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\title{Achievable \hinf-performance of multivariable systems with 
         unstable zeros and poles}
\author{
        K. Havre\thanks{Present address: ABB Corporate Research, 
        P.O.Box 90, N-1375 Billingstad, Norway. Fax: (+47) 66 84 35 41, 
        E-mail: Kjetil.Havre@no.abb.com} \ and  
        S. Skogestad\thanks{\KjTeN} \\
        \small Chemical Engineering, 
	Norwegian University of Science and Technology, \\
	\small N-7034 Trondheim, Norway. } % \\  \\ 


\date{}
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\begin{document}
   \maketitle

\begin{abstract}
   This paper examines the limitations imposed by 
   Right Half Plane (\Rhp) zeros 
   and poles in multivariable feedback systems.
   The main result is to provide lower bounds on 
   \mbox{\small $\l WXV(s)\r_\infty$} 
   where $X$ is the input or output sensitivity or 
   complementary sensitivity. $W$ and $V$ are matrix valued weights 
   who might depend on the plant and who also might be unstable. 
   Previously derived lower bounds on the \hinf-norm of the sensitivity 
   and the complementary sensitivity  
   are thus generalized to include bounds for reference tracking and 
   disturbance rejection. 
   Furthermore, new bounds which quantify the minimum {\em input\/} 
   usage for stabilization in the presence of measurement noise and 
   disturbances, are derived. 
   From the bounds we find that {\em output\/} performance 
   is {\em only\/} limited 
   if the plant has \Rhp-zeros. For a one degree-of-freedom 
   ($1$-{\Dof}) controller the presence of \Rhp-poles further 
   deteriorate the response, whereas there is no additional 
   penalty for having \Rhp-poles if we use a two degrees-of-freedom 
   ($2$-{\Dof}) controller (where the disturbance and/or reference 
   signal is measured). 
   For large classes of plants we {\em prove\/} that 
   the lower bounds given are {\em tight\/} in the sense that 
   there exist stable controllers (possible improper) that 
   achieve the bounds.  
\\ \\
{\bf Keywords:} System theory; Achievable \hinf-performance; Unstable systems;
\Rhp-zeros and poles; Stabilization.
\end{abstract}



\section{Introduction}
It is well known that the presence of {\Rhp} zeros and 
poles pose fundamental limitations on the achievable 
control performance. This was quantified for {\Siso} systems 
by \citeasnoun{Bode45} more than $50$ years ago, and most 
control engineers have an intuitive feeling of the limitations 
for scalar systems.
\citename{Ros66} \citeyear{Ros66,Ros70} was one of the first to point 
out that multivariable {\Rhp}-zeros pose similar limitations. 

%In Chapter~\ref{Ch:EffRHPZlowerBoundsSISO} %\cite{HS98a} 
%we derived tight lower bounds on Single Input Single Output 
%({\Siso}) closed-loop transfer functions on the forms $SV$ and $TV$ 
%when the plant has {\Rhp} zeros and/or poles. 
%Here $S$ is the {\Siso} sensitivity, $T$ is the {\Siso} complementary 
%sensitivity and $V$ may be any known rational transfer function. 
%In this paper we extend these results to multivariable closed-loop 
%transfer functions. 
%
%Two factors which complicate the {\Mimo} results compared to the {\Siso} 
%results, are:
%\begin{entry}
%   \item Zeros and poles in {\Mimo} systems have directions. 
%   \item The order of multiplication in multivariable transfer 
%         functions matter, {\ie} two multivariable transfer functions 
%         do not in general commute. 
%\end{entry}
%The main results in this paper are 
%lower bounds on the \hinf-norm of closed-loop transfer functions. 
%on the four forms $WSV$, $WS_IV$, $WTV$ and $WT_IV$, 
%where $S$ is the sensitivity, $T$ is the complementary sensitivity, 
%$S_I$ is the input sensitivity, $T_I$ is the input complementary 
%sensitivity, $W$ and $V$ are general known multivariable transfer 
%function matrices.  
The main results in this paper are 
explicit lower bounds on the \hinf-norm of closed-loop transfer functions. 
Of course, it is relatively straightforward to
compute the exact minimum value of the  \hinf-norm for a given case using
standard software, and a direct computation of the value of the \hinf-norm
is also possible, e.g. using the Hankel-norm as explained in
\cite{Francis87}.
%Francis (1987) 
%(legg inn:  Lecture Notes in Control and Information Sciences:
%A Course in \hinf Control Theory
%Bruce A. Francis
%Springer-Verlag, 1987).
Therefore, we want to stress that the objective is to derive
explicit (analytical) bounds that yield direct insight into the
limitations imposed by RHP-poles and zeros.


The basis of our results is the {\em important\/} work by 
\citeasnoun{Zames81}, who made use of the interpolation 
constraint $y_z^H S(z) = y_z^H$ and the maximum modulus theorem 
to derive bounds on \hinf-norm of $S$ for plants with one 
\Rhp-zero. 
The results by \citename{Zames81} were generalized to plants with 
\Rhp-poles by \citeasnoun{DFT92} in the {\Siso} case, and by 
\citeasnoun{SP96}, \citeasnoun{HS98} 
in the {\Mimo} case.  

In this paper we extend the work of \citeasnoun{Zames81} 
and 
%the work given in Chapters~\ref{Ch:EffRHPZPonPerf} and 
%\ref{Ch:EffRHPZlowerBoundsSISO},
\citeasnoun{HS98}
and quantify the 
fundamental limitations imposed by {\Rhp} zeros and poles in 
terms of lower bounds on the \hinf-norm of important 
closed-loop transfer functions. 
The main generalization of the previous result is that from the 
results in this paper we can derive lower bounds on \hinf-norm of 
closed-loop transfer functions other than sensitivity and 
complementary sensitivity. Further\footnote{
In order to accomplish lower bounds on \hinf-norm of general 
closed-loop transfer functions, it was necessary to generalize the 
previous results to include multivariable, unstable and non-minimum 
phase weights.} generalizations include multivariable weights and 
unstable and non-minimum phase weights.
%A further motivation and the basis for deriving these results, 
%are given in 
%% Chapter~\ref{Ch:EffRHPZlowerBoundsSISO}.
%\cite[Chapter~4]{Havre98}. 
%Consider for example an unstable plant $G$, and we want to obtain 
%a lower bound on the closed-loop transfer function $KS$ which 
%involves the input usage. 
%We can then use the relation $KS = T_IG^{-1}$ and the lower 
%bound on $WT_IV$ with $W=I$ and $V=G^{-1}$ to obtain a bound on 
%$\l KS(s)\r_\infty = \l T_IG^{-1}(s)\r_\infty$, 
%or we can use the relation $KS=G^{-1}T$ and the lower 
%bound on $WTV$ with $W=G^{-1}$ and $V=I$ to obtain a bound on 
%$\l KS(s)\r_\infty = \l G^{-1}T(s)\r_\infty$. 

One important application of the lower bounds, is that we can 
{\em quantify\/} the minimum usage needed to stabilize 
an unstable plant in the presence of the ``worst case'' 
disturbance, measurement noise and reference changes for the 
``best''\footnote{The best possible controller in the sense that 
the controller which minimizes the \hinf-norm of the closed-loop 
transfer function from the disturbances, measurement noise and 
references to the outputs.} 
possible controller. 
An additional important contribution of this paper is that we 
prove that the lower bounds are {\em tight\/} in a large number 
of cases. 
That is, we give analytical expressions for controllers 
which {\em achieve\/} an \hinf-norm of the closed-loop transfer 
function which is equal to the lower bound. 



