# Towards Agrobots: Trajectory Control of an Autonomous Tractor Using Type-2
Fuzzy Logic Controllers

Erdal Kayacan, Erkan Kayacan, Herman Ramon, Okyay Kaynak, and Wouter Saeys E.
Kayacan, E. Kayacan, H. Ramon and W. Saeys are with the Division of
Mechatronics, Biostatistics and Sensors, Department of Biosystems, University
of Leuven (KU Leuven), Kasteelpark Arenberg 30, B-3001 Leuven, Belgium.
e-mail: {erdal.kayacan, erkan.kayacan, herman.ramon,
<EMAIL_ADDRESS>Kaynak is with the Department of Electrical
and Electronics Engineering, Bogazici University, 34342, Istanbul, Turkey.
e-mail<EMAIL_ADDRESS>
###### Abstract

Provision of some autonomous functions to an agricultural vehicle would
lighten the job of the operator but in doing so, accuracy should not be lost
to still obtain an optimal yield. Autonomous navigation of an agricultural
vehicle involves the control of different dynamic subsystems, such as the yaw
angle dynamics and the longitudinal speed dynamics. In this study, a PID
controller is used to control the longitudinal velocity of the tractor. For
the control of the yaw angle dynamics, a PD controller works in parallel with
a type-2 fuzzy neural network. In such an arrangement, the former ensures the
stability of the related subsystem, while the latter learns the system
dynamics and becomes the leading controller. In this way, instead of modeling
the interactions between the subsystems prior to the design of a model-based
control, we develop a control algorithm which learns the interactions on-line
from the measured feedback error. In addition to the control of the stated
subsystems, a kinematic controller is needed to correct the errors in both the
$x$ and the $y$ axis for the trajectory tracking problem of the tractor. To
demonstrate the real-time abilities of the proposed control scheme, an
autonomous tractor is equipped with the use of reasonably priced sensors and
actuators. Experimental results show the efficacy and the efficiency of the
proposed learning algorithm.

###### Index Terms:

Fuzzy-neuro control, sliding mode learning algorithm, type-2 fuzzy logic
systems, autonomous tractor, agricultural vehicles.

##  I Introduction

In recent years, considerable efforts are being made to automate functions in
agricultural machinery and production machines that are still carried out by
humans, through the use of self-learning controllers and continuously adapt
the machine settings to the crop/animal variability and the environmental
conditions.

Agricultural production machines with higher efficiencies will be very
important in the future because of the limited agricultural areas in the world
and constantly rising energy and labor costs. One way to offset the rising
labor costs may be to increase the capacity by increasing the size of the
machines. However, the limits in this direction will soon be reached as there
is a maximum size that will still permit road transportation. On the other
hand, the requirement for the minimization of the energy costs may make the
multi-objective optimization problem in hand a difficult one to solve. A
different approach to increase effectiveness would be to increase the
operational efficiency by the use of advanced learning algorithms, which can
learn the operational dynamics online and adjust the operational parameters
accordingly. The motivation behind the use of self-learning controllers
instead of conventional controllers for the control of agricultural production
machines is that there are different subsystems interacting with each other in
these machines, and well tuning of the controller coefficients simultaneously
is a difficult task. Even if the operator becomes proficient in proper
adjustment of the different controller coefficients, crop/animal variability
and the environmental conditions force the operator to change the machine
settings continuously resulting in the fact that adaptability is a must. In
this study, the trajectory control of an autonomous tractor is considered and,
for the provision of such adaptability, a Takagi-Sugeno-Kang (TSK) type-2
fuzzy neural network (T2FNN) with a sliding mode control (SMC) theory-based
learning algorithm is proposed. Various uncertainties, disturbances and
nonlinearities that would inherently exist in such a system are thus handled.

As for the structure, the combination of a conventional controller and an
intelligent controller is chosen. The former is a proportional-derivative (PD)
controller and the latter is a T2FNN in this study. Such a structure is called
_feedback error learning_ (FEL) in literature; it was originally proposed in
[1] for robot control in which a neural network based controller works with a
PD controller. In this approach, the output of the conventional controller is
used as the learning error signal to train the intelligent controller. A
recent extension of this approach to fuzzy neural networks (FNNs) can be seen
in [2]. This study, presents a further extension by using SMC theory to train
T2FNNs. The novelty of the approach is that instead of trying to minimize an
error function, the learning parameters are tuned by the proposed algorithm in
a way to enforce the error to satisfy a stable equation. The parameter update
rules of T2FNNs are derived and the stability of the learning algorithm in the
Lyapunov sense is proved.

For the training of a FNN or a T2FNN, two methods are widely used in
literature. One of them is gradient-descent-based [3] and therefore requires
the computation of partial derivatives or sensitivity functions. It can be
considered as an extension of the commonly used learning algorithm with back
propagation of the error. There are some drawbacks of this method such as slow
speed of learning, long computational time, and difficulties are met in
determining the convergence and the stability of the learning scheme in an
analytical way. What is more, as is the case with all repetitive algorithms, a
number of numerical robustness issues may emerge when the algorithm is run
over a long period of time [4]. In addition to these drawbacks, the tuning
process can easily be trapped into a local minimum [5]. Another well known
learning algorithm is based on evolutionary computations with genetic
algorithms [6, 7]. Since genetic algorithms-based methods basically do a
random search, they are slower than gradient-descent algorithms and
computationally more intensive. Moreover, the stability of such approaches is
questionable and the optimal values for the stochastic operators are difficult
to derive. In order to overcome these difficulties, SMC theory-based
algorithms have been proposed in [8, 9] for the parameter update rules of
artificial neural networks (ANNs) and FNNs as robust learning algorithms. Even
though the main strength of SMC is its robustness, there are some
disadvantages of it as well. For instance, when the system dynamics are close
to the sliding surface, high frequency oscillations in the control input, also
known as chattering, occur. One of the most common methods to eliminate the
chattering is inserting a boundary layer to replace the corrective control by
an equivalent one when the system is inside this layer [10].

SMC is an approach that guarantees the robustness of a system in the case of
external disturbances, parameter variations and uncertainties and as such has
attracted the attention of many researchers to guarantee robustness in
computationally intelligent architectures [11, 12]. The main idea behind this
control scheme is to restrict the motion of the system in a plane referred to
as the _sliding surface_ , where the predefined function of the error is zero
[13]. SMC-based learning algorithms can, not only make the overall system more
robust, but also ensure faster convergence than the traditional learning
techniques in online tuning of ANNs and type-1 fuzzy neural networks (T1FNNs).
There are various studies in literature that aim to use the robustness
property of SMC in the learning process of ANNs and T1FNNs [14]. Conversely,
the robustness and the stability properties of soft computing-based control
strategies can also be analyzed through the use of SMC theory [15].

The most common tools used in the literature to implement model free designs
are ANNs and fuzzy logic systems (FLSs). On the fuzzy logic theory, Zadeh
argues that _”fuzzy logic is a precise conceptual system of reasoning,
deduction and computation in which the objects of discourse and analysis are,
or are allowed to be, associated with imperfect information. Imperfect
information is information which in one or more respects is imprecise,
uncertain, incomplete, unreliable, vague or partially true”_ [16]. ANNs are
well known for their representation capability, even in the case of highly
nonlinear systems. FNNs combine the advantages of both techniques, _i.e._ the
fuzzy reasoning ability of FLSs and the learning ability of ANNs.

The use of type-2 fuzzy sets has an advantage when it is difficult to
determine the place of the membership functions (MFs) precisely [17]. Since
there are infinite number of type-1 MFs within an interval in a type-2 MF,
type-2 fuzzy logic systems (T2FLSs) appear to be a more promising method than
their type-1 counterparts for handling uncertainties such as noisy data and
changing environments [18, 19]. In [20, 21], the effects of the measurement
noise in type-1 and type-2 fuzzy logic controllers and identifiers were
simulated to perform a comparative analysis. It was concluded that the use of
a type-2 fuzzy logic controller can be a better option than the use of its
type-1 counterpart. These claims have been verified in a real-time application
in [22]. Even if there exists a number of papers in literature that claim that
the performance of T2FLSs is better than T1FLSs under noisy conditions, this
claim is tried to be justified by simulation or real-time studies only for
some specific systems. However, in [23], a simpler T2FLS is considered with a
novel MF in which the effect of input noise in the rule base is shown
numerically in a general way, and it is concluded that T2FLSs should be used
when needed, i.e. in the presence of noise and uncertainties in the system.

Whereas the secondary MFs can take values in the interval of [0,1] in
generalized T2FLSs, they are uniform functions that only take on values of 1
in interval T2FLSs. Since the general T2FLSs are computationally very
demanding (this is because the type-reduction is computationally expensive),
the use of interval T2FLSs is more commonly seen in literature. In an interval
TSK T2FLS, there exists a design parameter that weights the sharing of lower
and upper firing levels of each fired rule. That parameter can either be fixed
or optimized online. While the least mean square method is generally used to
find the optimal value of this parameter, an SMC theory-based learning
algorithm is proposed for tuning it in [24].

