# Adaptive-Robust Control of a Class of Nonlinear Systems with Unknown Input
Delay

Spandan Roy and Indra Narayan Kar S. Roy and I. N. Kar are with the Department
of Electrical Engineering, Indian Institute of Technology-Delhi, New Delhi,
India e-mail<EMAIL_ADDRESS>ink@ee.iitd.ac.in).

###### Abstract

In this paper, the tracking control problem of a class of uncertain Euler-
Lagrange systems subjected to unknown input delay and bounded disturbances is
addressed. To this front, a novel delay dependent control law, referred as
Adaptive Robust Outer Loop Control (AROLC) is proposed. Compared to the
conventional predictor based approaches, the proposed controller is capable of
negotiating any input delay, within a stipulated range, without knowing the
delay or its variation. The maximum allowable input delay is computed through
Razumikhin-type stability analysis. AROLC also provides robustness against the
disturbances due to input delay, parametric variations and unmodelled dynamics
through switching control law. The novel adaptive law allows the switching
gain to modify itself online in accordance with the tracking error without any
prerequisite of the uncertainties. The uncertain system, employing AROLC, is
shown to be Uniformly Ultimately Bounded (UUB). As a proof of concept,
experimentation is carried out on a nonholonomic wheeled mobile robot with
various time varying as well as fixed input delay, and better tracking
accuracy of the proposed controller is noted compared to predictor based
methodology.

###### Index Terms:

Adaptive-robust control, Euler-Lagrange systems, Input delay, Razumikhin
theorem, Wheeled mobile robot.

##  I Introduction

###  I-A Background and Motivation

Time delays could be inherent in the system or may be a result of some sources
such as transportation and transmission lags, communication delays etc. A few
applications where delays are evident include, chemical and biological
processes, teleoperated robotic systems, rolling mills, delays due to sensor
response and input delay in mobile robots, engine cycle delays in internal
combustion engine, control over network etc. [1]-[3]. Left unattended, such
delay reduces performance of the system and may create potential instability
to the system. The survey articles [1]-[5] document the recent advances,
challenges and controllers developed to tackle the unwanted delay for linear
systems [1]-[4] and nonlinear systems [5]. The control strategies reported in
literature can broadly be classified in two categories, one for linear systems
([6]-[22]) and other for nonlinear systems ([23]-[33]).

Controllers developed for linear systems are first discussed. Control law,
that predicts the input delay to the system is one the most studied as well as
used method and has a root to the Smith predictor based approach [6]. However,
performance of this method depends on modelling accuracy of the system and can
be applied to stable systems only. Artstein model reduction [7], finite
spectrum assignment [8], on the contrary can be applied to unstable and
multivariable plants. The predictor based approach transforms the delayed
system into a delay free system using the control input history. The
prediction scheme of [8] was extended in [9] when state delay is also present
along with the input delay. A discrete time predictor based controller is
reported in [10] for uncertain linear systems. A reduction method to design a
delayed feedback based controller for uncertain linear systems with time
varying input delay is proposed by [11]. However, this approach requires exact
knowledge of the input delay to compensate its effect. [12] designed a robust
controller for time varying input delay combining Lyapunov-Krasvoskii
functional and a neutral transformation, assuming bound of the first
derivative of delay is available. A predictor based state feedback controller
is proposed in [13]-[14] for stabilization of uncertain discrete time systems
via LMI technique. In the predictor based approach of [15], an equivalent
transformation is used on the characteristic equation to convert the predictor
feedback control into a stabilization problem of neutral time delay system for
constant time delay. Predictor feedback control law, based on small gain
analysis, for linear time invariant systems with unknown time varying delay is
designed in [16].

Adaptive control techniques are addressed in [17]-[19] for linear systems.
[17] designed an adaptive control law for feedforward linear systems subjected
to both state and input delay, where the unknown time delay is predicted
adaptively using a projection function without considering plant
uncertainties. [18] considered linearly parametrized uncertainty in plants
with known bounds. Adaptive backstepping method is used in [19] for robust
stabilization problem of linear non-minimum phase systems subjected to unknown
dynamics and input delay. Sliding mode control strategies are considered in
[20]-[22]. However, [20] only considered constant time delay and uncertainties
in plant parameters were ignored. [22] used integral sliding mode control for
uncertain linear systems under time varying delay at the input but it requires
known bounds of the uncertainties as well as its first time derivative and can
only tackle slowly varying input delay. On the other hand, [22] used adaptive
sliding mode control for linear input delayed systems where bound on the
uncertainties are estimated online. However, estimation of the uncertainties
becomes a monotonically increasing function of states for non zero values of
states. This may cause very high switching gain and consequent chattering.

Nonlinear systems with input delay pose a greater challenge since linear
boundedness of plant model cannot be incorporated in the stability proof [5].
Compared to linear systems, results for nonlinear systems with time delay at
the input is limited. [23] provided a delay dependent Lyapunov-Razumikhin
based approach on controlling nonlinear systems. Delay independent Razumikhin
approach for nonlinear time delay systems with triangular structure is
proposed in [24]. The system was shown to be Uniformly Ultimately Bounded
(UUB) assuming the delay disturbances are bounded. However, no control input
was designed to negotiate the time delayed uncertainties and the delay
independent solution is conservative. [25] used Smith predictor based globally
linearizing controller for known nonlinear system. [26] proposed a
backstepping based predictor approach for delay compensation of forward
complete and strict-feedforward nonlinear systems utilizing ODE-PDE cascade
transformation. [27]-[28] developed controllers for rigid and flexible link
manipulators by linearizing the system. Here, the controllers developed for
delay free system is shown to be stable under some delay dependent conditions.
However, the above mentioned control laws require exact knowledge of the
models which is difficult for nonlinear systems. This issue was addressed in
[29] where a predictor based approach is proposed for constant input delay
based on Lyapunov-Krasvoskii method. In this case, uncertainty in the inertia
matrix was considered. The same approach was later extended for time varying
input delay in [30] and also for state delay [31]. A filtered tracking error
based control law is used in [32] to tackle the effect of known constant input
time delay and used fuzzy logic systems of first type to approximate the
unknown disturbances with predefined uncertainty bound. A stabilizing
controller for spacecraft with unknown but constant input delay is proposed in
[33] with approximated uncertainty bound. However, in practice delay may not
always be constant and defining prior uncertainty bounds for the uncertainties
is not always feasible for nonlinear uncertain systems due to unmodelled
dynamics and unknown disturbances.

