# Fault-Tolerant Formation Tracking of Heterogeneous Multi-Agent Systems with
Time-Varying Actuator Faults and Its Application to Task-Space Cooperative
Tracking of Manipulators

Zhi Feng and Guoqiang Hu  This work was supported by Singapore Ministry of
Education Academic Research Fund Tier 1 RG180/17 (2017-T1-002-158). Z. Feng
and G. Hu are with the School of Electrical and Electronic Engineering,
Nanyang Technological University, Singapore 639798 (E-mail: zhifeng,
gqhu@ntu.edu.sg). Part of results was presented in the IEEE Conference on
Decision and Control, Korea, 2020.

###### Abstract

This paper addresses a formation tracking problem for nonlinear multi-agent
systems with time-varying actuator faults, in which only a subset of agents
has access to the leader’s information over the directed leader-follower
network with a spanning tree. Both the amplitudes and signs of control
coefficients induced by actuator faults are unknown and time-varying. The
aforementioned setting improves the practical relevance of the problem to be
investigated, and meanwhile, it poses technical challenges to distributed
controller design and asymptotic stability analysis. By introducing a
distributed estimation and control framework, a novel distributed control law
based on a Nussbaum gain technique is developed to achieve robust fault-
tolerant formation tracking for heterogeneous nonlinear multi-agent systems
with time-varying actuator faults. It can be proved that the asymptotic
convergence is guaranteed. In addition, the proposed approach is applied to
task-space cooperative tracking of networked manipulators irrespective of the
uncertain kinematics, dynamics, and actuator faults. Numerical simulation
results are presented to verify the effectiveness of the proposed designs.

###### Index Terms:

Heterogeneous multi-agent system, Fault-tolerance, Formation tracking,
Directed graph, Task-space manipulation.

##  I Introduction

In coordination of multi-agent systems, formation tracking has attracted
considerable attention during the past decades due to its broad potential
applications such as cooperative localization [1], surveillance [2], target
enclosing [3], source seeking and mapping, etc. Although several classic
formation control strategies (e.g., the leader-follower, virtual structure,
and behavior-based ones in [4]) are proposed to drive the states of all the
agents to form a desired configuration, consensus-based formation control
frameworks are presented via the local neighboring relative interaction for
agents with single-/double-integrator [5], high-order linear [6, 7], and
nonholonomic [8] dynamics. Since many uncertainties and nonlinearities are
unavoidable in the practical physical system, the approaches presented in
aforementioned works cannot be directly applied to solve formation tracking
issues for unknown nonlinear multi-agent systems. On the other hand, the
increasingly equipped actuators, sensors, and other components of multi-agent
systems are inevitably subject to various faults that induce interruptions and
lead to performance degradation or even instability. Recently, there have been
some works reported on fault-tolerant consensus. Float actuator faults were
considered in [10, 11, 9, 12], while loss of effectiveness faults were studied
in [13, 14, 15]. However, uniformly ultimately bounded (UUB) results are
obtained for various faults. Overall, how to design algorithms to have fault-
tolerant formation tracking of nonlinear multi-agent systems is challenging,
and to the best of our knowledge, is still open.

One common feature in aforementioned works assumes that the amplitudes and
signs of control coefficients are known a priori for each agent. However, for
nonlinear multi-agent systems subject to time-varying actuator faults, the
control coefficient of each agent usually becomes unknown and time-varying
[34]. Moreover, both their amplitudes and signs might not be available a
priori in many applications [16]. For example, position of each robot
influences its own controlling effect in the robot formation tracking problem.
To handle this issue, there are mainly three systematic approaches: the
switching detection method [17], the nonlinear proportional-integral (PI)
scheme [18, 19], and the Nussbaum gain technique [20, 21, 22, 23, 24]. In
particular, the switching mechanism is proposed in [17] to deal with the
unknown control sign, where the nonsmooth design may bring undesired
chattering behaviors. The authors in [18, 19] develops nonlinear PI schemes to
solve this problem for a networked of single- and double-integrator agents
over strongly connected graphs, where the design relies on the assumption that
the control coefficients have to be constant. A class of Nussbaum-type
functions are proposed in [20] to achieve consensus for first- and second-
order systems with constant coefficients and identical signs. The piecewise
Nussbaum function is developed in [21] to allow nonidentical but partially
unknown control signs. Moreover, the design is extended in [22] to study
output-constrained consensus with partially unknown control signs. Distributed
cooperative output regulations are studied in [23, 24, 25], where internal
model schemes are used to allow each agent to have constant coefficients under
connected and undirected graphs.

This work focuses on the fault-tolerant formation tracking research of
heterogeneous nonlinear multi-agent systems with time-varying actuator faults
over the directed leader-follower network. In particular, the distributed
estimation and control framework is developed to solve this fault-tolerant
formation tracking problem. The main contributions of this paper are
summarized as follows. (a) Asymptotic fault-tolerant formation tracking is
achieved under the proposed distributed estimation and control algorithm over
a directed graph with a spanning tree. To the best of our knowledge, this
paper is the first attempt to solve this issue.

  * •

In contrast to existing formation/consensus works in [5, 6, 7, 8, 10, 11, 9,
12, 13, 14, 15] with known and constant control coefficients, the presence of
time-varying actuator faults in nonlinear multi-agent systems makes this
control coefficient time-varying with completely unknown nonidentical signs.

  * •

Although works in [20, 21, 22, 23, 25, 24] also adopt the Nussbaum gain
technique to achieve consensus, the designs require constant control
coefficients in [20, 23, 24, 25] or dynamic coefficients but with partially
known signs in [21, 22].

  * •

Unlike aforementioned leaderless consensus works, we consider a directed
leader-follower network to have fault-tolerant formation tracking
asymptotically. The proposed distributed algorithm does not need the upper
bound of faults as required in [10, 11, 9, 12, 13, 14, 15] where the UUB
consensus is obtained.

(b) Unlike existing works over undirected or strongly connected graphs, the
developed framework enables agents to communicate over the directed graph with
a spanning tree, which is nontrivial. Due to the time-varying control
coefficients with completely unknown nonidentical signs in the directed
leader-follower network, existing designs cannot be directly applied.

(c) Although the preliminary version of this paper was presented in [40], this
paper unifies and generalizes our conference contributions in [40]. In
particular, the proposed distributed algorithm is applied to solve the fault-
tolerant task-space coordinated tracking problem for manipulators. As compared
to most related works in [28, 30, 26, 27, 29, 31], the proposed algorithm can
bring several advantages. Specifically, the proposed schemes in [30, 27, 26,
28, 29, 31] require the global task reference for each robot to accomplish
cooperative tasks. On the contrary, the distributed framework is developed to
guarantee task-space coordination. The directed graph with a spanning tree is
more general. Moreover, the proposed design does not require the upper bounds
of unknown faults/uncertainties.

This paper is organized as follows. Section II gives the problem formulation.
The distributed estimation and control framework is developed in Section III
to provide the main results. Section IV presents its application to task-space
coordinated tracking control for networked manipulators. Simulation results
are presented in Section V followed by the conclusions provided in Section VI.

##  II Preliminaries and Problem Formulation

###  II-A Notation

Let $0_{N}$ ($1_{N}$) be the $N\times 1$ vector with all the zeros (ones). Let
col$(x_{1},...,x_{N})$ and diag$\\{a_{1},...,a_{N}\\}$ be the column vector
with entries $x_{i}$ and the diagonal matrix with entries $a_{i}$,
$i=1,\cdots,N$, respectively. $\otimes$ and $\left\|\cdot\right\|$ represent
the Kronecker product and the Euclidean norm, respectively. For
$x_{i}\in\mathbb{R}^{n}$, define
$\text{sig}^{\theta}(x_{i})=\text{col}(\text{sig}^{\theta}(x_{i1}),\cdots,\text{sig}^{\theta}(x_{in}))$,
where $\text{sig}^{\theta}(x_{ik})=\text{sgn}(x_{ik})|x_{ik}|^{\theta}$,
$k=1,\cdots,n$, $\text{sgn}(x_{ik})$ is a signum function, and $0<\theta<1$.
For a real symmetric matrix $M$, $M>0$ means that it is positive definite.
Moreover, $\lambda_{\min}(M)$ and $\lambda_{\max}(M)$ are its minimum and
maximum eigenvalues, respectively.

###  II-B Graph Theory

Let $\mathcal{G}$ $=$ $\left\\{\mathcal{V},\mathcal{E}\right\\}$ represent a
digraph, where the set of vertices is defined as $\mathcal{V}$ $\in$
$\left\\{1,...,N\right\\}$, and the set of edges is $\mathcal{E}$ $\subseteq$
$\mathcal{V\times V}$.
$\mathcal{N}_{i}=\left\\{j\in\mathcal{V}|(j,i)\in\mathcal{E}\right\\}$ denotes
the neighborhood set of vertex $i$. For a directed graph $\mathcal{G}$,
$(i,j)\in\mathcal{E}$ means that the information of node $i$ is accessible to
node $j$, but not conversely. $A$ $=$ $\left[a_{ij}\right]$ is the adjacency
matrix, where $a_{ij}>0$ if $(j,i)\in\mathcal{E}$, else $a_{ij}=0$. A matrix
$\mathcal{L}$ $\triangleq$ $D-A$ is called the Laplacian matrix, where
$D=\left[d_{ii}\right]$ is a diagonal matrix with
$d_{ii}=\sum\nolimits_{j=1}^{N}a_{ij}$. Let
$\mathcal{\bar{G}}=(\mathcal{\bar{V}},\mathcal{\bar{E}})$ be a directed graph
of a leader-follower network, where
$\bar{\mathcal{E}}\subseteq\bar{\mathcal{V}}\times\bar{\mathcal{V}}$,
$\mathcal{\bar{V}}=\\{0,\cdots,N\\}$, and the node $0$ is associated with the
leader. Clearly, $\mathcal{G}$ is a subgraph of $\mathcal{\bar{G}}$, where
$\mathcal{E}$ is obtained from $\mathcal{\bar{E}}$ by removing all the edges
between the node $0$ and the nodes in $\bar{\mathcal{V}}$. Define the
Laplacian matrix of $\mathcal{\bar{G}}$ as
$\mathcal{\bar{L}}=[0,0^{T}_{N};-\mathcal{B}1_{N},H]$ where $\mathcal{B}$ is a
diagonal matrix with its $i$-th diagonal element being $a_{i0}$, (similarly,
$a_{i0}>0$, if $(0,i)\in\mathcal{\bar{E}}$, and $a_{i0}=0$, otherwise), and
$H=[h_{ij}]\triangleq\mathcal{L}+\mathcal{B}$ is an information exchange
matrix.

###  II-C Problem Formulation

Consider a class of heterogeneous nonlinear multi-agent systems consisting of
$N$ followers labeled by agents $1,2,\cdots,N$ and one leader labeled by agent
$0$. The dynamics of follower $i$ are

$\left\\{\begin{array}[]{l}\dot{x}_{i,k}=x_{i,k+1},\ k=1,2,\cdots,m-1,\\\
\dot{x}_{i,m}=f^{T}_{i,m}(x_{i})\theta_{i}+g_{i,m}(x_{i})u_{ai}+d_{i,m}(x_{i},t),\\\
\ \ \ y_{i}=x_{i,1},\ i=1,2,\cdots,N,\end{array}\right.$ (1)

where $x_{i}=\text{col}(x_{i,1},\cdots,x_{i,m})\in\mathbb{R}^{nm}$ is the
state vector with $x_{i,k}\in\mathbb{R}^{n},k=1,\cdots,m$, $m$ is the system
order, $n$ is the system dimension, $u_{ai}\in\mathbb{R}^{n}$ denotes a
control input with actuator faults, $y_{i}\in\mathbb{R}^{n}$ is the system
output, $\theta_{i}\in\mathbb{R}^{r}$ is the unknown constant parameter,
$f_{i,m}:\mathbb{R}^{nm}\rightarrow\mathbb{R}^{r\times n}$ is the known
nonlinear function, $g_{i,m}:\mathbb{R}^{nm}\rightarrow\mathbb{R}$ is the
unknown coefficient, and $d_{i,m}:\mathbb{R}^{nm}\rightarrow\mathbb{R}^{n}$
are uncertainties/disturbances that can be upper bounded by certain unknown
constants. The sign of $g_{i,m}$ is unknown.

The leader evolves with the following dynamics

$\left\\{\begin{array}[]{l}\dot{x}_{0,k}=x_{0,k+1},\ k=1,2,\cdots,m-1,\\\
\dot{x}_{0,m}=o_{0,m}(x_{0},u_{0}),\ y_{0}=x_{0,1},\end{array}\right.$ (2)

where $x_{0}=\text{col}(x_{0,1},\cdots,x_{0,m})\in\mathbb{R}^{nm}$ is the
state vector with $x_{0,k}\in\mathbb{R}^{n},k=1,2,\cdots,m$,
$u_{0}\in\mathbb{R}^{n}$ is the control input of the leader,
$y_{0}\in\mathbb{R}^{n}$ is the leader’s output, and
$o_{0,m}:\mathbb{R}^{nm}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is an
unknown and nonlinear input function.

In contrast to many existing works in distributed coordination based on
healthy actuation of multi-agent systems, actuators with undetectable faults
are considered. When actuation faults occur, there exists the discrepancy
between the actual control input $u_{ai}$ and the designed control input
$u_{i}$ of the $i$-th actuator. Thus, two types of actuator faults that may
take place, are modeled as

$u_{ai}=\phi_{i}(t)u_{i}+\psi_{i}(t),\ i=1,2,\cdots,N,$ (3)

where $u_{i}\in\mathbb{R}^{n}$, and $\phi_{i}(t)\in\mathbb{R}$,
$\psi_{i}(t)\in\mathbb{R}^{n}$ denote the actuation loss of effectiveness
fault and the float fault, respectively.

###### Remark 1

The multi-agent model in (1) involves the heterogeneous nonlinear dynamics and
intrinsic unknown parameters. The system in (2) represents a class of non-
autonomous leaders with an unknown and nonlinear input function. The fault
formulation in (3) describes a more generalized form that captures more
failure processes than those in [10, 11, 9, 12, 13, 14, 15], where either
float faults or loss of effectiveness actuation faults are considered. It
follows from (1) and (3) that the control coefficients are thus time-varying
with completely unknown nonidentical signs.

As aforementioned, the main technical challenges of this paper lie in handling
the time-varying unknown control coefficient with nonidentical signs for
nonlinear multi-agent systems (1)-(2) under time-varying actuator faults (3)
over the directed graph.

The problem of this paper is stated as follows.

###### Problem 1

Given the nonlinear multi-agent system composed of (1)-(3) and a directed
graph $\bar{\mathcal{G}}$, design a distributed control law $u_{i}$ for an
auxiliary state $\hat{x}_{i}$ and two smooth functions $h_{i1},h_{i2}$,

$u_{i}=h_{i1}(x_{i},\hat{x}_{i}),\
\dot{\hat{x}}_{i}=h_{i2}(\hat{x}_{i},\hat{x}_{j}),i\in\mathcal{V},j\in\bar{\mathcal{N}}_{i},$
(4)

such that for each agent $i$, it can track the nonlinear leader while
maintaining a prescribed formation pattern in the sense that

$\underset{t\rightarrow\infty}{\text{lim}}(y_{i}(t)-y_{0}(t))=\varDelta_{i}\
\text{and}\
\underset{t\rightarrow\infty}{\text{lim}}(x_{i,k}(t)-x_{0,k}(t))=0_{n},$ (5)

where $k=2,\cdots,m$ and $\varDelta_{i}\in\mathbb{R}^{n}$ is a desired
formation offset.