The contributions of this paper are as follows:

  * •

An SMC theory-based learning algorithm is proposed for the parameter tuning of
T2FNNs, including the design parameter that weights the sharing of the lower
and the upper firing levels of each fired rule, and its stability in the
Lyapunov sense is proved.

  * •

The proposed learning algorithm is tested on the trajectory tracking problem
of an autonomous agricultural tractor in the presence of various
nonlinearities and uncertainties in real-time.

  * •

A practical mechatronic system, illustrating how control, sensing and
actuation can be integrated to achieve an intelligent system, is designed and
presented.

The body of the paper contains five sections: In Section II, the dynamic
equations of the autonomous tractor are presented. In Section III, the overall
control is scheme is given. The proposed sliding mode FEL approach is
presented and the parameter update rules for T2FNNs are proposed for the case
of triangular MFs in Section IV. In Section V, real-time results are given.
Finally, conclusions are presented in Section VI.

##  II Mathematical Description of the Tractor

###  II-A Kinematic Model

The schematic diagram of the autonomous tractor is presented in Fig. 1.

Figure 1: The autonomous tractor

The linear velocities $\dot{x}$, $\dot{y}$ and the yaw rate $\dot{\psi}$ at
the $R$ point are written as follows:

$\displaystyle\left[\begin{array}[]{c}\dot{x}\\\ \dot{y}\\\ \dot{\psi}\\\
\end{array}\right]=\left[\begin{array}[]{c}{u}\cos{\psi}\\\ {u}\sin{\psi}\\\
\frac{utan{\delta}}{L}\\\ \end{array}\right]$ (7)

where $u$, $\psi$, $\delta$ and $L$ represent the longitudinal velocity, the
yaw angle defined on the point $R$ on the tractor, the steering angle of the
front wheel, the distance between the front and the rear axles of the tractor,
respectively.

Considering the center of gravity (CG) shown in Fig. 1, the linear velocities
$\dot{x}$, $\dot{y}$ and the yaw rate $\dot{\psi}$ of CG can be written as
follows:

$\displaystyle\left[\begin{array}[]{c}\dot{x}\\\ \dot{y}\\\ \dot{\psi}\\\
\end{array}\right]=\left[\begin{array}[]{c}{u}\cos{\psi}-v\sin{\psi}\\\
{u}\sin{\psi}+v\cos{\psi}\\\ \frac{utan{\delta}}{L}\\\ \end{array}\right]$
(14)

where $v$ equals to the multiplication of $\dot{\psi}$ and $l_{r}$.

###  II-B The Yaw Dynamics Model

The velocities and the side-slip angles on the rigid body of the tractor are
presented in Fig. 2(a). Similarly, the forces on the rigid body of the tractor
are shown in Fig. 2(b).

(a)

(b)

Figure 2: Dynamic bicycle model for a tractor: (a) velocities and side-slip
angles (b) forces on the rigid body of the system

The lateral dynamics of the tractor can be written as follows:

$m(\dot{v}+u\gamma)=F_{t,f}\sin\delta+F_{l,f}\cos\delta+F_{l,r}$ (15)

where $m$, $v$, $u$, $\gamma$, $F_{t,f}$, $F_{l,f}$, $F_{l,r}$ and $\delta$
represent the mass of the tractor, the lateral velocity of the CG, the
longitudinal velocity of the CG, the yaw rate, the traction and the lateral
forces on the front wheel, the lateral force on the rear wheel and the
steering angle of the front wheel, respectively.

The yaw dynamics of the tractor are written as follows:

$\displaystyle I_{z}\dot{\gamma}$ $\displaystyle=$ $\displaystyle
l_{f}(F_{t,f}\sin\delta+F_{l,f}\cos\delta)-l_{r}F_{l,r}$ (16)

where $l_{f},l_{r}$ and $I_{z}$ respectively represent the distance between
the front axle and the CG of the tractor, the distance between the rear axle
and the CG of the tractor, and the inertial moment of the tractor.

The tire side-slip angles must be calculated in order to determine the forces
caused by the slip. It is assumed that the steering angle of the front wheel
is small, and this allows to make the following approximations:
$\sin\delta\approx\delta$ and $\cos\delta\approx 1$. The side-slip angles of
the front ($\alpha_{f}$) and the rear tires ($\alpha_{r}$) are written as
follows:

$\alpha_{f}=\frac{v+l_{f}\gamma}{u}-\delta\;\;\text{and}\;\;\alpha_{r}=\frac{v-l_{r}\gamma}{u}$
(17)

To determine the lateral force on the tire, there are many different
approaches in literature. In this study, the lateral tire forces are
calculated using a linear model which assumes these to be proportional to the
side-slip angles in [25, 26, 27]

$F_{l,i}=-C_{\alpha,i}\alpha_{i}\quad\quad i=\\{f,r\\}$ (18)

where $C_{\alpha,i}$, $i=\\{f,r\\}$, represents the cornering stiffness of the
tires of the tractor. The tire cornering stiffness parameters are the averaged
slopes of the lateral force characteristics in this method.

The equations of yaw motion of the autonomous tractor are written in state
space form by combining (15), (16), (17) and (18) as follows:

$\displaystyle\left[\begin{array}[]{c}\dot{v}\\\ \dot{\gamma}\\\
\end{array}\right]=\left[\begin{array}[]{ccc}A_{11}&A_{12}\\\ A_{21}&A_{22}\\\
\end{array}\right]\left[\begin{array}[]{c}v\\\ \gamma\\\
\end{array}\right]+\left[\begin{array}[]{c}B_{1}\\\ B_{2}\\\
\end{array}\right]\delta(t)$ (27)

where

$\displaystyle A_{11}$ $\displaystyle=$
$\displaystyle-\frac{C_{\alpha,f}+C_{\alpha,r}}{mu},$ $\displaystyle A_{12}$
$\displaystyle=$
$\displaystyle\frac{-l_{f}C_{\alpha,f}+l_{r}C_{\alpha,r}}{mu}-u,$
$\displaystyle A_{21}$ $\displaystyle=$
$\displaystyle\frac{-l_{f}C_{\alpha,f}+l_{r}C_{\alpha,r}}{I_{z}u},$
$\displaystyle A_{22}$ $\displaystyle=$
$\displaystyle-\frac{l_{f}^{2}C_{\alpha,f}+l_{r}^{2}C_{\alpha,r}}{I_{z}u},$
$\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle\frac{C_{\alpha,f}}{m}\;\;,\;\;B_{2}=\frac{l_{f}C_{\alpha,f}}{I_{z}}$
(28)

##  III Overall Control Scheme

###  III-A Kinematic Controller

The kinematic model is re-written in a state-space as follows:

$\displaystyle\left[\begin{array}[]{c}\dot{x}\\\ \dot{y}\\\ \dot{\psi}\\\
\end{array}\right]=\left[\begin{array}[]{cc}\cos{\psi}&-l_{r}\sin{\psi}\\\
\sin{\psi}&l_{r}\cos{\psi}\\\ 0&1\\\
\end{array}\right]\left[\begin{array}[]{c}{u}\\\ \gamma\\\ \end{array}\right]$
(37)

where lateral velocity $v$ equals to $\gamma l_{r}$.

An inverse kinematic model is needed to calculate the reference speed and the
yaw rate for the tractor. It is written as follows:

$\displaystyle\left[\begin{array}[]{c}u\\\ \gamma\\\
\end{array}\right]=\left[\begin{array}[]{cc}\cos{\psi}&\sin{\psi}\\\
-\frac{1}{l_{r}}\sin{\psi}&\frac{1}{l_{r}}\cos{\psi}\\\
\end{array}\right]\left[\begin{array}[]{c}\dot{x}\\\ \dot{y}\\\
\end{array}\right]$ (44)

and the kinematic control law proposed in [28] to be applied to the tractor
for trajectory tracking control is written as:

$\displaystyle\left[\begin{array}[]{c}u_{ref}\\\ \gamma_{ref}\\\
\end{array}\right]=\left[\begin{array}[]{cc}\cos{\psi}&\sin{\psi}\\\
-\frac{1}{l_{r}}\sin{\psi}&\frac{1}{l_{r}}\cos{\psi}\\\
\end{array}\right]\left[\begin{array}[]{c}\dot{x_{d}}+k_{s}\tanh{(k_{e}e_{x})}\\\
\dot{y_{d}}+k_{s}\tanh{(k_{e}e_{y})}\\\ \end{array}\right]$ (51)

where $e_{x}=x_{d}-x$ and $e_{y}=y_{d}-y$ are the current position errors in
the axes $X$ and $Y$, respectively. The parameter $k_{e}$ is the gain of the
controller and $k_{s}$ is the saturation constant. The coordinates $(x,y)$ and
$(x_{d},y_{d})$ are the current and the desired coordinates, respectively. The
parameters $u_{ref}$ and $\gamma_{ref}$ are the generated references for the
speed and the yaw rate controllers.

