]> 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  or Section 6.3 for a short presentation of the SISO case – we should factorize the expression
$\begin{array}{ccc}\hfill 1+ĝ\left(\lambda \right)ĝ\left(-\lambda \right)& \hfill =\hfill & 1+\frac{a{e}^{-\lambda r}}{1+b{e}^{-2\lambda r}}\phantom{\rule{0ex}{0ex}}\frac{a{e}^{\lambda r}}{1+b{e}^{2\lambda r}}=\frac{1+b{e}^{-2\lambda r}+b{e}^{2\lambda r}+{b}^{2}+{a}^{2}}{\left(1+b{e}^{-2\lambda r}\right)\left(1+b{e}^{2\lambda r}\right)}=\hfill \\ \hfill & \hfill =\hfill & \frac{p+q{e}^{-2\lambda r}}{1+b{e}^{-2\lambda r}}\phantom{\rule{0ex}{0ex}}\frac{p+q{e}^{2\lambda r}}{1+b{e}^{2\lambda r}},\hfill \end{array}$

where the pair $\left(p,q\right)$ is a solution of the equations

 $\left\{\begin{array}{ccc}\hfill pq& \hfill =\hfill & b\hfill \\ \hfill {p}^{2}+{q}^{2}& \hfill =\hfill & 1+{a}^{2}+{b}^{2}\hfill \end{array}\right\}$ (8.12)

in such a way that a spectral factor

 $\stackrel{̂}{\vartheta }\left(\lambda \right)=\frac{p+q{e}^{-2\lambda r}}{1+b{e}^{-2\lambda r}},\phantom{\rule{2em}{0ex}}\lambda \notin \sigma \left(\mathcal{𝒜}\right)$ (8.13)

should be invertible in the Callier–Desoer algebra ${\stackrel{̂}{\mathfrak{𝔄}}}_{-}$ (case of a countably many $\delta$–impulses). In other words its numerator should be a stable quasipolynomial, which holds for

 $\left\{\begin{array}{ccc}\hfill 2p& \hfill =\hfill & \sqrt{{\left(1+b\right)}^{2}+{a}^{2}}+\sqrt{{\left(1-b\right)}^{2}+{a}^{2}}\hfill \\ \hfill 2q& \hfill =\hfill & \sqrt{{\left(1+b\right)}^{2}+{a}^{2}}-\sqrt{{\left(1-b\right)}^{2}+{a}^{2}}\hfill \end{array}\right\}$ (8.14)

Indeed, it is easy to check that $\left|q∕p\right|<1$, 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 $j\mathbb{ℝ}$, as for the open–loop–system quasipolynomial $1+b{e}^{-2\lambda r}$).

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).