Adaptive-Robust Control of a Class of Uncertain Nonlinear Systems Utilizing Time-Delayed Input and Position Feedback
Spandan~Roy and~Indra~Narayan~Kar
~\IEEEmembership{Senior~Member
~IEEE
....lastname \thanks{...
In this paper, the tracking control problem of a class of Euler-Lagrange systems subjected to unknown uncertainties is addressed and an adaptive-robust control strategy, christened as Time-Delayed Adaptive Robust Control (TARC) is presented. The proposed control strategy approximates the unknown dynamics through time-delayed logic, and the switching logic provides robustness against the approximation error. The novel adaptation law for the switching gain, in contrast to the conventional adaptive-robust control methodologies, does not require either nominal modelling or predefined bounds of the uncertainties. Also, the proposed adaptive law circumvents the overestimation-underestimation problem of switching gain. The state derivatives in the proposed control law is estimated from past data of the state to alleviate the measurement error when state derivatives are not available directly. Moreover, a new stability notion for time-delayed control is proposed which in turn provides a selection criterion for controller gain and sampling interval. Experimental result of the proposed methodology using a nonholonomic wheeled mobile robot (WMR) is presented and improved tracking accuracy of the proposed control law is noted compared to time-delayed control and adaptive sliding mode control.
atrc_filter_in_elec_ver_4__2_.tex

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\title{Adaptive-Robust Control of a Class of Uncertain Nonlinear Systems Utilizing Time-Delayed Input and Position Feedback }
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        and~Indra~Narayan~Kar,~\IEEEmembership{Senior~Member,~IEEE}% <-this % stops a space
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\begin{abstract}
In this paper, the tracking control problem of a class of Euler-Lagrange systems subjected to unknown uncertainties is addressed and an adaptive-robust control strategy, christened as Time-Delayed Adaptive Robust Control (TARC) is presented. The proposed control strategy approximates the unknown dynamics through time-delayed logic, and the switching logic provides robustness against the approximation error. The novel adaptation law for the switching gain, in contrast to the conventional adaptive-robust control methodologies, does not require either nominal modelling or predefined bounds of the uncertainties. Also, the proposed adaptive law circumvents the overestimation-underestimation problem of switching gain. The state derivatives in the proposed control law is estimated from past data of the state to alleviate the measurement error when state derivatives are not available directly. Moreover, a new stability notion for time-delayed control is proposed which in turn provides a selection criterion for controller gain and sampling interval. Experimental result of the proposed methodology using a nonholonomic wheeled mobile robot (WMR) is presented and improved tracking accuracy of the proposed control law is noted compared to time-delayed control and adaptive sliding mode control.
\end{abstract}

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Adaptive-robust control, Euler-Lagrange system, time-delayed control, state derivative estimation, wheeled mobile robot.
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\section{Introduction}
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\subsection{Background and Motivation}\label{intro}
\IEEEPARstart{D}{esign} of an efficient controller for nonlinear systems subjected to parametric and nonparametric uncertainties has always been a challenging task. Among many other approaches, Adaptive control and Robust control are the two popular control strategies that researchers have extensively employed while dealing with uncertain nonlinear systems. In general, adaptive control uses predefined parameter adaptation laws and equivalence principle based control law which adjusts the parameters of the controller on the fly according to the pertaining uncertainties \cite{Ref:1}. However, this approach has poor transient performance and online calculation of the unknown system parameters and controller gains for complex systems is computationally intensive \cite{Ref:2}. Whereas, robust control aims at tackling the uncertainties of the system within an uncertainty bound defined a priori. It reduces computation complexity to a great extent for complex systems compared to adaptive control as exclusive online estimation of uncertain parameters is not required \cite{Ref:3}. 
However, nominal modelling of the uncertainties is necessary to decide upon their bounds, which is not always possible. Again, to increase the operating region of the controller, often higher uncertainty bounds are assumed. This in turn leads to problems like higher controller gain and consequent possibility of chattering for the switching law based robust controller like Sliding Mode Control (SMC). This in effect reduces controller accuracy \cite{Ref:4}. Higher order sliding mode \cite{Ref:5} can alleviate the chattering problem but prerequisite of uncertainty bound still exists. 
\par  Time-Delayed Control (TDC) is utilized in \cite{Ref:6-1} to implement state derivative feedback for enhancing stability margin of SISO linear time invariant (LTI) systems. In \cite{Ref:6}, \cite{Ref:6-2}, \cite{Ref:25}-\cite{Ref:26}, \cite{Ref:33}, \cite{Ref:34}, TDC is used to provide robustness against uncertainties. In this process, all the uncertain terms are represented by a single function which is then approximated using control input and state information of the immediate past time instant. The advantage of this robust control approach in uncertain systems is that it reduces the burden of tedious modelling of complex system to a great extent. In spite of this, the unattended approximation error, commonly termed as time-delayed error (TDE) causes detrimental effect to the performance of the closed system and its stability. In this front, a few work have been carried out to tackle TDE which includes internal model \cite{Ref:7}, gradient estimator \cite{Ref:8}, ideal velocity feedback \cite{Ref:9}, nonlinear damping \cite{Ref:10} and sliding mode based approach \cite{Ref:11}-\cite{Ref:12}.  The stability of the closed loop system \cite{Ref:7}-\cite{Ref:9}, \cite{Ref:25}-\cite{Ref:26}, depends on the boundedness of TDE as shown in \cite{Ref:6}. This method approximates the continuous time closed loop system in a discrete form without considering the effect of discretization error. Again, the stability criterion mentioned in \cite{Ref:6} restricts the allowable range of perturbation and thus limits controller working range. Stability of the system in \cite{Ref:11} is established in frequency domain, which makes the approach inapplicable to the nonlinear systems. Moreover, the controllers designed in \cite{Ref:10}, and \cite{Ref:12}, \cite{Ref:34} require nominal modelling and upper bound of the TDE respectively which is not always possible in practical circumstances. Also, to the best knowledge of the authors, controller design issues such as selection of controller gains and sampling interval to achieve efficient performance is still an open problem. In contrast to TDC, works reported in \cite{Ref:27}-\cite{Ref:29} use low pass filter to approximate the unknown uncertainties and disturbances. However, frequency range of system dynamics and external disturbances are required to determine the time constant of the filter. Furthermore, the order of the low pass filter needs to be adjusted according to order of the disturbance to maintain stability of the controller.
\par Considering the individual limitations of adaptive and robust control, recently global research is reoriented towards adaptive-robust control (ARC) where switching gain of the controller is adjusted online. The series of publications \cite{Ref:2}, \cite{Ref:13}-\cite{Ref:19} regarding ARC, estimates the uncertain terms online based on predefined projection function, but predefined bound on uncertainties is still a requirement. The work reported in \cite{Ref:20}, \cite{Ref:32} attempts to estimate the maximum uncertainty bound but the integral adaptive law makes the controller susceptible to very high switching gain and consequent chattering \cite{Ref:33}. The adaptive sliding mode control (ASMC) as presented in \cite{Ref:21}-\cite{Ref:22} proposed two laws for the switching gain to adapt itself online according to the incurred error. In the first adaptive law, the switching gain decreases or increases depending on a predefined threshold value. However, until the threshold value is achieved, the switching gain may still be increasing (resp. decreasing) even if tracking error decreases (resp. increases) and thus creates overestimation (resp. underestimation) problem of switching gain \cite{Ref:23}. Moreover, to decide the threshold value the maximum bound of the uncertainty is required. For the second adaptive law, the threshold value changes online according to switching gain. Yet, nominal model of the uncertainties is needed for defining the control law. This limits the adaptive nature of the control law and applicability of the controller. 
\subsection{Problem Definitions and Contributions}\label{sec 1.2}
In this paper, three specific related problems on TDC have been dealt with and the corresponding solutions to the same which are also the contributions of this paper are summarized below: 
\begin{itemize}
\item \textit{ \textbf{Problem 1:} The stability analysis of TDC, as provided in \cite{Ref:6}, \cite{Ref:25}-\cite{Ref:26}, \cite{Ref:33}, \cite{Ref:34}, approximates the continuous time system in discrete time domain without considering the effects of discretization error. Again, choice of the delay time and its relation with the controller gains is still an open problem.} 

  In this paper, a new stability analysis for TDC, based on the Lyapunov-Krasvoskii method, is provided in continuous time domain. Furthermore, through the proposed stability approach, a relation between the sampling interval and controller gain is established.  

\item \textit{\textbf{Problem 2:} %In many applications state derivatives are not directly available. They are calculated numerically which invokes measurement error. For TDC, first and second order state derivatives are required to formulate the control law.
The TDC reported in (\cite{Ref:6}, \cite{Ref:25}-\cite{Ref:26}, \cite{Ref:33}, \cite{Ref:34}), velocity and acceleration feedback are necessary to compute the control law. While in \cite{Ref:6-1}, only velocity feedback is required and acceleration term is approximated numerically using time delay. However, in many applications velocity and acceleration feedback are not available explicitly and numerical approximation of these terms invokes measurement error.}

%TDC is utilized to implement state derivative feedback for enhancing stability margin of SISO linear time invariant (LTI) systems. However, it still requires and .
 As a second contribution of this paper, Filtered Time-Delayed Control (F-TDC) control law is formulated where only position feedback is sufficient while velocity and acceleration terms are estimated using past and present position information to curb the effect measurement error. Stability analysis of the proposed F-TDC is provided which also maintains the relation between controller gains and sampling interval.  
\item \textit{ \textbf{Problem 3:} Robustness property against TDE is essential to achieve good tracking accuracy. The robust controllers reported in literature, either requires nominal model of the uncertainties (\cite{Ref:21}-\cite{Ref:22}) or its predefined bound (\cite{Ref:2}, \cite{Ref:10}, \cite{Ref:12}, \cite{Ref:13}-\cite{Ref:19}). So, it is required to devise a control law which would avoid any prior knowledge of the uncertainties while providing robustness against TDE.}

 Towards the last contribution of this article, an adaptive-robust control strategy, Time-Delayed Adaptive Robust Control (TARC) has been formulated for a class of uncertain Euler-Lagrange systems. The proposed control law approximates uncertainties by time-delayed logic and provides robustness against the TDE, arising from time-delayed logic based estimation, by switching control. The novel adaptive law, presented here, aims at overcoming the overestimation-underestimation problem of the switching gain without any prior knowledge of uncertainties. The proposed adaptive law provides flexibility to the control designer to select any suitable error function according to the application requirement while maintaining similar system stability notion. 
\end{itemize}
\par As a proof of concept, experimental validation of the proposed control methodology is provided using the "PIONEER-3" nonholonomic WMR in comparison to TDC \cite{Ref:6} and ASMC \cite{Ref:21}-\cite{Ref:22}.

\subsection{Organization}
The article is organized as follows: a new stability analysis of TDC along with its design issues is first discussed in Section II. This is followed by the proposed adaptive-robust control methodology and its detail analysis. Section III presents the experimental results of the proposed controller and its comparison with TDC and ASMC. Section IV concludes the entire work.

\subsection{Notations}
The following notations are assumed for the entirety of the paper: any variable $\mu$ delayed by an amount $h$ as $\mu(t-h)$, is denoted as $\mu_{h}$; $\lambda _{min}(\cdot)$ and $|| \cdot ||$ represent minimum eigen value and Euclidean norm of the argument respectively; $I$ represents identity matrix.
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%\hfill September 17, 2014
\section{Controller Design}\label{sec: 2}
\subsection{Time-Delayed Control: Revisited}\label{sec: 2.1}
In general, an Euler-Lagrange system with second order dynamics, devoid of any delay, can be written as,
\begin{equation}\label{sys}
M(q)\ddot{q}+N(q,\dot{q})=\tau(t),
\end{equation}
where, $q(t)\in\mathbb{R}^{n}$ is the system state, $\tau(t)\in\mathbb{R}^{n}$ is the control input, $M(q)\in\mathbb{R}^{n\times n}$  is the mass/inertia matrix and $N(q,\dot{q})\in\mathbb{R}^{n}$ denotes combination of other system dynamics terms based on system properties. In practice, it can be assumed that unmodelled dynamics and disturbances is subsumed by $N$. The control input is defined to be,
%denotes combination of other system dynamics terms based on system properties such as Coriolis, gravitational, friction, damping forces etc
\begin{equation}\label{input}
\tau=\hat{M}u+\hat{N},
\end{equation}
where, $u$ is the auxiliary control input, $\hat{M}$ and $\hat{N}$ are the nominal values of $M$ and $N$ respectively. To reduce the modelling effort of the complex systems, $\hat{N}$ can be approximated from the input-output data of previous instances using the time-delayed logic (\cite{Ref:6}, \cite{Ref:25}-\cite{Ref:26}) and the system definition (\ref{sys}) as,

\begin{equation}\label{approx}
\hat{N}( q,\dot{q} )\cong N(q_h,\dot{q}_h)=\tau_h-\hat{M}(q_h)\ddot{q}_h,
\end{equation}
where, $h>0$ is a fixed small delay time. Substituting (\ref{input}) and (\ref{approx}) in (\ref{sys}), the system dynamics is converted into an input as well state delayed dynamics as,
\begin{equation}\label{sys new}
\hat{M}(q)\ddot{q}+\bar{N}(q,\dot{q},\ddot{q},\ddot{q}_h)=\tau_h.
\end{equation}
where $\bar{N}=(M-\hat{M})\ddot{q}+\hat{M}_h\ddot{q}_h-\hat{M}u+ N$.







%\begin{equation}\label{approx}
%\hat{H}( t )\cong H_h=\tau_h-\hat{M}_h\ddot{q}_h,
%\end{equation}
%where, $h$  is a fixed small delay time. Putting (\ref{input}) and (\ref{approx}) into (\ref{sys}), the system dynamics is converted into an input as well state delayed dynamics as,
%\begin{equation}\label{sys new}
%\hat{M}\ddot{q}+\bar{H}=\tau_h.
%\end{equation}
%where $\bar{H}=(M-\hat{M})\ddot{q}+\hat{M}_h\ddot{q}_h-\hat{M}u+ H$. 

Let, $q^d(t)$ be the desired trajectory to be tracked and $e_1(t)=q(t)-q^d(t)$ is the tracking error. The auxiliary control input $u$ is defined in the following way,
\begin{equation}\label{aux}
u(t)=\ddot{q}^d(t)-K_2\dot{e}_1(t)-K_1e_1(t),
\end{equation}
where, $K_1$ and $K_2$ are two positive definite matrices with appropriate dimensions. Putting (\ref{aux}) and (\ref{input}) in (\ref{sys new}), following error dynamics is obtained,
\begin{equation}\label{error dyn delayed 2}
\ddot{e}_1=-K_2\dot{e}_{1h}-K_1e_{1h}+\sigma_1,
\end{equation}
where, $\sigma_1=(\hat{M}^{-1}\hat{M}_h-I)u_h+\hat{M}^{-1}(\hat{N}_h-\bar{N})+\ddot{q}^d_h-\ddot{q}^d$ and can be treated as overall uncertainty. Further, (\ref{error dyn delayed 2}) can be written in state space form as,
\begin{equation}\label{new err dyn 1}
\dot{e}=A_1e+B_1e_h+B\sigma_1,
\end{equation}
where, $e=\begin{bmatrix}
e_1\\
\dot{e}_1
\end{bmatrix}$, $ A_1=\begin{bmatrix}
0 & I \\
0 & 0
\end{bmatrix} 
, 
B_1=\begin{bmatrix}
0 & 0\\
-K_1 & -K_2
\end{bmatrix}$, $B=\begin{bmatrix}
0\\
I
\end{bmatrix}$. Noting that, $e(t-h)=e(t)-\int\limits_{-h}^0 \dot{e}(t+\theta)\mathrm{d}\theta$, where the derivative inside the integral is with respect to $\theta$, the error dynamics (\ref{new err dyn 1}) is modified as,
\begin{equation}\label{error dyn delayed}
\dot{e}(t)=Ae(t)-B_1\int\limits_{-h}^0 \dot{e}(t+\theta)\mathrm{d}\theta+B\sigma_1,
\end{equation}
%where, $A=A_1+B_1$.