%\section{Zeros and poles in multivariable systems}
\section{Elements from linear system theory}

\subsection{Zeros and zero directions.}
\label{Ch6:Sec:ZerPoles}
Zeros of a system arise when competing effects, internal to the 
system, are such that the output is zero even when the inputs 
and the states are not identically zero. 
Here we apply the following definition of zeros \cite{MFK76}. 
\begin{Definition}\textsc{(Zeros).}
%{\normalfont \citeaffixed{MFK76}{{\scshape Zeros},}}. 
   \label{Ch6:zeropol} 
   $z_i \in \mathbb{C}$ is a zero of $G(s)$ if the rank of $G(z_i)$ 
   is less than the normal rank of $G(s)$. 
%   The zero polynomial is 
%   defined as $ z(s) = \prod_{i=1}^{N_z} (s-z_i)$ where $N_z$ is 
%   the number of finite zeros of $G(s)$. 
\end{Definition}
The normal rank of $G(s)$ is defined as the rank of $G(s)$ at all $s$
except a finite number of singularities (which are the zeros).
\begin{Definition}{\scshape (Zero Directions)}.
   If $G(s)$ has a zero for $s=z \in \mathbb{C}$ then there exist 
   non-zero vectors, denoted the input zero direction 
   $\uz \in \mathbb{C}^{m}$ and the output zero direction 
   $\yz \in \mathbb{C}^{l}$, such that $\u_z^H \uz = 1$, 
   $\yz^H \yz = 1$ and 
   \begin{eqnarray} 
      G(z) u_z = 0; \q \q y_z^H G(z) = 0   \label{Ch6:defzd}
   \end{eqnarray}
\end{Definition}
For a system $G(s)$ with state-space realization 
\mbox{\scriptsize $ \left[\begin{array}{c|c} A & B \\ \hline C & D 
\end{array}\right]$}, 
the zeros $z$ of the system, the input zero  
directions $u_z$ and the state input zero vectors  
$\xziv \in \mathbb{C}^{n}$ ($n$ is the number of states) 
can all be computed from the generalized eigenvalue problem
\begin{equation}
  \mbox{\small $
      \pmatrix{A-sI & B \cr C & D\cr} 
      \pmatrix{\xziv\cr \uz\cr} = 
      \pmatrix{0\cr 0\cr}$} \label{Ch6:izde}
\end{equation}
Similarly one can compute the zeros $z$ and the output zero directions
$\yz$ from $G^T$.
%, see %Chapter~\ref{Ch:MIMOzerosPoles}  
%\cite[Section~2.3]{Havre98}
%%Section~\ref{Sec:ZerosAndZeroDir} 
%for further details.

\subsection{Poles and pole directions.}
\citeasnoun{Bode45} states that {\em the poles are the singular points 
at which the transfer function fails to be analytic\/}. 
In this work we replace ``fails to be analytic'' with ``is infinite'',
which certainly implies that the transfer function is {\em not
analytic\/}. 
When we evaluate\footnote{Strictly speaking, 
the transfer function $G(s)$ can {\em not\/} be evaluated at $s=p$, 
since $G(s)$ is not analytic at $s=p$.} 
the transfer function $G(s)$ at $s=p$, $G(p)$ 
is infinite in some directions at the input and the output. 
This is the basis for the following definition of input and 
output pole directions.
\begin{Definition}{\scshape (Pole Directions)}.\label{Ch6:Def:PolDirSimpl}
   If $s=p \in \mathbb{C}$ is a distinct pole of $G(s)$ then there exist an 
   input direction $\up \in \mathbb{C}^{m}$ and an output direction 
   $\yp \in \mathbb{C}^{l}$ with infinite gain for $s=p$.
\end{Definition}
For a system $G(s)$ with minimal state-space realization 
\mbox{\scriptsize $ \left[\begin{array}{c|c} A & B \\ \hline C & D 
\end{array}\right]$} the pole directions $\up$ and $\yp$ for a 
{\em distinct\/} pole $p$ can be computed from %\cite{HS96ech6} 
\cite[Section~2.4]{Havre98}
%(Section~\ref{Sec:PolesAndPoleDir})
\begin{equation}
   \up = B^Hx_{\ip}/\l B^Hx_\ip\r_2; \q \q \yp = Cx_\po/\l Cx_\po\r_2
   \label{Ch6:PoleDir}
\end{equation}
where $x_\ip \in \mathbb{C}^{n}$ and $x_\po \in \mathbb{C}^{n}$ are 
the eigenvectors corresponding to the two eigenvalue problems 
$$
    \xpi^H A = p \xpi^H; \q \q A \xpo = p \xpo 
$$
Note, that the pole directions are normalized, {\ie} $\l u_p\r_2=1$ and 
$\l y_p\r_2=1$. 
For the sake of simplicity we will only consider distinct poles 
in this paper. 
%For computation and definition of pole directions 
%in the case when the pole $p$ is not distinct refer to 
%\cite[Chapter~2]{Havre98}.
%%Chapter~\ref{Ch:MIMOzerosPoles}.
%%\cite{HS96ech6}.

\subsection{All-pass factorizations of {\Rhp} zeros and poles}
A transfer function matrix $B(s)$ is all-pass if 
$B^T(-s) B(s) = I$, which implies that all singular values 
of $B(j\w)$ are equal to one.    

A rational transfer function matrix $M(s)$ with \Rhp-poles 
$p_i\in \orhp$, can be factorized either at the 
input (subscript $i$) or at the output (subscript $o$) 
as follows\footnote{Note that the notation on the 
all-pass factorizations of {\Rhp} zeros and poles used in this paper 
is reversed compared to the notation used in \cite{GL95,SP96,HS96do}. 
The reason for this change of notation is to be consistent with 
what the literature generally defines as an all-pass filter.} 
\begin{equation}
   M(s) = M_{si}\, \Bpi^{-1}(M); \q \q     
   M(s) = \Bpo^{-1}(M)\, M_{so}(s)
   \label{Eq:PoleFact}
\end{equation} \vspace{-0.5cm}
\begin{entry}
   \item[$M_{si}$, $M_{so}$] -- \ Stable (subscript $s$) versions of 
          $M$ with the \Rhp-poles mirrored across the imaginary axis. 
   \item[$\Bpi(M)$, $\Bpo(M)$] -- \ Stable all-pass rational transfer 
          function matrices 
          %(all singular values are equal to $1$ for $s=j\w$) 
          containing the \Rhp-poles (subscript $p$) of $M$
          as \Rhp-zeros.   
\end{entry}
The all-pass filters are %\cite{HS95}
%\cite[Appendix~A]{Havre98}
%(Appendix~\ref{Ch:DirFactZerPol}) 
\begin{eqnarray}
    & \mbox{$ \Bpi(M(s)) = 
        \prod\limits_{i=1}^{N_p}(I - \frac{\mbox{\scriptsize$2\Re({p_i})$}}
                                   {\mbox{\footnotesize$s+\bar{p}_i$}}
                   \hat{u}_{p_i}\hat{u}_{p_i}^H) $}; \q 
    \mbox{$ \Bpi^{-1}(M(s)) = 
        \prod\limits_{i=N_p}^{1}(I + \frac{\mbox{\scriptsize$2\Re({p_i})$}}
                                   {\mbox{\footnotesize$s-p_i$}}
                   \hat{u}_{p_i}\hat{u}_{p_i}^H) $} \q \q &
    \label{Ch6:CombPoleFactInp} \\
    & \mbox{$ \Bpo(M(s)) = 
        \prod\limits_{i=N_p}^{1}(I - \frac{\mbox{\scriptsize$2 \Re({p_i})$}}
                                   {\mbox{\footnotesize$s+\bar{p}_i$}}
                   \hat{y}_{p_i}\hat{y}_{p_i}^H) $}; \q
    \mbox{$ \Bpo^{-1}(M(s)) = 
        \prod\limits_{i=1}^{N_p}(I + \frac{\mbox{\scriptsize$2 \Re({p_i})$}}
                                   {\mbox{\footnotesize$s-p_i$}}
                   \hat{y}_{p_i}\hat{y}_{p_i}^H) $} \q \q & 
    \label{Ch6:CombPoleFact} 
\end{eqnarray}
$\Bpo(M)$ is obtained by factorizing at the output one 
\Rhp-pole at a time, starting with 
$$
   M=\mathcal{B}_{p_1o}^{-1}(M)M_{p_1o}
$$
where 
$$ 
   \mathcal{B}_{p_1o}^{-1}(M(s)) = \mbox{\small $
      I + \frac{\mbox{\scriptsize$2\Re(p_1)$}}
               {\mbox{\footnotesize$s-p_1$}}
               \hat{y}_{p_1}\hat{y}_{p_1}^H $}
$$ 
and $\hat{y}_{p_1} = y_{p_1}$ is the output pole direction of $M$ 
for $p_1$. 
This procedure may be continued to factor out $p_2$ from 
$M_{p_1o}$ where $\hat{y}_{p_2}$ is the output pole 
direction of $M_{p_1o}$ (which need not coincide with 
$y_{p_2}$, the pole direction\footnote{In fact: 
$\hat{y}_{p_2}=\mathcal{B}_{p_1o}^{-H}(M)\ev{s=p_2}y_{p_2}$. Here 
$\B\ev{s=s_0}$ means the rational transfer function matrix $\B(s)$ 
evaluated at the complex number $s=s_0$. Thus, it provides an 
alternative to $\B(s_0)$, and it will mainly be used to avoid 
double parenthesis.} of $M$) and so on. 
A similar procedure may be used to factorize the poles at the input of
$M$. Note that the sequence get reversed in the input factorization
compared to the output factorization. 

