State and parameter estimation for nonlinear systems

This paper deals with the problem of joint state and parameter estimation based on a set adaptive observer design. The problem is formulated and solved for an LPV (linear parameter-varying) system. The resolution methodology avoids the exponential complexity obstruction usually encountered in the set-membership parameter estimation. A simulation example is presented to illustrate the efficiency of the proposed approach.

A kernel functional representation of linear time-varying systems is employed in conjunction with B-spline functional approximation techniques to construct non-asymptotic state and parameter estimators for LTV systems. Total observability of the estimated system must be assumed. Practical identifiability conditions for parametric estimation are also stated. In the absence of output measurement noise the observer provides almost exact reconstruction of the system state and delivers high fidelity functional estimates of the time varying system parameters. It also shares the usual superior features of algebraic observers such as independence of the initial conditions of the system and good noise attenuation properties. Other advantages of the kernel and B-spline based identification of linear time-varying systems are elucidated.

The primary objective of this study is to develop a mathematical framework for failure prognostics and uncertainty management based on a state-space degradation model, where unknown parameters are present in the model. Joint state and parameter estimation is carried out by the Sequential Monte Carlo methods. Then, a degradation prediction with uncertainty limits is made. Sequential Monte Carlo methods for joint state and parameter estimation are reviewed, and the effectiveness of the techniques for the prognostics task is assessed. A performance comparison of these algorithms is made based on a numerical study. The results presented in this study show that the state-of-art SMC techniques can provide satisfactory results of joint state and parameter estimation for the prognostics task. The Particle Markov Chain Monte Carlo method provides relatively more accurate estimation results compared with other SMC methods while demanding more computing resources.

In this paper, the set-membership state and parameter estimation problems are considered for linear time-varying systems. The noises in the system are unknown but bounded and the variation of parameters is considered as a bounded expansion. The entire estimator is an interactive estimation of parameters and states. In parameter estimation, the exact set of parameter estimates may be non-convex because of the uncertain states. A polytopic linear parameter varying (LPV) enclosure of the system regression expression is constructed as a convex relaxation. The result of parameter estimation is a set to include the true parameter of the model. Meanwhile, in the state estimation, a similar LPV enclosure of the state-space model is also used to bound the state matrix uncertainties. This LPV enclosure in the state estimation is narrowed by the parameter estimation, and then the set of state estimates is computed to include the true state. The constrained zonotope (CZ) is used to describe sets of interest and the necessary set operations are expanded to LPV mapping. Because of the highly adjustable accuracy of CZ, the proposed algorithm can make a better trade-off between computational accuracy and efficiency. Finally, a numerical example is given to illustrate the effectiveness of the proposed algorithm.

Multi-modal sensor fusion has become ubiquitous in the field of vehicle motion estimation. Achieving a consistent sensor fusion in such a set-up demands the precise knowledge of the misalignments between the coordinate systems in which the different information sources are expressed. In ego-motion estimation, even sub-degree misalignment errors lead to serious performance degradation. The present work addresses the extrinsic calibration of a land vehicle equipped with standard production car sensors and an automotive-grade inertial measurement unit (IMU). Specifically, the article presents a method for the estimation of the misalignment between the IMU and vehicle coordinate systems, while considering the IMU biases. The estimation problem is treated as a joint state and parameter estimation problem, and solved using an adaptive estimator that relies on the IMU measurements, a dynamic single-track model as well as the suspension and odometry systems. Additionally, we show that the validity of the misalignment estimates can be assessed by identifying the misalignment between a high-precision INS/GNSS and the IMU and vehicle coordinate systems. The effectiveness of the proposed calibration procedure is demonstrated using real sensor data. The results show that estimation accuracies below 0.1 degrees can be achieved in spite of moderate variations in the manoeuvre execution.