To solve Problem 1, the following assumptions and some useful lemmas are
introduced to facilitate the development of distributed algorithm and Lyapunov
stability analysis.

###### Assumption 1

The sign of $g_{i,m}\neq 0$ in (1) is unknown, but there exist two continuous
positive functions $g^{-}_{i,m}(x_{i})$, $g^{+}_{i,m}(x_{i})$ so that
$g^{-}_{i,m}(x_{i})\leq|g_{i,m}|\leq g^{+}_{i,m}(x_{i})$.

###### Assumption 2

The faults $\phi_{i}(t)$ in (3) satisfy: $0<\phi_{i}(t)\leq 1$ and
$\psi_{i}(t)$ is bounded by certain unknown constants.

###### Assumption 3

The leader’s input $o_{0,m}(x_{0},u_{0})$ in (2) and its time derivative are
bounded by certain unknown constants. Only $x_{0,k}$, $k=1,\cdots,m$ are
available to a subset of followers.

###### Assumption 4

The directed graph $\mathcal{\bar{G}}$ contains a spanning tree with the
leader $0$ being the root.

###### Remark 2

Assumption 1 implies that the sign of coefficients is completely unknown, and
allowed to be different for each agent, which is a much weaker assumption
compared to existing related works in [20, 21, 22, 23, 25, 24] considering
constant/time-varying coefficients with known or partially known signs.
Assumption 2 is adopted for the robust design and is conventional in a single
agent system with actuator faults. Assumption 3 is widely used in existing
works for a nonlinear leader system. Assumption 4 is a standard assumption for
consensus works in the existing literature.

###### Lemma 1

[10] Let $\xi_{1},\xi_{2},\cdots,\xi_{N}\geq 0$. Then, we can have
$(\sum_{i=1}^{N}\xi_{i})^{p}\leq\sum_{i=1}^{N}\xi^{p}_{i}\leq
N^{1-p}(\sum_{i=1}^{N}\xi_{i})^{p}$ for $0<p\leq 1$, and
$N^{1-p}(\sum_{i=1}^{N}\xi_{i})^{p}\leq\sum_{i=1}^{N}\xi^{p}_{i}\leq(\sum_{i=1}^{N}\xi_{i})^{p}$
for $p>1$.

###### Lemma 2

[33] Under Assumption 4, $H$ is positive definite, and there exists a positive
diagonal matrix $\Pi=\text{diag}\\{\pi_{1},\pi_{2},\cdots,$ $\pi_{N}\\}$ such
that $\Xi=(\Pi H+H^{T}\Pi)/2$ is symmetric and positive definite, where
$\pi=\text{col}(\pi_{1},\pi_{2},\cdots,\pi_{N})=(H^{T})^{-1}1_{N}$.

###### Lemma 3

[36] For $\xi\in\mathbb{R}$ and $\gamma>0$, the following inequality holds:
$0\leq|\xi|-\xi^{2}/\sqrt{\xi^{2}+\gamma^{2}}\leq\gamma$.

###### Lemma 4

[35] Consider a dynamic system $\dot{x}=f(x,t)$, $x\in\mathbb{R}^{n}$ with
$f(0,t)=0_{n}$. Suppose that there exists a positive definite
$\mathcal{C}^{1}$ Lyapunov function $V(x,t)$ defined on a neighborhood of the
origin ($\mathcal{D}\in\mathbb{R}^{n}$), and there are constants $a>0$,
$b\in(0,1)$, and an open neighborhood $\mathcal{U}\subseteq\mathcal{D}$ so
that $\dot{V}(x,t)+aV^{b}(x,t)\leq 0,\ x\in\mathcal{U}\setminus{0}$, then the
origin of this system is finite-time stable, and the settling time is
described by $T\leq V^{1-b}(x(0),t)/(a(1-b))$.

###### Lemma 5

(Barbalat’s Lemma, [39]): Let $f(t):\mathbb{R}\rightarrow\mathbb{R}$ be a
uniformly continuous function for $t\geq 0$. If
$\lim_{t\rightarrow\infty}\int_{0}^{t}f(\omega)d\omega$ exists and is finite,
then $\lim_{t\rightarrow\infty}f(t)=0$.

##  III Main Result

In this section, we first present a distributed nonlinear estimator to
cooperatively estimate the states of the nonlinear leader system for each
agent so that its output tracks the reference trajectory in a finite time. As
mentioned in Section I, this estimator design is significant as it can not
only handle the case when only a subset of followers has access to the
leader’s states, but also provide a distributed solution to adopt Nussbaum
gain technique for each agent having unknown control coefficients with
nonidentical signs. Each agent can use its associated estimator’s output as a
local reference. Then, this issue can be transformed into a simultaneous
tracking problem. As a result, we propose a distributed adaptive controller to
achieve fault-tolerant formation tracking for a nonlinear leader-follower
agent system subject to time-varying actuator faults and completely unknown
nonidentical control signs.

###  III-A Distributed Nonlinear Leader Estimator Design

Distributed Nonlinear Estimator: for the nonlinear leader, motivated by the
design in our preliminary work [35], a finite-time distributed estimator is
designed for $i\in\mathcal{V}$, $k=1,\cdots,m-1$,

$\displaystyle\dot{\hat{x}}_{i,k}$
$\displaystyle=\hat{x}_{i,k+1}+\kappa_{ki}\text{sig}^{\gamma}(\sum_{j=0}^{N}a_{ij}(\hat{x}_{j,k}-\hat{x}_{i,k})),\hat{x}_{0,k}=x_{0,k},$
(6a) $\displaystyle\dot{\hat{x}}_{i,m}$
$\displaystyle=\eta_{i}+\kappa_{mi}\text{sig}^{\beta}(\sum_{j=0}^{N}a_{ij}(\hat{x}_{j,m}-\hat{x}_{i,m})),\hat{x}_{0,m}=x_{0,m},$
(6b) $\displaystyle\dot{\eta}_{i}$ $\displaystyle=\kappa_{\eta
i}[\text{sig}^{\alpha}(e^{\eta}_{i})+\text{sgn}(e^{\eta}_{i})],e^{\eta}_{i}=\sum_{j=1}^{N}a_{ij}(\eta_{j}-\eta_{i})+e^{\varrho}_{i},$
(6c)

where $\kappa_{ki}$, $\kappa_{mi}$, $\kappa_{\eta i}\in\mathbb{R}$ are
positive gains, $\alpha$, $\beta$, $\gamma$ $\in(0.5,1)$, and $\hat{x}_{i,k}$,
$k=1,2,\cdots,m$, and $\eta_{i}$ are the state estimates of the leader’s
states $x_{0,k}$, and $\dot{x}_{0,m}$, respectively, for any
$\hat{x}_{i,k}(0)$, $\eta_{i}(0)\in\mathbb{R}^{n}$. In (6c),
$e^{\varrho}_{i}=a_{i0}(\varrho_{i}-\eta_{i})$ where $\varrho_{i}$, an
estimate of unavailable $\dot{x}_{0,m}$, is generated by the following
estimator that for $\kappa_{\xi i}$, $\kappa_{\varrho i}>0$,

$\displaystyle\dot{\varrho}_{i}$ $\displaystyle=\kappa_{\varrho
i}a_{i0}\text{sgn}\left(x_{0,m}-\xi_{i}\right),\ \varrho_{i}(0)=0_{n},$ (7a)
$\displaystyle\dot{\xi}_{i}$ $\displaystyle=\varrho_{i}+\kappa_{\xi
i}a_{i0}\text{sig}^{\frac{1}{2}}\left(x_{0,m}-\xi_{i}\right),\
\xi_{i}(0)=0_{n}.$ (7b)

Define the local estimate errors $\tilde{x}_{i,k}$, $\tilde{\eta}_{i}$,
$\tilde{\xi}_{i}$, $\tilde{\varrho}_{i}$ as

$\displaystyle\tilde{x}_{i,k}$ $\displaystyle=\hat{x}_{i,k}-x_{0,k},\
\tilde{\eta}_{i}=\eta_{i}-\dot{x}_{0,m},\ k=1,\cdots,m,$ (8a)
$\displaystyle\tilde{\xi}_{i}$ $\displaystyle=\xi_{i}-x_{0,m},\
\tilde{\varrho}_{i}=\varrho_{i}-\dot{x}_{0,m},\ i=1,\cdots,N.$ (8b)

Then, based on (8), the estimated error dynamics can be derived under (6) and
(7) for $k=1,2,\cdots,m-1$,

$\displaystyle\dot{\tilde{x}}_{i,k}$
$\displaystyle=\tilde{x}_{i,k+1}+\kappa_{ki}\text{sig}^{\gamma}(\sum_{j=1}^{N}a_{ij}(\tilde{x}_{j,k}-\tilde{x}_{i,k})-a_{i0}\tilde{x}_{i,k}),$
(9a) $\displaystyle\dot{\tilde{x}}_{i,m}$
$\displaystyle=\tilde{\eta}_{i}+\kappa_{mi}\text{sig}^{\beta}(\sum_{j=1}^{N}a_{ij}(\tilde{x}_{j,m}-\tilde{x}_{i,m})-a_{i0}\tilde{x}_{i,m}),$
(9b) $\displaystyle\dot{\tilde{\eta}}_{i}$ $\displaystyle=\kappa_{\eta
i}\text{sig}^{\alpha}(\sum_{j=1}^{N}a_{ij}(\tilde{\eta}_{j}-\tilde{\eta}_{i})+a_{i0}(\tilde{\varrho}_{i}-\tilde{\eta}_{i}))-\ddot{x}_{0,m}$
$\displaystyle\ \ \ +\kappa_{\eta
i}\text{sgn}(\sum_{j=1}^{N}a_{ij}(\tilde{\eta}_{j}-\tilde{\eta}_{i})+a_{i0}(\tilde{\varrho}_{i}-\tilde{\eta}_{i})),i\in\mathcal{V},$
(9c) $\displaystyle\dot{\tilde{\xi}}_{i}$
$\displaystyle=\tilde{\varrho}_{i}-\kappa_{\xi
i}a_{i0}\text{sig}^{\frac{1}{2}}(\tilde{\xi}_{i}),\
\dot{\tilde{\varrho}}_{i}=-\kappa_{\varrho
i}a_{i0}\text{sgn}(\tilde{\xi}_{i})-\ddot{x}_{0,m}.$ (9d)

Next, we define two disagreement estimate errors as

$\displaystyle\bar{x}_{i,k}$
$\displaystyle=\textstyle\sum_{j=1}^{N}a_{ij}(\tilde{x}_{j,k}-\tilde{x}_{i,k})-a_{i0}\tilde{x}_{i,k},k=1,\cdots,m,$
(10a) $\displaystyle\bar{\eta}_{i}$
$\displaystyle=\textstyle\sum_{j=1}^{N}a_{ij}(\tilde{\eta}_{j}-\tilde{\eta}_{i})+a_{i0}(\tilde{\varrho}_{i}-\tilde{\eta}_{i}),i=1,\cdots,N.$
(10b)

Substituting (10) into (9) gives the following disagreement error dynamics for
each order $k=1,\cdots,m-1$,

$\displaystyle\dot{\bar{x}}_{i,k}$
$\displaystyle=\bar{x}_{i,k+1}+\kappa_{ki}e^{\gamma}_{i,s},\
\dot{\bar{x}}_{i,m}=\bar{\eta}_{i}-a_{i0}\tilde{\varrho}_{i}+\kappa_{mi}e^{\beta}_{i,m},$
(11a) $\displaystyle\dot{\bar{\eta}}_{i}$ $\displaystyle=\kappa_{\eta
i}e^{\alpha}_{i,\eta}+\kappa_{\eta i}e^{0}_{i,\eta}-\kappa_{\varrho
i}a^{2}_{i0}\text{sgn}(\tilde{\xi}_{i}),\ i=1,\cdots,N,$ (11b)
$\displaystyle\dot{\tilde{\xi}}_{i}$
$\displaystyle=\tilde{\varrho}_{i}-\kappa_{\xi
i}a_{i0}\text{sig}^{\frac{1}{2}}(\tilde{\xi}_{i}),\
\dot{\tilde{\varrho}}_{i}=-\kappa_{\varrho
i}a_{i0}\text{sgn}(\tilde{\xi}_{i})-\ddot{x}_{0,m},$ (11c)

where
$e^{*}_{i,s}=\sum_{j=1}^{N}a_{ij}(\text{sig}^{*}(\bar{x}_{j,s})-\text{sig}^{*}(\bar{x}_{i,s}))-a_{i0}\text{sig}^{*}(\bar{x}_{i,s})$
with $*=\gamma$ for $s=1,2,\cdots,m-1$, $*=\beta$ for $s=m$, and $*=\alpha$
for $s=\eta$, and
$e^{0}_{i,\eta}=\sum_{j=1}^{N}a_{ij}(\text{sgn}(\bar{\eta}_{j})-\text{sgn}(\bar{\eta}_{i}))-a_{i0}\text{sgn}(\bar{\eta}_{i})$.

Next, finite-time convergence of the proposed distributed nonlinear estimator
in (6) and (7) is presented as follows.

###### Theorem 1

Suppose that Assumptions 3 and 4 hold. Under the proposed distributed
nonlinear estimator (6) and (7), all state estimates are uniformly bounded,
and the finite-time estimation is achieved in the sense that $\lim_{t\to
T_{1}}a_{i0}\tilde{\varrho}_{i}=0_{n}$, $\lim_{t\to
T_{2}}\tilde{\eta}_{i}=0_{n}$, $\lim_{t\to T_{3}}\tilde{x}_{i,m}=0_{n}$, and
$\lim_{t\to T_{4}}\tilde{x}_{i,s}=0_{n}$, $s=1,\cdots,m-1$ for certain
$T_{i}>0,i=1,2,3,4$.

