Ehsan Arabi
Department of Mechanical Engineering,
University of South Florida,
Tampa, FL 33620
e-mail: [email protected]
Tansel Yucelen1
Department of Mechanical Engineering,
University of South Florida,
Tampa, FL 33620
e-mail: [email protected]
Wassim M. Haddad
School of Aerospace Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: [email protected]
Mitigating the Effects of Sensor
Uncertainties in Networked
Multi-Agent Systems
Networked multi-agent systems consist of interacting agents that locally exchange information, energy, or matter. Since these systems do not in general have a centralized architecture to monitor the activity of each agent, resilient distributed control system design
for networked multi-agent systems is essential in providing high system performance,
reliability, and operation in the presence of system uncertainties. An important class of
such system uncertainties that can significantly deteriorate the achievable closed-loop
system performance is sensor uncertainties, which can arise due to low sensor quality,
sensor failure, sensor bias, or detrimental environmental conditions. This paper presents
a novel distributed adaptive control architecture for networked multi-agent systems with
undirected communication graph topologies to mitigate the effect of sensor uncertainties.
Specifically, we consider agents having identical high-order, linear dynamics with agent
interactions corrupted by unknown exogenous disturbances. We show that the proposed
adaptive control architecture guarantees asymptotic stability of the closed-loop dynamical system when the exogenous disturbances are time-invariant and uniform ultimate
boundedness when the exogenous disturbances are time-varying. Two numerical examples are provided to illustrate the efficacy of the proposed distributed adaptive control
architecture. [DOI: 10.1115/1.4035092]
Keywords: networked multi-agent systems, sensor uncertainty, distributed control, adaptive control, asymptotic stability, uniform ultimate boundedness
1
Introduction
Networked multi-agent systems (e.g., communication networks,
power systems, and process control systems) consist of interacting
agents that locally exchange information, energy, or matter [1–4].
These systems require a resilient distributed control system design
architecture for providing high system performance, reliability,
and operation in the presence of system uncertainties [5–9]. An
important class of such system uncertainties that can significantly
deteriorate achievable closed-loop dynamical system performance
is sensor uncertainties, which can arise due to low sensor quality,
sensor failure, sensor bias, or detrimental environmental conditions
[10–13]. If relatively cheap sensor suites are used for low-cost,
small-scale unmanned vehicle applications, then this can result in
inaccurate sensor measurements. Alternatively, sensor measurements can be corrupted by malicious attacks if these dynamical
systems are controlled through large-scale, multilayered communication networks as in the case of cyberphysical systems.
Early approaches that deal with sensor uncertainties focus on
classical fault detection, isolation, and recovery schemes (see, for
example, Refs. [14,15]). In these approaches, sensor measurements are compared with an analytical model of the dynamical
system by forming a residual signal and analyzing this signal to
determine if a fault has occurred. However, in practice it is difficult to identify a single residual signal per failure mode, and as
the number of failure modes increases, this becomes prohibitive.
In addition, a common underlying assumption of the classical
fault detection, isolation, and recovery schemes is that all the
dynamical system signals remain bounded during the fault detection process, which may not always be a valid assumption.
More recently, Pasqualetti et al. [16] considered the fundamental limitations of attack detection and identification methods for
linear systems. However, their approach is not only
1
Corresponding author.
Contributed by the Dynamic Systems Division of ASME for publication in the
JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received
October 27, 2015; final manuscript received October 15, 2016; published online
February 6, 2017. Assoc. Editor: Mazen Farhood.
computationally expensive but also it is not linked to the controller design. In Ref. [17], adversarial attacks on actuator and sensors
are modeled as exogenous disturbances. However, the presented
control methodology cannot address situations where more than
half of the sensors are compromised and the set of attacked nodes
changes over time. Finally, Sadikhov et al. Ref. [18] analyzed a
case where the interaction between networked agents is corrupted
by exogenous disturbances. However, their approach is limited to
agents having scalar dynamics and is not linked to the controller
design in order to mitigate the effect of such sensor uncertainties.
In this paper, we present a novel distributed adaptive control
architecture for networked multi-agent systems with undirected
communication graph topologies to mitigate the effect of sensor
uncertainties. Specifically, we consider multi-agent systems having identical high-order, linear dynamics with agent interactions
corrupted by unknown exogenous disturbances. We show that the
proposed adaptive control architecture guarantees asymptotic stability of the closed-loop dynamical system when the exogenous disturbances are time-invariant and uniform ultimate boundedness when
the exogenous disturbances are time-varying. A preliminary conference version of this paper appeared in Ref. [19]. The present paper
considerably expands on Ref. [19] by providing detailed proofs of all
the results along with additional examples and motivation.
The contents of the paper are as follows: In Sec. 2, we present
mathematical preliminaries and give the problem formulation. In
Sec. 3, we develop a distributed adaptive control architecture for
the case of time-invariant sensor uncertainties, whereas Sec. 4
extends this architecture to the case of time-varying sensor uncertainties. Two illustrative examples are provided in Sec. 5, and
conclusions are drawn in Sec. 6.
2 Mathematical Preliminaries and Problem
Formulation
2.1 Mathematical Preliminaries. The notation used in this
paper is fairly standard. Specifically, R denotes the set of real
numbers, Rn denotes the set of n 1 real column vectors, Rnm
denotes the set of n m real matrices, Rþ denotes the set of
Journal of Dynamic Systems, Measurement, and Control
C 2017 by ASME
Copyright V
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positive real numbers, Pn (respectively, Nn ) denotes the set of
n n positive definite (respectively, non-negative definite) real
matrices, 0 n denotes the n 1 zero vector, 1 n denotes the n 1
ones vector, 0 nm denotes the n m zero matrix, and “¢”
denotes equality by definition. In addition, we write ðÞT for the
transpose operator, ðÞ1 for the inverse operator, det() for the
determinant operator, ðÞ0 for the Frechet derivative, k k2 for the
Euclidean norm, and for the Kronecker product. Furthermore,
we write kmin ðAÞ (respectively, kmax ðAÞ) for the minimum (respectively, maximum) eigenvalue of the square matrix A, ki(A) for the
ith eigenvalue of the square matrix A (with eigenvalues ordered
from minimum to maximum value), spec(A) for the spectrum of
the square matrix A including multiplicity, [A]ij for the (i, j)th
entry of the matrix A, and x (respectively, x) for the lower bound
(respectively, upper bound) of a bounded signal xðtÞ 2 Rn ; t 0,
that is, x kxðtÞk2 ; t 0 (respectively, kxðtÞk2 x; t 0).