We propose an iterative method for joint state and parameter estimation using measurements on a time interval [0, T] for systems that are backward output stabilizable. Since this time interval is fixed, errors in initial state may have a big impact on the parameter estimate. We propose to use the back and forth nudging (BFN) method for estimating the system’s initial state and a Gauss–Newton step between BFN iterations for estimating the system parameters. Taking advantage of results on the optimality of the BFN method, we show that for systems with skew-adjoint generators, the initial state and parameter estimate minimizing an output error cost functional is an attractive fixed point for the proposed method. We treat both linear source estimation and bilinear parameter estimation problems.

The relationship between the parameters and the states of state-space systems is nonlinear, which makes the identification problems of state-space systems complicated. This paper considers the joint parameter and state estimation issues for a class of state-space systems in the observer canonical form with the process noises and the observation noises. By means of the least squares principle and the Kalman filtering, we derive a Kalman filtering based recursive extended least squares algorithm. For purpose of achieving the higher estimation accuracy, a Kalman filtering based multi-innovation recursive extended least squares algorithm is proposed by utilizing a range of available data and more information at each recursion. Finally, the effectiveness of the proposed algorithms is validated through a simulation example.

State estimation aims at approximately reconstructing the solution $u$ to a parametrized partial differential equation from $m$ linear measurements when the parameter vector $y$ is unknown. Fast numerical recovery methods have been proposed in Maday et al. [Internat. J. Numer. Methods Engrg., 102 (2015), pp. 933--965] based on reduced models which are linear spaces of moderate dimension $n$ that are tailored to approximate the solution manifold ${\cal M}$ where the solution sits. These methods can be viewed as deterministic counterparts to Bayesian estimation approaches and are proved to be optimal when the prior is expressed by approximability of the solution with respect to the reduced model [P. Binev et al., SIAM/ASA J. Uncertain. Quantif., 5 (2017), pp. 1--29]. However, they are inherently limited by their linear nature, which bounds from below their best possible performance by the Kolmogorov width $d_m({\cal M})$ of the solution manifold. In this paper, we propose to break this barrier by using simple nonlinear reduced models that consist of a finite union of linear spaces $V_k$, each having dimension at most $m$ and leading to different estimators $u_k^*$. A model selection mechanism based on minimizing the PDE residual over the parameter space is used to select from this collection the final estimator $u^*$. Our analysis shows that $u^*$ meets optimal recovery benchmarks that are inherent to the solution manifold and not tied to its Kolmogorov width. The residual minimization procedure is computationally simple in the relevant case of affine parameter dependence in the PDE. In addition, it results in an estimator $y^*$ for the unknown parameter vector. In this setting, we also discuss an alternating minimization (coordinate descent) algorithm for joint state and parameter estimation that potentially improves the quality of both estimators.

This manuscript deals with parametric and state estimation problem of discrete-time nonlinear polynomials systems. In this respect, the aim is to design a Recursive State and Parametric Estimation (RSPE) algorithm, based on least square principle, polynomial adjustable models and Kalman filter theory. Our simulation results show the validity of the proposed algorithm.

International audience

This paper considers the estimation problem of traffic state in freeway networks in light of the unknown input filtering (UIF) framework. The freeway traffic flow is modeled as a dynamic stochastic nonlinear system and is based on a recently developed speed-extended cell-transmission model of freeway traffic. Since the environmental conditions on a freeway may change over time, model parameters estimation is also considered. It is shown that the dual state and parameter estimation problem can be solved by applying the UIF to a nonlinear system with unknown inputs. Recently, a nonlinear version of the extended recursive three-step filter, named as the NERTSF, was employed to solve the problem. However, a numerical approximation method is used to calculate the model partial derivatives. To relax that restriction, in this paper a derivative-free versions of the NERTSF is further proposed to solve the addressed estimation problem of the freeway traffic flow.

