provides net gains in MPC. It compares over 2 months of simulated operation the representation power and control performance of two architectures: an RNN architecture and a linear state-space (LSS) model with a nonlinear regressor to estimate energy consumption. The results show that RNNs yield an identification error 69% lower than LSS, but the LSS models yield control laws that achieve 10% lower objective function with a computational time three times lower than the RNNs. Thus, on balance, well-designed LSS models with nonlinear regressors are best in most cases of MPC.

Predicting Alzheimer's disease progression using deep recurrent neural networks

Early identification of individuals at risk of developing Alzheimer's disease (AD) dementia is important for developing disease-modifying therapies. In this study, given multimodal AD markers and clinical diagnosis of an individual from one or more timepoints, we seek to predict the clinical diagnosis, cognition and ventricular volume of the individual for every month (indefinitely) into the future. We proposed and applied a minimal recurrent neural network (minimalRNN) model to data from The Alzheimer's Disease Prediction Of Longitudinal Evolution (TADPOLE) challenge, comprising longitudinal data of 1677 participants (Marinescu et al., 2018) from the Alzheimer's Disease Neuroimaging Initiative (ADNI). We compared the performance of the minimalRNN model and four baseline algorithms up to 6 years into the future. Most previous work on predicting AD progression ignore the issue of missing data, which is a prevalent issue in longitudinal data. Here, we explored three different strategies to handle missing data. Two of the strategies treated the missing data as a “preprocessing” issue, by imputing the missing data using the previous timepoint (“forward filling”) or linear interpolation (“linear filling). The third strategy utilized the minimalRNN model itself to fill in the missing data both during training and testing (“model filling”). Our analyses suggest that the minimalRNN with “model filling” compared favorably with baseline algorithms, including support vector machine/regression, linear state space (LSS) model, and long short-term memory (LSTM) model. Importantly, although the training procedure utilized longitudinal data, we found that the trained minimalRNN model exhibited similar performance, when using only 1 input timepoint or 4 input timepoints, suggesting that our approach might work well with just cross-sectional data. An earlier version of our approach was ranked 5th (out of 53 entries) in the TADPOLE challenge in 2019. The current approach is ranked 2nd out of 63 entries as of June 3rd, 2020.

Causality and Network Graph in General Bilinear State-Space Representations

This article proposes an extension of the well-known concept of Granger causality, called GB–Granger causality. GB–Granger causality is designed to relate the internal structure of bilinear state-space systems and statistical properties of their output processes. That is, if such a system generates two processes, where one does not GB–Granger cause the other, then it can be interpreted as the interconnection of two subsystems: one that sends information to the other, and one which does not send information back.This result is an extension of earlier obtained results on the relationship between Granger causality and the internal structure of linear time-invariant state-space representations.

Design of hierarchical control strategies for biological wastewater treatment plants to reduce operational costs

In this research, hierarchical control strategies are developed for biological wastewater treatment plants (WWTPs) to reduce the operational expenses. The benchmark simulation model no. 1 (BSM1) is used as the working platform which is developed based on activated sludge model no. 1 (ASM1) to control dissolved oxygen in aerobic reactors and nitrate levels in anoxic reactors. Fractional PI (FPI) controllers are designed at the lower level and model predictive control (MPC) and a fuzzy controller are designed at higher level in order to achieve enhanced set-point tracking. Initially, linear state space model is developed around the operating point using prediction-error method for lower level. For identification of the higher level model, the lower level control loop is closed in a feedback sense with the designed controller. Based on the identified model in the higher level, the controllers are designed. This paper presents two combinations of hierarchical control strategies: FPI-Fuzzy and FPI-MPC. It is observed that FPI controller at the lower level and MPC controller at the higher level results in better plant performance with better set-point tracking with reduced operational costs. It is observed that FPI-Fuzzy control strategy results in better EQI of 7041.7 for storm weather condition which also resulted in 48% reduction in total nitrogen violations and FPI-MPC results in better OCI of 17119.6 and this strategy resulted in nearly 38% reduction in total nitrogen violations. A significant improvement in the plant performance is observed for dry and rain weather conditions as well.

Polynomial nonlinear state space identification of an aero-engine structure

Most nonlinear identification problems often require prior knowledge or an initial assumption of the mathematical law (model structure) and data processing to estimate the nonlinear parameters present in a system, i.e. they require the functional form or depend on a proposition that the measured data obey a certain nonlinear function. However, obtaining prior knowledge or performing nonlinear characterisation can be difficult or impossible for certain identification problems due to the individualistic nature of practical nonlinearities. For example, joints between substructures of large aerospace design frequently feature complex physics at local regions of the structure, making a physically motivated identification in terms of nonlinear stiffness and damping impossible. As a result, black-box models which use no prior knowledge can be regarded as an effective method. This paper explores the pragmatism of a black-box approach based on Polynomial Nonlinear State Space (PNLSS) models to identify the nonlinear dynamics observed in a large aerospace component. As a first step, the Best Linear Approximation (BLA), noise and nonlinear distortion levels are estimated over different amplitudes of excitation using the Local Polynomial Method (LPM). Next, a linear state space model is estimated on the non-parametric BLA using the frequency domain subspace identification method. Nonlinear model terms are then constructed in the form of multivariate polynomials in the state variables while the parameters are estimated through a nonlinear optimisation routine. Further analyses were also conducted to determine the most suitable monomial degree and type required for the nonlinear identification procedure. Practical application is carried out on an Aero-Engine casing assembly with multiple joints, while model estimation and validation is achieved using measured sine-sweep and broadband data obtained from the experimental campaign.