Next, we recall some basic notions from graph theory, where
we refer the reader to Refs. [3,20] for further details. Specifically,
graphs are broadly adopted in the multi-agent systems literature to
encode interactions between networked systems. An undirected
graph G is defined by a set V G ¼ f1; …; Ng of nodes and a set
E G V G V G of edges. If ði; jÞ 2 E G , then nodes i and j are
neighbors and the neighboring relation is indicated by i j. The
degree of a node is given by the number of its neighbors. Letting
di denote the degree of node i, then the degree matrix of a graph
G, denoted by DðGÞ 2 RNN , is given by DðGÞ¢diag½d
, where
d ¼ ½d1 ; …; dN T . A path i0 i1 …iL of a graph G is a finite sequence
of nodes such that ik1 ik ; k ¼ 1; …; L, and if every pair of distinct nodes has a path, then the graph G is connected. We write
AðGÞ 2 RNN for the adjacency matrix of a graph G defined by
1; if ði; jÞ 2 E G
(1)
½AðGÞ
ij ¢
0; otherwise
and BðGÞ 2 RNM for the (node–edge) incidence matrix of a
graph G defined by
8
if node i is the head of edge j
< 1;
½BðGÞ
ij ¢ 1; if node i is the tail of edge j
(2)
:
0;
otherwise
where M is the number of edges, i is an index for the node set, and
j is an index for the edge set.
The graph Laplacian matrix, denoted by LðGÞ 2 NN ,
is defined by LðGÞ¢DðGÞ AðGÞ or, equivalently,
LðGÞ ¼ BðGÞBðGÞT , and the spectrum of the graph Laplacian of
a connected, undirected graph G can be ordered as
0 ¼ k1 ðLðGÞÞ < k2 ðLðGÞÞ … kN ðLðGÞÞ, with 1N being the
eigenvector corresponding to the zero eigenvalue k1 ðLðGÞÞ,
and LðGÞ1N ¼ 0N and eLðGÞ 1N ¼ 1N . Finally, we partition
the incidence matrix BðGÞ ¼ ½BL ðGÞT ; BF ðGÞT T , where
BL ðGÞ 2 RNL M ; BF ðGÞ 2 RNF M ; NL þ NF ¼ N, and NL and
NF, respectively, denote the cardinalities of the leader and follower groups [3]. Furthermore, without loss of generality, we
assume that the leader agents are indexed first and the follower
agents are indexed last in the graph G so that LðGÞ ¼
BðGÞBðGÞT is given by
"
#
LðGÞ
GðGÞT
LðGÞ ¼
(3)
GðGÞ
FðGÞ
x_ i ðtÞ ¼ Axi ðtÞ þ Bui ðtÞ;
xi ð0Þ ¼ xi0 ;
t0
(4)
where xi ðtÞ 2 Rn ; t 0, is the state vector of agent i,
ui ðtÞ 2 Rm ; t 0, is the control input of agent i, and A 2 Rnn
and B 2 Rnm are the system matrices. We assume that the pair
(A, B) is controllable, and the control input ui(), i ¼ 1,…, N, is
restricted to the class of admissible controls consisting of measurable functions such that ui ðtÞ 2 Rm ; t 0. In addition, we assume
that the agents can measure their own state and can locally
exchange information via a connected, undirected graph G with
nodes and edges representing agents and interagent information
exchange links, respectively, resulting in a static network topology; that is, the time evolution of the agents does not result in
edges appearing or disappearing in the network.
Here, we consider a networked multi-agent system, where the
agents lie on an agent layer and their local controllers lie on a control layer as depicted in Fig. 1. Furthermore, we assume that the
graph for the controller structure is the same as the graph for the
agents’ communication. Specifically, agent i 僆 {1,…, N} sends
its state measurement to its corresponding local controller at a
given control layer and this controller sends its control input to
agent i lying on the agent layer. In addition, we assume that the
compromised state measurement
x~i ðtÞ ¼ xi ðtÞ þ di ðtÞ;
i ¼ 1; …; N
(5)
is available to the local controller and the neighboring agents of
agent i 僆 {1,…, N}, where x~i ðtÞ 2 Rn ; t 0, and
di ðtÞ 2 Rn ; t 0, capture sensor uncertainties. In particular, if
di ðÞ is nonzero, then the state vector xi ðtÞ; t 0, of agent i 2
f1; …; Ng is corrupted with a faulty or malicious signal di ðÞ.
Alternatively, if di ðÞ is zero, then x~i ðtÞ ¼ xi ðtÞ; t 0, and the
uncompromised state measurement is available to the local controller of agent i 2 f1; …; Ng.
Given the two-layer networked multi-agent system hierarchy,
we are interested in the problem of asymptotically (or approximately) driving the state vector of each agent
xi ðtÞ; i ¼ 1; …; N; t 0, to the state vector of a (virtual) leader
x0 ðtÞ 2 Rn ; t 0, that lies on the control layer with dynamics
x_ 0 ðtÞ ¼ A0 x0 ðtÞ;
x0 ð0Þ ¼ x00 ;
t0
(6)
where A0 2 Rnn is Lyapunov stable and is given by A0 ¢A BK0 with K0 2 Rmn .
For the case where the uncompromised state measurement is
available to the local controller of agent i 2 f1; …; Ng, that is,
di ðtÞ 0, then the controller
and
where
LðGÞ¢BL ðGÞBL ðGÞT ; GðGÞ¢BF ðGÞBL ðGÞT ,
FðGÞ¢BF ðGÞBF ðGÞT . Note that FðGÞ 2 PNF for a connected,
undirected graph G and satisfies FðGÞ1NF ¼ GðGÞ1NL . This
implies that each row sum of FðGÞ1 GðGÞ is equal to 1.
