\chapter{Unknown Input Observer (UIO) Design}
\label{chap:UIO_chapter}

\begin{flushright}
\parbox{5in}{The best piece of advice I got before I wrote this book was from a colleague of mine, David Ignatius, who is editor of the ``Outlook'' section at {\it The Washington Post}. He had written a novel about the Middle East [SIRO], a very good novel. He said, ``When you're writing this book always imagine whether the reader is going to want to turn the page. Whether what you're writing, how you're writing, and the anecdotes you're telling will compel that reader to want to turn the page.'' That's what I tried to keep in mind. Hence the book is a really a combination of analysis of history, of autobiography, of anecdote, put into my own kind of Cuisinart.'' Thomas Friedman in~\cite[p.~288]{lamb97}}
\end{flushright}


\bigskip
%Design procedures are proposed for two different classes of observers for uncertain
%dynamical systems. In the first approach, the state of the system, whose state is to
%be estimated, is decomposed into known and unknown components. The unknown component
%is a projection, not necessarily orthogonal, of the whole state along the subspace
%in which the available state component resides. Then a dynamical system to estimate
%the unknown component is constructed. Combining the output of the dynamical system,
%which estimates the unknown state component, with the available state information
%results in the observer that estimates the whole state. It is shown that some
%previously proposed observer architectures can be obtained using the projection
%operator approach presented in this paper. The second approach combines sliding
%modes and the second method of Lyapunov.

In this Chapter, we present design procedures for full- and reduced-order observers
for a class of dynamical systems modeled by a triple $(\b A, \b B,\b C)$ where some
or all of the inputs are unknown. The unknown input can be a combination of
unmeasurable or unmeasured disturbances, unknown control action, and unmodeled
system dynamics. The first design method uses a projection operator approach to the
state estimation where the state of the system, whose state is to be estimated, is
decomposed into known and unknown components. The unknown component is, in general,
a skew projection, that is, not necessarily orthogonal, of the whole state along the
subspace in which the available state component resides. We then construct a
dynamical system to estimate the unknown component. Finally, we combine the output
of the dynamical system, which estimates the unknown state component, with the
available state information to obtain the observer that estimates the whole state.
In the second design method we employ a sliding mode approach combined with the
second method of Lyapunov. We include design algorithms and illustrate the results
with numerical examples.



\section{Introduction}
Observers use the plant input and output signals to generate an estimate of the
plant's state, which is then employed to close the control loop. Observers are
utilized to augment or replace sensors in a control system. The observer was first
proposed and developed by Luenberger in the early sixties of the last
century~\cite{luenberger66,luenberger71,luenberger79}. Franklin and
Powell~\cite[page~139]{franklin80} suggest using the term ``estimator'' rather than
the commonly used term ``observer'' because ``observer'' seems to imply a direct
measurement while, in fact, we estimate the state. On the other hand, the term
estimator seems to suggest the use of statistical methods when estimating the
uncertain system state. In this Chapter, we utilize only deterministic methods to
estimate the state of an uncertain dynamical system. In view of the above, we will
be using the term observer for the deterministic dynamical system whose goal is to
estimate the state of an uncertain dynamical system employing only the known system
input and output signals. Since the early developments, which concentrated on
observers for systems without uncertainties, observers for plants with both known
and unknown inputs have been developed resulting in the so-called unknown input
observer~(UIO) architectures, such as, for example,
in~\cite{hostetter73,wang75,sundareswaran77,bhattacharyya78,kudva80,
kurek83,yang88,hou92,hui_uio93,darouach94,chen96,corless98,chen99,hou99}. Barmish
and Galimidi~\cite{barmish86} showed that robust stabilization of uncertain systems
via the Luenberger observer is possible provided the uncertain parameters do not
exceed a certain computable threshold. More recently, observer architectures
utilizing the concept of sliding modes were proposed for uncertain systems, see, for
example,~\cite{walcott87,walcott87b,walcott88,hui90,zak90,zak93b,zak93c,edwards98,
utkin99,zak03,ha03,koshkouei04}. Other methods of the observer design for linear
systems developed prior 1983 are reported by O'Reilly in~\cite{o'reilly83}.



Observers for uncertain systems play an essential role in robust model-based fault
detection~\cite{edwards98,chen99,edwards00,saif03,jiang04}. The basic idea behind
the use of the observers for fault detection is to form residuals from the
difference between the actual system outputs and the estimated outputs using an
observer. Once a fault occurs, the residuals are expected to react by becoming
greater than a pre-specified threshold. When the system under consideration is
subject to unknown disturbances or unknown inputs, their effect has to be decoupled
from the residuals to avoid false alarms.












\section{Uncertain System Model}
\label{sec:uncertain_model}

The class of uncertain dynamical systems that we consider is modeled by
\begin{eqnarray}
\dot{\b x} &=& \b A\b x + \b B\b u\label{eq:uncert_sys}\\
\b y &=& \b C\b x, \label{eq_lin_system}
\end{eqnarray}
where the input matrix $\b B\in \mb{\Bb R}^{n\times m}$ and the output matrix $\b C
\in \mb{\Bb R}^{p\times n}$. We assume that the model parameters $(\b A, \b B, \b
C)$ are known. We further assume that some or all of the inputs are unknown and that
the last $m_2$ inputs are unknown and the remaining $m_1=m-m_2$ inputs are known. We
partition the input matrix $\b B$ corresponding to the known and unknown inputs as
\[
\b B=\left[\begin{array}{cc} \b B_1 & \b B_2
\end{array}\right],
\]
where $\b B_1\in \mb{\Bb R}^{n\times m_1}$ and $\b B_2\in \mb{\Bb R}^{n\times m_2}$.
Let
\[
\b u=\left[\begin{array}{c}
\b u_1\\
\b u_2
\end{array}\right].
\]
Then, the system model~(\ref{eq:uncert_sys}) can be represented as
\begin{equation}
\fbox{\displaymath$ \dot{\b x} = \b A\b x + \b B_1\b u_1 + \b B_2\b u_2 $}
\label{eq:two_inputs}
\end{equation}
The vector function $\b u_2$ may also model
lumped uncertainties or nonlinearities in the plant. The output matrix $\b C \in
\mb{\Bb R}^{p\times n}$. We assume that the pair $\left( \b A,\b C \right)$ is
detectable.





\section{A Projection Operator Approach}
In this section we employ a projection operator approach  to the problem of state
observation of uncertain systems. In our further discussion, we assume that the
matrix $\b B_2$ has full column rank. We begin our presentation by noticing that
because the system output $\b y$ is known, it would seem reasonable to decompose the
state $\b x$ as
\begin{eqnarray}
\b x &=&(\b I-\b M\b C)\b x+\b M\b C\b x\nonumber\\
&=&(\b I-\b M\b C)\b x+\b M\b y, \label{eq:state_decompose}
\end{eqnarray}
where $\b M$ is an $n\times p$ real matrix, and the unknown part of the
decomposition is $(\b I-\b M\b C)\b x$. Let $\b q=(\b I-\b M\b C)\b x$, then $\b
x=\b q+\b M\b y$, and we have
\begin{eqnarray*}
\dot{\b q}&=& (\b I-\b M\b C)\dot{\b x}\\
&=& (\b I-\b M\b C)(\b A\b x+\b B_1\b u_1+\b B_2\b u_2)\\
&=& (\b I-\b M\b C)(\b A\b x+\b B_1\b u_1)+(\b I-\b M\b C)\b B_2\b u_2\\
&=&(\b I-\b M\b C)(\b A\b q+\b A\b M\b y+\b B_1\b u_1)+(\b I-\b M\b C)\b B_2\b u_2.
\end{eqnarray*}
If $\b M$ is chosen so that $(\b I-\b M\b C)\b B_2=\b O$, then the dynamics of $\b
q$ depend only on the known quantities $\b u_1$ and $\b y$:
\begin{equation} \dot{\b q}=(\b I-\b M\b C)(\b A\b q+\b A\b M\b y+\b B_1\b u_1).
\label{eq:pre_obser}
\end{equation}
Note that if we start the above dynamical system with the initial condition $\b
q(0)=(\b I-\b M\b C)\b x(0)$, then $\b x=\b q+\b M\b C\b x=\b q+\b M\b y$ for all
$t\geq 0$. But since $\b x(0)$ is assumed to be unknown, then $\tilde{\b x}=\b q+\b
M\b y$ is only an approximation of $\b x$. To improve convergence rate or to ensure
convergence, we add an extra term to the right-hand side of~(\ref{eq:pre_obser}) to
obtain
\begin{eqnarray*} \dot{\b q}&=&(\b I-\b M\b C)(\b A\b q+\b A\b M\b y+\b B_1\b u_1 +
\b L(\b y-\b C\b q-\b C\b M\b y))\\%\label{eq:pre_obser}
&=&(\b I-\b M\b C)(\b A\b q+\b A\b M\b y+\b B_1\b u_1+\b L\b C(\b x-\b q-\b M\b
y)).\end{eqnarray*} Let $\b e=\b x-\tilde{\b x}$. We will show that
\[ \dot{\b e}=(\b I-\b M\b C)(\b A-\b L\b C)\b e\] and $\b e(t) \rightarrow{\b 0}$ as
$t\rightarrow \infty$ under mild conditions.



Since
\[ \mbox{rank }(\b M\b C\b B_2)\leq \mbox{rank }(\b C\b B_2)\leq \mbox{rank }(\b B_2),\]
the equality $(\b I-\b M\b C)\b B_2=\b O$ makes it necessary that $\mbox{rank }(\b
C\b B_2)=\mbox{rank}(\b B_2)$, which we assume throughout the Chapter. This rank
condition also implies that there must be at least as many independent outputs as
unknown inputs for the method to work.


%We will show that, in order to arrive at a reduced-order observer using the above
%presented approach, it is critical that the term $\b L(\b y-\b C\b q-\b C\b M\b y)$
%is premultiplied by $(\b I-\b M\b C)$, or equivalently, $\b L$ has $(\b I-\b M\b C)$
%as a left factor. Indeed, let $\tilde{\b P}=\b I-\b M\b C$. Then, if $\tilde{\b P}$
%is a projection, that is, $\tilde{\b P}^2=\tilde{\b P}$, then the subspace
%${\cal{V}} = {\tilde{\b P}}{\mb{\Bb R}^n}$ is invariant under $\tilde{\b P}$. It
%follows that $\dot{\b q}$ in equation~(\ref{eq:pre_obser}) lies in $\cal{V}$. If the
%initial condition $\b q(0)$ is also in $\cal{V}$, then the trajectories of the
%system will reside in $\cal{V}$ for $t\geq 0$. If the term $\b L(\b y-\b C\b q-\b
%C\b M\b y)$ is not premultiplied by $(\b I-\b M\b C)$ or $\b L$ does not have $(\b
%I-\b M\b C)$ as a left factor, then the trajectory will not stay in $\cal{V}$ and,
%in general, it would not be possible to transform the full order-observer into a
%reduced order-observer.
%
%
%The condition $\b q(0)\in \cal{V}$ alone is not sufficient to guarantee that the
%observation error $\b e$ tends to $\b 0$. The reason is that we do not know $\b
%x(0)$ and it is not obvious how to choose $\b q(0)$ so that the $\b e(t)$ converges
%to $\b 0$. Unless $\b q(0)$ is chosen appropriately, the observation error $\b e(t)$
%stays in a hyperplane not containing $\b 0$ and thus $\b e(t)$ cannot converge to
%$\b 0$.
%
%
%
%Another difficulty that must be overcome is that,
For the error dynamics  matrix
$(\b I-\b M\b C)(\b A-\b L\b C)$ to be asymptotically stable, it is not sufficient
for $(\b A-\b L\b C)$ to be asymptotically stable. It is possible for a product of a
projection matrix and an asymptotically stable matrix to be unstable as the
following simple example shows.
\begin{example}\rm
Let
\[\b{\Pi} =\left[\begin{array}{cc}1 & 0\\0& 0\end{array}\right]\mbox{   and   }
\b A=\left[\begin{array}{cc}1 & -3\\3& -2\end{array}\right].\] It is easy to check
that $\b A$ is asymptotically stable while $\b{\Pi}\b A$ is unstable. Furthermore, the
system $\dot{\b x}=\b A\b x$ restricted to the range of $\b{\Pi}$ is governed by
$\dot{z}=z$, which is also unstable in this case.
\end{example}
\markex