%\subsubsection*{Stability Analysis}
%\begin{prb}
%The stability analysis of TDC, as provided in \cite{Ref:6}, \cite{Ref:25}-\cite{Ref:26}, \cite{Ref:33}, approximates the continuous time system in discrete time domain without considering the effects of discretization error. Again, choice of the delay time $h$ and its relation with the controller gains $K_1$ and $K_2$ is still an open problem.  
%\end{prb}
%\subsubsection*{Solution}
where, $A=A_1+B_1$. It is assumed that the choice controller gains $K_1$ and $K_2$ makes $A$ Hurwitz which is always possible. Also, it is assumed that the unknown uncertainties are bounded. In this paper, a new stability criterion, based on the Lyapunov-Krasvoskii method, is presented through Theorem 1 which addresses the issues defined in Problem 1. 
%Before stating Theorem 1, the following assumptions are made: 
%\begin{assumption}\label{assum 1}
%
%\end{assumption}
%\begin{lemma}
%[\cite{Ref:31}]: Using Jensen's inequality, for a positive definite matrix $D>0$ the following inequality holds,
%\begin{align}
%\int_{-h}^{0}e^{T}(\psi)De(\psi)d\psi \geq \frac{1}{h}\int_{-h}^{0}e^{T}(\psi)d\psi D \int_{-h}^{0}e(\psi)d\psi.\label{lemma 1}
%\end{align}
%\end{lemma}
%\begin{assum}\label{assum 2}
%Let, $V(e)$ be a Lyapunov function candidate. Then  following the Razumikhin-type theorem \cite{Ref:36}, for some constant $r>1$, let the following inequality holds:
%\begin{equation}\label{razu}
%V(e(\nu ))<rV(e(t)),\qquad t-2h\leq \nu \leq t.
%\end{equation}
%\end{assum}
\begin{theorem}
The system (\ref{sys new}) employing the control input (\ref{input}), having auxiliary control input (\ref{aux}) is UUB if the controller gains and delay time is selected such that the following condition holds:
\begin{equation}\label{delay value}
\Psi=\begin{bmatrix}
Q-E-(1+\xi)\frac{h^2}{\beta}D & 0\\ 
0 & (\xi-1)\frac{h^2}{\beta}D
\end{bmatrix}>0
\end{equation}

where, $E=\beta PB_1( A_1D^{-1}A_1^T+B_1D^{-1}B_1^{T}+D^{-1} )B_1^{T}P$, $\xi>1$ and $\beta>0$ are scalar, and $P>0$ is the solution of the Lyapunov equation $A^TP+PA=-Q$ for some $Q>0$. 
\end{theorem}
\begin{proof}
Let us consider the following Lyapunov function:
\begin{equation}\label{lyapunov}
V(e)=V_1(e)+V_2(e)+V_3(e)+V_4(e),
\end{equation} 
where,
\begin{align}
V_1(e)&=e^TPe\\
V_2(e)&=\frac{h}{\beta}\int_{-h}^{0}\int_{t+\theta}^{t}e^T(\psi )De(\psi )d\psi d\theta\\
V_3(e)&=\frac{h}{\beta}\int_{-h}^{0}\int_{t+\theta}^{t}e^T(\psi-h )De(\psi-h )d\psi d\theta\\
V_4(e)&=\xi \frac{h^2}{\beta}\int_{t-h}^{t}e^T(\psi)De(\psi)d\psi
\end{align}
Using (\ref{error dyn delayed}), the time derivative  of $V_1(e)$ yields,
\begin{equation}\label{lya_dot for time delay}
\dot{V}_1(e)=-e^TQe-2e^{T}PB_1\int_{-h}^{0}\dot{e}(t+\theta)d\theta+2\hat{s}^T\sigma_1
\end{equation}
where, $\hat{s}=B^TPe$. Again using (\ref{new err dyn 1}),
\begin{align}
-2e^{T}PB_1\int_{-h}^{0}\dot{e} & (t+\theta)d\theta = - 2e^{T}PB_1\int_{-h}^{0}[ A_1e(t+\theta ) +  B_1e(t-h+\theta )+B\sigma _1(t+\theta ) ]d\theta,\label{relation 1} 
\end{align}
%Applying (\ref{razu}) to (\ref{lyapunov}) the following relation is achieved,
%\begin{equation}\label{razu 1}
%e^{T}(\nu )Pe(\nu )<re^{T}(t)Pe(t),
%\end{equation}  
For any two non zero vectors $z_1$ and $z_2$, there exists a scalar $\beta>0$ and matrix $D>0$ such that the following inequality holds,
\begin{equation}\label{ineq 2}
\pm 2z_1^{T}z_2\leq \beta z_1^{T}D^{-1}z_1+(1/\beta )z_2^{T}Dz_2.
\end{equation}
Again, using Jensen's inequality the following inequality holds \cite{Ref:31},
\begin{align}
\int_{-h}^{0}e^{T}(\psi)De(\psi)d\psi \geq \frac{1}{h}\int_{-h}^{0}e^{T}(\psi)d\psi D \int_{-h}^{0}e(\psi)d\psi.\label{lemma 1}
\end{align}
Applying (\ref{ineq 2}) and (\ref{lemma 1}) to (\ref{relation 1}) the following inequalities are obtained,
\begin{align}
&- 2e^{T}PB_1 A_1\int_{-h}^{0} e(t+\theta)d\theta \leq \beta e^{T}PB_1A_1D^{-1}A_1^{T}B_1^{T}Pe +\frac{1}{\beta}\int_{-h}^{0}e^{T}(t+\theta )d\theta D \int_{-h}^{0} e(t+\theta ) d\theta\nonumber\\
%& \qquad \qquad \qquad \ \qquad \qquad +\frac{1}{\beta}\int_{-h}^{0}e^{T}(t+\theta )d\theta D \int_{-h}^{0} e(t+\theta ) d\theta \nonumber\\
&\qquad \qquad \qquad \qquad \qquad \qquad \leq \beta e^{T}[PB_1A_1D^{-1}A_1^{T}B_1^{T}P]e +\frac{h}{\beta}\int_{-h}^{0}e^{T}(t+\theta ) D e(t+\theta ) d\theta \label{cond1}\\
%& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \times D e(t+\theta ) d\theta \label{cond1}\\
&- 2e^{T}PB_1B_1\int_{-h}^{0}   e(t-h+\theta)d\theta \leq \beta e^{T}PB_1B_1D^{-1} B_1^{T}B_1^{T}Pe+\frac{1}{\beta}\int_{-h}^{0}e^{T}(t-h+\theta )d\theta D \int_{-h}^{0} e(t-h+\theta )d\theta  \nonumber\\
%&\times B_1^{T}B_1^{T}Pe+\frac{1}{\beta}\int_{-h}^{0}e^{T}(t-h+\theta )d\theta D \int_{-h}^{0} e(t-h+\theta )d\theta \nonumber\\
& \qquad \qquad \qquad \qquad \qquad \qquad \leq \beta e^{T}[PB_1B_1D^{-1}B_1^{T}B_1^{T}P]e+\frac{h}{\beta}\int_{-h}^{0}e^{T}(t-h+\theta ) D e(t-h+\theta )d\theta \label{cond2} \\
%& \\qquad  \times D e(t-h+\theta )d\theta \label{cond2}\\
&- 2e^{T}PB_1\int_{-h}^{0} [ B \sigma _1(t+\theta)]d\theta \leq \beta e^{T}PB_1D^{-1}B_1^{T}Pe+ \frac{1}{\beta}\int_{-h}^{0}(B \sigma _1(t+\theta))^{T}d\theta D \int_{-h}^{0} B\sigma _1(t+\theta)]d\theta   \nonumber\\ 
%& \qquad \quad   \nonumber \\
& \qquad \qquad \qquad \qquad \qquad \qquad\leq \beta e^{T}\left [  PB_1D^{-1}B_1^{T}P \right ]e+\frac{h}{\beta}\int_{-h}^{0}(B \sigma _1(t+\theta))^{T} D \sigma _1(t+\theta)  d\theta \label{cond3}
%& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \times B \sigma _1(t+\theta)  d\theta \label{cond3} 
\end{align}
Since $D>0$, we can write $D=\bar{D}^T\bar{D}$ for some $\bar{D}>0$. Then, assuming the uncertainties to be square integrable within the delay, let there exists a scalar $\Gamma_1>0$ such that the following inequality holds:
\begin{align}\label{cond41}
\frac{h}{\beta }\left \| \int_{-h}^{0}\left [ (B \sigma _1(t+\theta))^{T}\bar{D}^T\bar{D}B \sigma _1(t+\theta) \right ]d\theta  \right \|\leq \Gamma_1. 
\end{align} 


%\begin{equation}\label{dot v1 new}
%\dot{V}_1(e) \leq -e^{T}\left [ Q-E \right]e+ \Gamma_1+ 2\hat{s}^{T}\sigma_1. 
%\end{equation}
Again,
\begin{align}
\dot{V}_2(e)&=\frac{h^2}{\beta}e^TDe-\frac{h}{\beta}\int_{-h}^{0}e^T(t+\theta) D e(t+\theta)d\theta \label{dot v2 new}\\ 
\dot{V}_3(e)&=\frac{h^2}{\beta}e^T_h D e_h-\frac{h}{\beta}\int_{-h}^{0}e^T(t-h+\theta) D e(t-h+\theta)d\theta \label{dot v3 new}\\  
\dot{V}_4(e)&=\xi \frac{h^2}{\beta}(e^TDe-e^T_hDe_h)\label{dot v4 new}
\end{align}
Substituting (\ref{cond1})-(\ref{cond41}) into (\ref{lya_dot for time delay}) and adding it with (\ref{dot v2 new})-(\ref{dot v4 new}) yields,
\begin{equation}
\dot{V}(e)\leq -\bar{e}^T \Psi \bar{e}+\Gamma_1+2\hat{s}^T\sigma_1,
\end{equation}
where, $\bar{e}=\begin{bmatrix}
e^T & e^T_h
\end{bmatrix}^T$. Let controller gains $K_1, K_2$ and delay time $h$ are selected to make $\Psi>0$.
One can find a positive scalar $\iota$ such that $|| \hat{s} || \leq \iota || || \bar{e}||$. Then, $\dot{V}(e)<0$ would be established if $\lambda _{min}(\Psi)|| \bar{e}||^{2} >  \Gamma_1 +2 \iota || \sigma_1 ||   || \bar{e}||$.
%Then, $\dot{V}(e)<0$ would be established if $\lambda _{min}(\Psi)|| \bar{e}||^{2} >  \Gamma_1 +2|| \sigma_1 || || \hat{s} ||=\varpi_0$. 
Thus (\ref{sys new}) would be UUB with the ultimate bound,
\begin{equation}\label{error bound}
%||\bar{e}||=\sqrt{\frac{2 \Gamma_1 +2|| \sigma_1 || || \hat{s} ||}{\lambda _{min}(\Psi)}}=\varpi_0.
%||\bar{e}||=\sqrt{\frac{\varpi_0}{\lambda _{min}(\Psi)}}.
||\bar{e}||=\gamma_1+\sqrt{\frac{\Gamma_1}{\lambda _{min}(\Psi)}+\gamma_1^2}=\varpi_0.
\end{equation}
where, $\gamma_1=\frac{\iota || \sigma_1 ||}{\lambda _{min}(\Psi)}$.
Let $\Xi$ denote the smallest level surface of $V$ containing the ball $B_{\varpi_0}$ with radius $\varpi_0$ centred at $\bar{e}=0$. For initial time $t_0$, if $\bar{e}(t_0)\in \Xi$ then the solution remains in $\Xi$. If  $\bar{e}(t_0)\notin \Xi$ then $V$ decreases as long as $\bar{e}(t)\notin \Xi$. The time required to reach $\varpi_0$ is zero when $\bar{e}(t_0)\in \Xi$, otherwise, while $\bar{e}(t_0)\notin \Xi$ the finite time $t_{r0}$ to reach $\varpi_0$, for some $c_0>0$, is given by \cite{Ref:24},
\begin{equation*}
t_{r0}-t_0\leq (||\bar{e}(t_0) ||-\varpi_0)/c_0 \quad \text{where} \quad \dot{V}(t)\leq-c_0
\end{equation*}
\end{proof}
\begin{remark}:
Since $E$ depends on the controller gains, (\ref{delay value}) provides a selection criterion for the choice of delay $h$ for given controller gains and $Q$. This design issue was previously unaddressed in the literature. Moreover, the approximation error $(\hat{N}-N)$, as in (\ref{approx}), would reduce for small values of $h$. However, $h$ cannot be selected smaller than the sampling interval because, the input output data is only available at sampling intervals. So, the lowest possible selection of $h$ is the sampling interval. Again, choice of sampling interval is governed by the corresponding hardware response time, computation time etc. Hence, the proposed stability approach provides a necessary step for the selection of sampling interval for given controller gains or vice-versa.
\end{remark}
\subsection{Filtered Time-Delayed Control (F-TDC)}\label{2.2}
%\begin{prb}
%It can be noticed from (\ref{approx}) and (\ref{aux}) the state derivatives are necessary to compute the control law and are calculated numerically when state derivatives are not available explicitly. This induces measurement error.    
%\end{prb}
%\begin{assumption}\label{assum 2}
%All the states i.e. $q$ are available, however $\dot{q}, \ddot{q}$ is not available.
%\end{assumption}

%\subsubsection*{Solution}
\par It can be noticed from (\ref{approx}) and (\ref{aux}) that state derivatives are necessary to compute the control law of TDC. However, in many circumstances, only $q$ is available amongst $q, \dot{q}, \ddot{q}$. Under this scenario, a new control strategy F-TDC is proposed, which estimates the state derivatives from the state information of past instances \cite{Ref:30}. Before proposing the control structure of F-TDC, the following two Lemmas are stated which are instrumental for formulation as well as stability analysis of F-TDC.

\begin{lemma}
[\cite{Ref:30}]: For time $t \geq \varsigma$, the $j$-th order time derivative of the $\Lambda$-th degree polynomial $q$ in (\ref{sys new}) can be computed in the following way,
\begin{equation}
\hat{q}^{(j)}(t)=\int_{-\varsigma}^{0}\Omega _{j}(\varsigma ,\psi)q(t+\psi)d\psi\label{lemma 2}
\end{equation}
where, 
$\varsigma>0$ is a prespecified scalar and 
\begin{align}
\Omega _{j}(\varsigma ,\psi) =\frac{(\Lambda+1+j)!}{\varsigma ^{(j+1)}j!(\Lambda-j)!}& \sum_{k=0}^{\Lambda}\frac{(-1)^{k}(\Lambda+1+k)!}{(j+k+1)(\Lambda-k)!(k!)^{2}} \left ( \frac{-\psi}{\varsigma} \right )^{k}.\label{omega}
\end{align}
%\begin{proof}
%A sketch of the proof is provided in Appendix A.
%\end{proof}
\end{lemma}
\begin{lemma}
For any non zero vector $\vartheta(\psi)$, constant matrix $F>0$ the following relation holds,
\begin{align}
&\int_{-h}^{0}\int_{-\varsigma}^{0}\vartheta ^T(\psi) F \vartheta(\psi) d\psi d\theta \geq \frac{1}{h \varsigma}\left \{  \int_{-h}^{0}\int_{-\varsigma}^{0}\vartheta ^T(\psi)d\psi d\theta\right \} F \left \{  \int_{-h}^{0}\int_{-\varsigma}^{0}\vartheta(\psi)d\psi d\theta\right \}\label{lemma 3}
\end{align}
%\begin{proof}
%Since $F>0$, we have
%\begin{equation}\label{1}
%\begin{bmatrix}
%\vartheta^T (\psi) F \vartheta (\psi) & \vartheta ^T(\psi) \\ 
% \vartheta (\psi) & F^{-1}
%\end{bmatrix} \geq 0
%\end{equation}
%Integrating (\ref{1}) from $-\varsigma$ to $0$ gives,
%\begin{equation}\label{2}
%\begin{bmatrix}
%\int_{-\varsigma}^{0}\vartheta^T (\psi)F \vartheta (\psi)d\psi  & \int_{-\varsigma}^{0}\vartheta ^T(\psi) d\psi \\ 
% \int_{-\varsigma}^{0} \vartheta (\psi) d\psi  & \varsigma F^{-1}
%\end{bmatrix} \geq 0 
%\end{equation}
%Again, integrating (\ref{2}) from $-h$ to $0$ gives,
%\begin{equation}\label{3}
%\begin{bmatrix}
%\int_{-h}^{0}\int_{-\varsigma}^{0}\vartheta^T (\psi) F \vartheta (\psi)d\psi d\theta  & \int_{-h}^{0}\int_{-\varsigma}^{0}\vartheta ^T(\psi)d\psi d\theta  \\ 
%\int_{-h}^{0} \int_{-\varsigma}^{0} \vartheta (\psi) d\psi d\theta & h \varsigma F^{-1}
%\end{bmatrix} \geq 0
%\end{equation}
%Applying Schur's complement to (\ref{3}), (\ref{lemma 3}) can be achieved.
%\end{proof}
\end{lemma}
The structure of F-TDC is similar to (\ref{input}), except, the auxiliary control input $u$ and $\hat{N}$ in (\ref{input}) selected in the following way,
\begin{align}
u(t)&=\ddot{q}^d(t)-K_1e_1(t)-K_2\dot{\hat{e}}_1(t)\label{u hat}\\
\hat{N}(t)& \cong N_{h}=u_{h}-\hat{M}_{h}\ddot{\hat{q}}_{h}, \label{h hat new}
\end{align}
where, $\dot{\hat{e}}_1=\dot{\hat{q}}-\dot{q}^d$. $\dot{\hat{q}}$ and $\ddot{\hat{q}}$ are evaluated from (\ref{lemma 2}) and (\ref{omega}).
%\subsubsection*{Stability Analysis for F-TDC}
The stability of the system (\ref{sys new}) employing F-TDC is derived in the sense of  Uniformly Ultimately Bounded (UUB) notion as stated in Theorem 2.

