Direct Adaptive Control of Nonlinear Systems with Uncertain Unstable Zero Dynamics

Adaptive control is an essential technology for many control applications that have high levels of modeling uncertainty due to complex physics or unpredictable changes. Modeling complex physics for control is often expensive and challenging, and in some cases, impossible due to a lack of understanding. Even if the physics are well understood, parametric uncertainty necessitates data-based system identification, which can be difficult and expensive. For example, obtaining aircraft flight parameters requires extensive wind-tunnel or flight testing. The modeling challenge is exacerbated in systems that are subject to unpredictable changes, because an accurate model may become inaccurate. A typical example is an aircraft that experiences damage or even something as common as icing. The challenge of modeling real-world engineering applications limits what can be achieved with model-based controllers. For example, if an exact model of a linear time-invariant (LTI) system is known, then LQG can be used to achieve closed-loop stability and optimal performance. However, LQG does not guarantee robustness to model uncertainty, and thus small modeling errors can lead to unpredictable and even unstable closed-loop behavior. Robust control techniques address model uncertainty; however, these techniques sacrifice performance in order to address uncertainty. In contrast, adaptive control overcomes uncertainty without sacrificing performance by adjusting to the actual plant and environment. Substantial progress on adaptive control has been made over the past several decades, but significant challenges remain. Many existing techniques are limited due to assumptions of matched uncertainty and matched disturbances; restrictions to full-state feedback or low-relativedegree plants; and the need to avoid systems with nonminimum-phase (NMP) zeros. The proposed research aims to not only overcome all of these obstacles but also achieve significant extensions to nonlinear systems. This project will focus on a fundamental, challenging, and important problem in systems theory, namely, control of systems with uncertain unstable zero dynamics. Task 1 provides the foundation for the project by developing almost global boundedness and output convergence (AGBOC) theory. Task 2 uses this theory to analyze the ability to adapt controllers for uncertain linear-time-varying (LTV) systems. Of particular interest is the case where an LTV system has an unknown transition from minimum-phase to nonminimum-phase zero dynamics. Task 3 extends these results to nonlinear systems that possess unmodeled static nonlinearities at the input of the plant (Hammerstein systems), at the output of the plant (Wiener systems), or interconnected through feedback (Lure systems). These models include systems with uncertain unstable zero dynamics, which are potentially time varying.