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\begin{frontmatter}

\title{Sliding Mode Control for a Class of Control-Affine Nonlinear Systems} 
% Title, preferably not more than 10 words.

%\thanks[footnoteinfo]{Sponsor and financial support acknowledgment
%goes here. Paper titles should be written in uppercase and lowercase
%letters, not all uppercase.}

\author[First]{Belem Saldivar} 
\author[Second]{Juan Carlos \'Avila Vilchis} 
\author[Second]{Adriana H. Vilchis Gonz\'alez}
\author[Second]{Edgar Mart\'inez Marb\'an}

\address[First]{CONACYT research fellow in the Facultad de Ingenier\'ia - Universidad Aut\'{o}noma del Estado de M\'{e}xico, Instituto Literario No. 100 Ote., 50130 Toluca, Edo. de M\'{e}xico, Mexico (e-mail: mbsaldivarma@conacyt.mx).}
\address[Second]{Facultad de Ingenier\'ia - Universidad Aut\'{o}noma del Estado de M\'{e}xico, Instituto Literario No. 100 Ote., 50130 Toluca, Edo. de M\'{e}xico, Mexico (e-mail: jcavilav@uaemex.mx, avilchisg@uaemex.mx, edgmarb.08@gmail.com)}


\begin{abstract}                % Abstract of not more than 250 words.
This paper concerns the synthesis of a sliding mode-based controller for a class of nonlinear control-affine systems where sufficient conditions for the system stabilization are provided. The effectiveness of the proposed approach is highlighted through a practical example: the regulation task of an aerodynamic system.
\end{abstract}

\begin{keyword}
Sliding mode control, nonlinear systems, stability conditions, aerodynamic system.
\end{keyword}

\end{frontmatter}
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\section{Introduction}
\label{intro}

The Sliding Mode Control (SMC) technique was introduced in the 1950s in the former Soviet Union as a variable
structure control system (\cite{Emelyanov}); two decades later, a book by \cite{Itks} and a survey paper by \cite{Utkin1} were published in English. Since then, several research studies were performed both from theoretical and practical perspectives, see for example \cite{DeCarlo}, \cite{Hung} and \cite{Utkin2}, where the fundamental principles of the SMC theory can be found, see also \cite{Azar}, \cite{Bandyopadhyay}, \cite{Bartolini}, \cite{LFridman} and \cite{Pisano} where recent results on SMC are summarized. 


It is important to point out that a wide range of control problems in engineering have been treated using the SMC framework, just to mention some of them: \cite{Young} provides an overview of the problems arising in the practical implementation of the SMC technique, \cite{Bartolini2} studies the SMC approach applied to mechanical systems, \cite{Piltan} and \cite{Othman} present a review of the application of SMC to robotic manipulators, and to electrohydraulic systems, respectively, \cite{Rossomando} applies the SMC technique to tackle the trajectory tracking problem in a mobile robot, \cite{Sefriti} propose a sliding mode based control for the robust tracking of a electrically-driven two-links robot 
manipulator, \cite{Rekioua} and \cite{Gonzalez} deal with the control of renewable energy generation systems via the SMC technique, \cite{Zhenga} and \cite{Khebbache} present the SMC approach applied to quadrotor helicopters.



%and \cite{Pai}, \cite{Niu} and \cite{Huo} where the classic SMC has been enhanced with the use of different control schemes. 

One can see that the SMC technique has been studied and applied, in distinct contexts, since its beginning to the present day. One of the reasons of the great success of the SMC approach is its robustness; with this kind of controllers, the system states are forced to reach and move through a predefined sliding surface, hence the system dynamics are determined by this surface instead of being influenced by uncertainties or disturbances. Once the sliding surface and the switching function are chosen, the dynamic performance of the system is fixed (\cite{Liu_Li}). Besides robustness, the sliding mode controllers feature other remarkable properties such as accuracy and easy tuning and implementation. 

This control method has been examined for a wide spectrum of system types including nonlinear systems, multi-input/multi-output systems, discrete-time models, large-scale and infinite-dimensional systems, and stochastic systems (\cite{Hung}). Furthermore, the control objectives have been extended from stabilization to other functions. This paper concerns the stabilization via the SMC technique of a special class of nonlinear systems which are affine in the control. As an application example, the regulation problem for a particular aerodynamic system is addressed.

