Data-driven control of nonlinear systems using dissipativity theory

Abstract

In the field of polynomial systems, control design is commonly related to finding Lyapunov conditions in the form of linear matrix inequalities and determining a solution to them through semidefinite programming algorithms. However, the bilinearities inherent to Lyapunov conditions require the use of often suboptimal strategies, such as the D-K iteration, which has no guarantees of solution convergence. In addition, the challenge of some plants that are difficult to model is solved through data collection experiments in order to identify the system. Yet, since the data is subject to disturbances or noise, it hinders the development of a reliable model. In this work, a data-driven approach was used to design control laws without an intermediate identification step and with robustness to data subject to unknown but limited disturbances. Using Lyapunov theory and QSR-dissipation theory, sum-of-squares conditions were constructed to design a control law for polynomial nonlinear systems that guarantees asymptotic stabilization locally around the origin. These conditions were implemented using sum-of-squares programming with the help of SOSTools and Mosek in iterative algorithms without bilinearities and constraints for solution convergence. Finally, the control law and the Lyapunov function were validated using formal verification tools such as Z3Prover to issue a Lyapunov certificate. Keywords: Data-driven control. Nonlinear systems. Dissipativity theory.