\section{Inner functions and antistable systems V:23:04:94} \label{S:inner} Inner functions play an important role both in system and control theory. In later sections we will often come across these functions in various contexts. In this section coprime factorizations will be constructed of antistable rational functions such that the denominators are square inner functions. We first define inner functions and give a state space description of them. \begin{definition} \label{D:inner:1} Let $B$ be a stable proper rational function. Then $B$ is called an {\em inner} function if $B^*B=I$ and $B$ is called a {\em co-inner} function if $BB^*=I$. \end{definition} Lemma~\ref{L:factintro:2} gave a state space characterization of $(J_1,J_2)$-allpass functions. Clearly square inner functions are $(J_1,J_2)$-allpass functions for $J_1=I$ and $J_2=I$. Lemma~\ref{L:factintro:2} could therefore also be used to derive a state space characterization for square inner functions. In the following lemma such a characterization is given also for the general case of non-square inner functions. \begin{lemma} \label{L:inner:1} Let $G$ be a not necessarily square proper rational function. Then \begin{enumerate} \item the following two statements are equivalent. \begin{enumerate} \item $G$ is inner, \item If $G \stackrel{r}= (A,B,C,D)$ is a minimal state space realization, then there exists $Y=Y^* >0$ such that \begin{enumerate} \item $A^*Y+YA=-C^*C $, \item $ C^*D + Y B=0$, \item $D^*D=I$. \end{enumerate} \end{enumerate} \item the following two statements are equivalent. \begin{enumerate} \item $G$ is co-inner, \item If $G \stackrel{r}= (A,B,C,D)$ is a minimal state space realization, then there exists $Z=Z^* >0$ such that \begin{enumerate} \item $AZ+ZA^*=-BB^*$, \item $CZ+DB^*=0$, \item $DD^*=I$. \end{enumerate} \end{enumerate} \end{enumerate} \end{lemma} \proof Consider first the case of inner functions. Assume that 1.) is true. Let $G \stackrel{r}=(A,B,C,D)$ be a minimal realization of the inner function $G$. Then $G^* \stackrel{r}= ( -A^*,-C^*,B^*,D^*)$ is also a minimal realization of $G^*$. Since $G$ is inner \[ I=G^*G \stackrel{r}= \left ( \begin{array}{cc|c} -A^* & -C^*C & -C^*D \\ 0 & A & B \\ \hline B^* & D^* C & D^*D \\ \end{array} \right ) . \] This implies that $D^*D=I$. Since $(A,B,C,D)$ is stable there exists $Y=Y^* > 0 $ such that \[ A^*Y + YA =-C^*C . \] Performing a state space transformation of this realization of $G^*G$ with $\left ( \begin{array}{cc} I & -Y \\ 0 & I \\ \end{array} \right )$ we obtain \[ I = G^*G \stackrel{r}= \left ( \begin{array}{cc|c} -A^* & -A^*Y -YA -C^*C & -C^*D - YB \\ 0 & A & B \\ \hline B^* & B^* Y + D^* C & I \\ \end{array} \right ) \] \[ = \left ( \begin{array}{cc|c} -A^* & 0 & -C^*D - YB \\ 0 & A & B \\ \hline B^* & B^* Y + D^* C & I \\ \end{array} \right ) \] \[ = \left ( \begin{array}{c|c} -A^* & -C^*D - YB \\ \hline B^* & I \\ \end{array} \right ) + \left ( \begin{array}{c|c} A & B \\ \hline B^* Y + D^* C & 0 \\ \end{array} \right ) . \] Since the first system in this decomposition is antistable and the second system is stable, the addition of these systems can only be $I$ if the first system is $I$ and the second system is the $0$ system. Since $(A,B,C,D)$ is minimal it follows that $(A,B,B^*Y+D^*C,0)$ is reachable. Hence this system is the $0$ system only if $B^*Y+D^*C=0$. But if $B^*Y+D^*C=0$ then also $(-A^*,-C^*D-YB, B^*,I) \stackrel{r}=I$. This shows 2.). That 2.) implies 1.) is easily verified. The statement concerning co-inner functions follows analogously. \eproof A result that will be of great importance later is the following lemma that establishes a connection of the solutions of the Lyapunov equations if the function square and inner. \begin{lemma} \label{L:inner:2} Let $(A,B,C,D)$ be a minimal realization of the square inner (co-inner) rational function $G$. Let $Y$, $Z$ be such that \[ A^*Y+YA=-C^*C, \ \ AZ+ZA^*=-BB^*. \] Then $ YZ=I$. \end{lemma} \proof Note that since $G$ is square, $G$ is both inner and co-inner. Since $G$ is co-inner, by Lemma \ref{L:inner:1} $CZ+DB^*=0$ and therefore $Z^{-1}B=-C^*D^{-*}$. Since $AZ+ZA^*=-BB^*$, and $DD^*=I$ we have \[ A^*Z^{-1}+Z^{-1}A=-Z^{-1}B(Z^{-1}B)^*=-C^*D^{-*}D^{-1}C=-C^*C. \] By the uniqueness of the situation to this Lyapunov equation it follows that $Z^{-1}=Y$, which implies the claim. The result follows analogously if $G$ is co-inner. \eproof The first class of systems for which we are going to consider normalized factorizations is the class of antistable functions. The Douglas-Shapiro-Shields factorization is a factorization that is normalized so that the denominator is inner. This factorization is amongst other applications particularly important in the theory of Hankel operators. Let $G$ be a proper antistable function, i.e. all poles of $G$ are in the open right half plane. A right (left) coprime factorization $G=NM^{-1}$ ($G=\tM^{-1}\tN$) is called a right (left) {\em Douglas-Shapiro-Shields (DSS)} factorization if $M^*M=I$ ($\tM\tM^*=I$.) Hence a right (left) Douglas-Shapiro-Shields factorization is a $J_s^r$-right ($J_s^l$-left) factorization with $J_s^r:=diag(I,0)$ ($J_s^l=diag(0,I)$). In the following theorem the existence of Douglas-Shapiro-Shields factorizations is established. \begin{theorem} \label{T:inner:1} Let $G$ be an antistable proper rational function. Then there exists a right and left Douglas-Shapiro-Shields factorization of $G$. Let $G \stackrel{r}=(A,B,C,D)$ be a minimal realization of $G$. Then \begin{enumerate} \item all right Douglas Shapiro Shields factorizations of $G$ are given by \[ \tbo{M}{N} \stackrel{r}= \left ( \begin{array}{c|c} A-BB^*Y & BU \\ \hline -B^*Y & U \\ C-DB^*Y & DU \\ \end{array} \right ), \] where $Y$ is the unique positive definite solution to the Riccati equation \[ A^*Y+YA -YBB^*Y=0, \] and $U$ is a constant matrix. \item all left Douglas Shapiro Shields factorizations of $G$ are given by \[ \obt{-\tN}{\tM} \stackrel{r}= \left ( \begin{array}{c|cc} A-ZC^*C & ZC^*D-B & -ZC^* \\ \hline \tU C & -\tU D & \tU \\ \end{array} \right ), \] where $Z$ is the unique positive definite solution to the Riccati equation \[ AZ+ZA^*-ZC^*CZ =0 \] and $\tU$ is a constant unitary matrix. \end{enumerate} \end{theorem} \proof 1.) First note that since $(-A,B,C,D)$ is a stable and reachable system, there exists a unique positive definite solution $Q$ to the equation \[ 0=(-A)Q+Q(-A)^*+BB^*=AQ+QA^*-BB^*. \] Setting $Y:=Q^{-1}$, this shows that $Y$ is the unique positive definite solution to the Riccati equation \[ 0=A^*Y+YA-YBB^*Y. \] Note that $F:=B^*Y$ is a stabilizing feedback for the system $(A,B,C,D)$ since \[ 0=(A-BB^*Y)^*Y+Y(A-BB^*Y)+YBB^*Y \] and hence \[ 0=Y^{-1}(A-BB^*Y)+(A-BB^*Y)Y^{-1}+BB^* . \] Since $Y=Y^*>0$ and $(A-BB^*Y,B,C,D)$ is reachable this shows that $A-BB^*Y$ is stable. Therefore by Corollary~\ref{C:existence:5} \[ \tbo{M}{N} \stackrel{r}= \left ( \begin{array}{c|c} A-BB^*Y & BU \\ \hline -B^*Y & U \\ C-DB^*Y & DU \\ \end{array} \right ), \] with $U$ unitary defines a minimal right coprime factorization of $G$. It is straightforward to check that $M \stackrel{r}= \left ( \begin{array}{c|c} A-BB^*Y & BU \\ \hline -B^* Y & Y \\ \end{array} \right )$ satisfies the properties of Lemma~\ref{L:inner:1}. Hence $M$ is inner and $G=NM^{-1}$ is a right Douglas Shapiro Shields factorization. The uniqueness of the representation up to right multiplication of the factors by a constant unitary matrix follows from Lemma~\ref{L:inner:1}. The proof of 2.) is analogous to the proof of 1.). \eproof In the following corollary a generalization of the existence of the Douglas Shapiro Shields factorization is established to functions with non-trivial stable part. \begin{corollary} \label{C:inner:1} Let $G$ be a proper rational function such that no poles of $G$ are on the imaginary axis. Then \begin{enumerate} \item there exists a right (left) coprime factorization $G=NM^{-1}$ ($G=\tM^{-1}\tN$) such that $M^*M=I$ ($\tM\tM^*$). \item all other such right (left) coprime factorizations are given by $G=NU (MU)^{-1}$ ($G=(\tU\tM)^{-1}(\tU\tN)$), where $U$ ($\tU$) is a constant unitary matrix. \end{enumerate} \end{corollary} \proof 1.) Let $G=:G_s+G_a$ be an additive decomposition of $G$ into a stable and antistable part, i.e. $G_s$ is proper stable and $G_a$ is proper and antistable. Let $G_a=N_aM_a^{-1}$ be a right Douglas-Shapiro-Shields factorization of $G_a$. Then there exist $\tU_a$, $\tV_a$ proper and stable such that $\tV_aM_a-\tU_aN_a=I$. Then $G=N_aM_a^{-1}+G_s=(N_a+G_sM_a)M_a^{-1}$. Set $N:=N_a+G_sM_a$ and $M:=M_a$. Clearly, $N$ and $M$ are stable and $M^*M=I$. Also with $\tU:=\tU_a$ and $\tV:=\tV_a+\tU_aG_s$, we have that $\tU$ and $\tV$ are proper stable and \[ \tV M-\tU N=(\tV_a + \tU_aG_s)M_a-\tU_a(N_a+G_sM_a) \] \[ \ \ =\tV_aM_a-\tU_aN_a+\tU_aG_sM_a-\tU_aG_sM_a=I. \] This shows that $G=NM^{-1}$ is the desired right coprime factorization. The existence of a left coprime factorization with the required properties follows similarly. 2.) Since the factorization in 1.) is $J_s^r$-right coprime ($J_s^r$-left coprime) the uniqueness of the factorization follows from Lemma~\ref{L:factintro:1}. \eproof In the following proposition the existence of a so-called {\em inner-outer} factorization will be established for square stable rational functions that have no poles or zeros (define!!!) on the imaginary axis. \begin{proposition} \label{P:inner:1} let $G$ be a square stable proper rational function with proper inverse. Assume also that $G^{-1}$ has no poles on the imaginary axis. Then there exist square inner functions $B$, $\tilde{B}$ and a stable proper rational function $Q$, $\tilde{Q}$ with proper stable inverse, such that \[ G=BQ=\tilde{Q}\tilde{B} . \] All other such factorizations are given by \[ G=BU(U^*Q)=(\tilde{Q}\tU)(\tU^*\tilde{B}) , \] where $U$ and $\tU$ are constant unitary matrices. \end{proposition} \proof Consider $G^{-1}$ which by assumption is a proper rational function with no poles on the imaginary axis. By Corollary~\ref{C:inner:1} there exists a right coprime factorization $G^{-1}=NM^{-1}$ with $M$ square and inner and $\tV M- \tU N =I$ for some proper stable $\tU$, $\tV$. Note that $N=G^{-1}M$ has proper inverse as $G$ and $M^{-1}=M$ are proper. Then $G=MN^{-1}$. Set $B:=M$ and $Q:=N^{-1}$. Since $\tV M - \tU N=I$ it follows that $Q=N^{-1}=\tV M N^{-1}-\tU=\tV G-N$, which is stable since $G$ is stable. As $Q^{-1}=N$ is proper and stable, $Q$ is a unit and therefore $G=BQ$ is the desired factorization. Let now $G=B_1Q_1$ be another such factorization, then $B_1Q_1=BQ$ and therefore $Q_1Q^{-1}=B_1^*B$ which is stable. Also, $(B_1^*B)^*=B^*B_1=QQ_1^{-1}$ which is stable. This is only possible if $QQ_1^{-1}=B^*B_1=U$ for some constant unitary matrix. Therefore $B_1=BU$ and $Q_1=U^*Q$. Obviously $G=(BU)(U^*Q)$ for all constant unitary matrices $U$. The result concerning the other factorizations are derived analogously. \eproof The following corollary shows that using square inner functions, unstable poles and unstable zeros can be factored out. \begin{corollary} \label{C:inner:2} Let $G$ be a square proper rational function with proper inverse and assume that $G$ and $G^{-1}$ have no poles on the imaginary axis. Then there exist square inner functions $B_1$, $B_2$ ($\tilde{B}_1$, $\tilde{B}_2$) and a proper stable rational function $Q$ ($\tilde{Q}$) with proper stable inverse such that \[ G=B_1^*B_2Q \ \ \ \ (G=\tilde{Q}\tilde{B}_2\tilde{B}^*_1). \] \end{corollary} \proof By Corollary~\ref{C:inner:1} there exists a left coprime factorization $G=\tM^{-1}\tN$ of $G$ such that $\tM$ is inner. Set $B_1:=\tM$. Then $\tN$ is proper stable with proper inverse such that both $\tN$ and $\tN^{-1}$ have no poles on the imaginary axis. By the proposition there exists an inner function $B_2$ and a proper stable $Q$ with proper stable inverse $Q^{-1}$ such that $\tN=B_2Q$. Hence $G=B_1^*B_2Q$ has the required properties. The existence of the other factorization is proved analogously. \eproof \subsection{Exercises} \begin{enumerate} \item Let $(A,B,C,D)$ be a realization of the transfer function $G$. Let $\hat G:\stackrel{r}=(A^*,C^*,B^*,D^*)$. Show that $G$ is inner if and only if $\hat G$ is co-inner. Derive part 2 of Lemma \ref{L:inner:1} directly from part 1. \item Let $g:={p \over q}$ be a scalar rational function where $p$ and $q$ are coprime polynomials. Show that $g$ is all pass if and only if $p(s)=\overline{q(-\bar{s})}$, $s \in \cal C$, and $q$ has all its zeroes in the open left half plane. \item In this section we only consider continuous-time systems and their transfer functions. Inner functions can also be defined for transfer functions of discrete-time systems. A proper rational function $B_d$ is called {\em discrete-time inner} if it is a discrete-time stable and $(B_d(z^{-1}))^*B_d(z)=I$, for all $z \in \cal C$. If $B_c$ is the inverse Moebius transform of $B_d$ (see Exercise 1.4.1.5) then show that $B_d$ is discrete-time inner if and only if $B_c$ is an inner function in the sense of Definition \ref{D:inner:1}. \end{enumerate} \subsection{Notes} The Douglas-Shapiro-Shields factorization has been introduced by Douglas, Shapiro and Shields \cite{...} for scalar functions and by Fuhrmamn \cite{fuhrmamn} for matrix-valued functions. State space formulae for Douglas-Shapiro-Shields factorizations appear in the control literature, see e.g. Doyle [1984]. The approach to the proof of Theorem \ref{T:inner:1} is due to Ober and Fuhrmamn \ref{..}.