Interest for the modeling and control of aerodynamic systems has been present for many years in research projects all around the world. Several researchers have studied different aerodynamic systems with different approaches;
% to analyze a significant number of problems in this branch. With the purpose of providing an overview of such variety, we slightly comment some of the works reported in literature. For 
for instance \cite{Bouguerra} and \cite{Lopes} study pedestal aerodynamic systems; in \cite{Bouguerra} a fault tolerant control for a 2 DoF (Degrees of Freedom) aerodynamic system is proposed while \cite{Lopes} considers an aerodynamic system with 3 DoF for which a predictive control is synthesized; \cite{Balas} focuses on the modeling and control of the position and the yaw angle of a quadrotor system; %\cite{James} takes into account an aerodynamical system that carries suspended loads; 
%a switching networked attitude controller for an unmanned quadrotor is presented in \cite{Nikola}; an optimal attitude control of a quadrotor subject to wind disturbances is developed in \cite{Alexis}; micro aerodynamic systems are studied by \cite{Schafroth}; 
%\cite{Avila} provides a nonlinear model and control of a Vario scale model helicopter where 7 DoF are initially considered; 
\cite{Schreck} points out the difficulties to the exact computing of aerodynamic forces, due to the lack of mathematical models;
% \cite{Hamdi} characterizes centrifugal and gravitational effects in a wind turbine blade;
 \cite{Bejar} proposes an illustrative reading about different aerodynamic platforms and control approaches. Optimization techniques applied to the design of rotor blades and applications that consider aerodynamic systems evolving at high altitude are reported in \cite{Leusink} and in \cite{Mueller}, respectively.


%This paper is organized as follows: Section 2 presents the SMC design for a general class of affine nonlinear systems; sufficient conditions for the system stability are provided. In Section 3 the AAS model is presented and the regulation problem is solved via the SMC; numerical simulations show the effectiveness of the proposed approach. Concluding remarks are provided in Section 4.

%In this paper, a sliding mode control strategy is applied to the Aerodynamic Angular System (AAS), shown in Fig. \ref{FigAAS}, consisting of the following elements: a pedestal (1) which provides support to the AAS bar (2) defining a planar angular movement with respect to a pivot (3) where a viscous friction joint takes place, at the extremities of the bar, two actuators are located (4), (5). Each actuator generates an aerodynamic force or lift force (F$_1$, F$_2$) thanks to the angular velocity ($\omega_1$, $\omega_2$) of each associated propeller driven by direct current motors. The pivots  at the bar ends (6),  (7) enable actuators to remain upright at all times. A difference between the magnitudes of these two aerodynamic forces produces a torque with respect to the pivot (3) and, consequently, the rotation of the bar (2). As it is illustrated in Fig. \ref{FigAAS}, an offset ($\delta$) is present in this system, so the bar does not rotate about its geometrical center. It is assumed that an angular movement sensor is located on the pivot (3) to measure the rotation angle of the bar ($\theta$).

In this paper, a sliding mode controller for a particular class of nonlinear control-affine systems is synthesized and the results are applied to the Aerodynamic Angular System (AAS), shown in Fig. \ref{FigAAS} consisting of the elements described below. 

The pedestal (1) provides support to the AAS bar (2) defining a planar angular movement with respect to the pivot (3) where a viscous friction torque that opposes the angular movement of the bar is assumed. Two actuators (4) and (5) are located at the extremities of the bar and generate the lift forces F$_1$ and F$_2$ thanks to the aerodynamic effect of the corresponding propellers that rotate with angular velocities $\omega_1$ and $\omega_2$, respectively. These actuators correspond to direct current motors. The pivots (6) and (7) at the bar ends, enable actuators to remain upright continuously in such a way that the lift forces will be vertical all the time. If the two aerodynamic forces F$_1$ and F$_2$ have the same magnitude, the bar (2) will remain horizontal; a difference between these two aerodynamic forces will produce a torque with respect to the pivot (3) and, consequently, the rotation of the bar (2). As it is illustrated in Fig. \ref{FigAAS}, an offset ($\delta$) is present in this system, so the bar does not rotate about its geometrical center; the bar rotation is, then, an asymmetrical movement. An angular movement sensor is located on the pivot (3) to measure the rotation angle of the bar ($\theta$) and to deduce its angular velocity ($\dot{\theta}$).  




%A difference between the magnitudes of these two aerodynamic forces produces a torque with respect to the pivot (3) and, consequently, the rotation of the bar (2). 

%where a viscous friction joint takes place. At the extremities of the bar, actuators (4) and (5) are located, each actuator generates an aerodynamic force or lift force (F$_1$, F$_2$) thanks to the angular velocity ($\omega_1$, $\omega_2$) of the corresponding propellers driven by direct current motors. 





\begin{figure}[h!]
\centering
\includegraphics[width=\columnwidth, height=7cm]{AAS}
\caption{Aerodynamic Angular System diagram.}\label{FigAAS}
\end{figure}

