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D. Properties of Continuous-Time Riccati Equations


D.2 Solutions of the CTARE

The Continuous Time Algebraic Riccati Equation (CTARE) has many solutions, because it is a matrix quadratic equation. The solutions can be characterized as follows.

Lemma D.3 Consider the following CTARE:


\begin{displaymath}0=\ensuremath{\boldsymbol{\Psi}} -\ensuremath{\mathbf{P}}\ens... ...{\mathbf{A}} +\ensuremath{\mathbf{A}} ^T\ensuremath{\mathbf{P}}\end{displaymath} (D.2.1)

(i) The CTARE can be expressed as
\begin{displaymath}\begin{bmatrix}-\ensuremath{\mathbf{P}} &\ensuremath{\mathbf{... ...\\ \ensuremath{\mathbf{P}}\end{bmatrix}=\ensuremath{\mathbf{0}}\end{displaymath} (D.2.2)
where $\ensuremath{\mathbf{H}} $ is defined in (D.1.8).
(ii) Let $\ensuremath{\mathbf{\overline{V}}} $ be defined so that
\begin{displaymath}\ensuremath{\mathbf{\overline{V}}}\;^{-1}\ensuremath{\mathbf{... ...mathbf{0}} & \ensuremath{\boldsymbol{\Lambda}} _b \end{bmatrix}\end{displaymath} (D.2.3)
where $\ensuremath{\boldsymbol{\Lambda}} _a,\ensuremath{\boldsymbol{\Lambda}} _b$ are any partitioning of the (generalized) eigenvalues of $\ensuremath{\mathbf{H}} $ such that, if $\lambda $ is equal to $(\Lambda _a)_i$ for same $i$, then $-\lambda ^{*}=(\Lambda _{b})_j$ for some $j$.
Let
\begin{displaymath}\ensuremath{\mathbf{\overline{V}}} = \begin{bmatrix} \ensurem... ... _{21} & \ensuremath{\mathbf{\overline{V}}} _{22} \end{bmatrix}\end{displaymath} (D.2.4)
Then $\ensuremath{\mathbf{\overline{P}}} =\ensuremath{\mathbf{\overline{V}}} _{21}\ensuremath{\mathbf{\overline{V}\;}} _{11}^{-1}$is a solution of the CTARE.
Proof
(i) This follows direct substitution.
(ii) The form of $\ensuremath{\mathbf{\overline{P}}} $ ensures that
\begin{displaymath}\begin{bmatrix}-\ensuremath{\mathbf{\overline{P}}} &\ensurema... ...}}} = \begin{bmatrix}\ensuremath{\mathbf{0}} &* \end{bmatrix} \end{displaymath} (D.2.5)
\begin{displaymath}\ensuremath{\mathbf{\overline{V}}} ^{-1} \begin{bmatrix}\;\en... ...trix}= \begin{bmatrix}*\\ \ensuremath{\mathbf{0}}\end{bmatrix}\end{displaymath} (D.2.6)

where * denotes a possible nonzero component.
Hence,

\begin{displaymath}\begin{bmatrix}-\ensuremath{\mathbf{\overline{P}}} &\ensurema... ...ambda}} \begin{bmatrix}*\\ \ensuremath{\mathbf{0}}\end{bmatrix}\end{displaymath} (D.2.7)
\begin{displaymath}=0 \end{displaymath} (D.2.8)

$\Box \Box \Box $