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Network–Decentralized Control Strategies for Stabilization
Franco Blanchini a, Elisa Franco band Giulia Giordano a
Abstract— We consider the problem of stabilizing a class
of systems formed by a set of decoupled subsystems (nodes)
interconnected through a set of controllers (arcs). Controllers
are network–decentralized, i.e. they use information exclusively
from the nodes they interconnect. This condition requires a
block–structured feedback matrix, having the same structure
as the transpose of the overall input matrix of the system. If
the subsystems do not have common unstable eigenvalues, we
demonstrate that the problem is solvable. In the general case, we
provide sufficient conditions for solvability. When subsystems
are identical and each input agent controls a pair of subsystems
with input matrices having opposite sign (flow networks), we
prove that stabilization is possible if and only if the system
is connected with the external environment. Our proofs are
constructive and lead to structured Linear Matrix Inequalities
(LMIs).
Index Terms—Linear matrix inequalities (LMIs), network analysis
and control, network decentralized control, state feedback.
I. INTRO DUC TIO N AND MOT IVATI ON
Control and coordination of independent units is rele-
vant in many applications, such as platoons of autonomous
vehicles [13], [16], [17], large data communication net-
works [22], [21], [14], [18], [19], inventory management
and production–distribution systems [5], [6], [7], [9], [10],
[27], [28] and network flows in general [5], [2], [25]. These
systems can be viewed as complex systems composed by
naturally independent subunits that interact through designed
control actions. In these networked control systems it is
often too expensive or physically impossible to implement a
centralized controller deciding an optimal strategy based on
information about all the subsystems. Therefore, controllers
have to be computed based on information about a limited
subset of agents/components. Literature on the topic of
decentralized networked control has flourished in the past
decades, yielding a variety of approaches to stabilize [13],
coordinate [12], or synchronize [24], [26] large sets of
systems using locally computed controllers.
In a wide class of applications, the same controller may
affect simultaneously several subsystems in the network. For
instance, in water distribution networks [20], [4] the flow
controlled in a pipe affects the upstream and the downstream
reservoirs simultaneously; in transportation networks, traffic
control in a communication route affects at once the density
of vehicles at both extremities of the route [3], [23]. If
we associate a graph with this kind of networked systems,
controllers are associated with the arcs connecting the nodes
aDipartimento di Matematica e Informatica, Universit`
a degli
Studi di Udine, 33100 Udine, Italy. blanchini@uniud.it,
giulia.giordano@uniud.it
bDepartment of Mechanical Engineering, University of California at
Riverside, Riverside, CA 92521. efranco@engr.ucr.edu
(dynamically independent subunits). The design and synthe-
sis of this type of controllers has been pioneered in [18],
[17], [19]; more recent work is due to [9], [4], although
essentially limited to the case in which subsystems are first–
order integrators.
In this paper we consider the case in which the nodes
are arbitrary subsystems with their own, possibly unstable,
dynamics. Under stabilizability assumptions, we seek lin-
ear network–decentralized [18], [17], [19], [9], [4] state–
feedback controllers in which each control agent (arc) can
use information only from the subsystems (nodes) it con-
nects. This is equivalent to imposing that the feedback
matrix has the same structure as the transpose of the input
matrix. This type of control is intrinsically different from
decentralized control frameworks where several naturally
interacting subsystems are equipped with their own local
controller [29], since we consider control agents which are
associated not with subsystems, but with flow arcs.
Our main results are:
•if the subsystems do not have common unstable eigen-
values, we show that the problem is solvable;
•in the case of (possibly) common eigenvalues, we dis-
cuss general structural sufficient conditions, including a
constrained LMI [8];
•in the case of a single common eigenvalue (typical in
distribution systems), the problem is solvable if and only
if the LMI is feasible;
•in the special case in which all subsystems are equal,
each control agent regulates at most two nodes, and
the input matrices in these nodes have opposite sign
(typical in flow and platoon problems), we prove that
a necessary and sufficient condition for solvability is
that the system is suitably connected with the external
environment.
A. Motivations
In most of the literature on network decentralized dynamic
flow, nodes are buffers modelled by simple integrators [17],
[19], [18]. Some exceptions are first–order node dynam-
ics [18], [9] and systems with a Laplacian state matrix [4].
The general equation for the class of buffer systems is
˙x(t) = Bu(t) + Ed(t)(1)
where Bis the flow matrix, uis the controlled flow and dis
an external signal. Yet, in many cases, the nodes have some
local processing dynamics which are more complex and have
to be taken into account.
Example 1: Consider the model of a water distribution
system, shown in Fig. 1, where each node (circled) represents
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Publisher version DOI: 10.1109/TAC.2014.2331415
Fig. 1: The water distribution network considered in Sections I-A and V
a subsystem with its internal dynamics. Precisely, each node
includes two reservoirs, where water exchange depends on
their relative levels. Different subsystems are connected by
pipes whose flow ucan be controlled. The network has a
constant demand vector d. In Fig. 1, supplementary integra-
tors are added to some reservoirs, so that they asymptotically
achieve exact desired levels. Indeed in previous work [9], [4]
it has been shown that, for systems described by (1), zero
steady state error cannot be assured using static continuous
controllers; yet, discontinuous controllers may not be appli-
cable in flow networks. In this example, zero steady state
error can be guaranteed for all the reservoirs equipped with
a supplementary integrator, for any demand vector d.