In a similar sequential manner, the \Rhp-zeros can be factorized 
either at the input or at the output of $M$
\begin{equation}
   M(s) = M_{mi}\Bzi(M(s)); \q \q  
   M(s) = \Bzo(M)M_{mo}(s)
   \label{Eq:ZeroFact}
\end{equation} % \vspace{-0.5cm}
\begin{entry}
   \item[$M_{mi}$, $M_{mo}$] -- \ Minimum phase (subscript $m$) 
          versions of $M$ with the \Rhp-zeros mirrored across 
          the imaginary axis. 
   \item[$\Bzi(M)$, $\Bzo(M)$] -- \ Stable all-pass rational transfer 
          function matrices 
          %(all singular values are equal to $1$ for $s=j\w$) 
          containing the \Rhp-zeros (subscript $z$) of $M$.   
\end{entry} 
We get %\cite{HS95}
%(Appendix~\ref{Ch:DirFactZerPol}) 
%\cite[Appendix~A]{Havre98}
\begin{eqnarray}
    & \mbox{$ \Bzi(M(s)) = 
      \prod\limits_{j=N_z}^{1} (I - \frac{\mbox{\scriptsize $2 \Re({z_j})$}}
                                   {\mbox{\footnotesize $s+\bar{z}_j$}}
                   \hat{u}_{z_j}\hat{u}_{z_j}^H)$}; \q  
      \mbox{$ \Bzi^{-1}(M(s)) = 
      \prod\limits_{j=1}^{N_z} (I + \frac{\mbox{\scriptsize $2 \Re({z_j})$}}
                                   {\mbox{\footnotesize $s-z_j$}}
                   \hat{u}_{z_j}\hat{u}_{z_j}^H)$} \q \q &
       \label{Ch6:CombZeroFactInp} \\ 
    & \mbox{$ \Bzo(M(s)) = 
      \prod\limits_{j=1}^{N_z} (I - \frac{\mbox{\scriptsize $2 \Re({z_j})$}}
                                   {\mbox{\footnotesize $s+\bar{z}_j$}}
                   \hat{y}_{z_j}\hat{y}_{z_j}^H) $}; \q    
      \mbox{$ \Bzo^{-1}(M(s)) = 
      \prod\limits_{j=N_z}^{1} (I + \frac{\mbox{\scriptsize $2 \Re({z_j})$}}
                                   {\mbox{\footnotesize $s-z_j$}}
                   \hat{y}_{z_j}\hat{y}_{z_j}^H) $} \q \q & 
    \label{Ch6:CombZeroFact} 
\end{eqnarray}

Alternative all-pass factorizations are in use, 
{\eg} the inner-outer factorizations used in \cite{MZ89} 
which are the same as (\ref{Ch6:CombZeroFactInp}) and 
(\ref{Ch6:CombZeroFact}) except for the
multiplication of a constant unitary matrix. 
Reasons for using the factorizations  
given here are:
\begin{entry}
   \item The factorizations of \Rhp-zeros given here are analytic 
         and in terms of the zeros and the zero directions, 
         whereas the inner-outer 
         factorizations in \cite{MZ89} are given in terms of the 
         solution to an algebraic Riccati equation. 
   \item To factorize \Rhp-poles using the inner-outer 
         factorization one needs to assume that $G^{-1}$ exist. 
\end{entry}



\subsection{Closing the loop}
In this paper we consider the general two degrees-of-freedom (2-{\Dof})
control configuration shown in Figure~\ref{Fig:GenTwoDegFre}.
\begin{figure}[htb]
  \centerline{\input{fig/wtdfc}}  
  \caption{Two degrees-of-freedom control configuration with 
   $K = \[K_1 \  K_2 \]$}
  \label{Fig:GenTwoDegFre}
\end{figure}
In the figure the performance weights are given in dashed lines.
We have included both references $r$ and measurement noise $n$
in addition to disturbances $d$ as external inputs. 
The three matrices $G_d$, $R$ and $N$ can be viewed as weights 
on the inputs, and the inputs $\tilde{d}$, $\tilde{r}$ and 
$\tilde{n}$ are normalized in magnitude. 
Normally, $N$ is diagonal and $[N]_{ii}$ is the inverse of signal 
to noise ratio. 
For most practical purposes, we can assume that $R$ and $N$ are 
stable. However, from a technical point of view it suffices 
that the unstable modes in $N$ and $R$ can be stabilized 
through the inputs $u$. For the disturbance plant $G_d$ we 
assume that all the unstable modes of $G_d$ also appears in $G$ 
(which is required if the unstable modes of $G_d$ are state 
controllable in $u$). 

The controller can be divided into a negative feedback part from $y$ 
($K_2$) and a feed forward part from $r$ ($K_1$)
\begin{equation}
   u = K_1 r - K_2 y_m = K_1 r - K_2(y+n)
\end{equation}
The closed-loop transfer function $F$ from 
$$
   v=\pmatrix{\tilde{r} \cr \tilde{d} \cr \tilde{n}} \q \mbox{to} \q  
   z=\pmatrix{z_1 \cr z_2 \cr z_3}^T = 
     \pmatrix{W_P (y-r) \cr W_T y \cr W_u u}
$$ 
is 
\begin{equation}
   F(s) = \pmatrix{
           W_P(SGK_1-I)R &   W_PSG_d   & -W_PTN \cr
             W_TSGK_1R   &   W_TSG_d   & -W_TTN \cr
             W_uS_IK_1R  & -W_uK_2SG_d & -W_uK_2SN }
\end{equation}
where the sensitivity $S$, the complementary sensitivity $T$ 
and the input sensitivity $S_I$ are defined by  
\begin{eqnarray} 
  S   & \triangleq & (I+GK_2)^{-1}            \\
  T   & \triangleq & I-S \ \ = \ \ GK_2(I+GK_2)^{-1} \\
  S_I & \triangleq & (I+K_2G)^{-1}
\end{eqnarray}
We also define the input complementary sensitivity  
\begin{equation}
   T_I \ \ \triangleq \ \ I-S_I \ \ = \ \ K_2G(I+K_2G)^{-1}
\end{equation}
By setting $K_1=K_2$ in the above equations, the one 
degree-of-freedom (1-{\Dof}) control configuration can be analyzed.  