We now analyze convergence properties of the proposed full-order observer and then
we use the results of our analysis to propose a new type of reduced-order observer
for uncertain systems. Consider the following dynamical system model,
\begin{eqnarray}
\dot{\b q}&=&(\b I-\b M\b C)(\b A\b q+\b A\b M\b y+\b B_1\b u_1 + \b L(\b y-\b C\b
q-\b C\b M\b y))\\\label{eq:p_obser} \tilde{\b x} &=& \b q + \b M\b
y.\label{eq:q_express}
\end{eqnarray}
We will now show that $\tilde{\b x} \rightarrow{\b x}$ as $t\rightarrow \infty$. To
this end let
\[
\b e(t)=\b x(t) - \tilde{\b x}(t)
\]
denote the estimation error. Then, using $(\b I -\b M \b C)\b B_2=\b O$ and $\b y=\b
C\b x$, we have
\begin{eqnarray}
\frac{d \b e}{dt} &=& \frac{d}{dt}\left( \b x-\tilde{\b x}\right)\nonumber\\
 &=& \frac{d}{dt}\left( \b x-\b q - \b
M\b C\b x\right)\nonumber\\&=& \frac{d}{dt}\left( (\b I -\b M \b C)\b x-\b q\right)
\nonumber\\ &=& (\b I -\b M \b C)(\b A\b x+\b B_1\b u_1+\b B_2\b u_2)\nonumber\\
&{}&\quad -(\b I-\b M\b C)(\b A\b q+\b A\b M\b y+\b B_1\b u_1 + \b L(\b y-\b C\b
q-\b C\b M\b y))\nonumber\\
&=& (\b I -\b M \b C)(\b A\b x + \b B_1\b u_1) +
(\b I -\b M \b C)\b B_2\b u_2\nonumber\\
&{}& \quad -(\b I-\b M\b C)(\b A\b q+\b A\b M \b C\b x+\b B_1\b u_1 + \b L(\b C\b x
- \b C\b q-\b C\b M\b C\b x))\nonumber\\
&=& (\b I-\b M\b C)(\b A-\b L\b C)(\b x - \b q-\b M\b C\b x)\nonumber\\
&=& (\b I-\b M\b C)\left(\b A-\b L\b C\right)\b e.\label{der_estierror}
\end{eqnarray}
Our objective is to specify $\b M$ and $\b L$ and a set of initial conditions so
that $\b e (t) \rightarrow \b 0$ as $t\rightarrow\infty$. A particular class of
solutions to $(\b I -\b M \b C)\b B_2=\b O$ is given by
\[ \b M = \b B_2 \left((\b C\b B_2)^{\dagger}+\b H_0\left(\b I_p - (\b C\b B_2)(\b C\b
B_2)^{\dagger}\right)\right),
\]
where the superscript $\dagger$ denotes the Moore-Penrose pseudo-inverse operation
and $\b H_0\in \mb{\Bb R}^{m_2\times p}$ is a design parameter matrix. (See, for
example, Kaczorek~\cite[Section~1.5]{kaczorek98} for more information on the subject
of pseudo-inverse matrices.) Because, by assumption, $\mbox{rank }(\b C\b
B_2)=\mbox{rank }\b B_2$ and $\b B_2$ has full rank, we have $(\b C\b
B_2)^{\dagger}(\b C\b B_2)=\b I_{m_2}$. If $\b C \b B_2$ is a square matrix, then
$\b C\b B_2$ is invertible by assumption and the above $\b M$ reduces to $\b B_2(\b
C\b B_2)^{-1}$. Furthermore, it is easy to check that for the above class of $\b M$,
the product $\b M\b C$ is a projection (not necessarily orthogonal):
\[ (\b M\b C)^2 =\b M\b C.\]
It follows that
\[ \tilde{\b{\Pi}} =\b I-\b M\b C\]
is also a projection.




To proceed further, we need the following lemma.
\begin{lemma}\label{projector}
Let $\tilde{\b{\Pi}}:\mb{\Bb R}^n \to \mb{\Bb R}^n$ be a projection, that is,
$\tilde{\b{\Pi}}^2=\tilde{\b{\Pi}}$, and let $\mb{rank}\,\tilde{\b{\Pi}}=n-m_2$. Then
$\tilde{\b{\Pi}}$ has $(n-m_2)$ eigenvalues equal to $1$ while the remaining $m_2$
eigenvalues are located at $0$ and there is a basis of $\mb{\Bb R}^n$ in which the
matrix $\tilde{\b{\Pi}}$ relative to this basis has the form
\[
\b \Pi=\left[ \begin{array}{cc} \b I_{n-m_2} & \b O\\\b O & \b O\end{array}\right],
\]
that is, there is an invertible matrix $\b T$ whose columns are eigenvectors of
$\tilde{\b P}$ such that
\[ \b T^{-1} \tilde{\b \Pi}\b T =\b \Pi=\left[ \begin{array}{cc} \b I_{n-m_2} & \b O\\\b O &
\b O\end{array}\right].\]
\end{lemma}
{\bf Proof}\quad See Smith~\cite[pp.~156--158 and pp.~194--195]{smith84}.\\
\hspace*{5.5in} $\Box$



\section{Constructing Full-Order UIO}
\label{fo_uio} We begin this section by introducing
the following coordinate transformation,
\begin{equation}
\tilde{\b e}=\b T^{-1}\b e, \label{error_p} \end{equation} where the transformation
matrix $\b T$ is obtained, using Lemma~\ref{projector}, from the representation of
the projection operator $\tilde{\b \Pi}$ in the form
\begin{equation}\label{defQ}
\tilde{\b \Pi}=\b T\b \Pi\b T^{-1}.
\end{equation}
Applying the coordinate transformation~(\ref{error_p}) to the error
equation~(\ref{der_estierror}) gives
\begin{eqnarray}
\dot{\tilde{\b e}}&=& \b \Pi\b T^{-1}(\b A-\b L\b C)\b T\tilde{\b e}\nonumber\\
&=& \b \Pi\left(\b T^{-1}\b A\b T - \left(\b T^{-1}\b L\right)\left(\b C\b T \right)
\right)\tilde{\b e}.\label{err_innew}
\end{eqnarray}
Let
\begin{equation}
\tilde{\b A}=\b T^{-1}\b A\b T =\left[\begin{array}{cc}
\tilde{\b A}_{11} & \tilde{\b A}_{12}\\
\tilde{\b A}_{21} & \tilde{\b A}_{21}
\end{array}\right],\;\; \tilde{\b L}=\b T^{-1}\b L=\left[\begin{array}{c}
\tilde{\b L}_1\\
\tilde{\b L}_2
\end{array}\right],\;\; \tilde{\b C}= \b
C\b T =\left[\begin{array}{cc} \tilde{\b C}_1 & \tilde{\b C}_2
\end{array}\right],
\label{matrices}
\end{equation}
where $\tilde{\b A}_{11}\in \mb{\Bb R}^{(n-m_2)\times (n-m_2)}$, $\tilde{\b
L}_{1}\in \mb{\Bb R}^{(n-m_2)\times p}$, $\tilde{\b C}_{1}\in \mb{\Bb R}^{p\times
(n-m_2)}$, and the remaining block submatrices are of appropriate dimensions. Using
the above notation, we represent~(\ref{err_innew}) in the form,
\begin{eqnarray}
\dot{\tilde{\b e}}&=& \b \Pi\left(\tilde{\b A} -
\tilde{\b L}\tilde{\b C} \right)\tilde{\b e}\nonumber\\
&=& \left[\begin{array}{cc} \b I_{n-m_2} & \b O\\
\b O & \b O
\end{array}\right]\left(\left[\begin{array}{cc}
\tilde{\b A}_{11} & \tilde{\b A}_{12}\\
\tilde{\b A}_{21} & \tilde{\b A}_{21}
\end{array}\right]-\left[\begin{array}{c}
\tilde{\b L}_1\\
\tilde{\b L}_2
\end{array}\right]\left[\begin{array}{cc}
\tilde{\b C}_1 & \tilde{\b C}_2
\end{array}\right]\right)\tilde{\b e}\nonumber\\
&=&  \left[\begin{array}{cc} \tilde{\b A}_{11}-\tilde{\b L}_1\tilde{\b C}_1 &
\tilde{\b A}_{12} -
\tilde{\b L}_1\tilde{\b C}_2\\
\b O & \b O
\end{array}\right]\tilde{\b e}\label{errindiffco}
\end{eqnarray}
Let
\begin{equation}
\tilde{\b e}=\left[\begin{array}{c}
\tilde{\b e}_1\\
\tilde{\b e}_2
\end{array}\right],
\end{equation}
where $\tilde{\b e}_1 \in \mb{\Bb R}^{n-m_2}$. Note that $\dot{\tilde{\b e}}_2 =\b
0$. Hence if $\tilde{\b e}_2(0)=\b 0$, then $\tilde{\b e}_2 = \b 0$ for all $t\ge
0$. Thus if $\tilde{\b e}_2=\b 0$ then
\[
\dot{\tilde{\b e}}_1=\left(\tilde{\b A}_{11}-\tilde{\b L}_1\tilde{\b
C}_1\right)\tilde{\b e}_1,
\]
and so if $\tilde{\b e}_2=\b 0$ and $\tilde{\b e}_1\to\b 0$ then $\tilde{\b e} \to
\b 0$. Obviously $\tilde{\b e}_1\to\b 0$ for arbitrary $\tilde{\b e}_1(0)$ if and
only if the matrix $\left(\tilde{\b A}_{11}-\tilde{\b L}_1\tilde{\b C}_1\right)$ is
asymptotically stable.



We now give a condition on $\b q(0)$ that guarantees that $\tilde{\b e}_2=\b 0$. We
have,
\begin{eqnarray*}
\b M\b C &=& \b I-\tilde{\b \Pi}\\
&=& \b T(\b I-\b \Pi)\b T^{-1}.
\end{eqnarray*}
Hence
\begin{eqnarray}
\left[\begin{array}{c}
\b 0\\
\tilde{\b e}_2
\end{array}\right] &=& (\b I_n - \b \Pi)\b T^{-1}\b e\nonumber\\
&=& \b T^{-1}\b M\b C\b e\nonumber\\
&=& \b T^{-1}\b M\b C(\b x-\b q-\b M\b C\b x)\nonumber\\
&=& \b T^{-1}(\b M\b C\b x -\b M\b C\b q - (\b M\b C)^2\b x)\nonumber\\
&=& - \b T^{-1}\b M\b C\b q.\label{e2toq}
\end{eqnarray}
Therefore $\tilde{\b e}_2(0)=\b 0$ if and only if $\b M\b C\b q(0)=\b 0$, which is
equivalent to
\[
\b q(0)=(\b I-\b M\b C)\b v
\]
for arbitrary $\b v\in \mb{\Bb R}^n$. In particular $\b q(0)=\b 0$ satisfies the
above condition.