###  III-B Dynamic Controllers

The proposed control scheme used in this study is illustrated in Fig. 4. The
arrow in Fig. 4 indicates that the output of the PD controller is used to tune
the parameters of the T2FNN. The output of the PD+T2FNN controller is the
steering angle of the front wheel. A low level controller is used to control
the steering mechanism. A proportional-integral-derivative (PID) controller is
used for the control of longitudinal velocity. For the control of yaw
dynamics, a conventional controller (such as a PD controller) works in
parallel with an intelligent controller. One of the main concerns of this
study is to implement a novel learning algorithm for T2FNNs by using some
novel SMC theory-based learning rules to a real time system. As a testing
environment, the yaw dynamics of the tractor in our laboratory has been
chosen. On the other hand, the longitudinal dynamics could, of course, be
selected. Moreover, there could be two T2FNNs running on the control of the
two subsystems (yaw dynamics and longitudinal dynamics) simultaneously. The
reason for such a selection in this study is that the yaw dynamics control of
the tractor is more common in agricultural machines. Even if a time-based
trajectory (both the yaw angle and the longitudinal speed of the tractor are
controlled simultaneously) is given to the system in this study, a space-based
trajectory (the longitudinal speed is fixed and only the yaw dynamics is
controlled) is also very common in agricultural applications. Based-on these
concerns, the yaw dynamics of the tractor has been chosen for the
implementation for the novel learning algorithm proposed in this study.

##  IV Type-2 Fuzzy-Neuro Control Approach

###  IV-A Type-2 Fuzzy Triangular Membership Functions

Among the existing MFs in literature (that include a novel one proposed by the
authors [23]), triangular MFs are preferred in this paper. Their advantages
over the others are mentioned in [29].

The mathematical expression for a type-1 fuzzy triangular MF can be written
as:

${{\mu}({{x}})}=\left\\{\begin{array}[]{l
l}1-\left|\frac{{{x}}-{{c}}}{{{d}}}\right|&\left|{{x}}-{{c}}\right|<{{d}}\\\
0&otherwise\\\ \end{array}\right.$ (52)

where $c$ and $d$ are the center and the width of the MF and $x$ is the input
vector. On the other hand, type-2 fuzzy triangular MFs with uncertain width
and uncertain center are shown in Figs. 3(a) and 3(b), respectively. In this
paper, MFs with uncertain width are preferred in the antecedent parts of the
fuzzy _if-then_ rules.

(a)

(b)

Figure 3: A type-2 fuzzy triangular MF with uncertain width (a) and uncertain
center (b).
Figure 4: Block diagram of the proposed adaptive fuzzy neuro scheme

Each MF in the antecedent part is represented by an _upper_ (red line in Fig.
3) and a _lower_ (blue line in Fig. 3) MF. The membership values for the upper
and the lower MFs are denoted as
$\overline{\mu}(x){\rm\;and\;\;}\underline{\mu}(x){\rm},$ respectively. The
strength of the rule $R_{ij}$ is calculated as a _T_ -norm of the MFs in the
premise part by using a multiplication operator:

$\underline{W_{ij}}=\underline{\mu_{1i}(x_{1})}\;\underline{\mu_{2j}(x_{2})}\;\;\text{and}\;\;\overline{W_{ij}}=\overline{\mu_{1i}(x_{1})}\;\overline{\mu_{2j}(x_{2})}$
(53)

The type-2 fuzzy triangular membership values $\underline{\mu_{1i}(x_{1})}$,
$\overline{\mu_{1i}(x_{1})}$, $\underline{\mu_{2j}(x_{2})}$, and
$\overline{\mu_{2j}(x_{2})}$ of the inputs $x_{1}$ and $x_{2}$ in the above
expression have the following appearance:

$\underline{{{\mu}_{1i}}({{x}_{1}})}=\left\\{\begin{array}[]{l
l}1-\left|\frac{{{x}_{1}}-{{{c}_{1i}}}}{\underline{{{d}_{1i}}}}\right|&\left|{{x}_{1}}-{{{c}_{1i}}}\right|<\underline{{{d}_{1i}}}\\\
0&otherwise\\\ \end{array}\right.$ (54)

$\overline{{{\mu}_{1i}}({{x}_{1}})}=\left\\{\begin{array}[]{l
l}1-\left|\frac{{{x}_{1}}-{{{c}_{1i}}}}{\overline{{{d}_{1i}}}}\right|&\left|{{x}_{1}}-{{{c}_{1i}}}\right|<\overline{{{d}_{1i}}}\\\
0&otherwise\\\ \end{array}\right.$

$\underline{{{\mu}_{2j}}({{x}_{2}})}=\left\\{\begin{array}[]{l
l}1-\left|\frac{{{x}_{2}}-{{{c}_{2j}}}}{\underline{{{d}_{2j}}}}\right|&\left|{{x}_{2}}-{{{c}_{2j}}}\right|<\underline{{{d}_{2j}}}\\\
0&otherwise\\\ \end{array}\right.$

$\overline{{{\mu}_{2j}}({{x}_{2}})}=\left\\{\begin{array}[]{l
l}1-\left|\frac{{{x}_{2}}-{{{c}_{2j}}}}{\overline{{{d}_{2j}}}}\right|&\left|{{x}_{2}}-{{{c}_{2j}}}\right|<\overline{{{d}_{2j}}}\\\
0&otherwise\\\ \end{array}\right.$

###  IV-B Interval Type-2 A2-CO TSK Model

The interval T2FLS considered in this paper uses type-2 triangular MFs in the
premise part and crisp numbers in the consequent part. This structure is
called A2-C0 fuzzy system [30], and it is shown in Fig. 5. The fuzzy _If-Then_
rule $R_{ij}$ of a $zero^{th}$-order type-2 TSK model with two input variables
in which the consequent part is a crisp number can be defined as follows:

$R_{ij}:\;\;\text{If}\;x_{1}\;\text{is}\;\;\tilde{A}_{1i}\;\;\text{and}\;x_{2}\;\text{is}\;\;\tilde{A}_{2j},\;\;\text{then}\;f_{ij}=d_{ij}$
(55)

The followings are the operations in each Layer in Fig. 5: In Layer 1, the
input signals feed the system. The related figure shows the system for two
inputs which are the error and the time derivative of the error. In Layer 2,
the membership degrees $\underline{\mu}$ and $\overline{\mu}$ are determined
for each input signal entering the system. Layer 3 calculates the firing
strengths of the rules which are realized using the _prod_ t-norm operator
using (53). Layer 4 determines the outputs of the linear functions $f_{ij}$
$(i=1,\dots,I$ and $j=1,\dots,J)$, in the consequent parts for the two inputs
case.

$f_{ij}=d_{ij}$ (56)

Layer 5 computes the product of the membership degrees
$\underline{W_{ij}}{\rm\;\;and\;\;}\overline{W_{ij}}{\rm\;}$ and linear
functions $f_{ij}$. Two summation blocks are in Layer 6. One of these blocks
computes the sum of the output signals from Layer 5 (the numerator part of
(57)) and the other block computes the sum of the output signal of Layer 3
(the denominator part of (57)). Finally, Layer 7 calculates the output of the
network using (58).

Figure 5: The structure of the proposed T2FNN for two inputs

The output of the network is calculated as follows:

$\tau_{n}=\int_{W_{11}\epsilon[\underline{W}_{11},\overline{W}_{11}]}\ldots\int_{W_{IJ}\epsilon[\underline{W}_{IJ},\overline{W}_{IJ}]}1\bigg{/}\frac{\sum_{i=1}^{I}\sum_{j=1}^{J}W_{ij}(x)f_{ij}}{\sum_{i=1}^{I}\sum_{j=1}^{J}W_{ij}(x)}$
(57)

where $f_{ij}$ is given by the _If-Then_ rule. The inference engine used in
this paper replaces the type-reduction which is given as:

$\tau_{n}=\frac{q(t)\sum_{i=1}^{I}\sum_{j=1}^{J}\underline{W}_{ij}f_{ij}}{\sum_{i=1}^{I}\sum_{j=1}^{J}\underline{W}_{ij}}+\frac{\big{(}1-q(t)\big{)}\sum_{i=1}^{I}\sum_{j=1}^{J}\overline{W}_{ij}f_{ij}}{\sum_{i=1}^{I}\sum_{j=1}^{J}\overline{W}_{ij}}$
(58)

The design parameter $q$, weights the sharing of the lower and the upper
firing levels of each fired rule [30]. This parameter can be a constant (equal
to $0.5$ in most cases) or a time varying parameter. In this paper, the latter
is preferred. In other words, the parameter update rules and the proof of the
stability of the learning process are given for the case of a time varying
$q$.