###  I-B Contribution

In this paper, a delay dependent control law, referred as Adaptive Robust
Outer Loop Control (AROLC), is proposed for tracking control of a class of
Euler-Lagrange systems subjected to unknown time-varying input delay and
bounded uncertainty. The two major advantages of the proposed control scheme
is stated below,

(i) The proposed control law does not require any explicit knowledge of the
delay or its time derivatives to design the control law unlike the ones
reported in [29]-[32]. It can negotiate any input delay within a maximum delay
range, obtained from the delay dependent stability criterion of the closed
loop system employing the Razumikhin-type stability approach [35].

(ii) As opposed to the controllers developed in [29]-[33], the proposed
adaptive-robust law does not necessitate the knowledge of any predefined
uncertainty bound or its modelling. The switching gain adapts itself with an
error function defined by the user. The proposed controller possesses the
flexibility that a user can specify any error function based on which the
switching gain would be modified while maintaining the same stability
criterion. The closed loop stability analysis of the system dynamics is
carried out in the sense of UUB.

(iii) Experimental validation of the proposed control law is also carried out
on PIONEER-3 wheeled mobile robot (WMR) and compared the results with the
predictive control approaches reported in [29]-[30].

###  I-C Notations

The following notations are used throughout the paper: any quantity $\rho$
delayed by an amount $h$ as $\rho(t-h)$, would be denoted as $\rho_{h}$;
$\lambda_{min}(\cdot)$ and $||\cdot||$ represent minimum eigen value and
Euclidean norm of the argument respectively; $I$ represents identity matrix.

###  I-D Organization

The article is organized as follows: the detailed problem formulation is first
carried out in Section 2. This is followed by the proposed adaptive-robust
control methodology and its detail analysis. Section 3 presents the
experimental results of the proposed controller and its comparison with [30].
Section 4 concludes the entire work.

##  II Controller Design

###  II-A Problem Formulation and Objective Definition

In general, an Euler-Lagrange system with second order dynamics, with input
delay, can be written as,

$M(q)\ddot{q}+N(q,\dot{q})=\tau(t-h(t)),$ (1)

where, $q(t)\in\Re^{n}$ is the system state, $\tau(t)\in\Re^{n}$ is the
control input, $M(q)\in\Re^{n\times n}$ is the mass/inertia matrix and
$N(q,\dot{q})\in\Re^{n}$ 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 $N$. $h(t)$ is a time varying delay
at the input $\tau$. The delay may be result of computation delay,
communication delay between the controller and the actuator etc. Let,
$q^{d}(t)$ be the desired trajectory to be tracked and
$e_{1}(t)=q^{d}(t)-q(t)$ is the tracking error. Before introducing the
proposed control law, brief overview of the control structure of predictive
control law, reported in [30] is provided below:

####  II-A1 Predictive Controller ([29]-[30])

######  Assumption 1.

All the disturbances and their first time derivative is bounded by some known
constant.

######  Assumption 2.

Input delay $h(t)$ and $\ddot{h}(t)$ is bounded by a known constant and
$\dot{h}(t)<1$.

Let $\varrho$ be a measurable filtered tracking error and defined as

$\displaystyle\varrho$ $\displaystyle=\dot{e}_{1}+\kappa e_{1}-\vartheta
e_{z}$ (2) $\displaystyle e_{z}$
$\displaystyle=\int_{t-h(t)}^{t}\tau(\theta)d\theta$ (3)

where, $\kappa$ is a positive constant and $\vartheta$ is a positive definite
matrix. Taking time derivative of (2) and multiplying it by $M$ and then using
(1) and (3) we get,

$M\dot{\varrho}=M\ddot{q}^{d}+N-M\eta(\tau-\tau_{h}+\dot{h}\tau_{h})-\tau-\dot{h}\tau_{h}+\kappa
M\dot{e}_{1},$ (4)

where, $\eta=\vartheta-M^{-1}$. The control input is selected as
$\tau=k_{b}\varrho$ to track the desired trajectory provided Assumptions 1 and
2 holds. Selection methods of $k_{b}$ and $\kappa$ are detailed in [29]-[30].

######  Remark 1.

In practice, proper bound estimation of unmodelled dynamics and unknown
disturbances is almost impossible. Moreover, according to Assumption 2,
knowledge of the bound of $\dot{h}$ and $\ddot{h}$ is required and the
controller can only tackle slowly varying time delay [30]. However, this is
difficult to attain since variation in time delay $h$ can be arbitrary in
practical scenario. Hence, it is not always possible to maintain Assumptions 1
and 2.