This paper is organized as follows: Section 2 is devoted to the sliding mode control design for a class of nonlinear systems and sufficient conditions for the system stability are provided; in Section 3 the regulation problem statement for the AAS is provided, numerical results concerning the regulation via the SMC are discussed and the effectiveness of the proposed approach is pointed out; the paper concludes and describes directions for the future work in Section 4.
%concluding remarks and an outline of future work are provided in Section 4.     
\section{Sliding mode control} \label{Sec:sliding}
Consider a $n$th-order nonlinear system of the form:\footnote{In the following, the time dependence symbol $(t)$ of dynamic variables will be omitted for simplification.}
\begin{equation}
\dot{x}=Ax+\bar{f}(x)+\sum_{i=1}^{m}\bar{g}_{i}(x)u_{i},\label{eq:nonlinear system}
\end{equation}
where $x\in\mathcal{R}^{n}$ is the state vector and $u_i\in\mathcal{R}$, $i=1,...,m$ are the control inputs.
In the linear part, matrix $A\in \mathcal{R}^{n \times n}$ is given by:
\[
A=\left[\begin{array}{ccccc}
0 & 1 & 0 & \cdots & 0\\
0 & 0 & 1 &  & 0\\
\vdots & \vdots &  & \ddots\\
0 & 0 & 0 &  & 1\\
-a_{n} & -a_{n-1} & -a_{n-2} & \cdots & -a_{1}
\end{array}\right],
\] \\
with constants $a_1,...,a_n$. The drift and control vector fields $\bar{f}$ and $\bar{g}_i$, respectively, are such that:\\
\[
\bar{f}(x)=\left[\begin{array}{c}
0\\
0\\
\vdots\\
0\\
f(x)
\end{array}\right], \quad
\bar{g}_i(x)=\left[\begin{array}{c}
0\\
0\\
\vdots\\
0\\
g_i(x)
\end{array}\right],
\] \\
where $f(x)$ and $g_i(x)$ are scalar nonlinear functions.

It is well known that the sliding mode control technique drives the
state of the system to a predefined surface allowing the reaching of the equilibrium. In general, this method consists of two elements:
the switching rule and the equivalent control. The switching
law constitutes a discontinuous control which is applied in order to
reach the sliding surface while the equivalent control is continuous and
aims keeping the system state on the sliding surface.

A sliding mode-based technique to control a nonlinear system of the form (\ref{eq:nonlinear system}) is proposed in what follows.


First of all, the switching part of the sliding mode control will be derived.

Consider a sliding surface defined by \\
\begin{equation}
S=Kx=0,\quad K\in\mathcal{R}^{1\times n},\label{eq:sliding surface}
\end{equation} \\
where $K$ is a row vector with constant elements.

The sliding surface is reached by the system state if the condition $$S\dot{S}<0$$ \\
%\begin{equation}
%\frac{d}{dt}(S^{T}S)<0
%\end{equation} 
is fulfilled (see for instance \cite{Sira});
in view of (\ref{eq:sliding surface}), this condition is rewritten
as \\ $$KxK\dot{x}<0.$$ \\ Introducing the system dynamics given by (\ref{eq:nonlinear system}),
one gets:
\begin{equation}
\label{inequality_switch}
KxK(Ax+\bar{f}(x))+\sum_{i=1}^{m}KxK\bar{g}_{i}(x)u_{i}<0.
\end{equation}
Inequality (\ref{inequality_switch}) is satisfied for a proper choice of $u_{i}$
for all $i=1,...,m$.

Consider the following switching law: \\
\begin{equation}
u_{i_{s}}=\left\{ \begin{array}{lll}
-\phi\frac{\left|KxK(Ax+\bar{f}(x))\right|}{KxK\bar{g}_{i}(x)} &  & \textrm{if }S\neq0\textrm{ and }KxK\bar{g}_{i}(x)\neq0,
%\quad\forall x\neq0
\\
\\
0 &  & \textrm{else,}
\end{array}\right.\label{eq:switching control-1previo}
\end{equation} 
where $\phi$ is a positive constant. Then, by taking \\ $$u_{i}=u_{i_{s}},$$ \\ inequality
(\ref{inequality_switch}) is reduced to: \\
\begin{equation}
\label{inequality_switch2}
KxK(Ax+\bar{f}(x))-m\phi\left|KxK(Ax+\bar{f}(x))\right|<0
\end{equation}\\
that is satisfied for all $$\phi>1/m.$$


The switching control (\ref{eq:switching control-1previo}) can be rewritten as follows: \\
\begin{equation}
u_{i_{s}}=\left\{ \begin{array}{lll}
-\phi\text{sgn}(Kx)\frac{\left|K(Ax+\bar{f}(x))\right|}{K\bar{g}_{i}(x)} &  & \textrm{if }S\neq0\textrm{ and }K\bar{g}_{i}(x)\neq0,
%\quad\forall x\neq0
\\
\\
0 &  & \textrm{else,}
\end{array}\right.\label{eq:switching control-1}
\end{equation} 

\textit{Remark 1.} The robustness property against matched uncertainties of the switching control can be ensured by a proper choice of the controller gain $\phi$. To see this, consider a system of the form: 
$$\dot{x}=Ax+\bar{f}(x)+\sum_{i=1}^{m}\bar{g}_{i}(x)(u_{i}+\xi_{i})$$ \\
where $\xi_{i}$ represents external disturbances or model uncertainties which are unknown but bounded in magnitude:  \\ $$\left|\xi_{i}\right|\leq\bar{\xi}_{i}, \quad i=1,...,m,$$ \\ with $\bar{\xi}_{i}$ known constant upper bounds.

