]> 8.5 Formal derivation of the optimal state feedback

8.5 Formal derivation of the optimal state feedback

The next step in the spectral–factorization method for the lq controller synthesis is finding the realization of ϑ̂(λ) 1 (where ϑ̂ is the spectral factor defined by (8.13)) as a transfer function of the system similar to (8.7) but with c# replaced by a new functional g# such that D(g#) C[r, 0] C[r, 0]. To be more precise, we look for a linear continuous functional g# on C[r, 0] C[r, 0] such that
g#𝒜(𝒜 λI)1d = ϑ̂(λ) 1 = (p 1) + (q b)e2λr 1 + be2λr ,λσ(𝒜) (8.15)

Now, such linear continuous functional can be represented by the Stieltjes integral

g#x =r0dΞ(θ)x(θ) (8.16)

where Ξ is a vector–valued function of bounded variation on [r, 0]. Using (8.16) and (8.10), we can find an equivalent representation of (8.15) in terms of entire functions, viz.

r0dΞ(θ)eλθ eλr 1 = p1+(qb)e2λr,λ (8.17)

Hence, by the injectivity of the finite Laplace–Stieltjes transform appearing in the left–hand side of (8.17), we obtain

g#x = (p 1)x 2(0) + (q b)x1(r),D(g#) C[r, 0] C[r, 0] (8.18)