Robust local stabilization and state estimation of nonlinear systems using quadratic constraints and convex optimization

Abstract

Nonlinear systems are challenging to model and control, particularly when accurate knowledge of the system’s nonlinearities is not available. This thesis presents convex-optimization-based frameworks for the synthesis of stabilizing controllers and the state estimation of nonlinear systems. Quadratic constraints are used to characterize the system’s nonlinearities when performing controller synthesis and bounding the covariance of the state estimation error, which enables the use of convex optimization for these purposes. The first part of this thesis develops a control synthesis method for a nonlinear system decomposed into a linear model and a data-driven description of the system’s nonlinearities. The second part of this thesis extends this synthesis method to account for the presence of exogenous harmonic excitation signals injected during system identification, resulting in the certification of a locally-bounded closed-loop response. Finally, the synthesized controllers developed in the first parts of the thesis rely on the availability of a full-state measurement, which is unrealistic in practice. To address this, quadratic constraints are used to bound the time-propagation of the covariance on the state estimation error of a nonlinear process model subject to white noise. This is incorporated within an extended Kalman filter to provide a certified bound on the state estimation error covariance during the time-update step.