###### Proof:

The proof includes three steps:
Step (i): prove that $\lim_{t\to T_{1}}a_{i0}\tilde{\varrho}_{i}=0_{n}$. Two
cases are studied:

When $a_{i0}=0$, we obtain that $\lim_{t\to
T_{1}}a_{i0}\tilde{\varrho}_{i}=0_{n}$; When $a_{i0}=1$, motivated by [38], we
can define an error variable $\zeta_{ik}=\\\
\text{col}(\text{sig}^{\frac{1}{2}}(\tilde{\xi}_{ik}),\tilde{\varrho}_{ik})$,
where $\tilde{\xi}_{ik},\tilde{\varrho}_{ik}$ are the $k$th element of
$\tilde{\xi}_{i}$, $\tilde{\varrho}_{i}$, $k=1,\cdots,n$, respectively. Select
a Lyapunov function candidate as
$V_{\zeta}(t)=\sum_{i=1}^{N}\sum_{k=1}^{n}V_{\zeta
ik}(t)=\sum_{i=1}^{N}\sum_{k=1}^{n}\zeta^{T}_{ik}P_{ik}\zeta_{ik}$, where
$P_{ik}>0$ is a constant matrix. Let $x_{0k,m}$ be the $k$th element of
$x_{0,m}$. Then, the time derivative of $\zeta_{ik}$ is given by [38]

$\dot{\zeta}_{ik}=\frac{1}{2}|\tilde{\xi}_{ik}|^{-\frac{1}{2}}\left[\begin{array}[]{c}-\kappa_{\xi
i}\text{sig}^{\frac{1}{2}}(\tilde{\xi}_{ik})+\tilde{\varrho}_{ik}\\\
-2[\kappa_{\varrho
i}-\ddot{x}_{0k,m}\text{sgn}(\tilde{\xi}_{ik})]\text{sig}^{\frac{1}{2}}(\tilde{\xi}_{ik})\end{array}\right].$

Then, the time derivative of $V_{\zeta}$ along the system (11c) is given by
$\dot{V}_{\zeta}=\sum_{i=1}^{N}\sum_{k=1}^{n}|\tilde{\xi}_{ik}|^{-\frac{1}{2}}\zeta^{T}_{ik}\left(R^{T}_{ik}P_{ik}+P_{ik}R_{ik}\right)\zeta_{ik}$,
where $R_{ik}=\begin{bmatrix}-\frac{1}{2}\kappa_{\xi i}&\frac{1}{2}\\\
-[\kappa_{\varrho i}-\ddot{x}_{0k,m}\text{sign}(\tilde{\xi}_{ik})]&0\\\
\end{bmatrix}$ is Hurwitz if and only if $\kappa_{\xi i}$ $>0$ and
$\kappa_{\varrho i}>\sup_{t\in(0,\infty)}\\{\|\ddot{x}_{0,m}\|_{\infty}\\}+1$
based on Assumption 3. Since $R_{ik}$ is Hurwitz, for each matrix
$\Gamma_{ik}>0$, we can find a matrix $P_{ik}>0$ to the following algebraic
Lyapunov inequality:
$R^{T}_{ik}P_{ik}+P_{ik}R_{ik}\leq-\Gamma_{ik},i=1,\cdots,N,k=1,\cdots,n$,
such that for the constructed strict Lyapunov function $V_{\zeta}$ [38], we
have that
$\dot{V}_{\zeta}=-\sum_{i=1}^{N}\sum_{k=1}^{n}|\tilde{\xi}_{ik}|^{-\frac{1}{2}}\zeta^{T}_{ik}\Gamma_{ik}\zeta_{ik}\leq
0$. On the other hand, we have
$|\tilde{\xi}_{ik}|^{\frac{1}{2}}=|\text{sig}^{\frac{1}{2}}(\tilde{\xi}_{ik})|\leq|\zeta_{ik}|\leq\lambda^{-\frac{1}{2}}_{min}(P_{ik})V^{\frac{1}{2}}_{1ik}$.
Then, defining
$\epsilon_{0}=\text{min}_{i,k}\\{\lambda^{\frac{1}{2}}_{min}(P_{ik})\lambda_{min}(\Gamma_{ik})/\lambda_{max}({P_{ik}})\\}>0$
and using Lemma 1 yield the following inequality

$\dot{V}_{\zeta}\leq-\sum_{i=1}^{N}\sum_{k=1}^{n}\lambda^{\frac{1}{2}}_{min}(P_{ik})V^{-\frac{1}{2}}_{\zeta
ik}\frac{\lambda_{min}(\Gamma_{ik})}{\lambda_{max}(P_{ik})}V_{\zeta
ik}\leq-\epsilon_{0}V^{\frac{1}{2}}_{\zeta}.$ (12)

Based on Lemma 4, $V_{\zeta}\in\mathcal{L}_{\infty}$ and $\tilde{\varrho}_{i}$
converges to zero in a finite time, e.g., $\lim_{t\to
T_{1}}\tilde{\varrho}_{i}=0_{n}$ with
$T_{1}=\frac{2}{\epsilon_{0}}V^{\frac{1}{2}}_{\zeta}(0)$. Overall, we obtain
that $\lim_{t\to T_{1}}a_{i0}\tilde{\varrho}_{i}=0_{n}$ for $a_{i0}=0\
\text{or}\ 1$.

Step (ii): prove that $\lim_{t\to T_{2}}\tilde{\eta}_{i}=0_{n}$.

Consider a nonnegative Lyapunov function candidate
$V_{\eta}(t)=\sum_{i=1}^{N}\sum_{k=1}^{n}\pi_{i}(|\bar{\eta}_{ik}|+\frac{1}{\alpha+1}|\bar{\eta}_{ik}|^{\alpha+1})$,
where $\bar{\eta}_{ik}$ is the $k$th entry of $\bar{\eta}_{i}$ in (10b) and
$\pi_{i}$ is given in Lemma 2. The upper bound of $V_{\eta}(t)$ is derived by
Lemma 1 in two cases:

Case a) For $\sum_{i=1}^{N}\sum_{k=1}^{n}\pi_{i}|\bar{\eta}_{ik}|>1$,
$V_{\eta}\leq\sum_{i=1}^{N}\sum_{k=1}^{n}\pi_{i}|\bar{\eta}_{ik}\\\
|+\frac{1}{\alpha+1}\sum_{i=1}^{N}\sum_{k=1}^{n}\pi_{i}|\bar{\eta}_{ik}|^{\alpha+1}\leq\bar{\pi}\frac{\alpha+2}{\alpha+1}(\sum_{i=1}^{N}\sum_{k=1}^{n}\bar{\eta}^{2}_{ik})^{\frac{\alpha+1}{2}}$
with $\bar{\pi}$$=\max\\{\pi_{i}\\}$. Thus, we have

$\sum_{i=1}^{N}\sum_{k=1}^{n}|\bar{\eta}_{ik}|\geq\left(\frac{(\alpha+1)V_{\eta}}{\bar{\pi}(\alpha+2)}\right)^{\frac{1}{\alpha+1}}.$
(13)

Case b) For $\sum_{i=1}^{N}\sum_{k=1}^{n}\pi_{i}|\bar{\eta}_{ik}|\leq 1$,
$V_{\eta}\leq\bar{\pi}(1+1/(\alpha+1))$
$\sum_{i=1}^{N}\sum_{k=1}^{n}|\bar{\eta}_{ik}|$. Thus, we obtain

$\sum_{i=1}^{N}\sum_{k=1}^{n}|\bar{\eta}_{ik}|\geq\left(\frac{(\alpha+1)V_{\eta}}{\bar{\pi}(\alpha+2)}\right).$
(14)

Let $\bar{\eta}$ be the stacked column vector of $\bar{\eta}_{i}$ and
$\kappa_{\eta}=\text{diag}\\{\kappa_{\eta i}\\}$. Then, the time derivative of
$V_{\eta}(t)$ along (11b) is given by

$\displaystyle\dot{V}_{\eta}$
$\displaystyle=\sum_{i=1}^{N}\sum_{k=1}^{n}\pi_{i}(\text{sig}^{\alpha}(\bar{\eta}_{ik})+\text{sgn}(\bar{\eta}_{ik}))\times[-a^{2}_{i0}\kappa_{\varrho
i}\text{sgn}(\tilde{\xi}_{ik})$ $\displaystyle\ \ \ +\kappa_{\eta
i}(\sum_{j=1}^{N}a_{ij}(\text{sig}^{\alpha}(\bar{\eta}_{jk})-\text{sig}^{\alpha}(\bar{\eta}_{ik}))-a_{i0}\text{sig}^{\alpha}(\bar{\eta}_{ik}))$
$\displaystyle\ \ \ +\kappa_{\eta
i}(\sum_{j=1}^{N}a_{ij}(\text{sgn}(\bar{\eta}_{jk})-\text{sgn}(\bar{\eta}_{ik}))-a_{i0}\text{sgn}(\bar{\eta}_{ik}))]$
$\displaystyle=-[\text{sig}^{\alpha}(\bar{\eta})+\text{sgn}(\bar{\eta})]^{T}(\kappa_{\eta}\Pi
H\otimes I_{n})[\text{sig}^{\alpha}(\bar{\eta})+\text{sgn}(\bar{\eta})]$
$\displaystyle\ \ \ -\sum_{i=1}^{N}\sum_{k=1}^{n}\kappa_{\varrho
i}\pi_{i}a^{2}_{i0}(\text{sig}^{\alpha}(\bar{\eta}_{ik})+\text{sgn}(\bar{\eta}_{ik}))\text{sgn}(\tilde{\xi}_{ik})$
$\displaystyle\leq-\lambda_{min}(\Xi)\underline{\kappa}^{2}_{\eta}\sum_{i=1}^{N}\sum_{k=1}^{n}\left(1+2|\bar{\eta}_{ik}|^{\alpha}+|\bar{\eta}_{ik}|^{2\alpha}\right)+Nn\bar{\pi}\bar{k}_{\varrho}$
$\displaystyle\ \ \
+Nn\bar{\pi}\bar{\kappa}_{\varrho}\sum_{i=1}^{N}\sum_{k=1}^{n}|\bar{\eta}_{ik}|^{\alpha}\leq-\bar{\pi}\bar{\kappa}_{\varrho}\sum_{i=1}^{N}\sum_{k=1}^{n}|\bar{\eta}_{ik}|^{\alpha},$
(15)

where $\Xi=(\Pi H+H^{T}\Pi)/2$ is given in Lemma 2,
$\underline{\kappa}_{\eta}=\min\\{\kappa_{\eta i}\\}$
, $\bar{\kappa}_{\varrho}=\max\\{\kappa_{\varrho i}\\}$ and
$\underline{\kappa}^{2}_{\eta}\geq\bar{\pi}\bar{\kappa}_{\varrho}/\lambda_{min}(\Xi)$.

By Lemma 1,
$\dot{V}_{\eta}\leq-\bar{\pi}\bar{\kappa}_{\varrho}\left(\sum_{i=1}^{N}\sum_{k=1}^{n}|\bar{\eta}_{ik}|\right)^{\alpha}\leq-\epsilon_{1}V^{\frac{\alpha}{\alpha+1}}_{\eta}$
by (13) and
$\dot{V}_{\eta}\leq-\bar{\pi}\bar{\kappa}_{\varrho}\left(\sum_{i=1}^{N}\sum_{k=1}^{n}|\bar{\eta}_{ik}|\right)^{\alpha}\leq-\epsilon_{2}V^{\alpha}_{\eta}$
by (14), where
$\epsilon_{1}=\bar{\pi}\bar{\kappa}_{\varrho}(\frac{\alpha+1}{\bar{\pi}(\alpha+2)})^{\frac{\alpha}{\alpha+1}}$
and
$\epsilon_{2}=\bar{\pi}\bar{\kappa}_{\varrho}(\frac{\alpha+1}{\bar{\pi}(\alpha+2)})^{\alpha}$.

Since $0.5<\alpha<1$, exploiting Lemma 4 yields
$V_{\eta}\in\mathcal{L}_{\infty}$ and $\bar{\eta}_{i}$ converges to zero in a
finite time, e.g., $\lim_{t\to T_{\eta}}\bar{\eta}_{i}=0_{n}$ with $T_{\eta}=$
$\max\\{\frac{1+\alpha}{\epsilon_{1}}V^{\frac{1}{1+\alpha}}_{\eta}(0),\frac{1}{\epsilon_{2}(1-\alpha)}V^{1-\alpha}_{\eta}(0)\\}$.
As $\lim_{t\to T_{1}}a_{i0}\tilde{\varrho}_{i}=0_{n}$ from Step (i), we get
that for $t\geq T_{1}$, it follows from (10b) and Assumption 4 that
$\lim_{t\to T_{2}}\tilde{\eta}_{i}=0_{n}$, $T_{2}=T_{\eta}+T_{1}$.

Step (iii): prove that $\lim_{t\to T_{3}}\tilde{x}_{i,m}=0_{n}$.

Consider a nonnegative Lyapunov function candidate
$V_{m}(t)=\sum_{i=1}^{N}\sum_{k=1}^{n}\pi_{i}\frac{1}{\beta+1}|\bar{x}_{ik,m}|^{\beta+1}$,
where $\bar{x}_{ik,m}$ is the $k$th element of $\bar{x}_{i,m}$. By Lemma 1,
$(Nn)^{-\beta}\frac{\underline{\pi}}{\beta+1}(\sum_{i=1}^{N}\sum_{k=1}^{n}|\bar{x}_{ik,m}|)^{\beta+1}\leq
V_{m}(t)\leq\frac{\bar{\pi}}{\beta+1}(\sum_{i=1}^{N}\sum_{k=1}^{n}|\bar{x}_{ik,m}|)^{\beta+1}$.

Let $\bar{x}_{m}$ be a stacked vector of $\bar{x}_{i,m}$ and
$\underline{\kappa}_{m}=\min\\{\kappa_{mi}\\}$. The time derivative of
$V_{m}(t)$ along (11a) can be expressed as

$\displaystyle\dot{V}_{m}$
$\displaystyle=\sum_{i=1}^{N}\sum_{k=1}^{n}\pi_{i}\text{sig}^{\beta}(\bar{x}_{ik,m})(\bar{\eta}_{ik}-a_{i0}\tilde{\varrho}_{ik})+\sum_{i=1}^{N}\sum_{k=1}^{n}\pi_{i}\text{sig}^{\beta}(\bar{x}_{ik,m})$
$\displaystyle\
\times\kappa_{mi}(\sum_{j=1}^{N}a_{ij}(\text{sig}^{\beta}(\bar{x}_{jk,m})-\text{sig}^{\beta}(\bar{x}_{ik,m}))-a_{i0}\text{sig}^{\beta}(\bar{x}_{ik,m}))$
$\displaystyle\leq\sum_{i=1}^{N}\sum_{k=1}^{n}[-\lambda_{min}(\Xi)\underline{\kappa}_{m}|\bar{x}_{ik,m}|^{2\beta}+\bar{\pi}|\bar{x}_{ik,m}|^{{}^{\beta}}(|\bar{\eta}_{ik}|+|\tilde{\varrho}_{ik}|)].$

By using the fact that $\lim_{t\to T_{1}}a_{i0}\tilde{\varrho}_{i}=0_{n}$ and
$\lim_{t\to T_{2}}\bar{\eta}_{i}=0_{n}$ in Steps (i) and (ii), respectively,
we get for $t\geq T_{2}$,

$\dot{V}_{m}\leq-\lambda_{min}(\Xi)\underline{\kappa}_{m}(\sum_{i=1}^{N}\sum_{k=1}^{n}|\bar{x}_{ik,m}|)^{2\beta}\leq-\epsilon_{3}V^{\frac{2\beta}{\beta+1}}_{m},$
(16)

where
$\epsilon_{3}=\lambda_{min}(\Xi)\underline{\kappa}_{m}(\frac{\beta+1}{\bar{\pi}})^{\frac{2\beta}{\beta+1}}$.
Thus, by Lemma 4, $V_{m}\in\mathcal{L}_{\infty}$ and $\bar{x}_{i,m}$ converges
to zero in a finite time, e.g., $\lim_{t\to T_{3}}\bar{x}_{i,m}=0_{n}$ with
$T_{3}=\frac{\beta+1}{\epsilon_{3}(1-\beta)}V^{\frac{1-\beta}{\beta+1}}_{m}(T_{2})+T_{2}$.
Further, it follows from (10a) that $\lim_{t\to T_{3}}\tilde{x}_{i,m}=0_{n}$
by Assumption 4.

Next, we show that $\tilde{x}_{i,m}$ is uniformly bounded for $t<T_{3}$.

