Nonlinear Consensus Strategies for Multi-Agent Networks in Presence of Communication Delays and Switching Topologies: Real-Time Receding Horizon Approach

2.3 MB·

by  Fei Sun

Nonlinear Consensus Strategies for Multi-Agent
Networks in Presence of Communication Delays and
Switching Topologies:
Real-Time Receding Horizon Approach
6
1 Fei Suna,∗, Kamran Turkoglua,∗
0
2 aAerospace Engineering, San Jos´e State University, San Jose, CA 95192

n
a
J

5
1 Abstract

] This paper presents a novel framework which combines a non-iterative solu-
C
O tion of Real-Time Nonlinear Receding Horizon Control (NRHC) methodology
.
h to achieve consensus within complex network topologies with existing time-
t
a delaysandinpresenceofswitchingtopologies. Inthisformulation,wesolvethe
m
distributednonlinearoptimizationproblemformulti-agentnetworksystemsdi-
[

rectly, in real-time, without any dependency on iterative processes, where the
1
v
stabilityandconvergenceguaranteesareprovidedforthesolution. Threebench-
7
9 mark examples on non-linear chaotic systems provide validated results which
0
4 demonstrate the significant outcomes of such methodology.
0
. Keywords: multi-agent consensus problems, nonlinear receding horizon
1
0 control, real-time optimization, switching topologies
6
1
:
v
1. INTRODUCTION
i
X
r The important problem of finding distributed control laws governing net-
a
work and/or corresponding multi-agent dynamics has been of interest within
the control research community the for the recent decade. One important as-
pect of this research effort is to answer the existence and uniqueness properties
of a distributed control law (and solution) that can lead each individual agent
∗Email: [email protected],[email protected]
Preprint submitted to Elsevier January 19, 2016
within the network to a common state value, which constitutes the core of the
(multi-agent) network consensus problem.
For several years, the complex and sophisticated nature of multi-agent con-
sensus problem has been serving as a fruitful research field for the application
of advanced control algorithms (such as formation flights, power grid networks,
flock dynamics, cooperative control ... etc.) ([1]-[13]).
So far, in existing literature, consensus methodologies have been widely ex-
ploredformulti-agentlinear dynamicalsystemsinmanyimportantandground-
breaking studies ([14] - [16]). However, there exists some important studies
which also investigate nonlinear consensus problems in multi agent systems.
For example, Bausso et al. (2006) investigated a nonlinear protocol design that
permits consensus on a general set of values [17]. In Qu et al. (2007) em-
phasis was put on nonlinear cooperative control for consensus of nonlinear and
heterogeneous systems [18]. Liu et al. (2013) demonstrates the applicability
of a variable transformation method to convert a general nonlinear consensus
problem to a partial stability problem [19]. Franco et al. (2008) investigates
an important connection between nonlinear consensus problems and receding
horizon methodology with existing time-delay information and constraints [20].
In recent years, with the fascinating results improvements obtained in mi-
croprocessor related technologies, the applicability of real-time based control
methodologies, and more specifically Receding Horizon Control based method-
ologiesbecamemoreofareality. Inthatsense,recedinghorizoncontrolemerged
asoneoftheexistingcontrolmethodologiesthatcouldbeadaptedtoconsensus
problem of multi-agent dynamics, for real-time solutions. Based on this ap-
proach, there have been many results developed for consensus problems and its
applications. The work of [22] presented a distributed receding horizon control
law for dynamically coupled nonlinear systems based on its linearization repre-
sentative. Therobustdistributedrecedinghorizoncontrolmethodswerestudied
in [23] for nonlinear systems with coupled constraints and communication de-
lays. [24] proposed a robust distributed model predictive control methods for
nonlinear systems subject to external disturbances. In a very recent study Qiu
2
and Duan (2014) investigated brain-storm type of optimization in combination
withrecedinghorizoncontrolstrategiesforUAVformationflightdynamics,[25].
