# A Self-Learning Disturbance Observer for Nonlinear Systems in Feedback-Error
Learning Scheme

Erkan Kayacan  Joshua M. Peschel  Girish Chowdhary  Coordinated Science
Laboratory, University of Illinois at Urbana-Champaign,
1206 W Gregory Drive, Urbana, Illinois 61801, USA. Tel. +1 217 721 8278
e-mail: { erkank, girishc<EMAIL_ADDRESS>Department of Agricultural and
Biosystems Engineering, Iowa State University, Ames, Iowa 50011, USA
e-mail<EMAIL_ADDRESS>
###### Abstract

This paper represents a novel online self-learning disturbance observer (SLDO)
by benefiting from the combination of a type-2 neuro-fuzzy structure (T2NFS),
feedback-error learning scheme and sliding mode control (SMC) theory. The SLDO
is developed within a framework of feedback-error learning scheme in which a
conventional estimation law and a T2NFS work in parallel. In this scheme, the
latter learns uncertainties and becomes the leading estimator whereas the
former provides the learning error to the T2NFS for learning system dynamics.
A learning algorithm established on SMC theory is derived for an interval
type-2 fuzzy logic system. In addition to the stability of the learning
algorithm, the stability of the SLDO and the stability of the overall system
are proven in the presence of time-varying disturbances. Thanks to learning
process by the T2NFS, the simulation results show that the SLDO is able to
estimate time-varying disturbances precisely as distinct from the basic
nonlinear disturbance observer (BNDO) so that the controller based on the SLDO
ensures robust control performance for systems with time-varying
uncertainties, and maintains nominal performance in the absence of
uncertainties.

###### keywords:

Disturbance observer, neural networks, neuro-fuzzy structure, online learning
algorithm, robustness, sliding mode control, type-2 fuzzy logic systems,
uncertainty.

††journal: Engineering Applications of Artificial Intelligence, vol. 62, pp.
276-285, 2017.

##  1 Introduction

One of the most essential requirements for controllers is to be insensitive to
uncertainties. Many control methods have been proposed to handle different
types of uncertainties, (e.g. $H_{\infty}$ control (Glover and Doyle, 1988;
Gahinet and Apkarian, 1994), sliding mode control (SMC) (Slotine et al., 1991;
Li et al., 2013; Xu et al., 2014), adaptive control (Sun et al., 2013a, b; He
et al., 2014; Yao et al., 2015), etc…). In $H_{\infty}$ control, uncertainties
must be bounded in $H_{\infty}$-norm. This implies that disturbances must
disappear suddenly and completely. However, this is not a realistic assumption
for real-time applications. In SMC theory, integral SMC controller is proposed
in the presence of uncertainties. It is a well known fact that the integral
action may cause unwanted effects such as, large settling time and overshoots.
Moreover, adaptive control systems may not have an ability to control
uncertain systems with highly changing parameters (Chen et al., 2015). As an
alternative method, disturbance observers (DOs) have been proposed since they
are very crucial for control of systems due to the fact that uncertainties
extensively exist in practice and are extremely difficult to be modeled. These
uncertainties, such as parameter variations, noise, unmodeled dynamics and
interactions between subsystems, must be taken for the controller design into
account to have a capability of getting robustness. For this purpose,
different DOs have been designed in literature to obtain robust control
performance for systems (Chen et al., 2000b, a, 2016).

In DO based control approaches, the model uncertainties and external
disturbances are merged into one term and a control law contains the estimated
disturbance value by a DO. The aims are to achieve performance specifications
while stabilizing the system considering the nominal model of the system and
remove the disturbance effect on the system (Chen, 2004; Yang et al., 2013;
Ginoya et al., 2014). A nonlinear dynamics inversion control method was
designed for the longitudinal autopilot of a missile in (Chen, 2003). It was
reported that the control method exhibit poor performance in case of unknown
uncertainties while a basic nonlinear DO (BNDO) based nonlinear dynamics
inversion control approach ensured robust performance against uncertainty. The
same BNDO has been used to design a robust SMC controller for systems with
mismatched uncertainties (Yang et al., 2013). The main drawback in these
studies, the BNDO is only be able to estimate time-invariant disturbances. If
disturbances are time-varying, then the BNDO gives bias estimates.
Furthermore, there are well known nonlinear observers, such as extended Kalman
filter (EKF), particle filtering and nonlinear moving horizon estimation
methods. EKF works well if linear approximation is valid and noise on
measurements is small (Haseltine et al., 2005). Besides, particle filter and
nonlinear moving horizon estimation methods require very large computation
time (Daum, 2005). For this reason, an observer, computationally cheap, is
required to be able to estimate time-varying disturbances.

Type-2 fuzzy logic systems (T2FLSs) are proposed as the extended versions of
type-1 fuzzy logic systems (T1FLSs) in literature (Kayacan et al., 2010;
Castillo et al., 2012; Lee et al., 2015). It allows us to have more degrees of
freedom for design than their type-1 counterparts so that it results in better
capability of handling uncertainty (Castro et al., 2011; Castro and Castillo,
2013; Lee et al., 2014; Rubio-Solis and Panoutsos, 2015; Zaheer et al., 2015).
Type-2 fuzzy sets are especially preferred as the decision of the position of
the membership functions (MFs) precisely is a very troublesome task (Liang et
al., 2000; Castillo et al., 2013). However, the computational complexity of
generalized type-2 fuzzy sets is very high. For this reason, the interval
type-2 fuzzy sets were proposed to decrease computation time and made it
feasible in real-time applications (Liang and Mendel, 2000; Maldonado et al.,
2013; Lin et al., 2014; Chang and Chan, 2014; Castillo et al., 2014; Zaheer et
al., 2015; Wagner et al., 2015).

Neuro-fuzzy structures as a model-free method have been widely used for
control and identification of systems in literature. It is well know that the
stability of systems controlled by model-free controllers cannot be proven.
Therefore, the feedback-error learning scheme has been proposed for neuro-
fuzzy structures to guarantee the global asymptotic stability of the system in
a compact space (Efe et al., 2000). SMC achieves robustness to parametric
uncertainty and high-frequency unmodeled dynamics; therefore, the SMC theory-
based learning algorithms for neuro-fuzzy structures have been proposed to
ensure the robustness of the overall system (Kaynak et al., 2001; Yu and
Kaynak, 2009). Moreover, they ensure faster convergence rate than the
traditional learning methods, such as gradient descent, Levenberg-Marquardt
and particle swarm optimization, because they are computationally simple.
There are numerous examples for SMC theory-based learning algorithms for
artificial neural networks, type-1 and type-2 neuro-fuzzy structures (Topalov
et al., 2009; Kayacan et al., 2012).

The main contribution of this paper is to develop a novel online SLDO, which
can learn the disturbance behavior of systems in time-varying case as distinct
from basic nonlinear DOs (BNDOs), be solved in the range of millisecond, and
robust against uncertainties. For this purpose, the T2NFS in feedback-error
learning scheme is proposed due to fact that they are very suitable techniques
for adaptive learning. Additionally, computationally efficient sliding mode
learning algorithm is used as the training algorithm of the T2NFS because it
is a powerful approach for the stability issue. Consequently, the use of the
combination of T2NFS, feedback-error learning scheme and sliding mode control
theory harmoniously allow to better handle uncertainties.