 
\section{Lower bounds on the \hinf-norm of closed-loop 
         transfer functions}\label{Ch6:Sec:MainResult}
\markright{\thesection\ \ \uppercase{Lower bounds on the} 
           \hinf\uppercase{-norm of closed-loop}\ldots}
In this section we derive general lower bounds on the \hinf-norm of 
closed-loop transfer functions when the plant $G$ has one or more 
{\Rhp} zeros and/or poles, by using the interpolation 
constraints and the maximum modulus principle. 
The bounds are applicable to closed-loop 
transfer functions on the form 
\begin{equation}
   W(s)X(s)V(s)
\end{equation}
where $X$ may be $S$, $T$, $S_I$ or $T_I$. 
The idea is to derive lower bounds on \mbox{\small $\l WXV(s)\r_\infty$} 
which are independent of the controller $K$. 
In general, we assume that $WXV$ is stable. 
The ``weights'' $W$ and $V$ must be independent of $K$, they may be 
unstable provided that the unstable modes can be stabilized 
by feedback conntrol of the plant $G$ (\eg unstable disturbance 
model $G_d$ or non-minimum phase plant $G$ with an unstable $G^{-1}$). 
This implies that the unstable modes of $W$ and $V$ also appear in 
$L=GK_2$. Otherwise, the system is not stabilizable. 
The results are stated in terms of four theorems. 

Theorems~\ref{Theo:LBWSV} and \ref{Theo:LBWSIV} provide lower bounds 
on the \hinf-norm of closed-loop transfer functions on the forms $WSV$ 
and $WS_IV$ caused by one or more \Rhp-zeros in $G$. 
%The bounds quantify the effect of any number of \Rhp-poles and one 
%\Rhp-zero at a time. 
By maximizing over all \Rhp-zeros, we find the largest 
lower bounds on $\l WSV(s)\r_\infty$ and 
$\l WS_IV(s)\r_\infty$ which takes into account one 
\Rhp-zero and all \Rhp-poles in the plant. 
\begin{Theorem}\textsc{(Lower Bound on 
\mbox{\small $\l WSV(s)\r_\infty$}).}
\label{Theo:LBWSV}
   Consider a plant $G$ with $N_z\ge 1$ \Rhp-zeros $z_j$, 
   output directions $y_{z_j}$ and $N_p\ge 0$ 
   \Rhp-poles $p_i\in\orhp$. Let $W$ and $V$ be rational 
   transfer function matrices, where $W$ is stable.
%   and assume that $W$ has no \Rhp-poles at locations corresponding 
%   to {\Rhp} poles or zeros in $G$.
%%   and let it be no  
%%   {\Rhp} pole/zero cancellation between zeros of $S$ due 
%%   to poles in $G$, and poles of $W$. 
   Assume that the closed-loop transfer function $WSV$ is 
   (internally) stable. 
   Then the following lower bound on $\l WSV(s)\r_\infty$ applies: 
   \begin{eqnarray} 
      \l WSV(s)\r_\infty   & \mms \ge \mms & 
          \max_{\mbox{\scriptsize\normalfont 
              \Rhp-zeros $z_j$ in $G$}}\,
          \l W_{mo}(z_j)\,y_{z_j}\r_2 \cdot \l y_{z_j}^HV
               \Bzi^{-1}(\Bpo(G)\, V)\ev{s=z_j}\r_2 \q \q
      \label{Eq:lbWSV} % \\
   \end{eqnarray}
\end{Theorem}
\begin{findproof}
%   The main steps in the proof of (\ref{Eq:lbWSV}) are given in 
%   Section~\ref{Sec:MainSteps}. 
   see Section~\ref{Ch6:App:Proofs}.  
\end{findproof}

\begin{Theorem}\textsc{(Lower Bound on 
\mbox{\small $\l WS_IV(s)\r_\infty$}).}
\label{Theo:LBWSIV}
   Consider a plant $G$ with $N_z\ge 1$ \Rhp-zeros $z_j$, 
   input directions $u_{z_j}$ and $N_p\ge 0$ 
   \Rhp-poles $p_i\in\orhp$. Let $W$ and $V$ be rational 
   transfer function matrices, where $V$ is stable.
%   and assume that $V$ has no \Rhp-poles at locations corresponding 
%   to {\Rhp} poles or zeros in $G$.
%%   and let it be no  
%%   {\Rhp} pole/zero cancellation between zeros of $S_I$ due 
%%   to poles in $G$, and poles of $V$. 
   Assume that the closed-loop 
   transfer function $WS_IV$ is (internally) stable. 
   Then the following lower bound on $\l WS_IV(s)\r_\infty$ applies:
   \begin{equation} 
      \l WS_IV(s) \r_\infty  \ge  
          \max_{\mbox{\scriptsize\normalfont\Rhp-zeros, $z_j$ in $G$}}\,
          \l\Bzo^{-1}(W\Bpi(G))\, W\ev{s=z_j}u_{z_j}\r_2 \cdot 
               \l u_{z_j}^HV_{mi}(z_j) \r_2 \q \q
      \label{Eq:lbWS_IV}
   \end{equation}
\end{Theorem}

Theorems~\ref{Theo:LBWTV} and \ref{Theo:LBWTIV} provide lower 
bounds on the \hinf-norm of closed-loop transfer functions on the 
forms $WTV$ and $WT_IV$ caused by one or more \Rhp-poles in $G$. 
%The bounds quantify the effect of any number of \Rhp-zeros and one 
%\Rhp-pole at a time. 
By maximizing over all \Rhp-poles, we find 
the largest lower bounds on $\l WTV(s)\r_\infty$ and 
$\l WT_IV(s)\r_\infty$ which takes into account one \Rhp-pole and 
all \Rhp-zeros in the plant.  
\begin{Theorem}\textsc{(Lower Bounds on 
\mbox{\small $\l WTV(s)\r_\infty$}).}
\label{Theo:LBWTV}
   Consider a plant $G$ with $N_p\ge 1$ \Rhp-poles $p_i$, 
   output directions $y_{p_i}$ and $N_z\ge 0$ 
   \Rhp-zeros $z_j\in\orhp$. Let $W$ and $V$ be rational 
   transfer function matrices, where $V$ is stable.
%   and assume that $V$ has no \Rhp-poles at locations corresponding 
%   to {\Rhp} zeros or poles in $G$.
%%   and let it be no  
%%   {\Rhp} pole/zero cancellation between zeros of $T$ due 
%%   to zeros in $G$, and poles of $V$. 
   Assume that the closed-loop 
   transfer function $WTV$ is (internally) stable. 
   Then the following lower bound on $\l WTV(s)\r_\infty$  applies:
   \begin{equation}
      \l WTV(s) \r_\infty \ge 
          \max_{\mbox{\normalfont\scriptsize\Rhp-poles, $p_i$ in $G$}}\,
          \l\Bzo^{-1}(W\Bzo(G))\, W\ev{s=p_i}y_{p_i}\r_2 \cdot 
            \l y_{p_i}^HV_{mi}(p_i)\r_2  \q \q    
      \label{Eq:lbWTV} \\
   \end{equation} 
\end{Theorem}
\begin{Theorem}\textsc{(Lower Bounds on 
\mbox{\small $\l WT_IV(s)\r_\infty$}).}
\label{Theo:LBWTIV}
   Consider a plant $G$ with $N_p\ge 1$ \Rhp-poles $p_i$, input
   directions $u_{p_i}$ and $N_z\ge 0$ 
   \Rhp-zeros $z_j\in\orhp$. Let $W$ and $V$ be rational 
   transfer function matrices, where $W$ is stable.
%   and assume that $W$ has no \Rhp-poles at locations corresponding 
%   to {\Rhp} zeros or poles in $G$.
%%   and let it be no  
%%   {\Rhp} pole/zero cancellation between zeros of $T_I$ due 
%%   to zeros in $G$, and poles of $W$.
   Assume that the closed-loop 
   transfer function $WT_IV$ is (internally) stable. 
   Then the following lower bound on $\l WT_IV(s)\r_\infty$ applies:
   \begin{equation}
      \l WT_IV(s)\r_\infty \ge 
          \max_{\mbox{\normalfont\scriptsize\Rhp-poles, $p_i$ in $G$}}\,
          \l W_{mo}(p_i)\, u_{p_i}\r_2\cdot 
          \l u_{p_i}^HV\Bzi^{-1}
                   (\Bzi(G)\, V)\ev{s=p_i}\r_2 \q \q 
          \label{Eq:lbWT_IV}
   \end{equation} 
\end{Theorem}
Remarks on Theorems~\ref{Theo:LBWSV}--\ref{Theo:LBWTIV}:
\begin{entry}
   \item The somewhat messy notation can easily be interpreted. 
         As an example take the last factor of (\ref{Eq:lbWSV}): 
         Factorize the \Rhp-poles at the output of $G$ into an
         all-pass filter $\Bpo(G)$ (yields \Rhp-zeros), multiply 
         on the right with 
         $V$ (may add \Rhp-zeros if $V$ is non-minimum 
         phase), then factorize at the input the \Rhp-zeros of the 
         product into an all-pass transfer function, take its inverse,
         multiply on the left with 
         $y_{z_j}^H V$ and finally evaluate the result for $s=z_j$.
   \item The lower bounds (\ref{Eq:lbWSV})--(\ref{Eq:lbWT_IV}) 
         are independent of the feedback controller $K_2$ if the 
         weights $W$ and $V$ are independent of $K_2$.  
   \item The internal stability assumption on the closed-loop transfer
         functions $WXV$, where $X\in \{S,\,S_I,\,T,\,T_I\}$,
         means that $WXV$ are stable, and we have 
         no {\Rhp} pole/zero cancellations between the plant $G$ and
         the feedback controller $K_2$. 
   \item The assumption on stability of $W$ and $V$ in 
         Theorems~\ref{Theo:LBWSV}--\ref{Theo:LBWTIV}
         is in practice {\em not\/} restrictive, since when the assumption 
          is {\em not\/} fulfilled we can generally rewrite 
          the transfer function and apply another theorem instead. 
          \begin{expl}{\hspace{-0.55em}.}
          Consider deriving a bound on \hinf-norm of the closed-loop 
          transfer function $K_2SG_d$ (input usage due to disturbances). 
          We can use the  relation $K_2SG_d = G^{-1} T G_d$ and apply 
          Theorem~\ref{Theo:LBWTV} with $W=G^{-1}$ and $V=G_d$,
          but we must assume that $G_d$ is {\em stable\/}. 
          However, we can use the relation 
          $K_2SG_d = T_I G^{-1} G_d$ and apply 
          Theorem~\ref{Theo:LBWTIV} with $W=I$ and $V=G^{-1}G_d$, 
          and in this case we can also allow $G_d$ to be 
          {\em unstable\/}. 
          \end{expl}
\end{entry}