After the normalization of (58), the output signal of the T2FNN will obtain
the following form:

$\tau_{n}=q(t)\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}\widetilde{\underline{W_{ij}}}+\big{(}1-q(t)\big{)}\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}\widetilde{\overline{W_{ij}}}$
(59)

where $\widetilde{\underline{W_{ij}}}$ and $\widetilde{\overline{W_{ij}}}$ are
the normalized values of the lower and the upper output signals of the neuron
$ij$ from the second hidden layer of the network:

$\displaystyle\widetilde{\underline{{W_{ij}}}}=\frac{\underline{W_{ij}}}{\sum_{i=1}^{I}\sum_{j=1}^{J}\underline{W_{ij}}}\mbox{
and
}\widetilde{\overline{{W_{ij}}}}=\frac{\overline{W_{ij}}}{\sum_{i=1}^{I}\sum_{j=1}^{J}\overline{W_{ij}}}$

The following vectors can be specified:

$\widetilde{\underline{{W}}}\left(t\right)=\left[\widetilde{\underline{{W_{11}}}}\left(t\right)\;\widetilde{\underline{{W_{12}}}}\left(t\right)...\;\widetilde{\underline{{W_{21}}}}\left(t\right)\;...\;\widetilde{\underline{{W_{ij}}}}\left(t\right)\;...\;\widetilde{\underline{W}_{IJ}}\left(t\right)\right]^{T}$

$\widetilde{\overline{{W}}}\left(t\right)=\left[\widetilde{\overline{{W_{11}}}}\left(t\right)\;\widetilde{\overline{{W_{12}}}}\left(t\right)...\;\widetilde{\overline{{W_{21}}}}\left(t\right)\;...\;\widetilde{\overline{{W_{ij}}}}\left(t\right)\;...\;\widetilde{\overline{W}_{IJ}}\left(t\right)\right]^{T}$

$F=[f_{11}\;f_{12}\;...\;f_{21}\;...\;f_{ij}\;...\;f_{IJ}]$

The following assumptions have been used in this investigation: Both the input
signals $x_{1}(t)$ and $x_{2}(t)$, and their time derivatives can be
considered bounded:

$|x_{1}(t)|\leq\widetilde{B_{x}},\hskip
7.11317pt|x_{2}(t)|\leq\widetilde{B_{x}}\hskip 14.22636pt\forall t$ (60)

$|\dot{x}_{1}(t)|\leq\widetilde{B_{\dot{x}}},\hskip
7.11317pt|\dot{x}_{2}(t)|\leq\widetilde{B_{\dot{x}}}\hskip 14.22636pt\forall
t$ (61)

where $\widetilde{B_{x}}$ and $\widetilde{B_{\dot{x}}}$ are assumed to be some
known positive constants. It is obvious that
$0<\widetilde{\underline{{W_{ij}}}}\leq 1$ and
$0<\widetilde{\overline{{W_{ij}}}}\leq 1$. In addition, it can be easily seen
that $\sum_{i=1}^{I}\sum_{j=1}^{J}\widetilde{\underline{{W_{ij}}}}=1$ and
$\sum_{i=1}^{I}\sum_{j=1}^{J}\widetilde{\overline{{W_{ij}}}}=1$. It is also
considered that, $\tau$ and $\dot{\tau}$ will be bounded signals too, _i.e._

$\left|\tau(t)\right|<B_{\tau},\hskip
7.11317pt\left|\dot{\tau}\left(t\right)\right|<B_{\dot{\tau}}\quad\forall t$
(62)

where $B_{\tau}$ and $B_{\dot{\tau}}$ are some known positive constants.

###  IV-C The SMC Theory-based Learning Algorithm

Using the principles of SMC theory [31] the zero value of the learning error
coordinate $\tau_{c}\left(t\right)$ can be defined as a time-varying sliding
surface, _i.e._ ,

$S_{c}\left(\tau_{n},\tau\right)=\tau_{c}\left(t\right)=\tau_{n}\left(t\right)+\tau\left(t\right)=0$
(63)

which is the condition that the T2FNN is trained to become a nonlinear
regulator to obtain the desired response during the tracking-error convergence
movement by compensating the nonlinearity of the controlled plant.

The sliding surface for the nonlinear system under control
$S_{p}\left(e,\dot{e}\right)$ is defined as:

$S_{p}\left(e,\dot{e}\right)=\dot{e}+\chi e$ (64)

with $\chi$ being a positive constant determining the slope of the sliding
surface.

Definition: A sliding motion will appear on the sliding manifold
$S_{c}\left(\tau_{n},\tau\right)=\tau_{c}\left(t\right)=0$ after a time
$t_{h}$, if the condition
$S_{c}(t)\dot{S}_{c}(t)=\tau_{c}\left(t\right)\dot{\tau}_{c}\left(t\right)<0$
is satisfied for all $t$ in some nontrivial semi-open subinterval of time of
the form $\left[t,t_{h}\right)\subset\left(0,t_{h}\right)$.

It is desired to devise a dynamical feedback adaptation mechanism, or an
online learning algorithm for the parameters of the T2FNN considered, such
that the sliding mode condition of the above definition is enforced.

###  IV-D The Proposed Parameter Update Rules for the T2FNN

The parameter update rules for the T2FNN proposed in this paper which has two
inputs are given by the following theorem.

_Theorem 1:_ If the adaptation laws for the parameters of the considered T2FNN
are chosen as:

$\underline{\dot{c}_{1i}}=\overline{\dot{c}_{1i}}={\dot{c}_{1i}}=\dot{x}_{1}$
(65)
$\underline{\dot{c}_{2j}}=\overline{\dot{c}_{2j}}={\dot{c}_{2j}}=\dot{x}_{2}$
(66)
$\underline{\dot{d}_{1i}}=\underline{\mu_{1i}}\frac{-\alpha\underline{d_{1i}}^{2}}{x_{1}-{c_{1i}}}sgn(\tau_{c})sgn\bigg{(}\frac{x_{1}-{c_{1i}}}{\underline{d_{1i}}}\bigg{)}$
(67)
$\overline{\dot{d}_{1i}}=\overline{\mu_{1i}}\frac{-\alpha\overline{d_{1i}}^{2}}{x_{1}-{c_{1i}}}sgn(\tau_{c})sgn\bigg{(}\frac{x_{1}-{c_{1i}}}{\overline{d_{1i}}}\bigg{)}$
(68)
$\underline{\dot{d}_{2j}}=\underline{\mu_{2j}}\frac{-\alpha\underline{d_{2j}}^{2}}{x_{2}-{c_{2j}}}sgn(\tau_{c})sgn\bigg{(}\frac{x_{2}-{c_{2j}}}{\underline{d_{2j}}}\bigg{)}$
(69)
$\overline{\dot{d}_{2j}}=\overline{\mu_{2j}}\frac{-\alpha\overline{d_{2j}}^{2}}{x_{2}-{c_{2j}}}sgn(\tau_{c})sgn\bigg{(}\frac{x_{2}-{c_{2j}}}{\overline{d_{2j}}}\bigg{)}$
(70)

$\dot{f_{ij}}=-\frac{\Big{(}q(t)\underline{\widetilde{W_{ij}}}+\big{(}1-q(t)\big{)}\widetilde{\overline{W_{ij}}}\Big{)}\alpha
sgn(\tau_{c})}{\Big{(}q(t)\underline{\widetilde{W}}+\big{(}1-q(t)\big{)}\widetilde{\overline{W}}\Big{)}^{T}\Big{(}q(t)\underline{\widetilde{W}}+\big{(}1-q(t)\big{)}\widetilde{\overline{W}}\Big{)}}$
(71) $\dot{q}(t)=-\frac{\alpha
sgn(\tau_{c})}{F(\underline{\widetilde{W}}-\widetilde{\overline{W}})^{T}}$
(72)

where $\alpha$ is a sufficiently large positive design constant satisfying the
inequality below:

$\alpha>B_{\dot{\tau}}$ (73)

Then, given an arbitrary initial condition $\tau_{c}(0)$, the learning error
$\tau_{c}(t)$ will converge to zero within a finite time $t_{h}$.

Proof: The reader is referred to Appendix A.

In order to avoid division by zero in the adaptation laws of (67)-(72) an
instruction is included in the algorithm to make the denominator equal to
$0.001$ when its calculated value is smaller than this threshold.

It is well-known that sliding mode control suffers from high-frequency
oscillations in the control input, which are called _chattering_. The
following are the two common methods used to eliminate chattering [32]:

  1. 1.

Using a saturation function to replace the signum function.

  2. 2.

Inserting a boundary layer so that an equivalent control replaces the
corrective one when the system is inside this layer.