2.2 Problem Formulation. Consider a networked multiagent system consisting of N agents with the dynamics of agent i,
i 僆 {1,…, N}, given by
041003-2 / Vol. 139, APRIL 2017
Fig. 1 A networked multi-agent system with agents lying on an
agent layer and their local controllers lying on a control layer
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X
ui ðtÞ ¼ K0 xi ðtÞ cK li ðxi ðtÞ x0 ðtÞÞ þ
ðxi ðtÞ xj ðtÞÞ (7)
ij
guarantees that limt!1 xi ðtÞ ¼ x0 ðtÞ for all i ¼ 1,…, N, where
li ¼ 1 for a set of NL agents that have access to the state of the
leader x0 ðtÞ; t 0, and li ¼ 0 for the remaining NF agents with
N ¼ NL þ NF , and where K 2 Rmn and c 2 Rþ denote an appropriate feedback gain matrix and coupling strength, respectively,
such that An ¢A0 gi cBK is Hurwitz for all gi 2 specðFðGÞÞ. To
see this, let ni ðtÞ¢xi ðtÞ x0 ðtÞ and note that using Eqs. (4), (6),
and (7)
X
ðni ðtÞ nj ðtÞÞ ;
n_ i ðtÞ ¼ A0 ni ðtÞ cBK li ni ðtÞ þ
ij
(8)
ni ð0Þ ¼ ni0 ; t 0
with ni0 ¢xi0 x0 . In addition, defining the augmented state
nðtÞ¢½nT1 ðtÞ; …; nTN ðtÞ
T , Eq. (8) can be written in compact form
as
_
nðtÞ
¼ ½IN A0 cFðGÞ BK
nðtÞ;
nð0Þ ¼ n0 ;
t 0 (9)
Now, using the results in Refs. [9,21,22], it can be shown that
IN A0 cFðGÞ BK is Hurwitz when An is Hurwitz for all
gi 2 specðFðGÞÞ. Hence, limt!1 nðtÞ ¼ 0, that is, limt!1 xi ðtÞ ¼
x0 ðtÞ for all i ¼ 1,…, N.
For d() 6¼ 0, our objective is to design a local controller for
each agent i 僆 {1,…, N} of the form
X
x i ðtÞ x0 ðtÞÞ þ
ð~
x i ðtÞ x~j ðtÞÞ þ vi ðtÞ
ui ðtÞ ¼ K0 x~i ðtÞ cK li ð~
ij
(10)
where vi ðtÞ 2 Rm ; t 0, is a local corrective signal that suppresses or counteracts the effect of di ðtÞ; t 0, to asymptotically
(or approximately) recover the ideal system performance (i.e.,
limt!1 xi ðtÞ ¼ x0 ðtÞ for all i ¼ 1,…, N) that is achieved when the
state vector is available for feedback. Thus, assuming that An is
Hurwitz for all gi 2 specðFðGÞÞ implies that there exists an ideal
system performance that can be recovered by designing the local
corrective signals vi ðtÞ 2 Rm ; t 0, for each agent i 僆 {1,…, N}.
Although we consider this specific problem in this paper, the proposed approach dealing with sensor uncertainties can be used in
many other problems that exist in the networked multi-agent systems literature [3,4].
3 Adaptive Leader Following With Time-Invariant
Sensor Uncertainties
^d i ðtÞ 2 Rn ; t 0, is the estimate of the sensor uncertainty
di ðtÞ; t 0, x^i ðtÞ 2 Rn ; t 0, is the state estimate of the uncompromised state vector xi ðtÞ; t 0; c 2 Rþ and l 2 Rþ are the
design gains, and P 2 Pn is a solution to the linear matrix inequality (LMI) given by
IN ðAT0 P þ PA0 2lPÞ cFðGÞ ðK T BT P þ PBKÞ < 0
(14)
For the statement of next result, we note from Eqs. (4) and (10)
that
x_ i ðtÞ ¼ Axi ðtÞ BK0 xi ðtÞ cBK ½li ðxi ðtÞ x0 ðtÞÞ
#
X
X
ðxi ðtÞ xj ðtÞÞ cBK
ðdi dj Þ
þ
ij
ij
Bðcli K þ K0 Þdi þ Bvi ðtÞ;
xi ð0Þ ¼ xi0 ;
Now, define xðtÞ¢½xT1 ðtÞ; …; xTN ðtÞ
T ; d¢½dT1 ; …; dTN T , and
vðtÞ¢½v1 ðtÞ; …; vN ðtÞ
T , and note that Eq. (15) can be written in a
compact form as
_ ¼ ½IN A0 cFðGÞ BK
xðtÞ þ ðcGðGÞ BKÞx0 ðtÞ
xðtÞ
ðcFðGÞ BK þ IN BK0 Þd
þðIN BÞvðtÞ; xð0Þ ¼ x0 ; t 0
(16)
Using Eqs. (5) and (16), the dynamics for x~ðtÞ; t 0, with
x~ðtÞ¢½~
x T1 ðtÞ; …; x~TN ðtÞ
T , can also be written in compact form as
x~_ ðtÞ ¼ ½IN A0 cFðGÞ BK
~
x ðtÞ þ ðcGðGÞ BKÞx0 ðtÞ
ðIN AÞdþðIN BÞvðtÞ; x~ð0Þ ¼ x~0 ; t 0
(17)
Next, letting x^ðtÞ¢½^
x T1 ðtÞ; …; x^TN ðtÞ
T , a compact form for the
dynamics of x^ðtÞ; t 0, is given by
x ðtÞ þ ðcGðGÞ BKÞx0 ðtÞ
x^_ ðtÞ ¼ ½IN A0 cFðGÞ BK
^
ðcFðGÞ BK þ IN BK0 Þ^dðtÞþðIN BÞvðtÞ þ /ðtÞ;
x^ð0Þ ¼ x^0 ;
t0
ð18Þ
_
where /ðtÞ¢½/T1 ðtÞ; …; /TN ðtÞ
T with /i ðtÞ¢ ^d i ðtÞ þ lei ðtÞ.
^
~
x i ðtÞ x^i ðtÞ d i ðtÞ and d i ðtÞ¢di ^d i ðtÞ,
Finally, define ei ðtÞ¢~
and note that
:
eðtÞ ¼ ðAr lInN ÞeðtÞ ðIN AÞ~dðtÞ;
In this section, we design the local corrective signal
vi ðtÞ; i ¼ 1; …; N; t 0, in Eq. (10) to achieve asymptotic adaptive leader following in the presence of time-invariant sensor
uncertainties, that is, di ðtÞ di ; i ¼ 1; …; N; t 0. For this problem, we propose the corrective signal
X
vi ðtÞ ¼ K0 ^d i ðtÞ þ cK li ^d i ðtÞ þ
ð^d i ðtÞ ^d j ðtÞÞ
(11)
t 0 (15)
~_ ¼ ðIN cAT PÞeðtÞ;
dðtÞ
~dð0Þ ¼ ~d 0 ;
eð0Þ ¼ e0 ;
t0
t0
(19)
(20)
(13)
where Ar ¢IN A0 cFðGÞ BK; eðtÞ¢½eT1 ðtÞ; …; eTN ðtÞ
T , and
~dðtÞ¢½~d T ðtÞ; …; ~d T ðtÞ
T .