\section{Tightness of lower bounds}\label{Sec:Tightness}
Theorems~\ref{Theo:LBWSV} to \ref{Theo:LBWTIV} 
provide lower bounds on $\l WXV(s)\r_\infty$ where 
$X\in \{S,\, S_I,\, T,\, T_I\}$. 
The question is whether these bounds are tight, meaning that there 
exist controllers which achieve these bounds? The answer is ``yes'' 
if there is only one \Rhp-zero or one \Rhp-pole. 
Specifically, we find that the bounds on  $\l WSV(s) \r_\infty$ and 
$\l WS_IV(s) \r_\infty$ are tight if the plant $G$ has one \Rhp-zero 
and any number of \Rhp-poles. 
Similarly, we find that the bounds on $\l WTV(s) \r_\infty$ and 
$\l WT_IV(s) \r_\infty$ are tight if the plant $G$ has one \Rhp-pole 
and any number of \Rhp-zeros. 
We prove tightness of the lower bounds by constructing controllers
which achieve the bounds. 


\begin{Theorem}\textsc{(Controller which Minimizes 
   \mbox{\small$\l WSV(s)\r_\infty$}).} \label{Theo:KWSVinf}
   Consider a plant $G$ with one \Rhp-zero $z$, output 
   direction $y_z$, and $N_p\ge 0$ \Rhp-poles $p_i\in\orhp$. 
   Let $W$ and $V$ be rational transfer function matrices, 
   where $W$ is stable.
%   and assume that $W$ has no \Rhp-poles at locations corresponding 
%   to {\Rhp} poles or zeros in $G$.
%%   and let it be no {\Rhp} pole/zero cancellation between 
%%   zeros of $S$ due to poles in $G$, and poles of $W$.
   A feedback controller (possible improper) 
   which stabilizes $WSV$, is given by 
   \begin{equation}
      K_2(s)  =  G_{smo}^{-1}(s) \, P(s)\, Q^{-1}(s)
   \end{equation}
   where 
   \begin{eqnarray}
      Q(s) & = & W_{mo}^{-1}(s)\,W_{mo}(z)\,V_0\,
                 \Bpo^{-1}(G)\ev{s=z}\, M_{mi}(z)\,M_{mi}^{-1}(s) \\ 
      P(s) & = & \Bzo^{-1}(G_{so})\, \left( I - \Bpo(G)\, Q\right)
                 \\ \nonumber 
      V_0  & = & y_z y_z^H + k_0^2 U_0 U_0^H \q \mbox{and} \q 
      M_{mi}(s) \ = \ (\Bpo(G)\, V(s))_{mi}
   \end{eqnarray}
   where the columns of the matrix $U_0\in \mathbb{R}^{l\times(l-1)}$ 
   together with $y_z$ forms an orthonormal basis for $\mathbb{R}^l$ 
   and $k_0$ is any constant. 
   $P(s)$ is stable since the \Rhp-zero for $s=z$ in 
   $I - \Bpo(G)\, Q$ cancels the \Rhp-pole for $s=z$ in 
   $\Bzo^{-1}(G_{so})$, in a minimal realization of $P$. 
   With this controller we have    
   \begin{equation}
      \lim_{k_0\rightarrow 0}\, \l WSV(s) \r_\infty = 
          \l W_{mo}(z)\,y_z\r_2 \cdot 
          \l y_z^H V \Bzi^{-1}(\Bpo(G)\, V)\ev{s=z}\r_2
   \end{equation}
\end{Theorem}
From Theorem~\ref{Theo:KWSVinf} it follows that 
the bound (\ref{Eq:lbWSV}) is tight when the plant has one 
\Rhp-zero. 

We can proove that the three other bounds in 
Theorems~\ref{Theo:LBWSIV}, \ref{Theo:LBWTV} and \ref{Theo:LBWTIV} 
are tight, under conditions similar to those given in 
Theorem~\ref{Theo:KWSVinf}. 

\section{Applications of lower bounds}\label{Ch6:Sec:Applications}
The lower bounds on \mbox{\small $\l WXV(s) \r_{\infty}$} in 
Theorems~\ref{Theo:LBWSV} and \ref{Theo:LBWTIV} 
can be used to derive a large number of interesting and useful bounds. 


\subsection{Output performance}
The previously derived bounds in terms of the \hinf-norms of $S$ 
and $T$ given in \cite{Zames81,SP96} and in 
%(Chapter~\ref{Ch:EffRHPZPonPerf}) 
\citeasnoun{HS98}  
follow easily, 
and further generalizations involving output performance can 
be derived. Here we assume that the performance weights $W_P$ and 
$W_T$ are stable and minimum phase. 


