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D. Properties of Continuous-Time Riccati Equations

D.3 The stabilizing solution of the CTARE

We see from Section D.2 that we have as many solutions to the CTARE as there are ways of partitioning the eigevalues of H into the groups $\ensuremath{\boldsymbol{\Lambda}} _a$ and $\ensuremath{\boldsymbol{\Lambda}} _b$. Provided that $(A,B)$ is stabilizable and that $(\ensuremath{\boldsymbol{\Psi}} ^{\frac{1}{2}}\,, \ensuremath{\mathbf{A}} )$ has no unobservable modes in the imaginary axis, then \ensuremath{\mathbf{H}} has no eigenvalues in the imaginary axis. In this case, there exists a unique way of partitioning the eigenvalues so that $\ensuremath{\boldsymbol{\Lambda}} _a$ contains only the stable eigenvalues of \ensuremath{\mathbf{H}}. We call the corresponding (unique) solution of the CTARE the stabilizing solution and denote it by $\ensuremath{\mathbf{P}} _{\infty}^{s}$.
Properties of the stabilizing solution are given in the following.

Lemma D.4

(a) The stabilizing solution has the property that the closed loop \ensuremath{\mathbf{A}} matrix,
\begin{displaymath}\ensuremath{\mathbf{A_{cl}}} =\ensuremath{\mathbf{A}} -\ensuremath{\mathbf{BK}} _{\infty}^{s} \end{displaymath} (D.3.1)
where
\begin{displaymath}K_{\infty}^{s}=\Phi^{-1}B^T\ensuremath{\mathbf{P}} _{\infty}^{s} \end{displaymath} (D.3.2)
has eigenvalues in the open left-half plane.
(b) If $(\ensuremath{\boldsymbol{\Psi}} ^{\frac{1}{2}}\,, \ensuremath{\mathbf{A}} )$ is detectable, then the stabilizing solution is the only nonnegative solution of the CTARE.
(c) If $(\ensuremath{\boldsymbol{\Psi}} ^{\frac{1}{2}}\,, \ensuremath{\mathbf{A}} )$ has no unobservable modes inside the stability boundary, then the stabilizing solution is positive definite, and conversely.
(d) If $(\Psi^{\frac{1}{2}}\,, A)$ has an unobservable mode outside the stabilizing region, then in addition to the stabilizing solution, there exists at least one other nonnegative solution of the CTARE. However, the stabilizing solution, $\ensuremath{\mathbf{P}} _{\infty}^{s}$ has the property that
\begin{displaymath}\ensuremath{\mathbf{P}} _{\infty}^{s} -\ensuremath{\mathbf{P}} _{\infty}^{'}\geq \ensuremath{\mathbf{0}} \end{displaymath} (D.3.3)
where $\ensuremath{\mathbf{P}} _{\infty}^{'}$ is any other solution of the CTARE.

Proof

For part (a), we argue as follows:

Consider (D.1.11) and (D.1.14). Then

\begin{displaymath}\ensuremath{\mathbf{H}} \begin{bmatrix} \ensuremath{\mathbf{... ...uremath{\mathbf{V_{21}}} \end{bmatrix}\ensuremath{\mathbf{H_s}}\end{displaymath} (D.3.4)

which implies that


\begin{displaymath} \ensuremath{\mathbf{H}} \begin{bmatrix} \ensuremath{\mathb... ...{\mathbf{H_s}}\ensuremath{\mathbf{V_{11}}} ^{-1} \end{bmatrix}\end{displaymath} (D.3.5)

If we consider only the first row in (D.3.5), then, using (D.1.8), we have


\begin{displaymath} \ensuremath{\mathbf{V_{11}}}\ensuremath{\mathbf{H_s}} \ensu... ...{P_\infty}} = \ensuremath{\mathbf{A}} -\ensuremath{\mathbf{BK}}\end{displaymath} (D.3.6)

Hence, the closed-loop poles are the eigenvalues of $\ensuremath{\mathbf{H_s}} $and, by construction, these are stable.

We leave the reader to pursue parts (b), (c), and (d) by studying the references given at the end of Chapter 24.

$\Box \Box \Box $