In summary, we proved the following theorem,
\begin{thm}\label{thm:UIO_full_order}
If the following conditions are satisfied:
\begin{enumerate}
\item $\mb{rank}\,\left(\b C\b B_2\right)= \mb{rank}\,\b B_2$; \item the pair
$(\tilde{\b A}_{11}, \tilde{\b C}_{1})$ defined in~(\ref{matrices}) is detectable;
\item $\b q(0)=(\b I-\b M\b C)\b v$ for arbitrary $\b v\in \mb{\Bb R}^n$,
\end{enumerate}
then there exists a gain matrix $\b L$ such that the estimation error, $\b e=\b
x-\tilde{\b x}$, of the full-order observer given by
\begin{eqnarray*}
\dot{\b q}&=&(\b I-\b M\b C)(\b A\b q+\b A\b M\b y+\b B_1\b u_1 + \b L(\b y-\b C\b
q-\b C\b M\b y))\\
\tilde{\b x} &=& \b q + \b M\b y
\end{eqnarray*}
converges to $\b 0$ as $t\to \infty$. \label{thm:fo_uio}
\end{thm}

In Figure~\ref{fig:full_order_UIO}, we show a block diagram of the full-order UIO observer of Theorem~\ref{thm:UIO_full_order}.
\begin{figure}
  \centering
  \includegraphics[width=5in]{zak_f_order}
  \caption{Full-order UIO described in Theorem~\ref{thm:UIO_full_order}.}
  \label{fig:full_order_UIO}
\end{figure}

\begin{thm}\label{thm:detect_condition}
The second condition of Theorem~\ref{thm:fo_uio} that states that the pair
$(\tilde{\b A}_{11}, \tilde{\b C}_{1})$ defined in~(\ref{matrices}) is detectable,
is equivalent to
\[
\mb{rank}\left[\begin{array}{cc}
s\b I_n - \b A & \b B_2\\
\b C & \b O
\end{array}\right]=n+m_2 \mb{  for all  } s \mb{  such that  } \mb{Re}(s)
\ge 0.
\]
\end{thm}
{\bf Proof}\quad We begin the proof by considering the projection matrix, $\tilde{\b
\Pi}=\b I-\b M\b C$, where
\[
\b M=\b B_2 \left((\b C\b B_2)^{\dagger}+\b H_0\left(\b I_p - (\b C\b B_2)(\b C\b
B_2)^{\dagger}\right)\right).
\]
To simplify further analysis, we let $\b F=(\b C\b B_2)^{\dagger}+\b H_0\left(\b I_p
- (\b C\b B_2)(\b C\b B_2)^{\dagger}\right)$ and $\b S=\b F\b C$. Note that
\begin{equation}
\b S\b B_2=\b I_{m_2}. \label{eq:sbtwoone}
\end{equation}
Thus, $\mb{rank}\,\b S=m_2$ and so we can find a full rank $n\times (n-m_2)$ matrix
$\b W$ such that
\begin{equation}\b S\b W=\b O.
\label{eq:swzero}
\end{equation}
Combining~(\ref{eq:sbtwoone}) and~(\ref{eq:swzero}) we conclude that $[\b W \,\, \b
B_2]$ is invertible. Let
\[
\left[\begin{array}{cc} \b W & \b B_2
\end{array}\right]^{-1}=\left[\begin{array}{c}
\b W^g\\
\b N
\end{array}\right],
\]
where $\b W^g$ is $(n-m_2)\times n$ and $\b N$ is $m_2\times n$. Then
\begin{eqnarray*}
\b N \left[\begin{array}{cc} \b W & \b B_2\end{array}\right]
=\left[\begin{array}{cc} \b O & \b I_{m_2} \end{array}\right] =\b S
\left[\begin{array}{cc} \b W & \b B_2\end{array}\right].
\end{eqnarray*}
Since $[\b W\,\,\b B_2]$ is full rank, we conclude that $\b N=\b S$. Since
\[
\left[\begin{array}{c} \b W^g\\ \b S \end{array}\right] \left[\begin{array}{cc} \b W
& \b B_2 \end{array}\right] =\b I_n,
\]
we have
\[
\b W^g\b W = \b I_{n-m_2}\quad\mb{and}\quad \b W^g\b B_2=\b O.
\]
Let
\begin{equation}
\b T = \left[\begin{array}{cc} \b W & \b B_2 \end{array}\right].
\label{eq:Proj_transfor}
\end{equation}
Then
\begin{eqnarray}
\b T^{-1}\tilde{\b \Pi}\b T &=& \left[\begin{array}{c}
\b W^g\\
\b S
\end{array}\right]\left(\b I_n-\b B\b S\right)\left[\begin{array}{cc} \b W & \b B_2
\end{array}\right]\nonumber\\
&=& \b I_n-\left[\begin{array}{c}
\b W^g\b B_2\b S\\
\b S\b B_2\b S
\end{array}\right]\left[\begin{array}{cc} \b W & \b B_2
\end{array}\right]\nonumber\\
&=& \left[\begin{array}{cc} \b I_{n-m_2} & \b O\\
\b O & \b I_{m_2}
\end{array}\right]-\left[\begin{array}{cc}
\b W^g\b B_2\b S\b W & \b W^g\b B_2\b S\b B_2\\
\b S\b B_2\b S\b W & \b S\b B_2\b S\b B_2
\end{array}\right]\nonumber\\
&=& \left[\begin{array}{cc} \b I_{n-m_2} & \b O\\
\b O & \b I_{m_2}
\end{array}\right]-\left[\begin{array}{cc} \b O & \b O\\
\b O & \b I_{m_2}
\end{array}\right]\nonumber\\
&=& \left[\begin{array}{cc} \b I_{n-m_2} & \b O\\
\b O & \b O
\end{array}\right].
\label{eq:qinptq}
\end{eqnarray}
We now apply the following coordinate transformation to the system modeled
by~(\ref{eq:uncert_sys}) and~(\ref{eq_lin_system}):
\[
\left[\begin{array}{c} \b z\\
\b{\sigma}
\end{array}\right]=\left[\begin{array}{c}
\b W^g\\
\b S
\end{array}\right]\b x=\b T^{-1}\b x.
\]
Then
\begin{eqnarray*}
\left[\begin{array}{c} \dot{\b z}\\
\dot{\b{\sigma}}
\end{array}\right]&=&\left[\begin{array}{cc}
\b W^g\b A\b W & \b W^g\b A\b B_2\\
\b S\b A\b W & \b S\b A\b B_2
\end{array}\right]\left[\begin{array}{c} \b z\\
\b{\sigma}
\end{array}\right]+\b T^{-1}\b B_1\b u_1 + \left[\begin{array}{c}
\b O\\
\b I_{m_2} \end{array}\right]\b u_2\\
&=& \left[\begin{array}{cc}
\tilde{\b A}_{11} & \tilde{\b A}_{12}\\
\tilde{\b A}_{21} & \tilde{\b A}_{22}
\end{array}\right]\left[\begin{array}{c} \b z\\
\b{\sigma}
\end{array}\right]+\b T^{-1}\b B_1\b u_1 + \left[\begin{array}{c}
\b O\\
\b I_{m_2} \end{array}\right]\b u_2\\
\b y &=& \left[\begin{array}{cc} \b C\b W & \b C\b B_2
\end{array}\right]\left[\begin{array}{c} \b z\\
\b{\sigma}
\end{array}\right]\\
&=&  \left[\begin{array}{cc} \tilde{\b C}_{1} & \tilde{\b C}_{2}
\end{array}\right]\left[\begin{array}{c} \b z\\
\b{\sigma}
\end{array}\right].
\end{eqnarray*}
If the trajectory of the system described by the triple $(\b A, \b B_2, \b F\b C)$
resides in the null space of $\b S$, then such a motion is described by
\[
\dot{\b z}= \tilde{\b A}_{11}\b z.
\]
It follows from~\cite[pages~328, 329]{zak03} that the poles of the above system are
the zeros of the system described by the triple $(\b A, \b B_2, \b F\b C)$, which
are the complex numbers $s$ for which the system matrix
\[
\left[\begin{array}{cc}
s\b I_n - \b A & \b B_2\\
\b F\b C & \b O
\end{array}\right]
\]
loses its full rank. On the other hand, the zeros of the triple $(\b A, \b B_2, \b
C)$ are also the zeros of the squared-down system $(\b A, \b B_2, \b F\b C)$, that
is, the zeros of $(\b A, \b B_2, \b C)$ form a subset of the set of the eigenvalues
of $\tilde{\b A}_{11}$. It is well known that zeros are invariant with respect to
similarity transformations. Therefore,
\[
\mb{rank}\left[\begin{array}{ccc}
s\b I_{n-m_2} - \tilde{\b A}_{11} & -\tilde{\b A}_{12} & \b O\\
-\tilde{\b A}_{21} & s\b I_{m_2}-\tilde{\b A}_{22}& \b I_{m_2}\\
\tilde{\b C}_{1} & \tilde{\b C}_{2} & \b O
\end{array}\right] = \mb{rank}\left[\begin{array}{c}
s\b I_{n-m_2} - \tilde{\b A}_{11}\\
\tilde{\b C}_{1}
\end{array}\right]+2m_2\quad\mb{for}\quad s\in\mb{\Bb C},
\]
which means that zeros of the system $(\b A, \b B_2, \b C)$ are in the open
left-half plane if and only if the pair $(\tilde{\b A}_{11},\tilde{\b C}_{1})$ is
detectable.\\
\hspace*{5.5in} $\Box$
%{\bf Proof}\quad See Hui and \.{Z}ak~\cite{hui04_uio}.\\
%\hspace*{5.5in} $\Box$



\section{Reduced-Order UIO}



The error dynamics of the full-order observer that we analyzed above are given by
equation~(\ref{errindiffco}):
\[
\dot{\tilde{\b e}}=  \left[\begin{array}{cc} \tilde{\b A}_{11}-\tilde{\b
L}_1\tilde{\b C}_1 & \tilde{\b A}_{12} -
\tilde{\b L}_1\tilde{\b C}_2\\
\b O & \b O
\end{array}\right]\tilde{\b e}
\]
The reader may have noticed that since we choose the initial condition for $\b q$ to
force $\tilde{\b e}_2(t)=\b 0$ for $t\geq 0$, the dynamics of the error are
completely determined by the dynamics of $\tilde{\b e}_1$, which are given by
\begin{equation}\label{eq:ro_uio_error}
\dot{\tilde{\b e}}_1=\left(\tilde{\b A}_{11}-\tilde{\b L}_1\tilde{\b
C}_1\right)\tilde{\b e}_1,
\end{equation}
an $(n-m_2)$-dimensional system. This motivates us to apply the transformation from
$\b e$ to $\tilde{\b e}$ to $\b q$:
\[ \tilde{\b q}=\b T^{-1} \b q.
\]
From equation~(\ref{eq:p_obser}),
\[
\dot{\b q}=(\b I-\b M\b C)(\b A\b q+\b A\b M\b y+\b B_1\b u_1 + \b L(\b y-\b C\b
q-\b C\b M\b y))
\]
and $\b I-\b M\b C=\b T\b \Pi\b T^{-1}$, we obtain
\begin{eqnarray}
\dot{\tilde{\b q}}&=&\b \Pi(\b T^{-1}\b A\b T\tilde{\b q}+\b T^{-1}\b A\b M\b y+\b
T^{-1}\b B_1\b u_1 + \b T^{-1}\b L(\b y-\b C\b T\tilde{\b q}-\b C\b M\b y))\nonumber\\
&=& \b \Pi\b T^{-1}\b A\b T\tilde{\b q}+ \b \Pi\b T^{-1}\b A\b M\b y + \b \Pi\b T^{-1}\b
B_1\b u_1 + \b \Pi\b T^{-1}\b L(\b y-\b C\b T\tilde{\b q} - \b C\b M\b
y)\nonumber\\
&=& \b \Pi\left(\b T^{-1}\b A\b T -\b T^{-1}\b L\b C\b T \right)\tilde{\b q} + \b
\Pi\left(\b T^{-1}\b A\b M  + \b T^{-1}\b L -\b T^{-1}\b L\b C\b M\right)\b y\nonumber\\
&{}&\quad +  \b \Pi\b T^{-1}\b B_1\b u_1.\label{red_ord_dyn}
\end{eqnarray}
Using the notation defined in~(\ref{matrices}), we have
\[
\dot{\tilde{\b q}}=\b \Pi\left(\tilde{\b A} -\tilde{\b L}\tilde{\b C} \right)\tilde{\b
q} + \b \Pi\b T^{-1}\left[\left(\b A\b M  + \b T\tilde{\b L}(\b I_p - \b C\b M)
\right)\b y + \b B_1\b u_1\right].
\]
Let
\[
\tilde{\b q}=\left[\begin{array}{c} \tilde{\b q}_1\\
\tilde{\b q}_2
\end{array}\right],
\]
where $\tilde{\b q}_1\in \mb{\Bb R}^{n-m_2}$ and $\tilde{\b q}_2\in \mb{\Bb
R}^{m_2}$. Since
\[
\b \Pi=\left[\begin{array}{cc}
\b I_{n-m_2} & \b O\\
\b O & \b O
\end{array}\right],
\]
we have $\dot{\tilde{\b q}}_2(t)=\b 0$. Therefore, setting $\tilde{\b q}_2(0)=\b 0$
ensures that $\tilde{\b q}_2(t)=\b 0$ for $t\geq 0$. We thus can remove $m_2$
observer states from the observer dynamics. Let $\tilde{\b G}=\b A\b M+\b T\tilde{\b
L}(\b I_p-\b C\b M)$. Then the resulting reduced-order observer takes the form,
\begin{eqnarray*}
\dot{\tilde{\b q}}_1&=&(\tilde{\b A}_{11}- \tilde{\b L}_{1}\tilde{\b
C}_{1})\tilde{\b q}_1 + \left[\begin{array}{cc} \b I_{n-m_2} & \b O_{m_2}
\end{array}\right]\b T^{-1}\left(\tilde{\b G}\b y
+ \b B_1\b u_1\right),\quad \tilde{\b q}_1(0)=\b 0\\
\tilde{\b x}&=&\b T\left[\begin{array}{c} \b I_{n-{m_2}}\\
\b O_{m_2\times(n-{m_2})}
\end{array}\right]\tilde{\b q}_1 + \b M\b y,
\end{eqnarray*}
where the vector $\tilde{\b x}$ is the estimate of the plant state $\b x$. In Figure~\ref{fig:reduced_order_UIO}, we show a block diagram of the above reduced-order UIO observer.
\begin{figure}
  \centering
  \includegraphics[width=5in]{zak_red_ord}
  \caption{Reduced-order UIO.}
  \label{fig:reduced_order_UIO}
\end{figure}

We now summarize the above considerations in the form of the following design
algorithm.
\begin{center}
{\large\bf Reduced-Order Unknown Input Observer Design Algorithm}
\end{center}
For a given a quadruple of matrices $(\b A, \b B,\b C)$ modeling the plant,
\begin{enumerate}
\item Check that $\mb{rank}\,\left(\b C\b B_2\right)= \mb{rank}\,\b B_2$\\
If $\mb{rank}\,\left(\b C\b B_2\right) < \mb{rank}\,\b B_2$, STOP. The observer does
not exist. \item Compute
\[
\b M=\b B_2\left( (\b C\b B_2)^{\dagger}+\b H_0\left(\b I_p - (\b C\b B_2)(\b C\b
B_2)^{\dagger}\right)  \right),
\]
where the superscript $\dagger$ denotes the Moore-Penrose pseudo-inverse operation
and $\b H_0\in \mb{\Bb R}^{m_2\times p}$ is a design parameter matrix.

