We develop an adaptive control architecture to achieve stabilization and command following of uncertain dynamical systems with improved transient performance. Our framework consists of a new reference system and an adaptive controller. The proposed reference system captures a desired closed-loop dynamical system behavior modified by a mismatch term representing the high-frequency content between the uncertain dynamical system and this reference system, i.e., the system error. In particular, this mismatch term allows to limit the frequency content of the system error dynamics, which is used to drive the adaptive controller. It is shown that this key feature of our framework yields fast adaptation without incurring high-frequency oscillations in the transient performance. We further show the effects of design parameters on the system performance, analyze closeness of the uncertain dynamical system to the unmodified (ideal) reference system, discuss robustness of the proposed approach with respect to time-varying uncertainties and disturbances, and make connections to gradient minimization and classical control theory.

Abstract-This paper explores the application of Koopman operator theory to the control of robotic systems. The operator is introduced as a method to generate data-driven models that have utility for model-based control methods. We then motivate the use of the Koopman operator towards augmenting modelbased control. Specifically, we illustrate how the operator can be used to obtain a linearizable data-driven model for an unknown dynamical process that is useful for model-based control synthesis. Simulated results show that with increasing complexity in the choice of the basis functions, a closed-loop controller is able to invert and stabilize a cart-and VTOL-pendulum systems. Furthermore, the specification of the basis function are shown to be of importance when generating a Koopman operator for specific robotic systems. Experimental results with the Sphero SPRK robot explore the utility of the Koopman operator in a reduced state representation setting where increased complexity in the basis function improve open-and closed-loop controller performance in various terrains, including sand. I. INTRODUCTIONModeling for complex dynamical systems has typically been the first step when designing, control, planning, or stateestimation algorithms. System design and specifications have been dependent on the use of high-fidelity models. However, any derivation of a dynamical model from first principles is typically a demanding task when the complexity of state interactions is high. Moreover, analytical models do not capture external disturbances. As a result, derived models, for use in model-based control settings, often have limited use or poor prediction over longer time spans. Nevertheless, a representation of the behavior of a dynamical system is central to most model-based engineering and scientific application.Within the field of systems and control theory, model uncertainty has typically been mitigated with the use of robust and adaptive control architectures. Typically, adaptive controllers are self tuning and reactive to incoming state information while robust controllers are designed to be invariant to model uncertainty [1]-[4]. Motion planning for uncertain dynamical systems have also been extensively investigated. Generally, in this approach, uncertainty is explicitly modeled and incorporated into the decision making process [5]- [7]. However, like robust and adaptive control approaches, the need for an explicit uncertainty model often limits its utility in general settings. Machine learning, offers a much more general approach [8]- [10]. In particular, recent advances have utilized large sets

This paper presents a new adaptive control architecture to achieve stabilization and command following of uncertain dynamical systems with improved transient performance. Our framework consists of a new reference system and an adaptive controller. The proposed reference system captures a desired closed-loop dynamical system behavior modified by a mismatch term representing the high-frequency content between the uncertain dynamical system and this reference system, i.e., the system error. In particular, this mismatch term allows one to limit the frequency content of the system error dynamics, which is used to drive the adaptive controller. It is shown that this key feature of our framework yields fast adaptation without incurring high-frequency oscillations in the transient performance.

A concurrent learning adaptive control architecture for uncertain linear switched dynamical systems is presented. Like other concurrent learning adaptive control architectures, the adaptive weight update law uses both recorded and current data concurrently for adaptation. In addition, a verifiable condition on the linear independence of the recorded data is shown to be sufficient to guarantee global exponential stability and adaptive weight convergence. Furthermore, it is shown that the recorded data eventually meets this condition after a system switch without any additional excitation from the exogenous reference input or knowledge of the switching signal if there is sufficient time in between switches. That is, after a switch, the system will be automatically excited and sufficiently rich data will be recorded. As a result, data that is irrelevant to the current subsystem will be overwritten. Thus, reference model tracking error and adaptive weight error will eventually become globally exponential stable for all switched subsystems. Numerical examples are presented to illustrate the effectiveness of the proposed architecture.

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