In Steps (i) and (ii), it has been shown that $\lim_{t\to
T_{1}}a_{i0}\tilde{\varrho}_{i}=0_{n}$ and $\lim_{t\to
T_{2}}\tilde{\eta}_{i}=0_{n}$. Besides, $\tilde{\varrho}_{i}$,
$\tilde{\eta}_{i}$ are uniformly bounded for $t<T_{2}$. Thus,
$|\tilde{\varrho}_{ik}|\leq\varrho_{0}$, $|\tilde{\eta}_{ik}|\leq\eta_{0}$ for
certain constants $\eta_{0}$, $\varrho_{0}$. Next, we verify that for
$t<T_{3}$ and certain constant $c_{0}>0$,

$\dot{V}_{m}\leq\sum_{i=1}^{N}\sum_{k=1}^{n}\bar{\pi}|\bar{x}_{ik,m}|^{{}^{\beta}}(|\bar{\eta}_{ik}|+|\tilde{\varrho}_{ik}|)\leq
c_{0}V^{\frac{\beta}{\beta+1}}_{m},$ (17)

which implies that $\tilde{x}_{i,m}$ cannot escape in a finite time.
Therefore, $\hat{x}_{i,m}$ is uniformly bounded at any finite time interval if
$x_{0,m}$ will not escape to infinity in a finite time by Assumption 3.

Step (iv): prove that $\lim_{t\to T_{4}}\tilde{x}_{i,s}=0_{n},s=1,\cdots,m-1$.

Select the nonnegative Lyapunov function candidate as
$V_{s}(t)=\sum_{i=1}^{N}\sum_{k=1}^{n}\pi_{i}\frac{1}{\gamma+1}|\bar{x}_{ik,s}|^{\gamma+1}$
where $\bar{x}_{ik,s}$ is the $k$th entry of $\bar{x}_{i,s}$. Similar to Step
(iii), we get $\dot{V}_{s}\leq-\epsilon_{4}V^{\frac{2\gamma}{\gamma+1}}_{s}$
and $\lim_{t\to T_{ts}}\bar{x}_{i,s}=0_{n}$ for certain positive constants
$\epsilon_{4}$ and $T_{ts}$, $s=1,\cdots,m-1$. Similarly, $\bar{x}_{i,s}$ is
uniformly bounded for $t<T_{4}$, $T_{4}=\max\\{T_{ts}\\}$. Thus, $\lim_{t\to
T_{4}}\tilde{x}_{i,s}=0_{n}$ by Assumption 4.

To conclude, estimates $\varrho_{i}$, $\eta_{i}$, $x_{i,s}$, $s=1,\cdots,m$
are bounded, and finite-time estimation is achieved, i.e., $\lim_{t\to
T_{1}}a_{i0}\tilde{\varrho}_{i}=0_{n}$, $\lim_{t\to
T_{2}}\tilde{\eta}_{i}=0_{n}$, $\lim_{t\to T_{3}}\tilde{x}_{i,m}=0_{n}$, and
$\lim_{t\to T_{4}}\tilde{x}_{i,s}=0_{n}$, $s=1,\cdots,m-1$ for certain
$T_{i}>0,i=1,2,3,4$. ∎

###### Remark 3

Theorem 1 implies that each agent can accurately estimate the state of the
nonlinear leader after a finite time under the proposed nonlinear distributed
estimator. Then, each agent can use its associated estimate as a local
reference in the distributed control based on a cascaded structure. As a
result, this distributed estimator will facilitate the following distributed
controller design so that Nussbaum gains are decoupled for each agent as only
the local reference instead of neighboring information is utilized. On the
other hand, with the adopted function $\text{sgn}(\cdot)$ in the proposed
estimator, the right-hand sides of $\dot{\tilde{\eta}}_{i}$ and
$\dot{\tilde{\varrho}}_{i}$ in (9c) and (9d) are discontinuous, and their
solutions can be investigated in terms of differential inclusions based on the
nonsmooth analysis. To avoid symbol redundancy, the differential inclusion is
not applied.

###  III-B Fault-Tolerant Formation Tracking Control

With the estimated information in Theorem 1, a new distributed adaptive
controller will be developed in this subsection to achieve formation tracking
of nonlinear multi-agent systems.

In particular, the leader’s states $x_{0,k},k=1,\cdots,m$ have been
reconstructed via distributed estimator (6) and (7) for each agent. The rest
is to propose a novel distributed adaptive control law to solve Problem 1.
Specifically, we can transform the coordinated trajectory tracking problem
into a simultaneous tracking problem. For $i\in\mathcal{V}$, we define the
following tracking errors as

$e_{i,1}=x_{i,1}-x_{0,1}-\varDelta_{i}=y_{i}-\hat{y}_{i}+\hat{y}_{i}-y_{0}-\varDelta_{i}=z_{i,1}+\tilde{y}_{i},$
(18)
$e_{i,k}=x_{i,k}-x_{0,k}=z_{i,k}+\tilde{x}_{i,k},z_{i,k}=x_{i,k}-\hat{x}_{i,k},k=2,\cdots,m,$

where $\hat{y}_{i}=\hat{x}_{i,1}$, $z_{i,1}=y_{i}-\hat{y}_{i}-\varDelta_{i}$,
$\tilde{y}_{i}=\hat{y}_{i}-y_{0}$, $\varDelta_{i}$ is the formation offest
defined in (5), and $\tilde{x}_{i,k},k=1,\cdots,m$ is the estimated error
defined in (8a). Then, based on the fact that
$\lim_{t\to\infty}\tilde{y}_{i}=0_{n}$ and
$\lim_{t\to\infty}\tilde{x}_{i,k}=0_{n}$ from Theorem 1, the coordinated
tracking control objective in (5) can be transformed into the simultaneous
tracking objective as

$\lim_{t\to\infty}z_{i,k}=0_{n},\ k=1,2,\cdots,m,\ i\in\mathcal{V}.$ (19)

From (9), the time derivative of $z_{i,k},\ k=1,\cdots,m-1$ is

$\displaystyle\dot{z}_{i,k}$
$\displaystyle=z_{i,k+1}-\kappa_{ki}\text{sig}^{\gamma}(\sum_{j=1}^{N}a_{ij}(\tilde{x}_{j,k}-\tilde{x}_{i,k})-a_{i0}\tilde{x}_{i,k}),$
(20a) $\displaystyle\dot{z}_{i,m}$
$\displaystyle=\dot{x}_{i,m}-\eta_{i}-\kappa_{mi}\text{sig}^{\beta}(\sum_{j=1}^{N}a_{ij}(\tilde{x}_{j,m}-\tilde{x}_{i,m})-a_{i0}\tilde{x}_{i,m})$
$\displaystyle=f^{T}_{i,m}(x_{i})\theta_{i}+g_{i,m}(x_{i})[\phi_{i}(t)u_{i}+\psi_{i}(t)]+d_{i,m}(x_{i},t)$
$\displaystyle\ \ \
-\eta_{i}-\kappa_{mi}\text{sig}^{\beta}(\sum_{j=1}^{N}a_{ij}(\tilde{x}_{j,m}-\tilde{x}_{i,m})-a_{i0}\tilde{x}_{i,m})$
$\displaystyle=f^{T}_{i,m}(x_{i})\theta_{i}+G_{i,m}(x_{i},t)u_{i}+D_{i,m}(x_{i},t)-\tilde{\eta}_{i}$
$\displaystyle\ \ \
-\kappa_{mi}\text{sig}^{\beta}(\sum_{j=1}^{N}a_{ij}(\tilde{x}_{j,m}-\tilde{x}_{i,m})-a_{i0}\tilde{x}_{i,m}),$
(20b)

where $G_{i,m}(x_{i},t)=g_{i,m}(x_{i})\phi_{i}(t)\in\mathbb{R}$ and
$D_{i,m}(x_{i},t)=d_{i,m}(\\\
x_{i},t)+g_{i,m}(x_{i},t)\psi_{i}(t)-\dot{x}_{0,m}\in\mathbb{R}^{n}$. It
follows from Assumptions 1 and 2 that the sign of the control coefficient
$G_{i,m}(x_{i},t)\neq 0$ is unknown and there exist positive functions
$G^{-}_{i,m}(x_{i})$, $G^{+}_{i,m}(x_{i})$ so that
$G^{-}_{i,m}(x_{i})\leq|G_{i,m}(x_{i},t)|\leq G^{+}_{i,m}(x_{i})$. Define the
positive constant $\varepsilon_{i}=\text{sup}_{t\geq
0}\\{\|D_{i,m}(x_{i},t)\|\\}$. Then, $\hat{\varepsilon}_{i}$, representing the
estimate of this unknown bound vector $\varepsilon_{i}1_{n}$, is to be
determined later, and
$\tilde{\varepsilon}_{i}=\varepsilon_{i}1_{n}-\hat{\varepsilon}_{i}$ is its
estimated error.

Next, we introduce a Nussbaum gain technique to deal with the time-varying
control coefficients with completely unknown signs via a smooth function
$N(k)$ that satisfies the following properties:
$\underset{k\rightarrow\infty}{\text{lim}}\text{sup}\frac{1}{k}\int^{k}_{0}N(s)ds=+\infty$,
$\underset{k\rightarrow\infty}{\text{lim}}\text{inf}\frac{1}{k}\int^{k}_{0}N(s)ds=-\infty$.
Throughout this paper, select
$N_{i}(\kappa_{i})=\text{exp}(\kappa^{2}_{i})\text{cos}((\pi/2)\kappa_{i})+1$
in [34] with nonidentical Nussbaum gains $\kappa_{i}$ for each agent
$i=1,\cdots,N$. Let $s_{ij}$, $i=1,\cdots,N$, $j=1,\cdots,m$ represent the
$j$-th element of certain vector $s_{i}\in\mathbb{R}^{n}$, and define
$\text{diag}\\{s_{i1},\cdots,s_{im}\\}$ as a diagonal matrix with its main
diagonal being $s_{ij}$. Similarly, define
$\text{diag}\\{\frac{s_{i1}}{\sqrt{s^{2}_{i1}+\delta^{2}_{i1}(t)}},\cdots,\frac{s_{im}}{\sqrt{s^{2}_{im}+\delta^{2}_{im}(t)}}\\}$
as a diagonal matrix with its main diagonal being
$\frac{s_{ij}}{\sqrt{s^{2}_{ij}+\delta^{2}_{ij}(t)}}$, where
$\delta_{ij}(t)>0$ is an integrable function so that
$\int_{0}^{\infty}\delta_{ij}(\omega)d\omega\leq\delta^{*}_{ij}$ for
$\delta^{*}_{ij}>0$.

Based on the selected Nussbaum function, we propose a novel distributed
adaptive controller as follows.

Fault-Tolerant Distributed Adaptive Controller: in light of the Nussbaum
function and local estimates $\hat{x}_{i,k}$ from (6) and (7), a fault-
tolerant adaptive controller is proposed as

$\displaystyle u_{i}$ $\displaystyle=N_{i}(\kappa_{i})\bar{u}_{i},\
N_{i}(\kappa_{i})=\text{exp}(\kappa^{2}_{i})\text{cos}((\pi/2)\kappa_{i})+1,$
(21a) $\displaystyle\bar{u}_{i}$
$\displaystyle=\bar{k}_{mi}\tilde{z}_{i,m}+\tilde{z}_{i,m-1}-\dot{z}^{*}_{i,m}+f^{T}_{i,m}\hat{\theta}_{i}+\text{diag}\\{\tilde{z}_{\delta
m}\\}\hat{\varepsilon}_{i},$ (21b) $\displaystyle\dot{\hat{\varepsilon}}_{i}$
$\displaystyle=\varGamma_{\varepsilon i}\text{diag}\\{\tilde{z}_{\delta
m}\\}\tilde{z}_{i,m},\tilde{z}_{\delta
m}=\tilde{z}_{ij,m}/\sqrt{\tilde{z}^{2}_{ij,m}+\delta^{2}_{ij}(t)},$ (21c)
$\displaystyle\dot{\kappa}_{i}$ $\displaystyle=k_{\kappa
i}\tilde{z}^{T}_{i,m}\bar{u}_{i},\ \dot{\hat{\theta}}_{i}=\varGamma_{\theta
i}f_{i,m}\tilde{z}_{i,m},\ i=1,2,\cdots,N,$ (21d)

where $N_{i}(\kappa_{i})$ is the agent $i$’ Nussbaum function,
$f_{i,m}=f_{i,m}(x_{i})$, $k_{\kappa i}$ is the positive constant,
$\varGamma_{\varepsilon i},\varGamma_{\theta i}>0$ are two constrant gain
matrices, $\hat{\theta}_{i}$ is the estimate of unknown vector $\theta_{i}$,
$\hat{\varepsilon}_{i}$ is an estimate of the unknown vector
$\varepsilon_{i}1_{n}$, and $\tilde{z}_{i,m},z^{*}_{i,m}$ are the subsequently
defined error terms to be determined.

Now, we are ready to present the robust fault-tolerant formation tracking
result for a class of nonlinear multi-agent systems under time-varying
actuator faults over the directed graph.

###### Theorem 2

Consider a class of nonlinear multi-agent systems consisting of $N$ followers
in (1) and the leader in (2) with time-varying actuator faults in (3). Under
Assumptions 1-4, the proposed controller in (21) combined with the distributed
estimator in (6) and (7) ensures that Problem 1 is solvable in the sense that:
$\underset{t\rightarrow\infty}{\text{lim}}(y_{i}(t)-y_{0}(t))=\varDelta_{i}\
\text{and}\
\underset{t\rightarrow\infty}{\text{lim}}(x_{i,k}(t)-x_{0,k}(t))=0_{n}$.

###### Proof:

Define an estimated error
$\hat{e}_{i,k}=\text{sig}^{\gamma}(\sum_{j=1}^{N}a_{ij}(\tilde{x}_{j,k}\\\
-\tilde{x}_{i,k})-a_{i0}\tilde{x}_{i,k}),k=1,\cdots,m$. Then, for clarity and
conciseness, a step by step procedure is presented as follows.

Step 1: Introduce two error variables

$\displaystyle\tilde{z}_{i,1}$
$\displaystyle=z_{i,1}=y_{i}-\hat{y}_{i}-\varDelta_{i},\ i=1,2,\cdots,N,$
(22a) $\displaystyle\tilde{z}_{i,2}$ $\displaystyle=z_{i,2}-z^{*}_{i,2},\
z^{*}_{i,2}=-\bar{k}_{1i}\tilde{z}_{i,1},$ (22b)

where $z^{*}_{i,2}$ is a virtual control input for the first subsystem in (20)
and $\bar{k}_{1i}>0$ is a constant gain.

Choose the Lyapunov candidate as
$V_{i,1}=\frac{1}{2}\tilde{z}^{T}_{i,1}\tilde{z}_{i,1}$. Then, the time
derivative of $V_{i,1}$ along the first subsystem in (20) is

$\dot{V}_{i,1}=\tilde{z}^{T}_{i,1}(z_{i,2}-\kappa_{1i}\hat{e}_{i,1})=\tilde{z}^{T}_{i,1}(\tilde{z}_{i,2}-\bar{k}_{1i}\tilde{z}_{i,1}-\kappa_{1i}\hat{e}_{i,1}).$
(23)

Step 2: In this step, we analyze the error $\tilde{z}_{i,2}$, i.e., $z_{i,2}$
tracks the virtual control input $z^{*}_{i,2}$. Definite two error variables

$\displaystyle\dot{\tilde{z}}_{i,2}$
$\displaystyle=z_{i,3}-\kappa_{2i}\hat{e}_{i,2}-\dot{z}^{*}_{i,2},\
\dot{z}^{*}_{i,2}=-\bar{k}_{1i}\dot{\tilde{z}}_{i,1},$ (24a)
$\displaystyle\tilde{z}_{i,3}$ $\displaystyle=z_{i,3}-z^{*}_{i,3},\
z^{*}_{i,3}=-\bar{k}_{2i}\tilde{z}_{i,2}-\tilde{z}_{i,1}+\dot{z}^{*}_{i,2},$
(24b)

where $z^{*}_{i,3}$ is a virtual control input and $\bar{k}_{2i}>0$.

