Hamzi - Eurandom

Review of Some Concepts for Linear Control Systems
• For the autonomous system ˙x = Ax, x(0) = x0 ∈ Cn ; y = Cx where A
is Hurwitz, the observability operator is defined as
Ψo : C n {
CeAtx0, for t ≥ 0
→ L2(0, ∞); x0 ↦→
0, otherwise
• The corresponding observability ellipsoid is given by
E := {W 1
2
o x0 : x0 ∈ C n and ∥x0∥ = 1}.
• The energy of the output signal y = Ψox0, for x0 ∈ C n can then be
computed as
∥y∥ 2 2 = ⟨Ψox0, Ψox0⟩ = ⟨x0, Ψ ∗ oΨox0⟩ = ⟨x0, W0x0⟩
where Ψ ∗ o : L2[0, ∞) → C n is the adjoint of Ψo.
. . . . . .
Boumediene Hamzi (Imperial College) On Control and RDS in RKHS June 4th, 2012 9 / 55