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

Adaptive Control Design and Evaluation for Multibody High-speed Train Dynamic Models

In this paper, the adaptive tracking control problem is investigated for multibody high-speed train dynamic model in the presence of unknown parameters, which is an open adaptive control problem. A 4-car train unit model with input signals acting on the 2nd and 3rd cars and output signals being the speeds of the 1st and 3rd cars is chosen as a benchmark model, in which the aerodynamic resistance force is also considered. To handel the nonlinear term, the feedback linearization method is employed to decompose the system into a control dynamics subsystem and a zero dynamics subsystem. A new and detailed stability analysis is conducted to show that such a zero dynamic system is Lyapunov stable and is also partially input-to-state stable under the condition that the speed error between the 1st and 3rd cars is exponentially convergent (to be ensured by a nominal controller) or belongs to the L1 signal space (to be achieved by a properly designed adaptive controller). The system configuration leads to a relative degree 1 subsystem and a relative degree 2 subsystem, for which different feedback linearization-based adaptive controllers and their nominal versions are developed to ensure the needed stabilization condition, the desired closed-loop system signal boundedness and asymptotic output speed tracking. Detailed closed-loop system stability and tracking performance analysis are given for the new control schemes. Simulation results from a realistic train dynamic model are presented to verify the desired adaptive control system performance

Observer Based Traction/Braking Control Design for High Speed Trains Considering Adhesion Nonlinearity

Train traction/braking control, one of the key enabling technologies for automatic train operation, literally takes its action through adhesion force. However, adhesion coefficient of high speed train (HST) is uncertain in general because it varies with wheel-rail surface condition and running speed; thus, it is extremely difficult to be measured, which makes traction/braking control design and implementation of HSTs greatly challenging. In this work, force observers are applied to estimate the adhesion force or/and the resistance, based on which simple traction/braking control schemes are established under the consideration of actual wheel-rail adhesion condition. It is shown that the proposed controllers have simple structure and can be easily implemented from real applications. Numerical simulation also validates the effectiveness of the proposed control scheme

A novel iterative learning approach for tracking control of high-speed trains subject to unknown time-varying delay

In this article, a novel iterative learning control scheme is proposed for high-speed trains, aiming to track the desired reference displacement and velocity, where the Krasovskii function is constructed to compensate for the negative influence of unknown time-varying speed delays. The main feature of the proposed approach is that the hyperbolic tangent function and the command filter are integrated into the learning controller to overcome the singularity problem that may occur during the control process and relax the requirement for the derivability of the desired velocity. The stability of control system is strictly proved through establishing the composite energy function, and the effectiveness is confirmed via numerical simulations. Compared with the existing works, the merits of the proposed control scheme lie in that more general nonlinear uncertainties are imposed on the dynamic model of train instead of the Lipschitz condition, and the reference acceleration assigned by the railway department is not required

Approximate Gaussian Conjugacy: Parametric Recursive Filtering Under Nonlinearity, Multimodal, Uncertainty, and Constraint, and Beyond

This is a post-peer-review, pre-copyedit version of an article published in Frontiers of Information Technology & Electronic Engineering. The final authenticated version is available online at: https://doi.org/10.1631/FITEE.1700379Since 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

Modeling and Monitoring of the Dynamic Response of Railroad Bridges using Wireless Smart Sensors

Railroad bridges form an integral part of railway infrastructure in the USA, carrying approximately 40 % of the ton-miles of freight. The US Department of Transportation (DOT) forecasts current rail tonnage to increase up to 88 % by 2035. Within the railway network, a bridge occurs every 1.4 miles of track, on average, making them critical elements. In an effort to accommodate safely the need for increased load carrying capacity, the Federal Railroad Association (FRA) announced a regulation in 2010 that the bridge owners must conduct and report annual inspection of all the bridges. The objective of this research is to develop appropriate modeling and monitoring techniques for railroad bridges toward understanding the dynamic responses under a moving train. To achieve the research objective, the following issues are considered specifically. For modeling, a simple, yet effective, model is developed to capture salient features of the bridge responses under a moving train. A new hybrid model is then proposed, which is a flexible and efficient tool for estimating bridge responses for arbitrary train configurations and speeds. For monitoring, measured field data is used to validate the performance of the numerical model. Further, interpretation of the proposed models showed that those models are efficient tools for predicting response of the bridge, such as fatigue and resonance. Finally, fundamental software, hardware, and algorithm components are developed for providing synchronized sensing for geographically distributed networks, as can be found in railroad bridges. The results of this research successfully demonstrate the potentials of using wirelessly measured data to perform model development and calibration that will lead to better understanding the dynamic responses of railroad bridges and to provide an effective tool for prediction of bridge response for arbitrary train configurations and speeds.National Science Foundation Grant No. CMS-0600433National Science Foundation Grant No. CMMI-0928886National Science Foundation Grant No. OISE-1107526National Science Foundation Grant No. CMMI- 0724172 (NEESR-SD)Federal Railroad Administration BAA 2010-1 projectOpe