Setting \\ $$\eta=\left|K(Ax+\bar{f}(x))\right|,$$  \\ the reaching condition (RC) stated in (\ref{inequality_switch2}) would be:
$$
RC:=KxK(Ax+\bar{f}(x))-m\phi\left|Kx\right|\eta+\sum_{i=1}^{m}KxK\bar{g}_{i}(x)\xi_{i}<0
$$
Note that
$$
RC\leq \left|Kx\right|\eta-m\phi\left|Kx\right|\eta+\sum_{i=1}^{m}\left|KxK\bar{g}_{i}(x)\xi_{i}\right|,
$$
$$
RC\leq\eta-m\phi\eta+\sum_{i=1}^{m}\left|K\bar{g}_{i}(x)\right|\bar{\xi_{i}},
$$
then, the reaching condition is satisfied for \\ $$\phi>\frac{1}{m} + \frac{1}{m\eta} \sum_{i=1}^{m}\left|K\bar{g}_{i}(x)\right|\bar{\xi_{i}}.$$ 

\vspace{0.5cm}


Next,  the equivalent control will be derived. To guarantee that the system
state remains on $S$ during the sliding phase, the following condition
must be satisfied: \\
\begin{equation}
\frac{d}{dt}S=0\quad\textrm{when}\quad S=0.\label{eq:Equivalent condition}
\end{equation}



Note that \\
\[
\frac{d}{dt}S=K\dot{x}=K(Ax+\bar{f}(x))+\sum_{i=1}^{m}K\bar{g}_{i}(x)u_{i}.
\] \\
Thus, condition (\ref{eq:Equivalent condition}) is satisfied by considering the following equivalent control for $i=1,...,m$: \\
\begin{equation}
u_{i_{eq}}=\left\{ \begin{array}{lll}
-\frac{1}{m}\frac{K(Ax+\bar{f}(x))}{K\bar{g}_{i}(x)} &  & \textrm{if }S=0\textrm{ and }K\bar{g}_{i}(x)\neq0,
%\quad\forall x\neq0
\\
\\
0 &  & \textrm{else.}
\end{array}\right.\label{eq:equivalent control-1}
\end{equation} \\

So, the nonlinear system (\ref{eq:nonlinear system}) can be controlled by
the sliding mode control defined by: \\
\begin{equation}
u_{i}=u_{i_{s}}+u_{i_{eq}},\label{eq:control-1}
\end{equation} \\
where $u_{i_{s}}$ and $u_{i_{eq}}$ are given by (\ref{eq:switching control-1})
and (\ref{eq:equivalent control-1}), respectively.

\subsection{Stability analysis}

The stability of the nonlinear system (\ref{eq:nonlinear system}) under the sliding mode controller
(\ref{eq:control-1}) depends on two stages: the reaching phase and
the sliding mode. The stability of the closed loop system is guaranteed
if the reaching condition is satisfied and the system remains stable
on the sliding surface. The stability during the reaching phase is verified since $\dot{S}S<0$
when $S\neq0$, but it is necessary to guarantee the stability during
the sliding mode.

On the sliding surface, the equality: \\$$S=Kx=0$$ holds. Let us set 
\[
x=\left[\begin{array}{c}
x_a\\
x_n
\end{array}\right], \quad
x_a=\left[\begin{array}{c}
x_1\\
\vdots\\
x_{n-1}\\
\end{array}\right],
\]
and \\
$$ K=\left[\begin{array}{cc}
\bar{K} & 1\end{array}\right], \quad \bar{K}=\left[\begin{array}{cccc}
k_1 & k_2 & \cdots & k_{n-1} \end{array}\right].$$ \\
Then, the equality $Kx=0$ implies: \\
\begin{equation}
\label{xn}
x_n=-\bar{K}x_a.
\end{equation} \\
Regarding the above relation, the convergence of $x_a$ to the zero equilibrium point can be proved trough the convergence of $x_n$.

From (\ref{eq:nonlinear system}), one can obtain: 
\begin{equation}
\label{xndot1}
\dot{x}_{n}=-a_nx_1-a_{n-1}x_2-\cdots-a_1x_n+f(x)+\sum_{i=1}^{m}g_{i}(x)u_{i}.
\end{equation}
The control $u_{i}$ on the sliding surface corresponds to the equivalent control
(\ref{eq:equivalent control-1}), where:
\begin{equation*}
\begin{split}
K(Ax+\bar{f}(x))=&[k_{1} \cdots k_{n-1} 1]\hspace{-0.1cm}\left[\begin{array}{c}
x_{2}\\
\vdots\\
x_{n}\\
\hspace{-0.1cm}-a_{n}x_{1}-\cdots-a_{1}x_{n}+f(x)
\end{array}\hspace{-0.1cm}\right]
\\\\
=&k_{1}x_{2}+\cdots+k_{n-1}x_{n}-a_{n}x_{1}-a_{n-1}x_{2}-
\\ \\
&\cdots-a_{1}x_{n}+f(x),
\end{split}
\end{equation*}
and \\
$$
K\bar{g}_i(x)=g_i(x).
$$ \\
Substituting the equivalent control into (\ref{xndot1}) yields: \\
\begin{equation}
\label{xndot}
\begin{split}
\hspace{-0.8cm}\dot{x}_{n}=&-k_{1}x_{2}-\cdots-k_{n-1}x_{n} \\ \\
=&-k_{1}x_{2}-\cdots-k_{n-1}(-k_{1}x_{1}-\cdots -k_{n-1}x_{n-1}).
\end{split}
\end{equation}\\


