Hamzi - Eurandom

Controllability and Observability Energies in Model
Reduction of Linear Control Systems
• Gramians have several uses in Linear Control Theory. For example, for
the purpose of model reduction.
• Balancing: find a representation where the system’s observable and
controllable subspaces are aligned so that reduction, if possible, consists of
eliminating uncontrollable states which are also the least observable.
• More formally, we would like to find a new coordinate system such that
Wc = Wo = Σ = diag{σ1, · · · , σn},
where σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0. If (F, G) is controllable and (F, H) is
observable, then there exists a transformation such that the state space
expressed in the transformed coordinates (T F T −1 , T G, HT −1 ) is balanced
and T WcT ⊤ = T −⊤ WoT −1 = Σ.
. . . . . .
Boumediene Hamzi (Imperial College) On Control and RDS in RKHS June 4th, 2012 12 / 55