Finite-horizon expected-energy bounds for the self-tuning regulator

Establish finite-horizon bounds on the expected control energy of the ordinary least-squares-based Åström–Wittenmark self-tuning regulator with unknown input gain, while retaining its almost-sure stability, asymptotically optimal tracking, and strong parameter-consistency properties.

Background

The paper proves almost-sure stability and optimal tracking for every bounded reference, together with strong consistency when the estimated parameter dimension satisfies d≥2. These are asymptotic guarantees and do not control the expected control energy over a fixed finite horizon.

Appendix B.2 gives a bounded continuous-noise example in which the controller produces an input with infinite second moment at a finite time, and the corresponding tracking residual also has infinite second moment. The unresolved problem is therefore to derive finite-horizon expected-energy bounds without sacrificing the asymptotic stability, tracking, and consistency conclusions established in the paper.