1
N
THEOREM 1. Consider the networked multi-agent system consisting of N agents on a connected, undirected graph G, where the
dynamics of agent i 僆 {1,…, N} is given by Eq. (4). In addition,
assume that the local controller ui ðtÞ; i ¼ 1; …; N; t 0, for each
agent is given by Eq. (10) with the corrective signal
vi ðtÞ; i ¼ 1; …; N; t 0, given by Eq. (11). Moreover, assume
that di ðtÞ di ; t 0, and detðAÞ 6¼ 0. Then, the zero solution
ðeðtÞ; ~dðtÞÞ ð0; 0Þ of the closed-loop system given by Eqs. (19)
and (20) is Lyapunov stable and limt!1 eðtÞ ¼ 0 and
limt!1 ~dðtÞ ¼ 0 for all ðe0 ; ~d 0 Þ 2 RnN RnN .
Proof. To show Lyapunov stability of the zero solution
ðeðtÞ; ~dðtÞÞ ð0; 0Þ of the closed-loop system given by Eqs. (19)
and (20), consider the Lyapunov function candidate given by
Journal of Dynamic Systems, Measurement, and Control
APRIL 2017, Vol. 139 / 041003-3
ij
where
_
x i ðtÞ x^i ðtÞ ^d i ðtÞÞ;
d^ i ðtÞ ¼ cAT Pð~
^d i ð0Þ ¼ ^d i0 ;
t0
(12)
"
i
X
x^_ i ðtÞ ¼ A0 x^i ðtÞ cBK li ð^
xi ðtÞ x0 ðtÞÞ þ
ð^
x i ðtÞ x^j ðtÞÞ
ij
þðcA P þ lIn Þð~
x i ðtÞ x^i ðtÞ ^d i ðtÞÞ; x^i ð0Þ ¼ x^i0 ; t 0
T
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Vðe; ~dÞ ¼
N
X
T
T
ðeTi Pei þ c1 ~d i ~d i Þ ¼ eT ðIN PÞe þ c1 ~d ~d (21)
i¼1
where P is a solution to the LMI given by Eq. (14). Note that
Vð0; 0Þ ¼ 0; Vðe; ~dÞ > 0 for all ðe; ~dÞ 6¼ ð0; 0Þ, and Vðe; ~dÞ is
radially unbounded. The time derivative of Eq. (21) along the trajectories of Eqs. (19) and (20) is given by
t0
(22)
Hence, the zero solution ðeðtÞ; ~dðtÞÞ ð0; 0Þ of the closed-loop
system given by Eqs. (19) and (20) is Lyapunov stable for all
ðe0 ; ~d 0 Þ 2 RnN RnN .
To show limt!1 eðtÞ ¼ 0, note that V€ðeðtÞ; ~dðtÞÞ ¼ 2eT ðtÞ
_
Now,
½IN ðAT0 P þ PA0 2PlÞ cFðGÞ ðK T BT P þ PBKÞ
eðtÞ.
it follows from the Lyapunov stability of the zero solution
ðeðtÞ; ~dðtÞÞ ð0; 0Þ of Eqs. (19) and (20), and the boundedness of
_
eðtÞ;
t 0, that V€ðeðtÞ; ~dðtÞÞ is bounded for all t 0. Thus,
V_ ðeðtÞ; ~dðtÞÞ; t 0, is uniformly continuous in t. Now, it follows
from Barbalat’s lemma [23] that limt!1 V_ ðeðtÞ; ~dðtÞÞ ¼ 0, and
hence, limt!1 eðtÞ ¼ 0. Finally, to show limt!1 ~dðtÞ ¼ 0, define
R¢fðe; ~dÞ : V_ ðe; ~dÞ ¼ 0g and let M be the largest invariant set
contained in R. In this case, it follows from Eq. (19) that
ðIN AÞ~d ¼ 0, and hence, ~d ¼ 0 since detðAÞ 6¼ 0. Thus,
ðeðtÞ; ~dðtÞÞ ! M ¼ fð0; 0Þg as t ! 1.
w
Remark 1. It follows from Eqs. (4), (10), and (11) that
X
ðxi ðtÞ xj ðtÞÞ
x_ i ðtÞ ¼ A0 xi ðtÞ cBK li ðxi ðtÞ x0 ðtÞÞ þ
ij
t0
ð23Þ
which, using the boundedness of ~d i ðtÞ; i ¼ 1; …; N; t 0, and the
assumption that An is Hurwitz for all gi 2 specðFðGÞÞ, implies
that xi(t) is bounded for all t 0 and i 僆 {1,…, N}. Hence, using
Eq. (5), x~i ðtÞ is bounded for all t 0 and i 2 f1; …; Ng. Furthermore, since ei ðtÞ; t 0; x~i ðtÞ; t 0, and ^d i ðtÞ; t 0, are bounded
for all i 2 f1; …; Ng, it follows that x^i ðtÞ is bounded for all t 0
and i 僆 {1,…, N}.
Remark 2. Since, by Theorem 1, limt!1 ~dðtÞ ¼ 0 it follows
from Eq. (23) that each agent subject to the dynamics given by
Eq. (4) asymptotically recovers the ideal system performance
(i.e., limt!1 xi ðtÞ ¼ x0 ðtÞ for all i ¼ 1,…, N), which is a direct
consequence of the discussion given in Sec. 2.2 and the assumption that An is Hurwitz for all gi 2 specðFðGÞÞ; i ¼ 1; …; N. In
addition, limt!1 eðtÞ ¼ 0 and limt!1 ~dðtÞ ¼ 0 imply that
limt!1 ðxðtÞ x^ðtÞÞ ¼ 0, which shows that the state estimate
x^ðtÞ; t 0, converges to the uncompromised state measurement
x(t), t 0.