Consider the Lyapunov function \\ $$V(x)=\frac{1}{2}x_{n}^{2}.$$ \\
%For the stability, we have to prove $\dot{V}<0$. 
Note that $$\dot{V}(x)=x_{n}\dot{x}_{n},$$\\ which, in view of (\ref{xn}) and (\ref{xndot}), can be written as \\$$\dot{V}(x)=x_a^T\Psi x_a,$$\\ where $\Psi$ is the symmetric matrix defined in (\ref{Psi}) (next page). 
Then $\dot{V}(x)<0$ is fulfilled for $k_{1}, \cdots, k_{n-1}$ satisfying $\Psi<0$ and the system is
stable during the sliding mode.

    \begin{figure*}[!t]
    % ensure that we have normalsize text
    \normalsize
    % Store the current equation number.
    % Set the equation number to one less than the one
    % desired for the first equation here.
    % The value here will have to changed if equations
    % are added or removed prior to the place these
    % equations are referenced in the main text.
    
\begin{gather}
\Psi =\left[ 
\begin{array}{ccccc}
-k_{1}^{2}k_{n-1} &\frac{1}{2}k_{1}^{2}-k_{1}k_{2}k_{n-1} & \frac{1}{2}k_{1}k_{2}-k_{1}k_{3}k_{n-1} & \cdots & \frac{1}{2}k_{1}k_{n-2}-k_{1}k_{n-1}^{2} \vspace{0.2cm} \\\vspace{0.2cm}
* & k_{1}k_{2}-k_{2}^{2}k_{n-1} & \frac{1}{2}k_{2}^{2}-k_{2}k_{3}k_{n-1}+\frac{1}{2}k_{1}k_{3}  & \cdots &  \frac{1}{2}k_{2}k_{n-2}-k_{2}k_{n-1}^{2}+\frac{1}{2}k_{1}k_{n-1}      \\ \vspace{0.2cm}
* & * & k_{2}k_{3}-k_{3}^{2}k_{n-1} & \cdots & \frac{1}{2}k_{3}k_{n-2}-k_{3}k_{n-1}^{2} +\frac{1}{2}k_{2}k_{n-1}\\ \vspace{0.2cm}
\vdots & \vdots & \vdots & \ddots & \vdots\\ \vspace{0.2cm}
* & * & * & * &  k_{n-2}k_{n-1}-k_{n-1}^{3}
\end{array}%
\right] ,  \label{Psi} \\
\nonumber 
\end{gather}

    % IEEE uses as a separator
    \hrulefill
    % The spacer can be tweaked to stop underfull vboxes.
    \vspace*{4pt}
    \end{figure*}


%the symbol $*$ is used as an ellipsis for the terms induced by symmetry.

The above result is summarized in the following theorem. \\
\begin{thm}
\label{Theo}
The nonlinear system (\ref{eq:nonlinear system}) is stabilizable by the sliding
mode control (\ref{eq:control-1}) with $\phi>1/m$ and \\ $$K=[k_1 \cdots k_{n-1} \enskip 1],$$ \\
%with $k_{1}, \cdots, k_{n-1}$ 
where $k_1, k_2, ... , k_{n-1}$ are such that the matrix inequality $\Psi<0$, with $\Psi$ given in (\ref{Psi}), is satisfied. 
\end{thm} 
\vspace{0.5cm}
\textit{Remark 2.}
Note that for $n=1$, the switching and the equivalent control are given by: \\
\begin{equation*}
u_{i_{s}}=\left\{ \begin{array}{lll}
-\phi\text{sgn}(x)\frac{\left|-a_1x+f(x))\right|}{g_{i}(x)} &  & \textrm{if }S\neq0\textrm{ and }g_{i}(x)\neq0,
%\quad\forall x\neq0
\\
\\
0 &  &\textrm{else,}
\end{array}\right.
\end{equation*} \\
\begin{equation*}
u_{i_{eq}}=\left\{ \begin{array}{lll}
-\frac{1}{m}\frac{-a_1x+f(x)}{g_{i}(x)} &  & \textrm{if }S=0\textrm{ and }g_{i}(x)\neq0,
%\quad\forall x\neq0
\\
\\
0 &  & \textrm{else.}
\end{array}\right.
\end{equation*} \\
For $n=2$, the condition on the controller gains is reduced to $-k_1^3<0$, i.e., $k_1>0$, and, for $n>2$, it corresponds to a nonlinear matrix inequality which can be solved using an appropriate computational package such that the PENLAB of MATLAB.





\section{Practical example: Aerodynamic Angular System regulation}

The effectiveness of the proposed approach is highlighted through a practical example: the regulation problem for  the AAS described in Section 1.