The major contributions of this paper are as follows:

  1. 1.

The first major contribution is that a novel estimation approach in the
feedback-error learning scheme is developed for disturbance observer design
for the first-time.

  2. 2.

The second major contribution is that the stability of the training algorithm
has been always proven for feedback-error learning methods in literature. In
this paper, in addition to the proof of the training algorithm, the overall
system stability is proven considering the dynamics of the proposed SLDO by
adding a robust term. To the best knowledge of the author, this is also the
first-time such a stability analysis is ever proven.

The minor contributions of this paper are as follows:

  1. 1.

The first minor contribution is that the developed SLDO is solved within
milliseconds; therefore, the required computation time for the SLDO is
significantly less than other methods, such as particle filter and nonlinear
moving horizon estimation methods.

  2. 2.

The second minor contribution is that the learning rate of SMC theory-based
learning algorithm for the T2NFS is adaptive so that it is possible to
estimate the disturbance without the knowledge about the upper bound of the
disturbance and its derivatives.

The paper consists of six sections: The formulation of the BNDO is given in
Section 2. The SLDO benefiting from the T2NFS is developed in Section 3. The
T2NFS and online learning algorithm established on SMC theory are respectively
represented in Sections 3.1 and 3.2. The stability of the SLDO is proven in
Section 3.3. The controller design and the stability of the system are given
in Section 4. The simulation results are represented in Sections 5. Finally,
the paper is summarized in Section 6.

##  2 Basic Nonlinear Disturbance Observer

A number of physical systems, such as robots, spacecrafts and mechanical
systems, are generally described by second-order differential equations. A
second-order nonlinear system is written in the following form:

$\dot{x}=g_{1}(x)+g_{2}(x)u+zd(t)$ (1)

where $x=[x_{1},x_{2}]^{T}$ is the state vector, $u$ is the control input,
$d(t)$ is the disturbance, $z=[z_{1},z_{2}]^{T}$ is the disturbance
coefficients vector, $g_{1}(x)=[x_{2},a(x)]^{T}$ and $g_{2}(x)=[0,b(x)]^{T}$
are the nonlinear system dynamics.

The disturbance $d(t)$ in (1) is not measurable in practice. Therefore, it is
required to be estimated in practice to obtain robust control performance of
systems. The following basic nonlinear disturbance observer (BNDO) dynamics
have been proposed in (Chen, 2003; Yang et al., 2013) as follows:

$\displaystyle\dot{p}$ $\displaystyle=$ $\displaystyle-l_{p}zp-
l_{p}\Big{(}zl_{p}x+g_{1}(x)+g_{2}(x)u\Big{)}$ $\displaystyle\hat{d}_{BN}$
$\displaystyle=$ $\displaystyle p+l_{p}x$ (2)

where $p$, $l_{p}$ and $\hat{d}_{BN}$ denote respectively the internal state,
proportional observer gain and estimated disturbance. By taking time
derivative of the estimated disturbance considering (2), the time derivative
of the estimated disturbance $\dot{\hat{d}}_{BN}$ is obtained as:

$\dot{\hat{d}}_{BN}=l_{p}ze_{d}$ (3)

If the time derivative of the actual disturbance $\dot{d}(t)$ is added into
(3), the error dynamics of the BNDO are obtained as follows:

$\displaystyle\dot{d}(t)-\dot{\hat{d}}_{BN}$ $\displaystyle=$ $\displaystyle-
l_{p}ze_{d}+\dot{d}(t)$ $\displaystyle\dot{e}_{d}$ $\displaystyle=$
$\displaystyle-l_{p}ze_{d}+\dot{d}(t)$ (4)

where $e_{d}=d(t)-\hat{d}_{BN}$ is the disturbance error.

###### Assumption 1

The time derivative of the actual disturbance is bounded and
$\displaystyle\lim_{t\to\infty}\dot{d}(t)=0$.

If Assumption 1 is satisfied, then (2) is obtained as follows:

$\dot{e}_{d}+l_{p}ze_{d}=0$ (5)

###### Lemma 1

(Chen, 2003): If $l_{p}z$ is positive, i.e. $l_{p}z>0$, the disturbance error
dynamics in (5) converge to zero asymptotically.

Lemma 1 implies that the estimated disturbance by the BNDO is able to track
the actual disturbance of the system in (1) asymptotically in case Assumption
1 is satisfied.

###### Remark 1

If the time derivative of the actual disturbance $\dot{d}(t)$ is not equal to
zero, the error dynamics of th BNDO cannot converge to zero so that BNDO gives
bias. Therefore, there exists always difference between the estimated and true
values of the disturbance. Similar observers have been designed in literature
and the same drawback has been reported in (Chen, 2003; Yang et al., 2013).

##  3 Self-learning Disturbance Observer

Figure 1: Schematic diagram of the self-learning disturbance observer (SLDO)

BNDOs cannot give unbiased estimation results in case of time-varying
disturbances; therefore, a fast, computationally efficient, adaptive and
robust disturbance observer is required. A novel estimation law is proposed as
follows:

$\dot{\hat{d}}_{SL}=\tau_{c}+\tau_{r}-\tau_{n}$ (6)

where $\tau_{c}$, $\tau_{r}$ and $\tau_{n}$ denote respectively the outputs of
the conventional estimation law, robust term and T2NFS. The schematic diagram
of the SLDO is illustrated in Fig 1. As seen, the BNDO is working in series
with the feedback-error learning algorithm in which the conventional and
robust estimation laws work in parallel with T2NFS.

The conventional estimation law used in this paper is defined as follows:

$\tau_{c}=\dot{\hat{d}}_{BN}+\frac{l_{d}z}{l_{p}z}\ddot{\hat{d}}_{BN}$ (7)

where $\dot{\hat{d}}_{BN}$ denotes the time derivative of the estimated value
of the disturbance by the BNDO, $l_{p}$ and $l_{d}$ denote the proportional
and derivative gain vectors, and $l_{p}z$ and $l_{d}z$ are positive, i.e.
$l_{p}z,l_{d}z>0$.

The robust estimation law is written as follows:

$\tau_{r}=\frac{l_{r}z}{l_{p}z}\dot{\hat{d}}_{BN}$ (8)

$l_{r}z$ is positive, i.e. $l_{r}z>0$, and $l_{r}$ denotes the robust gain
vector.

###  3.1 Type-2 Neuro-Fuzzy Structure

An interval type-2 Takagi-Sugeno-Kang (TSK) fuzzy _if-then_ rule $R_{ij}$ is
written as:

$R_{ij}:\;\;\textrm{If}\;\xi_{1}\;\textrm{is}\;\;\widetilde{1}_{i}\;\;\textrm{and}\;\xi_{2}\;\textrm{is}\;\;\widetilde{2}_{j},\;\;\textrm{then}\;f_{ij}=\Upsilon_{ij}$
(9)

where $\xi_{1}=\dot{\hat{d}}_{BN}$ and $\xi_{2}=\ddot{\hat{d}}_{BN}$ denote
the inputs while $\widetilde{1}_{i}$ and $\widetilde{2}_{j}$ denote type-2
fuzzy sets for inputs. The function $f_{ij}$ is the output of the rules and
the total number of the rules are equal to $K=I\times J$ in which $I$ and $J$
are the total number of the inputs.