So, the aim of this paper is to design a control law which simultaneously
fulfils the following two objectives:

O1: To maintain system stability of the input delayed system (1), within a
stipulated maximum time delay, while it follows a predefined desired path.
This objective is to be met irrespective and without any knowledge of the
nature of the variation delay $h(t)$.

O2: To provide robustness to system (1) against bounded but unknown
uncertainties and disturbances. without any knowledge of the uncertainties and
its bounds.

###  II-B Adaptive Robust Outer Loop Controller

Towards achieving the outlined objectives, a novel controller, named Adaptive
Robust Outer Loop Controller (AROLC) is proposed. The structure of the the
proposed control law is selected to be,

$\tau=\hat{M}u+\hat{N},$ (5)

where, $u$ is the auxiliary control input; $\hat{M}$ and $\hat{N}$ are the
nominal values of $M$ and $N$ respectively. $u$ is defined in the following
way,

$u=\hat{u}+\Delta u,$ (6)

where, $\hat{u}$ and $\Delta u$ are nominal and switching control input
respectively and they are evaluated as,

$\hat{u}=\ddot{q}^{d}(t)+K_{2}\dot{e_{1}}(t)+K_{1}e_{1}(t),$ (7) $\Delta
u=\begin{cases}\alpha\hat{c}(e,t)\frac{s}{\parallel s\parallel}&\quad\text{if
}\parallel s\parallel\geq\epsilon,\\\
\alpha\hat{c}(e,t)\frac{s}{\epsilon}&\quad\text{if }\parallel
s\parallel<\epsilon,\\\ \end{cases}$ (8)

where, $K_{1}$ and $K_{2}$ are two positive definite matrices with appropriate
dimensions, $s=B^{T}Pe$,
$e=\begin{bmatrix}e_{1}^{T}&\dot{e}_{1}^{T}\end{bmatrix}^{T}$, $\hat{c}$ is
the switching gain responsible to tackle the uncertainties, $\alpha>0$ is a
scalar adaptive gain and $\epsilon>0$ represents a small scalar. To accomplish
the second objective, the following novel adaptive control law for evaluation
of $\hat{c}$ is proposed which eliminates the requirement of any predefined
bound of uncertainties:

$\dot{\hat{c}}=\begin{cases}\quad\parallel
s\parallel&\quad\hat{c}>\gamma,s^{T}\dot{s}>0\\\ -\parallel
s\parallel&\quad\hat{c}>\gamma,s^{T}\dot{s}\leq 0\\\
\qquad\gamma&\quad\hat{c}\leq\gamma,\\\ \end{cases}$ (9)

where, $\gamma>0$ is a small scalar to keep $\hat{c}$ always positive.
According to the adaptive law (9), $\hat{c}$ increases (resp. decreases)
whenever the error trajectories move away from (resp. move closer to) $s=0$.
Nevertheless, due to the delay $h$ the error dynamics of (1) employing (5) and
(6) is found to be,

$\ddot{e}_{1}=-K_{2}\dot{e}_{1h}-K_{1}e_{1h}+\sigma-\Delta u_{h},$ (10)

where,
$\sigma=(I-M^{-1}(q)\hat{M}(q_{h}))u_{h}+M^{-1}(q)(N(q,\dot{q})-\hat{N}(q_{h},\dot{q}_{h}))+\ddot{q}^{d}(t)-\ddot{q}^{d}_{h}$.
Further, (10) can be formulated in state space as,

$\dot{e}=A_{1}e+B_{1}e_{h}+B(-\Delta u_{h}+\sigma),$ (11)

where, $A_{1}=\begin{bmatrix}0&I\\\
0&0\end{bmatrix},B_{1}=\begin{bmatrix}0&0\\\ -K_{1}&-K_{2}\end{bmatrix}$.
Noting that, $e_{h}=e(t)-\int\limits_{-h}^{0}\dot{e}(t+\theta)d\theta$, where
the derivative inside the integral is with respect to $\theta$, the error
dynamics (11) is modified as,

$\dot{e}(t)=Ae(t)-B_{1}\int\limits_{-h}^{0}\dot{e}(t+\theta)d\theta+B(-\Delta
u_{h}+\sigma),$ (12)

where, $A=A_{1}+B_{1}$. The controller gains $K_{1}$ and $K_{2}$ are selected
in a way such that $A$ is Hurwitz which is always possible by noting the
structure of $A$. The stability analysis of the proposed controller is carried
out in the sense of UUB and is detailed subsequently.

####  II-B1 Stability Analysis of AROLC

######  Assumption 3.

All the states i.e $q,\dot{q}$ are available.

######  Assumption 4.

The uncertainties are bounded as $||\sigma||\leq c$, where $c$ is an unknown
scalar quantity. Knowledge of $c$ is not required to compute the proposed
control law. However, consideration of $c$ is necessary for stability
analysis.

######  Assumption 5.

Let, V(e) be a Lyapunov function candidate. Then following the Razumikhin-type
theorem [35], for some constant $r>1$, let the following inequality holds:

$V(e(\xi))<rV(e(t)),\qquad t-2h\leq\xi\leq t.$ (13)

Consider a Lyapunov function of the following form:

$V(e)=V_{1}(e)+V_{2}(e),$ (14)

where, $V_{1}(e)=\frac{1}{2}e^{T}Pe$, $V_{2}(e)=\frac{1}{2}(\hat{c}-c)^{2}$,
$P>0$ is the solution of the Lyapunov equation $A^{T}P+PA=-Q$ for some $Q>0$.

