\chapter{Nonlinear stabilization via state feedback}
\minitoc
Stability theory plays an important role in systems theory. Stability is addressed from several
points of view depending on the problem in question. In this chapter we are concerned with
stability of equilibrium points and its application to output tracking. In general, stability of
the equilibrium points is characterized in the sense of Lyapunov. In this theory, an equilibrium
point is stable if all solutions starting at nearby points stay nearby; otherwise it is unstable.
Furthermore, it is asymptotically stable if all solutions starting at nearby points tend to the
equilibrium point as time elapses. An extension of this theory has been developed in LaSalle's
invariance principle. Definitions and theorems describing this theory will be stated in the next
section.
\section{Stability of nonlinear dynamic systems}
Throughout this section we will consider the following
non-autonomous system
\begin{equation}
    \label{lyapsys}
    \dot{x}(t)=f(t,x(t))~, \quad t\ge 0
\end{equation}
where $x(t)\in \mathbb{R}^n$ and $f:\mathbb{R}^+\times \mathbb{R}^n \rightarrow \mathbb{R}^n$ is
continuous. It is assumed that (\ref{lyapsys}) has a unique solution corresponding to each initial
condition. We will further assume that the origin constitutes the unique equilibrium point and that
$f(t,0)=0$, $\forall t\ge t_0$. In what follows, $x(t,t_0,x_0)$ denotes the solution of
(\ref{lyapsys}) corresponding to the initial condition $x(t_0)=x_0$ evaluated at time $t$.

Lyapunov theory of stability investigates the behaviour of the solution $x(t,t_0,x_0)$ when $x_0\ne
0$ but situated in its neighborhood. The following definitions and theorems is a brief summary of
Lyapunov theory, for detailed study the reader is referred to the books of Hahn \cite{hahn} and
Vidyasagar \cite{vidyasagar}.

\begin{definition}
\label{stability1} The origin of (\ref{lyapsys}) is stable (in the sense of Lyapunov) if, for each
$\epsilon >0$ and each $t_0\in \mathbb{R}^+$ there exists $\delta = \delta(\epsilon,t_0)$ such that
\begin{equation}
    \label{stability1eq}
    \|x_0\|<\delta(\epsilon ,t_0) \Rightarrow
    \|x(t,t_0,x_0)\|<\epsilon~, \quad \forall t\ge t_0
\end{equation}
The origin is unstable if it is not stable. \hfill $\Box$
\end{definition}
\begin{definition}
\label{attractive1} The origin of (\ref{lyapsys}) is attractive if
for each $t_0 \in \mathbb{R}^+$ there is an $\eta(t_0)>0$ such
that
\begin{equation}
    \label{attractive1eq}
    \|x_0\|<\eta(t_0) \Rightarrow x(t,t_0,x_0)\rightarrow 0 \text{
    as } t\rightarrow \infty
\end{equation}
\hfill $\Box$
\end{definition}
If in definitions \ref{stability1}, \ref{attractive1} we can find
constants $\delta(\epsilon)$ and $\eta$ independent off $t_0$ then
the origin is said to be uniformly stable respectively uniformly
attractive.