\subsection{AAS model}

The regulation problem for this system consists on driving the angle $\theta$ defined by the rotation of the bar (see Fig. \ref{FigAAS}) to a predefined constant reference value $\theta_{ref}$ by controlling the lift forces F$_1$ and F$_2$. To achieve this goal, an accurate model of the AAS is required.
%The first step to tackle this problem is the construction of a reliable model of the system. 
In \cite{MtzMarban}, a Lagrangian formulation is used to obtain  the following model that represents the one degree of freedom dynamics of the AAS: \\
\begin{equation}
\label{modelAAS}
\dot{x}=Ax+\bar{f}(x)+\bar{g}_1(x)u_1+\bar{g}_2(x)u_2,
\end{equation}
where \\
$$
x=\left[\begin{array}{c}
x_{1}\\
x_{2}
\end{array}\right]=\left[\begin{array}{c}
\theta\\
\dot{\theta}
\end{array}\right],
\quad
A=\left[\begin{array}{cc}
0 & 1\\
0 & \alpha_1 
\end{array}\right],
\quad
\bar{f}(x)=\left[\begin{array}{c}
0\\
f(x)
\end{array}\right],
$$ \\
$$
f(x)=\alpha_2\sin\theta, \quad \bar{g}_i(x)=\left[\begin{array}{c}
0\\
g_i(x)
\end{array}\right],
$$ \\
$$
g_1(x)=\alpha_3\sin\theta+\alpha_4\cos\theta,
\quad
g_2(x)=\alpha_3\sin\theta-\alpha_4\cos\theta.
$$ \\
The control inputs $u_1$ and $u_2$ correspond to the lift forces generated from the left and right propellers, respectively, and the constants $\alpha_i$, $i=1,...,4$ are defined by the parameters of the AAS.

Notice that this model constitutes a nonlinear system of the form (\ref{eq:nonlinear system}). To tackle the regulation problem stated before, the results presented in Section 2 are applied.


\subsection{Regulation task}

In view of the application under consideration, the sliding
surface is defined as follows: \\
\begin{equation}
\label{surface}
S=Kx_{s},
\end{equation}
where \\
\[
x_{s}=\left[\begin{array}{c}
x_{1}-x_{1}^{*}\\
x_{2}
\end{array}\right]=\left[\begin{array}{c}
\theta-\theta_{ref}\\
\dot{\theta}
\end{array}\right].
\] \\

In view of (\ref{surface}), the reaching condition is given by: \\
\begin{equation}
\label{reaching_condition}
%\frac{d}{dt}\left(S^{T}S\right)=
Kx_{s}K\dot{x}_{s}=Kx_{s}K\dot{x}<0.
\end{equation} \\
Then, from (\ref{reaching_condition}) and the system dynamics (\ref{modelAAS}), the switching control (\ref{eq:switching control-1}) can be rewritten as: \\
\begin{equation}
u_{i_{s}}=\left\{ \begin{array}{lll}
\hspace{-0.1cm}-\phi\text{sgn}(Kx_{s})\frac{\left|K(Ax+\bar{f}(x))\right|}{K\bar{g}_{i}(x)} &  & \textrm{\hspace{-0.2cm}if }S\neq0\textrm{ and }K\bar{g}_{i}(x)\neq0,
%\quad\forall x\neq0
\\
\\
0 &  & \textrm{\hspace{-0.2cm}else.}
\end{array}\right.\label{eq:switching control}
\end{equation}

Following (\ref{eq:equivalent control-1}), the equivalent control is given by: \\
\begin{equation}
u_{i_{eq}}=\left\{ \begin{array}{lll}
-\frac{1}{2}\frac{K(Ax+\bar{f}(x))}{K\bar{g}_{i}(x)} &  & \textrm{if }S=0\textrm{ and }K\bar{g}_{i}(x)\neq0,
%\quad\forall x\neq0
\\
\\
0 &  & \textrm{\hspace{-0.4cm}else.}
\end{array}\right.\label{eq:equivalent control}
\end{equation}


It is concluded that the regulation problem for the AAS modeled by (\ref{modelAAS}) can be solved through the application of the sliding mode control defined by (\ref{eq:control-1}) where $u_{i_{s}}$ and $u_{i_{eq}}$ are given by (\ref{eq:switching control}) and (\ref{eq:equivalent control}), respectively.

Next, it will be shown that, for this study case, the stability condition stated in Theorem \ref{Theo} is reduced to $k_1>0$.

Following the developments presented in Section 2,  the control gain vector is chosen as: \\ $$K=\left[\begin{array}{cc}
k_1 & 1\end{array}\right].$$ \\ Then, on the sliding surface, \\ $$S=Kx_{s}=0$$ \\ implies 
\begin{equation}
\label{x1}
x_2=-k_1(x_{1}-x_{1}^{*}).
\end{equation} \\
The following equation is derived from the system dynamics:
$$
\dot{x}_2=\alpha_1x_2+f(x)+g_1(x)u_1+g_2(x)u_2.
$$ \\
Substituting the equivalent control (acting during the sliding mode) given by:
\begin{equation}
u_{i}=u_{i_{eq}}=-\frac{1}{2}\frac{k_{1}x_2+\alpha_1x_2+f(x)}{g_i(x)},
\label{eq:ctrl_eq_x2}
\end{equation}
yields $$\dot{x}_{2}=-k_1x_2.$$


Consider the Lyapunov function \\ $$V(x)=\frac{1}{2}x_{2}^{2},$$ observe that \\
\[
\dot{V}(x)=x_{2}\dot{x}_{2}=-k_1x_{2}^{2},
\] \\
then $\dot{V}(x)<0$ is satisfied for $k_{1}>0$.