In order to reduce the chattering effect, the following function is used in
this paper with $\delta_{s}=0.05$ instead of the signum function in the
dynamic strategy described in (67)-(72).

$\textrm{sgn}(\tau_{c}):=\frac{\tau_{c}}{|\tau_{c}|+\delta_{s}}$ (74)

where $\delta_{s}=0.05$.

The relation between the sliding line $S_{p}$ and the zero adaptive learning
error level $S_{c}$ is determined by the following equation:

$S_{c}=\tau_{c}=k_{D}\dot{e}+k_{P}e=k_{D}\left(\dot{e}+\frac{k_{p}}{k_{D}}e\right)=k_{D}S_{p}$
(75)

The tracking performance of the feedback control system can be analyzed by
introducing the following Lyapunov function candidate:

$V_{p}=\frac{1}{2}S_{p}^{2}$ (76)

Theorem 2: If the adaptation strategy for the adjustable parameters of the
T2FNN is chosen as in (65)-(72), then the negative definiteness of the time
derivative of the Lyapunov function in (76) is ensured.

Proof: The reader is referred to Appendix B.

_Remark_ : The obtained result means that, assuming that the SMC task is
achievable, using $\tau_{c}$ as a learning error for the T2FNN together with
the adaptation laws (65)-(72) enforces the desired reaching mode followed by a
sliding regime for the system under control.

##  V Experimental Setup

The global objective in the following real-time experiments is to track a
time-based trajectory with the Case New Holland TZ25DA tractor shown in Fig.
6.

Figure 6: The autonomous tractor

###  V-A Global Positioning System

A global positioning system (GPS) antenna is located straight up the center of
the tractor rear axle to provide highly accurate positional information for
the autonomous tractor. The height of the antenna is 2m above ground level. It
is connected to a Septentrio AsteRx2eH RTK-DGPS receiver (Septentrio Satellite
Navigation NV, Belgium) with a specified position accuracy of 2cm at a 20-Hz
sampling frequency (sampling period $T_{s}=0.05$ s). The RTK correction
signals are obtained from the Flepos network through a wireless internet
connection established with a Digi Connect WAN 3G modem.

###  V-B Hardware, Software and Sensors

The block diagram of hardware is shown in Fig. 7. The GPS receiver and the
internet modem are connected to a real time operating system (PXI platform,
National Instrument Corporation, USA) via a RS232 serial communication. The
PXI system gathers the steering angles and the GPS data, and it controls the
tractor by sending signals to the actuators. A laptop connected via a wireless
network to the PXI system functions as the user interface of the autonomous
tractor. The control algorithms are implemented in $LabVIEW^{TM}$ version
2011, National Instrument, USA. They are executed in real time on the PXI and
updated at a rate of 20-Hz.

Figure 7: Block diagram of the hardware

The PD+T2FNN control algorithm calculates the desired steering angle for the
front wheels. In the inner closed loop, the steering mechanism is controlled
by using an electro-hydraulic valve from Sauer Danfoss with a flow of 12
liter/min. The electro-hydraulic valve characteristics are highly nonlinear
and include a saturation and a dead-band region. The voltage, limited between
$0-12$ volt and the steering angle, limited between $\pm 45^{\circ}$
constitute the input and the output for the steering system, respectively. The
position of the front wheels is measured using a potentiometer mounted on the
front axle, yielding a position measurement with a resolution of $1^{\circ}$.
The position of the front wheel is measured 100 times at every 50 ms. The data
are filtered by using a Savitzky-Golay filter which tends to preserve features
of the distribution such as relative maxima, minima and width. After filtering
process, the steering angle is found by taking the mean value of the filtered
data. This procedure is repeated every 50 ms.

The speed of the tractor is controlled by using an electro-mechanic valve. Two
PID type controllers are used in cascade fashion. The one in the outer loop
generations the desired pedal position with respect to the speed of the
tractor and the one in the inner loop generates the voltage value for the
electro-mechanic valve with respect to the pedal position. Like in the
measuring of the steering angle, the pedal position is measured 100 times at
every 50 ms and the data are filtered by using a Savitzky-Golay filter and
taking the mean value of the filtered data. Since the measured speed signal
coming from the GPS is noisy, a discrete Kalman filter (KF) is used to reduce
noise. A position-velocity model described in [33] is used where vehicle
velocity is assumed as a random-walk process. The KF assumes that the vehicle
moves with a constant velocity between discrete-time steps. The state vector
of the model used in the KF and the state transition matrix are as follows:

$\displaystyle\widehat{x}_{k+1}$ $\displaystyle=$
$\displaystyle\Phi(T_{s})\widehat{x}_{k}$ (85) $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}1&T_{s}&0&0\\\ 0&1&0&0\\\
0&0&1&T_{s}\\\ 0&0&0&1\\\ \end{array}\right]\left[\begin{array}[]{c}x_{k}\\\
v_{x,k}\\\ y_{k}\\\ v_{y,k}\\\ \end{array}\right]$

where $\Phi(T_{s})$, $v_{x,k}$ and $v_{y,k}$ are the state transition matrix,
easting and northing velocities coming from the GPS, respectively.

###  V-C State Estimation

Some states of the autonomous tractor cannot be measured. Even when the states
can be measured directly, the measurements will contain delay and noise.
Moreover, at some time steps, no useful position data are obtained from the
GPS receiver.

An extended Kalman filter (EKF) is used for state estimation. Since the GPS
antenna is located at the R point on the tractor, the kinematic model in (7)
is used. The discrete-time kinematic model used by the EKF is written with a
sampling interval $T_{s}$ as follows:

$\displaystyle x_{k+1}$ $\displaystyle=$ $\displaystyle
x_{k}+T_{s}u_{k}\cos{\psi_{k}}$ $\displaystyle y_{k+1}$ $\displaystyle=$
$\displaystyle y_{k}+T_{s}u_{k}\sin{\psi_{k}}$ $\displaystyle\psi_{k+1}$
$\displaystyle=$ $\displaystyle\psi_{k}+T_{s}u_{k}\frac{\tan{\delta_{k}}}{L}$
(86)

The general form of the estimated system model is:

$\displaystyle\widehat{x}_{k+1}$ $\displaystyle=$ $\displaystyle
f(\widehat{x}_{k},u_{k})+w_{k}$ $\displaystyle\widehat{y}_{k+1}$
$\displaystyle=$ $\displaystyle h(\widehat{x}_{k})+v_{k}$ (87)

where $f$ is the estimation model for the system and $h$ is the measurement
function. The differences between the kinematic model and the estimation model
are the process noise $w_{k}$ and the observation noise $v_{k}$ both in the
state and the measurement equations. They are both assumed to be independent
with zero mean multivariate Gaussian noises with covariance matrices $Q_{k}$
and $R_{k}$, respectively:

$\displaystyle w_{k}\backsim N(0,Q_{k})$ $\displaystyle v_{k}\backsim
N(0,R_{k})$ (88)

Since only one GPS antenna was mounted on the tractor, the yaw angle of the
tractor was not measured directly. It is to be noted that the knowledge of the
yaw angle of the tractor plays a very important role in the accuracy of
trajectory tracking control. The estimated value of the yaw angle is used in
the inverse kinematic model to generate the desired speed and the desired yaw
rate for the system. The inputs of the EKF are the position and the velocity,
the velocity values from GPS and the steering angle value from the
potentiometer in the front wheels. The outputs of the EKF are the position of
the tractor in the x- and the y-coordinates system and the yaw angle. These
estimated values are then used in the trajectory control.

###  V-D Experimental Results

An 8-shaped reference trajectory is applied to the system. The reference and
the actual trajectories of the system, both the longitudinal and the lateral
error values on the related trajectory are shown in Fig. 8, Fig. 9 and Fig. 10
for two different controllers, respectively. The results show that the control
scheme consisting of a T2FNN working in parallel with a PD controller gives a
better trajectory following accuracy than the one where only a PD controller
acts alone. It can here be argued that the performance of the conventional
controller acting alone can be improved by better tuning, but as it has
already been stated that in real life, this is a challenging task; because in
addition to the interactions of the subsystems, there exist unmodeled dynamics
and uncertainties in real world applications. Thus, the proposed control
structure, consisting of an intelligent controller and a conventional
controller would be preferable in real life.

In Fig. 8, while the dotted lines show the first turns, the solid lines
represent the second turns. The control accuracy for the PD controller is the
same for the first and second turn which is the expected case. However, the
control accuracy for the T2FNN working in parallel with a PD controller is
better for the second turn. We can observe similar behaviour in Fig. 9 and
Fig. 10. The mean square of the lateral error in Fig. 10 is equal to 0.2575
and 0.1803 for the case of the PD controller working alone and the case of the
PD controller working in parallel with the T2FNN, respectively. The results
show a performance improvement of 30% in the latter case.

