Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond

Since the landmark work of R. E. Kalman in the 1960s, considerable efforts have been devoted to time series state space models for a large variety of dynamic estimation problems. In particular, parametric filters that seek analytical estimates based on a closed-form Markov–Bayes recursion, e.g., recursion from a Gaussian or Gaussian mixture (GM) prior to a Gaussian/GM posterior (termed ‘Gaussian conjugacy’ in this paper), form the backbone for a general time series filter design. Due to challenges arising from nonlinearity, multimodality (including target maneuver), intractable uncertainties (such as unknown inputs and/or non-Gaussian noises) and constraints (including circular quantities), etc., new theories, algorithms, and technologies have been developed continuously to maintain such a conjugacy, or to approximate it as close as possible. They had contributed in large part to the prospective developments of time series parametric filters in the last six decades. In this paper, we review the state of the art in distinctive categories and highlight some insights that may otherwise be easily overlooked. In particular, specific attention is paid to nonlinear systems with an informative observation, multimodal systems including Gaussian mixture posterior and maneuvers, and intractable unknown inputs and constraints, to fill some gaps in existing reviews and surveys. In addition, we provide some new thoughts on alternatives to the first-order Markov transition model and on filter evaluation with regard to computing complexity

ESTIMATION AND CONTROL OF NONLINEAR SYSTEMS: MODEL-BASED AND MODEL-FREE APPROACHES

State estimation and subsequent controller design for a general nonlinear system is an important problem that have been studied over the past decades. Many applications, e.g., atmospheric and oceanic sampling or lift control of an airfoil, display strongly nonlinear dynamics with very high dimensionality. Some of these applications use smaller underwater or aerial sensing platforms with insufficient on-board computation power to use a Monte-Carlo approach of particle filters. Hence, they need a computationally efficient filtering method for state-estimation without a severe penalty on the performance. On the other hand, the difficulty of obtaining a reliable model of the underlying system, e.g., a high-dimensional fluid dynamical environment or vehicle flow in a complex traffic network, calls for the design of a data-driven estimation and controller when abundant measurements are present from a variety of sensors. This dissertation places these problems in two broad categories: model-based and model-free estimation and output feedback. In the first part of the dissertation, a semi-parametric method with Gaussian mixture model (GMM) is used to approximate the unknown density of states. Then a Kalman filter and its nonlinear variants are employed to propagate and update each Gaussian mode with a Bayesian update rule. The linear observation model permits a Kalman filter covariance update for each Gaussian mode. The estimation error is shown to be stochastically bounded and this is illustrated numerically. The estimate is used in an observer-based feedback control to stabilize a general closed-loop system. A transferoperator- based approach is then proposed for the motion update for Bayesian filtering of a nonlinear system. A finite-dimensional approximation of the Perron-Frobenius (PF) operator yields a method called constrained Ulam dynamic mode decomposition (CUDMD). This algorithm is applied for output feedback of a pitching airfoil in unsteady flow. For the second part, an echo-state network (ESN) based approach equipped with an ensemble Kalman filter is proposed for data-driven estimation of a nonlinear system from a time series. A random reservoir of recurrent neural connections with the echo-state property (ESP) is trained from a time-series data. It is then used as a model-predictor for an ensemble Kalman filter for sparse estimation. The proposed data-driven estimation method is applied to predict the traffic flow from a set of mobility data of the UMD campus. A data-driven model-identification and controller design is also developed for control-affine nonlinear systems that are ubiquitous in several aerospace applications. We seek to find an approximate linear/bilinear representation of these nonlinear systems from data using the extended dynamic mode decomposition algorithm (EDMD) and apply Liealgebraic methods to analyze the controllability and design a controller. The proposed method utilizes the Koopman canonical transform (KCT) to approximate the dynamics into a bilinear system (Koopman bilinear form) under certain assumptions. The accuracy of this approximation is then analytically justified with the universal approximation property of the Koopman eigenfunctions. The resulting bilinear system is then subjected to controllability analysis using the Myhill semigroup and Lie algebraic structures, and a fixed endpoint optimal controller is designed using the Pontryagin’s principle

Composite Disturbance Filtering: A Novel State Estimation Scheme for Systems With Multi-Source, Heterogeneous, and Isomeric Disturbances