Oneveryinteresting(andatthesametimenatural)extensiontotheproblem
istheconsensusofnonlinearmulti-agentnetworkswithswitchingtopologiesand
time delays. Li and Qu (2014) [34] provide an interesting approach to the finite
time consensus problem of distributed nonlinear systems under general setting
ofdirectedandswitchingtopologies. JiaandTang(2012)[35]workedonaspe-
cific directed graph topology to achieve consensus within nonlinear agents with
switching topology and communication delays. In their studies, Ding and Guo
(2015)[33]concentratedtheireffortonsampled-databasedleader-followingcon-
sensus for nonlinear multi-agent systems with Markovian switching topologies
and communication delays.
In all of those existing valuable studies, one common fact remains as the
iterative nature of the problem solution methodology, which enforces the itera-
tive nature of the solution to still exist. This heavily deters the functionality of
real-time solution methodologies.
In this paper we aim to address this issue, and for this purpose we propose
a novel framework which combines non-iterative solution of Real-Time Non-
linear Receding Horizon Control (NRHC) methodology to achieve consensus
within complex network topologies with existing time-delays and in presence of
switching topologies. With this we propose the following three novelties: (i)
non-iterative solution is achieved in real-time which eliminates any need for an
iterativealgorithm, (ii)multi-agentconsensusproblemissolvedinreal-timefor
switching topologies and existing time-delays, (iii) stability and convergence of
the consensus is guaranteed in existence of complex switching topologies, (iv)
stabilityandconvergenceoftheconsensusisguaranteedinexistenceofinherent
communication delays.
The paper is organized as follows: In Section-2, problem statement is de-
fined. Distributed NRHC protocol is presented in Section-3 . Section-4 pro-
vides important results on convergence and stability assessment of the pre-
sented methodology, where in Section-5 multi-agent network dynamics with
3
inherent communication delays are discussed. Section-6 provides three exam-
ple cases demonstrating significant outcomes of the proposed methodology, and
with Section-7, the paper is concluded.
2. PRELIMINARIES AND PROBLEM STATEMENT
Inthissection,weintroducethedefinitionsandnotationsfromgraphtheory
and matrix theory. Then we formulate the main problem to be studied.
Inthiswork,Rdenotestherealspace. ForarealmatrixA,itstransposeand
inverse are denoted as AT and A−1, respectively. The symbol ⊗ represents the
Kroneckerproduct. FormatricesX andP,theEuclideannormofX isdenoted
√
by (cid:107)X(cid:107) and the P-weighted norm of X is denoted by (cid:107)X(cid:107) = P XTPX.
P
I stands for the identity matrix of dimension n. Given a matrix P, P > 0
n
(P < 0) represents that the matrix is positive definite (or negative definite).
Here, we define the column operation col(x ,x ,··· ,x ) as (xT,xT,··· ,xT)T
1 2 n 1 2 n
where x ,x ,··· ,x are column vectors.
1 2 n
Consider a multi-agent system of M nonlinear agents. For each agent i, the
dynamic system is given by:
x˙ =f(x ,x ,t)=F(x )+u , (1)
i i −i i i
where x = (x ,x ,··· ,x )T is the state vector of the ith oscillator, x are
i i1 i2 in −i
the collection of agent i’s neighbor’s states, the function F(·) is the correspond-
ing nonlinear vector field, and u is the control input of agent i. Here, function
i
F(·)satisfiestheglobalLipschitzcondition. Thereforethereexistspositivecon-
stant β such that
i
(cid:107)F(x )−F(x )(cid:107)≤β (cid:107)x −x (cid:107).
i j i i j
This condition is satisfied if the Jacobians ∂Fi are uniformly bounded.