Figure 8: The trajectory of the autonomous tractor
Figure 9: The longitudinal error
Figure 10: The lateral error

Figure 11 shows the control signals coming from the conventional PD controller
and T2FNN when the PD controller works in parallel with the T2FNN. As can be
seen from Fig. 11, at the beginning (in the first turn), the dominating
control signal is the one coming from the PD controller. After the first turn
(starting from $120^{th}$ second), the T2FNN is able to take over the control,
thus becoming the leading controller. Moreover, when the reference signal
changes, the output of the PD controller increases. However, after a finite
time, the output of the PD controller again comes back to approximately zero.

Figure 11: The control signals

Although there exist two independent subsystem controllers in the autonomous
tractor control system, the T2FNN works in parallel with a PD controller only
for the control of the yaw rate of the system. Thus, the error signals for
both PD controller acting alone and in parallel with the T2FNN are shown in
Fig. 12. As can be seen from Fig. 12, the T2FNN significantly increases the
control accuracy of the yaw dynamics of the system.

Figure 12: The yaw rate error

Figure 13 represents the online tuning of the parameter $q$ in (59) which
weights the sharing of the lower and the upper firing levels of each fired
rule. The learning algorithms derived in this paper includes a parameter
adaptation rule for $q$ too and we can see the effect of this on this figure
when the reference trajectory changes abruptly. It can also be observed that
the change in this parameter is not very big in the second turn when compared
to the first turn. During the training of the parameter q, a constraint is put
to the parameter in which it is bounded between 0 and 1. When it reaches its
limits, the tuning is turned off for that parameter.

Figure 13: The parameter q

##  VI Conclusions

In an interval TSK T2FLS, there exists a design parameter that balances the
sharing of the lower and the upper firing levels of each fired rule, and this
parameter can be tuned during the real-time control of the system. One novelty
of this paper is that an SMC theory-based learning algorithm is proposed for
the tuning of that parameter, the effect of which is evaluated in real-time
for the control of the yaw dynamics of an autonomous tractor. Instead of
forcing the designer to choose a fixed value for the above mentioned
parameter, this study allows the system to tune it online providing an
additional degree of freedom for the overall controller. The real-time results
show that when the T2FNN is used in parallel with a conventional PD
controller, the overall system learns the system dynamics to perform a better
performance in terms of a smaller settling time and near zero steady state
error. The use of the combination of fuzzy logic control, artificial neural
networks and sliding mode control theory harmoniously allows us to better
handle the interactions in the subsystems, uncertainties and lack of modeling
information. In addition to its well-known robustness property, another
prominent feature of the proposed approach is its computational simplicity.

## Acknowledgment

This work has been carried out within the framework of projects IWT-SBO 80032
(LeCoPro) of the Institute for the Promotion of Innovation through Science and
Technology in Flanders (IWT-Vlaanderen). We would like to thank Mr. Soner
Akpinar for his technical support for the preparation of the experimental set
up.

##  Appendix A Proof of Theorem 1

The time derivatives of (54) are as follows:

$\underline{\dot{\mu}_{1i}(x_{1})}=\bigg{(}-\frac{\dot{x}_{1}-{\dot{c}_{1i}}}{\underline{d_{1i}}}+\frac{x_{1}-{c_{1i}}}{\underline{d_{1i}^{2}}}\underline{\dot{d}_{1i}}\bigg{)}sgn\bigg{(}\overbrace{\frac{x_{1}-{c_{1i}}}{\underline{d_{1i}}}}^{\text{$\underline{E_{1}}$}}\bigg{)}$
(89)
$\overline{\dot{\mu}_{1i}(x_{1})}=\bigg{(}-\frac{\dot{x}_{1}-{\dot{c}_{1i}}}{\overline{d_{1i}}}+\frac{x_{1}-{c_{1i}}}{\overline{d_{1i}^{2}}}\overline{\dot{d}_{1i}}\bigg{)}sgn\bigg{(}\overbrace{\frac{x_{1}-{c_{1i}}}{\overline{d_{1i}}}}^{\text{$\overline{E_{1}}$}}\bigg{)}$
(90)
$\underline{\dot{\mu}_{2j}(x_{2})}=\bigg{(}-\frac{\dot{x}_{2}-{\dot{c}_{2j}}}{\underline{d_{2j}}}+\frac{x_{2}-{c_{2j}}}{\underline{d_{2j}^{2}}}\underline{\dot{d}_{2j}}\bigg{)}sgn\bigg{(}\overbrace{\frac{x_{2}-{c_{2j}}}{\underline{d_{2j}}}}^{\text{$\underline{E_{2}}$}}\bigg{)}$
(91)
$\overline{\dot{\mu}_{2j}(x_{2})}=\bigg{(}-\frac{\dot{x}_{2}-{\dot{c}_{2j}}}{\overline{d_{2j}}}+\frac{x_{2}-{c_{2j}}}{\overline{d_{2j}^{2}}}\overline{\dot{d}_{2j}}\bigg{)}sgn\bigg{(}\overbrace{\frac{x_{2}-{c_{2j}}}{\overline{d_{2j}}}}^{\text{$\overline{E_{2}}$}}\bigg{)}$
(92)

The time derivative of the strength of the rule $R_{ij}$ for the lower MF in
(53) is as follows:

$\displaystyle\dot{\widetilde{\underline{W_{ij}}}}=\frac{\Big{(}\overbrace{\underline{\dot{\mu}_{1i}(x_{1})}\underline{\mu_{2j}(x_{2})}+\underline{\mu_{1i}(x_{1})}\underline{\dot{\mu}_{2j}(x_{2})}}^{\text{A}}\Big{)}}{\sum_{i=1}^{I}\sum_{j=1}^{J}\underline{W_{ij}}}$
$\displaystyle{}-\frac{\widetilde{\underline{W_{ij}}}\bigg{(}\sum_{i=1}^{I}\sum_{j=1}^{J}\Big{(}\overbrace{\underline{\dot{\mu}_{1i}(x_{1})}\underline{\mu_{2j}(x_{2})}+\underline{\mu_{1i}(x_{1})}\underline{\dot{\mu}_{2j}(x_{2})}}^{\text{A}}\Big{)}\bigg{)}}{\sum_{i=1}^{I}\sum_{j=1}^{J}\underline{W_{ij}}}$

If the parameter update rules in Theorem 1 are used, (94) is achieved:

$\displaystyle A$ $\displaystyle=$
$\displaystyle\bigg{(}\Big{(}\frac{x_{1}-{c_{1i}}}{\underline{d_{1i}^{2}}}\Big{)}\dot{\underline{d_{1i}}}-\cancelto{0}{\Big{(}\frac{\dot{x_{1}}-{\dot{c_{1i}}}}{\underline{\dot{d_{1i}}}}}\Big{)}\bigg{)}sgn(\underline{E_{1}})\big{(}\overbrace{1-\underline{E}_{2}sgn(\underline{E_{2}})}^{\text{$\underline{\mu_{2j}}$}}\big{)}$
(94) $\displaystyle+$
$\displaystyle{}\bigg{(}\Big{(}\frac{x_{2}-{c_{2j}}}{\underline{d_{2j}^{2}}}\Big{)}\dot{\underline{d_{2j}}}-\cancelto{0}{\Big{(}\frac{\dot{x_{2}}-{\dot{c_{2j}}}}{\underline{\dot{d_{2j}}}}}\Big{)}\bigg{)}sgn(\underline{E_{2}})\big{(}\overbrace{1-\underline{E_{1}}sgn(\underline{E_{1}})}^{\text{$\underline{\mu_{1i}}$}}\big{)}$
$\displaystyle=$ $\displaystyle{}-2\alpha
sgn(\tau_{c})\underline{\mu_{1i}}\underline{\mu_{2j}}$ $\displaystyle=$
$\displaystyle-2\alpha sgn(\tau_{c})\underline{W_{ij}}$

If (94) is inserted into (A), (95) can be achieved:

$\displaystyle\dot{\widetilde{\underline{W_{ij}}}}$ $\displaystyle=$
$\displaystyle-2\alpha\widetilde{\underline{W_{ij}}}sgn(\tau_{c})-\frac{\widetilde{\underline{W_{ij}}}\sum_{i=1}^{I}\sum_{j=1}^{J}\Big{(}-2\alpha\underline{W_{ij}}sgn(\tau_{c})\Big{)}}{\sum_{i=1}^{I}\sum_{j=1}^{J}\underline{W_{ij}}}$
(95) $\displaystyle=$
$\displaystyle{}-2\alpha\widetilde{\underline{W_{ij}}}sgn(\tau_{c})+\widetilde{\underline{W_{ij}}}\sum_{i=1}^{I}\sum_{j=1}^{J}\Big{(}2\alpha\widetilde{\underline{W_{ij}}}sgn(\tau_{c})\Big{)}$
$\displaystyle=$
$\displaystyle{}-2\alpha\widetilde{\underline{W_{ij}}}sgn(\tau_{c})+2\alpha
sgn(\tau_{c})\widetilde{\underline{W_{ij}}}\overbrace{\sum_{i=1}^{I}\sum_{j=1}^{J}\widetilde{\underline{W_{ij}}}}^{\text{1}}$
$\displaystyle=$
$\displaystyle{}-2\alpha\widetilde{\underline{W_{ij}}}sgn(\tau_{c})+2\alpha\widetilde{\underline{W_{ij}}}sgn(\tau_{c})$
$\displaystyle=$ $\displaystyle 0$

