Estimation and control in discrete-time nonlinear systems.

Abstract

This thesis addresses a set of problems arising in the application of adaptive control to discrete-time nonlinear systems. First, we extend the class of discrete-time nonlinear systems to which a linear parameter estimation scheme and certainty equivalence control can be applied. That is, it is shown that direct model reference adaptive control (MRAC) via output-feedback is possible for SISO, minimum phase, linear discrete-time systems followed by a nonlinear element, with further nonlinearities acting on the output. A few assumptions that are needed in the case of continuous-time systems of the same form are relaxed. Simulations are performed with two examples: a first order system with Coulomb friction and a simple manipulator. A heuristic indirect MRAC scheme with the extended Kalman filter (EKF) as a parameter estimator is also simulated on the simple manipulator. Motivated by the simulation studies with the EKF, we then focus our attention on the EKF as a parameter estimator. Since the EKF is actually used as an approximate state estimator in this context, we make a detailed analysis of the EKF when it is used as an observer for discrete-time nonlinear systems. Based on our new proof of the fact that the Kalman filter is a global observer for linear (discrete-time) time-varying systems, it is shown that the EKF is indeed a local asymptotic observer for general discrete-time nonlinear systems. Finally, we address one of the most common assumptions in adaptive control or tracking: a well-defined relative degree. It is shown that this is indeed a necessary condition for asymptotic tracking in discrete-time nonlinear systems. To show this, tracking problems are defined, and a local linear input-output behavior of a discrete-time system is introduced in relation to a well-defined relative degree. It is then shown that if a plant is invertible and accessible from the origin and a compensator solves the local asymptotic tracking problem while keeping the closed-loop system invertible, then the plant necessarily has a well-defined relative degree at the origin. This result confirms that a well-defined relative degree is one of the basic requirements for model reference type control, which includes most adaptive control.