Remark 3. It is important to note that the local controller ui(t)
and corrective signal vi ðtÞ; i ¼ 1; …; N; t 0, are independent
of the system initial conditions. The same remark holds for
Theorem 2.
Let Q 2 Pm , set K ¼ Q1 BT P, and assume that An is Hurwitz
for all gi 2 specðFðGÞÞ; i ¼ 1; …; N for the given selection of K.
Substituting K ¼ Q1 BT P in Eq. (14) yields
IN ðAT0 P þ PA0 2lPÞ 2cFðGÞ ðPBQ1 BT PÞ < 0 (24)
Let T be such that TFðGÞT 1 ¼ J, where J is the Jordan form of
FðGÞ. Multiplying Eq. (24) by T IN from the left and by T 1 IN from the right yields
041003-4 / Vol. 139, APRIL 2017
(25)
which can be equivalently written as
IN ðAT0 P þ PA0 2lPÞ 2cJ ðPBQ1 BT PÞ < 0
AT0 P þ PA0 2lP 2cgi PBQ1 BT P < 0
(26)
(27)
where gi 2 specðFðGÞÞ; i ¼ 1; …; N. Now, letting the coupling
strength be such that c 1=minfgi g; i ¼ 1; …; N, it follows from
Eq. (27), using the results in Ref. [24], that
AT0 P þ PA0 2lP 2cgi PBQ1 BT P AT0 P þ PA0 2lP
(28)
2PBQ1 BT P
since cgi 1; i ¼ 1; …; N, and hence, one can equivalently
solve the LMI
ðA0 lIn ÞT P þ PðA0 lIn Þ 2PBQ1 BT P < 0
(29)
to obtain P 2 Pn rather than the LMI given by Eq. (14).
Remark 4. By setting S¢P1 [25], the LMI given by Eq. (29)
can be further simplified to
ðA0 lIn ÞS þ SðA0 lIn ÞT 2BQ1 BT < 0
ij
X
BK0 ~d i ðtÞ cBK li ~d i ðtÞ þ
ð~d i ðtÞ ~d j ðtÞÞ ;
xi ð0Þ ¼ xi0 ;
2ðT IN ÞðcFðGÞ ðPBQ1 BT PÞÞðT 1 IN Þ < 0
Note that since FðGÞ is symmetric, FðGÞ is real diagonalizable.
Hence, the diagonal form of J allows one to rewrite Eq. (26) as
V_ ðeðtÞ; ~dðtÞÞ ¼ eT ðtÞðIN ðAT0 P þ PA0 2PlÞ cFðGÞ
ðK T BT P þ PBKÞÞeðtÞ 0;
ðT IN ÞðIN ðAT0 P þ PA0 2lPÞÞðT 1 IN Þ
(30)
Thus, one can alternatively solve Eq. (30) for S 2 Pn and then set
P ¼ S–1 to obtain P 2 Pn for Eqs. (12) and (13). Note that since
one can solve Eq. (30) instead of Eq. (14), the computational complexity does not increase as the number of agents gets large.
Finally, note that in this paper we consider the case where
each agent is subject to sensor uncertainties. If, however, only a
fraction of the agents are subject to sensor uncertainties, then the
proposed corrective signal in Eq. (11) can be applied to
only those agents subject to sensor uncertainty and not to all the
agents.
4 Adaptive Leader Following With Time-Varying
Sensor Uncertainties
In this section, we generalize the results of Sec. 3 by designing
the local corrective signal vi ðtÞ; i ¼ 1; …; N; t 0, in Eq. (10) to
achieve approximate adaptive leader following in the presence of
time-varying sensor uncertainties di ðtÞ; i ¼ 1; …; N; t 0. We
assume that the time-varying sensor uncertainties are bounded and
have bounded time rates of change; that is, kdi ðtÞk2 d; t 0,
_
and kd_ i ðtÞk2 d;
t 0, for all i ¼ 1,…, N.
For the statement of our next result, it is necessary to introduce
the projection operator [26]. Specifically, let / : Rn ! R be a
continuously differentiable convex function given by
/ðhÞ¢ðeh þ 1ÞhT h h2max =eh h2max , where hmax 2 R is a projection norm bound imposed on h 2 Rn and eh > 0 is a projection
tolerance bound. Then, the projection operator Proj :
Rn Rn ! Rn is defined by
8
>
y;
if /ðhÞ < 0
>
>
< y;
if /ðhÞ 0and /0 ðhÞy 0
T
Projðh;yÞ ¢
/0 ðhÞ/0 ðhÞy
>
>
/ðhÞ; if /ðhÞ 0and /0 ðhÞy > 0
y
>
:
T
/0 ðhÞ/0 ðhÞ
(31)
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where y 2 Rn and /0 ðhÞ¢@/ðhÞ=@h. Note that it follows from
the
definition
of
the
projection
operator
that
ðh hÞT ½Projðh; yÞ y
0; h 2 Rn .
Next, for the controller given by Eq. (10), we use the local corrective signal
X
ð^d i ðtÞ ^d j ðtÞÞ
(32)
vi ðtÞ ¼ K0 ^d i ðtÞ þ cK li ^d i ðtÞ þ
ij
where
^d_ i ðtÞ ¼ cProjð^d i ðtÞ;AT Pð~
x i ðtÞ x^i ðtÞ ^d i ðtÞÞÞ; ^d i ð0Þ ¼ ^d i0 ; t 0
(33)
2
xi ðtÞ x0 ðtÞÞ þ
x^_ i ðtÞ ¼ A0 x^i ðtÞ cBK 4li ð^
#
X
ð^
xi ðtÞ x^j ðtÞÞ
ij
þlIn ð~
x i ðtÞ x^i ðtÞ ^d i ðtÞÞcProjð^d i ðtÞ;
^ i ðtÞÞÞ; x^i ð0Þ ¼ x^i0 ; t 0
^
x i ðtÞ d
AT Pð~
x i ðtÞ
:
_
eð0Þ ¼ e0 ; t 0
eðtÞ ¼ ðAr lInN ÞeðtÞ ðIN AÞ~dðtÞ þ dðtÞ;
(37)
~_ ¼ dðtÞc
_
^d P ðtÞ; ^d P ðtÞ¢½^d T ðtÞ;…; ^d T ðtÞ
T ; ~dð0Þ ¼ ~d 0 ; t 0
dðtÞ
P1
PN
(38)
where ^d Pi ðtÞ¢Projð^d i ðtÞ; AT Pei ðtÞÞ; i ¼ 1; …; N; t 0.