Choose the augmented Lyapunov candidate as
$V_{i,2}=V_{i,1}+\frac{1}{2}\tilde{z}^{T}_{i,2}\tilde{z}_{i,2}$. Then, the
time derivative of $V_{i,2}$ is derived as

$\displaystyle\dot{V}_{i,2}$
$\displaystyle=-\bar{k}_{1i}\tilde{z}^{T}_{i,1}\tilde{z}_{i,1}+\tilde{z}^{T}_{i,1}\tilde{z}_{i,2}-\kappa_{1i}\tilde{z}^{T}_{i,1}\hat{e}_{i,1}+\tilde{z}^{T}_{i,2}\dot{\tilde{z}}_{i,2}$
(25)
$\displaystyle=-\bar{k}_{1i}\tilde{z}^{2}_{i,1}-\bar{k}_{2i}\tilde{z}^{2}_{i,2}+\tilde{z}^{T}_{i,2}\tilde{z}_{i,3}-\kappa_{1i}\tilde{z}^{T}_{i,1}\hat{e}_{i,1}-\kappa_{2i}\tilde{z}^{T}_{i,2}\hat{e}_{i,2}.$

Step $\mathbf{q\ (3\leq q\leq m-1)}$: Similarly, define error variables

$\displaystyle\dot{\tilde{z}}_{i,s}$
$\displaystyle=z_{i,q+1}-\kappa_{qi}\hat{e}_{i,q}-\dot{z}^{*}_{i,q},\dot{z}^{*}_{i,q}=-\sum_{s=1}^{q-1}\bar{k}_{si}\dot{\tilde{z}}^{*}_{i,s},i\in\mathcal{V},$
(26a) $\displaystyle\tilde{z}_{i,q+1}$
$\displaystyle=z_{i,q+1}-z^{*}_{i,q+1},z^{*}_{i,q+1}=-\bar{k}_{qi}\tilde{z}_{i,q}-\tilde{z}_{i,q-1}+\dot{z}^{*}_{i,q},$
(26b)

where $z^{*}_{i,q+1}$ is a virtual control input and $\bar{k}_{qi}>0$.

Choose the augmented Lyapunov candidate as
$V_{i,q}=V_{i,q-1}+\frac{1}{2}\tilde{z}^{T}_{i,q}\tilde{z}_{i,q}$. Then, the
time derivative of $V_{i,q}$ can be derived as

$\dot{V}_{i,q}=-\sum_{s=1}^{q-1}\bar{k}_{si}\tilde{z}^{T}_{i,s}\tilde{z}_{i,s}+\tilde{z}^{T}_{i,q}\tilde{z}_{i,q+1}-\sum_{s=1}^{q-1}\kappa_{si}\tilde{z}^{T}_{i,s}\hat{e}_{i,s}.$
(27)

Step m: According to the analysis in Step $q$, we have that for this final
step, the time derivative of $\tilde{z}_{i,m}$ by (20) is

$\displaystyle\dot{\tilde{z}}_{i,m}$
$\displaystyle=f^{T}_{i,m}\theta_{i}+G_{i,m}(x_{i},t)u_{i}+D_{i,m}(x_{i},t)-\tilde{\eta}_{i}-\kappa_{mi}$
$\displaystyle\ \
\times\text{sig}^{\beta}(\sum_{j=1}^{N}a_{ij}(\tilde{x}_{j,m}-\tilde{x}_{i,m})-a_{i0}\tilde{x}_{i,m})-\dot{z}^{*}_{i,m}.$
(28)

Next, we select the augmented Lyapunov candidate as
$V_{i,m}=V_{i,m-1}+\frac{1}{2}\tilde{z}^{T}_{i,m}\tilde{z}_{i,m}$. Then, the
time derivative of $V_{i,m}$ using (28) can be further expressed as

$\displaystyle\dot{V}_{i,m}$
$\displaystyle=-\sum_{s=1}^{m-1}\bar{k}_{si}\tilde{z}^{T}_{i,s}\tilde{z}_{i,s}-\sum_{s=1}^{m-1}\kappa_{si}\tilde{z}^{T}_{i,s}\hat{e}_{i,s}+\tilde{z}^{T}_{i,m}[f^{T}_{i,m}\theta_{i}$
(29) $\displaystyle\
+G_{i,m}(x_{i},t)u_{i}+D_{i,m}(x_{i},t)+\tilde{z}_{i,m-1}-\tilde{\eta}_{i}-\dot{z}^{*}_{i,m}].$

Substituting the control law (21a) and (21b) into (28) yields

$\displaystyle\dot{\tilde{z}}_{i,m}$
$\displaystyle=f^{T}_{i,m}\tilde{\theta}_{i}-\bar{k}_{m}\tilde{z}_{i,m}-\tilde{z}_{i,m-1}-\text{diag}\\{\tilde{z}_{\delta
m}\\}\hat{\varepsilon}_{i}-\tilde{\eta}_{i}$ (30) $\displaystyle\ \
+(G_{i,m}(x_{i},t)N_{i}(\kappa_{i})+1)\bar{u}_{i}+D_{i,m}(x_{i},t)-\kappa_{mi}\hat{e}_{i,m}.$

Select $V_{i}=V_{i,m}+\frac{1}{2}\tilde{\theta}^{T}_{i}\varGamma^{-1}_{\theta
i}\tilde{\theta}_{i}+\frac{1}{2}\tilde{\varepsilon}^{T}_{i}\varGamma^{-1}_{\varepsilon
i}\tilde{\varepsilon}_{i}$ for each agent $i$, the time derivative of $V_{i}$
using (29) is written as

$\displaystyle\dot{V}_{i}$
$\displaystyle=-\sum_{s=1}^{m-1}\bar{k}_{si}\tilde{z}^{T}_{i,s}\tilde{z}_{i,s}-\sum_{s=1}^{m-1}\kappa_{si}\tilde{z}^{T}_{i,s}\hat{e}_{i,s}+\tilde{z}^{T}_{i,m}f^{T}_{i,m}\tilde{\theta}_{i}$
$\displaystyle\ \ \
+\tilde{z}^{T}_{i,m}(G_{i,m}(x_{i},t)N_{i}(\kappa_{i})+1)\bar{u}_{i}-\tilde{\theta}^{T}_{i}f_{i,m}\tilde{z}_{i,m}$
$\displaystyle\ \ \
+\tilde{z}^{T}_{i,m}(D_{i,m}(x_{i},t)-\text{diag}\\{\tilde{z}_{\delta
m}\\}\hat{\varepsilon}_{i})-\tilde{\varepsilon}^{T}_{i}\text{diag}\\{\tilde{z}_{\delta
m}\\}\tilde{z}_{i,m}$ $\displaystyle\ \ \
+\tilde{z}^{T}_{i,m}(-\bar{k}_{mi}\tilde{z}_{i,m}-\kappa_{mi}\hat{e}_{i,m}-\tilde{\eta}_{i}).$
(31)

Notice that it follows from Lemma 3 that
$|\tilde{z}_{ij,m}|-\tilde{z}_{ij,m}\tilde{z}_{ij,m}/\\\
\sqrt{\tilde{z}^{2}_{ij,m}+\delta^{2}_{ij}(t)}\leq\delta_{ij}(t)$, where
$\tilde{z}_{ij,m}$ denotes the $j$-th element of $\tilde{z}_{i,m}$,
$i=1,\cdots,N,j=1,\cdots,n$. Let $\varepsilon_{ij}$ be the $j$th element of
$\varepsilon_{i}=\text{sup}_{t\geq 0}\\{\|D_{i,m}(x_{i},t)\|\\}$. Then, we
have

$\displaystyle\dot{V}_{i}$
$\displaystyle\leq-\sum_{s=1}^{m-1}\bar{k}_{si}\tilde{z}^{T}_{i,s}\tilde{z}_{i,s}-\sum_{s=1}^{m-1}\kappa_{si}\tilde{z}^{T}_{i,s}\hat{e}_{i,s}-\tilde{z}^{T}_{i,m}(\bar{k}_{mi}\tilde{z}_{i,m}+\tilde{\eta}_{i}$
$\displaystyle\ \ \
+\kappa_{mi}\hat{e}_{i,m})+\sum^{n}_{j=1}\left[|\tilde{z}_{ij,m}|\varepsilon_{ij}-\frac{\tilde{z}^{2}_{ij,m}\varepsilon_{ij}}{\sqrt{\tilde{z}^{2}_{ij,m}+\delta^{2}_{ij}(t)}}\right]$
$\displaystyle\ \ \
+\tilde{z}^{T}_{i,m}(G_{i,m}(x_{i},t)N_{i}(\kappa_{i})+1)\bar{u}_{i}$
$\displaystyle\leq-\sum_{s=1}^{m}\bar{k}_{si}\tilde{z}^{T}_{i,s}\tilde{z}_{i,s}-\sum_{s=1}^{m}\kappa_{si}\tilde{z}^{T}_{i,s}\hat{e}_{i,s}+\sum^{n}_{j=1}\delta_{ij}(t)\varepsilon_{ij}$
$\displaystyle\ \ \
-\tilde{z}^{T}_{i,m}\tilde{\eta}_{i}+G_{i,m}(x_{i},t)(N_{i}(\kappa_{i})+1)\dot{\kappa}_{i}/k_{\kappa
i}.$ (32)

Thus, the overall Lyapunov function candidate is selected as
$V=\sum_{i=1}^{N}V_{i}$. Then, its time derivative is described by

$\displaystyle\dot{V}$
$\displaystyle\leq-\sum_{i=1}^{N}(\sum_{s=1}^{m}\bar{k}_{si}\tilde{z}^{T}_{i,s}\tilde{z}_{i,s}+\sum_{s=1}^{m}\kappa_{si}\tilde{z}^{T}_{i,s}\hat{e}_{i,s}+\tilde{z}^{T}_{i,m}\tilde{\eta}_{i})$
(33) $\displaystyle\ \ \
+\sum_{i=1}^{N}\sum^{n}_{j=1}\delta_{ij}(t)\varepsilon_{ij}+\sum_{i=1}^{N}\frac{1}{k_{\kappa
i}}G_{i,m}(x_{i},t)(N_{i}(\kappa_{i})+1)\dot{\kappa}_{i}.$

Then, integrating both sides of (33) gives rise to $V(t)\leq
V(0)-\int_{0}^{t}W(s)ds+I_{A}+I_{B}$, where
$W(t)=\sum_{i=1}^{N}(\sum_{s=1}^{m}\bar{k}_{si}\tilde{z}^{T}_{i,s}\tilde{z}_{i,s}+\sum_{s=1}^{m}\kappa_{si}\tilde{z}^{T}_{i,s}\hat{e}_{i,s}+\tilde{z}^{T}_{i,m}\tilde{\eta}_{i})$,
$I_{A}=\int_{0}^{t}\sum_{i=1}^{N}\sum^{n}_{j=1}\delta_{ij}(\omega)|\varepsilon_{ij}|d\omega\leq\sum_{i=1}^{N}\sum^{n}_{j=1}\delta^{*}_{ij}|\varepsilon_{ij}|<\delta^{*}$
for certain positive scalarss $\delta^{*}$, and
$I_{B}=\int_{0}^{t}\sum_{i=1}^{N}k^{-1}_{\kappa
i}|(G_{i,m}N_{i}(\kappa_{i}(\omega))+1)\dot{\kappa}_{i}(\omega)|d\omega$ can
be bounded by seeking a contradiction under Assumptions 1 and 2 with a similar
spirit of arguments in [37]. Then, $\int_{0}^{t}W(s)ds$ can be upper bounded.
Thus, the existence of $\lim_{t\rightarrow\infty}\int_{0}^{t}W(s)ds$ can be
guaranteed and it is finite. According to Theorem 1 and the input-to-state
stability theory, we can obtain that $\hat{e}_{i,s},\tilde{\eta}_{i}$ are
bounded and moreover, all signals in $V_{i}$ are bounded. Hence, the linear
analysis can be applied to show that $\tilde{z}_{i,s},s=1,\cdots,m$, and
$\dot{\tilde{z}}_{i,m}$ are bounded. Since $\int_{0}^{t}W(s)ds$ is bounded, we
further get that
$\lim_{t\rightarrow\infty}\int_{0}^{t}\sum_{i=1}^{N}\sum_{s=1}^{m}\bar{k}_{si}\tilde{z}^{T}_{i,s}(\tau)\tilde{z}_{i,s}(\tau)d\tau$
can be upper bounded as
$\int_{0}^{t}\sum_{i=1}^{N}(\sum_{s=1}^{m}\tilde{z}^{T}_{i,s}(\tau)\hat{e}_{i,s}(\tau)+\tilde{z}^{T}_{i,m}(\tau)\tilde{\eta}_{i}(\tau))d\tau$
are bounded due to $\lim_{t\rightarrow T_{2}}\tilde{\eta}_{i}=0_{n}$ and
$\lim_{t\rightarrow T_{3}}\hat{e}_{i,s}=0_{n}$. Thus, $\tilde{z}_{i,s}$ is
square integrable. The Barbalat’s Lemma in Lemma 5 is used to conclude
$\lim_{t\to\infty}\tilde{z}_{i,s}=0_{n}$. That is,
$\lim_{t\to\infty}z_{i,s}=0_{n}$ via the defined virtual control inputs.
Hence,
$\underset{t\rightarrow\infty}{\text{lim}}(y_{i}(t)-y_{0}(t))=\varDelta_{i}\
\text{and}\
\underset{t\rightarrow\infty}{\text{lim}}(x_{i,k}(t)-x_{0,k}(t))=0_{n}$,
$k=2,\cdots,m$. ∎

##  IV Application to Task-space Cooperative Tracking

###  IV-A Networked Manipulator Model

Consider a group of $N$ manipulators, where the kinematics and dynamics of
each manipulator $i\in\mathcal{V}$ are governed by

$\displaystyle x_{i}=S_{i}(q_{i}),\ \dot{x}_{i}=J_{i}(q_{i})\dot{q}_{i},\
i=1,2,\cdots,N,$ (34) $\displaystyle
M_{i}(q_{i})\ddot{q}_{i}+C_{i}(q_{i},\dot{q}_{i})\dot{q}_{i}+G_{i}(q_{i})+F_{i}(q_{i},\dot{q}_{i})=g_{i}(t)\tau_{ai}+d_{i}(t),$

where $x_{i}\in\mathbb{R}^{n}$ is a generalized end-effector configuration,
$S_{i}(q_{i})\\\ :\mathbb{R}^{l}\rightarrow\mathbb{R}^{n}$ is a nonlinear
mapping from the joint space to the task space, and
$J_{i}(q_{i})=\partial{S_{i}(q_{i})}/\partial{q_{i}}\in\mathbb{R}^{n\times l}$
is its Jacobian matrix, $q_{i}$ and $\dot{q}_{i}\in\mathbb{R}^{l}$ denote
generalized position and velocity vectors, respectively, $M_{i}(q_{i})$ $\in$
$\mathbb{R}^{l\times l}$ is an inertia matrix,
$C_{i}(q_{i},\dot{q}_{i})\dot{q}_{i}\in\mathbb{R}^{l}\\\ $ is a Coriolis
centrifugal force vector, $G_{i}(q_{i})\in\mathbb{R}^{l}$ is a gravity vector,
$F_{i}(q_{i},\dot{q}_{i})\in\mathbb{R}^{l}$ are uncertain dynamics (e.g.,
$F_{i}(q_{i},\dot{q}_{i})=F_{vi}\tanh(\dot{q}_{i})+F_{ci}\text{sgn}(\dot{q}_{i})$
for matrices $F_{vi}$ and $F_{ci}$), $\tau_{ai}\in\mathbb{R}^{l}$ are control
torques with faults, $d_{i}\in\mathbb{R}^{l}$ are external disturbances, and
$g_{i}(t)\neq 0$ are unknown control coefficients relating $\tau_{ai}$ to
torques $\tau_{i}$ to be designed. The sign of $g_{i}(t)$ is unknown.