State estimation has long been a fundamental problem in signal processing and control areas. The main challenge is to design filters with ability to reject or attenuate various disturbances. With the arrival of big data era, the disturbances of complicated systems are physically multi-source, mathematically heterogenous, affecting the system dynamics via isomeric (additive, multiplicative and recessive) channels, and deeply coupled with each other. In traditional filtering schemes, the multi-source heterogenous disturbances are usually simplified as a lumped one so that the "single" disturbance can be either rejected or attenuated. Since the pioneering work in 2012, a novel state estimation methodology called {\it composite disturbance filtering} (CDF) has been proposed, which deals with the multi-source, heterogenous, and isomeric disturbances based on their specific characteristics. With the CDF, enhanced anti-disturbance capability can be achieved via refined quantification, effective separation, and simultaneous rejection and attenuation of the disturbances. In this paper, an overview of the CDF scheme is provided, which includes the basic principle, general design procedure, application scenarios (e.g. alignment, localization and navigation), and future research directions. In summary, it is expected that the CDF offers an effective tool for state estimation, especially in the presence of multi-source heterogeneous disturbances

Feedback-control of quantum systems using continuous state-estimation

We present a formulation of feedback in quantum systems in which the best estimates of the dynamical variables are obtained continuously from the measurement record, and fed back to control the system. We apply this method to the problem of cooling and confining a single quantum degree of freedom, and compare it to current schemes in which the measurement signal is fed back directly in the manner usually considered in existing treatments of quantum feedback. Direct feedback may be combined with feedback by estimation, and the resulting combination, performed on a linear system, is closely analogous to classical LQG control theory with residual feedback.Comment: 12 pages, multicol revtex, revised and extende

Use of Bridging Strategy between the Ensemble Kalman Filter and Particle Filter for the Measurements with Various Quasi-Gaussian Noise

Filtering and estimation are two important tools of engineering. Whenever the state of the system needs to be estimated from the noisy sensor measurements, some kind of state estimator is used. If the dynamics of the system and observation model are linear under Gaussian conditions, the root mean squared error can be computed using the Kalman Filter. But practically, noise frequently enters the system as not strictly Gaussian. Therefore, the Kalman Filter does not necessarily provide the better estimate. Hence the estimation of the nonlinear system under non-Gaussian or quasi-Gaussian noise is of an acute interest. There are many versions of the Kalman filter such as the Extended Kalman filter, the Unscented Kalman filter, the Ensemble Kalman filter, the Particle filter, etc., each having their own disadvantages. In this thesis work I used a bridging strategy between the Ensemble Kalman filter and Particle filter called an Ensemble Kalman Particle filter. This filter works well in nonlinear system and non-Gaussian measurements as well. I analyzed this filter using MATLAB simulation and also applied Gaussian Noise, non-zero mean Gaussian Noise, quasi-Gaussian noise (with drift), random walk and Laplacian Noise. I applied these noises and compared the performances of the Particle filter and the Ensemble Kalman Particle filter in the presence of linear and nonlinear observations which leads to the conclusion that the Ensemble Kalman Particle filter yields the minimum error estimate. I also found the optimum value for the tuning parameter which is used to bridge the two filters using Monte Carlo Simulation

Generalized Multi-kernel Maximum Correntropy Kalman Filter for Disturbance Estimation

Disturbance observers have been attracting continuing research efforts and are widely used in many applications. Among them, the Kalman filter-based disturbance observer is an attractive one since it estimates both the state and the disturbance simultaneously, and is optimal for a linear system with Gaussian noises. Unfortunately, The noise in the disturbance channel typically exhibits a heavy-tailed distribution because the nominal disturbance dynamics usually do not align with the practical ones. To handle this issue, we propose a generalized multi-kernel maximum correntropy Kalman filter for disturbance estimation, which is less conservative by adopting different kernel bandwidths for different channels and exhibits excellent performance both with and without external disturbance. The convergence of the fixed point iteration and the complexity of the proposed algorithm are given. Simulations on a robotic manipulator reveal that the proposed algorithm is very efficient in disturbance estimation with moderate algorithm complexity.Comment: in IEEE Transactions on Automatic Control (2023

Optimal state estimation for cavity optomechanical systems

We demonstrate optimal state estimation for a cavity optomechanical system through Kalman filtering. By taking into account nontrivial experimental noise sources, such as colored laser noise and spurious mechanical modes, we implement a realistic state-space model. This allows us to obtain the conditional system state, i.e., conditioned on previous measurements, with minimal least-square estimation error. We apply this method for estimating the mechanical state, as well as optomechanical correlations both in the weak and strong coupling regime. The application of the Kalman filter is an important next step for achieving real-time optimal (classical and quantum) control of cavity optomechanical systems.Comment: replaced with published version, 5+12 page