The upper and lower Gaussian membership functions for type-2 fuzzy logic
systems are written as follows:

$\underline{\mu}_{1i}(\xi_{1})=\exp\Bigg{(}-\bigg{(}\frac{\xi_{1}-\underline{c}_{1i}}{\underline{\sigma}_{1i}}\bigg{)}^{2}\Bigg{)}$
(10)
$\overline{\mu}_{1i}(\xi_{1})=\exp\Bigg{(}-\bigg{(}\frac{\xi_{1}-\overline{c}_{1i}}{\overline{\sigma}_{1i}}\bigg{)}^{2}\Bigg{)}$
(11)
$\underline{\mu}_{2j}(\xi_{2})=\exp\Bigg{(}-\bigg{(}\frac{\xi_{2}-\underline{c}_{2j}}{\underline{\sigma}_{2j}}\bigg{)}^{2}\Bigg{)}$
(12)
$\overline{\mu}_{2j}(\xi_{2})=\exp\Bigg{(}-\bigg{(}\frac{\xi_{2}-\overline{c}_{2j}}{\overline{\sigma}_{2j}}\bigg{)}^{2}\Bigg{)}$
(13)

where $\underline{c},\overline{c},\underline{\sigma},\overline{\sigma}$ denote
respectively the lower and upper mean, and the lower and upper standard
deviation of the membership functions. These parameters are adjustable for the
T2NFS.

The lower and upper membership functions $\underline{\mu}$ and
$\overline{\mu}$ of A2-C0 fuzzy system employed in this paper are determined
for every signal. Then, the firing strength of rules are calculated as
follows:

$\underline{w}_{ij}=\underline{\mu}_{1i}(\xi_{1})\underline{\mu}_{2j}(\xi_{2})\quad\textrm{and}\quad\overline{w}_{ij}=\overline{\mu}_{1i}(\xi_{1})\overline{\mu}_{2j}(\xi_{2})$
(14)

The output of the every fuzzy rule is a linear function $f_{ij}$ formulated in
(9). The output of the network is formulated below:

$\tau_{n}=q\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}\widetilde{\underline{w}}_{ij}+(1-q)\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}\widetilde{\overline{w}}_{ij}$
(15)

where $\widetilde{\underline{{w}}}_{ij}$ and $\widetilde{\overline{{w}}}_{ij}$
are the normalized firing strengths of the lower and upper output signals of
the neuron $ij$ are written as follows:

$\widetilde{\underline{w}}_{ij}=\frac{\underline{w}_{ij}}{\sum_{i=1}^{I}\sum_{j=1}^{J}\underline{w}_{ij}}\;\;\;\textrm{and}\;\;\;\widetilde{\overline{w}}_{ij}=\frac{\overline{w}_{ij}}{\sum_{i=1}^{I}\sum_{j=1}^{J}\overline{w}_{ij}}$
(16)

The design parameter $q$ weights the participation of the lower and upper
firing levels and is generally set to $0.5$. In this paper, it is formulated
as a time-varying parameter in the next subsection.

The vectors are defined as::

$\displaystyle\widetilde{\underline{W}}(t)$ $\displaystyle=$
$\displaystyle[\widetilde{\underline{w}}_{11}(t)\;\widetilde{\underline{w}}_{12}(t)\dots\widetilde{\underline{w}}_{21}(t)\dots\widetilde{\underline{w}}_{ij}(t)\dots\widetilde{\underline{w}}_{IJ}(t)]^{T}$
$\displaystyle\widetilde{\overline{W}}(t)$ $\displaystyle=$
$\displaystyle[\widetilde{\overline{w}}_{11}(t)\;\widetilde{\overline{w}}_{12}(t)\dots\widetilde{\overline{w}}_{21}(t)\dots\widetilde{\overline{w}}_{ij}(t)\dots\widetilde{\overline{w}}_{IJ}(t)]^{T}$
$\displaystyle F$ $\displaystyle=$ $\displaystyle[f_{11}\;f_{12}\dots
f_{21}\dots f_{ij}\dots f_{IJ}]$

where these normalized firing strengths are between $0$ and $1$, i.e.
$0<\widetilde{\underline{w}}_{ij}\leq 1$ and
$0<\widetilde{\overline{w}}_{ij}\leq 1$. In addition,
$\sum_{i=1}^{I}\sum_{j=1}^{J}\widetilde{\underline{w}}_{ij}=1$ and
$\sum_{i=1}^{I}\sum_{j=1}^{J}\widetilde{\overline{w}}_{ij}=1$.

###  3.2 SMC Theory-based Learning Algorithm

The sliding surface $s$ is formulated as follows:

$s\left(\tau_{c}\right)=\tau_{c}$ (17)

where $\tau_{c}$ is the output of the conventional estimation law and used as
a sliding surface. It is to be noted that the sliding surface is used as
learning error to train the SMC theory-based learning algorithm.

The adaptation rules of the T2NFS parameters are given by the following
equations:

$\dot{\underline{c}}_{1i}=\dot{\xi}_{1}+(\xi_{1}-\underline{c}_{1i})\alpha\textrm{sgn}\left(s\right)$
(18)
$\dot{\overline{c}}_{1i}=\dot{\xi}_{1}+(\xi_{1}-\overline{c}_{1i})\alpha\textrm{sgn}\left(s\right)$
(19)
$\dot{\underline{c}}_{2j}=\dot{\xi}_{2}+(\xi_{2}-\underline{c}_{2j})\alpha\textrm{sgn}\left(s\right)$
(20)
$\dot{\overline{c}}_{2j}=\dot{\xi}_{2}+(\xi_{2}-\overline{c}_{2j})\alpha\textrm{sgn}\left(s\right)$
(21)
$\dot{\underline{\sigma}}_{1i}=-\bigg{(}\underline{\sigma}_{1i}+\frac{(\underline{\sigma}_{1i})^{3}}{(\xi_{1}-\underline{c}_{1i})^{2}}\bigg{)}\alpha\textrm{sgn}\left(s\right)$
(22)
$\dot{\overline{\sigma}}_{1i}=-\bigg{(}\overline{\sigma}_{1i}+\frac{(\overline{\sigma}_{1i})^{3}}{(\xi_{1}-\overline{c}_{1i})^{2}}\bigg{)}\alpha\textrm{sgn}\left(s\right)$
(23)
$\dot{\underline{\sigma}}_{2j}=-\bigg{(}\underline{\sigma}_{2j}+\frac{(\underline{\sigma}_{2j})^{3}}{(\xi_{2}-\underline{c}_{2j})^{2}}\bigg{)}\alpha\textrm{sgn}\left(s\right)$
(24)
$\dot{\overline{\sigma}}_{2j}=-\bigg{(}\overline{\sigma}_{2j}+\frac{(\overline{\sigma}_{2j})^{3}}{(\xi_{2}-\overline{c}_{2j})^{2}}\bigg{)}\alpha\textrm{sgn}\left(s\right)$
(25)
$\dot{f}_{ij}=-\frac{q\widetilde{\underline{w}}_{ij}+(1-q)\widetilde{\overline{w}}_{ij}}{(q\widetilde{\underline{W}}+(1-q)\widetilde{\overline{W}})^{T}(q\widetilde{\underline{W}}+(1-q)\widetilde{\overline{W}})}\alpha
sgn(s)$ (26)
$\dot{q}=-\frac{1}{F(\widetilde{\underline{W}}-\widetilde{\overline{W}})^{T}}\alpha
sgn(s)$ (27) $\dot{\alpha}=\gamma_{\alpha}\mid s\mid$ (28)

where $\alpha$ and $\gamma_{\alpha}$ denote respectively the learning rate and
the coefficient of the adaptation for the learning rate, and they must be
positive, i.e. $\alpha,\gamma_{\alpha}>0$.

