Model-Free Stabilization by Extremum Seeking pp 117-117 | Cite as

# Conclusions

## Abstract

This book presents a new application of Extremum Seeking, as a method for stabilization of unknown systems as well as trajectory tracking and optimization. The stabilization of unknown systems is possible due to the controller’s ability to perform a gradient descent of unknown functions or of purposely chosen Lyapunov-like functions.

This book presents a new application of Extremum Seeking, as a method for stabilization of unknown systems as well as trajectory tracking and optimization. The stabilization of unknown systems is possible due to the controller’s ability to perform a gradient descent of unknown functions or of purposely chosen Lyapunov-like functions.

The extremum seeking algorithm creates a closed loop system that is independent of the control vector’s direction. This is a useful property which allows us to stabilize and perform trajectory tracking with unknown, unstable, control direction-varying systems using a particular form of time-varying nonlinear high-gain feedback.

In Chap. 5, we showed that the ES controllers may be chosen such that they turn themselves off as equilibrium is approached, in Chap. 6 we showed that they may be chosen so that they are bounded, with all unknowns entering the ES scheme as arguments of a-priori known, bounded functions, in Chap. 7, we showed that we are not limited to systems affine in control, and in Chap. 8 we concluded that we may replace smooth perturbing functions such as \(\sin (\cdot )\) or \(\cos (\cdot )\), with a general class of integrable functions, such as square waves with dead time or triangle waves, signals which are more naturally implemented in digital logic. Finally, in Chap. 9 we described a few cases of implementation of these methods, in hardware, for tuning various components of a particle accelerator.

One of the main challenges of ESS, as with all other ES approaches, is that it is an existence result. For any system, for any given compact set and any desired degree of accuracy, there exists some \(\omega ^\star \), such that for all \(\omega >\omega ^\star \), the desired results hold. Finding concrete bounds on \(\omega ^\star \) based on a given system’s parameters and a set of initial conditions would be an interesting result. Another possible topic of future research may be in creating an auto-tuning ESC controller, which iteratively or dynamically adjusts its own *k*, \(\alpha \), and \(\omega \) values, in order to stabilize or optimize an unknown system.