]> 5.2 Representation of the orthoprojector

5.2 Representation of the orthoprojector

Another way of determining the approximants follows from the observation that any element of the subspace Mn can be regarded as an output Y n of the observed linear system
(t) =Ax(t) x(0) =b n Y n(t) =cx(t) = x(t),cn ,

where

A = diag{λ1,λ2,,λn},σ(A) = {λk}k=1n Π,cT = 111 .

From the orthogonal projection theorem we get

f PnfL2(0,)2 = min bn f Y n L2(0,)2 (5.1)

Observe that

f Y n L2(0,)2 =f Y n,f Y nL2(0,) = =f,fL2(0,) Y n,fL2(0,) f,Y nL2(0,) + Y n,Y nL2(0,) = = fL2(0,)2 2 Ref,Y nL2(0,) + Y n L2(0,)2 .

The term Y n L2(0,)2 can be calculated as follows

Y n L2(0,)2 =0cetAb2dt = b0etA ccetAdtb = bHb ,

where

H =0eλ1¯t eλ2¯t eλn¯t eλ1teλ2teλnt dt = 1 λi¯ + λj i,j=1,2,,n = H > 0

since the pair (A,c) is observable. Furthermore, H is a unique solution of the Lyapunov matrix equation

AH + HA = cc (5.2)

and H is the Gram matrix of the system {eλk¯()} k=1n L2(0,).

We now show a way to determine the scalar product f,Y nL2(0,).

f,Y nL2(0,) =0f(t)Y n(t)¯dt =0f(t)cetAb¯dt =0f(t)betA cdt = =b0f(t)etA dtc = b0f(t) diag eλ1¯t,eλ2¯t,,eλn¯t dtc = =b diag f̂(λ 1¯),f̂(λ2¯),,f̂(λn¯) c ,

where f̂ denotes the Laplace transform of f,

f̂(s) =0f(t)estdt .

In virtue of the Paley–Wiener theory, f L2(0,) iff f̂ belongs to H2(Π+), the Hardy space of functions ϕ analytic on the right complex half–plane Π+ = {s : Re s > 0}, such that

sup x>0ϕ(x + iy) 2dy < .

Since σ(A) is located in Π+, the domain of analyticity of f̂, we have (see [57, Theorem 5.3.2])

0f(t)etA dt =0f(t) diag eλ1¯t,eλ2¯t,,eλn¯t dt = = diag f̂(λ1¯),f̂(λ2¯),,f̂(λn¯) := f̂(A) .

Finally,

f Y n L2(0,)2 = f L2(0,)2 2 Re bf̂(A)c + bHb

and the minimal value in (5.1) is achieved on b n being a solution of the equation

Hb = f̂(A)c (5.3)

If b0 is the solution of (5.3) then

PnfL2(0,)2 = b 0Hb 0 = b0f̂(A)c (5.4)