\subsubsection{Weighted sensitivity, $W_PS$.} 
Select $W=W_P$, $V=I$, and apply the bound 
(\ref{Eq:lbWSV}) to obtain
\begin{equation}
    \l W_PS(s)\r_\infty \ge 
       \max_{\mbox{\scriptsize\Rhp-zeros, $z_j$}}\, 
           \l W_P(z_j)\, y_{z_j}\r_2 \cdot 
              \l y_{z_j}^H \Bpo^{-1}(G)\ev{s=z_j} \r_2
\end{equation}
Note, this generalizes the previously found bounds to the
case with a matrix valued weight. 
%For the special case $W_P=I$, we derive 
%\begin{equation}
%   \l S(s)\r_{\infty} \geq \max_{\mbox{\scriptsize\, 
%            \Rhp-zeros, $z_j$}} \,
%            \l y_{z_j}^H \Bpo^{-1}(G)\ev{s=z_j}\r_2 \ge 1  
%   \label{Ch6:ZPpeakS}
%\end{equation}

%\subsubsection{Weighted complementary sensitivity, $W_T T$.} 
%Select $W=W_T$, $V=I$, and apply the bound (\ref{Eq:lbWTV}) 
%to obtain 
%\begin{equation}
%    \l W_TT(s)\r_\infty \ge 
%       \max_{\mbox{\scriptsize\Rhp-poles, $p_i$}}\, 
%           \l \Bzo^{-1}(W_T\Bzo(G))\,W_T\ev{s=p_i} y_{p_i}\r_2 
%\end{equation}
%For the special case $W_T(s)=I$, we derive %from (\ref{Eq:lbWTV}) 
%\begin{equation}
%   \l T(s)\r_{\infty} \geq 
%         \max_{\mbox{\scriptsize\,\Rhp-poles, $p_i$}}
%          \l \Bzo^{-1}(G)\ev{s=p_i}y_{p_i}\r_2 \ge 1  
%   \label{Ch6:ZPpeakT}
%\end{equation}

%\subsubsection{Closed-loop transfer function from disturbances 
%to weighted outputs, $W_PSG_d$.} 
%(see Figure~\ref{Fig:GenTwoDegFre}).}
\subsubsection{Disturbance rejection.}  
Select $W = W_P$, $V=G_d$, and apply the bound (\ref{Eq:lbWSV}) to obtain 
\begin{equation}
    \l W_PSG_d(s)\r_\infty \ge 
       \max_{\mbox{\scriptsize\Rhp-zeros, $z_j$}}\, 
           \l W_P(z_j)\, y_{z_j}\r_2 \cdot 
              \l y_{z_j}^H G_d\,\Bzi^{-1}(\Bpo(G)\,G_d)\ev{s=z_j} \r_2
\end{equation}

%\subsubsection{Closed-loop transfer function from references of magnitude 
%$R$ to weighted outputs for $1$-{\Dof} controller, 
%$W_P(SGK_2-I)R = -W_PSR$.} 
\subsubsection{Reference tracking.}  
Select $W = W_P$, $V=R$, and apply the bound (\ref{Eq:lbWSV}) to obtain 
\begin{equation}
    \l W_PSR(s)\r_\infty \ge 
       \max_{\mbox{\scriptsize\Rhp-zeros, $z_j$}}\, 
           \l W_P(z_j)\, y_{z_j}\r_2 \cdot 
              \l y_{z_j}^H R\,\Bzi^{-1}(\Bpo(G)\,R)\ev{s=z_j} \r_2
    \label{Eq:lbWpSR}
\end{equation}


\vspace{1ex}
\noindent 
Note that we can also look at the combined effect of disturbances and
references by  selecting $V = \mbox{\small $\pmatrix{G_d & R}$}$.

%\subsubsection{Closed-loop transfer function from measurement noise of 
%magnitude $N$ to weighted outputs, $-W_PTN$.} 
\subsubsection{Measurement noise rejection.}  
Select $W=W_P$, $V=N$, and apply the bound (\ref{Eq:lbWTV}) to obtain  
\begin{equation}
    \l W_PTN(s)\r_\infty \ge 
       \max_{\mbox{\scriptsize\Rhp-poles, $p_i$}}\, 
           \l \Bzo^{-1}(W_P\Bzo(G))\, W_P\ev{s=p_i} y_{p_i}\r_2 \cdot 
              \l y_{p_i}^H N_{mi}(p_i) \r_2
\end{equation}
where we must assume that $N$ has no \Rhp-poles corresponding to {\Rhp} 
zeros or poles in $G$. Normally $N$ is stable.
%We assume that none (if there is any) of the unstable modes in $N$ 
%cancel against zeros in $T$ which comes from zeros in $G$.  
%However, if this is the case we can always replace the corresponding 
%poles by their stable counterparts, {\ie} mirror them into the {\Lhp}. 

\subsection{Input usage}
The above provide generalizations of previous results, 
but we can also derive some new bounds in terms of input usage 
from Theorems~\ref{Theo:LBWTV} and \ref{Theo:LBWTIV}. 
These new bounds provide very interesting insights, for example, 
into the possibility of stabilizing an unstable plant with inputs of
bounded magnitude.

The basis of these new bounds is to note that the transfer function
from the outputs to the inputs, $K_2 S$, can be rewritten as 
$K_2 S = T_I G^{-1}$ or $K_2 S = G^{-1} T$. 
When $G$ is unstable, $G^{-1}$ has one or more \Rhp-zeros, so 
it is important that the bounds in Theorem~\ref{Theo:LBWTIV} 
can handle the case when $V=G^{-1}$ has \Rhp-zeros. 
Otherwise, $G^{-1}$ evaluated at the pole of $G$, would be zero 
in a certain direction, and we would not derive any useful bounds.
Here we assume that the weight $W_u$ on the  input $u$ is stable 
and minimum phase. 
%The two alternative expressions for $K_2S$, give rise to two 
%bounds on the closed-loop transfer functions involving 
%$K_2S$. The bounds which rely on $K_2S=T_I G^{-1}$ 
%are given in terms of the input pole directions, and 
%the bounds which rely on $K_2S = G^{-1}T$ 
%are given in terms of output pole directions. 
%
%Some examples of bounds involving the input magnitude are 
%(see Figure~\ref{Fig:GenTwoDegFre}).

\subsubsection{Disturbance rejection.} 
Apply the equality $K_2S=T_IG^{-1}$, select $W=W_u$, $V=G^{-1}G_d$, 
and use the bound (\ref{Eq:lbWT_IV}) to obtain
\begin{eqnarray} \nonumber  
   \lefteqn{\l W_uK_2SG_d(s)\r_\infty \ge 
      \max_{\mbox{\scriptsize\Rhp-poles, $p_i$}}\,} \\
      & \l W_u(p_i)\, u_{p_i}\r_2 \cdot 
      \l u_{p_i}^H G^{-1}G_d \Bzi^{-1}(G_{mi}^{-1}G_d)\ev{s=p_i}\r_2 
      \q\q &
      \label{Eq:LBWuK2SGd}        
\end{eqnarray}
where we have used the identity $\Bzi(G)\,G^{-1} = G_{mi}^{-1}$. 
Again, reference tracking is included by replacing $G_d$ by $R$. 

\subsubsection{Measurement noise rejection.} 
Apply the equality $K_2S=T_IG^{-1}$, select $W=W_u$, $V=G^{-1}N$, 
and use the bound (\ref{Eq:lbWT_IV}) to obtain 
\begin{equation}
   \l W_uK_2SN(s)\r_\infty  \ge 
      \max_{\mbox{\scriptsize\Rhp-poles, $p_i$}}\, 
      \l W_u(p_i)\, u_{p_i}\r_2 \cdot 
      \l u_{p_i}^H G^{-1}N \Bzi^{-1}(G_{mi}^{-1}N)\ev{s=p_i}\r_2
    \label{Eq:LBWuK2SNI}
\end{equation}

We may look at the combined effect of reference tracking, 
disturbance rejection and measurement noise by using 
(\ref{Eq:lbWT_IV}) with  
$W=W_u$ and $V = G^{-1}\pmatrix{G_d &  R & N}$.
% or using (\ref{Eq:lbWTV}) with $W=W_uG^{-1}$ and 
% $V = \pmatrix{G_d &  R & N}$. 

