Omkar Sudhir Patil

Omkar Sudhir Patil, Emily J. Griffis, Wanjiku A. Makumi et al.

Although deep neural network (DNN)-based controllers are popularly used to control uncertain nonlinear dynamic systems, most results use DNNs that are pretrained offline and the corresponding controller is implemented post-training. Recent advancements in adaptive control have developed controllers with Lyapunov-based update laws (i.e., control and update laws derived from a Lyapunov-based stability analysis) for updating the DNN weights online to ensure the system states track a desired trajectory. However, the update laws are based on the tracking error, and offer guarantees on only the tracking error convergence, without providing any guarantees on system identification. This paper provides the first result on simultaneous online system identification and trajectory tracking control of nonlinear systems using adaptive updates for all layers of the DNN. A combined Lyapunov-based stability analysis is provided, which guarantees that the tracking error, state-derivative estimation error, and DNN weight estimation errors are uniformly ultimately bounded. Under the persistence of excitation (PE) condition, the tracking and weight estimation errors are shown to exponentially converge to a neighborhood of the origin, where the rate of convergence and the size of this neighborhood depends on the gains and a factor quantifying PE, thus achieving system identification and enhanced trajectory tracking performance. As an outcome of the system identification, the DNN model can be propagated forward to predict and compensate for the uncertainty in dynamics under intermittent loss of state feedback. Comparative simulation results are provided on a two-link manipulator system and an unmanned underwater vehicle system with intermittent loss of state feedback, where the developed method yields significant performance improvement compared to baseline methods.

Omkar Sudhir Patil, Brandon C. Fallin, Cristian F. Nino et al.

Deep neural networks (DNNs) have emerged as a powerful tool with a growing body of literature exploring Lyapunov-based approaches for real-time system identification and control. These methods depend on establishing bounds for the second partial derivatives of DNNs with respect to their parameters, a requirement often assumed but rarely addressed explicitly. This paper provides rigorous mathematical formulations of polynomial bounds on both the first and second partial derivatives of DNNs with respect to their parameters. We present lemmas that characterize these bounds for fully-connected DNNs, while accommodating various classes of activation function including sigmoidal and ReLU-like functions. Our analysis yields closed-form expressions that enable precise stability guarantees for Lyapunov-based deep neural networks (Lb-DNNs). Furthermore, we extend our results to bound the higher-order terms in first-order Taylor approximations of DNNs, providing important tools for convergence analysis in gradient-based learning algorithms. The developed theoretical framework develops explicit, computable expressions, for previously assumed bounds, thereby strengthening the mathematical foundation of neural network applications in safety-critical control systems.

Saiedeh Akbari, Emily J. Griffis, Omkar Sudhir Patil et al.

Deep neural network (DNN)-based adaptive controllers can be used to compensate for unstructured uncertainties in nonlinear dynamic systems. However, DNNs are also very susceptible to overfitting and co-adaptation. Dropout regularization is an approach where nodes are randomly dropped during training to alleviate issues such as overfitting and co-adaptation. In this paper, a dropout DNN-based adaptive controller is developed. The developed dropout technique allows the deactivation of weights that are stochastically selected for each individual layer within the DNN. Simultaneously, a Lyapunov-based real-time weight adaptation law is introduced to update the weights of all layers of the DNN for online unsupervised learning. A non-smooth Lyapunov-based stability analysis is performed to ensure asymptotic convergence of the tracking error. Simulation results of the developed dropout DNN-based adaptive controller indicate a 38.32% improvement in the tracking error, a 53.67% improvement in the function approximation error, and 50.44% lower control effort when compared to a baseline adaptive DNN-based controller without dropout regularization.

Yixuan Wang, Omkar Sudhir Patil, Warren E. Dixon

We introduce Riemannian Lyapunov Optimizers (RLOs), a family of optimization algorithms that unifies classic optimizers within one geometric framework. Unlike heuristic improvements to existing optimizers, RLOs are systematically derived from a novel control-theoretic framework that reinterprets optimization as an extended state discrete-time controlled dynamical system on a Riemannian parameter manifold. Central to this framework is the identification of a Normally Attracting Invariant Manifold (NAIM), which organizes training dynamics into two distinct stages: rapid alignment of the speed state to a target graph, followed by controlled evolution within it. We formalize this by constructing a strict Lyapunov function that certifies convergence to a target manifold. This perspective yields a constructive ``optimizer generator" that not only recovers classic algorithms but enables the principled design of RLOs. We validate our theory via geometric diagnostics and demonstrate that grounding optimizer design in control theory yields state-of-the-art performance in large-scale benchmarks. Overall, RLOs bridge control theory and modern machine learning optimization, providing a unified language and a systematic toolkit for designing stable, effective optimizers.

Rebecca G. Hart, Omkar Sudhir Patil, Zachary I. Bell et al.

Deep Neural Networks (DNNs) are increasingly used in control applications due to their powerful function approximation capabilities. However, many existing formulations focus primarily on tracking error convergence, often neglecting the challenge of identifying the system dynamics using the DNN. This paper presents the first result on simultaneous trajectory tracking and online system identification using a DNN-based controller, without requiring persistent excitation. Two new concurrent learning adaptation laws are constructed for the weights of all the layers of the DNN, achieving convergence of the DNN's parameter estimates to a neighborhood of their ideal values, provided the DNN's Jacobian satisfies a finite-time excitation condition. A Lyapunov-based stability analysis is conducted to ensure convergence of the tracking error, weight estimation errors, and observer errors to a neighborhood of the origin. Simulations performed on a range of systems and trajectories, with the same initial and operating conditions, demonstrated 40.5% to 73.6% improvement in function approximation performance compared to the baseline, while maintaining a similar tracking error and control effort. Simulations evaluating function approximation capabilities on data points outside of the trajectory resulted in 58.88% and 74.75% improvement in function approximation compared to the baseline.