On Data-Driven Modeling, Robust Control, and Analysis for Complex Dynamical Systems

This dissertation advances tools for robust control and analysis of complex nonlinear dynamical systems. Specifically, it leverages standard synthesis and robustness analysis techniques developed for linear systems and provides additional results to design robust controllers for nonlinear systems over the considered operating envelopes. To facilitate the application of these linear techniques, nonlinear systems are represented as uncertain linear models. A significant contribution of this dissertation is the development of data-driven approaches to generate these uncertain linear models, which capture the behavior of nonlinear systems reasonably well over the considered operating envelopes without being overly conservative.

We propose two approaches where a nominal linear time-invariant (LTI) approximation of a nonlinear system is first obtained using traditional linearization techniques, and data-driven methods are then applied to model the discrepancies arising from this simplification. In the first approach, the discrepancies are modeled using polynomials, resulting in an improved linear parameter-varying (LPV) approximation that can be expressed as a linear fractional transformation (LFT) on uncertainties. The second approach utilizes coprime factorization and a data-driven lifting technique to approximate the nonlinear discrepancy model with an LTI state-space system in a higher-dimensional state space. Additionally, a purely data-driven modeling approach is proposed for nonlinear systems with uncertain initial conditions. In this approach, a deep learning framework is developed to approximate nonlinear dynamical systems with LPV state-space models in higher-dimensional spaces while simultaneously characterizing the uncertain initial states within the lifted state space.

Another contribution is the development of a systematic method for identifying critical attack points in cyber-physical systems using integral quadratic constraints (IQCs). IQC analysis is also used in developing a framework focused on the design and analysis of robust path-following controllers for an autonomous underwater vehicle (AUV). In this framework, the AUV is modeled as an LFT on uncertainties and is affected by exogenous inputs such as measurement noise and ocean currents. A tuning routine is developed for robust control design, using the robust performance level derived from IQC analysis to guide the tuning process. This framework is applied to design ( H_\infty ), ( H_2 ), and LPV controllers for the AUV, with the results validated through extensive nonlinear simulations and underwater experiments.

Finally, this dissertation presents novel controller synthesis and IQC analysis techniques for LPV systems with uncertain initial conditions. These methods, combined with the lifting-based LPV modeling approach, enable the design of static, nonstationary LPV controllers for nonlinear systems in a higher-dimensional space. When interpreted in the original state space, these controllers become nonlinear with explicit dependence on both the scheduling parameters and time. Through examples, it is demonstrated that these controllers outperform those designed using nominal linearized models.