]> 8.4 Spectral factorization

8.4 Spectral factorization

According to the spectral–factorization method for finding the optimal lq controller due to Callier and Winkin – see [35] or Section 6.3 for a short presentation of the SISO case – we should factorize the expression
1 + ĝ(λ)ĝ(λ) =1 + aeλr 1 + be2λr aeλr 1 + be2λr = 1 + be2λr + be2λr + b2 + a2 (1 + be2λr)(1 + be2λr) = =p + qe2λr 1 + be2λrp + qe2λr 1 + be2λr ,

where the pair (p,q) is a solution of the equations

pq =b p2 + q2 =1 + a2 + b2 (8.12)

in such a way that a spectral factor

ϑ̂(λ) = p + qe2λr 1 + be2λr,λσ(𝒜) (8.13)

should be invertible in the Callier–Desoer algebra 𝔄̂ (case of a countably many δ–impulses). In other words its numerator should be a stable quasipolynomial, which holds for

2p = (1 + b)2 + a2 + (1 b)2 + a2 2q = (1 + b)2 + a2 (1 b)2 + a2 (8.14)

Indeed, it is easy to check that qp < 1, which is a necessary and sufficient condition for this quasipolynomial to have roots in the open left half complex plane (roots are located on a straight line parallel to j, as for the open–loop–system quasipolynomial 1 + be2λr).

In what follows, we assume that p and q in (8.13), and in other formulae in which they will appear, are determined by (8.14).