\begin{definition}
\label{sympstab1} The origin of (\ref{lyapsys}) is (uniformly)
asymptotically stable if it is (uniformly) stable and (uniformly)
attractive.\hfill $\Box$
\end{definition}
Asymptotic stability guaranties that trajectories initiated close
to the origin , eventually tends to it in the future. Another
stronger notion of stability that defines the speed of convergence
is the following
\begin{definition}
\label{expstab1} The origin of system (\ref{lyapsys}) is
exponentially stable if there exist constants $r~,a~,~b>0$ such
that
\begin{equation}
    \label{expstab1eq}
    \|x(t,t_0,x_0)\| \le a e^{-b(t-t_0)}~, \quad \forall t\ge t_0
    \quad \forall x_0\in B_r
\end{equation}
where $B_r$ is a ball of radius $r$ centered at the origin.\hfill
$\Box$
\end{definition}
All the forgoing properties were defined locally in a neighborhood
of the origin. However, if these properties still pertain for all
initial state, that is $B_r$ is extended to $\mathbb{R}^n$, we can
state the following definitions
\begin{definition}
\label{guas} The origin of (\ref{lyapsys}) is globally uniformly
asymptotically stable (GUAS) if
\begin{itemize}
    \item[(i)] it is uniformly stable
    \item[(ii)] for each $M>0$ arbitrarily large and $\epsilon>0$
    arbitrarily small, there exists a finite
    $T=T(M,\epsilon)<\infty$ such that
\end{itemize}
\begin{equation}
    \label{guaseq}
    \|x_0\|<M \Rightarrow \|x(t,t_0,x_0)\| <\epsilon~, \quad
    \forall t\ge T(M,\epsilon)
\end{equation}
\hfill $\Box$
\end{definition}
\begin{definition}
\label{ges} The origin of (\ref{lyapsys}) is globally
exponentially stable (GES) if there exist constants $a,~b>0$ such
that
\begin{equation}
    \label{geseq}
    \|x(t,t_0,x_0)\|\le a e^{-b(t-t_0)}~,\quad \forall t\ge t_0
    \quad \forall x_0\in \mathbb{R}^n
\end{equation}
\hfill $\Box$
\end{definition}
The problem with the above properties in stability analysis is that they require an explicit
knowledge of the solution $x(t,t_0,x_0)$ of (\ref{lyapsys}). Unfortunately, in general finding an
explicit expression for a solution of a nonlinear differential equation represents a challenge
itself. For these reasons Lyapunov direct methods are considered to be valuable. Definitions of the
so-called functions of class $\mathcal{K}$ and class $\mathcal{L}$ are necessary prior to stating
theorems of Lyapunov's direct method.
\begin{definition}
\label{classk} A function $\phi:\mathbb{R}^+\rightarrow \mathbb{R}^+$ is said to belong to class
$\mathcal{K}$ ($\phi \in \mathcal{K}$) if it is continuous, strictly increasing, and $\phi(0)=0$.
\hfill $\Box$
\end{definition}
\begin{definition}
\label{classl} A function $\sigma:\mathbb{R}^+\rightarrow \mathbb{R}^+$ is said to belong to class
$\mathcal{L}$ ($\sigma \in \mathcal{L}$) if it is continuous, strictly decreasing,
$\sigma(0)<\infty$, and $\sigma(r)\rightarrow 0$ as $r\rightarrow \infty$ . \hfill $\Box$
\end{definition}
Let us now recast the various stability definitions in terms of class $\mathcal{K}$ and class
$\mathcal{L}$ functions
\begin{definition}
\label{stability2} The origin of (\ref{lyapsys}) is stable if and only if for each $t_0\in
\mathbb{R}^+$ there exist a number $d(t_0)>0$ and a function $\phi_{t_0}\in \mathcal{K}$ such that
\begin{equation}
    \label{stability2eq}
    \|x(t,t_0,x_0)\|\le \phi_{t_0}(\|x_0\|)~, \quad \forall x_0\in
    B_{d(t_0)}
\end{equation}
\hfill $\Box$
\end{definition}
\begin{definition}
\label{attractive2} The origin of (\ref{lyapsys}) is attractive if and only if for each $t_0\in
\mathbb{R}^+$ there exist a number $r(t_0)>0$ and a function $\sigma_{t_0}\in \mathcal{L}$ such
that
\begin{equation}
    \label{attractive2eq}
    \|x(t,t_0,x_0)\|\le \sigma_{t_0}(t-t_0)~, \quad \forall t\ge t_0
    \quad \forall x_0\in B_{r(t_0)}
\end{equation}
\hfill $\Box$
\end{definition}
Again the uniform stability (attractivity) can be deduced if we can find a positive constant $d>0$
($r>0$) and a function $\phi\in \mathcal{K}$ ($\sigma \in \mathcal{L}$) independent off the initial
time $t_0$ such that the inequalities (\ref{stability2eq}) respectively (\ref{attractive2eq}) hold.