\begin{theorem}
The system (\ref{sys new}) employing the control input (\ref{input}), having the auxiliary input (\ref{u hat}) and (\ref{h hat new}) is UUB if $K_1, K_2, h$ and $\varsigma$ are selected such that the following condition holds:
%\begin{align}
%&\begin{bmatrix}
%Q-\bar{E}-(1+\xi)\frac{h^2}{\beta}D& 0 & P\bar{B}\\ 
%0 & (\xi-1)\frac{h^2}{\beta}D-\bar{F} &0\\
%\bar{B}^TP & 0 & R
%\end{bmatrix} =\Theta>0 \label{delay value 1}
%\end{align}
%
%where, $\bar{E}=\beta P \bar{B}_1( \bar{A}_1D^{-1}\bar{A}_1^T+\bar{B}_1D^{-1}\bar{B}_1^{T}+D^{-1}+\bar{B}D^{-1}\bar{B}^{T} )\bar{B}_1^{T}P$, $\bar{F}=(\frac{h^2}{\beta}D+R) \varsigma \int_{-\varsigma}^{0}A_d^2(\psi)d\psi$, $A_d(\psi)=\Omega _{1}(\varsigma ,\psi)$, $\bar{B}=B\begin{bmatrix}
% K_2 & 0
%\end{bmatrix}$.
\begin{align}
&\begin{bmatrix}
Q-\bar{E}-(1+\xi)\frac{h^2}{\beta}D& P \breve{B} & P\bar{B}\\ 
 \breve{B}^T P & (\xi-1)\frac{h^2}{\beta}D-\bar{F} &0\\
\bar{B}^TP & 0 & L
\end{bmatrix} \nonumber \\
& \qquad \qquad \qquad \qquad \qquad =\Theta>0 \label{delay value 1}
\end{align}

where, $\bar{E}=\beta P {B}_1( {A}_1D^{-1}{A}_1^T+{B}_1D^{-1}{B}_1^{T}+D^{-1}+\bar{B}D^{-1}\bar{B}^{T} ){B}_1^{T}P$, $\bar{F}=(\frac{h^2}{\beta}D+L) \varsigma \int_{-\varsigma}^{0}A_d^2(\psi)d\psi$, $L>0$, $A_d(\psi)=\Omega _{1}(\varsigma ,\psi)$, $\bar{B}=B\begin{bmatrix}
 K_2 & 0
\end{bmatrix}$, $\breve{B}=B\begin{bmatrix}
 0 & K_2
\end{bmatrix}$.
\end{theorem}
\begin{proof}
The proof is provided in Appendix B.
%Let us define the Lyapunov functional as,
%\begin{align}
%V_f(e)&=V(e)+V_{f1}(e)+V_{f2}(e)+V_{f3}(e),\label{ftdc lya}\\
%V_{f1}(e)&=\frac{h \varsigma}{\beta}\int_{-h}^{0}\int_{-\varsigma}^{0}\int_{t-h+\psi}^{t-h}e^T(\eta +\theta)D\times \nonumber\\
%& \qquad \qquad \qquad \qquad \qquad \times A_d^2 (\psi)e(\eta+\theta)d\eta d\psi d\theta\nonumber\\
%V_{f2}(e)&=\frac{h \varsigma}{\beta}\int_{-h}^{0}\int_{-\varsigma}^{0}\int_{t+\theta}^{t}e^T(\eta-h)D \times \nonumber\\
%& \qquad \qquad \qquad \qquad \qquad \times A_d^2(\psi)e(\eta-h)d\eta d\psi d\theta\nonumber\\
%V_{f3}(e)&=\varsigma \int_{-\varsigma}^{0}\int_{t+\psi}^{t}e^T(\eta-h)(\psi)RA_d^2(\psi)e(\eta-h)d\eta d\psi.\nonumber
%\end{align}
%where, $V$ is given in (\ref{lyapunov}) and putting (\ref{u hat}) into (\ref{sys new}) we obtain,
%%\begin{align}\label{error new}
%%\ddot{e}_1 &=-K_1e_{1h}-K_2\dot{e}_1-\begin{bmatrix}
%% K_2  &  0
%%\end{bmatrix} \times \nonumber\\
%%& \quad \times \int_{-\tau}^{0}A_d(\psi)\begin{bmatrix}
%%e_1(t-h+\psi )\\ 
%%\dot{e}_1(t-h+\psi )
%%\end{bmatrix}d\psi+\sigma_2.
%%\end{align}
%%with $\sigma_2=K_2\dot{e}_1+\sigma_1$. Further the error dynamics (\ref{error new}) can be written in sate space form as,
%\begin{equation}\label{error new}
%\dot{e}=\bar{A}_1e+\bar{B}_1e_h-\bar{B}\int_{-\varsigma}^{0}A_d(\psi )e(t-h+\psi )d\psi+B\sigma_2,
%%Ae-\bar{B}\int_{-\varsigma}^{0}A_d(\psi )e(t+\psi )d\psi+B\sigma_2,
%\end{equation}
%where, $\bar{A}_1=\begin{bmatrix}
%0 & I \\
%0 & -K_2
%\end{bmatrix} 
%, 
%\bar{B}_1=\begin{bmatrix}
%0 & 0\\
%-K_1 & 0
%\end{bmatrix}$, $\sigma_2=K_2\dot{e}_1+\sigma_1$. Further the error dynamics (\ref{error new}) can be written as,  
%\begin{equation}\label{error dyn state new 1}
%\dot{e}=Ae-\bar{B}_1 \int_{-h}^{0} \dot{e}(t+\theta)d \theta-\bar{B}\int_{-\varsigma}^{0}A_d(\psi )e(t-h+\psi )d\psi+B\sigma_2.
%\end{equation}
%
%
%
%
%Following the similar procedure for proving Theorem 1 and and applying Lemma 1 and Lemma 2 to the time derivative of $V_{f1}, V_{f3}$ and $V_{f3}$ yields
%
%\begin{align}
%&\dot{V}(e)\leq -e^{T}\left [ Q-\bar{E} -(1+\xi)\frac{h^2}{\beta}D \right]e+ \Gamma_2+\frac{h \varsigma}{\beta} \times \nonumber\\
%&\int_{-h}^{0}\int_{-\varsigma}^{0} e^T(t-h+\theta+\psi)D A_d^2(\psi)e(t-h+\theta+\psi)d\psi d\theta \nonumber\\
%& \qquad + 2\hat{s}^{T}\sigma_2 -2e^TP\bar{B}\int_{-\varsigma}^{0}A_d(\psi)e(t-h+\psi)d\psi,\label{v dot ftdc1}
%\end{align}
%where, $\frac{h}{\beta }\left \| \int_{-h}^{0}\left [ (B \sigma _2(t+\theta))^{T}\bar{D}^T\bar{D}B \sigma _2(t+\theta) \right ]d\theta  \right \|\leq \Gamma_2.$
%\begin{align}
%&\dot{V}_{f1}\leq \frac{h \varsigma}{\beta}\int_{-h}^{0} \int_{-\varsigma}^{0}e^T(t-h+\theta)DA_d^2(\psi)e(t-h+\theta)d\psi d\theta\nonumber\\
%& \qquad \qquad \quad -\frac{h \varsigma}{\beta}\int_{-h}^{0} \int_{-\varsigma}^{0}e^T(t-h+\theta+\psi)DA_d^2(\psi) \times \nonumber\\
%& \qquad \qquad \qquad \qquad \qquad \times e(t-h+\theta+\psi)d\psi d\theta\label{v1 dot ftdc}
%\end{align}
%\begin{align}
%\dot{V} & _{f2}  \leq \frac{h^2 \varsigma}{\beta} e^T(t-h)D \int_{-\varsigma}^{0}A_d^2(\psi)d\psi e(t-h)-\frac{h \varsigma}{\beta} \times \nonumber \\
%& \times \int_{-h}^{0} \int_{-\varsigma}^{0} e^T(t-h+\theta)DA_d^2(\psi)e(t-h+\theta)d\psi d\theta\label{v2 dot ftdc}
%\end{align}
%\begin{align}
%&\dot{V}_{f3}(e) \leq \varsigma e^T(t-h)\int_{-\varsigma}^{0}RA_d^2(\psi)d\psi e(t-h)-\int_{-\varsigma}^{0}A_{d}(\psi) \times \nonumber\\
%& \times e^T(t-h+\psi)d\psi R \int_{-\varsigma}^{0}A_d(\psi)e(t-h+\psi)d\psi \label{v3 dot ftdc}
%\end{align}
%%Now, $2\hat{s}^T K_2\dot{e}_1=2e^TP \breve{B}e$. 
%Combining (\ref{v dot ftdc1})-(\ref{v3 dot ftdc})
%\begin{align}
%\dot{V}_f(e) & \leq -e_f^T \Theta e_f + \Gamma_2 +2\hat{s}^T\sigma_2,
%\end{align}
%where $e_f=\begin{bmatrix}
%\bar{e}^T & \int_{-\varsigma}^{0}A_d(\psi)e^T(t-h+\psi)d\psi
%\end{bmatrix}^T$.
%%\begin{corollary}
%%Since $A$ entirely depends on controller gains, the solution of the Lyapunov equation $P$ is also governed by the controller gains and choice of $Q$. Hence, the sampling time $\tau$ and controller gains are to be selected in such a manner so that for a given $R>0$ and $Q>0$, $\exists$ $\Theta>0$ so that the following inequality holds:
%%\begin{equation}\label{sample 2}
%%\begin{bmatrix}
%%-Q+\tau \int_{-\tau}^{0}A_{d}^{T}(\psi)RA_d(\psi)d\psi & \quad-P\bar{B}\\ 
%%-\bar{B}^TP & \quad-R
%%\end{bmatrix}=-\Theta<0,\quad  \quad 
%%\end{equation}
%%\end{corollary}
%%%\begin{remark}
%%%The relation (\ref{sample 2}) provides a selection criterion for $\tau$ for given controller gains and vice-versa. This solves an important design issue for TDC which was previously not addressed in the literature. 
%%%\end{remark}
%%Thus, 
%%\begin{equation}
%%\dot{V}_f(e) \leq -\lambda_{min}(\Theta)||\bar{e}||^2+2||\sigma_1||||\hat{s}||\label{v dot ftdc}
%%\end{equation}
%%Here, $\lambda_{min}(\cdot)$ and $||\cdot||$ represent the minimum eigen value and Euclidean norm of the argument respectively.
%Let choice of $K_1, K_2$, $h, \varsigma$ make $\Theta>0$. So, $\dot{V}_f(e)<0$ would be achieved if $\lambda_{min}(\Theta)||e_f||^2> \Gamma_2+ 2||\sigma_2||||\hat{s}||=\varpi$. Thus, the system would be UUB with the following ultimate bound
%\begin{equation}\label{error bound f-tdc}
%%||e_f||=\sqrt{\frac{2\Gamma+ 2||\sigma_2||||\hat{s}||}{\lambda_{min}(\Theta)}}=\varpi
%||e_f||=\sqrt{\frac{\varpi}{\lambda_{min}(\Theta)}}
%\end{equation}
%The time $t_{r}$ to reach $\varpi$ is given by 
%%Let, $R_1$ denote the smallest level surface of $V_f$ containing the ball $B_{\varpi}$ with radius $\varpi$ centred at $\bar{e}=0$. For initial time $t_0$, if $\bar{e}(t_0)\in R$ then the solution remains in $R_1$. If  $\bar{e}(t_0)\notin R_1$ then $V_f$ decreases as long as $\bar{e}(t)\notin R_1$. The time required to reach $\varpi$ is zero when $\bar{e}(t_0)\in R_1$, otherwise, while $\bar{e}(t_0)\notin R_1$ the finite time $t_{r}$ to reach $\varpi$ is given by \cite{Ref:24},
%\begin{equation*}
%t_{r}-t_0\leq (||e_f(t_0)||-\varpi)/c \quad \text{where} \quad \dot{V}_f(e)\leq-c.
%\end{equation*}
\end{proof}

\subsection{Adaptive-Robust Control: Related Work}
%\subsubsection*{Problem Formulation}
It can be observed that TDE degrades tracking performance of both the TDC and F-TDC in the face of uncertainties. The control methods that attempt to counter uncertainties, as reported in \cite{Ref:2}, \cite{Ref:10}, \cite{Ref:12}, \cite{Ref:13}-\cite{Ref:19}, requires predefined bound on the uncertainties which is not always possible in practical circumstances. To circumvent this situation Adaptive Sliding Mode Control (ASMC) was proposed in \cite{Ref:21}-\cite{Ref:22}. The control input of ASMC is given by,
\begin{equation}\label{input asmc}
\tau=\Sigma_n^{-1}(-\kappa_n+\Delta u_s),
\end{equation}
where, $\Sigma_n$ and $\kappa_n$ is the nominal values of $\Sigma$ and $\kappa$, and $\Delta u_s$ is the switching control input. For a choice of sliding surface $\bar{s}$, $\Sigma$ and $\kappa$ is defined as follows:
\begin{equation}\label{s asmc}
\dot{\bar{s}}=\Sigma+\kappa\Delta u_s,
\end{equation}
%\begin{equation}\label{eq u}
%\hat{u}(t)=\ddot{q}^d(t)-\Omega \dot{e}_1-\hat{f}.
%\end{equation}
The switching control $\Delta u_s$ is calculated as
\begin{equation}\label{delta u asmc}
 \Delta u_s=-\hat{c}\frac{\bar{s}}{||\bar{s}||}
%  \begin{cases}
%    -\hat{c}(e,t)\frac{\bar{s}}{|| \bar{s} ||}       & \quad \text{if } ||\bar{s} ||\geq \epsilon,\\
%    -\hat{c}(e,t)\frac{\bar{s}}{\epsilon}        & \quad \text{if } || \bar{s} || < \epsilon,\\
%  \end{cases}
\end{equation}
\begin{equation}\label{ASMC}
 \dot{\hat{c}} =
  \begin{cases}
   \bar{c}|| \bar{s}|| sgn(|| \bar{s}||-\rho)      & \quad \hat{c}>\gamma,\\
    \qquad \qquad \gamma        & \quad \hat{c}\leq\gamma,\\
  \end{cases}
\end{equation}
%$\hat{f}$ is the nominal value of $f=-\hat{M}^{-1}((M-\hat{M})\ddot{q}+H), \hat{b}=\hat{M}^{-1}$, $\Omega$ is a positive definite matrix,
where, $\hat{c}$ is the switching gain, $\bar{c}>0$ is a scalar adaptive gain, $\rho>0$ is a threshold value, $\gamma >0$ is small scalar to always keep $\hat{c}$ positive. Evaluation of $\rho$ can be done in two ways \cite{Ref:21}:
\begin{gather}
\rho=\varrho \quad \text{or},\label{ASMC 1}\\
 \rho(t)=4\hat{c}(t)t_s,\label{ASMC 2}
\end{gather}
where, $\varrho>0$ is a scalar, $t_s$ is the sampling interval. The choice (\ref{ASMC 1}) requires predefined bound of uncertainties. It can be noted from (\ref{ASMC}) that even if $|| \bar{s}||$ decreases (resp. increases), unless it falls below (resp. goes above) $\rho$  switching gain does not decrease (resp. increase). This causes overestimation (resp. underestimation) of switching gain and controller accuracy is compromised. Again, improper and low choice of $\rho$ may lead to very high switching gain and consequent chattering. On the other hand, method (\ref{ASMC 2}) assumes that the nominal value of the uncertainties are always greater than the perturbations. This assumption may not hold due to the effect of unmodelled dynamics and thus,  necessitates rigorous nominal modelling of the uncertainties in $N$ to design the control law. Either of the two situations, i.e. bound estimation or uncertainty modelling, is not always feasible in practical circumstances and consequently compromises the adaptive nature of the controller.
\subsection{Time-Delayed Adaptive Robust Control}\label{sec: 2.3}
%\begin{prb}
%Considering the limitations of the controllers that aim at negotiating the uncertainties, as discussed earlier, it is required to devise a control law which would neither require the nominal model or any predefined bound of the uncertainties. Simultaneously, the controller also needs to aim at elimination the overestimation-underestimation problem of the switching gain.
%\end{prb}
Considering the limitations of the existing controllers that aim at negotiating the uncertainties, as discussed earlier, a novel adaptive-robust control law, named Time-Delayed Adaptive Robust Control (TARC) is proposed in this endeavour, which neither requires the nominal model nor any predefined bound of the uncertainties as well as eliminates the overestimation-underestimation problem of switching gain. The structure of the control input of TARC is similar to (\ref{input}) and $\hat{N}$ is also evaluated according to (\ref{h hat new}). However, the auxiliary control input $u$ is selected as below,
\begin{equation}\label{tarc input}
u = \hat{u}+\Delta u.
\end{equation}
$\hat{u}$ is the nominal control input and selected as similar to (\ref{u hat}). $\Delta u$ is the switching control law which is responsible for negotiating the TDE and it is defined as below,
\begin{align}
\Delta u=
  \begin{cases}
    -\alpha\hat{c}(e,t)\frac{s}{\parallel s \parallel}       & \quad \text{if } \parallel s \parallel\geq \epsilon,\\
    -\alpha\hat{c}(e,t)\frac{s}{\epsilon}        & \quad \text{if } \parallel s \parallel< \epsilon,\\
  \end{cases}\label{delta u}
\end{align}
where, $s=B^TP\begin{bmatrix}
e_1 & \dot{\hat{e}}_1
\end{bmatrix}^T$ and $\epsilon>0$ is a small scalar.
%\begin{equation}\label{s}
%s=B^TP\begin{bmatrix}
%e_1 & \dot{\hat{e}}_1
%\end{bmatrix}^T.
%\end{equation}
The following novel adaptive control law for evaluation of $\hat{c}$ is proposed:
\begin{equation}\label{ATRC}
 \dot{\hat{c}} =
  \begin{cases}
    \quad || s ||      & \quad \hat{c}>\gamma,f(e)>0\\
    -|| s ||              & \quad \hat{c}>\gamma,f(e) \leq 0 \\
    \quad  \gamma        & \quad \hat{c}\leq\gamma,\\
  \end{cases}
\end{equation}
%The present choice of $f(e)$  attempts to bring the error trajectories closer to $|| s || =0$ by increasing (resp. decreasing) the switching gain whenever $|| s ||$ moves away from (resp. closer to) $|| s || =0$.
where, $\alpha>0$ is a scalar adaptive gain  and $\epsilon>0$ represents a small scalar, $f(e)$ is a suitable function of error defined by the designer. Here, it is selected as $ f(e)=|| s(t)||-|| s_h  ||$. According to the adaptive law (\ref{ATRC}) and present choice of $f(e)$, $\hat{c}$ increases (resp. decreases) whenever error trajectories move away from (resp. close to ) $|| s ||=0$  The advantages of the proposed TARC can be summarized as follows:
\begin{itemize}
  \item TARC reduces complex system modelling effort as only the knowledge of $\hat{M}$ suffices the controller design since $N$ along with the uncertainties is approximated using the time-delayed logic as in (\ref{h hat new}). This in turn reduces the tedious modelling effort of complex nonlinear systems.
   %To illustrate the fact with an example, friction, slip, skid etc. for wheeled mobile robot and these terms can be treated as uncertainties.  
  \item Evaluation of switching gain does not require either of the nominal model or predefined bound of the uncertainties and also removes the overestimation-underestimation problem.
  \item State derivatives are not required to compute the control law explicitly, as they are evaluated from the past state information using (\ref{lemma 2}) and (\ref{omega}).
\end{itemize}