∂xi
There exists a communication network among these agents and the network
can be described as an undirected or directed graph G = (V,E,C). Here V =
{1,2,··· ,M} denotes the node set and E ⊂ V ×V denotes the edge set. A =
[a ] ∈ RM×M is the adjacency matrix. In this framework, if there exists a
ij
4
connection between i and j nodes(agents), then a >0; otherwise, a =0. We
ij ij
assumethereisnoself-circleinthegraphG,i.e.,a =0. Apathisasequenceof
ii
connectededgesinagraph. Ifthereisapathbetweenanytwonodes,thegraph
is said to be connected. If A is a symmetric matrix, G is called an undirected
graph. The set of neighbors of node i is denoted by N = {j|(i,j) ⊂ E}. The
i
in-degree of agent i is denoted as deg = (cid:80)M−1a and the degree matrix is
i j=1 ij
denoted as D =diag(deg ,··· ,deg ). The Laplacian matrix of G is described
1 M
as L=D−A.
In this paper, we consider the multi-agent network systems with switching
topologies, where interconnected structures of the network vary with respect to
time. For given formulation, denote P as an index set and σ(t):[0,∞)→P be
a switching signal that is defined as a piecewise function. At each time t, the
graph is represented as G and {G¯ |p ∈ P} include all possible graph on the
σ(t) p
node set V. G¯ is denoted as union of all possible subgraphs G . Here, each
σ(t)
subgraphmaybedisconnected, wherethenetworkbecomesjointlyconnectedif
the collection G¯ contains a spanning tree. In the light of these, we provide the
following definition:
Definition-1: The nonlinear multi-agent network system, given in (1), is
said to achieve consensus if,
lim =(cid:107)x (t)−x (t)(cid:107)=0, j =1,··· ,M, (2)
i j
t→∞
is satisfied under switching topology G , with a distributed control protocol
σ(t)
u =µ(x ,x ). Herex ’sarethecollectionofagenti’sneighbor’sstates(i.e.,
i i −i −i
x ={x ,j ∈N }).
−i j i
In such formulation, the main goal of this study becomes to design a real-
time nonlinear receding horizon control based distributed control strategy u =
i
µ(x ,x ),foreachagenti,thatwillachieveconsensus,withinthegivenswitch-
i −i
ing network topology and geometric constraints.
5
3. DISTRIBUTEDNONLINEARRECEDINGHORIZONCONTROL
PROTOCOL
Inthisframework,thefollowingoptimizationproblemisutilizedtogenerate
the local consensus protocol within the given network, for each specific agent i:
Problem-1:
u∗(t)=argminJ (x (t),u (t),x (t)) (3)
i i i i −i
ui(t)
subject to
x˙ (t)=F(x (t))+u (t),
i i i
where the performance index is designed as follows:
1(cid:90) t+T
J = ϕ + L (x ,x ,u ),
i i 2 i i −i i
t
(cid:88)
= a (cid:107)x (t+T)−x (t+T)(cid:107)2
ij i j QiN (4)
j∈Ni
1(cid:90) t+T (cid:88)
+ ( a (cid:107)x (τ)−x (τ)(cid:107)2 +(cid:107)u (cid:107)2 )dτ,
2 ij i j Qi i Ri
t j∈Ni
Here Q > 0, Q > 0 and R > 0 are symmetric matrices, and T defines the
iN i i
horizon. In addition, ϕ is used to describe(and define) the terminal cost for
i
each agent.
In this problem formulation, a control scheme is utilized to be able to deal
with the nonlinear nature of the topological graph under scrutiny. With this,
it is desired to solve the nonlinear optimization problem directly, in real-time,
without any dependency on iterative processes.
With the construction of the cost function, as given in (4), the consen-
sus problem is converted into an optimization procedure. For this purpose,
we utilize the powerful nature of real-time nonlinear receding horizon control
algorithm to generate the distributed consensus protocol by minimizing the as-
sociated cost function. In this context, each agent only needs to obtain its
neighbors’ information once via the given network which is more efficient than
the centralized control strategy (and the other distributed strategies) that in-
volve multiple information exchanges and predicted trajectories of states. The
6
performance index evaluates the performance from the present time t to the
finite future t+T, and then is minimized for each time segment t starting from
x (t). With this structure, it is possible to convert the present receding horizon
i
control problem into a family of finite horizon optimal control problems on the
artificial τ axis parametrized by time t.