######  Theorem 1.

The maximum allowable delay time for system (1) employing the control input
(5)-(8) is found to be,

$h<\frac{\lambda_{min}(Q)}{||\beta
PB_{1}(A_{1}P^{-1}A_{1}^{T}+B_{1}P^{-1}B_{1}^{T}+P^{-1})B_{1}^{T}P+2(r/\beta)P)||},$
(15)

provided gains $K_{1}$, $K_{2}$ and scalar design parameter $r>1,\beta>0$ are
selected in a manner which satisfies

$\lambda_{min}(Q)>h||\beta
PB_{1}(A_{1}P^{-1}A_{1}^{T}+B_{1}P^{-1}B_{1}^{T}+P^{-1})B_{1}^{T}P+2(r/\beta)P)||\quad\forall
h.$ (16)

###### Proof.

: Using (12), the time derivative of $V_{1}(e)$ yields,

$\dot{V}_{1}=-\frac{1}{2}e^{T}Qe-e^{T}PB_{1}\int_{-h}^{0}\dot{e}(t+\theta)d\theta+s^{T}(-\Delta
u_{h}+\sigma)$ (17)

Again using (11),

$\displaystyle-e^{T}PB_{1}\int_{-h}^{0}\dot{e}(t+\theta)d\theta=-\int_{-h}^{0}e^{T}PB_{1}$
$\displaystyle[A_{1}e(t+\theta)+B_{1}e(t-h+\theta)$
$\displaystyle+B\sigma_{2}(t+\theta)]d\theta,$ (18)

where, $\sigma_{2}(t)=-\Delta u_{h}+\sigma(t)$. Applying (13) to (14), the
following relation is obtained,

$e^{T}(\xi)Pe(\xi)<re^{T}(t)Pe(t)+\varphi(\xi),$ (19)

where, $\varphi(\xi)=r(\hat{c}(t)-c)^{2}-(\hat{c}(\xi)-c)^{2}$. For any two
non zero vectors $z_{1}$ and $z_{2}$, there exists a constant $\beta>0$ and
matrix $D>0$ such that the following inequality holds,

$-2z_{1}^{T}z_{2}\leq\beta z_{1}^{T}D^{-1}z_{1}+(1/\beta)z_{2}^{T}Dz_{2}.$
(20)

Applying (20) to (18) and taking $D=P$ the following inequalities are
obtained,

$\displaystyle-2\int_{-h}^{0}e^{T}PB_{1}A_{1}\left[e(t+\theta)\right]d\theta$
$\displaystyle\leq\int_{-h}^{0}[\beta
e^{T}PB_{1}A_{1}P^{-1}A_{1}^{T}B_{1}^{T}Pe$
$\displaystyle\qquad\qquad\qquad+(1/\beta)e^{T}(t+\theta)Pe(t+\theta)]d\theta$
$\displaystyle\leq he^{T}\left[\beta
PB_{1}A_{1}P^{-1}A_{1}^{T}B_{1}^{T}P+(r/\beta)P)\right]e$
$\displaystyle\qquad\qquad\qquad+\int_{-h}^{0}(1/\beta)\varphi(t+\theta)d\theta$
(21)
$\displaystyle-2\int_{-h}^{0}e^{T}PB_{1}B_{1}\left[e(t-h+\theta)\right]d\theta$
$\displaystyle\leq\int_{-h}^{0}[\beta
e^{T}PB_{1}B_{1}P^{-1}B_{1}^{T}B_{1}^{T}Pe$
$\displaystyle\qquad\quad+\frac{1}{\beta}e^{T}(t-h+\theta)Pe(t-h+\theta)]d\theta$
$\displaystyle\leq he^{T}\left[\beta
PB_{1}B_{1}P^{-1}B_{1}^{T}B_{1}^{T}P+\frac{r}{\beta}P)\right]e$
$\displaystyle\qquad\qquad+\int_{-h}^{0}(1/\beta)\varphi(t-h+\theta)d\theta$
(22)
$\displaystyle-2\int_{-h}^{0}e^{T}PB_{1}\left[B\sigma_{2}(t+\theta)\right]d\theta$
$\displaystyle\leq\int_{-h}^{0}[\beta e^{T}PB_{1}P^{-1}B_{1}^{T}Pe$
$\displaystyle\qquad\qquad+\frac{1}{\beta}(B\sigma_{2}(t+\theta))^{T}PB\sigma_{2}(t+\theta)]d\theta$
$\displaystyle\leq he^{T}\left[\beta PB_{1}P^{-1}B_{1}^{T}P\right]e$
$\displaystyle+\int_{-h}^{0}\frac{1}{\beta}(B\sigma_{2}(t+\theta))^{T}PB\sigma_{2}(t+\theta)d\theta$
(23)

Since $P>0$, we can write $P=\bar{P}^{T}\bar{P}$ for some $\bar{P}>0$. Then,
assuming the uncertainties to be square integrable within the delay, there
exists a scalar $\Gamma>0$ such that, the following inequality holds over the
stipulated time delay:

$\displaystyle\frac{1}{2\beta}\left\|\int_{-h}^{0}\left[\varphi(t+\theta)+\varphi(t-h+\theta)+(B\sigma_{2}(t+\theta))^{T}\bar{P}^{T}\bar{P}B\sigma_{2}(t+\theta)\right]d\theta\right\|\leq\Gamma.$
(24)