THEOREM 2. Consider the networked multi-agent system consisting of N agents on a connected, undirected graph G, where the
dynamics of agent i 僆 {1,…, N} is given by Eq. (4). In addition,
assume that the local controller ui ðtÞ; i ¼ 1; …; N; t 0, for each
agent is given by Eq. (10) with the corrective signal
vi ðtÞ; i ¼ 1; …; N; t 0, given by Eq. (32). Moreover, assume
that the sensor uncertainties are time-varying and detðAÞ 6¼ 0.
Then, the closed-loop system dynamics given by Eqs. (37) and
(38) are uniformly bounded for all ðe0 ; ~d 0 Þ 2 RnN RnN with the
ultimate bounds
"
(34)
n
and P 2 P is the solution to the LMI given by
IN ðAT0 P þ PA0 2lPÞ cFðGÞ ðK T BT P þ PBKÞ < 0
(35)
(see Remark 4 for solving the LMI given by Eq. (35)).
For the statement of the next theorem, define
R¢IN ðAT0 P þ PA0 2lPÞ cFðGÞ ðK T BT P þ PBKÞ < 0
(36)
where R 2 PNn , and note that, using similar steps as given in Sec.
3, the dynamics for eðtÞ ¼ x~ðtÞ x^ðtÞ ^dðtÞ and ~dðtÞ ¼
dðtÞ ^dðtÞ are given by
#12
kmax ðPÞ 2
1
g þ
g2 ;
keðtÞk2 kmin ðPÞ 1 ckmin ðPÞ 2
1
k~d ðtÞk2 ckmax ðPÞg21 þ g22 2 ;
tT
tT
(39)
(40)
pffiffiffiffiffi
pffiffiffiffiffi
where g1 ¢1= d1 ½d2 =2
d1 þ ðd22 =4d1 þ d3 Þ1 2
; g2 ¢^d max
_ ^
_
þd; d1 ¢kmin ðRÞ; d2 ¢2Nkmax ðPÞd,
d max þ dÞ.
and d3 ¢2Nc1 dð
Proof. To show uniform boundedness of the system
dynamics given by Eqs. (37) and (38), consider the Lyapunovlike function given by Eq. (21), where P satisfies Eq. (35).
Note that Vð0; 0Þ ¼ 0; Vðe; ~dÞ > 0 for all ðe; ~dÞ 6¼ ð0; 0Þ, and
Vðe; ~dÞ is radially unbounded. The time derivative of Eq. (21)
along the closed-loop system trajectories of Eqs. (37) and (38) is
given by
N X
X
2eTi ðtÞPA0 ei ðtÞ 2eTi ðtÞPBK
ei ðtÞ ej ðtÞ 2eTi ðtÞPA~d i ðtÞþ2eTi ðtÞPd_ i ðtÞ 2eTi ðtÞPBKli ei ðtÞ 2eTi ðtÞPlei ðtÞ
V_ eðtÞ; ~d ðtÞ ¼
ij
i¼1
i
T
þ2c1 ~d ðtÞ d_ i ðtÞ cProj ^d i ðtÞ; AT Pei ðtÞ
i
¼ eT ðtÞReðtÞ þ
i
N h
X
T
T
2eTi ðtÞPA~d i ðtÞ þ 2eTi ðtÞPd_ i ðtÞ þ 2c1 ~d i ðtÞd_ i ðtÞ2~d i ðtÞProj ^d i ðtÞ; AT Pei ðtÞ
i¼1
T N
X
T
2 ^d i ðtÞ di ðtÞ
Proj ^d i ðtÞ; AT Pei ðtÞ AT Pei ðtÞ þ2eTi ðtÞPd_ i ðtÞ þ 2c1 ~d i ðtÞd_ i ðtÞ
¼ eT ðtÞReðtÞ þ
i¼1
eT ðtÞReðtÞ þ
N h
X
T
2eTi ðtÞPd_ i ðtÞ þ 2c1 ~d i ðtÞd_ i ðtÞ
i
i¼1
d1 keðtÞk22
þ d2 keðtÞk2 þ d3
pffiffiffiffiffi
d2 2
d2
¼
d1 keðtÞk2 pffiffiffiffiffi þ 2 þ d3 ;
4d1
2 d1
t0
ð41Þ
and hence, V_ ðeðtÞ; ~dðtÞÞ < 0 outside the compact set X¢fðe; ~dÞ :
kek2 g1 and k~dk2 g2 g: This proves the uniform boundedness
of the solution ðeðtÞ; ~dðtÞÞ of the system dynamics given by Eqs.
(37) and (38) for all ðe0 ; ~d 0 Þ 2 RnN RnN [27].
To show the ultimate bounds for e(t), t T, and ~dðtÞ; t T,
given by Eqs. (39) and (40), respectively, note that
or,
kmin ðPÞkeðtÞk22 þ c1 k~dðtÞk22 kmax ðPÞg21 þ c1 g22 ; t T;
equivalently, kmin ðPÞkeðtÞk22 kmax ðPÞg21 þ c1 g22 ; t T; and
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suppress the effect of d~ i ðtÞ; i ¼ 1; …; N; t T. To elucidate the
effect of the controller design parameters on Eqs. (39) and (40),
let A ¼ 1, B ¼ 1, K ¼ 1, A0 ¼ 0, K0 ¼ 1, N ¼ 1, c ¼ 1,
^d max ¼ 1; d ¼ 1, and d_ ¼ 2. In this case, it follows from Eq. (36)
that P ¼ 0:5ð1 þ lÞ1 R, where we set R ¼ 1. Figure 2 shows the
effect of l 僆 [0, 5] and c 僆 [0.1, 100] on the ultimate bounds
given by Eqs. (39) and (40). Specifically, as expected, increasing
both l and c yields smaller ultimate bounds for keðtÞk2 and
k~dðtÞk2 for t T.
5
Illustrative Numerical Examples
In this section, we present two numerical examples to demonstrate the utility and efficacy of the proposed distributed control
architectures for networked multi-agent systems to mitigate the
effect of time-invariant and time-varying sensor uncertainties.
Fig. 2 Effect of l and c on the ultimate bounds given by Eqs.