######  Theorem 1 (Stability of the learning algorithm)

If adaptations rules are proposed as in (18)-(28) and the final value of the
learning rate $\alpha^{*}$ is large enough, i.e.
$\alpha^{*}>\dot{\tau}_{r}^{*}+\ddot{\hat{d}}^{*}_{SL}$ where
$\dot{\tau}_{r}^{*}$ and $\ddot{\hat{d}}^{*}_{SL}$ are respectively the upper
bounds of $\dot{\tau}_{r}$ and $\ddot{\hat{d}}_{SL}$, this ensures that
$\tau_{c}$ will converge to zero in finite time for a given arbitrary initial
condition $\tau_{c}(0)$.

###### Proof 1

The Lyapunov function is written as follows:

$\displaystyle
V=\frac{1}{2}\tau_{c}^{2}+\frac{1}{\gamma_{\alpha}}(\alpha-\alpha^{*})^{2}$
(29)

By taking the time derivative of the Lyapunov function in (29), it is obtained
as follows:

$\displaystyle\dot{V}=\tau_{c}\dot{\tau}_{c}+\frac{2\dot{\alpha}}{\gamma_{\alpha}}(\alpha-\alpha^{*})$
(30)

If the (28) is inserted into the equation above:

$\displaystyle\dot{V}=\tau_{c}(\dot{\tau}_{n}-\dot{\tau}_{r}+\ddot{\hat{d}}_{SL})+2\mid
s\mid(\alpha-\alpha^{*})$ (31)

The calculation of $\dot{\tau}_{n}$ in (63) is inserted into (31), it is
obtained as follows:

$\displaystyle\dot{V}=\tau_{c}\Big{(}-2\alpha\textrm{sgn}\left(s\right)-\dot{\tau}_{r}+\ddot{\hat{d}}_{SL}\Big{)}+2\mid
s\mid(\alpha-\alpha^{*})$ (32)

If (17) is inserted into the equation above:

$\displaystyle\dot{V}=\tau_{c}\Big{(}-2\alpha\textrm{sgn}\left(\tau_{c}\right)-\dot{\tau}_{r}+\ddot{\hat{d}}_{SL}\Big{)}+2\mid\tau_{c}\mid(\alpha-\alpha^{*})$
(33)

If it is assumed that $\ddot{\hat{d}}_{SL}$ and $\dot{\tau}_{r}$ are upper
bounded by $\ddot{\hat{d}}^{*}_{SL}$ and $\dot{\tau}_{r}^{*}$, (33) is
obtained as follows:

$\displaystyle\dot{V}$ $\displaystyle=$
$\displaystyle\mid\tau_{c}\mid(-2\alpha+\dot{\tau}_{r}^{*}+\ddot{\hat{d}}^{*}_{SL})+2\mid\tau_{c}\mid(\alpha-\alpha^{*})$
(34) $\displaystyle=$
$\displaystyle\mid\tau_{c}\mid(-2\alpha^{*}+\dot{\tau}_{r}^{*}+\ddot{\hat{d}}^{*}_{SL})$

As stated in Theorem 1, if the final value of the learning rate $\alpha^{*}$
is large enough, i.e. $\alpha^{*}>\dot{\tau}_{r}^{*}+\ddot{\hat{d}}^{*}_{SL}$,
then the time derivative of the Lyapunov function is negative, i.e.
$\dot{V}<0$ so that the SMC theory-based learning algorithm is stable and
$\tau_{c}$ will converge to zero in finite time.

###### Remark 2

Since the adaptation rules in (28) are enforced, the final value of the
learning rate of the T2NFS is determined during the adaptation of learning
rate, and it is able to reach large values to make learning algorithm stable.
This is a superiority of the proposed approach in this study as distinct from
previous studies in which the upper bounds are needed to be foreknown.

SMC theory endures high-frequency oscillations, i.e. chattering. In this
paper, the function in (35) has been proposed to remove the chattering effect
as the sign function in (18)-(28).

$\textrm{sgn}(s):=\frac{s}{\mid s\mid+\delta}$ (35)

where $\delta=0.05$.

###### Remark 3

The usage of the sliding surface $s$ in (17) as learning error for the T2NFS
with the adaptation laws in (18)-(28) accomplishes the desired sliding regime
for the observer.

###  3.3 Stability Analysis

The proposed SLDO law in (6) is re-written considering (7) and (8) as:

$\displaystyle\dot{\hat{d}}_{SL}=(1+\frac{l_{r}z}{l_{p}z})\dot{\hat{d}}_{BN}+\frac{l_{d}z}{l_{p}z}\ddot{\hat{d}}_{BN}-\tau_{n}$
(36)

The error dynamics for the SLDO are obtained by adding the actual disturbance
rate $\dot{d}$ into the estimated disturbance rate in (36) and considering the
calculated time derivative of the estimated disturbance by BNDO in (3):

$\displaystyle\dot{d}-\dot{\hat{d}}_{SL}$ $\displaystyle=$
$\displaystyle-(l_{p}+l_{r})ze_{d}-l_{d}z\dot{e}_{d}+\tau_{n}+\dot{d}$
$\displaystyle\dot{e}_{d}$ $\displaystyle=$
$\displaystyle\frac{-(l_{p}+l_{r})ze_{d}+\tau_{n}+\dot{d}}{1+l_{d}z}$ (37)

By taking the time derivative of (3.3), it is obtained as follows:

$\ddot{e}_{d}=\frac{-(l_{p}+l_{r})z\dot{e}_{d}+\dot{\tau}_{n}+\ddot{d}}{1+l_{d}z}$
(38)

As calculated in (63), $\dot{\tau}_{n}=-2\alpha sgn(s)$ is inserted into (38);

$\ddot{e}_{d}=\frac{-(l_{p}+l_{r})z\dot{e}_{d}-2\alpha
sgn(s)+\ddot{d}}{1+l_{d}z}$ (39)

If $\tau_{c}$ in (7) is inserted into (17), the sliding surface is obtained as
follows:

$\displaystyle
s\left(\dot{\hat{d}}_{BN},\ddot{\hat{d}}_{BN}\right)=\frac{l_{d}z}{l_{p}z}\Big{(}\ddot{\hat{d}}_{BN}+\lambda\dot{\hat{d}}_{BN}\Big{)}$
(40)

where $\lambda=\frac{l_{p}z}{l_{d}z}$ is the slope of the sliding surface. The
time derivative of the sliding surface is obtained as

$\dot{s}=\frac{l_{d}z}{l_{p}z}\Big{(}\dddot{\hat{d}}_{BN}+\frac{l_{p}z}{l_{d}z}\ddot{\hat{d}}_{BN}\Big{)}$
(41)

######  Theorem 2 (Stability of the SLDO)

The estimation law in (6) is employed as a DO, the closed-loop error dynamics
for the SLDO are stable if the robust gain $l_{r}$ is equal to
$\frac{l_{p}}{l_{d}z}$, i.e. $l_{r}=\frac{l_{p}}{l_{d}z}$, and the final value
of the learning rate of T2NFS $\alpha^{*}$ is large enough,
$\alpha^{*}>\ddot{d}^{*}$ where the acceleration of the actual disturbance
$\ddot{d}$ is upper bounded by $\ddot{d}^{*}$.

