Fast Control Systems: Nonlinear Approach

Abstract

This chapter treats the problem of fast control design for nonlinear systems. First, we discusses the question: which nonlinear system can be called fast? Next, we develop some tools for analysis and design of such control systems. The method generalized homogeneity is mainly utilized for these purposes. Finally, we survey possible research directions of the fast control systems.

A homogeneity property of discrete‐time systems: Stability and convergence rates

“…In this section, we show that, as in the CT case, the D r ‐homogeneity degree of a DT system determines the convergence rate of its trajectories. For the definition of hyper‐exponential stability, see definitions 12.5 and 12.6 in the work of Polyakov …”

“…According to definition 12.6 in the work of Polyakov, the convergence rate for ν =1 is exponential or hyper‐exponential of degree 0, and the convergence rate for ν > 1 is hyper‐exponential of degree 1. Note that, in any case of Theorem , g is a strictly decreasing function of k .…”