Mikkola, Kalle Mikael (Helsinki Univ. of Tech.)

The LQR Optimal Control Is Weakly Coprime

Scheduled for presentation during the Mini-Symposium "Distributed Parameter Systems-IV: Frequency Domain Approaches" (ThA06), Thursday, July 27, 2006, 11:15−11:40, Room G

17th International Symposium on Mathematical Theory of Networks and Systems, July 24-28, 2006, Kyoto, Japan

This information is tentative and subject to change. Compiled on May 19, 2024

Abstract

The LQ-optimal state feedback of a finite-dimensional system determines a coprime factorization N/M of the transfer function. We show that the same is true also for infinite-dimensional systems, in the sense that the factorization is "weakly coprime", i.e., if Nf,Mf are H2, then f is H2, for every function f.

We prove that every proper quotient of two bounded holomorphic operator-valued functions can be presented as the quotient of two bounded holomorphic weakly coprime functions. This result was already known for matrix-valued functions with the classical definition gcd(N,M)=I, which we show equivalent to our definition.

We give necessary and sufficient conditions for weak coprimeness and for Bézout coprimeness. We also establish a variant of the inner outer factorization with the inner factor being "weakly left-invertible". Our results hold for both discrete- and continuous-time linear systems over arbitrary Hilbert spaces, but most of them are new also in the scalar case.