Similar to (3), two types of actuator faults are described by

$\tau_{ai}=\phi_{i}(t)\tau_{i}+\psi_{i}(t),\ i=1,2,\cdots,N,$ (35)

where $0<\phi_{i}(t)\leq 1$ and $\psi_{i}(t)\in\mathbb{R}^{l}$ represent the
actuation loss of effectiveness fault and the float fault, respectively.

Property 1: $M_{i}(q_{i})$ is symmetric and positive definite.

Property 2: $\dot{M}_{i}(q_{i})-2C_{i}(q_{i},\dot{q}_{i})$ is skew symmetric.

Property 3: For $x,y\in\mathbb{R}^{n}$,
$M_{i}(q_{i})y+C_{i}(q_{i},\dot{q}_{i})x+G_{i}(q_{i})=Y_{i}(q_{i},\dot{q}_{i},x,y)\theta_{i}$,
where $Y_{i}(\cdot)\in\mathbb{R}^{l\times p}$ denotes a known dynamic
regression matrix, and $\theta_{i}\in\mathbb{R}^{p}$ is an unknown parameter
vector.

Property 4: The kinematics (34) relies linearly on a kinematic parameter
vector $a_{i}\in\mathbb{R}^{r}$, i.e.,
$\dot{x}_{i}=J_{i}(q_{i})\dot{q}_{i}=Z_{i}(q_{i},\dot{q}_{i})a_{i}$, where
$Z_{i}(q_{i},\dot{q}_{i})\in\mathbb{R}^{n\times r}$ is a known kinematic
regression matrix.

###### Problem 2

Consider the robots’ dynamics and kinematics in (34) with actuator faults in
(35). Given a directed graph $\mathcal{\bar{G}}$, design a distributed
controller $\tau_{i}$ so that each robot achieves

$\underset{t\rightarrow\infty}{\text{lim}}(x_{i}-x_{d})=0_{n}\ \text{and}\
\underset{t\rightarrow\infty}{\text{lim}}(\dot{x}_{i}-\dot{x}_{d})=0_{n},\
i\in\mathcal{V},$ (36)

where $x_{d}$ is a desired global task reference and $\dot{x}_{d}$ is its
velocity.

###  IV-B Task-space Coordinated Tracking of Networked Manipulators

Unlike works in [30, 27, 26, 28, 29] requiring available task information
($x_{d}$, $\dot{x}_{d}$, and $\ddot{x}_{d}$) to all robots, we present a
distributed estimation framework to reconstruct this global information for
each robot.

Distributed Nonlinear Estimator: similar to (6), the following finite-time
distributed estimator is developed for each robot $i$

$\displaystyle\dot{\chi}_{i}$ $\displaystyle=\vartheta_{i}+k_{\chi
i}\text{sig}^{\gamma}(\sum_{j=1}^{N}a_{ij}(\chi_{j}-\chi_{i})+a_{i0}(x_{d}-\chi_{i})),$
(37a) $\displaystyle\dot{\vartheta}_{i}$ $\displaystyle=\eta_{i}+k_{\vartheta
i}\text{sig}^{\beta}(\sum_{j=1}^{N}a_{ij}(\vartheta_{j}-\vartheta_{i})+a_{i0}(\dot{x}_{d}-\vartheta_{i})),$
(37b) $\displaystyle\dot{\eta}_{i}$ $\displaystyle=k_{\eta
i}[\text{sig}^{\alpha}(e^{\eta}_{i})+\text{sgn}(e^{\eta}_{i})],e^{\eta}_{i}=\sum_{j=1}^{N}a_{ij}(\eta_{j}-\eta_{i})+e^{\varrho}_{i},$
(37c) $\displaystyle\dot{\xi}_{i}$ $\displaystyle=\varrho_{i}+k_{\xi
i}a_{i0}\text{sig}^{\frac{1}{2}}\left(\dot{x}_{d}-\xi_{i}\right),\
e^{\varrho}_{i}=a_{i0}(\varrho_{i}-\eta_{i}),$ (37d)
$\displaystyle\dot{\varrho}_{i}$ $\displaystyle=k_{\varrho
i}a_{i0}\text{sgn}\left(\dot{x}_{d}-\xi_{i}\right),\xi_{i}(0)=0_{n},\varrho_{i}(0)=0_{n},$
(37e)

where $k_{\chi i},k_{\vartheta i},k_{\eta i},k_{\xi i},k_{\varrho i}>0$,
$\alpha,\beta,\gamma\in(0.5,1)$, $\chi_{i}$, $\vartheta_{i}$, $\eta_{i}$ are
the estimates of $x_{d}$, $\dot{x}_{d}$, $\ddot{x}_{d}$, respectively.

Define $\tilde{\chi}_{i}=\chi_{i}-x_{d}$,
$\tilde{\vartheta}_{i}=\vartheta_{i}-\dot{x}_{d}$,
$\tilde{\eta}_{i}=\eta_{i}-\ddot{x}_{d}$,
$\tilde{\xi}_{i}=\xi_{i}-\dot{x}_{d}$, and
$\tilde{\varrho}_{i}=\varrho_{i}-\ddot{x}_{d}$. Then, we get the estimation
result below.

###### Theorem 3

Under the proposed distributed estimator in (37), all these state estimates
are uniformly bounded, and the finite-time estimation is achieved: $\lim_{t\to
T_{1}}a_{i0}\tilde{\varrho}_{i}=0_{n}$, $\lim_{t\to
T_{2}}\tilde{\eta}_{i}=0_{n}$, $\lim_{t\to T_{3}}\tilde{\vartheta}_{i}=0_{n}$,
$\lim_{t\to T_{4}}\tilde{\chi}_{i}=0_{n}$ for $T_{i},i=1,2,3,4$.

###### Proof:

It is similar to that in Theorem 1 and is omitted. ∎

Next, we can transform the coordinated tracking problem into the simultaneous
tracking problem for the decoupled robot group. Specifically, denote the
position tracking error $\bar{x}_{i}=x_{i}-x_{d}$ and velocity tracking error
$\dot{\bar{x}}_{i}=\dot{x}_{i}-\dot{x}_{d}$, $i\in\mathcal{V}$. Then, based on
the estimations $\chi_{i}$ and $\vartheta_{i}$ obtained from the distributed
estimator in (37), we have $\bar{x}_{i}=x_{i}-\chi_{i}+\tilde{\chi}_{i}$ and
$\dot{\bar{x}}_{i}=\dot{x}_{i}-\vartheta_{i}+\tilde{\vartheta}_{i}$ where
$\tilde{\chi}_{i}=\chi_{i}-x_{d}$ and
$\tilde{\vartheta}_{i}=\vartheta_{i}-\dot{x}_{d}$. As it follows from Theorem
3 that $\lim_{t\to\infty}\tilde{\chi}_{i}=0_{n}$ and
$\lim_{t\to\infty}\tilde{\vartheta}_{i}=0_{n}$, $\forall i\in\mathcal{V}$, the
coordinated tracking control objective in (36) can be transformed into the
simultaneous tracking objective in the sense that

$\lim_{t\to\infty}(x_{i}-\chi_{i})=0_{n}\ \text{and}\
\lim_{t\to\infty}(\dot{x}_{i}-\vartheta_{i})=0_{n}.$ (38)

First, let us define a task-space sliding variable $s_{xi}$

$s_{xi}=e_{vi}+\alpha_{xi}e_{xi},\ e_{xi}=x_{i}-\chi_{i},\
e_{vi}=\dot{x}_{i}-\vartheta_{i},\ i\in\mathcal{V},$ (39)

where $e_{xi}$ and $e_{vi}$ are the task-space position and velocity tracking
errors of the $i$th manipulator and $\alpha_{xi}>0$ is a scalar.

In the presence of uncertain kinematics, $J_{i}(q_{i})$ becomes unknown and
satisfies Property 4. Using the estimate of $J_{i}(q_{i})$ and the sliding
vector $s_{xi}$, we define a joint-space reference velocity

$\dot{q}_{ri}=\hat{J}^{+}_{i}(q_{i})(\vartheta_{i}-\alpha_{xi}e_{xi}-\alpha_{ri}\int_{0}^{t}s_{xi}(\omega)d\omega),\
i\in\mathcal{V},$ (40)

where
$\hat{J}^{+}_{i}(q_{i})=\hat{J}^{T}_{i}(q_{i})(\hat{J}_{i}(q_{i})\hat{J}^{T}_{i}(q_{i}))^{-1}$
is a generalized inverse of the approximate Jacobian matrix, $\alpha_{ri}>0$
is a scalar, $\vartheta_{i}$ is obtained from (37), and $\hat{J}_{i}(q_{i})$
is the estimate of $J_{i}(q_{i})$, which is obtained by replacing $a_{i}$ in
$J_{i}(q_{i})$ with $\hat{a}_{i}$ that defines the estimate of the unknown
kinematic parameter $a_{i}$ in Property 4.

Differentiating (40) gives the joint-space reference acceleration

$\ddot{q}_{ri}=\hat{J}^{+}_{i}(q_{i})(\dot{\vartheta}_{i}-\alpha_{xi}\dot{e}_{xi}-\alpha_{ri}s_{xi})+\dot{\hat{J}}^{+}_{i}(q_{i})\hat{J}_{i}(q_{i})\dot{q}_{ri}.$
(41)

Based on the reference velocity given in (40), we define a joint-space sliding
vector $s_{i}=\dot{q}_{i}-\dot{q}_{ri}$ and an estimated end-effector velocity
$\dot{\hat{x}}_{i}=\hat{J}_{i}(q_{i})\dot{q}_{i}$. Then, it follows from (40)
that

$\displaystyle s_{i}$
$\displaystyle=\dot{q}_{i}-\dot{q}_{ri}=\hat{J}^{+}_{i}(q_{i})(\dot{\hat{x}}_{i}-\dot{x}_{i}+\dot{x}_{i})-\dot{q}_{ri}$
(42)
$\displaystyle=\hat{J}^{+}_{i}(q_{i})[(\hat{J}_{i}(q_{i})-J_{i}(q_{i}))\dot{q}_{i}+s_{xi}+\alpha_{ri}\int_{0}^{t}s_{xi}(\omega)d\omega],$

which yields the following relation between the joint-space sliding vector
$s_{i}$ and the task-space sliding vector $s_{xi}$

$\hat{J}_{i}(q_{i})s_{i}=s_{xi}+\alpha_{ri}\int_{0}^{t}s_{xi}(\omega)d\omega+Z_{i}(q_{i},\dot{q}_{i})\tilde{a}_{i},\
i\in\mathcal{V},$ (43)

where $\tilde{a}_{i}=\hat{a}_{i}-a_{i}$ is the kinematic parameter estimation
error.

By Property 3,
$M_{i}(q_{i})\ddot{q}_{ri}+C_{i}(q_{i},\dot{q}_{i})\dot{q}_{ri}+G_{i}(q_{i})$$=Y_{i}\theta_{i}$.
Then, for $b_{i}(t)=g_{i}\phi_{i}\in\mathbb{R}$ and
$D_{i}(t)=g_{i}\psi_{i}-F_{i}(q_{i},\dot{q}_{i})+d_{i}\in\mathbb{R}^{l}$,

$M_{i}(q_{i})\dot{s}_{i}+C_{i}(q_{i},\dot{q}_{i})s_{i}=b_{i}(t)\tau_{i}+D_{i}(t)-Y_{i}\theta_{i}.$
(44)

Similarly, let $\varepsilon_{i}=\text{sup}_{t\geq 0}\\{\|D_{i}(t)\|\\}$ and
$\hat{\varepsilon}_{i}$ denotes an estimate of this unknown bound vector
$\varepsilon_{i}1_{n}$. Then,
$\tilde{\varepsilon}_{i}=\varepsilon_{i}1_{n}-\hat{\varepsilon}_{i}$.

Fault-Tolerant Distributed Adaptive Controller: in light of the Nussbaum
function and local estimates $\chi_{i}$ and $\vartheta_{i}$ from (37), a
fault-tolerant adaptive controller is proposed as

$\displaystyle\tau_{i}$ $\displaystyle=N_{i}(\kappa_{i})u_{i},\
N_{i}(\kappa_{i})=\text{exp}(\kappa^{2}_{i})\text{cos}((\pi/2)\kappa_{i})+1,$
(45a) $\displaystyle u_{i}$
$\displaystyle=Y_{i}\hat{\theta}_{i}-\hat{J}^{T}_{i}K_{si}\hat{J}_{i}s_{i}-\text{diag}\\{s_{\delta
ij}\\}\hat{\varepsilon}_{i},\ \dot{\kappa}_{i}=-k_{\kappa i}s^{T}_{i}u_{i},$
(45b) $\displaystyle\dot{\hat{\varepsilon}}_{i}$
$\displaystyle=\varGamma_{\varepsilon i}\text{diag}\\{s_{\delta ij}\\}s_{i},\
s_{\delta ij}=s_{ij}/\sqrt{s^{2}_{ij}+\delta^{2}_{ij}(t)},\ i\in\mathcal{V},$
(45c)

where $k_{\kappa i}>0$ is a constant, and $K_{si},\varGamma_{\varepsilon i}>0$
are matrices. Then, for matrices $\varGamma_{\theta i},\varLambda_{i}>0$, the
dynamic adaptation law for $\hat{\theta}_{i}$ and the kinematic adaptation law
for $\hat{a}_{i}$ are updated by

$\displaystyle\dot{\hat{\theta}}_{i}$ $\displaystyle=-\varGamma_{\theta
i}Y^{T}_{i}(q_{i},\dot{q}_{i},\dot{q}_{ri},\ddot{q}_{ri})s_{i},\
i\in\mathcal{V},$ (46a) $\displaystyle\dot{\hat{a}}_{i}$
$\displaystyle=\varLambda_{i}Z^{T}_{i}(q_{i},\dot{q}_{i})(s_{xi}+\alpha_{ri}\int_{0}^{t}s_{xi}(\omega)d\omega-\hat{J}_{i}s_{i}).$
(46b)

Let $\tilde{\theta}_{i}=\hat{\theta}_{i}-\theta_{i}$. Then, substituting (45)
into (44) yields

$\displaystyle M_{i}(q_{i})\dot{s}_{i}$
$\displaystyle=-C_{i}(q_{i},\dot{q}_{i})s_{i}+(b_{i}(t)N_{i}(\kappa_{i})-1)u_{i}+Y_{i}\tilde{\theta}_{i}$
$\displaystyle\ \ \
-\hat{J}^{T}_{i}K_{si}\hat{J}_{i}s_{i}+D_{i}(t)-\text{diag}\\{s_{\delta
ij}\\}\hat{\varepsilon}_{i}.$ (47)

The combination of (39), (43) and (47) yields a cascade system

$\left\\{\begin{array}[]{l}\ \ s_{xi}=e_{vi}+\alpha_{xi}e_{xi},\
i\in\mathcal{V},\\\ \
\hat{J}_{i}s_{i}=s_{xi}+\alpha_{ri}\int_{0}^{t}s_{xi}(\omega)d\omega+Z_{i}(q_{i},\dot{q}_{i})\tilde{a}_{i},\\\
M_{i}\dot{s}_{i}=-C_{i}s_{i}+(b_{i}(t)N_{i}(\kappa_{i})-1)u_{i}-\hat{J}^{T}_{i}K_{si}\hat{J}_{i}s_{i}\\\
\ \ \ \ \ \ \ \ \ \ +Y_{i}\tilde{\theta}_{i}+D_{i}(t)-\text{diag}\\{s_{\delta
ij}\\}\hat{\varepsilon}_{i}.\end{array}\right.$ (48)

Now, we have the fault-tolerant task-space coordination result.