%\subsubsection*{Stability Analysis for TARC}
The stability of the system (\ref{sys new}) employing TARC is analysed in the sense of UUB as stated in Theorem 3.
\begin{assumption}\label{assum 3}
Let, $|| \sigma_1|| \leq c$. Here, $c$ is an unknown scalar quantity. Knowledge of $c$, however, is only required for stability analysis but not to compute control law. 
\end{assumption}
\begin{theorem}
The system (\ref{sys new}) employing (\ref{input}), (\ref{tarc input}) and having the adaptive law (\ref{ATRC}) is UUB, provided the selection of $K_1, K_2, h$ and $\varsigma$ holds condition (\ref{delay value 1}).
% for various cases as stated below:
%\par Case (i): $\left \| s \right \|>\left \| s_{\tau} \right \|,\quad \hat{c}>\gamma,\quad \left \| s \right \|\geq \epsilon$
%\par Case (ii): $\left \| s \right \|<\left \| s_{\tau} \right \|,\quad \hat{c}>\gamma,\quad \left \| s \right \|\geq \epsilon$
%\par Case (iii): $\hat{c}\leq \gamma,\quad \left \| s \right \|\geq \epsilon$
%\par Case (iv): $\left \| s \right \|>\left \| s_{\tau} \right \|,\quad \hat{c}>\gamma,\quad \left \| s \right \|< \epsilon $
%\par Case (v):  $\left \| s \right \|<\left \| s_{\tau} \right \|,\quad \hat{c}>\gamma,\quad \left \| s \right \|<\epsilon$
%\par Case (vi):  $\hat{c}\leq \gamma,\quad \left \| s \right \|< \epsilon$
\end{theorem}
\begin{proof}:
Let us define the Lyapunov functional as,
\begin{align}
V_r(e)&=V_f(e)+(\hat{c}-c)^2,\label{tarc lya}
\end{align}
where, $V_f(e)$ is defined in (\ref{ftdc lya}). Again, putting (\ref{tarc input}) in (\ref{sys new}) the  the error dynamics becomes,
\begin{equation}\label{error dyn state new}
%\dot{e}=Ae-\bar{B}\int_{-\varsigma}^{0}A_d(\psi )e(t+\psi )d\psi+B(\Delta u_a+\sigma_1).
\dot{e}={A}_1e+{B}_1e_h-\bar{B}\int_{-\varsigma}^{0}A_d(\psi )e(t-h+\psi )d\psi+B\sigma,
\end{equation}
where, $\sigma=\Delta u_h+\sigma_1$. Also following similar steps while proving Theorem 2 (provided in Appendix B) we have,
\begin{equation}\label{vf dot}
%\dot{e}=Ae-\bar{B}\int_{-\varsigma}^{0}A_d(\psi )e(t+\psi )d\psi+B(\Delta u_a+\sigma_2).
\dot{V}_f(e)\leq -e_f^T \Theta e_f + \Gamma +2\hat{s}^T(\Delta u+\sigma_1)+2\hat{s}^T \Upsilon,
\end{equation}
where, $ e_f$ is defined in Appendix B, $\Gamma \geq \frac{h}{\beta }\left \| \int_{-h}^{0}\left [ (B \sigma(t+\theta))^{T}\bar{D}^T\bar{D}B \sigma(t+\theta) \right ]d\theta  \right \|$ is a positive scalar, $\Upsilon=\Delta u_h-\Delta u$. Let us define the following,
\begin{equation}
\hat{s}=s+\Delta s \quad \text{where} \quad \Delta s=B^TP\begin{bmatrix}
0 & (\dot{e}_1-\dot{\hat{e}}_1)^T
\end{bmatrix}^T.
\end{equation}
Evaluating the structure of $s$ and $\Delta s$ one can find two positive scalars $\iota_2, \iota_3$ such that $||{s}||\leq \iota_2 ||e_f||, ||\Delta {s}||\leq \iota_3 ||e_f|| $.
Using (\ref{tarc lya}) the stability analysis for (\ref{sys new}) employing TARC is carried out for the following various cases.
\par \textbf{Case (i):} $f(e)>0,\quad \hat{c}>\gamma,\quad || s ||\geq \epsilon$. \\
Utilizing (\ref{delta u}), (\ref{ATRC}) and (\ref{vf dot}) we have,
\begin{align}
\dot{V}_r(e)&\leq -e_f^T \Theta e_f+\Gamma+2\hat{s}^T(-\alpha \hat{c} \frac{s}{|| s ||}+\sigma_1)+2\hat{s}^T \Upsilon+2(\hat{c}-c)||s||\nonumber\\
& =-e_f^T \Theta e_f+\Gamma-2\alpha\hat{c}\frac{s^Ts}{||s||}-2\alpha\hat{c}\frac{\Delta s^Ts}{||s||}+2\hat{s}^T \sigma_1 +2\hat{s}^T \Upsilon +2(\hat{c}-c)||s||\nonumber\\
& \leq -\lambda_{min}(\Theta)||e_f||^2-2(\alpha-1)\hat{c}||s||+\Gamma+2||s||||\Upsilon|| +2(\alpha\hat{c} +c+||\Upsilon||)||\Delta s||\label{case 1}
\end{align}
So, for $\alpha>1$, $\dot{V}_r(e)<0$ would be established if $\lambda_{min}(\Theta)||e_f||^2>\Gamma+2||s||||\Upsilon||+2(\alpha\hat{c} +c+||\Upsilon||)||\Delta s||$. Thus, using the relation $||{s}||\leq \iota_2 ||e_f||, ||\Delta {s}||\leq \iota_3 ||e_f|| $, the system would be UUB with the following ultimate bound
\begin{equation}\label{error bound 1}
%||e_f||=\sqrt{\frac{\Gamma+2||\hat{s}||||\Upsilon||+2(\alpha\hat{c} +c)||\Delta s||}{\lambda_{min}(\Theta)}}=\varpi_1
%||e_f||=\sqrt{\frac{\varpi_1}{\lambda_{min}(\Theta)}}.
||e_f||=\mu_1+\sqrt{\frac{\Gamma}{\lambda _{min}(\Theta)}+\mu_1^2}=\varpi_1.
\end{equation}
where, $\mu_1=\frac{\iota_2 || \Upsilon ||+\iota_3(\alpha\hat{c} +c+||\Upsilon||)}{\lambda _{min}(\Theta)}$.
%The finite time required to reach $\varpi_1$ is given by,
%\begin{equation*}
%t_{r1}-t_0\leq (||e_f(t_0)||-\varpi_1)/c_1 \quad \text{where} \quad \dot{V}_r(e)\leq-c_1.
%\end{equation*}
\par \textbf{Case (ii):} $f(e) \leq 0,\quad \hat{c}>\gamma,\quad \left \| s \right \|\geq \epsilon$. \\
Again, utilizing (\ref{ATRC}) for Case (ii),
\begin{align}
\dot{V}_r(e)&\leq -e_f^T \Theta e_f+\Gamma+2\hat{s}^T(-\alpha \hat{c} \frac{s}{|| s ||}+\sigma_1)+2\hat{s}^T \Upsilon -2(\hat{c}-c)||s||\nonumber\\
%& =-e_f^T \Theta e_f-2\alpha\hat{c}\frac{s^Ts}{||s||}-2\alpha\hat{c}\frac{\Delta s^Ts}{||s||}+2\hat{s}^T \sigma_1\nonumber\\
%& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad -2(\hat{c}-c)||s||\nonumber\\
& \leq -\lambda_{min}(\Theta)||e_f||^2+( 4c-2(\alpha+1)\hat{c}+2||\Upsilon|| )||s|| +\Gamma +2(\alpha\hat{c} +c+||\Upsilon||)||\Delta s||.\label{case 2}
\end{align}
$\dot{V}_r(e)<0$ would be achieved if $\lambda_{min}(\Theta)||e_f||^2> \Gamma +( 4c-2(\alpha+1)\hat{c}+2||\Upsilon|| )||s||+2(\alpha\hat{c} +c+||\Upsilon||)||\Delta s||$ and system would be UUB having following ultimate bound,
\begin{equation}\label{error bound 2}
%||e_f||=\sqrt{\frac{\Gamma +( 4c-2(\alpha+1)\hat{c} )||s||+2||\hat{s}||||\Upsilon||+2(\alpha\hat{c} +c)||\Delta s||}{\lambda_{min}(\Theta)}}=\varpi_2
%||e_f||=\sqrt{\frac{\varpi_2}{\lambda_{min}(\Theta)}}
||e_f||=\mu_2+\sqrt{\frac{\Gamma}{\lambda _{min}(\Theta)}+\mu_2^2}=\varpi_2.
\end{equation}
where, $\mu_2=\frac{\iota_2 ( 2c-(\alpha+1)\hat{c}+||\Upsilon|| )+\iota_3(\alpha\hat{c} +c+||\Upsilon||)}{\lambda _{min}(\Theta)}$.
%The finite time to reach $\varpi_2$ is obtained to be $t_{r2}\leq t_0+ (||e_f(t_0)||-\varpi_2)/c_1$.

\par \textbf{Case (iii):} $\hat{c}\leq \gamma,\quad \left \| s \right \|\geq \epsilon$.\\
Since $\hat{c}\leq \gamma$ we have $(\hat{c}-c )\gamma \leq \gamma^2-c\gamma \leq \gamma^2$. Using the adaptive law (\ref{ATRC}), for Case (iii) we have,
\begin{align}
\dot{V}_r(e)& \leq -e_f^T \Theta e_f+\Gamma+2\hat{s}^T(-\alpha \hat{c} \frac{s}{|| s ||}+\sigma_1)+2(\hat{c}-c)\gamma +2\hat{s}^T \Upsilon\nonumber\\
%& \leq -e_f^T \Theta e_f-2\alpha\hat{c}\frac{s^Ts}{||s||}-2\alpha\hat{c}\frac{\Delta s^Ts}{||s||}+2\hat{s}^T \sigma_1+2\gamma^2\nonumber\\
%& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad -2|\hat{c}-c|\gamma\nonumber\\
& \leq -\lambda_{min}(\Theta)||e_f||^2+\Gamma+2( c-\alpha\hat{c}+||\Upsilon||)||s|| +2(\alpha\hat{c} +c+||\Upsilon||)||\Delta s||+2\gamma^2.\label{case 3}
\end{align}
Similarly, as argued earlier the system would be UUB with the following ultimate bound,
\begin{equation}\label{error bound 3}
%||e_f||=\sqrt{\frac{\varpi_3}{\lambda_{min}(\Theta)}}
||e_f||=\mu_3+\sqrt{\frac{(\Gamma+2\gamma^2)}{\lambda _{min}(\Theta)}+\mu_3^2}=\varpi_3.
\end{equation}
where, $\mu_3=\frac{\iota_2 ( c-\alpha\hat{c}+||\Upsilon|| )+\iota_3(\alpha\hat{c} +c+||\Upsilon||)}{\lambda _{min}(\Theta)}$.

%where, $\varpi_3=\Gamma+2( c-\alpha\hat{c}+||\Upsilon||)||s||+2(\alpha\hat{c} +c+||\Upsilon||)||\Delta s||+2\gamma^2$. 
%The finite time to reach $\varpi_3$ is obtained to be $t_{r3}\leq t_0+(||e_f(t_0)||-\varpi_3)/c_1$.

