Topics in State and Parameter Estimation for Nonlinear and Uncertain Systems

Abstract

This thesis is submitted in partial fulfillment of the requirements for the degree of Philosphiae Doctor (PhD) at the Norwegian University of Science and Technology (NTNU). The thesis is divided into three parts, each of which addresses a specific topic within state and parameter estimation for nonlinear and uncertain systems.

Part I of the thesis is concerned with state and parameter estimation for automotive vehicles. The main focus is on estimation of the angle between the orientation of the vehicle and its direction of travel, known as the vehicle sideslip angle. The lateral dynamics of a car is accurately described by a linear model during normal operation, but the dynamics becomes highly nonlinear if one or more of the tires begin to lose road grip. Existing observers for estimating the vehicle sideslip angle are typically constructed as extended Kalman filters, without nonlinear stability analysis and with the drawback of being computationally expensive. The goal of Part I is to construct a vehicle sideslip observer that is less computationally expensive than the extended Kalman filter and that is based on explicit nonlinear stability analysis. The observer produces estimates of the longitudinal and lateral velocities of the vehicle, which are used to calculate the vehicle sideslip angle. In addition, estimates of the road inclination and bank angle, as well as a roadtire friction parameter, are introduced to improve accuracy. The observer is tested using measurements from production passenger cars.

Part II of the thesis is concerned with state and parameter estimation for systems perturbed by a nonlinear function of the system state, external time-varying signals, and a set of unknown, constant parameters. The majority of techniques for estimating unknown parameters in dynamic equations are based on the assumption that the parameters enter the equations linearly, which is not always accurate. Part II starts by considering the problem of parameter estimation in the case of full-state measurement, by using a modular design consisting of a perturbation estimator and a parameter estimator. The perturbation estimator estimates the full perturbation that depends on the unknown parameters, whereas the parameter estimator inverts the perturbation estimate dynamically with respect to the unknown parameters. The parameter estimates are in turn fed back to the perturbation estimator leading to an exponentially stable interconnection when the gains are chosen appropriately. The case of partial-state measurement is then considered for a class of systems where the perturbation is separated from the output by a left-invertible, minimum-phase linear system. To handle this case, the perturbation estimator is replaced by a modified high-gain observer, similar to extended high-gain observers found in the literature, which estimates both the state of the system and the perturbation. Part of the high-gain design procedure consists of transforming the linear part of the system to a special coordinate basis that explicitly reveals the system’s zero structure and invertibility properties. Numerical software tools for carrying out such transformations already exist. To facilitate the transformation of systems with symbolic representations as well, Part II presents a software tool written in the mathematics software suite Maple.

Part III is concerned with output-feedback stabilization of two classes of systems containing saturations. First, global stabilization of multivariable linear timeinvariant systems with saturated outputs is considered. An output-feedback design for single-input single-output systems of this type, which ensures global asymptotic stability if the linear system is controllable and observable, can already be found in the literature. This design is based on deactivating the saturation by using the sign of the output, and identifying the state of the system in a deadbeat manner once the saturation becomes inactive. The most logical extension of this design to systems with multiple outputs would be to deactivate all the saturations simultaneously; however, this is a difficult, if not impossible, task. The design presented in Part III is instead based on deactivating all of the saturations at least once, but not necessarily at the same time. The state of the system can then be identified in a deadbeat manner and controlled to the origin. Next, semiglobal stabilization of sandwich systems, consisting of two linear subsystems in a saturated cascade connection, is considered under the assumption that a partial-state measurement is available only from the second subsystem in the cascade. This problem is challenging because the first subsystem is separated from the output by the saturation, and the second subsystem is separated from the input by the saturation. The design presented in Part III therefore combines techniques for stabilizing systems with saturated inputs with techniques for stabilizing systems with saturated outputs. A key element of the design is a detection algorithm that determines whether the saturation is active or inactive. This detection algorithm operates on time intervals that must be chosen sufficiently small depending on the size of a compact set of admissible initial conditions