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Abstract
Many practical engineering systems exhibit nonlinear behaviors that make it difficult to predict the output response for a given input. One approach to compensate for nonlinear effects is to feed forward a model of the nonlinearities in the control design as a means to cancel destabilizing effects: feedback linearization is a simplified example of such an approach. However, the system model typically has uncertainties such as parametric uncertainty or more general uncertainty that is often modeled as a bounded exogeneous disturbance. Adaptive and learning-based control strategies are motivated by the desire to include an approximate feedforward model in the controller or to implicitly learn the model through the feedback structure. This dissertation is focused on the development of such adaptive and learning-based controllers. Since the dissertation focuses on general nonlinear systems, a constructive Lyapunov-based design and analysis approach is used.
In Chapter 2, a deep neural network (DNN)-based adaptive controller is developed to compensate for uncertainty in a nonlinear dynamic system. Although Lyapunov-based real-time update laws are well-known for neural network (NN)-based adaptive controllers with a single-hidden-layer, developing real-time weight update laws for DNNs remains an open question. This dissertation presents the first result with Lyapunov-based real-time weight adaptation laws for each layer of a feedforward DNN-based control architecture, with stability guarantees. Additionally, the developed method allows nonsmooth activation functions to be used in the DNN to facilitate improved transient performance. A nonsmooth Lyapunov-based stability analysis proves global asymptotic tracking error convergence. Simulation results are provided for a nonlinear system using DNNs with leaky rectified linear unit (LReLU) and hyperbolic tangent activation functions to demonstrate the efficacy and performance of the developed method. Although Chapter 2 provides weight adaptation laws for DNNs, the development is restricted for fully-connected DNNs, whereas deriving weight adaptation laws has been an open problem for deep residual neural networks (ResNets).
Chapter 3 provides the first result on Lyapunov-derived weight adaptation for a ResNet-based adaptive controller. A nonsmooth Lyapunov-based analysis is provided to guarantee global asymptotic tracking error convergence. Comparative Monte Carlo simulations are provided to demonstrate the performance of the developed ResNet-based adaptive controller. The ResNet-based adaptive controller shows a 49.52% and 54.38% improvement in the tracking and function approximation performance, respectively, in comparison to a fully-connected DNN-based adaptive controller.
Chapter 4 addresses the problem of adaptive control of systems with uncertain time-varying parameters. A continuous adaptive controller is developed for nonlinear dynamical systems with linearly parameterizable uncertainty involving time-varying uncertain parameters. Through a unique stability analysis strategy, a new adaptive feedforward term is developed, along with specialized feedback terms, to yield asymptotic tracking error convergence by compensating for the time-varying nature of the uncertain parameters. A Lyapunov-based stability analysis is shown for Euler-Lagrange systems, which ensures asymptotic tracking error convergence and boundedness of the closed-loop signals. Additionally, the time-varying uncertain function approximation error is shown to converge to zero. A simulation example of a two-link manipulator is provided to demonstrate the asymptotic tracking result.
Chapter 5 provides new stability results for a class of implicit learning controllers called Robust Integral of the Sign of the Error (RISE) controllers. RISE controllers have been published over the past two decades as a means to yield asymptotic tracking error convergence and implicit asymptotic identification of time-varying uncertainties, for classes of nonlinear systems that are subject to sufficiently smooth bounded exogenous disturbances and/or modeling uncertainties. Despite the wide application of RISE-based techniques, an open question that has eluded researchers during this time-span is whether the asymptotic tracking error convergence is also uniform or exponential. This question has remained open due to certain limitations in the traditional construction of a Lyapunov function for RISE-based error systems. In this dissertation, new insights on the construction of a Lyapunov function are used that result in an exponential stability result for RISE-based controllers. As an outcome of this breakthrough, the inherent learning capability of RISE-based controllers is shown to yield exponential identification of state-dependent disturbances/uncertainty.
