Practical sample-and-hold stabilization of nonlinear systems under approximate optimizers

TL;DR

This paper investigates how approximate solutions to optimization problems affect the stability of nonlinear systems under sample-and-hold control, providing bounds on optimization accuracy needed for desired stability margins.

Contribution

It introduces bounds on optimization accuracy necessary for practical stability in sample-and-hold stabilization of nonlinear systems with approximate optimizers.

Findings

Optimization accuracy critically influences state convergence.

Bounds on optimization errors ensure prescribed stability margins.

Simulations confirm the impact of optimization precision on stability.

Abstract

It is a known fact that not all controllable systems can be asymptotically stabilized by a continuous static feedback. Several approaches have been developed throughout the last decades, including time-varying, dynamical and even discontinuous feedbacks. In the latter case, the sample-and-hold framework is widely used, in which the control input is held constant during sampling periods. Consequently, only practical stability can be achieved at best. Existing approaches often require solving optimization problems for finding stabilizing control actions exactly. In practice, each optimization routine has a finite accuracy which might influence the state convergence. This work shows, what bounds on optimization accuracy are required to achieve prescribed stability margins. Simulation studies support the claim that optimization accuracy has high influence on the state convergence.