Substituting (1)-(24) into (17) yields,

$\dot{V}_{1}(e)\leq-\frac{1}{2}e^{T}\left[Q-hE\right]e+\Gamma+s^{T}(-\Delta
u+\sigma)+s^{T}\Theta,$ (25)

where, $E=\left(\beta
PB_{1}\left(A_{1}P^{-1}A_{1}^{T}+B_{1}P^{-1}B_{1}^{T}+P^{-1}\right)B_{1}^{T}P+2(r/\beta)P\right)$,
$\Theta=\Delta u(t)-\Delta u_{h}$. To achieve negativeness of the first term
of (25), the controller gains $K_{1},K_{2}$ and the scalar variables $r,\beta$
are to be selected in a manner such that $\lambda_{min}(Q)>h||E||\quad\forall
h$. Hence, the maximum allowable delay can be found from (25) as,

$h<\frac{\lambda_{min}(Q)}{||E||}.$ (26)

Since $E$ is governed by the controller gains, (15) provides the maximum input
delay that system (1) can sustain for some choice of $K_{1},K_{2},Q,r$ and
$\beta$. ∎

Exploring the various possible combinations of $\Delta u$ and $\hat{c}$ in (8)
and (9) respectively, six different cases have been identified. Stability
analysis of (1) employing AROLC is stated in Theorem 1. In each case, UUB is
established using a common Lyapunov function.

######  Theorem 2.

The system (1) employing the control input (5), (6), having nominal control
input (7) and switching control input (8) with adaptation law (9) is UUB.

###### Proof.

: Taking $\Psi=Q-hE>0$ and utilizing result of Theorem 1, (6)-(9) and (14),
each of the six cases is analysed as follows:

Case (i): $\hat{c}>\gamma,\quad||s||\geq\epsilon\quad\text{and}\quad
s^{T}\dot{s}>0$
Utilizing (9) and (25), the time derivative of (14) yields,

$\displaystyle\dot{V}(e)$ $\displaystyle\leq-\frac{1}{2}e^{T}\Psi
e+\Gamma+s^{T}(-\Delta u+\sigma)+s^{T}\Theta+(\hat{c}-c)||s||$
$\displaystyle=-\frac{1}{2}e^{T}\Psi
e+\Gamma+s^{T}(-\alpha\hat{c}\frac{s}{||s||}+\sigma)+s^{T}\Theta+(\hat{c}-c)||s||.$
(27)

Again,

$s^{T}(-\alpha\hat{c}\frac{s}{||s||}+\sigma)=-\alpha\hat{c}\frac{s^{T}s}{||s||}+s^{T}\sigma\leq(-\alpha\hat{c}+c)||s||$
(28)

Using (27) and (28) we have,

$\displaystyle\dot{V}(e)$
$\displaystyle\leq-\frac{1}{2}\lambda_{min}(\Psi)||e||^{2}-(\alpha-1)\hat{c}||s||+||\Theta||||s||+\Gamma$

So, for a choice of $\alpha>1$, $\dot{V}(e)<0$ would be established if
$\lambda_{min}(\Psi)||e||^{2}>2\Gamma+2||\Theta||||s||$. Again, we have
$||s||\leq||B^{T}P||||e||$. Thus (1) would be UUB with the ultimate bound,

$||e||=\mu_{1}+\sqrt{\frac{2\Gamma}{\lambda_{min}(\Psi)}+\mu_{1}^{2}}=\varpi_{1}.$
(29)

where, $\mu_{1}=\frac{||\Theta||||B^{T}P||}{\lambda_{min}(\Psi)}$. Let $\Xi$
denote the smallest level surface of $V$ containing the ball $B_{\varpi_{1}}$
with radius $\varpi_{1}$ centred at $e=0$. For initial time $t_{0}$, if
$e(t_{0})\in\Xi$ then the solution remains in $\Xi$. If $e(t_{0})\notin\Xi$
then $V$ decreases as long as $e(t)\notin\Xi$. The time required to reach
$\varpi_{1}$ is zero when $e(t_{0})\in\Xi$, otherwise, while
$e(t_{0})\notin\Xi$ the finite time $t_{r1}$ to reach $\varpi_{1}$ is given by
[34],

$t_{r1}-t_{0}\leq(\parallel e(t_{0})\parallel-\varpi_{1})/c_{0},$ (30)

where, $\dot{V}(t)\leq-c_{0}$ for some $c_{0}>0$.

Case (ii): $\hat{c}>\gamma,\quad||s||\geq\epsilon\quad\text{and}\quad
s^{T}\dot{s}<0$
Exploring (25), (9) and (28) the time derivative of (14) yields,

$\displaystyle\dot{V}(e)$ $\displaystyle\leq-\frac{1}{2}e^{T}\Psi
e+(c-\alpha\hat{c})||s||+s^{T}\Theta+\Gamma-(\hat{c}-c)||s||$
$\displaystyle\leq-\frac{1}{2}\lambda_{min}(\Psi)||e||^{2}+(2c-(\alpha+1)\hat{c}+||\Theta||)||s||+\Gamma$

So, $\dot{V}(e)<0$ would be achieved if
$\lambda_{min}(\Psi)||e||^{2}>2(2c-(\alpha+1)\hat{c}+||\Theta||)||s||+2\Gamma$.
Thus (1) would be UUB with the ultimate bound,

$||e||=\mu_{2}+\sqrt{\frac{2\Gamma}{\lambda_{min}(\Psi)}+\mu_{2}^{2}}=\varpi_{2}.$
(31)

where,
$\mu_{2}=\frac{(2c-(\alpha+1)\hat{c}+||\Theta||)||B^{T}P||}{\lambda_{min}(\Psi)}$.

