Extremum-Seeking for PWM-Controlled Double-Integrator System via Nonlinear Small-Gain Approach

This study considers the extremum-seeking control problem for double-integrator system that is controlled by using the pulse-width-modulation (PWM) technique. A new class of PWM control law is presented that the state trajectories of the closed-loop system are held within a desired range. A trajectory-based analysis approach is also proposed that characterize the influence of the hybrid PWM control input. A novel hierarchical structure is developed that separates the closed-loop extremum-seeking system into the interconnected networked subsystems, which are characterized in the sense of input-to-state practical stability (ISpS) and input-to-output stability (IOpS). By using the nonlinear small-gain theorem, the stability of the ES closed-loop system is ensured so that the PWM-controlled plant can gravitate towards an appropriate region of the maximum (minimum) of the nonlinear map (which can be considered as a gradient field). Finally, numerical examples demonstrate the validity and effectiveness of our proposed design.