\item Compute the projector
\[
\tilde{\b \Pi}=\b I_n-\b M\b C.
\]
\item Represent $\tilde{\b \Pi}$ as
\[
\tilde{\b \Pi}=\b T\b \Pi\b T^{-1},
\]
where
\[
\b \Pi=\left[\begin{array}{cc}
\b I_{n-m_2} & \b O\\
\b O & \b O
\end{array}\right].
\]
\item Compute
\[
\tilde{\b A}=\b T^{-1}\b A\b T=\left[\begin{array}{cc}
\tilde{\b A}_{11} & \tilde{\b A}_{12}\\
\tilde{\b A}_{21} & \tilde{\b A}_{22}
\end{array}\right]\;\;\mb{and}\;\;\tilde{\b C}=\b C\b T =\left[\begin{array}{cc}
\tilde{\b C}_{1} & \tilde{\b C}_{2}
\end{array}\right],
\]
where $\tilde{\b A}_{11} \in \mb{\Bb R}^{(n-m_2)\times (n-m_2)}$ and $\tilde{\b
C}_{1} \in \mb{\Bb R}^{p\times (n-m_2)}$
\item Check detectability of the pair $(\tilde{\b A}_{11}, \tilde{\b C}_{1})$\\
If the pair $(\tilde{\b A}_{11}, \tilde{\b C}_{1})$ is not detectable, STOP. The
observer does not exist.\\
Note that if the matrix $\tilde{\b A}_{11}$ is asymptotically stable, then the pair
$(\tilde{\b A}_{11}, \tilde{\b C}_{1})$ is detectable for an arbitrary matrix
$\tilde{\b C}_{1}$.



\item If there are eigenvalues of $\tilde{\b A}_{11}$ that are not asymptotically
stable, construct $\tilde{\b L}_1$ so that the matrix $(\tilde{\b A}_{11}-\tilde{\b
L}_1\tilde{\b C}_1)$ has its eigenvalues in locations as close to the desired
eigenvalues as possible. \item Form
\[
\tilde{\b L}=\left[\begin{array}{c}
\tilde{\b L}_{1}\\
\b O_{m_2\times p}
\end{array}\right],
\]
where $\b O_{m_2\times p}$ is an $m_2\times p$ zero matrix. \item Compute matrix
\[
\tilde{\b G}=\b A\b M+\b T\tilde{\b L}(\b I_p-\b C\b M).
\]
\item Construct the observer,
\begin{eqnarray*}
\dot{\tilde{\b q}}_1&=&(\tilde{\b A}_{11}- \tilde{\b L}_{1}\tilde{\b
C}_{1})\tilde{\b q}_1 + \left[\begin{array}{cc} \b I_{n-m_2} & \b O_{m_2}
\end{array}\right]\b T^{-1}\left(\tilde{\b G}\b y
+ \b B_1\b u_1\right),\quad \tilde{\b q}_1(0)=\b 0\\
\tilde{\b x}&=&\b T\left[\begin{array}{c} \b I_{n-{m_2}}\\
\b O_{m_2\times(n-{m_2})}
\end{array}\right]\tilde{\b q}_1 + \b M\b y.
\end{eqnarray*}
The vector $\tilde{\b x}$ is the estimate of the state $\b x$.
%\item\label{WZstop} STOP
\end{enumerate}
\markex



\begin{example}\label{ex:eg2}\rm We consider the same fifth-order lateral axis model of an
L-1011 fixed-wing aircraft that we considered in Example~\ref{ex:smo_L-1011}. We
first check that $\mb{rank}\,\left(\b C\b B_2\right)= \mb{rank}\,\b B_2$. We then
compute the matrix $\b M$, where in this example we set $\b H_0=\b O$,
\[
\b M=\b B_2(\b C\b B_2)^{\dagger}= \left[ \begin{array}{rrrr}
0.0000 & 0.0000 & 0.0000 & 0.0000 \\
0.9993 & 0.0000 & -0.0265 & 0.0000 \\
0.0000 & 1.0000 & 0.0008 & 0.0000 \\
-0.0265 & 0.0008 & 0.0007 & 0.0000 \\
0.0000 & 0.0000 & 0.0000 & 0.0000
\end{array} \right].
\]
Then, the projector $\tilde{\b \Pi}$ is
\[
\tilde{\b \Pi}=\b I_5-\b M\b C=\left[\begin {array}{rrrrr}
  1.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
  0.0000 & 0.0007 & -0.0000 & 0.0265 & 0.9993 \\
  0.0000 & -0.0000 & 0.0000 & -0.0008 & 0.0000 \\
  0.0000 & 0.0265 & -0.0008 & 0.9993 & -0.0265 \\
  0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.0000
\end{array}\right].
\]
We next use~(\ref{eq:Proj_transfor}) to compute $\b T$ such that
\[
\tilde{\b \Pi}=\b T\b \Pi\b T^{-1}=\b T\left[\begin{array}{cc} \b I_3 & \b O_{3\times 2}\\
\b O^{\top}_{3\times 2} & \b O_{2\times 2}
\end{array}\right]\b T^{-1}.
\]
We have
\[
\b T =\left[\begin {array}{rrrrr}
-0.7169  & -0.0190 & -0.6969  &      0  &      0\\
-0.4930  &  0.0131  &  0.5069 &  -0.7440 &  -0.0320\\
0.0000 &  -0.0008  &  0.0000  &  0.3370  & -1.1200\\
-0.0138 &   0.9996 &  -0.0131  &  0.0200    &  0\\
-0.4927 &  -0.0135  &  0.5072   &      0    &     0
\end{array}\right].
\]
Hence,
\[
\tilde{\b A}_{11}=\left[\begin {array}{rrr}
-0.0124 &  -0.0102  &  0.0145\\
0.4662 & -0.0863 &  -0.5326\\
-0.0000  &  0.0139 &  -0.0004
\end{array}\right]\mb{ and } \tilde{\b C}_{1}=
\left[\begin {array}{rrr}
-0.0004  &  0.0265 &  -0.0003\\
0.0000 &  -0.0008  &  0.0000\\
-0.0138 &  0.9996 & -0.0131\\
-0.7169 &  -0.0190 &  -0.6969
\end{array}\right].
\]
Note that the pair $(\tilde{\b A}_{11}, \tilde{\b C}_1)$ is detectable and that the
matrix $\tilde{\b A}_{11}$ is unstable with its eigenvalues at $\{0.0000,-0.0495 +
0.1041j, -0.0495 - 0.1041j\}$. We proceed to stabilize $\tilde{\b A}_{11}$. We
select the desired eigenvalues to be located at $-3, -4, -5$. The gain matrix
$\tilde{\b L}_{1}$ such that $\mb{eig}(\tilde{\b A}_{11}-\tilde{\b L}_{1}\tilde{\b
C}_{1})=\{-3, -4, -5\}$ is
\[
\tilde{\b L}_{1}=\left[\begin{array}{rrrr}
0.3866 & -0.0110 &  14.5695 &  -3.5473\\
0.2097 &  -0.0060 &  7.9007 &  -0.2190\\
-0.4028 &  0.0115 & -15.1794 &  -2.0853
\end{array}\right].
\]
We obtained the above gain matrix using MATLAB's command \verb!place!. We next form
the matrix $\tilde{\b L}$ by adding two zero rows to $\tilde{\b L}_{1}$ and compute
\[
\left[\begin{array}{cc} \b I_{3} & \b O_{3\times 2}
\end{array}\right]\b T^{-1}\tilde{\b G}=\left[\begin{array}{rrrr}
-0.0787 &  -0.7280 &  14.5813 &  -3.5473\\
-0.8007 &  -0.0248 &   7.9275 &  -0.2190\\
0.1035 &  -0.6854 & -15.1934 &  -2.0853
\end{array}\right].
\]
\begin{figure}
\centering\epsfig{file=eg2.eps,width=6.5in} \caption{Plots of $x_i$s and their
estimates versus time for Example~\ref{ex:eg2}.} \label{fig:eg2}
\end{figure}
The reduced-order UIO has the form
\begin{eqnarray*}
\dot{\tilde{\b q}}_1 &=& (\tilde{\b A}_{11}- \tilde{\b L}_{1}\tilde{\b
C}_{1})\tilde{\b q}_1 + \left[\begin{array}{cc} \b I_{3} & \b 0
\end{array}\right]\b T^{-1}\tilde{\b G}\b y\\
&=&\left[\begin{array}{rrr}
-2.3538 & -14.6523 &  -2.2668\\
0.4186 &  -7.9939 &  -0.5818\\
-1.7052 &  15.1589  & -1.6523
\end{array}\right]\tilde{\b q}_1 + \left[\begin{array}{rrrr}
-0.0787 &  -0.7280 &  14.5813 &  -3.5473\\
-0.8007 &  -0.0248 &   7.9275 &  -0.2190\\
0.1035 &  -0.6854 & -15.1934 &  -2.0853
\end{array}\right]\b y\\
\tilde{\b x} &=& \b T\left[\begin{array}{c} \b I_{3}\\
\b O_{2\times{3}}
\end{array}\right]\tilde{\b q}_1 + \b M\b y\\
&=& \left[\begin{array}{rrr}
-0.7169  & -0.0190 & -0.6969\\
-0.4930  &  0.0131  &  0.5069\\
0.0000 &  -0.0008  &  0.0000\\
-0.0138 &   0.9996 &  -0.0131\\
-0.4927 &  -0.0135  &  0.5072
\end{array}\right]\tilde{\b q}_1   + \left [\begin {array}{rrrr}
0.0000 & 0.0000 & 0.0000 & 0.0000 \\
0.9993 & 0.0000 & -0.0265 & 0.0000 \\
0.0000 & 1.0000 & 0.0008 & 0.0000 \\
-0.0265 & 0.0008 & 0.0007 & 0.0000 \\
0.0000 & 0.0000 & 0.0000 & 0.0000
\end{array}\right]\b y.
\end{eqnarray*}
In Figure~\ref{fig:eg2}, we show plots of the system state variables and their
estimates versus time. The initial conditions of the plant were selected randomly to
be equal to
\[
\b x(0)=\left[\begin{array}{ccccc}
 0.0342 & 0.3200 & 0.0178 & -0.0287 & -0.2497
\end{array}\right]^{\top}.
\]
The initial conditions of the observer were set to zero.
\end{example}
\markex







\section{Relation With Other UIO Architectures}
In this Chapter, we concentrated on the analysis and the design of full-order
observers that can be used to construct reduced-order observers. Our analysis can be
extended to cover the case
\begin{equation}
\dot{\b q}=(\b I-\b M\b C)(\b A\b q+\b A\b M\b y+\b B_1\b u_1) + \b L(\b y-\b C\b
q-\b C\b M\b y), \label{eq:full_order_out}
\end{equation}
where the term $\b L(\b y-\b C\b q-\b C\b M\b y)$ is not premultiplied by $(\b I-\b
M\b C)$. However, this case leads to the observer analyzed in \cite{chen96, chen99}
even though the approach there is quite different. Indeed, we can equivalently
represent the dynamics of the proposed full-order observer as follows:
\begin{eqnarray*}
\dot{\b q}&=&\left(\left(\b I-\b M\b C\right)\b A - \b L\b C\right)\b q +
\left(\left[\left(\b I-\b M\b C\right)\b A -\b L\b C\right]\b M + \b L \right)\b
y+\left(\b I-\b M\b C\right)\b B_1\b u_1\\
&=& (\b T\b A -\b L\b C)\b q + \b K\b y + \b T\b B_1\b u_1,\\
\tilde{\b x} &=& \b q + \b M\b y,
\end{eqnarray*}
where, using the notation similar to that in~\cite{chen96, chen99},
\begin{eqnarray*}
\b T &=& \b I-\b M\b C,\quad \b K_1 = \b L,\\
\b K_2 &=& \left[\b T\b A -\b L\b C\right]\b M, \quad \b K =\b K_1+\b K_2.
\end{eqnarray*}
In addition, the conditions for the existence of  the full-order observer presented
in~\cite{chen96, chen99} and our observers are equivalent.