%\par \textbf{Case (iv):} $f(e)>0,\quad \hat{c}>\gamma,\quad \left \| s \right \|< \epsilon $\\
%Utilizing (\ref{ATRC}) time derivative of (\ref{tarc lya}) yields,
%\begin{align}
%\dot{V}_r(e)&\leq -e_f^T \Theta e_f+\Gamma+2\hat{s}^T(-\alpha \hat{c} \frac{s}{\epsilon}+\sigma_1)+2(\hat{c}-c)||s||+2\hat{s}^T \Upsilon\nonumber\\
%& =-e_f^T \Theta e_f+\Gamma-2\alpha\hat{c}\frac{s^Ts}{\epsilon }-2\alpha\hat{c}\frac{\Delta s^Ts}{\epsilon }+2\hat{s}^T \sigma_1 +2(\hat{c}-c)||s||+2\hat{s}^T \Upsilon\nonumber\\
%& \leq -\lambda_{min}(\Theta)||e_f||^2-2\alpha\hat{c}\frac{||s||^2}{\epsilon }+2\alpha\hat{c}||\Delta s||\frac{||s||}{\epsilon } +2(\hat{c}+||\Upsilon||)||s|| +2(c+||\Upsilon||)||\Delta s||+\Gamma.\label{case 41}
%\end{align}
%Since $||s||<\epsilon$, from (\ref{case 41}) we get,
%\begin{align}
%\dot{V}_r(e) &\leq -\lambda_{min}(\Theta)||e_f||^2-2\alpha\hat{c}\frac{||s||^2}{\epsilon }+2(\hat{c}+||\Upsilon||)||s|| +2(\alpha\hat{c}+c+||\Upsilon||)||\Delta s||+\Gamma.\label{case 4}
%\end{align} 
%The combination of second and third term of (\ref{case 4}) takes the maximum value $\mu_{41}=\frac{\alpha\hat{c} }{2\epsilon}\left (\frac{\epsilon \left (\hat{c}+||\Upsilon|| \right ) }{\alpha \hat{c} } \right )^{2}$ for $||s||=\frac{\epsilon(\hat{c}+||\Upsilon||) }{2\alpha \hat{c} } $. So, the system would be UUB with the ultimate bound determined to be,
%\begin{equation}\label{error bound 4}
%||e_f||=\mu_4+\sqrt{\frac{(\Gamma+\mu_{41})}{\lambda _{min}(\Theta)}+\mu_4^2}=\varpi_4.
%\end{equation}
%where, $\mu_4=\frac{\iota_3(\alpha\hat{c} +c+||\Upsilon||)}{\lambda _{min}(\Theta)}$.
%
%%where, $\varpi_4=2\epsilon \Gamma+\alpha\hat{c}\left (|| \Delta s|| +\frac{\epsilon \left (\hat{c}+||\Upsilon|| \right ) }{\alpha \hat{c} } \right )^{2}+4\epsilon(c+||\Upsilon||) || \Delta s||$. 
%%The finite time to reach $\varpi_4$ is obtained to be, $t_{r4}\leq t_0+ (||e_f(t_0)||-\varpi_4)/c_1.$
%\par \textbf{Case (v):}  $f(e) \leq 0,\quad \hat{c}>\gamma,\quad \left \| s \right \|<\epsilon$.\\
%For this case the following is obtained:
%\begin{align}
%\dot{V}_r(e)&\leq -e_f^T \Theta e_f+\Gamma+2\hat{s}^T(-\alpha \hat{c} \frac{s}{\epsilon}+\sigma_1)-2(\hat{c}-c)||s||+2\hat{s}^T \Upsilon\nonumber\\
%%& =-\bar{e}^T\Theta \bar{e}-2\alpha\hat{c}\frac{s^Ts}{\epsilon }-2\alpha\hat{c}\frac{\Delta s^Ts}{\epsilon }+2\hat{s}^T \sigma_1\nonumber\\
%%&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad-2(\hat{c}-c)||s||\nonumber\\
%& \leq -\lambda_{min}(\Theta)||e_f||^2+\Gamma-2\alpha\hat{c}\frac{||s||^2}{\epsilon }+2(2c-\hat{c})||s|| +2||\Upsilon||||s||+2(\alpha\hat{c}+c+||\Upsilon||)||\Delta s||.\label{case 5}
%\end{align}
%The combination of third, fourth and fifth term of (\ref{case 5}) takes the maximum value $\mu_{51}=\frac{\left ( \left ( 2c-\hat{c}+||\Upsilon|| \right )\epsilon  \right )^{2}}{2\epsilon \alpha \hat{c}}$ for $||s||=\frac{ \left ( 2c-\hat{c}+||\Upsilon|| \right )\epsilon  }{2\alpha \hat{c}}$. Therefore, for Case (v) the system would be UUB with the ultimate bound obtained as,
%\begin{equation}\label{error bound 5}
%%||e_f||=\sqrt{\frac{\varpi_5}{2\epsilon \alpha \hat{c} \lambda_{min}(\Theta)}},
%||e_f||=\mu_4+\sqrt{\frac{(\Gamma+\mu_{51})}{\lambda _{min}(\Theta)}+\mu_4^2}=\varpi_5.
%\end{equation}
%%where, $\mu_5=\frac{\iota_3(\alpha\hat{c} +c+||\Upsilon||)}{\lambda _{min}(\Theta)}$.
%%\end{equation}
%%where, $\varpi_5=2\epsilon \alpha \hat{c} \Gamma+\left ( \alpha\hat{c}|| \Delta s||+\left ( 2c-\hat{c}+||\Upsilon|| \right )\epsilon  \right )^{2}+4\epsilon \alpha \hat{c}(c+||\Upsilon||)|| \Delta s ||$.
%% The finite time to reach $\varpi_5$ is obtained to be, $t_{r5} \leq t_0+(||e_f(t_0)||-\varpi_5)/c_1.$
%\par \textbf{Case (vi):}  $\hat{c}\leq \gamma,\quad \left \| s \right \|< \epsilon$.\\
%Again, time derivative of (\ref{tarc lya}) for Case (vi) is evaluated as,
%\begin{align}
%\dot{V}_r(e)& \leq -e_f^T \Theta e_f+\Gamma+2\hat{s}^T(-\alpha \hat{c} \frac{s}{|| s ||}+\sigma_1)+2(\hat{c}-c)\gamma +2\hat{s}^T \Upsilon\nonumber\\
%%& \leq-\bar{e}^T\Theta \bar{e}-2\alpha\hat{c}\frac{s^Ts}{\epsilon }-2\alpha\hat{c}\frac{\Delta s^Ts}{\epsilon }+2\hat{s}^T \sigma_1+2\gamma^2 \nonumber\\
%& \leq -\lambda_{min}(\Theta)||e_f||^2+\Gamma-2\alpha\hat{c}\frac{||s||^2}{\epsilon }+2(c+||\Upsilon||)||s|| +2(\alpha\hat{c}+c+||\Upsilon||)||\Delta s||+2\gamma^2.\label{case 6}
%\end{align}
%The combination of third and fourth term of (\ref{case 6}) takes the maximum value $\mu_{61}=\frac{\left(\epsilon \left (c+||\Upsilon|| \right ) \right )^{2}}{2\epsilon \alpha \hat{c}}$ for $||s||=\frac{\epsilon (c+||\Upsilon||) }{2 \alpha \hat{c}}$. So, the system would be UUB with the ultimate bound as,
%\begin{equation}\label{error bound 6}
%%||e_f||=\sqrt{\frac{\varpi_6}{2\epsilon \alpha \hat{c} \lambda_{min}(\Theta)}}
%||e_f||=\mu_4+\sqrt{\frac{(\Gamma+2\gamma^2+\mu_{61})}{\lambda _{min}(\Theta)}+\mu_4^2}=\varpi_6.
%\end{equation}
%%\end{equation}
%%where, $\varpi_6=2\epsilon \alpha \hat{c} \Gamma+\left(\alpha\hat{c}|| \Delta s ||+\epsilon \left (c+||\Upsilon|| \right ) \right )^{2}+4\epsilon \alpha\hat{c}((c+||\Upsilon||)|| \Delta s ||+\gamma^2)$.\\
%The finite time $t_{ri}$ to reach each error bound $\varpi_i$ is obtained to be \cite{Ref:24},
%\begin{equation}
%t_{ri} \leq t_0+ (||e_f(t_0)||-\varpi_i)/c_1, \quad \forall i=1,\cdots,6,
%\end{equation} 
%where, $\dot{V}_r(t)\leq-c_1, c_1>0$.
\end{proof}
\begin{remark}
The performance of TARC can be characterized by the various error bounds under various conditions. It can be noticed that low value of $h$ and high value of $\alpha$ would result in better accuracy. However, too large $\alpha$ may result in high control input. Also, one may choose different values of $\alpha$ for $|| s || > || s_h||$ and $|| s || \leq || s_h||$. Moreover, it is to be noticed that the stability notion of TARC is invariant to the choice of $f(e)$ and thus provides the designer the flexibility to select a suitable $f(e)$ according to the application requirement.   
\end{remark}
%Since, the adaptive sliding mode control (ASMC), reported in \cite{Ref:21}-\cite{Ref:22}, also provides robustness against the uncertain system without having any knowledge of the uncertainty bounds, it would be prudent to compare the performance of the ASMC with the proposed controllers. The adaptive law of ASMC can be computed either using (\ref{ASMC 1}) or (\ref{ASMC 2}). Henceforth, the ASMC law that follows (\ref{ASMC 1}) and (\ref{ASMC 2}) would be termed as ASMC 1 and ASMC 2 respectively for the remainder of the paper. However, the switching input in the form (\ref{delta u asmc}) induces chattering. Hence, it is modified as below{
%\begin{equation}\label{delta u asmc new}
% \Delta u_s=
%  \begin{cases}
%    -\hat{c}\frac{\bar{s}}{|| \bar{s} ||}       & \quad \text{if } ||\bar{s} ||\geq \epsilon,\\
%    -\hat{c}\frac{\bar{s}}{\epsilon}        & \quad \text{if } || \bar{s} || < \epsilon,\\
%  \end{cases}
%\end{equation}
 %The control structure of the various controllers, which would further be studied in Section III, are provided in Table \ref{table 3}. 

%\begin{table}[h]
%% increase table row spacing, adjust to taste
%\renewcommand{\arraystretch}{1.3}
%% if using array.sty, it might be a good idea to tweak the value of
%% \extrarowheight as needed to properly center the text within the cells
%\caption{Structure of Various Controllers}
%\label{table 3}
%\centering
%
%\begin{tabular}{|c||c||}
%\hline
%Controller & Control structure \\
%\hline
%TDC (\cite{Ref:6}), (\cite{Ref:25})-(\cite{Ref:26}) & (\ref{input})-(\ref{aux}) \\
%\hline
%F-TDC (proposed)& (\ref{input}), (\ref{u hat}), (\ref{h hat new})\\
%\hline
%ASMC 1 (\cite{Ref:21}) & (\ref{input asmc}), (\ref{s asmc}), (\ref{ASMC}), (\ref{ASMC 1}), (\ref{delta u asmc new}) \\
%\hline
%ASMC 2 (\cite{Ref:21}-\cite{Ref:22}) & (\ref{input asmc}), (\ref{s asmc}), (\ref{ASMC}), (\ref{ASMC 2}), (\ref{delta u asmc new}) \\
%\hline
%TARC (proposed)& (\ref{input}), (\ref{u hat}), (\ref{h hat new}), (\ref{tarc input}),  (\ref{delta u}), (\ref{ATRC})\\
%\hline
%\end{tabular}
%\end{table}

%\begin{figure*}[t]
%\begin{center}
%\includegraphics[width=6.5 in,height=2.5 in]{b_1.jpg}
%\caption{Control architecture for TARC while employing on PIONEER-3.}\label{fig:block}
%\end{center}
%\end{figure*}




%\section{Application: Nonholonomic WMR}
%In this section, we have focused on Nonholonomic WMR which provides a unique platform to test the proposed control law since under practical circumstances, a WMR is always subjected to uncertainties like friction, slip, skid etc. These terms are difficult to model and in many cases they are not considered while modelling. The dynamic equation of a WMR after solving the Lagrange multiplier as in \cite{Ref:12} can be written as follows,
%\begin{equation}
%\bar{M}(q)\ddot{q}+\bar{V}(q,\dot{q})=Gu,
%\end{equation}
%where,
%\begin{align*}
%\bar{M}=&\begin{bmatrix}
%m & 0 & Ksin\phi & k_1 & k_2 \\ 
%0 & m & -Kcos\phi & k_3 & k_4\\ 
%Ksin\phi & -Kcos\phi & \bar{I} & -k_5 & k_5 \\ 
%k_1 & k_3 & -k_5 & I_w & 0 \\ 
%k_2 & k_4 & k_5 & 0 & I_w
%\end{bmatrix}
%\bar{V}=\begin{bmatrix}
%md\dot{\phi}^2cos\phi+mr^2sin\phi(\dot{\theta }_{r}^{2}-\dot{\theta }_{l}^{2})/2b\\ 
%md\dot{\phi}^2cos\phi-mr^2sin\phi(\dot{\theta }_{r}^{2}-\dot{\theta }_{l}^{2})/2b\\ 
%Kr^2(\dot{\theta }_{r}^{2}-\dot{\theta }_{l}^{2})/2b\\ 
%-Kr\dot{\phi}^2/2\\ 
%-Kr\dot{\phi}^2/2\\ 
%\end{bmatrix},
%G=&\begin{bmatrix}
%0 &0 \\ 
%0 &0 \\ 
%0 & 0\\ 
%1 & 0\\ 
%0 & 1
%\end{bmatrix},
%,u=\begin{bmatrix}
%u_r\\ 
%u_l
%\end{bmatrix}\\
%k_1=&sin\phi (mdr-Kr)/b-mrcos\phi /2,
%k_2=sin\phi (Kr-mdr)/b-mrcos\phi /2,\\
%k_3=&cos\phi (Kr-mdr)/b-mrsin\phi /2,
%k_4=cos\phi (mdr-Kr)/b-mrsin\phi /2,
%k_5=r(I-Kd)/b,
%\end{align*}
%Here $q\in\mathbb{R}^5=\left\lbrace x_c,y_c, \phi, \theta_r, \theta_l \right\rbrace$ is the generalized coordinate vector of the system. The position of the WMR can be specified by three generalized coordinates $(x_c,y_c, \phi)$ where, $(x_c,y_c)$ are the coordinates of the center of mass of the system and $\phi$ is the heading angle. $(\theta_r, \theta_l)$ and $(u_r,u_l)$  are rotation and torque inputs of the right and left wheels respectively. $m$ and $\bar{I}$ represents the mass and inertia of the overall system. Definition of other system parameters are detailed in \cite{Ref:12}.
%
%\subsection{Experimental Results and Comparison}
%TARC is employed in "PIONEER-3" WMR while the robot is directed to track a predefined circular path. The control architecture of the proposed TARC is depicted in Fig. \ref{fig:block}. The 'State derivative estimator' block has a memory which can store state feedback history upto $(t-\varsigma)$ for computing the state derivatives. Tracking performance of TARC is compared with TDC, ASMC 1, ASMC 2 and the proposed F-TDC. The necessary controller parameters are selected as $K_1=4I, K_2=Q=D=I$, $L=0.3I$, $\xi=11$, $\varsigma=3 s$, $\beta=0.1, \Lambda=2$. The delay time and thus the sampling interval is selected as $h=30ms$  which satisfy both the conditions (\ref{delay value}) and (\ref{delay value 1}).
%\par The threshold value ($\rho$) for ASMC 1 is selected based on maximum uncertainty bound. There is no mean to determine exactly its value. However, $\rho=0.05$ is selected for ASMC 1. Other design parameters are defined as $\alpha=\bar{c}=2, \bar{s}=\hat{s}, \epsilon=0.1, \gamma=0.001$. The desired trajectories for circular path is defined as,
%\begin{align*}
%x_c^d&=1.25sin(0.35t)+.1,\\
%y_c^d&=1.25cos(0.35t)+1.35,\\
%\phi^d&=0.25t, \theta_r^d=3t, \theta_l^d=2t.
%\end{align*}
%\par To create a dynamic parametric variation, a further $3.5 kg$ payload is added and kept for $5 sec$ and then removed. This process is carried out for the entire duration of experimentation. A time gap $5 sec$ is maintained between two successive instances of addition of the payload. However, the payload was added randomly at different places on the robotic platform every time to create variation in center of mass and inertia.
%% Encoder is the only available sensor here to measure the states. Though "PIONEER-3" provides $(x_c,y_c,\phi)$, their time derivatives are computed numerically. However, TARC employs (\ref{lemma 1}) to evaluate all the state derivatives from state information of the past instance. 
%\par The trajectory tracking performance of TARC is depicted in Fig. \ref{fig:1} while following the desired circular path. Tracking performance comparison of TARC against TDC and F-TDC is illustrated in Fig. \ref{fig:2} and Fig. \ref{fig:3} in terms of $x_c$ and $y_c$ position respectively. All the error plots are in absolute value. According to Lemma 1, $\dot{q}, \ddot{q}$ cannot be computed using (\ref{lemma 2})-(\ref{omega}) for $t<\varsigma$. So, they are computed numerically for $t<\varsigma$. Hence, it can be observed from the error plots that tracking performance of TDC and F-TDC is similar during this time period. However, from $t \geq \varsigma$, immediate improvement in controller performance for F-TDC over TDC can be noticed due to the inclusion of sate derivative estimator for F-TDC. On the other hand, F-TDC is not embedded with any robust term to counter the approximation errors. TARC, on this front, provides robustness against all the uncertainties with its adaptive-robust control law, beside utilizing (\ref{lemma 2})-(\ref{omega}) for state derivative estimation. Thus, TARC provides better tracking accuracy than TDC and F-TDC. Yet again, tracking performance of TARC in comparison to ASMC 1 and ASMC 2 for $x_c$ and $y_c$ position is demonstrated through Fig. \ref{fig:4} and Fig. \ref{fig:5} respectively. Superior performance of TARC over ASMC 1 and ASMC 2 can easily be comprehended from the error plots. It is easier to infer the performance of the individual controllers in terms of average $x_c$ position error (AE-$x_c$) and average $y_c$ position error (AE-$y_c$) which are evaluated from the absolute value of the error. These data are represented in Table \ref{table 1} where the percentage error is calculated with respect to the diameter of the circular path. Due to the robustness property, ASMC 1 and ASMC 2 provides better accuracy over TDC and F-TDC through their switching control logic. 
%
%
%\begin{figure}[t]
%\begin{center}
%\includegraphics[width=3 in,height=2.7 in]{f-1.jpg}
%\caption{Trajectory tracking performance of TARC.}\label{fig:1}
%\end{center}
%\end{figure}
%
%\begin{figure}[!t]
%\begin{center}
%\includegraphics[width=3 in,height=2 in]{f-3-1.jpg}
%\caption{$x_c$ position tracking error performance comparision of TARC with TDC and F-TDC.}\label{fig:2}
%\end{center}
%\end{figure} 
%
%\begin{figure}[!t]
%\begin{center}
%\includegraphics[width=3 in,height=2 in]{f-4-2.jpg}
%\caption{$y_c$ position tracking error performance comparision of TARC with TDC and F-TDC.}\label{fig:3}
%\end{center}
%\end{figure}
%
%\begin{figure}[!t]
%\begin{center}
%\includegraphics[width=3 in,height=2 in]{f-5.jpg}
%\caption{$x_c$ position tracking error performance comparision of TARC with ASMC 1 and ASMC 2.}\label{fig:4}
%\end{center}
%\end{figure}
%
%\begin{figure}[!t]
%\begin{center}
%\includegraphics[width=3 in,height=2 in]{f-6-3.jpg}
%\caption{$y_c$ position tracking error performance comparision of TARC with ASMC 1 and ASMC 2.}\label{fig:5}
%\end{center}
%\end{figure}
%
%\begin{table}[!t]
%% increase table row spacing, adjust to taste
%\renewcommand{\arraystretch}{1.3}
%% if using array.sty, it might be a good idea to tweak the value of
%% \extrarowheight as needed to properly center the text within the cells
%\caption{$x_c$ and $y_c$ Position Error (mm) Comparison}
%\label{table 1}
%\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
%%\begin{tabular}{|c||c|}
%%\hline
%%One & Two\\
%%\hline
%%Three & Four\\
%%\hline
%%\end{tabular}
%\begin{tabular}{|c||c||c||c||c|}
%\hline
%Controller & AE-$x_c$ & $\%$ AE-$x_c$ & AE-$y_c$ & $\%$ AE-$y_c$ \\
%\hline
%TDC (\cite{Ref:6}), (\cite{Ref:25})-(\cite{Ref:26}) & 105.09& 4.20 & 99.55 & 3.98\\
%\hline
%F-TDC (proposed) & 75.99 & 3.04 & 70.80  & 2.83\\
%\hline
%ASMC 1 (\cite{Ref:21}) & 65.78 & 2.63 & 63.97 & 2.56\\
%\hline
%ASMC 2 (\cite{Ref:21})-(\cite{Ref:22}) & 58.15 & 2.33 & 56.89  & 2.28\\
%\hline
%TARC (proposed) & 37.57 & 1.50 & 13.00 & 0.52\\
%\hline
%\end{tabular}
%\end{table}
%
%%\begin{table}[!t]
%%% increase table row spacing, adjust to taste
%%\renewcommand{\arraystretch}{1.3}
%%% if using array.sty, it might be a good idea to tweak the value of
%%% \extrarowheight as needed to properly center the text within the cells
%%\caption{$y_c$ Position Error (mm) Comparison}
%%\label{table 2}
%%\centering
%%% Some packages, such as MDW tools, offer better commands for making tables
%%% than the plain LaTeX2e tabular which is used here.
%%%\begin{tabular}{|c||c|}
%%%\hline
%%%One & Two\\
%%%\hline
%%%Three & Four\\
%%%\hline
%%%\end{tabular}
%%\begin{tabular}{|c||c||c||c||c|}
%%\hline
%%Controller & A.E. & $\%$ A.E. & M.E. & $\%$ M.E. \\
%%\hline
%%TDC (\cite{Ref:6}), (\cite{Ref:25})-(\cite{Ref:26} & 99.55& 3.98 & 211.38& 8.46\\
%%\hline
%%F-TDC (proposed) & 70.80 & 2.83 & 169.64 & 6.79\\
%%\hline
%%ASMC 1 (\cite{Ref:21}) & 63.97 & 2.56 & 150.68 & 6.03\\
%%\hline
%%ASMC 2 (\cite{Ref:21})-(\cite{Ref:22}) & 56.89 & 2.28 & 128.56 & 5.14\\
%%\hline
%%TARC (proposed) & 13.00 & 0.52 & 112.91 & 4.52\\
%%\hline
%%\end{tabular}
%%\end{table}
%
%
%\par Another important attribute for an adaptive-robust control law is the evaluation of switching gain which tackles the uncertainties. The switching gain for both the wheels are provided in Fig. \ref{fig:6}, Fig. \ref{fig:7}, and Fig. \ref{fig:8} for ASMC 2, ASMC 1 and TARC respectively. Switching gain for ASMC 1 depends on the choice of $\rho$. In the current form, the nature of switching gain for ASMC 1 is monotonically increasing. This may cause actuator saturation or even chattering due to high control input in longer run. Choice of a high threshold value can circumvent this situation but too high a value will further enhance the overestimation problem of switching gain and reduce accuracy. Though ASMC 2 does not suffer from this problem, however, it assumes that the perturbations in any parameter is always less than its nominal values. This further necessitates nominal modelling of the uncertainties in $N(q,\dot{q},\ddot{q})$. Modelling of the uncertainties is tedious and thus ignored here. Thus the performance of ASMC 2 gets degraded. TARC is devoid of any such issues and this augments its superior performance. Th required control input for ASMC 1, ASMC 2 and TARC is illustrated through the Figs. \ref{fig:10}-\ref{fig:12} respectively. 
%%To verify the effect of $\tau$ on the controller performance, similar experiment was carried out for various sampling interval and the tracking performance is tabulated in Table \ref{table 4} for TARC in terms of path error. Path error is defined as the Euclidean error distance in both $x_c$ and $y_c$ position. The percentage path error is calculated with respect to the diameter of the circular path. It is evident from Table Table \ref{table 4} that higher choice of sampling interval has pervasive effect on the controller performance and results in poorer tracking accuracy.
%
%% These phenomena can be substantiated from the error bounds of both TDC and ATSMC where higher values of delay has pervasive effect on the error bounds.
%
%
%
%
%
% %as long as it is less than or equal to $\gamma=0.001$. The moment it crosses $\gamma$, it provides a zig-zag nature with minimum value $\gamma$. This situation arises, according to the law (\ref{ASMC 1}), as $\hat{c}$ never goes above $\rho$. This creates an overestimation problem of switching gain as $\hat{c}$ always tries to decrease irrespective of the nature of uncertainty. This phenomenon happens due to conservative selection of $\rho$. Choice of a lower threshold value can circumvent this situation and improve tracking accuracy. However, a very low value may cause the switching gain to increase monotonically (underestimation problem) and this may in turn cause actuator saturation or even chattering due to high control input in longer run.  For ASMC 2, $\rho$ is varied according to switching gain.  
%%Relatively more tracking error for ASMC 1 and ASMC 2, due to the inability of the switching gain to negotiate the uncertainties, results in more control input as $\hat{u}_a$ directly depends on tracking error (\ref{u hat}). This is illustrated through the control input plots for ASMC 1, ASMC 2 and TARC in Figs. \ref{fig:10}-\ref{fig:12} respectively and tabulated, further, in Table \ref{table 4}. 
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%\begin{figure}[!t]
%\begin{center}
%\includegraphics[width=3 in,height=2 in]{f-8.jpg}
%\caption{Switching gain evaluation for ASMC 2.}\label{fig:6}
%\end{center}
%\end{figure}
%
%
%\begin{figure}[!t]
%\begin{center}
%\includegraphics[width=3 in,height=2 in]{f-7_1.jpg}
%\caption{Switching gain evaluation for ASMC 1.}\label{fig:7}
%\end{center}
%\end{figure}
%
%\begin{figure}[!t]
%\begin{center}
%\includegraphics[width=3 in,height=2 in]{f-9_1.jpg}
%\caption{Switching gain evaluation for TARC.}\label{fig:8}
%\end{center}
%\end{figure}
%
%\begin{figure}[!t]
%\begin{center}
%\includegraphics[width=3 in,height=2 in]{f-10.jpg}
%\caption{Control input for ASMC 1.}\label{fig:10}
%\end{center}
%\end{figure}
%
%\begin{figure}[!t]
%\begin{center}
%\includegraphics[width=3 in,height=2 in]{f-11.jpg}
%\caption{Control input for ASMC 2.}\label{fig:11}
%\end{center}
%\end{figure}
%
%\begin{figure}[!t]
%\begin{center}
%\includegraphics[width=3 in,height=2 in]{f-12_1.jpg}
%\caption{Control input for TARC.}\label{fig:12}
%\end{center}
%\end{figure}