\subsubsection{Simplified lower bound on 
$\left\| K_2S(s)\right\|_\infty$.} 
Two useful simplified lower bounds on $\l K_2 S(s)\r_\infty$ 
%(which will be used in Chapter~\ref{Ch:SelRegContPoleDir}) 
can easily be derived. 
First, apply the equality $K_2S=T_IG^{-1}$, select $W=I$, $V=G^{-1}$, 
and use the bound (\ref{Eq:lbWT_IV}) to obtain 
\begin{equation}
   \l K_2S(s)\r_\infty \ge 
      \max_{\mbox{\scriptsize\Rhp-poles, $p_i$}}\, 
      \l u_{p_i}^H G^{-1} \Bzi^{-1}(G_{mi}^{-1})\ev{s=p_i}\r_2 =
      \l u_{p_i}^H G_{so}^{-1}\ev{s=p_i}\r_2  
   \label{Eq:InplbHinfK2S} 
\end{equation}
where the last identity follows from 
$\Bzi(G_{mi}^{-1}) = \Bzi(G^{-1}) = \Bpo(G)$.

Similarly, we obtain from (\ref{Eq:lbWTV}), with $W=G^{-1}$ 
and $V=I$ 
\begin{equation}
   \l K_2S(s)\r_\infty \ge 
      \max_{\mbox{\scriptsize\Rhp-poles, $p_i$}}\, 
      \l \Bzo^{-1}(G^{-1}_{mo})\, G^{-1}\ev{s=p_i}\, y_{p_i}\r_2 
      = \l G_{si}^{-1}\ev{s=p_i}\, y_{p_i}\r_2 
   \label{Eq:OutlbHinfK2S} 
\end{equation}
where the last identity follows from 
$\Bzo(G_{mo}^{-1}) = \Bzo(G^{-1}) = \Bpi(G)$.


\section{Two degrees-of-freedom control}\label{Sec:TwoDof}
For a 2-{\Dof} controller the closed-loop transfer function from 
references $\tilde{r}$ to outputs $z_1=W_p(y-r)$ becomes 
\begin{equation}
   W_P (SGK_1 - I) R
\end{equation}
We then have the following ``special'' lower bound on this transfer 
function.
\begin{Theorem}{\hspace{-0.6em}.}\label{Theo:lb2Dof}
   Consider a plant $G$ with $N_z\ge 1$ \Rhp-zeros $z_j$ and 
   $N_p\ge 0$ \Rhp-poles $p_i\in\orhp$. Let the performance weight 
   $W_P$ be minimum phase and let (for simplicity) $R$ be stable. 
   Assume that the closed-loop 
   transfer function $W_P (SGK_1-I) R$ is stable. 
   Then the following lower bound on 
   $\l W_P(SGK_1 -I )R(s) \r_\infty$ applies:
   \begin{equation}
      \l W_P(SGK_1-I)R(s) \r_\infty \ge 
          \max_{\mbox{\scriptsize\normalfont 
              \Rhp-zeros $z_j$ in $G$}}\, 
          \l W_P(z_j)y_{z_j}\r_2 \cdot 
          \l y_{z_j}^H R_{mi}(z_j)\r_2 \label{Eq:rhpzRefT} 
   \end{equation}
   The bound (\ref{Eq:rhpzRefT}) is tight if the plant
   has one \Rhp-zero $z$. 
\end{Theorem} 
%\begin{findproof}
%   See Section~\ref{Ch6:App:Proofs}. 
%\end{findproof}
Note that this bound does not follow directly from 
Theorems~\ref{Theo:LBWSV}--\ref{Theo:LBWTIV}. 
The bound in (\ref{Eq:rhpzRefT}) should be compared to the 
following bound for a 1-{\Dof} controller 
(which follows from Theorem~\ref{Theo:LBWSV}, assuming that $W_P$ is 
minimum phase).  
\begin{equation} 
   \l W_PSR(s)\r_\infty \ge 
          \max_{\mbox{\scriptsize\normalfont 
              \Rhp-zeros $z_j$ in $G$}}\, 
    \l W_P(z_j)y_{z_j}\r_2 \cdot 
       \l y_{z_j}^HR\Bzi^{-1}(\Bpo(G)\, R)\ev{s=z_j}\r_2 \label{Eq:lbO-Dof} 
\end{equation}
We see that for the 2-{\Dof} 
controller only the \Rhp-zeros pose limitations.

\section{Example}\label{Sec:LBMIMOExample}
\small 
In this section we consider the following multivariable plant $G$
$$
   G(s) = \pmatrix{\frac{s-z}{s-p}    & -\frac{0.1s+1}{s-p} \cr
                   \frac{s-z}{0.1s+1} & 1}, \q \mbox{with} \q 
   z = 2.5 \q \mbox{and} \q p = 2
$$
%One state-space realization of $G$ is given by 
%$$
%   G(s) \ssequiv \left[ \begin{array}{cc|cc}
% 2   & 11.180 &    10   &  -1  \cr
% 0   &  -10   & -11.180 &   0  \cr \hline 
% 1.2 &  1.118 &    1    & -0.1 \cr
%  0  & 11.180 &    10   &  1   
%            \end{array} \right]
%$$
The plant $G$ has one multivariable \Rhp-zero $z = 2.5$ and one 
\Rhp-pole $p = 2$. 
The corresponding input and output zero and pole directions are 
$$
   u_z = \pmatrix{1 \cr 0}, \q \q 
   y_z = \pmatrix{ 0.371 \cr
                   0.928 }, \q \q
   u_p = \pmatrix{ 0.385 \cr
                   0.923 }, \q \q 
   y_p = \pmatrix{1\cr 0}
$$
The \Rhp-pole $p$ can be factorized into $G(s) = 
\Bpo^{-1}(G) G_{so}(s)$ where  
$$
   \Bpo(G) = \pmatrix{\frac{s-p}{s+p} & 0 \cr 0 & 1} 
   \q \mbox{and} \q 
   G_{so}(s) = \pmatrix{\frac{s-z}{s+p}    & -\frac{0.1s+1}{s+p} \cr
                        \frac{s-z}{0.1s+1} & 1}    
$$
From the lower bound (\ref{Eq:lbWSV}), with $W = I$ 
and $V = I$, we find 
$$
   \l S(s)\r_\infty \ge \l y_z^H \Bpo^{-1}(G)\ev{s=z}\r_2 = 
   \left\|\pmatrix{0.371 & 0.928} \pmatrix{9 & 0 \cr 0 & 1}\right\|_2 
   %= \left\| \pmatrix{3.343 &  0.928} \right\|_2 
   = 3.4691
$$
Next, we use Theorem~\ref{Theo:KWSVinf} (with $W=I$ and $V=I$) 
to find the feedback controller which minimizes 
$\l S(s)\r_\infty$. 
With $k_0=10^{-2}$ we get the following balanced minimal 
state-space realization of the feedback controller $K_2$  
$$
   K_2(s) = G_{smo}^{-1} P Q^{-1}(s) \ssequiv 
       \mbox{\small $\left[ \begin{array}{c|cc} 
     -10 &    188.4 &   -75.49 \cr \hline 
       0 &      306 &   -122.6 \cr 
     203 &    -6508 &     2605  
       \end{array}\right]$} \q \mbox{which achieves} \q 
   \l S(s)\r_\infty = 3.4691
$$
Note the large gain in the controller (large elements in the $D$ 
matrix). The reason is the small value of $k_0 = 10^{-2}$,  
$k_0$ must be small to get the \hinf-norm of $S$ close to 
the lower bound. 
Note, it is not surprising that we get large gains in the controller 
(and large input usage) since no weight has been put on  
the transfer function $K_2S$. 

Next, consider minimizing the input usage, {\ie} 
to minimize the \hinf-norm of $K_2S$. 
We have two lower bounds on $\l K_2S(s)\r_\infty$, but they are 
identical since the bounds are tight. 
We use the equality $K_2S = T_I G^{-1}$ and the lower bound 
(\ref{Eq:lbWT_IV}) with $W=I$ and $V=G^{-1}$, 
to obtain\footnote{We use the following relations:  
$\Bzi(G^{-1}_{mi}) = \Bpo(G)$ and $G^{-1}\Bpo^{-1}(G)=G_{so}^{-1}$. 
The first, follows since the input factorization of \Rhp-zeros 
in $G$ does {\em not\/} change the output pole directions.} 
$$
   \l K_2S(s)\r_\infty \ge 
   \l u_{p}^H G^{-1} \Bzi^{-1}(G_{mi}^{-1})\ev{s=p}\,\r_2 = 
   \l u_p^H G_{so}^{-1}(p)\r_2 = 3.077
$$

In \cite[Section~5.7]{Havre98} reference tracking is also 
considered, and the benefit of applying $2$-{\Dof} controller 
when the plant is unstable is illustated. 