Accordingtothefirst-ordernecessaryconditionsofoptimality(i.e. forδJ =
i
0),alocal two-pointboundary-valueproblem(TPBVP)[36]isformedasfollows:
Λ ∗(τ,t)=−HT,
iτ xi
Λ∗(T,t)=ϕT [x∗(T,t)],
i xi (5)
x ∗(τ,t)=HT ,x∗(0,t)=x (t),
iτ Λi i i
H =0.
ui
where Λ denotes the costate of each agent i and H is the Hamiltonian which
i i
is defined as
H =L +Λ∗Tx˙
i i i i
1 (cid:88) (6)
= ( a (cid:107)x (τ)−x (τ)(cid:107)2 +(cid:107)u (cid:107)2 )+Λ∗T(F(x )+u ).
2 ij i j Qi i Ri i i i
j∈Ni
Then we have
(cid:88)
Λ ∗(τ,t)=−[ Q a (x (τ)−x (τ))+ΛTF (x )],
iτ i ij i j i xi i
j∈Ni (7)
(cid:88)
Λ∗(T,t)= Q a (x (τ)−x (τ)).
i iN ij i j
j∈Ni
In (5)-(7), ( )∗ denotes a variable in the optimal control problem so as to
distinguish it from its correspondence in the original problem. In this notation,
H denotes the partial derivative of H with respect to x , and so on.
xi i
In this methodology, since the state and co-state at τ = T are determined
by the TPBVP in Eq.(5) from the state and co-state at τ =0, the TPBVP can
be regarded as a nonlinear algebraic equation with respect to the co-state at
τ =0 as
P (Λ (t),x (t),T,t)=Λ∗(T,t)−ϕT [x∗(T,t)]=0, (8)
i i i i xi i
7
where Λ (t) denotes the co-state at τ = 0. The actual local control input for
i
each agent is then given by
u (t)=arg{H [x (t),Λ (t),u (t)]=0}. (9)
i ui i i i
Inthisformulation,theoptimalcontrolu (t)canbecalculateddirectlyfrom
i
Eq.(9) based on x (t) and Λ (t) information, where the ordinary differential
i i
equation of λ(t) can be solved numerically from Eq.(8), in real-time, without
any need of an iterative optimization routine. Since the nonlinear equation
Pi(Λi(t),xi(t),T,t) has to be satisfied at any time t, ddPti = 0 holds along
the trajectory of the closed-loop system of the receding horizon control. If T
is a smooth function of time t, it becomes possible to track the solution of
P (Λ (t),x (t),T,t) with respect to time. However, numerical errors associated
i i i
with the solution may accumulate as the integration proceeds in practice, and
therefore some correction techniques are required to correct such errors in the
solution. To address this problem, a stabilized continuation method [37, 38, 39,
40] is used. According to this method, it is possible to rewrite the statement as
dP
i =A P , (10)
dt s i
where A is a Hurwitz matrix, that helps the solution converge to zero, expo-
s
nentially.