The observer given by~(\ref{eq:full_order_out}) is also the same as the one proposed
by Yang and Wilde~\cite{yang88} and further analyzed by Darouach, Zasadzinski, and
Xu~\cite{darouach94}. The connections are as follows: (i) $\b M$ is called $-\b E$
in~\cite{yang88,darouach94}, (ii) $(\b I-\b M\b C)$ corresponds to $\b P$ there,
(iii) $\b B_1$ is $\b B$ and $\b B_2$ is $\b D$ in~\cite{yang88,darouach94}, (iv)
$(\b I-\b M\b C)(\b A-\b L\b C)$ corresponds to $\b N$.

We now compare the reduced-order UIO proposed by Hou and M\"{u}ller~\cite{hou92}
with our reduced-order UIO. Somewhat similar approach is proposed by Kudva et
al.~\cite{kudva80}. Hou and M\"{u}ller first transform the
system~(\ref{eq:two_inputs}) into the form
\begin{eqnarray*}
\dot{\b x}&=&\b A\b x+\b B_1\b u_1+\left[\begin{array}{c} \b O\\
\b I_{m_2}
\end{array}\right]\b u_2\\
&=&\left[\begin{array}{cc} \b A_{11} & \b A_{12}\\
\b A_{21} & \b A_{22}
\end{array}\right]\left[\begin{array}{c}
\b x_1\\
\b x_2
\end{array}\right]+\left[\begin{array}{c}
\b B_{11}\\
\b B_{12}
\end{array}\right]\b u_1\\
&{}&\quad +\left[\begin{array}{c} \b O\\
\b I_{m_2}
\end{array}\right]\b u_2.
\end{eqnarray*}
Note that $\b x_1$ in the new coordinates is independent of $\b u_2$ and we have
\begin{equation}
\dot{\b x}_1 = \b A_{11}\b x_1+\b A_{12}\b x_2 + \b B_{11}\b u_1. \label{eq:onlyx1}
\end{equation}
Let
\[
\b y=\b C\b x=\left[\begin{array}{cc} \b C_1 & \b C_2
\end{array}\right]\left[\begin{array}{c}
\b x_1\\
\b x_2 \end{array}\right],
\]
where $\b C_2\in \mb{\Bb R}^{p\times m_2}$. Because by assumption, $\mb{rank} (\b
C\b B_2)=\mb{rank} \b B_2 = m$, the submatrix $\b C_2$ has a left inverse, $\b
C_2^{\dag}$. Hence we can compute
\begin{equation}
\b x_2=-\b C_2^{\dag}\b C_1\b x_1 + \b C_2^{\dag}\b y. \label{eq:x2usingx1andy}
\end{equation}
Substituting the above into~(\ref{eq:onlyx1}) gives
\[
\dot{\b x}_1 = \left(\b A_{11}-\b A_{12}\b C_2^{\dag}\b C_1\right)\b x_1 + \b
A_{12}\b C_2^{\dag}\b y +  \b B_{11}\b u_1.
\]
Hou and M\"{u}ller~\cite{hou92} propose now to construct an observer for $\b x_1$
using only known signals and then substitute the estimate of $\b x_1$
into~(\ref{eq:x2usingx1andy}) to obtain an estimate of $\b x_2$. Thus, the resulting
architecture of the reduced-order UIO proposed by Hou and M\"{u}ller~\cite{hou92},
as well as their approach, differs from our design. Yet another approach to
constructing reduced-order UIOs can be found in~\cite{hui_uio93}.

\section{Sliding Mode Observer (SMO)}\label{sec:uncertain}

In this approach, we assume that $\b u_2$ is bounded, that is, there exists a
nonnegative real number, $\rho$, such that
\[
\|\b u_2\| \le \rho.
\]
Let $\hat{\b x}$ be an estimate of $\b x$. Let $\b e$ denote the estimation error,
that is,
\[
\b e(t)=\b x(t) - \hat{\b x}(t).
\]
The observability of $\left( \b A,\b C \right)$ implies the existence of a matrix
$\b L \in \mb{\Bb R}^{n \times p}$ such that the matrix $\left(\b A-\b L\b C\right)$
has prescribed (symmetric with respect to the real axis) eigenvalues in the open
left-half plane. Because $\left(\b A-\b L\b C\right)$ is asymptotically stable, for
any $\b Q=\b Q^{\top}>0$, there is a unique $\b P=\b P^{\top}>0$ such that
\begin{equation}
\left(\b A - \b L\b C \right)^{\top}\b P+\b P\left(\b A - \b L\b C \right)=-\b Q.
\label{eq:lyap_esti}
\end{equation}
We choose $\b Q$, if possible, so that for some $\b F \in \mb{\Bb R}^{m_2 \times
p}$,
\begin{equation}
\b F\b C=\b B_2^{\top}\b P. \label{eq:tech_cond}
\end{equation}
We need this technical condition to ensure the realizablity of the observer.

To proceed, we define the vector function
\[
\b E\left(\b e, \eta \right)=\left\{\begin{array}{lcc}
\eta\frac{\b F\b C\b e}{\|\b F\b C\b e\|_2} & \mb{for} & \b F\b C\b e\ne \b 0\\
\b 0  & \mb{for} & \b F\b C\b e=\b 0,
\end{array}\right.
\]
where $\eta \ge \rho$ is a design parameter. In the case of single-input
single-output plant, we can write
\[
\b E(\b e,\eta)=\eta \,\mb{sign}\left(\b F\b C\b e\right).
\]
We note that
\[
\b C\b e=\b C(\b x - \hat{\b x})=\b y - \hat{\b y}.
\]
The vector function $\b E(\b e,\eta)$ is an essential ingredient of the sliding mode
observer that we present next. When implementing the function $\b E$, we use the
output measurements $\hat{\b y}$ and $\b y$, that is, instead of using $E(\b
e,\eta)$, we utilize
\[
\b E(\b y, \hat{\b y}, \eta) = \left\{\begin{array}{lcc}
\eta\frac{\b F(\b y - \hat{\b y} )}{\|\b F(\b y - \hat{\b y})\|_2} & \mb{for} & \b F( \b y - \hat{\b y})\ne \b 0\\
\b 0  & \mb{for} & \b F(\b y - \hat{\b y})=\b 0.
\end{array}\right.
\]
Hence for the case of single-input single-output plant, we have
\[
\b E(\hat{y}, \b e, y)=\eta \,\mb{sign}\left(F(y - \hat{y})\right).
\]

Using arguments similar to those found in~\cite{walcott87}, we can show that the
state $\hat{\b x}$ of the dynamical system
\begin{equation}
\fbox{\displaymath$ \dot{\hat{\b x}}=\b A \hat{\b x}+\b B_1\b u_1 + \b L( \b y - \hat{\b y}) + \b B_2\b E (\b y,\hat{\b y}, \eta)$} \label{eq:sm_observer}
\end{equation}
for $\eta \ge \rho$, is an asymptotic estimate of the state $\b x$ of the system
described by~(\ref{eq:uncert_sys}) and (\ref{eq_lin_system}), that is,
\[
\lim_{t \to \infty}\b e(t)=\lim_{t \to \infty} \left(\b x(t) - \hat{\b x}(t)\right)=\b 0.
\]
In Figure~\ref{fig:wzsm_obs}, we show a block diagram of the sliding mode observer given by~(\ref{eq:sm_observer}).
\begin{figure}
  \centering
  \includegraphics[width=3.5in]{zak_11_20091}
  \caption{The sliding mode observer~(\ref{eq:sm_observer}) as an unknown input estimator.}
  \label{fig:wzsm_obs}
\end{figure}
To prove the above statement using Lyapunov's type of arguments, first
represent~(\ref{eq:sm_observer}) as
\[
\dot{\hat{\b x}} = \left(\b A-\b L\b C \right)\hat{\b x} + \b L\b y+\b B_1\b u_1 + \b
B_2\b E (\b e,\eta).
\]
Then construct the differential equation describing the dynamics of the estimation
error $\b e$,
\begin{equation}
\dot{\b e} = \dot{\b x} - \dot{\hat{\b x}}= \left(\b A-\b L\b C \right)\b e +\b B_2\b
u_2-\b B_2\b E(\b e,\eta). \label{eq:errodiffeqn}
\end{equation}
Then, show that
\[
\frac{d}{dt}\left(\b e^{\top}\b P\b e \right)=-\b e^{\top}\b Q\b e<0,
\]
which implies
\[
\lim_{t \to \infty}\b e(t)=\b 0.
\]
It follows from the above that the estimation error is insensitive to the
uncertainty modeled by the term $\b B_2\b u_2$. In summary, the design of the
observer proposed by Walcott and \.{Z}ak~\cite{walcott87} for a system modeled by
the quadruple $(\b A,\b B_1, \b B_2, \b C)$ can be thought of as finding a pair of
matrices $(\b P, \b F)$ satisfying~(\ref{eq:lyap_esti}) and~(\ref{eq:tech_cond}) for
some $\b L$ and $\b Q$. Edwards and Spurgeon~\cite{edwards98} (see also Saif and
Xiong~\cite{saif03}) present necessary and sufficient conditions for the existence
of the above observer, which are:
\begin{itemize}
\item[{\bf (i)}] $\mb{rank}\, \b B_2=\mb{rank}\,\b C\b B_2=r$; \item[{\bf (ii)}] the
system zeros of the triple $(\b A,\b B_2, \b C)$ are in the open left-hand complex
plane, that is,
\[
\mb{rank}\left[\begin{array}{cc}
s\b I_n - \b A & \b B_2\\
\b C & \b O
\end{array}\right]=n+r\;\;\mb{for all}\;\; s\;\;\mb{such that}\;\; \mb{Re}(s)
\ge 0.
\]
\end{itemize}

It is interesting to note that the above conditions are also necessary and
sufficient for the existence of the observers with unknown inputs of Hui and
\.{Z}ak~\cite{hui_uio93} as well as the unknown input observers (UIOs) analyzed by
us in the previous sections.