\subsection{Numerical results}

In terms of the model parameters, for $K=\left[\begin{array}{cc}
k_{1} & 1\end{array}\right]$, the switching and the equivalent control can be written as follows: \\
\begin{equation}
u_{i_{s}}=\left\{ \begin{array}{lll}
-\phi\frac{\gamma}{g_i} &  & \textrm{if }S\neq0\textrm{ and }g_i\neq0,\\
\\
0 &  & \textrm{else,}
\end{array}\right.\label{eq:switching control-model parameters}
\end{equation} \\ \\
with $\phi>1/2$ and \\
\begin{equation*}
\begin{split}
\gamma=&\text{sgn}((k_1(\theta-\theta_{ref})+\dot{\theta}))\left|\dot{\theta}(k_1+\alpha_1)+f\right|,
%\\
%\eta=&(k_1(\theta-\theta_{ref})+\dot{\theta})g_i.
\end{split}
\end{equation*}\\
\begin{equation}
u_{i_{eq}}=\left\{ \begin{array}{lll}
-\frac{1}{2}\frac{\dot{\theta}(k_1+\alpha_1)+f}{g_i} &  & \textrm{if }S=0\textrm{ and }g_i\neq0,\\
\\
0 &  & \textrm{else.}
\end{array}\right.\label{eq:equivalent control-model parameters}
\end{equation}\\
The numerical values of the proposed system parameters are given in Table~\ref{tab:1}.\\

\begin{table}[hb]
\begin{center}
\caption{AAS parameters.}\label{tab:1}
\begin{tabular}{ccc}
\textbf{Parameter}&\textbf{Value}&\textbf{Units} \\ \hline
 $\alpha_1$ & $-0.01764$ & $s^{-1}$ \\ 
$\alpha_2$ & $0.4079$ & $s^{-2}$ \\ 
$\alpha_3$ & $-0.02352$ & $(kgm)^{-1}$ \\ 
$\alpha_4$ & $-1.435$ & $(kgm)^{-1}$ \\ \hline
\end{tabular}
\end{center}
\end{table}





In this section one can notice that the regulation task is achieved for any controller gains such that $\phi>1/2$ and $k_1 > 0$; however, the response velocity and the controllers size are directly related with the choice of $\phi$ and $k1$.

Fig. \ref{Fig. 1.} shows the evolution of the state for $\theta_{ref}=-5^{\circ}$ and gains $\phi=25$ and $k_1=0.2$. As one can see, the position of the AAS bar starts at the horizontal position with $x_1=0$ and then reaches the reference value, the variable $x_2$, corresponding to the angular velocity of the bar, exhibits the expected behavior: its magnitude increases allowing the regulation task and once the reference value is reached, it goes to zero. In this case, the angular velocity of the bar takes negative values before reaching the stable condition ($x_2=0$) since the reference is negative too.
\begin{figure}[h!]
\centering
\leftskip-0.4cm
\includegraphics[width=10cm, height=5.5cm]{Fig1a}
\caption{Closed loop response of the AAS state with $\theta_{ref}=-5^{\circ}$, $\phi=25$ and $k_1=0.2$.}
\label{Fig. 1.}
\end{figure}

%Parte sup de fig 3

The behavior of the control inputs $u_1$ and $u_2$, under the previous conditions ($\theta_{ref}=-5^{\circ}$ and gains $\phi=25$ and $k_1=0.2$), is showed in Fig. \ref{Fig. 2.}  where the chatter phenomenon, associated to this kind of strategy, is noted; nevertheless, these two control signals remain bounded over the whole regulation task.

%[Aqu’ vendr’a la parte inferior de la Fig. 3]
\begin{figure}[h!]
\centering
\leftskip-0.4cm
\includegraphics[width=10cm, height=5.5cm]{Fig2a}
\caption{Control behavior of the AAS with $\theta_{ref}=-5^{\circ}$, $\phi=25$ and $k_1=0.2$.}
\label{Fig. 2.}
\end{figure}
 

The next paragraphs are devoted to the numerical results obtained for a positive value reference $\theta_{ref}=10^{\circ}$ where the effectiveness of the strategy as well as the effect of the gains $\phi$ and $k_1$ on its performance are highlighted.

Fig. \ref{Fig. 3.} illustrates the evolution of the states $x_1$ and $x_2$ for $\phi=25$ and $k_1=0.2$; in the same way as for the case showed in Fig. \ref{Fig. 1.}, $x_2$ exhibits an expected behavior going through increasing and decreasing positive values before reaching zero that corresponds to the stable angular position of the bar.  

%[Aqu’ vendr’a la parte superior de la Fig. 2]
\begin{figure}[h!]
\centering
\leftskip-0.4cm
\includegraphics[width=10cm, height=5.5cm]{Fig3a}
\caption{Closed loop response of the AAS state with $\theta_{ref}=10^{\circ}$, $\phi=25$ and $k_1=0.2$.}
\label{Fig. 3.}
\end{figure}

A smaller value of the $\phi$ gain generates a slower response as it can be observed in Fig. \ref{Fig. 4.}, where stabilization is reached approximately ten seconds after what is reached with a higher gain value; here $\phi=1.3$ is considered instead of 25, the other parameters remain invariant.

