A Time-delay Approach for Stabilization of Linear Systems with Unknown Control Direction

We propose a novel stability analysis for linear systems with unknown control directions, focusing on stabilization through extremum seeking to minimize a Lyapunov function. Our approach draws inspiration from Lie bracket systems, employing a transformation using time-delay analysis to convert the closed-loop system into a nominal time-delay system. Leveraging techniques from Input-to-State Stability (ISS), we provide a proof of semiglobal practical asymptotic stability. Compared to the preceding analysis based on Lie bracket systems, the use of time-delay methods transforms intricate mathematical discussions and proofs into a structured computational framework, facilitating a more intuitive understanding. Also, we propose a method for selecting controller parameters and can provide a more comprehensive explanation for the current high-gain parameter choices. Finally, we present a new control parameter to reduce the requirements of the frequency of the sinusoidal signals.