%\begin{table}[!t]
%\renewcommand{\arraystretch}{1.3}
%\caption{Path Error (mm) Comparison of TARC for Various Sampling Interval}
%\label{table 4}
%\begin{center}
%\begin{tabular}{|c||c||c||c||c|}
%\hline
%$\tau$ (ms) & A.E.  & $\%$ A.E. & M.E.  & $\%$ M.E. \\
%\hline
%30 & 43.37 & 1.73 & 131.29 & 5.25\\
%\hline
%60 & 52.65 & 2.11 & 141.19 & 5.65\\
%\hline
%90 & 62.87 & 2.51 & 149.25 & 5.97\\
%\hline
%120 & 73.21 & 2.93 & 158.98 & 6.36\\
%\hline
%\end{tabular}
%\end{center}
%\end{table}
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%\subfloat[Case I]{\includegraphics[width=2.5in]{box}%
%\label{fig_first_case}}
%\hfil
%\subfloat[Case II]{\includegraphics[width=2.5in]{box}%
%\label{fig_second_case}}
%\caption{Simulation results for the network.}
%\label{fig_sim}
%\end{figure*}
%
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% captions (using the optional argument to \subfloat[]), but instead will
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%\begin{table}[!t]
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%\caption{Average Control Input (Absolute Value)}
%\label{table 4}
%\centering
%
%\begin{tabular}{|c||c||c|}
%\hline
%Controller & Right Wheel (Nm) & Left Wheel (Nm)\\
%\hline
%ASMC 1 (\cite{Ref:21}) & 0.1450 & 0.0970\\
%\hline
%ASMC 2 (\cite{Ref:21})-(\cite{Ref:22}) & 0.1427 & 0.0965\\
%\hline
%TARC (proposed) & 0.0517 & 0.0245\\
%\hline
%\end{tabular}
%\end{table}


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\section{Conclusion}
Selection of the controller gain and sampling interval is crucial for the performance of TDC and this design issue is addressed in this paper through a new stability approach. A bound on the delay is derived to select a suitable sampling interval. A new control approach, F-TDC is devised where the state derivatives are estimated from the previous state information. Moreover, a novel adaptive-robust control law, TARC has been proposed for a class of uncertain nonlinear systems subjected to unknown uncertainties. The proposed controller approximates unknown dynamics through time-delayed law and negotiates the approximation error, that surfaces due to the time-delayed approximation of uncertainties and state derivatives, by switching logic. The adaptive law eliminates the overestimation-underestimation problem for online evaluation of switching gain without any prior knowledge of uncertainties. Experimentation with a WMR shows improved path tracking performance of TARC compared to TDC and conventional ASMC. The proposed framework can also be extended for other systems such as Autonomous Underwater Vehicle, Unmanned Aerial Vehicle, Robotic manipulator etc. 
%As a summarizing note, it would be interesting to use the past state values to compute current states and analyse the effect of the same on controller performance. This is left as a future course of research. %  pcomputation of current sate value from its delayed past values As a future work it is intended to carry out analysis where past state values would also be considered to compute current satestates are derived from%, %consideration of similar approach for states also.  Here, explicit effect of noise on state information is noise is not considered and %significant improvement of f-TDC over TDC in terms of performance paves the way for future research where past state values would also be considered along with the current state values to curb effect of noise.
 