(39) and (40) (arrow directions denote the increase of c from 0.1
to 100)
c1 k~dðtÞk22 kmax ðPÞg21 þ c1 g22 ; t T; which proves Eqs. (39)
and (40).
w
Remark 5. A similar remark to Remark 1 holds for Theorem 2.
Namely, all the signals used to construct the local controller
ui ðtÞ; i ¼ 1; …; N; t 0, for each agent given by Eq. (10) with
the local corrective signal vi ðtÞ; i ¼ 1; …; N; t 0, given by Eqs.
(32)–(34) are bounded.
Note that the projection operator is used in order to guarantee a
bounded estimate of the unknown parameter, since otherwise one
cannot conclude uniform ultimate boundedness from Eq. (41).
The ultimate bounds given by Eqs. (39) and (40) characterize the
controller design parameters that need to be chosen in order to
achieve small excursions of keðtÞk2 and k~dðtÞk2 for t T. This is
particularly important to obtain accurate estimates for
x^i ðtÞ; i ¼ 1; …; N; t T, and ^d i ðtÞ; i ¼ 1; …; N; t T, as well as
5.1 Example 1: Time-Invariant Sensor Uncertainties Case.
To illustrate the key ideas presented in Sec. 3, consider a group of
N ¼ 4 agents subject to a connected, undirected graph G given in
Fig. 1, where the dynamics of agent i satisfy
32
3 2 3
3 2
2
0
0 1 0
x1i ðtÞ
x_ 1i ðtÞ
76
7 6 7
7 6
6
76 2 7 6 7
6 2 7 6
6 x_ i ðtÞ 7 ¼ 6 0 0 1 76 xi ðtÞ 7 þ 6 0 7uðtÞ; i ¼ 1;…;4; t 0
54
5 4 5
5 4
4
1
4 4 1 x3i ðtÞ
x_ 3i ðtÞ
(42)
with initial conditions x1 ð0Þ ¼ ½0; 0; 0
T ; x2 ð0Þ ¼ ½1; 2; 1
T ;
x3 ð0Þ ¼ ½3; 1; 0
T , and x4 ð0Þ ¼ ½2; 2; 2
T . Note that detðAÞ 6¼ 0,
where A is the system matrix of Eq. (42). For this example, we are
interested in the problem of asymptotically driving the state vector
of each agent xi ðtÞ; i ¼ 1; …; 4; t 0, to the state vector of a
leader x0 ðtÞ; t 0, having dynamics given by
Fig. 3 Nominal system performance for a group of agents in Example 1 with the
local controller given by Eq. (7) (i.e., vi ðtÞ 0; i 5 1; . . . ; 4) when the uncompromised state measurement is available for feedback
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Fig. 4 System performance for a group of agents in Example 1 with the local controller given by Eq. (7) (i.e., vi ðtÞ 0; i 5 1; . . . ; 4) when the compromised state measurement is available for feedback
Fig. 5 System performance for a group of agents in Example 1 with the proposed
local controller given by Eq. (10) and the local corrective signal given by Eq. (11)
when the compromised state measurement is available for feedback
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Fig. 6 Time evolution of e(t), t 0, in Example 1 with the proposed local controller given by Eq. (10) and the local corrective
signal given by Eq. (11) when the compromised state measurement is available for feedback
2
x_ 10 ðtÞ
3
2
0
1
0
32
x10 ðtÞ
3
76 2 7
6 2 7 6
6 x_ ðtÞ 7 ¼ 6 0
6
7
0
1 7
54 x0 ðtÞ 5;
4 0 5 4
4 4 1
x30 ðtÞ
x_ 30 ðtÞ
2 3
1
6 7
6
x0 ð0Þ ¼ 4 1 7
5;
3
Fig. 7 Time evolution of ~
dðtÞ; t 0, in Example 1 with the proposed local controller given by Eq. (10) and the local corrective
signal given by Eq. (11) when the compromised state measurement is available for feedback
2
t0
(43)
For this problem, A0 ¼ A BK0 holds with K0 ¼ [0, 0, 0] as a
direct consequence of Eqs. (42) and (43).
To design the proposed local controllers, let K ¼ Q1 BT P and
set Q ¼ 0.1I3, l ¼ 0.2, and c ¼ 6 1=minfgi g; i ¼ 1; …; 4, so
that the LMI given by Eq. (30) for P ¼ S1 (see Remark 4) is satisfied with
3
0:0850 0:0316 0:0224
P ¼ 4 0:0316 0:0467 0:0090 5
0:0224 0:0090 0:0187
(44)
and hence, K ¼ [0.22, 0.09, 0.19]. Note that with this selection
for K, An is Hurwitz for all gi 2 specðFðGÞÞ; i ¼ 1; …; 4. The
nominal system performance for the case when the uncompromised state measurement is available to the local controller of
agent i, i 僆 {1,…,4}, (i.e., di() ¼ 0), is shown in Fig. 3 using
Eq. (7).
Next, consider a time-invariant sensor uncertainty given by Eq.
(5) with
Fig. 8 Nominal system performance for a group of agents in Example 2 with the
local controller given by Eq. (7) (i.e., vi ðtÞ 0; i 5 1; . . . ; 4) when the uncompromised state measurement is available for feedback
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Fig. 9 System trajectories of each agent in Fig. 8
2
3
5
d1 ¼ 4 7 5;
3
2
3
4
d2 ¼ 4 5 5;
4
2
3
6
d3 ¼ 4 3 5;
5
2
3
4
d4 ¼ 4 1 5
2
(45)
The system performance for the case when the compromised state
measurement is available to the local controller of agent i 僆
{1,…, 4} (i.e., di() 6¼ 0) is shown in Fig. 4 using Eq. (7) (i.e.,
vi ðtÞ 0; i ¼ 1; …; 4). Now, to illustrate the results of Theorem
1, we use the proposed local controller ui ðtÞ; i ¼ 1; …; 4; t 0,
for each agent given by Eq. (10) and the local corrective signal
vi ðtÞ; i ¼ 1; …; 4; t 0, given by Eq. (11) with c ¼ 100. For this
case, the system performance in the presence of time-invariant
sensor uncertainties is shown in Fig. 5. As expected, the proposed
distributed control architecture of Theorem 1 allows the state vector of each agent xi ðtÞ; i ¼ 1; …; 4; t 0, to asymptotically track
the state vector of the leader x0 ðtÞ; t 0. Finally, the time evolution of the error signals given by eðtÞ; t 0, and ~dðtÞ; t 0, is
shown, respectively, in Figs. 6 and 7 for the closed-loop system
with the proposed distributed control architecture. The values of
Fig. 11 System trajectories of each agent in Fig. 10
the design gains c and l are chosen to obtain acceptable system
performance without excessive control oscillations.