Using the preceding stability definitions and the definition of class $\mathcal{K}$ functions, the
following theorems can be proved (see \cite{vidyasagar}). We will denote by Lyapunov function
candidate or shortly Lyapunov function any continuously differentiable function ($\mathcal{C}^1$
function) $V(t,x)$ satisfying for all $t\in \mathbb{R}^+$
\[
\begin{cases}
    V(t,0)=0 \\
    V(t,x)>0 & \forall x\ne 0
\end{cases}
\]
\begin{theorem}
\label{stabtheo} The origin of system (\ref{lyapsys}) is (globally) stable if there exist a
Lyapunov function $V(t,x)$ and functions $\alpha \in \mathcal{K}$ such that $\forall x\in B_r~,r>0$
($\forall x \in \mathbb{R}^n$) we have
\begin{itemize}
    \item[(i)]$\alpha(\|x\|)\le V(t,x)$
    \item[(ii)]$\dot{V}(t,x)=\frac{\partial V(t,x)}{\partial
x}f(t,x(t))+\frac{\partial V(t,x)}{\partial t}\le 0 $\hfill $\Box$
\end{itemize}
\end{theorem}
\begin{theorem}
\label{guastheo} The origin of system (\ref{lyapsys}) is (globally) uniformly asymptotically stable
if there exist a Lyapunov function $V(t,x)$ and  functions $\alpha_i \in \mathcal{K}$ ($i=1,2,3$)
such that $\forall x\in B_r~,r>0$ ($\forall x \in \mathbb{R}^n$) we have
\begin{itemize}
    \item[(i)]$\alpha_1(\|x\|)\le V(t,x)\le \alpha_2(\|x\|)$
    \item[(ii)]$\dot{V}(t,x)\le -\alpha_3(\|x\|)$\hfill $\Box$
\end{itemize}
\end{theorem}
\begin{theorem}
\label{gestheo} The origin of system (\ref{lyapsys}) is (globally) exponentially stable if there
exists a Lyapunov function $V(t,x)$ and positive constants $a_i>0$ ($i=1,2,3$) and $p>0$ such that
$\forall x\in B_r~,r>0$ ($\forall x \in \mathbb{R}^n$) we have
\begin{itemize}
    \item[(i)]$a_1\|x\|^p\le V(t,x)\le a_2\|x\|^p$
    \item[(ii)]$\dot{V}(t,x)\le -a_3\|x\|^p$\hfill $\Box$
\end{itemize}
\end{theorem}
Conversely we have
\begin{theorem}
\label{guasconv} Suppose that $f(t,0)=0$ and that $f$ is $\mathcal{C}^k$ ($k\ge 1$) and Lipschitz
with respect to $x$ uniformly in $t$. Suppose further that the origin of (\ref{lyapsys}) is
uniformly asymptotically stable in the neighborhood of the origin ($B_r$). Under these conditions
there exist a $\mathcal{C}^k$ Lyapunov function $V(t,x)$ and functions $\alpha_i\in \mathcal{K}$
($i=1,2,3$) such that $\forall x\in B_r$
\begin{itemize}
    \item[(i)] $\alpha_1(\|x\|)\le V(t,x)\le \alpha_2(\|x\|)$
    \item[(ii)] $\dot{V}(t,x)\le -\alpha_3(\|x\|)$
    \item[(iii)] $\underset{x\in B_r}{\sup}\Big\|\frac{\partial V(t,x)}{\partial
    x}\Big\| < \infty $ \hfill $\Box$
\end{itemize}
\end{theorem}
\begin{theorem}
\label{gesconv} Suppose that $f(t,0)=0$ and that $f$ is $\mathcal{C}^k$ ($k\ge 1$) and Lipschitz
with respect to $x$ uniformly in $t$. Suppose further that the origin of (\ref{lyapsys}) is
exponentially stable in the neighborhood of the origin ($B_r$). Under these conditions there exist
a $\mathcal{C}^k$ Lyapunov function $V(t,x)$ and constants $a_i>0$ ($i=1,2,3$) such that the
following hold $\forall x\in B_r$
\begin{itemize}
    \item[(i)] $a_1\|x\|^2\le V(t,x)\le a_2\|x\|^2$
    \item[(ii)] $\dot{V}(t,x)\le -a_3\|x\|^2$
    \item[(iii)] $\Big\|\frac{\partial V(t,x)}{\partial
    x}\Big\| < a_4\|x\|$ \hfill $\Box$
\end{itemize}
\end{theorem}

As an extension to these theorems and especially Theorem \ref{stabtheo} LaSalle has developed
further results
\begin{theorem}
\label{invariance} Let $\Omega$ be a compact (closed and bounded) set with the property that every
solution of \eqref{lyapsys} which starts in $\Omega$ remains for all future time in $\Omega$. Let
$V:\Omega\rightarrow \mathbb{R}$ be a $\mathcal{C}^1$ function such that $\dot{V}(x)\le 0$ in
$\Omega$. Let $E$ be the set of all points in $\Omega$ where $\dot{V}(x)= 0$. Let $M$ be the
largest invariant set in $E$. Then every solution starting in $\Omega$ approaches $M$ as