%[Aqu’ vendr’a la parte superior de la Fig. 4]
\begin{figure}[h!]
\centering
\leftskip-0.4cm
\includegraphics[width=10cm, height=5.5cm]{Fig4a}
\caption{Closed loop response of the AAS state with $\theta_{ref}=10^{\circ}$, $\phi=1.3$ and $k_1=0.2$.}
\label{Fig. 4.}
\end{figure}


The behavior of the control inputs for the two choices illustrated previously are displayed in Fig. \ref{Fig. 5.} where one can see that when $\phi=1.3$ the controller takes about 10 seconds to start following the reference angle; however, the controllers magnitude is smaller compared to the case when $\phi=25$.

%[Aqu’ vendr’an las partes inferiores de las Figs. 2 y 4]
\begin{figure}[h!]
\centering
\leftskip-0.4cm
\includegraphics[width=10cm, height=9cm]{Fig5}
\caption{Control signals with $\theta_{ref}=10^{\circ}$, $k_1=0.2$, $\phi=25$ (top) and $\phi=1.3$ (bottom).}
\label{Fig. 5.}
\end{figure}


Concerning the $k_1$ gain, Fig. \ref{Fig. 6.} shows that the higher the value of $k_1$, the faster the system response. In this case, $\phi=1.3$ and $k_1$ was increased from 0.2 to 3.2 allowing the convergence of the trajectories to the desired values in the first few seconds.

%[Aqu’ vendr’a la Fig. 5]
\begin{figure}[h!]
\centering
\leftskip-0.4cm
\includegraphics[width=10cm, height=9cm]{regulation1a}
\caption{Closed loop response of the AAS with $\theta_{ref} = 10^{\circ}$, $\phi = 1.3$ and $k_1 = 3.2$; state behavior (top) and control behavior (bottom).}
\label{Fig. 6.}
\end{figure}

\begin{figure}[h!]
\centering
\leftskip-0.4cm
\includegraphics[width=10cm, height=9cm]{regulation5}
\caption{Closed loop response of the AAS using a smooth approximation of the discontinuity with $c=100$, $\theta_{ref}=10^{\circ}$, $\phi=1.3$ and $k_1=0.2$.}
\label{reg5}
\end{figure}

The price paid for the velocity of the response is a considerable increase in the controllers magnitude. In practice, the controllers size take importance due to the presence of physical limitations of the actuators in a real plant. Usually, it is required that the control inputs satisfy certain constraints related to its magnitude ($|u|<u_{max}$) to be implemented, then, an optimal trade-off between the required response velocity and the controller size must be established.


The discontinuity associated with the nonlinear switching control is the main difficulty in a practical implementation, especially in mechanical systems. Usually, this has been avoided by ``smoothing'' the discontinuity. After doing this, the state trajectories no longer slide on the sliding surface, and instead they evolve in the vicinity of the sliding surface: this is termed as pseudo-sliding (\cite{Hamayun}). 


The discontinuity can be avoided by using the following approximation: \\ $$\text{sgn}(x)\approx\tanh(cx),$$ \\ where the design scalar $c$ controls the smoothness of the sign function (as $c\rightarrow \infty $, the function $\tanh(cx)$ converges to the standard sign function).

Fig. \ref{reg5} shows that the chattering or high frequency switching of the control signal has been eliminated by replacing the term $\text{sgn}(Kx_s)$ by $\tanh(100Kx_s)$ in the switching control. Due to this approximation, the sliding motion will be in the vicinity of the sliding surface. As it is illustrated in Fig. \ref{reg5}, the state behaves in the same way that the response showed in Fig. \ref{Fig. 4.}.












\section{Conclusions}

In this paper a sliding mode control for a class of nonlinear control-affine systems is proposed. The strategy allows providing an efficient sliding mode control consisting of two elements: a switching control law which guarantees the system stability during the reaching phase, and an equivalent control law which aims at keeping the system state on the sliding surface once reached. The conditions on the controller gains for the closed loop system stability are formulated in terms of a nonlinear matrix inequality. The efficacy of the proposed approach was illustrated by tackling the regulation problem in the Aerodynamic Angular System; with this approach, the bar position can be driven to a prescribed angle through the manipulation of the lift forces generated by the aerodynamic effect in the system. An appropriate choice of the control gains allows adjusting the convergence rate and the controllers size.  

Future work will be oriented to the construction of the Aerodynamic Angular System, to its electronic instrumentation, to the establishment of communication between the AAS and a computer and to perform several experimental test to validate the theoretical results presented in this paper. Besides, the trajectory tracking problem will be addressed.



\bibliography{references}     



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%\appendix
%\section{A summary of Latin grammar}    % Each appendix must have a short title.
%\section{Some Latin vocabulary}              % Sections and subsections are supported  
                                                                         % in the appendices.
\end{document}