###### Theorem 4

Consider a group of networked manipulators subject to faults in (35). Under
the proposed distributed controller in (45) with the nonlinear estimator in
(37), Problem 2 is solvable, i.e.,
$\underset{t\rightarrow\infty}{\text{lim}}(x_{i}-x_{d})=0_{n}$ and
$\underset{t\rightarrow\infty}{\text{lim}}(\dot{x}_{i}-\dot{x}_{d})=0_{n}$.

###### Proof:

The proof includes three steps:

Step (i): prove $\hat{J}_{i}s_{i}\in\mathcal{L}_{2}$ and
$\underset{t\rightarrow\infty}{\text{lim}}\hat{J}_{i}s_{i}=0_{n}$,
$i\in\mathcal{V}$.

Construct the following Lyapunov function candidate:

$V_{si}=\frac{1}{2}\left(s_{i}^{T}M_{i}(q_{i})s_{i}+\tilde{\theta}_{i}^{T}\varGamma^{-1}_{\theta
i}\tilde{\theta}_{i}+\tilde{\varepsilon}^{T}_{i}\varGamma^{-1}_{\varepsilon
i}\tilde{\varepsilon}_{i}\right).$ (49)

Then, the time derivative of $V_{si}$ along (48) is given by

$\displaystyle\dot{V}_{si}$
$\displaystyle=s^{T}_{i}M_{i}(q_{i})\dot{s}_{i}+\frac{1}{2}s^{T}_{i}\dot{M}_{i}(q_{i})s_{i}+\tilde{\theta}^{T}_{i}\varGamma^{-1}_{\theta
i}\dot{\tilde{\theta}}_{i}+\tilde{\varepsilon}^{T}_{i}\varGamma^{-1}_{\varepsilon
i}\dot{\tilde{\varepsilon}}_{i}$
$\displaystyle=s^{T}_{i}(Y_{i}\tilde{\theta}_{i}-\hat{J}^{T}_{i}K_{si}\hat{J}_{i}s_{i})+s^{T}_{i}[\frac{1}{2}\dot{M}_{i}(q_{i})-C_{i}(q_{i},\dot{q}_{i})]s_{i}$
$\displaystyle\ \ \ +s^{T}_{i}[D_{i}(t)-\text{diag}\\{s_{\delta
ij}\\}\hat{\varepsilon}_{i}]-\tilde{\varepsilon}^{T}_{i}\text{diag}\\{s_{\delta
ij}\\}s_{i}$ $\displaystyle\ \ \
-\tilde{\theta}^{T}_{i}Y^{T}_{i}s_{i}+s^{T}_{i}\left(b_{i}(t)N(\kappa_{i})-1\right)u_{i}.$
(50)

Since
$\sum^{l}_{j=1}(|s_{ij}|\varepsilon_{ij}-s_{ij}s_{ij}\varepsilon_{ij}/\sqrt{s^{2}_{ij}+\delta^{2}_{ij}(t)})\leq\sum^{l}_{j=1}\delta_{ij}(t)\\\
\varepsilon_{ij}$ by Lemma 3, we have $\dot{V}_{si}\leq-\frac{1}{k_{\kappa
i}}(b_{i}(t)N_{i}(\kappa_{i}(t))$
$-1)\dot{\kappa}_{i}(t)-s^{T}_{i}\hat{J}^{T}_{i}K_{si}\hat{J}_{i}s_{i}+\sum^{l}_{j=1}\delta_{ij}(t)\varepsilon_{ij}$.
Then, integrating this inequality yields $V_{si}(t)\leq
V_{si}(0)-\int_{0}^{t}s^{T}_{i}(\omega)\hat{J}^{T}_{i}(\omega)K_{si}(\omega)\hat{J}_{i}(\omega)s_{i}(\omega)d\omega+I^{{}^{\prime}}_{A}+I^{{}^{\prime}}_{B},$
where $I^{{}^{\prime}}_{A}=\int_{0}^{t}k^{-1}_{\kappa
i}|(1-b_{i}(\omega)N_{i}(\kappa_{i}(\omega)))\dot{\kappa}_{i}(\omega)|d\omega$
and
$I^{{}^{\prime}}_{B}=\int_{0}^{t}\sum^{l}_{j=1}\delta_{ij}(\omega)|\varepsilon_{ij}|d\omega$.
Since $\int_{0}^{t}\delta_{ij}(\omega)d\omega\leq\delta^{*}_{ij}$, it is not
difficult to obtain that
$I^{{}^{\prime}}_{B}\leq\sum^{l}_{j=1}\delta^{*}_{ij}|\varepsilon_{ij}|$,
which can be upper bounded by certain constant. Moreover, with the similar
spirit of the argument in [37], the boundness of $I^{{}^{\prime}}_{A}$ can be
obtained by seeking a contradiction. Thus, it can be concluded from (49) that
$s_{i}\in\mathcal{L}_{\infty}$, $\tilde{\theta}_{i}\in\mathcal{L}_{\infty}$,
and $\tilde{\varepsilon}_{i}\in\mathcal{L}_{\infty}$.

In addition, define $W_{si}=s^{T}_{i}\hat{J}^{T}_{i}K_{si}\hat{J}_{i}s_{i}$
and we can obtain

$\int_{0}^{t}W_{si}(\omega)d\omega\leq
V_{si}(0)-V_{si}(t)+I^{{}^{\prime}}_{A}+I^{{}^{\prime}}_{B},\
i\in\mathcal{V},$ (51)

which means that $\int_{0}^{t}W_{si}(\omega)d\omega$ can be upper bounded and
the existence of
$\underset{t\rightarrow\infty}{\text{lim}}\int_{0}^{t}W_{si}(\omega)d\omega$
can be guaranteed and it is finite. Then, according to the definition of
$W_{si}$, we have that $\hat{J}_{i}s_{i}$ is square integrable. Hence, the
Barbalat’s Lemma in Lemma 5 is used to conclude that
$\hat{J}_{i}s_{i}\in\mathcal{L}_{2}$ and
$\underset{t\rightarrow\infty}{\text{lim}}\hat{J}_{i}s_{i}=0_{n}$.

Step (ii):  prove
$\underset{t\rightarrow\infty}{\text{lim}}(Z_{i}(q_{i},\dot{q}_{i})\tilde{a}_{i})=0_{n}$,
$\underset{t\rightarrow\infty}{\text{lim}}e_{xi}=\underset{t\rightarrow\infty}{\text{lim}}(x_{i}-\chi_{i})=0_{n}$,
and
$\underset{t\rightarrow\infty}{\text{lim}}e_{vi}=\underset{t\rightarrow\infty}{\text{lim}}(\dot{x}_{i}-\vartheta_{i})=0_{n}$.

Since $\hat{J}_{i}s_{i}\in\mathcal{L}_{2}$ by Step (i), there exists a
constant $I_{si}$ so that
$I_{si}=2\int_{0}^{\infty}s^{T}_{i}(\omega)\hat{J}^{T}_{i}(\omega)\hat{J}_{i}(\omega)s_{i}(\omega)d\omega<\infty$,
$i\in\mathcal{V}$. Then, for the second subsystem in (48), we can select

$\displaystyle V_{zi}$
$\displaystyle=\frac{1}{2}e^{T}_{xi}e_{xi}+(I_{si}-2\int_{0}^{t}s^{T}_{i}(\omega)\hat{J}^{T}_{i}(\omega)\hat{J}_{i}(\omega)s_{i}(\omega)d\omega)$
$\displaystyle+2\tilde{a}^{T}_{i}\Lambda^{-1}_{i}\tilde{a}_{i}+\alpha_{ri}(\int_{0}^{t}s_{xi}(\omega)d\omega)^{T}(\int_{0}^{t}s_{xi}(\omega)d\omega).$
(52)

By using
$\hat{J}_{i}s_{i}=s_{xi}+\alpha_{ri}\int_{0}^{t}s_{xi}(\omega)d\omega+Z_{i}(q_{i},\dot{q}_{i})\tilde{a}_{i}$
from the second subsystem of (48), the time derivative of $V_{zi}$ is

$\displaystyle\dot{V}_{zi}$
$\displaystyle=-\alpha_{xi}e^{T}_{xi}e_{xi}+e^{T}_{xi}(\hat{J}_{i}s_{i}-\alpha_{ri}\int_{0}^{t}s_{xi}(\omega)d\omega-
Z_{i}(q_{i},\dot{q}_{i})\tilde{a}_{i})$ $\displaystyle\ \ \
-2(s_{xi}+\alpha_{ri}\int_{0}^{t}s_{xi}d\omega)^{T}(s_{xi}+\alpha_{ri}\int_{0}^{t}s_{xi}d\omega)$
$\displaystyle\ \ \
-2\tilde{a}^{T}_{i}Z^{T}_{i}(q_{i},\dot{q}_{i})Z_{i}(q_{i},\dot{q}_{i})\tilde{a}_{i}-2\alpha_{ri}s^{T}_{xi}(\int_{0}^{t}s_{xi}(\omega)d\omega).$
(53)

Let $V_{szi}=V_{si}+V_{zi}$. Then, using Young inequality and (53),

$\displaystyle\dot{V}_{szi}$
$\displaystyle\leq-(\alpha_{xi}-\frac{3}{4})e^{T}_{xi}e_{xi}-s^{T}_{i}\hat{J}^{T}_{i}(K_{si}-I_{l})\hat{J}_{i}s_{i}-s^{T}_{xi}s_{xi}$
$\displaystyle\ \ \
-\left(s_{xi}+\alpha_{ri}\int_{0}^{t}s_{xi}d\omega\right)^{T}\left(s_{xi}+\alpha_{ri}\int_{0}^{t}s_{xi}d\omega\right)$
$\displaystyle\ \ \
+\sum^{l}_{j=1}\delta_{ij}(t)\varepsilon_{ij}-\frac{1}{k_{\kappa
i}}(b_{i}(t)N_{i}(\kappa_{i})-1)\dot{\kappa}_{i}$ $\displaystyle\ \ \
-2\tilde{a}^{T}_{i}Z^{T}_{i}(q_{i},\dot{q}_{i})Z_{i}(q_{i},\dot{q}_{i})\tilde{a}_{i}.$
(54)

Following the similar analysis in (51),
$\underset{t\rightarrow\infty}{\text{lim}}Z_{i}(q_{i},\dot{q}_{i})\tilde{a}_{i}=0_{n}$,
$\underset{t\rightarrow\infty}{\text{lim}}\hat{J}_{i}(q_{i})s_{i}=0_{n}$, and
$\underset{t\rightarrow\infty}{\text{lim}}e_{xi}=0_{n}$, provided that
$\alpha_{xi}>\frac{3}{4}$ and $K_{si}>I_{l}$. By (48),
$s_{xi}+\alpha_{ri}\int_{0}^{t}s_{xi}(\omega)d\omega=0_{n}$. From the input-
output property of exponentially stable and strictly proper linear systems,
$\underset{t\rightarrow\infty}{\text{lim}}\int_{0}^{t}s_{xi}(\omega)d\omega=0_{n}$
and $\underset{t\rightarrow\infty}{\text{lim}}s_{xi}=0_{n}$. By Step (ii),
$\underset{t\rightarrow\infty}{\text{lim}}e_{xi}=0_{n}$. Hence,
$\underset{t\rightarrow\infty}{\text{lim}}e_{vi}=0_{n}$.

Step (iii): prove
$\underset{t\rightarrow\infty}{\text{lim}}(x_{i}-x_{d})=0_{n}$ and
$\underset{t\rightarrow\infty}{\text{lim}}(\dot{x}_{i}-\dot{x}_{d})=0_{n}$.

Based on Theorem 3, we have
$\underset{t\rightarrow\infty}{\text{lim}}\tilde{\chi}_{i}=0_{n}$ and
$\underset{t\rightarrow\infty}{\text{lim}}\tilde{\vartheta}_{i}=0_{n}$. From
Step (ii), we have $\underset{t\rightarrow\infty}{\text{lim}}e_{xi}=0_{n}$,
and $\underset{t\rightarrow\infty}{\text{lim}}e_{vi}=0_{n}$,
$i\in\mathcal{V}$. Hence, we get the following expression

$\displaystyle\underset{t\rightarrow\infty}{\text{lim}}\bar{x}_{i}$
$\displaystyle=\underset{t\rightarrow\infty}{\text{lim}}(x_{i}-x_{d})=\underset{t\rightarrow\infty}{\text{lim}}(x_{i}-\chi_{i}+\chi_{i}-x_{d})$
$\displaystyle=\underset{t\rightarrow\infty}{\text{lim}}e_{xi}+\underset{t\rightarrow\infty}{\text{lim}}\tilde{\chi}_{i}=0_{n},$
(55) $\displaystyle\underset{t\rightarrow\infty}{\text{lim}}\dot{\bar{x}}_{i}$
$\displaystyle=\underset{t\rightarrow\infty}{\text{lim}}(\dot{x}_{i}-\dot{x}_{d})=\underset{t\rightarrow\infty}{\text{lim}}(\dot{x}_{i}-\vartheta_{i}+\vartheta_{i}-\dot{x}_{d})$
$\displaystyle=\underset{t\rightarrow\infty}{\text{lim}}e_{vi}+\underset{t\rightarrow\infty}{\text{lim}}\tilde{\vartheta}_{i}=0_{n},$
(56)

and the proof is thus completed. ∎

##  V Numerical Simulations

In this section, two examples and numerical simulation results are provided to
show the effectiveness of the proposed methods.