###### Proof 2

The Lyapunov function is written as follows:

$V=\frac{1}{2}s^{2}+\frac{l_{d}z}{\gamma_{\alpha}(1+l_{d}z)}(\alpha-\alpha^{*})^{2}$
(42)

By taking the time derivative of the Lyapunov function above considering (28),
it is obtained as

$\displaystyle\dot{V}$ $\displaystyle=$ $\displaystyle
s\dot{s}+l_{d}z\frac{2\mid s\mid}{1+l_{d}z}(\alpha-\alpha^{*})$ (43)

If the time derivative of the sliding surface is inserted into the
aforementioned equation, it is obtained as follows:

$\displaystyle\dot{V}$ $\displaystyle=$ $\displaystyle
s\frac{l_{d}z}{l_{p}z}\Big{(}\dddot{\hat{d}}_{BN}+\frac{l_{p}z}{l_{d}z}\ddot{\hat{d}}_{BN}\Big{)}+l_{d}z\frac{2\mid
s\mid}{1+l_{d}z}(\alpha-\alpha^{*})$ (44)

It is obtained considering (3)

$\displaystyle\dot{V}$ $\displaystyle=$ $\displaystyle
s\Big{(}l_{d}z\ddot{e}_{d}+l_{p}z\dot{e}_{d}\Big{)}+l_{d}z\frac{2\mid
s\mid}{1+l_{d}z}(\alpha-\alpha^{*})$ (45)

(39) is inserted into (45), it is obtained as follows:

$\displaystyle\dot{V}$ $\displaystyle=$ $\displaystyle
s\Big{(}l_{d}z\frac{-(l_{p}+l_{r})z\dot{e}_{d}-2\alpha
sgn(s)+\ddot{d}}{1+l_{d}z}+l_{p}z\dot{e}_{d}\Big{)}+l_{d}z\frac{2\mid
s\mid}{1+l_{d}z}(\alpha-\alpha^{*})$ (46)

If it is assumed that $\ddot{d}$ is upper bounded by $\ddot{d}^{*}$:

$\displaystyle\dot{V}$ $\displaystyle=$ $\displaystyle\mid s\mid
l_{d}z\frac{(-2\alpha+\ddot{d}^{*})}{1+l_{d}z}+s\dot{e}_{d}\Big{(}l_{d}z\frac{-(l_{p}+l_{r})z}{1+l_{d}z}+l_{p}z\Big{)}+l_{d}z\frac{2\mid
s\mid}{1+l_{d}z}(\alpha-\alpha^{*})$ (47) $\displaystyle=$ $\displaystyle\mid
s\mid
l_{d}z\frac{(-2\alpha^{*}+\ddot{d}^{*})}{1+l_{d}z}+s\dot{e}_{d}\underbrace{\Big{(}l_{d}z\frac{-(l_{p}+l_{r})z}{1+l_{d}z}+l_{p}z\Big{)}}_{0}$
(48)

As stated in Theorem 2, if $l_{r}$ is equal to $\frac{l_{p}}{l_{d}z}$, i.e.
$l_{r}=\frac{l_{p}}{l_{d}z}$, and the final value of the learning algorithm
$\alpha^{*}$ is large enough, i.e. $\alpha^{*}>\ddot{d}^{*}$, then the time
derivative of the Lyapunov function is negative, i.e. $\dot{V}<0$, so that the
SLDO is stable.

###### Remark 4

The main advantage of the SLDO is to be able to prove the stability in case of
not only time-invariant disturbances, such as BNDOs, but also time-varying
disturbances.

##  4 Controller Design

The control objective is to find a control law so that the system states can
track a desired trajectory. One of the most commonly used method for nonlinear
systems is feedback linearization control (FLC). The traditional FLC method
for nonlinear systems is formulated considering a second-order nonlinear
system in (1) where there exists no disturbance:

$\displaystyle
u=-b^{-1}(x)\Big{(}\ddot{x}_{d}+a(x)-k_{2}(\dot{x}_{d}-x_{2})-k_{1}(x_{d}-x_{1})\Big{)}$
(49)

where the controller coefficients $k_{1},k_{2}$ are positive, i.e.
$k_{1},k_{2}>0$. If the control law in (49) is applied to the system in (1),
the closed-loop error dynamics are obtained as follows:

$\displaystyle\ddot{e}+k_{2}\dot{e}+k_{1}e=-zd(t)$ (50)

where $e=x_{d}-x_{1}$ and $\dot{e}=\dot{x}_{d}-x_{2}$.

###### Lemma 2

(Khalil and Grizzle, 1996): If a nonlinear system $F(x,u)$ is input-to-state
stable and the input satisfies $\displaystyle\lim_{t\to\infty}u(t)=0$, then
the state satisfies $\displaystyle\lim_{t\to\infty}x(t)=0$.

###### Remark 5

As seen in (50), if the disturbance $d(t)$ is different from zero, the closed-
loop error dynamics cannot converge to zero in finite time. This shows that
the traditional FLC is sensitive to disturbances.

The FLC based on the SLDO by taking the estimated disturbance value into
account is formulated as follows:

$\displaystyle
u=-b^{-1}(x)\Big{(}\ddot{x}_{d}+a(x)-k_{2}(\dot{x}_{d}-x_{2})-k_{1}(x_{d}-x_{1})+z\hat{d}_{SL}\Big{)}$
(51)

If the control law in (51) is applied to the system in (1), the closed-loop
error dynamics are obtained as follows:

$\displaystyle\ddot{e}+k_{2}\dot{e}+k_{1}e=-ze_{d}$ (52)

where $e_{d}=d(t)-\hat{d}_{SL}(t)$. As stated in Theorem 2, the disturbance
error dynamics for the SLDO can converge to zero asymptotically. As stated in
Lemma 2, if the disturbance error $e_{d}$ satisfies
$\displaystyle\lim_{t\to\infty}e_{d}(t)=0$, then the system error satisfies
$\displaystyle\lim_{t\to\infty}e(t)=0$. As a result, the closed-loop error
dynamics of the system can converge to zero asymptotically in finite time
under the control law in (51) if the controller coefficients $k_{1}$ and
$k_{2}$ are positive, i.e. $k_{1},k_{2}>0$.

###### Remark 6

If there exists no disturbance, i.e. $d(t)=0$, then the estimated value of the
disturbance in the control law (51) will be zero, i.e. $\hat{d}_{SL}=0$. This
results in the traditional FLC in (49) so that it maintains the nominal
performance the absence of disturbances.

##  5 Simulation Studies

The following nonlinear system, i.e., chaotic Duffing oscillator, is
considered for the simulation studies (Hsu, 2012):

$\displaystyle\dot{x}_{1}$ $\displaystyle=$ $\displaystyle x_{2}$
$\displaystyle\dot{x}_{2}$ $\displaystyle=$ $\displaystyle
1.1x_{1}-0.4x_{2}-x_{1}^{3}+2.1\cos{(1.8t)}+u+d$ (53)

where $a(x)=1.1x_{1}-0.4x_{2}-x_{1}^{3}+2.1\cos{(1.8t)}$, $b(x)=1$ and
$z=[0,1]^{T}$ as can be seen from (1).