Case (iii): $\hat{c}\leq\gamma,\quad||s||\geq\epsilon\quad\text{and any}\quad
s^{T}\dot{s}$
Since $\hat{c}\leq\gamma$ we have
$(\hat{c}-c)\gamma\leq\gamma^{2}-c\gamma\leq\gamma^{2}$. So, using (9) and
following similar procedure we have,

$\displaystyle\dot{V}(e)$ $\displaystyle\leq-\frac{1}{2}e^{T}\Psi
e+\Gamma+s^{T}(-\Delta u+\sigma)+s^{T}\Theta+(\hat{c}-c)\gamma$
$\displaystyle\leq-\frac{1}{2}e^{T}\Psi
e+\Gamma+s^{T}(-\alpha\hat{c}\frac{s}{||s||}+\sigma)+||s||||\Theta||+\gamma^{2}.$

Using (28) we have,

$\dot{V}(e)\leq-\frac{1}{2}\lambda_{min}(\Psi)||e||^{2}+(c-\alpha\hat{c}+||\Theta||)||s||+\Gamma+\gamma^{2}$

So, system would be UUB with the following ultimate bound,

$||e||=\mu_{3}+\sqrt{\frac{2(\Gamma+\gamma^{2})}{\lambda_{min}(\Psi)}+\mu_{3}^{2}}=\varpi_{3}.$
(32)

where, $\mu_{3}=\frac{(c-\alpha\hat{c}+||\Theta||)}{\lambda_{min}(\Psi)}$.

Case (iv): $\hat{c}>\gamma,\quad||s||<\epsilon\quad\text{and}\quad
s^{T}\dot{s}>0$
The term $s^{T}\sigma$ turns out to be:

$s^{T}\sigma\leq\|s\|\|\sigma\|\leq c\|s\|=c\frac{s^{T}s}{\|s\|}$ (33)

Using (9), (25) and (33), the time derivative of (14) gives,

$\displaystyle\dot{V}(e)$ $\displaystyle\leq-\frac{1}{2}e^{T}\Psi
e+\Gamma+s^{T}(-\Delta u+\sigma)+s^{T}\Theta+(\hat{c}-c)||s||$
$\displaystyle\leq-\frac{1}{2}\lambda_{min}(\Psi)||e||^{2}+\Gamma+s^{T}(-\alpha\hat{c}\frac{s}{\epsilon}+c\frac{s}{||s||})+||s||||\Theta||+(\hat{c}-c)||s||.$
(34)

The combination of third, fourth and fifth term of (34) takes the maximum
value of $(\epsilon(\hat{c}+||\Theta||)^{2})/4\alpha\hat{c}$ for
$||s||=(\epsilon(\hat{c}+||\Theta||))/2\alpha\hat{c}$. Thus $\dot{V}(e)<0$
would be achieved if
$\lambda_{min}(\Psi)||e||^{2}>2\Gamma+(\epsilon(\hat{c}+||\Theta||)^{2})/2\alpha\hat{c}$.
So, the system is UUB with the ultimate bound is calculated to be,

$||e||=\sqrt{\frac{4\alpha\Gamma\hat{c}+\epsilon(\hat{c}+||\Theta||)^{2}}{2\alpha\hat{c}\lambda_{min}(\Psi)}}=\varpi_{4}$
(35)

Case (v): $\hat{c}>\gamma,\quad||s||<\epsilon\quad\text{and}\quad
s^{T}\dot{s}<0$
Using (9), (25) and (33),

$\displaystyle\dot{V}(e)$ $\displaystyle\leq-\frac{1}{2}e^{T}\Psi
e+\Gamma+s^{T}(-\Delta u+\sigma)+s^{T}\Theta-(\hat{c}-c)||s||$
$\displaystyle\leq-\frac{1}{2}\lambda_{min}(\Psi)||e||^{2}+\Gamma+s^{T}(-\alpha\hat{c}\frac{s}{\epsilon}+c\frac{s}{||s||})+||s||||\Theta||-(\hat{c}-c)||s||$
(36)

The combination of third, fourth and fifth term of (36) takes the maximum
value of $(\epsilon(2c-\hat{c}+||\Theta||)^{2})/4\alpha\hat{c}$ for
$||s||=\left(\epsilon(2c-\hat{c}+||\Theta||)\right)/2\alpha\hat{c}$. Thus
$\dot{V}(e)<0$ would be achieved if
$\lambda_{min}(\Psi)||e||^{2}>2\Gamma+(\epsilon(2c-\hat{c}+||\Theta||)^{2})/2\alpha\hat{c}$.
So, the system is UUB and the ultimate bound is calculated to be,

$||e||=\sqrt{\frac{4\alpha\Gamma\hat{c}+\epsilon(2c-\hat{c}+||\Theta||)^{2}}{2\alpha\hat{c}\lambda_{min}(\Psi)}}=\varpi_{5}$
(37)

Case (vi): $\hat{c}\leq\gamma,\quad||s||<\epsilon\quad\text{and any}\quad
s^{T}\dot{s}$

Following similar procedure, time derivative of (14) for Case (vi) yields,

$\displaystyle\dot{V}(e)$ $\displaystyle\leq-\frac{1}{2}e^{T}\Psi
e+\Gamma+s^{T}(-\Delta u+\sigma)+s^{T}\Theta+\gamma^{2}$
$\displaystyle\leq-\frac{1}{2}\lambda_{min}(\Psi)||e||^{2}+\Gamma+s^{T}(-\alpha\hat{c}\frac{s}{\epsilon}+c\frac{s}{||s||})+||s||||\Theta||+\gamma^{2}.$
(38)

The combination of third and fourth term of (38) takes the maximum value of
$(\epsilon(c+||\Theta||)^{2})/4\alpha\hat{c}$ for
$\|s\|=\left(\epsilon\left(c+||\Theta||\right)\right)/2\alpha\hat{c}$. Thus
$\dot{V}(e)<0$ would be achieved if
$\lambda_{min}(\Psi)||e||^{2}>2\Gamma+(\epsilon(c+||\Theta||)^{2})/2\alpha\hat{c}+2\gamma^{2}$.
So, the system is UUB with the ultimate bound is defined as,

$||e||=\sqrt{\frac{4\alpha\hat{c}(\Gamma+\gamma^{2})+\epsilon(c+||\Theta||)^{2}}{2\alpha\hat{c}\lambda_{min}(\Psi)}}=\varpi_{6}$
(39)

The finite reaching time to the error bounds $\varpi_{i},i=2,\cdots,6$ can be
computed similarly from (30).