Linear–quadratic optimal steady state controllers for engineering students and practicing engineers

This paper is an overview of fundamental linear–quadratic optimal control techniques used for linear dynamic systems. The presentation is suitable for undergraduate and graduate students and practicing engineers. The paper can be used by class instructors as supplemental material for undergraduate and graduate control system courses. The paper shows how to find the solution to a dynamic optimization problem: optimize an integral quadratic performance criterion along trajectories of a linear dynamic system over an infinite time period (steady-state linear–quadratic optimal control problem). The solution is obtained by solving a static optimization problem. All derivations done in the paper require only elementary knowledge of linear algebra and state space linear system analysis. The results are presented also for the observer-driven linear–quadratic steady-state optimal controller, output feedback-based linear–quadratic optimal controller, and the Kalman filter-driven linear–quadratic stochastic optimal controller. Having full understanding of derivations of the linear–quadratic optimal controller, observer-driven linear–quadratic optimal controller, optimal linear–quadratic output feedback controller, and optimal linear–quadratic stochastic controller, students and engineers will feel confident to use these controllers in numerous engineering and scientific applications. Several optimal linear–quadratic control case studies involving models of real physical systems, with the corresponding Simulink block diagrams and MATLAB codes, are included in the paper.

Bayesian approach to linear state space model with unknown measurement matrix

Bayesian approach to linear state space model with unknown measurement matrix

Constrained expectation maximisation algorithm for estimating ARMA models in state space representation

This paper discusses the fitting of linear state space models to given multivariate time series in the presence of constraints imposed on the four main parameter matrices of these models. Constraints arise partly from the assumption that the models have a block-diagonal structure, with each block corresponding to an ARMA process, that allows the reconstruction of independent source components from linear mixtures, and partly from the need to keep models identifiable. The first stage of parameter fitting is performed by the expectation maximisation (EM) algorithm. Due to the identifiability constraint, a subset of the diagonal elements of the dynamical noise covariance matrix needs to be constrained to fixed values (usually unity). For this kind of constraints, so far, no closed-form update rules were available. We present new update rules for this situation, both for updating the dynamical noise covariance matrix directly and for updating a matrix square-root of this matrix. The practical applicability of the proposed algorithm is demonstrated by a low-dimensional simulation example. The behaviour of the EM algorithm, as observed in this example, illustrates the well-known fact that in practical applications, the EM algorithm should be combined with a different algorithm for numerical optimisation, such as a quasi-Newton algorithm.

Likelihood ratio testing in linear state space models: An application to dynamic stochastic general equilibrium models

This paper considers the problem of hypothesis testing in linear Gaussian state space models. We consider two hypotheses of interest: a simple null and a hypothesis of explicit parameter restrictions. We derive the asymptotic distributions of the corresponding likelihood ratio test statistics and compute the Bartlett adjustments. The results are non-trivial because the unrestricted state space model is not (even locally) identified. We apply our analysis to test the validity of the Dynamic Stochastic General Equilibrium (DSGE) models. A Monte Carlo exercise illustrates our findings and confirms the importance of Bartlett corrections at sample sizes typically encountered in macroeconomics.

Conditionally structured variational Gaussian approximation with importance weights

We develop flexible methods of deriving variational inference for models with complex latent variable structure. By splitting the variables in these models into “global” parameters and “local” latent variables, we define a class of variational approximations that exploit this partitioning and go beyond Gaussian variational approximation. This approximation is motivated by the fact that in many hierarchical models, there are global variance parameters which determine the scale of local latent variables in their posterior conditional on the global parameters. We also consider parsimonious parametrizations by using conditional independence structure and improved estimation of the log marginal likelihood and variational density using importance weights. These methods are shown to improve significantly on Gaussian variational approximation methods for a similar computational cost. Application of the methodology is illustrated using generalized linear mixed models and state space models.

Semi-supervised optimal recursive filtering and smoothing in non-Gaussian Markov switching models

Filtering and smoothing in switching state-space models are important in numerous applications. The classic family of conditionally Gaussian linear state space models (CGLSSMs) is a natural extension of the Gaussian linear system by introducing its dependence on switches. In spite of their simplicity, recursive filtering and smoothing are no longer feasible in CGLSSMs and approximate methods must be used. Conditionally Markov switching hidden linear models (CMSHLMs) are alternative models which allow recursive optimal exact filtering and smoothing. We introduce an original family of CMSHLMs defined with copulas and we address the problem of their identification. The proposed identification method chooses a model in a family of admissible parametric models and estimates the parameters. It is applied to a learning sample containing observations and states, while the switches are unknown. The interest of the proposed ”semi-unsupervised” filtering and smoothing is validated via experiments on simulated data.

JOINT ESTIMATION OF STATES AND PARAMETERS OF LINEAR SYSTEMS WITH PARAMETER FAULTS UNDER NON-GAUSSIAN NOISES

Joint estimation of states and time-varying parameters of linear state space models is of practical importance for the fault diagnosis and fault tolerant control. Previous works on this topic consider the joint estimation in the Gaussian noise environment, but not in the presence of outliers. The known fact is that the measurements have inconsistent observations with the largest part of the observation population (outliers). They can significantly make worse the properties of linearly recursive algorithms which are designed to work in the presence of Gaussian noises. This paper proposes the strategy of the joint parameter-state robust estimation of linear state space models in the presence of non-Gaussian noises. The case of parameter-dependent matrices is considered. Because of its good features in robust filtering, the extended Masreliez-Martin filter represents a cornerstone for realization of the robust algorithms for joint state-parameter estimation of linear time-varying stochastic systems in the presence of non-Gaussian noises. The good features of the proposed robust algorithm for joint estimation of linear time-varying stochastic systems are illustrated by intensive simulations.