% if have a single appendix:
\appendices
\section{Proof of Theorem 2} 
%% or
%%\appendix  % for no appendix heading
%% do not use \section anymore after \appendix, only \section*
%% is possibly needed
%\begin{proof}
%Let us consider a linear time invariant system,
%\begin{align}
%\dot{x}(t)&=\Sigma x(t) \label{x}\\
%y(t)&=Cx(t)\label{y}
%\end{align}
%where, $x\in \mathbb{R}^{\Lambda+1}, \Sigma \in \mathbb{R}^{(\Lambda+1) \times (\Lambda+1)}, y \in \mathbb{R}$. If $(\Sigma,C)$ is observable, then from the definition of observability, we can detect $x(t_1)$ from the measurement $y(t)$ in $\upsilon \in [t_0 \quad t_1]$. So,
%\begin{equation} \label{transit}
%y(\upsilon )=C\Upsilon (\upsilon ,t_1)x(t_1),
%\end{equation} 
%where, $\Upsilon (\upsilon ,t_1)$ is state transition matrix. Again, from (\ref{transit}) we have
%\begin{align}\label{int}
%\int_{t_0}^{t_1}\Upsilon^T (\upsilon,t_1) C^T y(\upsilon )d \upsilon & = \int_{t_0}^{t_1}\Upsilon ^T(\upsilon ,t_1)  C^T \times \nonumber \\
%& \qquad \times  C \Upsilon(\upsilon ,t_1)d\upsilon
%x(t_1).
%\end{align}
%An estimate of $x(t)$ can be obtained from (\ref{int}) as,
%\begin{equation}\label{est}
%\hat{x}(t_1)=\omega_{r}^{-1}(t_0,t_1)\int_{t_0}^{t_1}\Upsilon ^T(\upsilon ,t_1)C^Ty(\upsilon )d\upsilon,
%\end{equation}
%where, $\omega_{r}(t_0,t_1)=\left ( \int_{t_0}^{t_1}\Upsilon ^T(\upsilon ,t_1)C^TC\Upsilon(\upsilon ,t_1)d\upsilon  \right )$ is the observability Gramian. As, $(\Sigma, C)$ is observable, $\omega_{r}^{-1}$ would always exist. Now, taking $x=\begin{bmatrix}
%y, & \dot{y}, & \cdots , & y^{(j)}, & \cdots , & y^{(\Lambda)}
%\end{bmatrix}^T, C=\begin{bmatrix}
%1 & 0 & \cdots & 0
%\end{bmatrix}, \Sigma=\begin{bmatrix}
%0 & 1 &0 &\cdots & 0 \\ 
%0 & 0 &1  &\cdots & 0\\ 
%\vdots & \vdots & \vdots & \ddots & 0\\
%0 & 0 & 0 & \cdots & 0
%\end{bmatrix}, t_0=t-\varsigma,t_1=t$, (\ref{est}) can be obtained as (\ref{lemma 2})-(\ref{omega}).
%\end{proof}
%\section{Proof of Theorem 2}
%\begin{proof}
%
Let us define the Lyapunov functional as,
\begin{align}
V_f(e)&=V(e)+V_{f1}(e)+V_{f2}(e)+V_{f3}(e),\label{ftdc lya}\\
V_{f1}(e)&=\frac{h \varsigma}{\beta}\int_{-h}^{0}\int_{-\varsigma}^{0}\int_{t-h+\psi}^{t-h}e^T(\eta +\theta)D\times \nonumber\\
& \qquad \qquad \qquad \qquad \qquad \times A_d^2 (\psi)e(\eta+\theta)d\eta d\psi d\theta\nonumber\\
V_{f2}(e)&=\frac{h \varsigma}{\beta}\int_{-h}^{0}\int_{-\varsigma}^{0}\int_{t+\theta}^{t}e^T(\eta-h)D \times \nonumber\\
& \qquad \qquad \qquad \qquad \qquad \times A_d^2(\psi)e(\eta-h)d\eta d\psi d\theta\nonumber\\
V_{f3}(e)&=\varsigma \int_{-\varsigma}^{0}\int_{t+\psi}^{t}e^T(\eta-h)(\psi)RA_d^2(\psi)e(\eta-h)d\eta d\psi.\nonumber
\end{align}
%%where, $V$ is given in (\ref{lyapunov}) and putting (\ref{u hat}) into (\ref{sys new}) we obtain,
%%%\begin{align}\label{error new}
%%%\ddot{e}_1 &=-K_1e_{1h}-K_2\dot{e}_1-\begin{bmatrix}
%%% K_2  &  0
%%%\end{bmatrix} \times \nonumber\\
%%%& \quad \times \int_{-\tau}^{0}A_d(\psi)\begin{bmatrix}
%%%e_1(t-h+\psi )\\ 
%%%\dot{e}_1(t-h+\psi )
%%%\end{bmatrix}d\psi+\sigma_2.
%%%\end{align}
%%%with $\sigma_2=K_2\dot{e}_1+\sigma_1$. Further the error dynamics (\ref{error new}) can be written in sate space form as,
%%\begin{equation}\label{error new}
%%\dot{e}=\bar{A}_1e+\bar{B}_1e_h-\bar{B}\int_{-\varsigma}^{0}A_d(\psi )e(t-h+\psi )d\psi+B\sigma_2,
%%%Ae-\bar{B}\int_{-\varsigma}^{0}A_d(\psi )e(t+\psi )d\psi+B\sigma_2,
%%\end{equation}
%%where, $\bar{A}_1=\begin{bmatrix}
%%0 & I \\
%%0 & -K_2
%%\end{bmatrix} 
%%, 
%%\bar{B}_1=\begin{bmatrix}
%%0 & 0\\
%%-K_1 & 0
%%\end{bmatrix}$, $\sigma_2=K_2\dot{e}_1+\sigma_1$. Further the error dynamics (\ref{error new}) can be written as,  
%%\begin{equation}\label{error dyn state new 1}
%%\dot{e}=Ae-\bar{B}_1 \int_{-h}^{0} \dot{e}(t+\theta)d \theta-\bar{B}\int_{-\varsigma}^{0}A_d(\psi )e(t-h+\psi )d\psi+B\sigma_2.
%%\end{equation}
%%
%%
%%
%%
%%Following the similar procedure for proving Theorem 1 and and applying Lemma 1 and Lemma 2 to the time derivative of $V_{f1}, V_{f3}$ and $V_{f3}$ yields
%%
%%\begin{align}
%%&\dot{V}(e)\leq -e^{T}\left [ Q-\bar{E} -(1+\xi)\frac{h^2}{\beta}D \right]e+ \Gamma_2+\frac{h \varsigma}{\beta} \times \nonumber\\
%%&\int_{-h}^{0}\int_{-\varsigma}^{0} e^T(t-h+\theta+\psi)D A_d^2(\psi)e(t-h+\theta+\psi)d\psi d\theta \nonumber\\
%%& \qquad + 2\hat{s}^{T}\sigma_2 -2e^TP\bar{B}\int_{-\varsigma}^{0}A_d(\psi)e(t-h+\psi)d\psi,\label{v dot ftdc1}
%%\end{align}
%%where, $\frac{h}{\beta }\left \| \int_{-h}^{0}\left [ (B \sigma _2(t+\theta))^{T}\bar{D}^T\bar{D}B \sigma _2(t+\theta) \right ]d\theta  \right \|\leq \Gamma_2.$
%%\begin{align}
%%&\dot{V}_{f1}\leq \frac{h \varsigma}{\beta}\int_{-h}^{0} \int_{-\varsigma}^{0}e^T(t-h+\theta)DA_d^2(\psi)e(t-h+\theta)d\psi d\theta\nonumber\\
%%& \qquad \qquad \quad -\frac{h \varsigma}{\beta}\int_{-h}^{0} \int_{-\varsigma}^{0}e^T(t-h+\theta+\psi)DA_d^2(\psi) \times \nonumber\\
%%& \qquad \qquad \qquad \qquad \qquad \times e(t-h+\theta+\psi)d\psi d\theta\label{v1 dot ftdc}
%%\end{align}
%%\begin{align}
%%\dot{V} & _{f2}  \leq \frac{h^2 \varsigma}{\beta} e^T(t-h)D \int_{-\varsigma}^{0}A_d^2(\psi)d\psi e(t-h)-\frac{h \varsigma}{\beta} \times \nonumber \\
%%& \times \int_{-h}^{0} \int_{-\varsigma}^{0} e^T(t-h+\theta)DA_d^2(\psi)e(t-h+\theta)d\psi d\theta\label{v2 dot ftdc}
%%\end{align}
%%\begin{align}
%%&\dot{V}_{f3}(e) \leq \varsigma e^T(t-h)\int_{-\varsigma}^{0}RA_d^2(\psi)d\psi e(t-h)-\int_{-\varsigma}^{0}A_{d}(\psi) \times \nonumber\\
%%& \times e^T(t-h+\psi)d\psi R \int_{-\varsigma}^{0}A_d(\psi)e(t-h+\psi)d\psi \label{v3 dot ftdc}
%%\end{align}
%%%Now, $2\hat{s}^T K_2\dot{e}_1=2e^TP \breve{B}e$. 
%%Combining (\ref{v dot ftdc1})-(\ref{v3 dot ftdc})
%%\begin{align}
%%\dot{V}_f(e) & \leq -e_f^T \Theta e_f + \Gamma_2 +2\hat{s}^T\sigma_2, \label{ef}
%%\end{align}
%%where $e_f=\begin{bmatrix}
%%\bar{e}^T & \int_{-\varsigma}^{0}A_d(\psi)e^T(t-h+\psi)d\psi
%%\end{bmatrix}^T$.
%%%\begin{corollary}
%%%Since $A$ entirely depends on controller gains, the solution of the Lyapunov equation $P$ is also governed by the controller gains and choice of $Q$. Hence, the sampling time $\tau$ and controller gains are to be selected in such a manner so that for a given $R>0$ and $Q>0$, $\exists$ $\Theta>0$ so that the following inequality holds:
%%%\begin{equation}\label{sample 2}
%%%\begin{bmatrix}
%%%-Q+\tau \int_{-\tau}^{0}A_{d}^{T}(\psi)RA_d(\psi)d\psi & \quad-P\bar{B}\\ 
%%%-\bar{B}^TP & \quad-R
%%%\end{bmatrix}=-\Theta<0,\quad  \quad 
%%%\end{equation}
%%%\end{corollary}
%%%%\begin{remark}
%%%%The relation (\ref{sample 2}) provides a selection criterion for $\tau$ for given controller gains and vice-versa. This solves an important design issue for TDC which was previously not addressed in the literature. 
%%%%\end{remark}
%%%Thus, 
%%%\begin{equation}
%%%\dot{V}_f(e) \leq -\lambda_{min}(\Theta)||\bar{e}||^2+2||\sigma_1||||\hat{s}||\label{v dot ftdc}
%%%\end{equation}
%%%Here, $\lambda_{min}(\cdot)$ and $||\cdot||$ represent the minimum eigen value and Euclidean norm of the argument respectively.
%%Let choice of $K_1, K_2$, $h, \varsigma$ make $\Theta>0$. So, $\dot{V}_f(e)<0$ would be achieved if $\lambda_{min}(\Theta)||e_f||^2> \Gamma_2+ 2||\sigma_2||||\hat{s}||=\varpi$. Thus, the system would be UUB with the following ultimate bound
%%\begin{equation}\label{error bound f-tdc}
%%%||e_f||=\sqrt{\frac{2\Gamma+ 2||\sigma_2||||\hat{s}||}{\lambda_{min}(\Theta)}}=\varpi
%%||e_f||=\sqrt{\frac{\varpi}{\lambda_{min}(\Theta)}}
%%\end{equation}
%%The time $t_{r}$ to reach $\varpi$ is given by 
%%%Let, $R_1$ denote the smallest level surface of $V_f$ containing the ball $B_{\varpi}$ with radius $\varpi$ centred at $\bar{e}=0$. For initial time $t_0$, if $\bar{e}(t_0)\in R$ then the solution remains in $R_1$. If  $\bar{e}(t_0)\notin R_1$ then $V_f$ decreases as long as $\bar{e}(t)\notin R_1$. The time required to reach $\varpi$ is zero when $\bar{e}(t_0)\in R_1$, otherwise, while $\bar{e}(t_0)\notin R_1$ the finite time $t_{r}$ to reach $\varpi$ is given by \cite{Ref:24},
%%\begin{equation*}
%%t_{r}-t_0\leq (||e_f(t_0)||-\varpi)/c \quad \text{where} \quad \dot{V}_f(e)\leq-c.
%%\end{equation*}
%Let us define the Lyapunov functional as,
%\begin{align}
%V_f(e)&=V(e)+V_{f1}(e)+V_{f2}(e)+V_{f3}(e),\label{ftdc lya}\\
%V_{f1}(e)&=\frac{h \varsigma}{\beta}\int_{-h}^{0}\int_{-\varsigma}^{0}\int_{t-h+\psi}^{t-h}e^T(\eta +\theta)D A_d^2 (\psi)e(\eta+\theta)d\eta d\psi d\theta\nonumber\\
%V_{f2}(e)&=\frac{h \varsigma}{\beta}\int_{-h}^{0}\int_{-\varsigma}^{0}\int_{t+\theta}^{t}e^T(\eta-h)D  A_d^2(\psi)e(\eta-h)d\eta d\psi d\theta\nonumber\\
%V_{f3}(e)&=\varsigma \int_{-\varsigma}^{0}\int_{t+\psi}^{t}e^T(\eta-h) L A_d^2(\psi)e(\eta-h)d\eta d\psi.\nonumber
%\end{align}
%where, $V$ is given in (\ref{lyapunov}) and putting (\ref{u hat}) into (\ref{sys new}) we obtain,
%%\begin{align}\label{error new}
%%\ddot{e}_1 &=-K_1e_{1h}-K_2\dot{e}_1-\begin{bmatrix}
%% K_2  &  0
%%\end{bmatrix} \times \nonumber\\
%%& \quad \times \int_{-\tau}^{0}A_d(\psi)\begin{bmatrix}
%%e_1(t-h+\psi )\\ 
%%\dot{e}_1(t-h+\psi )
%%\end{bmatrix}d\psi+\sigma_2.
%%\end{align}
%%with $\sigma_2=K_2\dot{e}_1+\sigma_1$. Further the error dynamics (\ref{error new}) can be written in sate space form as,
%\begin{equation}\label{error new}
%\dot{e}={A}_1e+{B}_1e_h-\bar{B}\int_{-\varsigma}^{0}A_d(\psi )e(t-h+\psi )d\psi+B\sigma_2,
%%Ae-\bar{B}\int_{-\varsigma}^{0}A_d(\psi )e(t+\psi )d\psi+B\sigma_2,
%\end{equation}
%where, $\sigma_2=K_2\dot{e}_{1h}+\sigma_1$. Further the error dynamics (\ref{error new}) can be written as,  
%\begin{equation}\label{error dyn state new 1}
%\dot{e}=Ae-{B}_1 \int_{-h}^{0} \dot{e}(t+\theta)d \theta-\bar{B}\int_{-\varsigma}^{0}A_d(\psi )e(t-h+\psi )d\psi+B\sigma_2.
%\end{equation}
%Applying (\ref{ineq 2}) and Lemma 2 to the term $-2e^TPB_1\bar{B}\int_{-h}^{0}\int_{-\varsigma}^{0}A_d(\psi )e(t-h+\theta+\psi )d\psi$ we get,
%\begin{align}
%-2e^TPB_1\bar{B}\int_{-h}^{0}\int_{-\varsigma}^{0}A_d(\psi )e(t-h+\theta+\psi )d\psi &\leq \beta e^{T}PB_1\bar{B}D^{-1}\bar{B}^{T}B_1^{T}Pe +\frac{1}{\beta}\int_{-h}^{0}\int_{-\varsigma}^{0}e^{T}(t-h+\theta+\psi )A_d(\psi )d\psi D \times \nonumber\\ 
%& \qquad \qquad \qquad \qquad \qquad \times \int_{-h}^{0}\int_{-\varsigma}^{0}A_d(\psi )e(t-h+\theta+\psi )d\psi \nonumber\\
%&\leq \beta e^{T}PB_1\bar{B}D^{-1}\bar{B}^{T}B_1^{T}Pe +\frac{h \varsigma}{\beta}\int_{-h}^{0}\int_{-\varsigma}^{0}e^{T}(t-h+\theta+\psi )D \times \nonumber\\
%& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \times A_d^2(\psi ) e(t-h+\theta+\psi )d\psi \label{new jensen}
%\end{align}
%
%
%
%Following the similar procedure for proving Theorem 1 and utilizing (\ref{new jensen}) the time derivative of $V_{f1}, V_{f3}$ and $V_{f3}$ yields
%
%\begin{align}
%&\dot{V}(e)\leq -e^{T}\left [ Q-\bar{E} -(1+\xi)\frac{h^2}{\beta}D \right]e+ \Gamma_2+\frac{h \varsigma}{\beta}\int_{-h}^{0}\int_{-\varsigma}^{0} e^T(t-h+\theta+\psi)D A_d^2(\psi)e(t-h+\theta+\psi)d\psi d\theta \nonumber\\
%& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + 2\hat{s}^{T}\sigma_2 -2e^TP\bar{B}\int_{-\varsigma}^{0}A_d(\psi)e(t-h+\psi)d\psi,\label{v dot ftdc1}
%\end{align}
%where, $\frac{h}{\beta }\left \| \int_{-h}^{0}\left [ (B \sigma _2(t+\theta))^{T}\bar{D}^T\bar{D}B \sigma _2(t+\theta) \right ]d\theta  \right \|\leq \Gamma_2.$
%\begin{align}
%&\dot{V}_{f1}\leq \frac{h \varsigma}{\beta}\int_{-h}^{0} \int_{-\varsigma}^{0}e^T(t-h+\theta)DA_d^2(\psi)e(t-h+\theta)d\psi d\theta -\frac{h \varsigma}{\beta}\int_{-h}^{0} \int_{-\varsigma}^{0}e^T(t-h+\theta+\psi)DA_d^2(\psi) e(t-h+\theta+\psi)d\psi d\theta\label{v1 dot ftdc}
%\end{align}
%\begin{align}
%\dot{V} & _{f2}  \leq \frac{h^2 \varsigma}{\beta} e^T(t-h)D \int_{-\varsigma}^{0}A_d^2(\psi)d\psi e(t-h)-\frac{h \varsigma}{\beta}  \int_{-h}^{0} \int_{-\varsigma}^{0} e^T(t-h+\theta)DA_d^2(\psi)e(t-h+\theta)d\psi d\theta\label{v2 dot ftdc}
%\end{align}
%\begin{align}
%&\dot{V}_{f3}(e) \leq \varsigma e^T(t-h)\int_{-\varsigma}^{0} L A_d^2(\psi)d\psi e(t-h)-\int_{-\varsigma}^{0}A_{d}(\psi)  e^T(t-h+\psi)d\psi L \int_{-\varsigma}^{0}A_d(\psi)e(t-h+\psi)d\psi \label{v3 dot ftdc}
%\end{align}
%Now, $2\hat{s}^T K_2\dot{e}_{1h}=2e^TP \breve{B}e_h$. Combining (\ref{v dot ftdc1})-(\ref{v3 dot ftdc})
%\begin{align}
%\dot{V}_f(e) & \leq -e_f^T \Theta e_f + \Gamma_2 +2\hat{s}^T\sigma_1,
%\end{align}
%where $e_f=\begin{bmatrix}
%\bar{e}^T & \int_{-\varsigma}^{0}A_d(\psi)e^T(t-h+\psi)d\psi
%\end{bmatrix}^T$.
%%\begin{corollary}
%%Since $A$ entirely depends on controller gains, the solution of the Lyapunov equation $P$ is also governed by the controller gains and choice of $Q$. Hence, the sampling time $\tau$ and controller gains are to be selected in such a manner so that for a given $R>0$ and $Q>0$, $\exists$ $\Theta>0$ so that the following inequality holds:
%%\begin{equation}\label{sample 2}
%%\begin{bmatrix}
%%-Q+\tau \int_{-\tau}^{0}A_{d}^{T}(\psi)RA_d(\psi)d\psi & \quad-P\bar{B}\\ 
%%-\bar{B}^TP & \quad-R
%%\end{bmatrix}=-\Theta<0,\quad  \quad 
%%\end{equation}
%%\end{corollary}
%%%\begin{remark}
%%%The relation (\ref{sample 2}) provides a selection criterion for $\tau$ for given controller gains and vice-versa. This solves an important design issue for TDC which was previously not addressed in the literature. 
%%%\end{remark}
%%Thus, 
%%\begin{equation}
%%\dot{V}_f(e) \leq -\lambda_{min}(\Theta)||\bar{e}||^2+2||\sigma_1||||\hat{s}||\label{v dot ftdc}
%%\end{equation}
%%Here, $\lambda_{min}(\cdot)$ and $||\cdot||$ represent the minimum eigen value and Euclidean norm of the argument respectively.
%Let choice of $K_1, K_2$, $h, \varsigma$ make $\Theta>0$. 
%Again, a positive scalar $\iota_1$ can be found such that $||\hat{s}||\leq \iota_1 ||e_f||$. So, $\dot{V}_f(e)<0$ would be achieved if $\lambda_{min}(\Theta)||e_f||^2> \Gamma_2+ 2\iota_1||\sigma_1||||e_f||$.
%
%%So, $\dot{V}_f(e)<0$ would be achieved if $\lambda_{min}(\Theta)||e_f||^2> \Gamma_2+ 2||\sigma_1||||\hat{s}||=\varpi$. 
%Thus, the system would be UUB with the following ultimate bound
%\begin{equation}\label{error bound f-tdc}
%%||e_f||=\sqrt{\frac{2\Gamma+ 2||\sigma_2||||\hat{s}||}{\lambda_{min}(\Theta)}}=\varpi
%%||e_f||=\sqrt{\frac{\varpi}{\lambda_{min}(\Theta)}}
%||e_f||=\gamma_2+\sqrt{\frac{\Gamma_2}{\lambda _{min}(\Theta)}+\gamma_2^2}=\varpi.
%\end{equation}
%where, $\gamma_2=\frac{\iota_1 || \sigma_1 ||}{\lambda _{min}(\Theta)}$. The time $t_{r}$ to reach $\varpi$, for some $c_2>0$, is given by 
%%Let, $R_1$ denote the smallest level surface of $V_f$ containing the ball $B_{\varpi}$ with radius $\varpi$ centred at $\bar{e}=0$. For initial time $t_0$, if $\bar{e}(t_0)\in R$ then the solution remains in $R_1$. If  $\bar{e}(t_0)\notin R_1$ then $V_f$ decreases as long as $\bar{e}(t)\notin R_1$. The time required to reach $\varpi$ is zero when $\bar{e}(t_0)\in R_1$, otherwise, while $\bar{e}(t_0)\notin R_1$ the finite time $t_{r}$ to reach $\varpi$ is given by \cite{Ref:24},
%\begin{equation*}
%t_{r}-t_0\leq (||e_f(t_0)||-\varpi)/c_2 \quad \text{where} \quad \dot{V}_f(e)\leq-c_2.
%\end{equation*}
%\end{proof}
% use appendices with more than one appendix
% then use \section to start each appendix
% you must declare a \section before using any
% \subsection or using \label (\appendices by itself
% starts a section numbered zero.)
%


%\appendices
%\section{Proof of the First Zonklar Equation}
%Appendix one text goes here.
%
%% you can choose not to have a title for an appendix
%% if you want by leaving the argument blank
%\section{}
%Appendix two text goes here.
%
%
%% use section* for acknowledgment
%\section*{Acknowledgment}


%The authors would like to thank...


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% argument is your BibTeX string definitions and bibliography database(s)
%\bibliography{IEEEabrv,../bib/paper}
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% <OR> manually copy in the resultant .bbl file
% set second argument of \begin to the number of references
% (used to reserve space for the reference number labels box)
\begin{thebibliography}{1}
%P. Tallapragada and N. Chopra, On event triggered tracking for nonlinear systems, 
%\emph{IEEE Transactions on Automatic Control}, vol. 58(9), pp. 2343-2348, 2013.
%Avimanyu Sahoo, Hao Xu and S. Jagannathan. Adaptive event-triggered control of a uncertain linear discrete time system using measured input and output data, \emph{American Control Conference}, pp.5672 -5677, 2013 

\bibitem{Ref:1}
M. Kristic, I. Kanellakopoulos and P. V. Kokotovic, \emph{Nonlinear and Adaptive Control Design}, Wiley, New York, 1995.
\bibitem{Ref:2}
X. Liu, H. Su, B. Yao and J.Chu, Adaptive robust control of a class of uncertain nonlinear systems with unknown sinusoidal disturbances, \emph{47th  IEEE Conference on  Decision and Control}, pp. 2594-2599, 2008.
\bibitem{Ref:3}
M. J. Corless and G. Leitmann, Continuous state feedback guaranteeing uniform ultimate boundness for uncertain dynamic system, \emph{IEEE Transactions on Automatic Control}, vol. 26(10), pp. 1139-1144, 1981.
\bibitem{Ref:4}
H. Lee and V. I. Utkin, Chattering suppression methods in sliding mode control systems, \emph{Annual Reviews in Control}, vol. 31, pp. 179-188, 2007.
\bibitem{Ref:5}
A. Levant, Higher-order sliding modes, differentiation and output-feedback control, \emph{International Journal of Control}, vol. 76(9/10), pp. 924-941, 2003.
\bibitem{Ref:6-1}
A. G. Ulsoy, Time-delayed control of siso systems for improved stability margins, \emph{ASME Journal of Dynamic Systems, Measurement, and Control}, vol. 137(4), pp. 1-12, 2015.
\bibitem{Ref:6}
T. C. Hsia and L. S. Gao, Robot manipulator control using decentralized linear time-invariant time-delayed joint controllers, \emph{IEEE International Conference on  Robotics and Automation}, pp. 2070-2075, 1990.
\bibitem{Ref:6-2}
K. Youcef-Toumi and O. Ito, A time delay controller for systems with unknown dynamics, \emph{ASME Journal of Dynamic Systems, Measurement, and Control}, vol. 112(1), pp. 133-142, 1990.