∎

######  Remark 2.

The performance of AROLC can be characterized by the various error bounds
($\varpi_{i},i=1,\cdots,6$) under various conditions. The scalars
$\varpi_{i}^{\prime}s$ are function of $\alpha$ and delay $h$. It can be seen
that as $\alpha$ increases and $h$ drops, better tracking accuracy can be
achieved. However, too large a $\alpha$ may result in high control input.
Also, one may choose different values of $\alpha$ for $s^{T}\dot{s}\leq 0$ and
$s^{T}\dot{s}>0$. Again, it is to be noticed that, instead of $s^{T}\dot{s}$
user can select any suitable error function while keeping the same stability
notion.

Comparison with the existing results:

  * •

It can be observed from (15) that, proposed stability approach is independent
of the rate of change in $h$ and thus can negotiate any arbitrarily time-
varying delay within the stipulated maximum bound. On the contrary, [30]-[31]
can negotiate only slowly varying time delay and also bound of $\ddot{h}$ is
required.

  * •

Computation of switching gain $\hat{c}$ of AROLC using (9), unlike [29]-[33],
does not require predefined bound of the uncertainties and helps to to attain
the tracking objective. Accomplishment of this objective in turn reduces the
tedious modelling effort of complex nonlinear systems. To illustrate the fact
with an example, $\hat{N}$ does not need to include friction, slip, skid etc.
for wheeled mobile robot and these terms can be treated as uncertainties.

##  III Application: Nonholonomic WMR

Nonholonomic WMR 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 extremely
difficult to model and in many cases they are not considered while modelling.
The dynamic equation of a WMR after solving the Lagrange multiplier can be
written as follows [35],

$\bar{M}(q)\ddot{q}+\bar{V}(q,\dot{q})=Gu,$ (40)

where,

$\displaystyle\bar{M}=$
$\displaystyle\begin{bmatrix}m&0&Ksin\phi&k_{1}&k_{2}\\\
0&m&-Kcos\phi&k_{3}&k_{4}\\\ Ksin\phi&-Kcos\phi&\bar{I}&-k_{5}&k_{5}\\\
k_{1}&k_{3}&-k_{5}&I_{w}&0\\\
k_{2}&k_{4}&k_{5}&0&I_{w}\end{bmatrix},u=\begin{bmatrix}u_{r}\\\
u_{l}\end{bmatrix}$ $\displaystyle G=$ $\displaystyle\begin{bmatrix}0&0\\\
0&0\\\ 0&0\\\ 1&0\\\
0&1\end{bmatrix},\bar{V}=\begin{bmatrix}md\dot{\phi}^{2}cos\phi+m\bar{r}^{2}sin\phi(\dot{\theta}_{\bar{r}}^{2}-\dot{\theta}_{l}^{2})/2b\\\
md\dot{\phi}^{2}cos\phi-m\bar{r}^{2}sin\phi(\dot{\theta}_{\bar{r}}^{2}-\dot{\theta}_{l}^{2})/2b\\\
K\bar{r}^{2}(\dot{\theta}_{\bar{r}}^{2}-\dot{\theta}_{l}^{2})/2b\\\
-K\bar{r}\dot{\phi}^{2}/2\\\ -K\bar{r}\dot{\phi}^{2}/2\\\ \end{bmatrix}$
$\displaystyle k_{1}=$ $\displaystyle
sin\phi(md\bar{r}-K\bar{r})/b-m\bar{r}cos\phi/2,$ $\displaystyle k_{2}=$
$\displaystyle sin\phi(K\bar{r}-md\bar{r})/b-m\bar{r}cos\phi/2,$
$\displaystyle k_{3}=$ $\displaystyle
cos\phi(K\bar{r}-md\bar{r})/b-m\bar{r}sin\phi/2,$ $\displaystyle k_{4}=$
$\displaystyle
cos\phi(md\bar{r}-K\bar{r})/b-m\bar{r}sin\phi/2,k_{5}=\bar{r}(I-Kd)/b,$

Here $q\in\Re^{5}=\\{x_{c},y_{c},\phi,\theta_{r},\theta_{l}\\}$ 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 [36].

###  III-A Experimental Results and Comparison

Figure 1: Control architecture for TARC while employing on PIONEER-3.

AROLC is employed in "PIONEER-3" WMR while the robot is directed to track the
following circular path:

$\displaystyle x_{c}^{d}$ $\displaystyle=1.25sin(0.35t)+.1,$ $\displaystyle
y_{c}^{d}$ $\displaystyle=1.25cos(0.35t)+1.35,$ $\displaystyle\phi^{d}$
$\displaystyle=0.25t,\theta_{r}^{d}=3t,\theta_{l}^{d}=2t.$

The control architecture of the proposed AROLC is depicted in Fig. 1. Tracking
performance of AROLC is compared with the Predictive Controller (PCON)reported
in [30]. For a choice of $K_{1}=K_{2}=Q=I,\beta=1,r=1.1$, the maximum
allowable delay is found to be $h_{m}=125ms$. Other parameters to design AROLC
are defined as $\alpha=2,\epsilon=0.1,\gamma=0.001$. The following time-
varying input delay was induced into the system for both the controllers:

S1: $h(t)=20+80$ $abs(sin(t))$ ms

S2: $h(t)=5+120$ $abs(sin(0.1t))$ ms.