Similarly, it can easily be shown that:

$\dot{\widetilde{\overline{W_{ij}}}}=0$ (96)

By using the following Lyapunov function, the stability condition is checked
as follows:

$V_{c}=\frac{1}{2}\tau_{c}^{2}(t)$ (97)
$\dot{V_{c}}=\tau_{c}\dot{\tau_{c}}=\tau_{c}(\dot{\tau_{n}}+\dot{\tau})$ (98)
$\displaystyle\tau_{n}$ $\displaystyle=$
$\displaystyle\frac{q(t)\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}\underline{W_{ij}}}{\sum_{i=1}^{I}\sum_{j=1}^{J}\underline{W_{ij}}}+\frac{\big{(}1-q(t)\big{)}\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}\overline{W_{ij}}}{\sum_{i=1}^{I}\sum_{j=1}^{J}\overline{W_{ij}}}$
(99) $\displaystyle=$
$\displaystyle{}q(t)\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}\widetilde{\underline{W_{ij}}}+\big{(}1-q(t)\big{)}\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}\widetilde{\overline{W_{ij}}}$

The time derivative of (99) is as follows:

$\displaystyle\dot{\tau_{n}}=q(t)\sum_{i=1}^{I}\sum_{j=1}^{J}(\dot{f_{ij}}\widetilde{\underline{W_{ij}}}+f_{ij}\dot{\widetilde{\underline{W_{ij}}}})$
(100)
$\displaystyle+(1-q(t))\sum_{i=1}^{I}\sum_{j=1}^{J}(\dot{f_{ij}}\widetilde{\overline{W_{ij}}}+f_{ij}\dot{\widetilde{\overline{W_{ij}}}})$
$\displaystyle+\dot{q}(t)\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}\widetilde{\underline{W_{ij}}}-\dot{q}(t)\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}\widetilde{\overline{W_{ij}}}$

If (100) is inserted into (98), the following is obtained:

$\displaystyle\dot{V_{c}}$ $\displaystyle=$
$\displaystyle\tau_{c}\bigg{(}q(t)\sum_{i=1}^{I}\sum_{j=1}^{J}(\dot{f_{ij}}\widetilde{\underline{W_{ij}}}+f_{ij}\cancelto{0}{\dot{\widetilde{\underline{W_{ij}}}}}){}$
(101)
$\displaystyle+(1-q(t))\sum_{i=1}^{I}\sum_{j=1}^{J}(\dot{f_{ij}}\widetilde{\overline{W_{ij}}}+f_{ij}\cancelto{0}{\dot{\widetilde{\overline{W_{ij}}}}}){}$
$\displaystyle+\dot{q}(t)\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}\big{(}\widetilde{\underline{W_{ij}}}-\widetilde{\overline{W_{ij}}}\big{)}+\dot{\tau}\bigg{)}$
$\displaystyle=$
$\displaystyle\tau_{c}\bigg{(}\sum_{i=1}^{I}\sum_{j=1}^{J}\bigg{(}q(t)\widetilde{\underline{W_{ij}}}\dot{f_{ij}}+\big{(}1-q(t)\big{)}\widetilde{\overline{W_{ij}}}\dot{f_{ij}}\bigg{)}{}$
$\displaystyle+\dot{q}(t)\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}\big{(}\widetilde{\underline{W_{ij}}}-\widetilde{\overline{W_{ij}}}\big{)}+\dot{\tau}\bigg{)}$
$\displaystyle=$
$\displaystyle\tau_{c}\bigg{(}\sum_{i=1}^{I}\sum_{j=1}^{J}\dot{f_{ij}}\bigg{(}q(t)\widetilde{\underline{W_{ij}}}+\big{(}1-q(t)\big{)}\widetilde{\overline{W_{ij}}}\bigg{)}{}$
$\displaystyle+\dot{q}(t)\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}\big{(}\widetilde{\underline{W_{ij}}}-\widetilde{\overline{W_{ij}}}\big{)}+\dot{\tau}\bigg{)}$

If Eqns. (71) and (72) are inserted to the equation above, the following can
be obtained:

$\displaystyle\dot{V_{c}}=\tau_{c}\Big{(}-2\alpha
sgn{(\tau_{c})}+\dot{\tau}\Big{)}<\Big{(}-2\alpha|\tau_{c}|+|\tau_{c}|B_{\dot{\tau}}\Big{)}<0$
(102)

##  Appendix B Proof of Theorem 2

Evaluating the time derivative of the Lyapunov function in (76) yields:

$\begin{split}\dot{V}_{p}&=\dot{S}_{p}S_{p}=\frac{1}{k_{D}^{2}}\dot{S}_{c}S_{c}\\\
&\leq\frac{\left|\tau_{c}\right|}{k_{D}^{2}}\Big{(}-\alpha+B_{\dot{\tau}}\Big{)}<0,\hskip
17.34518pt\forall S_{c},S_{p}\neq 0\end{split}$ (103)