The desired states values are defined as $\ddot{x}_{d}=\dot{x}_{d}=x_{d}=0$.
The initial conditions on the states of the system and the controller
coefficients are respectively selected as $x(0)=[1,-1]^{T}$ and $k_{1}=50$,
$k_{2}=25$. Since the disturbance coefficient vector in (5) is equal to
$z=[0,1]^{T}$, the proportional, derivative and robust gains for DOs must be
positive, i.e., $l_{p},l_{d},l_{r}>0$. The proportional gain is selected as
$l_{p}=[0,3]^{T}$ while the derivative gain is selected as
$l_{d}=[0,1.2]^{T}$. As stated in Theorem 2, since the robust gain must be
equal to $\frac{l_{p}}{l_{d}z}$, i.e. $l_{r}=\frac{l_{p}}{l_{d}z}=2.5$, the
robust gain is selected as $l_{r}=[0,2.5]^{T}$. The coefficient
$\gamma_{\alpha}$ to adjust the learning rate $\alpha$ for the SLDO is
selected as $0.001$. The initial conditions on the learning rate $\alpha$ and
parameter $q$ are set to $0.05$ and $0.5$, respectively. To benchmark
different disturbance observers in the presence and absence of uncertainties,
no disturbance is imposed on the system at the beginning, a step external
disturbance $d=3$ is imposed on the system at $t=10$ second and a sinusoidal
external disturbance $d=3sin(t)$ is imposed on the system at $t=20$ second as
formulated below:

$d(t)=\Bigg{\\{}\begin{array}[]{rclcl}0\leq&t&<10&d=&0\\\ 10\leq&t&<20&d=&3\\\
20\leq&t&<30&d=&3\sin(t)\\\ \end{array}$ (54)

In simulation studies, the control performance of the FLC based on the SLDO is
firstly compared with the traditional FLC and the FLC based on the BNDO. Then,
the SLDO is analyzed under noisy conditions and compared with its type-1
counterpart. Throughout simulation studies, the sampling time is set to
$0.001$ second while the number of membership functions are selected as
$I=J=3$. In the presence of plant uncertainties, the adaptation of the
learning rate of sliding mode learning algorithm must be a robust adaptation
law to avoid having infinite values. Therefore, a dead-zone has been proposed
in literature to handle this problem. In this paper, if the sliding surface is
smaller than the dead-zone parameter $\epsilon=0.05$, i.e., $s<\epsilon$, then
the learning rate $\alpha$ is not updated.

The states responses $x_{1},x_{2}$ are shown in Figs. 2(a) and 2(b). Firstly,
the FLC controller can control the system without steady-state error while
there exists no disturbance on the system. However, after the disturbances are
imposed on the system, it is observed that it is not robust against any
external disturbance and gives steady-state error as seen in Fig. 2(a) and
stated in Remark 5. Secondly, the FLC based on the BNDO can control the system
without any steady-state error while there exist no disturbance and a time-
invariant disturbance. However, it is seen that it is not robust against a
time-varying disturbance as stated in Remark 1. Thirdly, the FLC based on the
SLDO can control system without steady-state error and it is robust against
time-invariant and time-varying disturbances. Moreover, the FLCs based on the
BNDO and SLDO maintain the nominal performance while there exists no
disturbance between t=0-10 seconds as stated in Remark 6.

The actual and estimated disturbances are shown in Fig. 2(c). As can be
observed, the SLDO can estimate time-varying disturbances while BNDO is only
able to estimate only time-invariant disturbances as stated in Remark 4. This
fact results in the robust control performance of the FLC based on the SLDO
against time-varying uncertainties. Thanks to learning process by the
feedback-error learning algorithm, the T2NFS takes the overall estimation
signal while the conventional estimation signal converges to zero in finite
time as shown in Fig. 2(d). Inasmuch as the total generated estimation signal
by the feedback-error learning structure is equal to $\tau_{c}-\tau_{n}$, the
output of the T2NFS $\tau_{n}$ is multiplied by $-1$ in Fig. 2(d) not to cause
the reader to become perplexed. The T2NFS becomes the leading estimator after
a short time period. The output of the conventional estimation law $\tau_{c}$
becomes nonzero only during the time intervals when the T2NFS is learning.

The adaptation of the learning rate $\alpha$ is shown in Fig 2(e). As seen,
the initial condition on the learning rate $\alpha$ is set to $0.05$ and the
learning rate is constant while the output of the conventional estimation law
$\tau_{c}$ is equal to zero due to the fact that learning is not required.
When the disturbances are imposed on the system, the learning rate is
increasing for a short time period till $\tau_{c}$ becomes zero. Moreover, the
adaptation of the parameter $q$ is shown in Fig. 2(f). The initial condition
on the parameter $q$ is set to $0.5$, which is the general case. Thanks to the
adaptation rule in (27), the proportion of the upper and lower membership
functions is adjusted throughout the simulations.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2:  (a) Responses of state $x_{1}$(b) Responses of state $x_{2}$ (c)
True and estimated values of the disturbance (d) Estimation signals (e)
Learning parameter $\alpha$ (f) Parameter $q$

Type-2 fuzzy membership functions are used in the proposed estimation
structure and it is possible to downgrade them to type-1 counterparts by
equalizing the upper and lower values of parameters in (18)-(25). In
literature, it is claimed that the type-2 fuzzy logic system gives better
performance than its type-1 counterpart in the presence of noise and
uncertainty in the system. The initial conditions on the states of the system
are set to $x(0)=[1,1]^{T}$. In order to compare the performance of T2NFS with
its type-1 counterpart under noisy conditions, the actual disturbance $d$ with
different noise levels $SNR$, which are equal to $20$ $dB$, $40$ $dB$ and $60$
$dB$, is applied the system. The mean squared errors for the different noise
levels are given in Table 1. As seen from this table, the T2NFS gives less
error than the type-1 neuro-fuzzy structure (T1NFS) and the performance of
T2NFS is more remarkable while the noise level is increasing. Figure 3(a)
shows the absolute disturbance error responses with a noise level of $20$ $dB$
for T1NFS and T2NFS. As can be observed, T2NFS gives less disturbance error
when compared to its type-1 counterpart. Moreover, the system states under
noisy condition are shown in Figs. 3(b) and 3(c). The FLC with the SLDO based
on T2NFS exhibits better control performance. These results verify previous
results seen in (Mendel, 2000; Hsiao et al., 2008; Khanesar et al., 2011).
Type-2 fuzzy systems have more degrees of freedom so that they have capability
of dealing with noisy measurements and uncertainties in the system more
effectively.