Toevaluatetheoptimalcontrol(bycomputingderivativeofΛ (t)=Λ∗(0,t))
i i
in real time, we consider the partial differentiation of (5) with respect to time
t,
δx˙ =f δx +f δu ,
i xi i ui i
δΛ˙ =−H δx −H δΛ −H δu (11)
i xixi i xiΛi i xiui i
0=H δx +fTδΛ +H δu .
uixi i ui i uiui i
Since δu =−H−1 (H δx +fTδΛ ), we have
i uiui uixi i ui i
δx˙ =(f −f H−1 H )δx −f H−1 fTδΛ ,
i xi ui uiui uiui i ui uiui ui i
δΛ˙ =−(H −H H−1 H )δx −(fT −H H−1 fT)δΛ ,
i xixi xiui uiui uixi i xi xiui uiui ui i
8
which leads to the following form of a linear differential equation:
    
∂ xi∗t −xi∗τ = Ai −Bixi∗t −xi∗τ  (12)
∂τ Λ ∗−Λ ∗ −C −AT Λ ∗−Λ ∗
it iτ i i it iτ
where
A =f −f H−1 H ,
i xi ui uiui uixi
B =f H−1 fT,
i ui uiui ui
C =H −H H−1 H .
i xixi xiui uiui uixi
And the matrix H should be non-singular.
uiui
In order to reduce the computational cost without resorting to any approxi-
mationtechnique,thebackward-sweepmethod[36,39,40]isimplemented. The
derivative of the function P with respect to time is given by
i
dP dT
i =Λ ∗(T,t)−ϕ x ∗(T,t)+[Λ ∗(T,t)−ϕ x ∗(T,t)] , (13)
dt it xixi it iτ xixi iτ dt
where x∗ and Λ ∗ are given by (5).
τ iτ
The relationship between the co-state and other variables is assumed as
follows:
Λ ∗−Λ ∗ =S (τ,t)(x ∗−x ∗)+c (τ,t). (14)
it iτ i it iτ i
which leads to
S (T,t)=ϕ | ,
i xixi τ=T
(15)
dT
c (T,t)=(HT +ϕ f)| (1+ )+A P .
i xi xixi τ=T dt s i
Accordingto(14)and(12),itbecomespossibletoformthefollowingdifferential
equations:
∂S
i =−ATS −S A +S B S −C ,
∂τ i i i i i i i i
(16)
∂c
i =−(AT −S B )c .
∂τ i i i i
Based on (14), the differential equation of the co-state to be integrated in real
time is obtained as:
dΛ (t)
i =−HT +c (0,t). (17)
dt xi i
9
The NRHC method for computing the distributed optimal control u (t) is
i
summarized in Algorithm-1, where t >0 denotes the sampling time.
s
Algorithm 1 u∗(t)=argmin J (x (t),u (t),x (t))
i ui(t) i i i −i
(1) Set t=0 and initial state x =x (0).
i i
(2) For t(cid:48) ∈ [t,t+t ], integrate the defined TPBVP in (7) forward from t to
s
t+T, then integrate (16) backward with terminal conditions provided in (15)
from t+T to t.
(3) Integrate the differential equation of Λ (t), from t to t+t .
i s
(4) At time t+t , compute u∗ by Eq.(9) with the terminal values of x (t) and
s i i
Λ (t).
i
(5) Set t=t+t , return to Step-(2).
s
Lemma-1: The cost function, defined in Eq. (4), is strictly convex and
guarantees the global minimum.
Proof: Since all weighting functions maintain positive definite nature in
their structure (such as Q > 0, Q > 0, R > 0), from the Karush-Kuhn-
iN i i
Tucker(KKT) conditions [41], the proposed method guarantees the global min-
ima. (cid:4)
4. CONVERGENCE AND STABILITY ANALYSIS
For the sake of clarity, and without loss of generality, here we define the
consensus error as
δ (t)=x (t)−x (t)
1 i 1
for all i and ∆(t) = col(δ (t),δ (t),··· ,δ (t)). The optimal control U is de-
1 2 M
noted as U=col(u ,··· ,u ) for all i.
1 M
The cost function can be written as
1(cid:90) t+T
J =Φ+ [∆∗TQ∆∗+U∗TRU∗]dτ, (18)
2
t
(cid:80)
where Φ= ϕ , Q=col(Q ,··· ,Q ) and R=col(R ,··· ,R ).
i∈M i 1 M 1 M
10