# Classical Discrete-Time Adaptive Control Revisited: Exponential Stabilization

## TL;DR

This paper proves that classical pole placement adaptive controllers with projection achieve exponential stability and bounded noise gain without persistent excitation, improving robustness and performance guarantees.

## Contribution

It demonstrates exponential stabilization and bounded noise gain for classical adaptive controllers using the original projection algorithm without persistent excitation.

## Key findings

- Exponential stability of the closed-loop system.
- Bounded noise gain in the presence of disturbances.
- Tolerance to unmodelled dynamics and parameter variations.

## Abstract

Classical discrete-time adaptive controllers provide asymptotic stabilization. While the original adaptive controllers did not handle noise or unmodelled dynamics well, redesigned versions were proven to have some tolerance; however, exponential stabilization and a bounded gain on the noise was rarely proven. Here we consider a classical pole placement adaptive controller using the original projection algorithm rather than the commonly modifed version; we impose the assumption that the plant parameters lie in a convex, compact set and that the parameter estimates are projected onto that set at every step. We demonstrate that the closed-loop system exhibits very desireable closed-loop behaviour: there are linear-like convolution bounds on the closed loop behaviour, which implies exponential stability and a bounded noise gain, as well an easily proven tolerance to unmodelled dynamics and plant parameter variation. We emphasize that there is no persistent excitation requirement of any sort.