(a)

(b)

(c)

Figure 3:  (a) Disturbance error when the actual disturbance with a noise level $SNR=20$ $db$ (b) Responses of state $x_{1}$ when the actual disturbance with a noise level $SNR=20$ $db$ (c) Responses of state $x_{2}$ when the actual disturbance with a noise level $SNR=20$ $db$ Table 1: Mean Squared Error. | 20 (dB) | 40 (dB) | 60 (dB)
---|---|---|---
T1NFS | 1.3112 | 0.0288 | 0.0034
T2NFS | 1.2352 | 0.0276 | 0.0033
Decrease in Disturbance Error | 6.14 % | 4.35 % | 3.03 %

The simulations are executed on the computer, which is equipped with $3.1$ GHz
Intel Core i$7-5557U$ CPU and $16$ GB of RAM. It is to be noted that the
sampling and simulation times are set to $0.001$ and $30$ seconds in the sense
that total number of sample is equal to $30000$. The total required
computation time is calculated as $5.531993$ seconds so that the required
computation time for each sampling time instant is around $0.18$ millisecond.
The required computation time of the sliding mode learning algorithm is
significantly lower than the one of the other methods, such as gradient
descent, Levenberg-Marquardt, particle swarm optimization and extended Kalman
filter (Kayacan et al., 2015a). The reason is that the sliding mode learning
algorithm does not contain any high-order matrices, matrix manipulations or
calculations of the partial derivatives. Moreover, particle filter becomes
infeasible in real-time due to large number of states and moving horizon
estimation requires solving nonlinear optimization problem which results in
large computation times (Daum, 2005). Recent developed numerical methods have
reduced required computation times for solving nonlinear optimization problems
around 5 milliseconds. However, even this is 25 times more than our proposed
algorithms in the paper (Kayacan et al., 2015b). Therefore, it can be
concluded that the proposed method in this paper is more practical in real-
time applications.

##  6 Conclusion

A novel SLDO is developed by benefiting from the T2NFS with online sliding
mode learning algorithm in feedback-error learning scheme. In addition to the
stability of the SMC-theory learning algorithm, the stability of the SLDO by
taking the system dynamics into account and the stability of the FLC based on
the SLDO are proven by using separation principle. The simulations results
show that the traditional FLC is sensitive to disturbances and the FLC based
on the BNDO is only robust to time-invariant disturbance while the FLC based
on the SLDO is robust against any kind of disturbances by performing precise
online estimation of the immeasurable time-varying disturbances. Moreover, the
FLC based on the SLDO maintains the nominal control performance in the absence
of uncertainties. Thanks to online sliding mode learning algorithm, the
parameters of the T2NFS are spontaneously adjusted to learn disturbances and
this makes systems robust to cope with uncertainties. Moreover, the developed
SMC-theory based learning algorithm requires significantly less computation
time than the traditional ones, e.g. gradient descent and evolutionary
training algorithms, so that it is more practical in real-time applications.

##  Appendix A  Calculation of $\dot{\tau}_{n}$

By taking the time derivative of (10)-(13), the following equations are
obtained as:

$\displaystyle\dot{\underline{\mu}}_{1i}(\xi_{1})$ $\displaystyle=$
$\displaystyle-2\underline{N}_{1i}\dot{\underline{N}}_{1i}\underline{\mu}_{1i}(\xi_{1})$
$\displaystyle\dot{\overline{\mu}}_{1i}(\xi_{1})$ $\displaystyle=$
$\displaystyle-2\overline{N}_{1i}\dot{\overline{N}}_{1i}\overline{\mu}_{1i}(\xi_{1})$
$\displaystyle\dot{\underline{\mu}}_{2j}(\xi_{2})$ $\displaystyle=$
$\displaystyle-2\underline{N}_{2j}\dot{\underline{N}}_{2j}\underline{\mu}_{2j}(\xi_{2})$
$\displaystyle\dot{\overline{\mu}}_{2j}(\xi_{2})$ $\displaystyle=$
$\displaystyle-2\overline{N}_{2j}\dot{\overline{N}}_{2j}\overline{\mu}_{2j}(\xi_{2})$
(55)

where

$\displaystyle\underline{N}_{1i}=\Big{(}\frac{\xi_{1}-\underline{c}_{1i}}{\underline{\sigma}_{1i}}\Big{)},\quad\dot{\underline{N}}_{1i}=\frac{(\dot{\xi_{1}}-\dot{\underline{c}}_{{1i}})\underline{\sigma}_{1i}-(\xi_{1}-\underline{c}_{1i})\dot{\underline{\sigma}}_{1i}}{\underline{\sigma}^{2}_{1i}}$
$\displaystyle\underline{N}_{2j}=\Big{(}\frac{\xi_{2}-\underline{c}_{2j}}{\underline{\sigma}_{2j}}\Big{)},\quad\dot{\underline{N}}_{2j}=\frac{(\dot{\xi_{2}}-\dot{\underline{c}}_{{2i}})\underline{\sigma}_{2i}-(\xi_{2}-\underline{c}_{2i})\dot{\underline{\sigma}}_{2i}}{\underline{\sigma}^{2}_{2i}}$
$\displaystyle\overline{N}_{1i}=\Big{(}\frac{\xi_{1}-\overline{c}_{1i}}{\overline{\sigma}_{1i}}\Big{)},\quad\dot{\overline{N}}_{1i}=\frac{(\dot{\xi_{1}}-\dot{\overline{c}}_{{1i}})\overline{\sigma}_{1i}-(\xi_{1}-\overline{c}_{1i})\dot{\overline{\sigma}}_{1i}}{\overline{\sigma}^{2}_{1i}}$
$\displaystyle\overline{N}_{2j}=\Big{(}\frac{\xi_{2}-\overline{c}_{2j}}{\overline{\sigma}_{2j}}\Big{)}\quad\dot{\overline{N}}_{2j}=\frac{(\dot{\xi_{2}}-\dot{\overline{c}}_{{2i}})\overline{\sigma}_{2i}-(\xi_{2}-\overline{c}_{2i})\dot{\overline{\sigma}}_{2i}}{\overline{\sigma}^{2}_{2i}}$
(56)

It is obtained from (A)

$\underline{N}_{1i}\dot{\underline{N}}_{1i}=\underline{N}_{2j}\dot{\underline{N}}_{2j}=\overline{N}_{1i}\dot{\overline{N}}_{1i}=\overline{N}_{2j}\dot{\overline{N}}_{2j}=\alpha
sgn{(s)}$ (57)

By taking the time derivative of (16), the following equations are obtained as
follows:

$\displaystyle\dot{\widetilde{\underline{w}}}_{ij}=\frac{\Big{(}\underline{\mu}_{1i}(\xi_{1})\underline{\mu}_{2j}(\xi_{2})\Big{)}^{\prime}\Big{(}\sum_{i=1}^{I}\sum_{j=1}^{J}\underline{w}_{ij}\Big{)}-\underline{w}_{ij}\Big{(}\sum_{i=1}^{I}\sum_{j=1}^{J}\underline{\mu}_{1i}(\xi_{1})\underline{\mu}_{2j}(\xi_{2})\Big{)}^{\prime}}{(\sum_{i=1}^{I}\sum_{j=1}^{J}\underline{w}_{ij})^{2}}$
(58)

Since
$\widetilde{\underline{w}}_{ij}=\frac{\underline{w}_{ij}}{\sum_{i=1}^{I}\sum_{j=1}^{J}\underline{w}_{ij}}$,