\section{SMO Construction}
We first present a lemma that will serve us as a platform for the design of the
sliding mode observer for uncertain systems. The lemma is a minor modification of
Lemma~1 of Corless and Tu~\cite{corless98} who proved it constructively using a
singular value decomposition approach. We offer a different constructive proof using
the Q-R decomposition.
\begin{lemma}\label{lem_transform} For a triple $(\b A, \b B_2, \b C)
 \in \mb{\Bb R}^{n\times n}\times
\mb{\Bb R}^{n\times m_2}\times \mb{\Bb R}^{p\times n}$,
\begin{equation}
\mb{rank}\, \b B_2 = \mb{rank}\, (\b C\b B_2) = r, \label{eq:distribute_con}
\end{equation}
if and only if there exist nonsingular matrices $\b T$ and $\b S$ such that
\begin{equation}\label{eq:transAGC}
\b T\b A\b T^{-1}=\left[\begin{array}{cc} \b A_{11} & \b A_{12}\\
\b A_{21} & \b A_{22}
\end{array}\right],\quad \b T\b B_2=\left[\begin{array}{c}
\b B_{2_1}\\
\b O
\end{array}\right], \quad \b S\b C\b T^{-1}=\left[\begin{array}{cc} \b I_r & \b O\\
\b O & \b C_{22}
\end{array}\right],
\end{equation}
where $\b A_{11}\in \mb{\Bb R}^{r\times r}$, $\b A_{22}\in \mb{\Bb R}^{(n-r)\times
(n-r)}$, $\b B_{2_1} \in \mb{\Bb R}^{r\times m_2}$, $\mb{rank}\, \b B_{2_1} = r$,
and $\b C_{22}\in \mb{\Bb R}^{(p-r)\times (n-r)}$.
\end{lemma}
{\bf Proof}\quad ($\Longrightarrow$) The proof is constructive. Using the Q-R
decomposition applied to $\b G$, we obtain
\[
\b B_2=\b Q_{B_2}\b R_{B_2},
\]
where $\b Q_{B_2}\in \mb{\Bb R}^{n\times n}$ is an unitary matrix and the matrix $\b
R_{B_2} \in \mb{\Bb R}^{n\times m_2}$ is upper triangular where
\[
\mb{rank}\, \b R_{B_2} =  r.
\]
Let $\b T_1=\b Q_{B_2}^{-1}$. Then, we obtain
\[
\b T_1\b B_2=\left[\begin{array}{c} \tilde{\b B}_{2_1}\\
\b O
\end{array}\right],
\]
where $\tilde{\b G}_1 \in \mb{\Bb R}^{r\times m_2}$. We next partition the matrix
$\b C\b T_1^{-1}$ as follows,
\[
\b C\b T_1^{-1}=\left[\begin{array}{cc} \tilde{\b C}_1 & \tilde{\b C}_2
\end{array}\right],
\]
where $\tilde{\b C}_1 \in \mb{\Bb R}^{p\times r}$. Note that \[ \b C\b B_2 =
\left(\b C\b T_1^{-1} \right)\left(\b T_1\b B_2 \right)=\tilde{\b C}_1\tilde{\b
B}_{2_1}.
\]
By the lemma hypothesis, $\mb{rank}\, \b B_2 = \mb{rank}\, (\b C\b B_2) = r$. Hence
\[
\mb{rank}\, \tilde{\b C}_1=  r.
\]
Applying the Q-R decomposition to $\tilde{\b C}_1$ yields
\[
\tilde{\b C}_1=\b Q_{\tilde{C}_1}\b R_{\tilde{C}_1},
\]
where
\[
\b R_{\tilde{C}_1}=\left[\begin{array}{c} \b C_{11}\\
\b O
\end{array}\right]\quad\mb{and}\quad \det \b C_{11} \ne 0.
\]
Note that $\b C_{11} \in \mb{\Bb R}^{r\times r}$. Let $\b S=\b
Q_{\tilde{C}_1}^{-1}$, then,
\[
\b S\b C\b T_1^{-1}=\left[\begin{array}{cc}
\b C_{11} & \b C_{12}\\
\b O & \b C_{22}
\end{array}\right].
\]
Postmultiplying $\b S\b C\b T_1^{-1}$ by
\[
\b T_2^{-1}=\left[\begin{array}{cc}
\b C_{11}^{-1} & -\b C_{11}^{-1}\b C_{12}\\
\b O & \b I_{n-r}
\end{array}\right]
\]
gives
\[
\b S\b C\b T_1^{-1}\b T_2^{-1}=\left[\begin{array}{cc}
\b C_{11} & \b C_{12}\\
\b O & \b C_{22}
\end{array}\right]\left[\begin{array}{cc}
\b C_{11}^{-1} & -\b C_{11}^{-1}\b C_{12}\\
\b O & \b I_{n-r}
\end{array}\right]=\left[\begin{array}{cc}
\b I_r & \b O\\
\b O & \b C_{22}
\end{array}\right].
\]
We then have $\b T=\b T_2\b T_1$.

($\Longleftarrow$) By inspection.\\
\hspace*{6in}$\Box$

We will now show that the systems
\[
\left.\begin{array}{lll}
\dot{\b x} &=& \b A\b x + \b B_2\b u_2\\
\b y &=& \b C\b x
\end{array}\right\}
\]
and
\[
\left.\begin{array}{lll}
\dot{\tilde{\b x}} &=& \b T\b A\b T^{-1}\tilde{\b x} + \b T\b B_2\b u_2\\
\tilde{\b y} &=& \b S\b C\b T^{-1}\b x
\end{array}\right\}
\]
have the same system zeros, that is, their system matrices have the same rank for
all $s\in \mb{\Bb C}$, where $\mb{\Bb C}$ is the set of complex numbers. To prove
the above statement, we note the following relation between the system matrices of
the above models:
\begin{equation}
\left[\begin{array}{cc}
s\b I_n - \b T\b A\b T^{-1} & \b T\b B_2\\
\b S\b C\b T^{-1} & \b O
\end{array}\right]=\left[\begin{array}{cc}
\b T & \b O\\
\b O & \b S
\end{array}\right]\left[\begin{array}{cc}
s\b I_n - \b A & \b B_2\\
\b C & \b O
\end{array}\right]\left[\begin{array}{cc}
\b T^{-1} & \b O\\
\b O & \b I_q
\end{array}\right].
\label{eq:invariant_z}
\end{equation}
Suppose now that we have two matrices, $\b X\in \mb{\Bb R}^{r\times s}$ and $\b Y\in
\mb{\Bb R}^{s\times t}$. Then, we have the following bounds on the rank of the
product of these matrices:
\[
\mb{rank}\,\b X + \mb{rank}\, \b Y - s \le \mb{rank}\, (\b X\b Y) \le \min (
\mb{rank}\, \b X, \mb{rank}\, \b Y ).
\]
The above are referred to as the Sylvester's inequalities---see, for example,
Gantmacher~\cite[pages 65, 66]{gantmacher59v1} for their proof. Applying the
Sylvester's inequalities to~(\ref{eq:invariant_z}), we conclude that
\begin{equation}
\mb{rank}\,\left[\begin{array}{cc}
s\b I_n - \b T\b A\b T^{-1} & \b T\b B_2\\
\b S\b C\b T^{-1} & \b O
\end{array}\right]=\mb{rank}\,\left[\begin{array}{cc}
s\b I_n - \b A & \b B_2\\
\b C & \b O
\end{array}\right]\;\;\mb{for all}\;\; s\in \mb{\Bb C}.
\label{eq:same_rank}
\end{equation}

\begin{lemma}\label{lem_detect}
Assume that $\mb{rank}\,\b B_2 = \mb{rank}\,(\b C\b B_2)=r$. Then, the pair $(\b
A_{22}, \b C_{22})$ is detectable if and only if
\begin{equation}
\mb{rank}\left[\begin{array}{cc}
s\b I_n - \b A & \b B_2\\
\b C & \b O
\end{array}\right]=n+r\;\;\mb{for all}\;\; s\;\;\mb{such that}\;\; \mb{Re}(s)
\ge 0. \label{eq:rank_condition}
\end{equation}
\end{lemma}
{\bf Proof}\quad By assumption, $\mb{rank}\,\b B_2 = \mb{rank}\,(\b C\b B_2)=r$. By
Lemma~\ref{lem_transform}, the above condition is equivalent to the existence of
nonsingular matrices $\b T$ and $\b S$ such that
\[
\b T\b A\b T^{-1}=\left[\begin{array}{cc} \b A_{11} & \b A_{12}\\
\b A_{21} & \b A_{22}
\end{array}\right],\quad \b T\b B_2=\left[\begin{array}{c}
\b B_{2_1}\\
\b O
\end{array}\right], \quad \b S\b C\b T^{-1}=\left[\begin{array}{cc} \b I_r & \b O\\
\b O & \b C_{22}
\end{array}\right],
\]
where $\b B_{2_1} \in \mb{\Bb R}^{r\times m_2}$ and $\mb{rank}\, \b B_{2_1} = r$.
Then, by~(\ref{eq:same_rank}) for any $s\in \mb{\Bb C}$,
\begin{eqnarray*}
\mb{rank}\left[\begin{array}{ccc}
s\b I_r - \b A_{11} &-\b A_{12} & \b B_{2_1}\\
-\b A_{21} & s\b I_{n-r} - \b A_{22} & \b O\\
\b I_r & \b O & \b O\\
\b O &\b C_{22} & \b O
\end{array}\right] &=& \mb{rank}\left[\begin{array}{cc}
-\b A_{12} & \b B_{2_1}\\
s\b I_{n-r} - \b A_{22} & \b O\\
\b C_{22} & \b O
\end{array}\right] + r\\
&=& \mb{rank}\left[\begin{array}{c}
s\b I_{n-r} - \b A_{22}\\
\b C_{22}
\end{array}\right]+2r.
\end{eqnarray*}
It follows from the above that the pair $(\b A_{22},\b C_{22})$ is detectable if and
only if the rank condition~(\ref{eq:rank_condition}) holds.\\
\hspace*{6in}$\Box$

Note that if $m_2=p=r$, then $\mb{rank}\left[\begin{array}{cc}
s\b I_n - \b A & \b B_2\\
\b C & \b O
\end{array}\right]=n+r$ for all $s$ such that $\mb{Re}(s)
\ge 0$ if and only if the matrix $\b A_{22}$ is asymptotically stable.


To proceed, we need the following technical lemma.
\begin{lemma}\label{lemma:same_kers}
For a given matrix $\b V \in \mb{\Bb R}^{n\times m}$,
\[
\ker(\b V)=\ker(\b V^{\top}\b V).
\]
\end{lemma}
{\bf Proof}\quad If $\b x\in \ker(\b V)$, that is, if $\b x$ is such that $\b V\b
x=\b 0$, then obviously $\b V^{\top}\b V\b x=\b 0$, that is, $\b x\in \ker(\b V^{\top}\b V)$.
Thus, we showed that if $\b x\in \ker(\b V)$ then $\b x\in \ker(\b V^{\top}\b V)$.

Suppose now that $\b x\in \ker(\b V^{\top}\b V)$, that is, $\b V^{\top}\b V\b x=\b 0$. Then,
$\b y= \b V\b x \in \ker(\b V^{\top})$. Thus, $\b y\in \mb{range}\,(\b V)\cap\ker(\b
V^{\top})$. However, it is well know from linear algebra that
\[
\mb{range}\,(\b V)\cap\ker(\b V^{\top})=\{\b 0\},
\]
that is, $\b y=\b V\b x=\b 0$ which means that $\b x\in \ker(\b V)$. Thus, we showed
that if $\b x\in \ker(\b V^{\top}\b V)$ then $\b x\in \ker(\b V)$.