## References

  * [1] M. Kawato, Y. Uno, M. Isobe, and R. Suzuki, “Hierarchical neural network model for voluntary movement with application to robotics,” _Control Systems Magazine, IEEE_ , vol. 8, no. 2, pp. 8 –15, apr 1988.
  * [2] A. Topalov, E. Kayacan, Y. Oniz, and O. Kaynak, “Adaptive neuro-fuzzy control with sliding mode learning algorithm: Application to antilock braking system,” in _Asian Control Conference, 2009. ASCC 2009. 7th_ , 27-29 2009, pp. 784 –789.
  * [3] S. Beyhan and M. Alci, “Extended fuzzy function model with stable learning methods for online system identification,” _International Journal of Adaptive Control and Signal Processing_ , vol. 25, no. 2, pp. 168–182, 2011.
  * [4] K. J. Astrom and B. Wittenmark, _Adaptive Control_. Addison-Wesley, 1995.
  * [5] A. Topalov and O. Kaynak, “Online learning in adaptive neurocontrol schemes with a sliding mode algorithm,” _IEEE Transactions on Systems, Man, and Cybernetics Part B: Cybernetics_ , vol. 31, no. 3, pp. 445 –450, december 2001\.
  * [6] A. Celikyilmaz and I. B. Turksen, “Uncertainty modeling of improved fuzzy functions with evolutionary systems,” _IEEE Transactions on Systems, Man, and Cybernetics Part B: Cybernetics_ , vol. 38, pp. 1098 – 1110, 2008.
  * [7] S. P. Panigrahi, S. K. Nayak, and S. K. Padhy, “A genetic-based neuro-fuzzy controller for blind equalization of time-varying channels,” _International Journal of Adaptive Control and Signal Processing_ , vol. 22, no. 7, pp. 705–716, 2008.
  * [8] G. Parma, B. Menezes, and A. Braga, “Sliding mode algorithm for training multilayer artificial neural networks,” _Electronics Letters_ , vol. 34, no. 1, pp. 97 –98, 8 1998.
  * [9] Y. Shuanghe, X. Y., and Z. M., “Fuzzy sets and systems,” _A fuzzy neural network approximator with fast terminal sliding mode and its applications_ , vol. 148, no. 3, pp. 469–486, 11 2004.
  * [10] J. J. Slotine and W. Li, _Applied Nonlinear Control_. Englewood Cliffs, NJ: Prentice-Hall, 1991.
  * [11] X. Yu and O. Kaynak, “Sliding-mode control with soft computing: A survey,” _Industrial Electronics, IEEE Transactions on_ , vol. 56, no. 9, pp. 3275 –3285, 2009.
  * [12] O. Kaynak, K. Erbatur, and M. Ertugrul, “The fusion of computationally intelligent methodologies and sliding-mode control-a survey,” _Industrial Electronics, IEEE Transactions on_ , vol. 48, no. 1, pp. 4 –17, Feb. 2001.
  * [13] Y. Oniz, E. Kayacan, and O. Kaynak, “A dynamic method to forecast the wheel slip for antilock braking system and its experimental evaluation,” _Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on_ , vol. 39, no. 2, pp. 551 –560, april 2009.
  * [14] M. O. Efe, O. Kaynak, and X. Yu, “Sliding mode control of a three degrees of freedom anthropoid robot by driving the controller parameters to an equivalent regime,” _ASME Journal of Dynamic Systems, Measurement, and Control_ , vol. 122, no. 4, pp. 632 –640, December 2000.
  * [15] B. Yoo and W. Ham, “Adaptive fuzzy sliding mode control of nonlinear system,” _Fuzzy Systems, IEEE Transactions on_ , vol. 6, no. 2, pp. 315 –321, may 1998\.
  * [16] L. Zadeh, “Toward extended fuzzy logic-A first step,” _Fuzzy Sets and Systems_ , vol. 160, pp. 3175 – 3181, 2009.
  * [17] Q. Liang, N. Karnik, and J. Mendel, “Connection admission control in ATM networks using survey-based type-2 fuzzy logic systems,” _IEEE Transactions on Systems, Man, and Cybernetics Part C_ , vol. 38, pp. 329 – 339, 2000.
  * [18] C.-F. Juang and C.-H. Hsu, “Reinforcement interval type-2 fuzzy controller design by online rule generation and q-value-aided ant colony optimization,” _Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on_ , vol. 39, no. 6, pp. 1528 –1542, dec. 2009.
  * [19] ——, “Reinforcement ant optimized fuzzy controller for mobile-robot wall-following control,” _Industrial Electronics, IEEE Transactions on_ , vol. 56, no. 10, pp. 3931–3940, October 2009.
  * [20] R. Sepulveda, P. Melin, A. Rodriguez, A. Mancilla, and O. Montiel, “Analyzing the effects of the footprint of uncertainty in type-2 fuzzy logic controllers,” _Engineering Letters_ , vol. 13, pp. 138 –147, 2006.
  * [21] M. Roopaei, B. Sahraei, T.-C. Lin, and M.-C. Chen, “Synchronization of two different chaotic systems using chattering-free adaptive interval type-2 fuzzy sliding mode control,” in _Industrial Electronics and Applications (ICIEA), 2010 the 5th IEEE Conference on_ , jun. 2010, pp. 121 –126.
  * [22] M. Biglarbegian, W. Melek, and J. Mendel, “Design of novel interval type-2 fuzzy controllers for modular and reconfigurable robots: Theory and experiments,” _Industrial Electronics, IEEE Transactions on_ , vol. 58, no. 4, pp. 1371 –1384, april 2011.
  * [23] M. Khanesar, E. Kayacan, M. Teshnehlab, and O. Kaynak, “Analysis of the noise reduction property of type-2 fuzzy logic systems using a novel type-2 membership function,” _Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on_ , vol. 41, no. 5, pp. 1395–1406, 2011.
  * [24] E. Kayacan, W. Saeys, E. Kayacan, H. Ramon, and O. Kaynak, “Intelligent control of a tractor-implement system using type-2 fuzzy neural networks,” in _IEEE WCCI 2012 - the 2012 IEEE World Congress on Computational Intelligence_ , Brisbane, Australia, 10-15 June, 2012, pp. 171–178.
  * [25] M. Karkee and B. L. Steward, “Study of the open and closed loop characteristics of a tractor and a single axle towed implement system,” _Journal of Terramechanics_ , vol. 47, no. 6, pp. 379 – 393, 2010.
  * [26] D. Piyabongkarn, R. Rajamani, J. A. Grogg, and J. Y. Lew, “Development and experimental evaluation of a slip angle estimator for vehicle stability control,” _IEEE Transactions on Control Systems Technologhy_ , vol. 17, no. 1, pp. 78 – 88, 2009.
  * [27] C. Geng, L. Mostefai, M. Denai, and Y. Hori, “Direct yaw-moment control of an in-wheel-motored electric vehicle based on body slip angle fuzzy observer,” _IEEE Transactions on Industrial Electronics_ , vol. 56, no. 5, pp. 1411 – 1419, 2009.
  * [28] F. N. Martins, W. C. Celeste, R. Carelli, M. Sarcinelli-Filho, and T. F. Bastos-Filho, “An adaptive dynamic controller for autonomous mobile robot trajectory tracking,” _Control Engineering Practice_ , vol. 16, no. 11, pp. 1354 – 1363, 2008.
  * [29] E. Kayacan and O. Kaynak, “Sliding mode control theory-based algorithm for online learning in type-2 fuzzy neural networks: application to velocity control of an electro hydraulic servo system,” _International Journal of Adaptive Control and Signal Processing_ , vol. 26, no. 7, pp. 645–659, 2012\.
  * [30] M. B. Begian, W. W. Melek, and J. M. Mendel, “Parametric design of stable type-2 TSK fuzzy systems,” in _Proceedings of the North American Fuzzy Information Processing Systems_ , 2008, pp. 1 –6.
  * [31] V. I. Utkin, _Sliding Modes in Control Optimization_. Springer-Verlag, 1992.
  * [32] J.-J. Slotine and W. Li, _Applied Nonlinear Control_. Prentice Hall, 1991.
  * [33] R. G. Brown and P. Y. C. Hwang, _Introduction to Random Signals and Applied Kalman Filtering (4th Edition)_. John Wiley and Sons, 2012.

|  Erdal Kayacan (S’06-SM’12) was born in Istanbul, Turkey on January 7, 1980.
He received a B.Sc. degree in electrical engineering from in 2003 from
Istanbul Technical University in Istanbul, Turkey as well as a M.Sc. degree in
systems and control engineering in 2006 from Bogazici University in Istanbul,
Turkey. In September 2011, he received a Ph.D. degree in electrical and
electronic engineering at Bogazici University in Istanbul, Turkey. He is
currently pursuing his post-doctoral degree at University of Leuven (KU
Leuven) in the division of mechatronics, biostatistics and sensors (MeBioS).
His research areas are robotics, mechatronics, soft computing methods, sliding
mode control and model predictive control. Dr. Kayacan is active in the IEEE
CIS Student Activities Subcommittee, the IEEE CIS Social Media Subcommittee,
the IEEE CIS GOLD Subcommittee and the IEEE SMC Technical Committee on Grey
Systems. He has been serving as an editor in Journal on Automation and Control
Engineering (JACE) and editorial advisory board in Grey Systems Theory and
Application.
---|---
|  Erkan Kayacan (S’12) was born in Istanbul, Turkey, on April 17, 1985. He
received the B.Sc. and the M.Sc. degrees in mechanical engineering from
Istanbul Technical University, Istanbul, in 2008 and 2010, respectively. He is
a PhD student and research assistant at University of Leuven (KU Leuven) in
the division of mechatronics, biostatistics and sensors (MeBioS). His research
interests include model predictive control, moving horizon estimation,
distributed and decentralized control, intelligent control and vehicle
dynamics. He is active in the IEEE CIS GOLD Subcommittee and the IEEE Webinars
subcommittee.
---|---
|  Herman Ramon graduated as an agricultural engineer from Gent University. In
1993 he obtained a Ph.D. in applied biological sciences at the Katholieke
Universiteit Leuven. He is currently Professor at the Faculty of Agricultural
and Applied Biological Sciences of the Katholieke Universiteit Leuven,
lecturing on agricultural machinery and mechatronic systems for agricultural
machinery. He has a strong research interest in precision technologies and
advanced mechatronic systems for processes involved in the production chain of
food and nonfood materials, from the field to the end user.He is author or co-
author of more than 40 papers.
---|---
|  Okyay Kaynak (M’80-SM’90-F’03) received the B.Sc. degree with first class
honors and Ph.D. degrees in Electronic and Electrical Engineering from the
University of Birmingham, UK, in 1969 and 1972 respectively. From 1972 to
1979, he held various positions within the industry. In 1979, he joined the
Department of Electrical and Electronics Engineering, Bogazici University,
Istanbul, Turkey, where he is presently a full professor, holding the UNESCO
Chair on Mechatronics. He has hold long-term (near to or more than a year)
Visiting Professor / Scholar positions at various institutions in Japan,
Germany, U.S. and Singapore. His current research interests are in the fields
of intelligent control and mechatronics. He has authored three books and
edited five and authored or coauthored almost 400 papers that have appeared in
various journals and conference proceedings. Currently, he is the Editor in
Chief of IEEE/ASME Trans. on Mechatronics. Dr. Kaynak is a fellow of IEEE. He
has served on many committees of IEEE and was the president of IEEE Industrial
Electronics Society during 2002-2003.
---|---
|  Wouter Saeys is currently Assistant Professor in Biosystems Engineering at
the Department of Biosystems of the University of Leuven in Belgium. He
obtained his Ph.D. at the same institute and was a visiting postdoc at the
School for Chemical Engineering and Advanced Materials of the University of
Newcastle upon Tyne, UK and at the Norwegian Food Research Institute - Nofima
Mat in Norway. His main research interests are optical sensing, process
monitoring and control with applications in food and agriculture. He is author
of 50 articles (ISI) and member of the editorial board of Biosystems
Engineering.
---|---