$\displaystyle\dot{\widetilde{\underline{w}}}_{ij}$ $\displaystyle=$
$\displaystyle\frac{\Big{(}\dot{\underline{\mu}}_{1i}(\xi_{1})\underline{\mu}_{2j}(\xi_{2})+\underline{\mu}_{1i}(\xi_{1})\dot{\underline{\mu}}_{2j}(\xi_{2})\Big{)}}{\sum_{i=1}^{I}\sum_{j=1}^{J}\underline{w}_{ij}}$
(59)
$\displaystyle-\frac{\widetilde{\underline{w}}_{ij}\bigg{(}\sum_{i=1}^{I}\sum_{j=1}^{J}\Big{(}\dot{\underline{\mu}}_{1i}(\xi_{1})\underline{\mu}_{2j}(\xi_{2})+\underline{\mu}_{1i}(\xi_{1})\dot{\underline{\mu}}_{2j}(\xi_{2})\Big{)}\bigg{)}}{\sum_{i=1}^{I}\sum_{j=1}^{J}\underline{w}_{ij}}$
$\displaystyle=$
$\displaystyle\frac{-2\underline{\mu}_{1i}(\xi_{1})\underline{\mu}_{2j}(\xi_{2})+2\widetilde{\underline{w}}_{ij}\sum_{i=1}^{I}\sum_{j=1}^{J}\underline{\mu}_{1i}(\xi_{1})\underline{\mu}_{2j}(\xi_{2})}{\sum_{i=1}^{I}\sum_{j=1}^{J}\underline{w}_{ij}}\Big{(}\underline{N}_{1i}\dot{\underline{N}}_{1i}+\underline{N}_{2j}\dot{\underline{N}}_{2j}\Big{)}$
$\displaystyle=$
$\displaystyle-\widetilde{\underline{w}}_{ij}\dot{\underline{K}}_{ij}+\widetilde{\underline{w}}_{ij}\sum_{i=1}^{I}\sum_{j=1}^{J}\widetilde{\underline{w}}_{ij}\dot{\underline{K}}_{ij}$

where

$\dot{\underline{K}}_{ij}=2\Big{(}\underline{N}_{1i}\dot{\underline{N}}_{1i}+\underline{N}_{2j}\dot{\underline{N}}_{2j}\Big{)}=4\alpha
sgn{(s)}$

Similarly, it is readily obtained that:

$\dot{\widetilde{\overline{w}}}_{ij}=-\widetilde{\overline{w}}_{ij}\dot{\overline{K}}_{ij}+\widetilde{\overline{w}}_{ij}\sum_{i=1}^{I}\sum_{j=1}^{J}\widetilde{\overline{w}}_{ij}\dot{\overline{K}}_{ij}$
(60)

where

$\dot{\overline{K}}_{ij}=2\Big{(}\overline{N}_{1i}\dot{\overline{N}}_{1i}+\overline{N}_{2j}\dot{\overline{N}}_{2j}\Big{)}=4\alpha
sgn{(s)}$

The time derivative of (15) is obtained to find $\dot{\tau}_{n}$ as follows:

$\displaystyle\dot{\tau}_{n}$ $\displaystyle=$ $\displaystyle
q\sum_{i=1}^{I}\sum_{j=1}^{J}(\dot{f}_{ij}\widetilde{\underline{w}}_{ij}+f_{ij}\dot{\widetilde{\underline{w}}}_{ij})+(1-q)\sum_{i=1}^{I}\sum_{j=1}^{J}(\dot{f}_{ij}\widetilde{\overline{w}}_{ij}+f_{ij}\dot{\widetilde{\overline{w}}}_{ij})$
(61)
$\displaystyle+\dot{q}\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}\widetilde{\underline{w}}_{ij}-\dot{q}\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}\widetilde{\overline{w}}_{ij}$

If (59) and (60) are inserted into the aforementioned equation:

$\displaystyle\dot{\tau}_{n}$ $\displaystyle=$ $\displaystyle
q\sum_{i=1}^{I}\sum_{j=1}^{J}\bigg{(}\Big{(}-\widetilde{\underline{w}}_{ij}\dot{\underline{K}}_{ij}+\widetilde{\underline{w}}_{ij}\sum_{i=1}^{I}\sum_{j=1}^{J}\widetilde{\underline{w}}_{ij}\dot{\underline{K}}_{ij}\Big{)}f_{ij}+\widetilde{\underline{w}}_{ij}\dot{f}_{ij}\bigg{)}$
(62)
$\displaystyle+(1-q)\sum_{i=1}^{I}\sum_{j=1}^{J}\bigg{(}\Big{(}-\widetilde{\overline{w}}_{ij}\dot{\overline{K}}_{ij}+\widetilde{\overline{w}}_{ij}\sum_{i=1}^{I}\sum_{j=1}^{J}\widetilde{\overline{w}}_{ij}\dot{\overline{K}}_{ij}\Big{)}f_{ij}+\widetilde{\overline{w}}_{ij}\dot{f}_{ij}\bigg{)}$
$\displaystyle+\dot{q}\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}(\widetilde{\underline{w}}_{ij}-\widetilde{\overline{w}}_{ij})$
$\displaystyle=$ $\displaystyle 4\alpha
sgn{(s)}\Bigg{(}q\sum_{i=1}^{I}\sum_{j=1}^{J}\bigg{(}\Big{(}-\widetilde{\underline{w}}_{ij}+\widetilde{\underline{w}}_{ij}\sum_{i=1}^{I}\sum_{j=1}^{J}\widetilde{\underline{w}}_{ij}\Big{)}f_{ij}+\widetilde{\underline{w}}_{ij}\dot{f}_{ij}\bigg{)}$
$\displaystyle+(1-q)\sum_{i=1}^{I}\sum_{j=1}^{J}\bigg{(}\Big{(}-\widetilde{\overline{w}}_{ij}+\widetilde{\overline{w}}_{ij}\sum_{i=1}^{I}\sum_{j=1}^{J}\widetilde{\overline{w}}_{ij}\Big{)}f_{ij}+\widetilde{\overline{w}}_{ij}\dot{f}_{ij}\bigg{)}\Bigg{)}$
$\displaystyle+\dot{q}\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}(\widetilde{\underline{w}}_{ij}-\widetilde{\overline{w}}_{ij})$

Since $\sum_{i=1}^{I}\sum_{j=1}^{J}\widetilde{\overline{w}}_{ij}=1$ and
$\sum_{i=1}^{I}\sum_{j=1}^{J}\widetilde{\underline{w}}_{ij}=1$, the
aforementioned equation becomes by using (26) and (27) as follows:

$\displaystyle\dot{\tau}_{n}$ $\displaystyle=$ $\displaystyle
q\sum_{i=1}^{I}\sum_{j=1}^{J}\widetilde{\underline{w}}_{ij}\dot{f}_{ij}+(1-q)\sum_{i=1}^{I}\sum_{j=1}^{J}\widetilde{\overline{w}}_{ij}\dot{f}_{ij}+\dot{q}\sum_{i=1}^{I}\sum_{j=1}^{J}f_{ij}(\widetilde{\underline{w}}_{ij}-\widetilde{\overline{w}}_{ij})$
(63) $\displaystyle=$ $\displaystyle-2\alpha sgn{(s)}$

## Acknowledgement

The information, data, or work presented herein was funded in part by the
Advanced Research Projects Agency-Energy (ARPA-E), U.S. Department of Energy,
under Award Number DE-AR0000598.

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