In summary, $\ker(\b V)=\ker(\b V^{\top}\b V)$.\\
\hspace*{6in}$\Box$

The following theorem appears in Corless and Tu~\cite[Lemma 3, page 760]{corless98}.
A related result was obtained by Edwards and Spurgeon~\cite[Proposition 6.2, page
138]{edwards98}. An algorithm for constructing matrices $\b L$, $\b F$ and $\b P$
that are essential ingredients of the sliding mode observer for the uncertain
systems is contained in the proof of the theorem.
\begin{thm}
There exists a triple of matrices $(\b L, \b F, \b P) \in \mb{\Bb R}^{n\times
p}\times \mb{\Bb R}^{m_2\times p}\times \mb{\Bb R}^{n\times n}$ such that
\begin{equation}
(\b A-\b L\b C)^{\top}\b P + \b P(\b A-\b L\b C) < 0 \label{eq:pexistence}
\end{equation}
and
\begin{equation}
\b F\b C=\b B_2^{\top}\b P \label{eq:tech_condit}
\end{equation}
if and only if
\begin{itemize}
\item[{\bf (i)}] $\mb{rank}\, \b B_2=\mb{rank}\,(\b C\b B_2)=r$; \item[{\bf (ii)}]
the system zeros of the triple $(\b A,\b B_2, \b C)$ are in the open left-hand
complex plane, that is,
\[
\mb{rank}\left[\begin{array}{cc}
s\b I_n - \b A & \b B_2\\
\b C & \b O
\end{array}\right]=n+r\;\;\mb{for all}\;\; s\;\;\mb{such that}\;\; \mb{Re}(s)
\ge 0.
\]
\end{itemize}
\end{thm}
{\bf Proof}\quad $(\Longleftarrow)$ We follow the arguments of Corless and
Tu~\cite{corless98}. By Lemma~\ref{lem_transform},  the condition, $\mb{rank}\,\b
B_2 = \mb{rank}\,(\b C\b B_2)=r$, is equivalent to the existence of nonsingular
matrices $\b T$ and $\b S$ such that
\[
\hat{\b A}=\b T\b A\b T^{-1}=\left[\begin{array}{cc} \b A_{11} & \b A_{12}\\
\b A_{21} & \b A_{22}
\end{array}\right],\hat{\b B}_2=\b T\b {\b B}_2=\left[\begin{array}{c}
\b B_{2_1}\\
\b O
\end{array}\right], \quad \hat{\b C}=\b S\b C\b T^{-1}=\left[\begin{array}{cc} \b I_r & \b O\\
\b O & \b C_{22}
\end{array}\right],
\]
where $\b B_{2_1} \in \mb{\Bb R}^{r\times m_2}$ and $\mb{rank}\, \b B_{2_1} = r$.
Let
\begin{equation}\label{eq:PLFinnewcor}
\hat{\b P}=\b T^{-{\top}}\b P\b T^{-1},\;\;  \hat{\b L}=\b T\b L\b
S^{-1},\;\;\mb{and}\;\;\hat{\b F}=\b F\b S^{-1}.
\end{equation}
To proceed, note that condition {\bf (ii)} is equivalent to the existence of a
matrix $\b L_{22}$ such that the eigenvalues of $(\b A_{22} - \b L_{22}\b C_{22})$
are all in the open left-half complex plane. Then, for any symmetric positive
positive definite $\b Q_{22}$, the symmetric solution $\b P_{22}$ to the Lyapunov
matrix equation,
\[
(\b A_{22} - \b L_{22}\b C_{22})^{\top}\b P_{22}+\b P_{22}(\b A_{22} - \b L_{22}\b
C_{22})=-\b Q_{22},
\]
is also positive definite. Let
\[
\hat{\b L}=\left[\begin{array}{cc}
\kappa\b I_r & \b O\\
\b O & \b L_{22}
\end{array}\right],
\]
where $\kappa >0$ is a design parameter whose lower bound is determined in the
following considerations. We have
\begin{eqnarray*}
\hat{\b A}-\hat{\b L}\hat{\b C} &=& \left[\begin{array}{cc} \b A_{11} & \b A_{12}\\
\b A_{21} & \b A_{22}
\end{array}\right]-\left[\begin{array}{cc}
\kappa\b I_r & \b O\\
\b O & \b L_{22}
\end{array}\right]\left[\begin{array}{cc}
\b I_r & \b O\\
\b O & \b C_{22}
\end{array}\right]\\
&=& \left[\begin{array}{cc} \b A_{11}-\kappa\b I_r & \b A_{12}\\
\b A_{21} & \b A_{22}-\b L_{22}\b C_{22}
\end{array}\right].
\end{eqnarray*}
Let
\[
\hat{\b P}=\left[\begin{array}{cc} \b I_r & \b O\\
\b O &\b P_{22}
\end{array}\right]\quad\mb{and}\quad\hat{\b Q}=\left[\begin{array}{cc} \b Q_{11} & \b Q_{12}\\
\b Q_{12}^{\top} & \b Q_{22}
\end{array}\right].
\]
Using the above, we obtain
\begin{eqnarray}
-\hat{\b Q} &=& (\hat{\b A}-\hat{\b L}\hat{\b C})^{\top}\hat{\b P} + \hat{\b P}(\hat{\b
A}-\hat{\b L}\hat{\b C})\nonumber\\
&=& \left[\begin{array}{cc} \b A_{11}^{\top}-\kappa\b I_r & \b A_{21}^{\top}\\
\b A_{12}^{\top} & (\b A_{22}-\b L_{22}\b C_{22})^{\top}
\end{array}\right]\left[\begin{array}{cc} \b I_r & \b O\\
\b O &\b P_{22}
\end{array}\right]\nonumber\\
&{}&\quad\quad+\left[\begin{array}{cc} \b I_r & \b O\\
\b O &\b P_{22}
\end{array}\right]\left[\begin{array}{cc} \b A_{11}-\kappa\b I_r & \b A_{12}\\
\b A_{21} & \b A_{22}-\b L_{22}\b C_{22}
\end{array}\right]\nonumber\\
&=& \left[\begin{array}{cc} \b A_{11}^{\top}-\kappa\b I_r & \b A_{21}^{\top}\b P_{22}\\
\b A_{12}^{\top} & (\b A_{22}-\b L_{22}\b C_{22})^{\top}\b P_{22}
\end{array}\right]+\left[\begin{array}{cc} \b A_{11}-\kappa\b I_r & \b A_{12}\\
\b P_{22}\b A_{21} & \b P_{22}(\b A_{22}-\b L_{22}\b C_{22})
\end{array}\right]\nonumber\\
&=& \left[\begin{array}{cc} \b A_{11}^{\top}+\b A_{11}-2\kappa\b I_r & \b A_{21}^{\top}\b
P_{22}
+ \b A_{12}\\
\b A_{12}^{\top} + \b P_{22}\b A_{21} & (\b A_{22}-\b L_{22}\b C_{22})^{\top}\b P_{22}+\b
P_{22}(\b A_{22}-\b L_{22}\b C_{22})
\end{array}\right]\nonumber\\
&=&\left[\begin{array}{cc} -\b Q_{11} & -\b Q_{12}\\
-\b Q_{12}^{\top} & -\b Q_{22} \end{array}\right], \label{eq:exp_forqhat}
\end{eqnarray}
where $\b Q_{22}=-\left((\b A_{22}-\b L_{22}\b C_{22})^{\top}\b P_{22}+\b P_{22}(\b
A_{22}-\b L_{22}\b C_{22}) \right)$ is positive definite by construction. Our goal
is to obtain a lower bound on the parameter $\kappa$ that would yield a positive
definite $\hat{\b Q}$. Using the Schur complement of the positive definite $\b
Q_{22}$, we have that $\hat{\b Q}$ is positive definite if and only if
\[
\b Q_{11} > \b Q_{12}\b Q_{22}^{-1}\b Q_{12}^{\top}.
\]
Employing the above in~(\ref{eq:exp_forqhat}), we obtain
\[
2\kappa\b I_r -\left(\b A_{11}^{\top} + \b A_{11}\right) > \left(\b A_{21}^{\top}\b P_{22} +
\b A_{12}\right)\b Q^{-1}_{22}\left( \b A_{12}^{\top} + \b P_{22}\b A_{21}\right).
\]
Hence, $\hat{\b Q}$ is positive definite if
\[
\kappa >\frac{1}{2}\lambda_{\max}\left(\b A_{11}^{\top}+\b A_{11} + \left(\b A_{21}^{\top}\b
P_{22} + \b A_{12}\right)\b Q^{-1}_{22}\left( \b A_{12}^{\top} + \b P_{22}\b
A_{21}\right) \right).
\]
Let
\[
\hat{\b F}=\left[\begin{array}{cc} \b B_{2_1}^{\top} & \b O
\end{array}\right].
\]
Then, it is easy to see that
\begin{equation}\label{eq:layaptrans}
(\hat{\b A}-\hat{\b L}\hat{\b C})^{\top}\hat{\b P} + \hat{\b P}(\hat{\b A}-\hat{\b
L}\hat{\b C}) < 0,
\end{equation}
and
\begin{equation}\label{eq:techtransfor}
\hat{\b F}\hat{\b C} = \hat{\b B}_2^{\top}\hat{\b P},
\end{equation}
which are conditions (\ref{eq:pexistence}) and ({\ref{eq:tech_condit}) in the new
basis. Hence, the proof of the sufficiency conditions for the existence of the
desired triple of matrices $(\b L, \b F, \b P)$ is complete.

\noindent$(\Longrightarrow)$ We assume that there exists a triple of matrices $(\b
L, \b F, \b P)$ such that the conditions
\[
(\b A-\b L\b C)^{\top}\b P + \b P(\b A-\b L\b C) < 0\quad\mb{and}\quad \b F\b C = \b
B_2^{\top}\b P
\]
are satisfied. Because $\b P$ is symmetric positive definite, it can be represented
as
\[
\b P=\b P^{\frac{1}{2}}\b P^{\frac{1}{2}},
\]
where $\b P^{\frac{1}{2}}\in \mb{\Bb R}^{n \times n}$ is also symmetric positive
definite and hence nonsingular. We refer to $\b P^{\frac{1}{2}}$ as a square root of
$\b P$---see, for example, Luenberger~\cite[page 459]{luenberger84}. We then have
\[
\b B_2^{\top}\b P\b B_2=\b B_2^{\top}\b P^{\frac{1}{2}}\b P^{\frac{1}{2}}\b B_2=\left(\b
P^{\frac{1}{2}}\b B_2 \right)^{\top}\left( \b P^{\frac{1}{2}}\b B_2\right).
\]
By Lemma~\ref{lemma:same_kers} and the fact that $\b P^{\frac{1}{2}}$ is
nonsingular,
\[
\ker\left(\b B_2^{\top}\b P\b B_2\right)=\ker\left(\b P^{\frac{1}{2}}\b B_2
\right)=\ker(\b B_2).
\]
Therefore,
\[
\mb{rank}\, \left(\b B_2^{\top}\b P\b B_2\right)=\mb{rank}\,\b B_2.
\]
Using the above, the condition, $\b B_2^{\top}\b P=\b F\b C$, and the Sylvester's
inequality, we obtain
\[
\mb{rank}\,\b B_2=\mb{rank}\,\left(\b B_2^{\top}\b P\b B_2\right)=\mb{rank}\,\left( \b
F\b C\b B_2\right)\le \mb{rank}\,\left(\b C\b B_2\right)\le \mb{rank}\,\b B_2.
\]
Hence,
\[
\mb{rank}\,\left(\b C\b B_2\right)= \mb{rank}\,\b B_2.
\]

Our goal now is to show that conditions~(\ref{eq:pexistence})
and~(\ref{eq:tech_condit}) imply that the system zeros of the triple $(\b A,\b B_2,
\b C)$ are in the open left-hand complex plane. Because $\mb{rank}\,\left(\b C\b
B_2\right)= \mb{rank}\,\b B_2$, by Lemma~\ref{lem_transform} that we can construct
transformation matrices $\b T$ and $\b S$ so that the transformed matrices $\b A$,
$\b B_2$, and $\b C$ take the form given by~(\ref{eq:transAGC}). By
Lemma~\ref{lem_detect}, the system zeros of the triple $(\b A,\b B_2, \b C)$ are in
the open left-hand complex plane if and only the pair $(\b A_{22}, \b C_{22})$ is
detectable. Let matrices $\hat{\b P}$, $\hat{\b L}$, and $\hat{\b F}$ be defined
by~(\ref{eq:PLFinnewcor}), where
\[
\hat{\b P}=\left[\begin{array}{cc}
\b P_{11} & \b P_{12}\\
\b P_{21} & \b P_{22}
\end{array}\right],\quad \hat{\b L}=\left[\begin{array}{cc}
\b L_{11} & \b L_{12}\\
\b L_{21} & \b L_{22}
\end{array}\right],\quad\mb{and}\quad \hat{\b F}=\left[\begin{array}{cc}
\b F_1 & \b F_2 \end{array}\right].
\]
Because $\hat{\b P}$ is symmetric positive definite, the submatrix $\b P_{22}\in
\mb{\Bb R}^{(n-r)\times (n-r)}$ is symmetric positive definite and hence invertible.
We write down the corresponding $2-2$ block equation of
inequality~(\ref{eq:layaptrans}),
\begin{equation}\label{eq:22eqn}
(\b A_{12} - \b L_{12}\b C_{22})^{\top}\b P_{12} + (\b A_{22} - \b L_{22}\b C_{22})^{\top}\b
P_{22} + \b P_{21}(\b A_{12} - \b L_{12}\b C_{22}) + \b P_{22}(\b A_{22} - \b L_{22}
\b C_{22})<0.
\end{equation}
Equation~(\ref{eq:techtransfor}) yields
\[
\b F_2\b C_{22}=\b B_{2_1}^{\top}\b P_{12},
\]
where $\b B_{2_1}$ has a full row rank. Therefore, $\b B_{2_1}\b B_{2_1}^{\top}\in
\mb{\Bb R}^{r\times r}$ is invertible and we have
\begin{equation}\label{eq:submatp12}
\b P_{12}=\left( \b B_{2_1}\b B_{2_1}^{\top}\right)^{-1}\b B_{2_1}\b F_2\b C_{22}.
\end{equation}
Note that
\[
\b P_{21}=\b P_{12}^{\top}=\b C_{22}^{\top}\b F_2^{\top}\b B_{2_1}^{\top}\left( \b B_{2_1}\b
B_{2_1}^{\top}\right)^{-1}.
\]
Substituting the above into~(\ref{eq:22eqn}), performing manipulations and denoting
\[
\tilde{\b L}_{22}= \b L_{22}-\b P_{22}^{-1}(\b A_{12}-\b L_{12}\b C_{22})^{\top}\left(\b
B_{2_1}\b B_{2_1}^{\top} \right)^{-1}\b B_{2_1}\b F_2,
\]
we obtain
\[
(\b A_{22}-\tilde{\b L}_{22}\b C_{22})^{\top}\b P_{22}+\b P_{22}(\b A_{22}-\tilde{\b
L}_{22}\b C_{22})<0.
\]
Because $\b P_{22}$ is symmetric and positive definite, the above implies that $(\b
A_{22}-\tilde{\b L}_{22}\b C_{22})$ is asymptotically stable, which in turn implies
that the pair $(\b A_{22}, \b C_{22})$ is detectable. The proof of the
theorem is now complete.\\
\hspace*{6in}$\Box$