\normalsize 

\section{Conclusion}
\begin{bentry}
   \item We have derived tight lower bounds on closed-loop 
         transfer functions valid for multivariable plants. 
         The bounds are independent of the controller and 
         therefore reflects the controllability of the plant. 
   \item The bounds extend and generalize the results 
         by \citeasnoun{Zames81}, \citeasnoun{DFT92}, 
         \citeasnoun{SP96} and the results given in 
         \citeasnoun{HS98},
         to also handle non-minimum phase and unstable weights. 
         This allows us to derive {\em new\/} lower bounds on 
         input usage due to disturbances, measurement noise 
         and reference changes. 
   \item The new lower bounds on input usage make it possible 
         to {\em quantify\/} the minimum input usage for 
         stabilization of unstable plants in the presence 
         of worst case disturbances, measurement noise and 
         reference changes. 
   \item It is proved that the lower bounds are {\em tight\/}, 
         by deriving analytical expressions for stable 
         controllers which achieves an \hinf-norm of the 
         closed-loop transfer functions equal to the lower 
         bound for large classes of systems.  
   \item Theorem~\ref{Theo:lb2Dof} expresses the benefit of 
         applying a $2$-{\Dof} controller compared to 
         a $1$-{\Dof} controller when the plant is unstable 
         and has a \Rhp-zero. 
\end{bentry}


\nocite{BD85,Chen95}
%\nocite{HS98a} 
\nocite{HS98} 


\bibliographystyle{thesis}
\bibliography{thesis}

\vspace{1ex}\small 
\noindent PhD thesis and \Matlab-software to compute the bounds in 
Theorems~\ref{Theo:LBWSV} to \ref{Theo:LBWTIV} 
are available on the internet: 
{\ttfamily http://www.chembio.ntnu.no/users/skoge}. 
%{\ttfamily http://www.kjemi.unit.no/\hspace{2pt}\mbox{\LARGE$_\mathbf{\tilde{ }}$}\hspace{2pt}skoge}.


\normalsize

\vspace{1cm}

\appendix 

\section{Proofs of the results}\label{Ch6:App:Proofs}

\begin{proofof}{Theorem~\ref{Theo:LBWSV}}
   We prove (\ref{Eq:lbWSV}) by applying the following six steps:
   \begin{entry}
      \item Factor out {\Rhp}-zeros in $WSV$:
            \Rhp-poles in $G$ appears as \Rhp-zeros in 
            $S$. 
% (\ref{Ch6:mimointp}).
 Factor out 
            $S=\widetilde{S}\,\Bpo(G)$ to obtain 
            \begin{eqnarray*}
               WSV(s) & = & \Bzo(W)\,W_{mo}\,\widetilde{S}\,\Bpo(G)\,V \\ 
                      & = & \Bzo(W)\,\underbrace{W_{mo}\,\widetilde{S}\, 
                            (\Bpo(G)\,V)_{mi}}_{(WSV)_{m}}\,\Bzi(\Bpo(G)\,V)
            \end{eqnarray*}
            $WSV$ is stable by assumption. 
            From the assumption on internal stability it follows that 
            $S$ is stable (if one closed-loop transfer function 
            is stable then internal stability implies that  
            all the other closed-loop transfer functions are stable).
            Then it is only the \Rhp-zeros in $S$ which can cancel 
            \Rhp-poles in $V$ and $W$. 
            So, factorizing the zeros in $\orhp$ of $W$ does not 
            introduce instability in $(WSV)_m$, since none of these 
            cancel unstable modes in $S$ or $V$ . Similarly, we can 
            factorize the zeros in $\orhp$ of $V$. 
            However, when factorizing the zeros in $S$ we must avoid 
            factorizing the zeros which cancel poles in $\orhp$ of $V$. 
            Otherwise, $(WSV)_m$ becomes unstable. 
            By factorizing only the zeros in a minimal realization 
            of $\Bpo(G)\, V$ we accomplish this. 
            Since $W$ is stable there are no canncellations 
            against the zeros in $S$ due to poles in $G$.
            It then follows that $(WSV)_m$ is stable. 
%            By assumption none of the modes in $\orhp$ of $W$ 
%            cancel against the zeros in $S$ due to poles in $G$.
%            It then follows that $(WSV)_m$ is stable. 
      \item Introduce 
            $f(s) = \max\limits_{\left\|x_1\right\|_2=1,\,
                                 \left\|x_2\right\|_2=1}\, 
                       x_1^H (WSV)_{m}\, x_2$, 
            then 
            $$
               \l WSV(s)\r_\infty = \l (WSV(s))_m\r_\infty \ge 
               \l f(s)\r_\infty 
            $$
      \item Apply the maximum modulus theorem to $f(s)$ at the 
            \Rhp-zeros $z_j$ of $G$  
            $$
               \l f(s)\r_\infty \ge |f(z_j)| 
            $$ 
      \item Resubstitute the factorization of \Rhp-zeros in $S$, 
            {\ie} use $\widetilde{S} = S\,\Bpo^{-1}(G)$ 
            \begin{eqnarray*}
               f(z_j) & = & \max_{\left\|x_1\right\|_2=1,\,
                                 \left\|x_2\right\|_2=1}\, 
               x_1^H W_{mo}\, S\, \Bpo^{-1}(G)\, 
                     (\Bpo(G)\,V)_{mi} \ev{s=z_j} x_2 \\  
                      & = & \max_{\left\|x_1\right\|_2=1,\,
                                 \left\|x_2\right\|_2=1}\, 
               x_1^H W_{mo}\, S\, V\, 
                   \Bzi^{-1}(\Bpo(G)\,V) \ev{s=z_j} x_2         
            \end{eqnarray*}
      \item Use the interpolation constraint for \Rhp-zeros $z_j$ 
            in $G$, {\ie} use $y_{z_j}^H S(z_j) = y_{z_j}^H$ 
            \begin{eqnarray*}
               f(z_j) & = & \max_{\left\|x_1\right\|_2=1,\,
                                 \left\|x_2\right\|_2=1}\, 
               x_1^H W_{mo}\,S\,V \Bzi^{-1}(\Bpo(G)\,V) \ev{s=z_j} x_2 \\
                     & \ge & \max_{\left\|x_1\right\|_2=1,\,
                                 \left\|x_2\right\|_2=1}\,        
               x_1^H W_{mo}\, y_{z_j}y_{z_j}^H 
                           S\, V\, \Bzi^{-1}(\Bpo(G)\,V) \ev{s=z_j} x_2   \\ 
                     &  =  & \max_{\left\|x_1\right\|_2=1,\,
                                 \left\|x_2\right\|_2=1}\, 
               x_1^H W_{mo}\, y_{z_j}y_{z_j}^H 
                          V\, \Bzi^{-1}(\Bpo(G)\,V) \ev{s=z_j} x_2 
            \end{eqnarray*}
      \item Evaluate the lower bound 
            $$
               \l WSV(s)\r_\infty \ge |f(z_j)| \ge 
                  \l W_{mo}(z_j) y_{z_j} \r_2 \cdot 
                  \l y_{z_j}^H V \Bzi^{-1}(\Bpo(G)\,V) \ev{s=z_j}\r_2
            $$
   \end{entry}
   Since these steps apply to all \Rhp-zeros in $G$, the 
   bound (\ref{Eq:lbWSV}) follows. 
\end{proofof}



\end{document}