In this article, we address the problem of estimating the state and learning of the parameters in a linear dynamic system with generalized L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -regularization. Assuming a sparsity prior on the state, the joint state estimation and parameter learning problem is cast as an unconstrained optimization problem. However, when the dimensionality of state or parameters is large, memory requirements and computation of learning algorithms are generally prohibitive. Here, we develop a new augmented Lagrangian Kalman smoother method for solving this problem, where the primal variable update is reformulated as Kalman smoother. The effectiveness of the proposed method for state estimation and parameter learning is demonstrated in spectro-temporal estimation tasks using both synthetic and real data.

COVID-19 pandemic response with non-pharmaceutical interventions is an intrinsic control problem. Governments weigh social distancing policies to avoid overload in the health system without significant economic impact. The mutability of the SARS-CoV-2 virus, vaccination coverage, and mobility restriction measures change epidemic dynamics over time. A model-based control strategy requires reliable predictions to be efficient on a long-term basis. In this paper, a SEIR-based model is proposed considering dynamic feedback estimation. State and parameter estimations are performed on state estimators using augmented states. Three methods were implemented: constrained extended Kalman filter (CEKF), CEKF and smoother (CEKF & S), and moving horizon estimator (MHE). The parameters estimation was based on vaccine efficacy studies regarding transmissibility, severity of the disease, and lethality. Social distancing was assumed as a measured disturbance calculated using Google mobility data. Data from six federative units from Brazil were used to evaluate the proposed strategy. State and parameter estimations were performed from 1 October 2020 to 1 July 2021, during which Zeta and Gamma variants emerged. Simulation results showed that lethality increased between 11 and 30% for Zeta mutations and between 44 and 107% for Gamma mutations. In addition, transmissibility increased between 10 and 37% for the Zeta variant and between 43 and 119% for the Gamma variant. Furthermore, parameter estimation indicated temporal underreporting changes in hospitalized and deceased individuals. Overall, the estimation strategy showed to be suitable for dynamic feedback as simulation results presented an efficient detection and dynamic characterization of circulating variants.

This paper develops a hybrid strategy for identifying the nonlinear errors‐in‐variables (EIV) Wiener state‐space systems subjected to the unknown states. The Wiener nonlinear system with the noisy input is investigated, and a state‐space subsystem affected by the process disturbance is considered. The unbiased parameter estimation of the EIV Wiener systems with nonideal observations and polynomial nonlinearities is realized using the proposed bias‐compensated recursive least squares algorithm. Specifically, the estimated bias caused by the noisy measurements is eliminated, and the analytic formulas for estimating unmeasurable nonlinear intermediate variables are derived. Furthermore, based on the adaptive Kalman filtering principle, a state observer is designed to simultaneously estimate the subsystem states and the unknown noise variances in the estimated bias. The presented algorithms in the hybrid framework achieve the joint estimation of the parameters and states for EIV Wiener state‐space systems. The convergence analysis shows that the parameter estimates can converge to their true values. Finally, a numerical simulation and a quarter‐car model simulation demonstrate the effectiveness of the proposed algorithm.

Growing environmental concerns and shrinking water resources require methods beyond conventional wastewater treatment. Membrane bioreactor (MBR) is a technology that has become a ubiquitous choice for high quality treatment and reuse of wastewater. One of the key challenges in wastewater treatment is the high energy cost associated with aeration. MBR systems use feedback control to regulate the measured dissolved oxygen level at a predetermined set point by manipulating the blower throughputs. However, for high dynamic loads, feedback control may not result in the best performance and energy efficiency. Any attempt to optimize performance and power consumption beyond a simple controls strategy requires a proper trade‐off analysis between investments on additional sensors and the long‐term benefits. This article proposes a joint state and parameter estimation methodology, which measures and controls the MBR system using available measurements. Thus, limitations of feedback strategies can be overcome by predicting the impact of time varying disturbances on the outputs. The novelty in this approach is the ability to reconstruct the unknown states and parameters with available measurements. The unknown parameters are adaptively estimated online. Lyapunov's direct method is employed to show boundedness of state and parameter estimation errors. Simulation results illustrate the efficacy of the approach. Copyright © 2011 Curtin University of Technology and John Wiley &amp; Sons, Ltd.