We now summarize the above considerations in the form of the following design
algorithm.
\begin{center}
{\bf Sliding-Mode Observer Design Algorithm}
\end{center}
Given a quadruple of matrices $(\b A, \b B_1, \b B_2, \b C)$ modeling the plant.
\begin{enumerate}
\item Check that the rank condition, $\mb{rank}\,\left(\b C\b B_2\right)
= \mb{rank}\,\b B_2$, is satisfied.\\
If $\mb{rank}\,\left(\b C\b B_2\right)\ne \mb{rank}\,\b B_2$, the sling-mode
observer cannot be constructed, STOP.
\item Transform the triple $(\b A, \b C, \b B_2)$.\\
Use the method of the proof of Lemma~\ref{lem_transform} to construct nonsingular
matrices $\b T$ and $\b S$ and compute
\[
\b T\b A\b T^{-1}=\left[\begin{array}{cc} \b A_{11} & \b A_{12}\\
\b A_{21} & \b A_{22}
\end{array}\right],\;\; \b T\b B_2=\left[\begin{array}{c}
\b B_{2_1}\\
\b O
\end{array}\right],\;\;  \b S\b C\b T^{-1}=\left[\begin{array}{cc} \b I_r & \b O\\
\b O & \b C_{22}
\end{array}\right],
\]
where $\b B_{2_1} \in \mb{\Bb R}^{r\times m_2}$ and $\mb{rank}\, \b B_{2_1} = r$.
\item Check detectability of $(\b A_{22}, \b C_{22})$.\\
If the pair $(\b A_{22}, \b C_{22})$ is not detectable, the sliding-mode observer
cannot be constructed, STOP. \item Construct a matrix $\b L_{22}$ so that the
eigenvalues of $(\b A_{22}-\b L_{22}\b C_{22})$ are in the open left-half plane.
\item Choose a positive definite $\b Q_{22} \in \mb{\Bb R}^{(n-r)\times (n-r)}$ and
solve for positive definite $\b P_{22}$ the Lyapunov matrix equation,
\[
(\b A_{22} - \b L_{22}\b C_{22})^{\top}\b P_{22}+\b P_{22}(\b A_{22} - \b L_{22}\b
C_{22})=-\b Q_{22}.
\]
\item Choose $\kappa$ that satisfies the condition,
\[
\kappa >\frac{1}{2}\lambda_{\max}\left(\b A_{11}^{\top}+\b A_{11} + \left(\b A_{21}^{\top}\b
P_{22} + \b A_{12}\right)\b Q^{-1}_{22}\left( \b A_{12}^{\top} + \b P_{22}\b
A_{21}\right) \right).
\]
\item Construct,
\[
\hat{\b L}=\left[\begin{array}{cc}
\kappa\b I_r & \b O\\
\b O & \b L_{22}
\end{array}\right],\quad \hat{\b F}=\left[\begin{array}{cc} \b B_{2_1}^{\top} & \b O
\end{array}\right].
\]
\item Compute
\[
\b L=\b T^{-1}\hat{\b L}\b S,\quad \b F=\hat{\b F}\b S.
\]
\item Construct the observer,
\[
\dot{\hat{\b x}}=\b A \hat{\b x}+\b B_1\b u_1 + \b L(\b y - \hat{\b y})+\b B_2\b E
(\b y, \hat{\b y}, \eta),
\]
where
\[
\b E(\b y, \hat{\b y}, \eta) = \left\{\begin{array}{lcc}
\eta\frac{\b F(\b y - \hat{\b y})}{\|\b F(\b y - \hat{\b y})\|_2} & \mb{for} & \b F(\b y - \hat{\b y})\ne \b 0\\
\b 0  & \mb{for} & \b F(\b y - \hat{\b y}) =
\b 0.
\end{array}\right.
\]
Note that $\b C\b e = \b C(\b x - \hat{\b x})=\b y - \hat{\b y}$.
\end{enumerate}
\markex
\begin{example}\label{ex:smo_L-1011}\rm We consider the fifth-order lateral axis model of an
L-1011 fixed-wing aircraft, with the actuator dynamics neglected, at cruise flight
conditions. This model can be found in the book by Edwards and Spurgeon~\cite[pages
122, 123 and pages 179, 180]{edwards98}. We assume that the inputs to the system are
unknown and there are no known inputs. We have,
\[
\b A= \left[ \begin{array}{rrrrr}
  0.0000 & 0.0000 & 1.0000 & 0.0000 & 0.0000 \\
  0.0000 & -0.1540 & -0.0042 & 1.5400 & 0.0000 \\
  0.0000 & 0.2490 & -1.0000 & -5.2000 & 0.0000 \\
  0.0386 & -0.9960 & -0.0003 & -0.1170 & 0.0000 \\
  0.0000 & 0.5000 & 0.0000 & 0.0000 & -0.5000
\end{array} \right]
\]
and
\[\b B_2=
\left[ \begin{array}{rr}
  0.0000 & 0.0000 \\
  -0.7440 & -0.0320 \\
  0.3370 & -1.1200 \\
  0.0200 & 0.0000 \\
  0.0000 & 0.0000
\end{array} \right]
,\quad \b C = \left[ \begin{array}{rrrrr}
  0 & 1 & 0 & 0 & -1 \\
  0 & 0 & 1 & 0 & 0 \\
  0 & 0 & 0 & 1 & 0 \\
  1 & 0 & 0 & 0 & 0
\end{array} \right],
\]
%\end{eqnarray*}
where $\b u_2=\left[\begin{array}{cc} \cos(t) & \sin(t)
\end{array}\right]^{\top}$.
We first check that $\mb{rank}\,\left(\b C\b B_2\right)= \mb{rank}\,\b B_2$. We then
use Lemma~\ref{lem_transform} to compute the transformation matrices $T$ and $S$,
\[
\b T=\left[ \begin{array}{rrrrr}
-0.0000 & -0.9106 & 0.4125 & 0.0245 & 0.9106\\
0.0000 & -0.4124 & -0.9110 & 0.0103 & 0.4124\\
-0.9998 & -0.0005 & 0.0000 & -0.0179 & 0\\
-0.0179 & 0.0265 & -0.0008 & 0.9995 & 0\\
0 & 0 & 0 & 0 & 1.0000
\end{array} \right]
\]
and
\[
\b S=\left[\begin {array}{rrrr}
-0.9106  &  0.4125  &  0.0245  & -0.0000\\
-0.4124 &  -0.9110  &  0.0103  &  0.0000\\
0.0265  & -0.0008  &  0.9996 &  -0.0000\\
0.0000  &  0.0000  & -0.0000 &   1.0000
\end{array}\right].
\]
We next transform the given model into the new coordinates and find the pair$(\b
A_{22}, \b C_{22})$, where
\[
\b A_{22}=\left[\begin {array}{rrr}
0.0006  &  0.0025  &  0.0178\\
-0.0368  &  -0.0993 &  -0.9971\\
-0.0002  &  0.0133 &  -0.0004
\end{array}\right]\;\;\;\mb{and}\;\;\;\b C_{22}=\left[\begin {array}{rrr}
-0.0179  &  0.9998  & -0.0265\\
-0.9998 &  -0.0179  & -0.0000
\end{array}\right].
\]
The pair $(\b A_{22}, \b C_{22})$ is detectable and we use MATLAB's \verb!place!
function to construct the gain matrix $\b L_{22}$ so that the eigenvalues of the
matrix $(\b A_{22}-\b L_{22}\b C_{22})$ are located at $-3, -4, -5$, where
\[
\b L_{22}=\left[\begin {array}{rr}
-0.1441 & -4.0022\\
7.5018  &  0.0744\\
-14.9883 &  -0.4586
\end{array}\right].
\]
We then solve the Lyapunov matrix equation,
\[
(\b A_{22} - \b L_{22}\b C_{22})^{\top}\b P_{22}+\b P_{22}(\b A_{22} - \b L_{22}\b
C_{22})=-\b I_3,
\]
to obtain
\[
\b P_{22}=\left[\begin {array}{rrr}
0.1273 &  -0.0445 &  -0.0236\\
-0.0445  &  0.9995  &  0.4735\\
-0.0236  &  0.4735  &  0.3059
\end{array}\right].
\]
After that we compute
\[
\frac{1}{2}\lambda_{\max}\left(\b A_{11}^{\top}+\b A_{11} + \left(\b A_{21}^{\top}\b P_{22} +
\b A_{12}\right)\b Q^{-1}_{22}\left( \b A_{12}^{\top} + \b P_{22}\b A_{21}\right)
\right)=12.6790
\]
and select
\[
\kappa=13.6790.
\]
Finally we construct
\[
\b L=\b T^{-1}\hat{\b L}\b S=\left[\begin {array}{rrrr}
0.0003 & -0.0000 & 0.0100 & 4.0002\\
13.2773  &  0.0115 & -15.1363  & -0.4544\\
0.0001 &  13.6790  &  0.0044  & -0.0001\\
-0.1532  &  0.0044  &  7.9048  &  0.1580\\
-0.3976 &   0.0114 & -14.9831  & -0.4586
\end{array}\right]
\]
and
\[
\b F=\hat{\b F}\b S=\left[\begin {array}{rrrr}
-0.7440  &  0.3370 &   0.0200 & -0.0000\\
-0.0320 & -1.1200  & -0.0000  &  0.0000
\end{array}\right].
\]
We selected $\eta=7$. We then constructed the observer,
\[
\dot{\hat{\b x}}=\b A \hat{\b x} + \b L(\b y - \hat{\b y}) + \b B_2\b E (\hat{\b y},
\b y, \eta),
\]
where
\[
\b E( \b y, \hat{\b y}, \eta) = \left\{\begin{array}{lcc}
\eta\frac{\b F(\b y - \hat{\b y})}{\|\b F(\b y - \hat{\b y})\|_2+\mu} & \mb{for} & \b F(\b y - \hat{\b y})\ne \b 0\\
\b 0  & \mb{for} & \b F(\b y - \hat{\b y}) =
\b 0,
\end{array}\right.
\]
where $\mu=0.0005$. We the parameter $\mu$ was introduced to smooth out the
discontinuity to facilitate the simulations. In Figure~\ref{fig:smo_L-1011}, we show
plots of the system state variables and their estimates versus time.
\begin{figure}
\centering\epsfig{file=smo_L-1011.eps,width=6.5in} \caption{Plots of $x_i$s and
their estimates versus time for Example~\ref{ex:smo_L-1011}.} \label{fig:smo_L-1011}
\end{figure}
The initial conditions of the plant were selected randomly to be equal to
\[
\b x(0)=\left[\begin{array}{ccccc}
 0.0342 & 0.3200 & 0.0178 & -0.0287 & -0.2497
\end{array}\right]^{\top}.
\]
The initial conditions of the observer were set to zero.
\end{example}
\markex


\section{Conclusions}
In his practical guide for the selection and installation of observers in control
systems, Ellis~\cite[page~3]{ellis02} writes: ``Observers add complexity to the
system and require computational resources. They may be less robust than physical
sensors, especially when plant parameters change substantially during operation.
Still, an observer applied with skill can bring substantial performance benefits and
do so, in many cases, while reducing cost or increasing reliability.'' Examples of
impressive applications of nonlinear observers to the control of electric machinery
can be found in~\cite{dawson98, utkin99,solsona03}. The above applications should
serve as a motivation to generalize our technique to nonlinear plants. R\"{o}benack
and Lynch~\cite{robenack04} presented a method to the observer design for a class of
nonlinear plants which yields almost linear observation error dynamics. This method
looks like a promising tool to be used to extend our approach to a class of
nonlinear plants.


\section{Exercises}
\begin{exercise}\rm 

\end{exercise}

\begin{exercise}\rm

\end{exercise}

\begin{exercise}\rm

\end{exercise}

\begin{exercise}\rm

\end{exercise}

\begin{exercise}\rm

\end{exercise}

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