###  V-A Fault-Tolerant Formation Tracking of Second-Order Nonlinear Multi-
Agent Systems with Unknown Control Signs

Consider a second-order multi-agent system consisting of six followers and one
leader with the followers’ dynamics given by

$\left\\{\begin{array}[]{l}\dot{x}_{i,1}=x_{i,2},\ y_{i}=x_{i,1},\
i=1,2,\cdots,6,\\\
\dot{x}_{i,2}=f^{T}_{i}(x_{i})\theta_{i}+g_{i}(x_{i})u_{ai}+d_{i}(x_{i},t),\\\
\end{array}\right.$ (57)

where $x_{i}=\text{col}(x_{i,1},x_{i,2})\in\mathbb{R}^{4}$, the nonlinear
function $f_{i}(x_{i})\in\mathbb{R}^{2\times 2}$ and the unknown parameter
$\theta_{i}\in\mathbb{R}^{2}$ are described by

$f_{i}(x_{i})=\left[\begin{array}[]{cc}-\sin(x^{1}_{i,1})&x^{2}_{i,2}\\\
x^{1}_{i,2}&-x^{2}_{i,1}\end{array}\right],\
\theta_{i}=\left[\begin{array}[]{l}\theta_{i,1}\\\
\theta_{i,2}\end{array}\right],$ (58)

where $x^{k}_{i,1}$, $x^{k}_{i,2}$, $k=1,2$ denote the $k$-th elements of
their states, respectively, $\theta_{i,1}=0.3i$, $\theta_{i,1}=0.5i$,
$i=1,2,\cdots,6$, the control coefficients are
$g_{i}(x_{i})=p_{i}(\cos(x^{T}_{i,1}x_{i,1}+x^{T}_{i,2}x_{i,2}))$ with
$p_{i}=(-1)^{i}*0.1i$, the time-varying actuator faults $u_{ai}$ are given by

$u_{ai}=\left\\{\begin{array}[]{l}(0.2\sin(t)+0.4)u_{i}+[2,2\cos(t)]^{T},\
3\leq t<6,\\\ (0.3\cos(t)+0.6)u_{i}+[\sin(0.1t),3]^{T},\ 6\leq
t,\end{array}\right.$ (59)

and the bounded uncertainties/disturbances are provided as

$d_{i}(x_{i},t)=\left[\begin{array}[]{l}0.1\sin(x^{1}_{i,1}+x^{2}_{i,1})-0.3\cos(0.3t)\\\
0.2\cos(x^{1}_{i,2}x^{2}_{i,2})+0.5\sin(0.5t)\end{array}\right].$ (60)

The leader’s dynamics are described by

$\dot{x}_{0,1}=x_{0,2},\ \dot{x}_{0,2}=o_{0}(x_{0},u_{0},t),\ y_{0}=x_{0,1},$
(61)

where $x_{0}=\text{col}(x_{0,1},x_{0,2})$,
$u_{0}=\text{col}(0.8\sin(t),0.8\cos(t))$ and

$o_{0}(x_{0},u_{0},t)=\left[\begin{array}[]{l}0.1\cos(0.1x^{1}_{0,1}+x^{2}_{0,2})+0.8\sin(t)\\\
0.2\sin(x^{2}_{0,1}+0.2x^{1}_{0,2})+0.8\cos(t)\end{array}\right].$ (62)

Figure 1: A directed communication topology for the leader-following network.

The initial states are set as $x_{1,1}=\text{col}(-0.3,-0.5)$,
$x_{2,1}=\text{col}(-2,-1.6)$, $x_{3,1}=\text{col}(1,-3)$,
$x_{4,1}=\text{col}(0.2,0.8)$, $x_{5,1}=\text{col}(2,-1.5)$,
$x_{6,1}=\text{col}(2.5,1.8)$, $x_{i,2}=\text{col}(0,0)$,
$x_{01}=\text{col}(0,-2)$ and $x_{02}=\text{col}(1,0)$. The communication
graph is depicted in Fig. 1. The prescribed hexagonal formation is
$\varDelta=\text{col}(-1,0,-\frac{1}{2},\frac{\sqrt{3}}{2},\frac{1}{2},\frac{\sqrt{3}}{2},1,0,\frac{1}{2},-\frac{\sqrt{3}}{2},-\frac{1}{2},-\frac{\sqrt{3}}{2})$.
The proposed distributed algorithm in (21) with estimator in (6)-(7) is
performed with parameters selected as $\kappa_{1i}=15$, $\kappa_{2i}=5$,
$\kappa_{\eta i}=8$, $\kappa_{\xi i}=6$, $\kappa_{\varrho i}=4$,
$\bar{k}_{1i}=0.8$, $\bar{k}_{2i}=80$, $\delta_{i}=0.05$, $k_{\kappa i}=1$,
$\varGamma_{\kappa i}=I_{2}$, $\varGamma_{\varepsilon i}=I_{2}$, and
$\varGamma_{\theta i}=10I_{2}$. The simulation results are obtained as
illustrated in Figs. 2-5. In particular, Fig. 2 depicts the position and
velocity estimates of the leader’s states, respectively. Then, the formation
trajectory of the six agents is shown in Fig. 3, where the agents’ initial
positions are marked by circles and their final positions make the hexagonal
formation. In addition, the position and velocity tracking errors between the
leader and the followers are shown in Fig. 4. From Fig. 5, the adaptive
parameters $\hat{\varepsilon}_{i}$ and $\kappa_{i}$ are bounded. Thus, it can
be seen that the formation tracking is achieved for nonlinear multi-agent
systems with time-varying actuator faults over the directed graph.

---

(a) Agents’ estimates $\hat{x}_{i,1}$

(b) Agents’ estimated errors $\tilde{x}_{i,1}$.

(c) Agents’ estimates $\hat{x}_{i,2}$

(d) Agents’ estimated errors $\tilde{x}_{i,2}$.

Figure 2: The estimates of the leader’s states under the estimator (6)-(7).
Figure 3: The formation trajectory of the six agents under the controller (21)

---

(a) Formation tracking error $e_{i,1}$

(b) Velocity tracking error $e_{i,2}$

Figure 4: The trajectories of position and velocity tracking errors under
(21).

---

(a) $\hat{\varepsilon}_{i}$

(b) $\kappa_{i}$

Figure 5: The trajectories of adaptive parameters under (21).

###  V-B Fault-Tolerant Task-Space Coordination of Manipulators

In this simulation, six networked two-link planar manipulators are
investigated. Since all robots are planar manipulators, the main objective is
to synchronize the end-effectors, while ensuring that they follow the desired
trajectory in the task space [30, 27, 26, 28, 29].

The classic manipulator’s dynamics are described by

$M_{i}(q_{i})\ddot{q}_{i}+C_{i}(q_{i},\dot{q}_{i})\dot{q}_{i}+G_{i}(q_{i})+F_{i}(q_{i},\dot{q}_{i})=g_{i}(t)\tau_{ai}+d_{i},$

where $q_{i}=\text{col}(q_{1i},q_{2i})$ denote the joint angles, and

$M_{i}(q_{i})=\left(\begin{array}[]{cc}\theta_{1i}+\theta_{2i}+2\theta_{3i}\cos(q_{2i})&\theta_{2i}+\theta_{3i}\cos(q_{2i})\\\
\theta_{2i}+\theta_{3i}\cos(q_{2i})&\theta_{2i}\end{array}\right),$
$C_{i}(q_{i},\dot{q}_{i})=\left(\begin{array}[]{cc}-\theta_{3i}\sin(q_{2i})\dot{q}_{2i}&-\theta_{3i}\sin(q_{2i})(\dot{q}_{1i}+\dot{q}_{2i})\\\
\theta_{3i}\sin(q_{2i})\dot{q}_{1i}&0\end{array}\right),$
$G_{i}(q_{i})=\left(\begin{array}[]{c}\theta_{4i}\mathrm{g}\cos(q_{1i})+\theta_{5i}\mathrm{g}\cos(q_{1i}+q_{2i})\\\
\theta_{5i}\mathrm{g}\cos(q_{1i}+q_{2i})\end{array}\right),$

with
$\theta_{i}=\text{col}(\theta_{1i},\theta_{2i},\theta_{3i},\theta_{4i},\theta_{5i})$,
where $\theta_{1i}=I_{1i}+m_{1i}l^{2}_{c1i}+m_{2i}l^{2}_{1i}$,
$\theta_{2i}=I_{2i}+m_{2i}l^{2}_{c2i}$, $\theta_{3i}=m_{2i}l_{1i}l_{c2i}$,
$\theta_{4i}=(m_{1i}+m_{2i})l_{1i}$, and $\theta_{5i}=m_{2i}l_{2i}$. The
physical parameters of the six robotic manipulators are listed in Table I. The
term
$F_{i}(q_{i},\dot{q}_{i})=F_{vi}\tanh(\dot{q}_{i})+F_{ci}\text{sgn}(\dot{q}_{i})$
with $F_{vi}=[1\ 0;0\ 1]$, $F_{ci}=[1\ 1;1\ 1]$ are friction forces, and
$d_{i}$ are disturbances described by

$\displaystyle d_{i}(t)$
$\displaystyle=\left[\begin{array}[]{c}0.2\text{sin}(\frac{\pi}{10i}t)+0.4\text{sin}(\frac{\pi}{20i}t)\\\
0.4\text{cos}(\frac{\pi}{10i}t)+0.8\text{cos}(\frac{\pi}{20i}t)\end{array}\right],i=1,2,\cdots,6.$
(65)

In addition, the control coefficients are $g_{i}(t)=p_{i}(\cos(t)+1.2)$ with
$p_{i}=(-1)^{i}*0.1i$ and the actuator faults $\tau_{ai}$ are given by

$\tau_{ai}=\left\\{\begin{array}[]{l}(0.2\sin(t)+0.4)\tau_{i}+[2,2\cos(t)]^{T},\
3\leq t<6,\\\ (0.3\cos(t)+0.6)\tau_{i}+[\sin(0.1t),3]^{T},\ 6\leq
t.\end{array}\right.$ (66)

In this simulation, the desired end-effector reference trajectory for the
manipulators is a circle in the task space [26],

$x_{d}(t)=\left[\begin{array}[]{c}1.2+0.5\text{sin}(0.6t)\\\
1.0+0.5\text{cos}(0.6t)\end{array}\right].$ (67)

TABLE I: The physical dynamic parameters of robotic manipulators [27]. Robot | $m_{1i},m_{2i}\ \text{(kg)}$ | $I_{1i},I_{2i}\ (\text{kg}\text{m}^{2})$ | $l_{1i},l_{2i}\ \text{(m)}$ | $l_{c1i},l_{c2i}\ \text{(m)}$
---|---|---|---|---
$1$ | 1.5, 1.3 | 0.50, 0.43 | 2.0, 2.0 | 1.00, 1.00
$2$ | 1.2, 1.5 | 0.53, 0.36 | 2.3, 1.7 | 1.15, 0.85
$3$ | 1.2, 1.3 | 0.32, 0.52 | 1.8, 2.2 | 0.90, 1.10
$4$ | 1.8, 1.5 | 0.66, 0.45 | 2.1, 1.9 | 1.05, 0.95
$5$ | 1.7, 1.6 | 0.56, 0.43 | 2.0, 1.8 | 1.00, 0.90
$6$ | 1.9, 1.3 | 0.46, 0.48 | 1.7, 2.1 | 0.85, 1.05

Then, the forward kinematics for the $i$th robot are

$x_{i}=S_{i}(q_{i})=\left[\begin{array}[]{c}c_{12i}l_{2i}v_{1i}+c_{1i}l_{1i}v_{1i}\\\
s_{12i}l_{2i}v_{2i}+s_{1i}l_{1i}v_{2i}\end{array}\right],\
\dot{x}_{i}=J_{i}(q_{i})\dot{q}_{i},$ (68)

where $c_{12i}=\text{cos}(q_{1i}+q_{2i})$, $c_{1i}=\text{cos}(q_{1i})$,
$s_{12i}=\text{sin}(q_{1i}+q_{2i})$, $s_{1i}=\text{sin}(q_{1i})$, and
$v_{1i}$, $v_{2i}$ are scaling factors [29]. The overall Jacobian matrix from
joint space to task space is expressed as

$J_{i}(q_{i})=\left[\begin{array}[]{cc}-s_{12i}l_{2i}v_{1i}-s_{1i}l_{1i}v_{1i}&-s_{12i}l_{2i}v_{1i}\\\
c_{12i}l_{2i}v_{2i}+c_{1i}l_{1i}v_{2i}&c_{12i}l_{2i}v_{2i}\end{array}\right].$
(69)

Thus, $\dot{x}_{i}=J_{i}(q_{i})\dot{q}_{i}$ in (68) is written as the product
of a known regressor matrix $Z_{i}(q_{i},\dot{q}_{i})$ and an unknown constant
vector $a_{i}$

$\displaystyle\dot{x}_{i}$
$\displaystyle=\left[\begin{array}[]{cccc}-s_{1i}\dot{q}_{1i}&-s_{12i}(\dot{q}_{1i}+\dot{q}_{2i})&0&0\\\
0&0&c_{1i}\dot{q}_{1i}&c_{12i}(\dot{q}_{1i}+\dot{q}_{2i})\end{array}\right]$
(72) $\displaystyle\ \ \ \times\left[\begin{array}[]{c}l_{1i}v_{1i}\\\
l_{1i}v_{2i}\\\ l_{2i}v_{1i}\\\
l_{2i}v_{2i}\end{array}\right]=Z_{i}(q_{i},\dot{q}_{i})a_{i},$ (77)

where $a_{i1}=l_{1i}v_{1i}$, $a_{i2}=l_{1i}v_{2i}$, $a_{i3}=l_{2i}v_{1i}$, and
$a_{i4}=l_{2i}v_{2i}$.

The simulation results are presented in Figs. 6-9. Under the proposed finite-
time distributed estimator in (37), Fig. 6 depicts the trajectories of the
estimated position $\chi_{i}$ and their tracking errors $\tilde{\chi}_{i}$.
Fig. 7 shows the position trajectories of the end-effectors $x_{i}$ and its
tracking errors $\tilde{x}_{i}$ under the proposed distributed controller (45)
with (37). The path of the six end-effectors is depicted in Fig. 8, where all
the robots can track the circle path in the task space. Fig. 9 shows that all
adaptive parameters
$\hat{a}_{i},\hat{\theta}_{i},\hat{\varepsilon}_{i},\hat{\kappa}_{i}$ are
bounded. It can be concluded from Figs. 6-9 that the task-space cooperative
tracking can be achieved under the proposed distributed algorithm for
networked manipulators irrespective of uncertain kinematics, dynamics, and
time-varying actuator faults.

---

(a) Estimated position $\chi_{i}$

(b) Estimated position error $\tilde{\chi}_{i}$

Figure 6: The estimated position trajectories of the global task reference
under the proposed distributed estimator (37).

---

(a) Position $x_{i}$

(b) Position tracking error $\tilde{x}_{i}$

Figure 7: The position trajectories of end-effectors under the proposed
distributed control algorithm (45) with the distributed estimator (37). Figure
8: The path of end-effectors $x_{i}$ under (45) with (37).

---

(a) $\hat{a}_{i}$

(b) $\hat{\theta}_{i}$

(c) $\hat{\varepsilon}_{i}$

(d) $\hat{\kappa}_{i}$

Figure 9: The trajectories of parameters under the proposed distributed
algorithm (45) with (37).

##  VI Conclusion

In this paper, we considered the fault-tolerant formation tracking problem for
nonlinear multi-agent systems with time-varying actuator faults over the
directed graph. We proposed a distributed estimation and control framework by
incorporating a distributed nonlinear estimator and a Nussbaum gain technique.
Based on the proposed algorithm, the problem was solvable under completely
unknown control coefficients. The proposed strategy was applied to task-space
cooperative tracking of networked manipulators with unknown kinematics,
dynamics, and actuator faults.

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