\bibitem{Ref:7}
G. R. Cho, P. H. Chang, S. H. Park and M. Jin, Robust tracking under nonlinear friction using time delay control with internal model, \emph{IEEE Transactions on Control System Technology}, vol. 17(6), pp. 1406-1414, 2009.
\bibitem{Ref:8}
D. K.Han and P. H. Chang, Robust tracking of robot manipulator with nonlinear friction using time delay control with gradient estimator, \emph{Journal of Mechanical Science and Technology}, vol. 24(8), pp. 1743-1752, 2010.
\bibitem{Ref:9}
M. Jin, S. H. Kang and P. H. Chang, Robust compliant motion control of robot with nonlinear friction using time-delay estimation, \emph{IEEE Transactions on Industrial Electronics}, vol. 55(1), pp. 258-269, 2008.
\bibitem{Ref:10}
Y. Jin, P. H.  Chang, M. Jin and D. G. Gweon, Stability guaranteed time-delay control of manipulators using nonlinear damping and terminal sliding mode, \emph{IEEE Transactions on Industrial Electronics}, vol. 60(8), pp. 3304-3318, 2013.
\bibitem{Ref:11}
P. H. Chang and S. H. Park, On improving time-delay control under certain hard nonlinearities, \emph{Mechatronics}, vol. 13(4), pp. 393-412, 2003.
\bibitem{Ref:12}
S. Roy, S. Nandy, R. Ray and S. N. Shome, Time delay sliding mode control nonholonomic wheeled mobile robot: experimental validation, \emph{IEEE International Conference on  Robotics and Automation}, pp. 2886-2892, 2014.
\bibitem{Ref:13}
X. Zhu, G. Tao, B. Yao and J. Cao, Adaptive robust posture control of parallel manipulator driven by pneumatic muscles with redundancy, \emph{IEEE/ASME Transactions on Mechatronics}, vol. 13(4), pp. 441-450, 2008.
\bibitem{Ref:14}
X. Zhu, G. Tao, B. Yao and J. Cao, Integrated direct/indirect adaptive robust posture control of parallel manipulator driven by pneumatic muscles, \emph{IEEE Transactions on Control System Technology}, vol. 17(3), pp. 576-588, 2009.
%\bibitem{Ref:15}
%G. Zhang, J. Chen and Z. Lee, Adaptive robust control of servo mechanisms with partially known states via dynamic surface control approach, \emph{IEEE Transactions on Control System Technology}, vol. 18(3), pp. 723-731, 2010.
\bibitem{Ref:16}
W. Sun, Z. Zhao and H. Gao, Saturated adaptive robust control for active suspension systems, \emph{IEEE Transactions on Industrial Electronics}, vol. 60(9), pp.  3889-3896, 2013.
\bibitem{Ref:17}
S. Islam, P. X. Liu and A. E. Saddik, Robust control of four rotor unmanned aerial vehicle with disturbance uncertainty, \emph{IEEE Transactions on Industrial Electronics}, vol. 63(3), pp. 1563-1571, 2015.
\bibitem{Ref:18}
Z. Chen, B. Yao and Q. Wang, $\mu$-synthesis based adaptive robust posture control of linear motor driven stages with high frequency dynamics: a case study, \emph{IEEE/ASME Transactions on Mechatronics}, vol. 20(3), pp. 1482-1490, 2015.
\bibitem{Ref:19}
Z. Liu, H. Su and S. Pan, A new adaptive sliding mode control of uncertain nonlinear dynamics, \emph{Asian Journal of Control}, vol. 16(1), pp. 198-208, 2014.
\bibitem{Ref:20}
C. Chen, T. S. Li, Y. Yeh and C. Chang, Design and implementation of an adaptive sliding-mode dynamic controller for wheeled mobile robots, \emph{Mechatronics}, vol. 19, pp. 156-166, 2009.
\bibitem{Ref:21}
F. Plestan, Y. Shtessel, V. Bregeault and A. Poznyak, New methodologies for adaptive sliding mode control, \emph{International Journal of Control}, vol. 83(9), pp. 1907-1919, 2010.
\bibitem{Ref:22}
F. Plestan, Y. Shtessel, V. Bregeault and A. Poznyak, Sliding mode control with gain adaptation - application to an electropneumatic actuator state, \emph{Control Engineering Practice}, vol. 21, pp. 679-688, 2013.
\bibitem{Ref:23}
B. Bandyopadhayay, S. Janardhanan and  S. K. Spurgeon, \emph{Advances in Sliding Mode Control}, Springer-Verlag, New York, 2013.
\bibitem{Ref:24}
G. Leitmann, On the efficiency of nonlinear control in uncertain linear systems, \emph{ASME Journal of Dynamic Systems, Measurement and Control}, vol. 103, pp. 95-102, 1981.
\bibitem{Ref:25}
P. H. Chang and J. W. Lee, A model  reference observer for time-delay control and its application to robot trajectory control, \emph{IEEE Transactions on Control System Technology}, vol. 4(1), pp. 2-10, 1996.
\bibitem{Ref:26}
Y. H. Shin and K. J. Kim, Performance enhancement of pneumatic vibration isolation tables in low frequency range by time delay control, \emph{Journal of Sound and Vibration}, vol. 321, pp. 537-553, 2009.

\bibitem{Ref:27}
A. Kuperman and Q. C. Zhong, Robust control of uncertain nonlinear systems based on an uncertainty and disturbance estimator, \emph{International Journal of Robust and Nonlinear Control}, vol. 21, pp. 79-92, 2011.
\bibitem{Ref:28}
S. E. Talole and S. B. Phadke, Robust input-output linearization using uncertainty and disturbance estimator, \emph{International Journal of Control}, vol. 82(10), pp. 1794-1803, 2009.
\bibitem{Ref:29}
P. V. Suryawanshi, P. D. Shengde, and S. B. Phadke, Robust sliding mode control for a class of nonlinear systems using inertial delay control, \emph{Nonlinear Dynamics}, vol. 78(3), pp. 1921-1932, 2014.
\bibitem{Ref:30}
J. Reger and J. Jouffroy, On algebraic time-derivative estimation and deadbeat state reconstruction, \emph{ Joint 47th  IEEE Conference on  Decision and Control and 28th Chinese Control Conference}, pp. 1740-1745, 2009.
\bibitem{Ref:31}
K. Gu, V. Khartionov and J. Chen, \emph{Stability of time-delay systems}, Birkh\"{a}user, Boston (2003).
\bibitem{Ref:32}
A. Nasiri, S. K. Nguang and A. Swain, Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainty, \emph{Journal of The Franklin Institute}, vol. 351, pp. 2048-2061, 2014.
\bibitem{Ref:33}
S. Roy, S. Nandy, R. Ray and S. N. Shome, Robust path tracking control of nonholonomic wheeled mobile robot, \emph{International Journal of Control, Automation and Systems}, vol. 13(4), pp. 897-905, 2015.
\bibitem{Ref:34}
J. Kim, H. Joe, S-C Yu, J. S. Lee and M. Kim, Time delay controller design for position control of autonomous underwater vehicle under disturbances, \emph{IEEE Transactions on Industrial Electronics},  DOI 10.1109/TIE.2015.2477270, 2015.
\end{thebibliography}

% biography section
% 
% If you have an EPS/PDF photo (graphicx package needed) extra braces are
% needed around the contents of the optional argument to biography to prevent
% the LaTeX parser from getting confused when it sees the complicated
% \includegraphics command within an optional argument. (You could create
% your own custom macro containing the \includegraphics command to make things
% simpler here.)
%\begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{mshell}}]{Michael Shell}
% or if you just want to reserve a space for a photo:

%\begin{IEEEbiography}{Michael Shell}
%Biography text here.
%\end{IEEEbiography}
%
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%\begin{IEEEbiographynophoto}{John Doe}
%Biography text here.
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%\begin{IEEEbiographynophoto}{Jane Doe}
%Biography text here.
%\end{IEEEbiographynophoto}

% You can push biographies down or up by placing
% a \vfill before or after them. The appropriate
% use of \vfill depends on what kind of text is
% on the last page and whether or not the columns
% are being equalized.

%\vfill

% Can be used to pull up biographies so that the bottom of the last one
% is flush with the other column.
%\enlargethispage{-5in}



% that's all folks
\end{document}



1509.07314.extracted_bib
{Ref:1 M. Kristic, I. Kanellakopoulos and P. V. Kokotovic, Nonlinear and Adaptive Control Design, Wiley, New York, 1995.
1509.07314.extracted_bib
{Ref:2 X. Liu, H. Su, B. Yao and J.Chu, Adaptive robust control of a class of uncertain nonlinear systems with unknown sinusoidal disturbances, 47th IEEE Conference on Decision and Control, pp. 2594-2599, 2008.
1509.07314.extracted_bib
{Ref:3 M. J. Corless and G. Leitmann, Continuous state feedback guaranteeing uniform ultimate boundness for uncertain dynamic system, IEEE Transactions on Automatic Control, vol. 26(10), pp. 1139-1144, 1981.
1509.07314.extracted_bib
{Ref:4 H. Lee and V. I. Utkin, Chattering suppression methods in sliding mode control systems, Annual Reviews in Control, vol. 31, pp. 179-188, 2007.
1509.07314.extracted_bib
{Ref:5 A. Levant, Higher-order sliding modes, differentiation and output-feedback control, International Journal of Control, vol. 76(9/10), pp. 924-941, 2003.
1509.07314.extracted_bib
{Ref:6-1 A. G. Ulsoy, Time-delayed control of siso systems for improved stability margins, ASME Journal of Dynamic Systems, Measurement, and Control, vol. 137(4), pp. 1-12, 2015.
1509.07314.extracted_bib
{Ref:6 T. C. Hsia and L. S. Gao, Robot manipulator control using decentralized linear time-invariant time-delayed joint controllers, IEEE International Conference on Robotics and Automation, pp. 2070-2075, 1990.
1509.07314.extracted_bib
{Ref:6-2 K. Youcef-Toumi and O. Ito, A time delay controller for systems with unknown dynamics, ASME Journal of Dynamic Systems, Measurement, and Control, vol. 112(1), pp. 133-142, 1990.
1509.07314.extracted_bib
{Ref:7 G. R. Cho, P. H. Chang, S. H. Park and M. Jin, Robust tracking under nonlinear friction using time delay control with internal model, IEEE Transactions on Control System Technology, vol. 17(6), pp. 1406-1414, 2009.
1509.07314.extracted_bib
{Ref:8 D. K.Han and P. H. Chang, Robust tracking of robot manipulator with nonlinear friction using time delay control with gradient estimator, Journal of Mechanical Science and Technology, vol. 24(8), pp. 1743-1752, 2010.
1509.07314.extracted_bib
{Ref:9 M. Jin, S. H. Kang and P. H. Chang, Robust compliant motion control of robot with nonlinear friction using time-delay estimation, IEEE Transactions on Industrial Electronics, vol. 55(1), pp. 258-269, 2008.
1509.07314.extracted_bib
{Ref:10 Y. Jin, P. H. Chang, M. Jin and D. G. Gweon, Stability guaranteed time-delay control of manipulators using nonlinear damping and terminal sliding mode, IEEE Transactions on Industrial Electronics, vol. 60(8), pp. 3304-3318, 2013.
1509.07314.extracted_bib
{Ref:11 P. H. Chang and S. H. Park, On improving time-delay control under certain hard nonlinearities, Mechatronics, vol. 13(4), pp. 393-412, 2003.
1509.07314.extracted_bib
{Ref:12 S. Roy, S. Nandy, R. Ray and S. N. Shome, Time delay sliding mode control nonholonomic wheeled mobile robot: experimental validation, IEEE International Conference on Robotics and Automation, pp. 2886-2892, 2014.
1509.07314.extracted_bib
{Ref:13 X. Zhu, G. Tao, B. Yao and J. Cao, Adaptive robust posture control of parallel manipulator driven by pneumatic muscles with redundancy, IEEE/ASME Transactions on Mechatronics, vol. 13(4), pp. 441-450, 2008.
1509.07314.extracted_bib
{Ref:14 X. Zhu, G. Tao, B. Yao and J. Cao, Integrated direct/indirect adaptive robust posture control of parallel manipulator driven by pneumatic muscles, IEEE Transactions on Control System Technology, vol. 17(3), pp. 576-588, 2009. %
1509.07314.extracted_bib
{Ref:15 %G. Zhang, J. Chen and Z. Lee, Adaptive robust control of servo mechanisms with partially known states via dynamic surface control approach, IEEE Transactions on Control System Technology, vol. 18(3), pp. 723-731, 2010.
1509.07314.extracted_bib
{Ref:16 W. Sun, Z. Zhao and H. Gao, Saturated adaptive robust control for active suspension systems, IEEE Transactions on Industrial Electronics, vol. 60(9), pp. 3889-3896, 2013.
1509.07314.extracted_bib
{Ref:17 S. Islam, P. X. Liu and A. E. Saddik, Robust control of four rotor unmanned aerial vehicle with disturbance uncertainty, IEEE Transactions on Industrial Electronics, vol. 63(3), pp. 1563-1571, 2015.
1509.07314.extracted_bib
{Ref:18 Z. Chen, B. Yao and Q. Wang, $\mu$-synthesis based adaptive robust posture control of linear motor driven stages with high frequency dynamics: a case study, IEEE/ASME Transactions on Mechatronics, vol. 20(3), pp. 1482-1490, 2015.
1509.07314.extracted_bib
{Ref:19 Z. Liu, H. Su and S. Pan, A new adaptive sliding mode control of uncertain nonlinear dynamics, Asian Journal of Control, vol. 16(1), pp. 198-208, 2014.
1509.07314.extracted_bib
{Ref:20 C. Chen, T. S. Li, Y. Yeh and C. Chang, Design and implementation of an adaptive sliding-mode dynamic controller for wheeled mobile robots, Mechatronics, vol. 19, pp. 156-166, 2009.
1509.07314.extracted_bib
{Ref:21 F. Plestan, Y. Shtessel, V. Bregeault and A. Poznyak, New methodologies for adaptive sliding mode control, International Journal of Control, vol. 83(9), pp. 1907-1919, 2010.
1509.07314.extracted_bib
{Ref:22 F. Plestan, Y. Shtessel, V. Bregeault and A. Poznyak, Sliding mode control with gain adaptation - application to an electropneumatic actuator state, Control Engineering Practice, vol. 21, pp. 679-688, 2013.
1509.07314.extracted_bib
{Ref:23 B. Bandyopadhayay, S. Janardhanan and S. K. Spurgeon, Advances in Sliding Mode Control, Springer-Verlag, New York, 2013.
1509.07314.extracted_bib
{Ref:24 G. Leitmann, On the efficiency of nonlinear control in uncertain linear systems, ASME Journal of Dynamic Systems, Measurement and Control, vol. 103, pp. 95-102, 1981.
1509.07314.extracted_bib
{Ref:25 P. H. Chang and J. W. Lee, A model reference observer for time-delay control and its application to robot trajectory control, IEEE Transactions on Control System Technology, vol. 4(1), pp. 2-10, 1996.
1509.07314.extracted_bib
{Ref:26 Y. H. Shin and K. J. Kim, Performance enhancement of pneumatic vibration isolation tables in low frequency range by time delay control, Journal of Sound and Vibration, vol. 321, pp. 537-553, 2009.
1509.07314.extracted_bib
{Ref:27 A. Kuperman and Q. C. Zhong, Robust control of uncertain nonlinear systems based on an uncertainty and disturbance estimator, International Journal of Robust and Nonlinear Control, vol. 21, pp. 79-92, 2011.
1509.07314.extracted_bib
{Ref:28 S. E. Talole and S. B. Phadke, Robust input-output linearization using uncertainty and disturbance estimator, International Journal of Control, vol. 82(10), pp. 1794-1803, 2009.
1509.07314.extracted_bib
{Ref:29 P. V. Suryawanshi, P. D. Shengde, and S. B. Phadke, Robust sliding mode control for a class of nonlinear systems using inertial delay control, Nonlinear Dynamics, vol. 78(3), pp. 1921-1932, 2014.
1509.07314.extracted_bib
{Ref:30 J. Reger and J. Jouffroy, On algebraic time-derivative estimation and deadbeat state reconstruction, Joint 47th IEEE Conference on Decision and Control and 28th Chinese Control Conference, pp. 1740-1745, 2009.
1509.07314.extracted_bib
{Ref:31 K. Gu, V. Khartionov and J. Chen, Stability of time-delay systems, Birkh\"{auser, Boston (2003).
1509.07314.extracted_bib
{Ref:32 A. Nasiri, S. K. Nguang and A. Swain, Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainty, Journal of The Franklin Institute, vol. 351, pp. 2048-2061, 2014.
1509.07314.extracted_bib
{Ref:33 S. Roy, S. Nandy, R. Ray and S. N. Shome, Robust path tracking control of nonholonomic wheeled mobile robot, International Journal of Control, Automation and Systems, vol. 13(4), pp. 897-905, 2015.
1509.07314.extracted_bib
{Ref:34 J. Kim, H. Joe, S-C Yu, J. S. Lee and M. Kim, Time delay controller design for position control of autonomous underwater vehicle under disturbances, IEEE Transactions on Industrial Electronics, DOI 10.1109/TIE.2015.2477270, 2015.