$t\rightarrow \infty$.\hfill $\Box$
\end{theorem}
\begin{theorem}
Consider system \eqref{lyapsys} with its origin as an equilibrium point. Let $V$ be a
$\mathcal{C}^1$ function defined on a neighborhood of the origin $ U \subseteq \mathbb{R}^n$ such
that $\dot{V}(x)\le 0$ in $ U \subseteq \mathbb{R}^n$. Let $S=\{x\in U \subseteq \mathbb{R}^n |
\dot{V}(x)= 0\}$, and suppose that no solution can stay forever in $S$, other than the trivial
solution. Then the origin is asymptotically stable. It is globally asymptotically stable if $U=
\mathbb{R}^n$ and in addition $V$ is radially unbounded.\hfill $\Box$
\end{theorem}
The above corollary shows formally that if in a domain about the origin we can find a Lyapunov
function whose derivative along the trajectories of the system is negative semidefinite, and if we
can establish that no system trajectory can stay forever at points where $\dot{V}(x)= 0$ except at
the origin, then the origin is asymptotically stable.
\section{Stabilization via state feedback}
The problem of controlling a dynamical system has been addressed from two directions, namely the
open loop and the closed loop control. Applications have shown that closed loop control is more
accurate whenever stabilization is reached.

In closed loop technique, the control law is designed using the state variables. Thus we can
describe the feedback stabilization problem for a nonlinear dynamic system
\begin{equation}
    \label{gensys}
    \dot{x}=f(x,u)
\end{equation}
to be the problem of designing a feedback control law
\begin{equation}
    \label{gencontrol}
    u=\Gamma(x)
\end{equation}
such that the origin $x=0$ is an asymptotically stable equilibrium point of the closed-loop system
\begin{equation}
    \label{gencloop}
    \dot{x}=f(x,\Gamma(x))
\end{equation}
In a typical control problem, additional constraint other than asymptotic stability might be
required. Therefrom several stabilizing techniques replying to different requirements has been
designed. To name a few, we state the tangential linearization of the nonlinear dynamical system
around the origin, therefore one can use the linear theory results to design a stabilizing
controller. An other approach to the synthesis of a feedback control law is the so-called exact
linearization. It consists in finding a nonlinear diffeomorphism that transforms the nonlinear
system into a linear one see \cite{isidori} \cite{ieee_con14}. Away from the linearization
techniques, a third method consists in designing a stabilizing control law directly from the
nonlinear model without any kind of linearization. The method concerns nonlinear systems,
non-affine in the control, with dissipative drift see Jurdjevic and Quinn \cite{misc_con11},
Gauthier and Bonard \cite{misc_con8}, Byrnes \emph{et. al.} Outbib and Sallet \cite{scl_con31},
Wei Lin \cite{scl_con8} \cite{auto_con7} and Outbib and Richard \cite{outbib2}. The common feature
in all the previous approaches is that the nonlinear system is supposed to be perfectly known.
However in general, mathematical models which describe real systems contain uncertain elements due
to modelling error and parameter variation to name a few. Sometimes these uncertainties yield to
degrading the performance of the stabilizing scheme or even to instabilities in some other times.
Therefore, a new approach to design robust stabilizing controller is required. Sliding mode
approach is one that was adapted by many researchers see for instance Utkin \cite{utkin}, Slotine
\cite{ijc_con6}, Ramirez \cite{ijc_con10}, Spurgeon \cite{scl_con1}.

Figure \ref{gbf} shows a feedback control system.
\begin{figure}[!ht]
	\centering
	\includegraphics[width=10cm]{figure10}
	\caption{Feedback control of linear system.}
	\label{gbf}
\end{figure}