The input delay is envisaged by halting the programme for $h(t)$ amount of
time in the VC++ programming environment between the control input computation
and feeding it to the system. Again, to create a dynamic payload variation, a
further $3.5kg$ payload is added and kept for $5sec$ and then removed. This
process is carried out for the entire duration of experimentation. A time gap
$5sec$ 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 dynamic variation in center of mass and
inertia.

The trajectory tracking performance between AROLC and PCON is depicted in Fig.
2 for the condition S1. The corresponding $x_{c}$ and $y_{c}$ position error
comparison is illustrated through Fig. 3 and Fig. 4 respectively. All the
error plots in this paper are in absolute value. In the situation S2, Fig. 5
and Fig. 6 depict the $x_{c}$ and $y_{c}$ position error comparison
respectively. Due to the robustness property against the unmodelled dynamics,
AROLC provides better accuracy over PCON through its switching control logic.
This can easily be comprehended from these error plots. To infer the
performance of the individual controllers, absolute average error in $x_{c}$
(AE-$x_{c}$) and $y_{c}$ position (AE-$y_{c}$) is provided in Table I for S1
and S2. The percentage error is calculated with respect to the diameter of the
circular path. The tabulated data further establishes the superior performance
of AROLC over PCON.

Figure 2: Trajectory tracking performance comparison for input delay S1. Figure 3:  $x_{c}$ position error comparison for input delay S1. Figure 4: $y_{c}$ position error comparison for input delay S1. Table I: $x_{c}$ and $y_{c}$ Position Error (mm) Comparison for S1 and S2 Delay (ms) | Controller | AE-$x_{c}$ |  $\%$ AE-$x_{c}$. | AE-$y_{c}$ |  $\%$ AE-$y_{c}$
---|---|---|---|---|---
S1 | PCON ([30]) | 58.30 | 2.33 | 53.09 | 2.12
AROLC (proposed) | 23.33 | 0.93 | 22.92 | 0.92
S2 | PCON ([30]) | 78.46 | 3.14 | 81.37 | 3.25
AROLC (proposed) | 36.42 | 1.46 | 39.66 | 1.59
Figure 5: $x_{c}$ position error comparison for input delay S2. Figure 6:
$y_{c}$ position error comparison for input delay S2.

The performance of AROLC is also tested for the following two cases when the
input delay is fixed and compared with the predictive controller reported in
[29] (denoted here as PCONf):

S3: $h(t)=60$ ms

S4: $h(t)=120$ ms

The comparative performance of AROLC and PCONf is provided in Table II for S3
and S4. Superior performance of the proposed controller over PCONf is clearly
evident from the tabulated data. However, tracking accuracy of AROLC degrades
as $h$ increases and this is commensurate with the fact that the error bands
for AROLC increases with high input delay. The total variation (TV), a measure
of smoothness of input, is denoted as [37],

$\text{TV}=\sum_{i=1}^{n-1}|u_{r}(i+1)-u_{r}(i))|+|u_{l}(i+1)-u_{l}(i))|$ (41)

where, $n$ is the length of the samples in $u_{r}$ and $u_{l}$ accumulated
during experimentation. High value of TV denotes excessive usage of control
input ([37]). TV for the three controllers under various cases are provided in
Table III. It can be noticed that fixed time delay resulted in more control
input requirement than the time varying delay for all the controllers.
However, AROLC consumed the least control input for all the cases, which
further augment its superior performance compared to PCON and PCONf. Some
boxes in Table III are left blank since those particular controllers were not
used for the corresponding conditions.

Table II: $x_{c}$ and $y_{c}$ Position Error (mm) Comparison for S3 and S4 Delay (ms) | Controller | AE-$x_{c}$ |  $\%$ AE-$x_{c}$ | AE-$y_{c}$ |  $\%$ AE-$y_{c}$
---|---|---|---|---|---
S3 | PCONf ([29]) | 99.63 | 3.99 | 85.79 | 3.43
AROLC (proposed) | 45.66 | 1.82 | 51.84 | 2.07
S4 | PCONf ([29]) | 136.91 | 5.48 | 102.19 | 4.79
AROLC (proposed) | 82.29 | 3.29 | 69.05 | 2.76
Table III: TV of Controllers for S1, S2, S3 S4 Controller | S1 |  S2. | S3 | S4
---|---|---|---|---
PCONf ([29]) | - | - | 8.74 | 10.73
PCON ([30]) | 5.45 | 6.39 | - | -
AROLC (proposed) | 0.42 | 0.55 | 2.23 | 3.36

##  IV Conclusion

In this paper, the tracking problem for a class of uncertain nonlinear systems
in presence of unknown input delay is solved. The proposed delay dependent
control law does not require any knowledge of the input delay to compute the
control law and it is also insensitive towards the variation of the delay. The
maximum sustainable delay is determined by utilizing the Razumikhin approach.
AROLC, through its novel adaptive law, is further able to provide robustness
against the parametric and unmodelled uncertainties without any prior
knowledge of their bounds. Experimental results using a nonholonomic wheeled
mobile robot, validates the superiority of the proposed controller compared to
the predictor based control approach.

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