5.2 Example 2: Time-Varying Sensor Uncertainties Case.
To illustrate the key ideas presented in Sec. 4, consider the same
group of agents as in Example 1 given in Fig. 1, where the dynamics of agent i satisfy
" 1 # "
#"
# " #
0 1:5 x1i ðtÞ
0
x_ i ðtÞ
¼
þ
uðtÞ; i ¼ 1;…; 4; t 0
2 0
1
x2i ðtÞ
x_ 2i ðtÞ
(46)
with
initial
conditions
x1 ð0Þ ¼ ½0; 0
T ; x2 ð0Þ ¼ ½1; 2
T ;
x3 ð0Þ ¼ ½3; 1
T , and x4 ð0Þ ¼ ½2; 2
T . Note that detðAÞ 6¼ 0, where
A is the system matrix of Eq. (46). For this example, we are interested in the problem of approximately driving the state vector of
each agent xi ðtÞ; i ¼ 1; …; 4; t 0, to the state vector of a leader
x0 ðtÞ; t 0, having dynamics given by
Fig. 10 System performance for a group of agents in Example 2 with the local controller given by Eq. (7) (i.e., vi ðtÞ 0; i 5 1; . . . ; 4) when the compromised state measurement is available for feedback
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Fig. 12 System performance for a group of agents in Example 2 with the proposed
local controller given by Eq. (10) and the local corrective signal given by Eq. (32)
when the compromised state measurement is available for feedback
Fig. 13 System trajectories of each agent in Fig. 11
"
x_ 10 ðtÞ
x_ 20 ðtÞ
#
"
¼
0
1:5
2
0
#"
x10 ðtÞ
x20 ðtÞ
#
;
x0 ð0Þ ¼
" #
1
1
;
t 0 (47)
For this problem, A0 ¼ A BK0 holds with K0 ¼ [0, 0] as a direct
consequence of Eqs. (46) and (47).
To design the proposed local controllers, let K ¼ Q1 BT P and
set Q ¼ 0.1I2, l ¼ 1, and c ¼ 6 1=minfgi g; i ¼ 1; …; 4, so that
the LMI given by Eq. (30) for P ¼ S1 (see Remark 4) is satisfied
with
0:0398 0:0037
P¼
(48)
0:0037 0:0360
and hence, K ¼ [0.04, 0.36]. Note that with this selection for K, An
is Hurwitz for all gi 2 specðFðGÞÞ; i ¼ 1; …; 4. The nominal system performance for the case when the uncompromised state
041003-10 / Vol. 139, APRIL 2017
Fig. 14 Time evolution of Eq. (37) in Example 2 with the proposed local controller given by Eq. (10) and the local corrective
signal given by Eq. (32) when the compromised state measurement is available for feedback
measurement is available to the local controller of agent i, i 僆
{1,…, 4}, (i.e., di() ¼ 0), is shown in Figs. 8 and 9 using Eq. (7).
Next, consider the time-varying sensor uncertainty given by Eq.
(5) with
0:1
0:2
; d2 ðtÞ ¼ 1 þ sinð0:85tÞ
d1 ðtÞ ¼ 2 þ sinð0:6tÞ
0:2
0:4
0:3
;
d3 ðtÞ ¼ 4 þ sinð0:25tÞ
0:6
(49)
0:4
d4 ðtÞ ¼ 1 þ sinð0:25tÞ
0:8
(50)
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Acknowledgment
This research was supported in part by the University of Missouri Research Board and the Air Force Office of Scientific
Research under Grant No. FA9550-16-1-0100.
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achievable closed-loop system performance. In this paper, we presented two distributed adaptive control architectures to mitigate
the effect of time-invariant and time-varying sensor uncertainties
in networked multi-agent systems with agents having high-order,
linear dynamics. Specifically, we modeled agent uncertainty
between networked agents using unknown exogenous disturbances and showed that the proposed adaptive control architectures
guarantee asymptotic stability of the closed-loop dynamical system when the exogenous disturbances are time-invariant and uniform ultimate boundedness when the exogenous disturbances are
time-varying. Future extensions will focus on output feedback distributed adaptive control strategies that can suppress the effect of
simultaneous sensor and actuator uncertainties. Generalizations to
agents having nonlinear and uncertain dynamics as well as timevarying communication graph topologies will also be developed.
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Journal of Dynamic Systems, Measurement, and Control
APRIL 2017, Vol. 139 / 041003-11
Fig. 15 Time evolution of Eq. (38) in Example 2 with the proposed local controller given by Eq. (10) and the local corrective
signal given by Eq. (32) when the compromised state measurement is available for feedback
The system performance for the case when the compromised state
measurement is available to the local controller of agent i 僆
{1,…, 4} (i.e., di(t) 6¼ 0) is shown in Figs. 10 and 11 using Eq. (7)
(i.e., vi ðtÞ 0; i ¼ 1; …; 4). Now, to illustrate the results of Theorem
2,
we
use
the
proposed
local
controller
ui ðtÞ; i ¼ 1; …; 4; t 0, for each agent given by Eq. (10) and the
local corrective signal vi ðtÞ; i ¼ 1; …; 4; t 0, given by Eq. (32)
with c ¼ 100, and ^d max ¼ 100. For this case, the system performance in the presence of time-varying sensor uncertainties is shown
in Figs. 12 and 13. As expected, the proposed distributed control
architecture of Theorem 2 allows the state vector of each agent
xi ðtÞ; i ¼ 1; …; 4; t 0, to approximately track the state vector of
the leader x0 ðtÞ; t 0. Finally, the time evolution of the error signals given by Eqs. (37) and (38) is, respectively, shown in Figs. 14
and 15. As for the previous example, the values of the design gains
c and l are chosen to obtain acceptable system performance with
minimal control oscillations. In particular, after choosing c ¼ 100 to
achieve an acceptable small ultimate bound on keðtÞk2 , we chose
l ¼ 1 to achieve an acceptable small ultimate bound on k~dðtÞk2 .
6
Conclusion
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