II. DEC ENTRALIZED CONTROL O F NETW ORK S:
PROB LEM FO RMU LATIO N
We consider a class of linear, interconnected systems:
˙xi(t) = Aixi(t) + X
j∈Ci
Bij uj(t) + Eid(t)
where xi(t)∈Rniis the state of the i–th subsystem; Ci
is the set that indexes the control subvectors uj∈Rmj,
j= 1, . . . , M , named agents, affecting the i–th subsystem;
Bij represents the effect of control ujon the i–th subsystem;
dis an external signal affecting the i–th subsystem through
matrix Ei. The overall system can be written as
˙x(t) = Ax(t) + Bu(t) + Ed(t)(2)
where x(t)∈Rnincludes the state variables associated with
each subsystem, u(t)∈Rmis the control vector, d(t)∈Rn
is the vector representing an external, non–controllable signal
affecting the system, Eis a generic matrix, while Aand B
are block–structured: A∈Rn×nis a block–diagonal matrix
A=blockdiag{A1, A2, A3, . . . , AN}(3)
while matrix B∈Rn×mis a suitably structured matrix.
Assumption 1: (A, B)is stabilizable.
System (2) can be naturally represented with a hyper-
graph, where the Nsubsystems are associated with nodes
and control agents are associated with hyperarcs. In the
following, for simplicity, hypergraphs and hyperarcs will
be referred to as graphs and arcs. Each control component
uj,j= 1, . . . , M is a vector in Rmjassociated with a
block column of B. Such a block column has zero blocks
Bij ∈Rni×mjcorresponding to all the nodes not directly
affected by agent uj: formally, Bij = 0 if and only if j6∈ Ci.
Denoting by Njthe set that indexes the nodes affected by
agent j, we also have Bij = 0 if and only if i6∈ Nj. All
the block dimensions must be compatible with the block
structure of A, namely PN
i=1 ni=nand PM
i=1 mi=m.
Example 2: For illustrative purposes, let us consider a
system with 4nodes and 6agents, where
A=blockdiag{A1, A2, A3, A4},
B=
B11 B12 0 0 B15 0
0B22 B23 B24 0 0
000B34 0B36
0 0 B43 B44 B45 B46
,
E=blockdiag{0,0,−I, −I}.
We have C1={1,2,5},C2={2,3,4},C3={4,6}and
C4={3,4,5,6}. The agents control the following nodes:
N1={1},N2={1,2},N3={2,4},N4={2,3,4},
N5={1,4},N6={3,4}. The graph corresponding to B
(and E) is shown in Fig. 2.
1 2
34
Fig. 2: The graph corresponding to Example 2: pink squares on solid arcs
indicate controllers, dashed arcs represent external, non–controllable signals
We consider controls restricted to the class:
uj=φ(xi, i ∈ Nj).
Thus each agent ujcan have information from nodes in Nj
only. In the case of linear feedback, we can give the following
equivalent definition.
Definition 1: A control of the form u=−Kx is decen-
tralized in the sense of networks if Khas the same structural
zero blocks as B⊤.
In Example 2, Khas the following structure:
K=
K⊤
11 K⊤
12 0 0 K⊤
15 0
0K⊤
22 K⊤
23 K⊤
24 0 0
000K⊤
34 0K⊤
36
0 0 K⊤
43 K⊤
44 K⊤
45 K⊤
46
⊤
.
III. CAS E OF DI S TI N CT UN STABLE EI GEN VALU E S
In this section we show that under the following assump-
tion, which is a generic property, decentralized stabilizability
is always possible. We refer to all the eigenvalues whose real
part is not strictly negative as “unstable eigenvalues”.
Assumption 2: Two different subsystems do not share
unstable eigenvalues.
Definition 2: The system is node–stabilizable if any sub-
system ican be stabilized using the control inputs in Ci,
namely (Ai,[Bi1Bi2. . . BiM ]) are stabilizable ∀i.
In Example 2 we would have that (A1,[B11 B12 B15]),
(A2,[B22 B23 B24]),(A3,[B34 B36 ]) and
(A4,[B43 B44 B45 B46]) are stabilizable.
We now provide two preliminary results, before we state
our main findings.
Claim 1: Given any state–input matrix pair (F, G), there
exists a Kalman–like transformation (KL-transformation for
short) such that
T−1F T =S R
0U, T −1G=V
0,
where (S, V )is a stabilizable pair and Ucontains only
unreachable unstable eigenvalues.
Claim 2: Consider any system of the form
F=blockdiag{F1, . . . , Fr}, G = [G⊤
1. . . G⊤
r]⊤,
where Gihave the same number of columns and Fi,Fjdo
not share unstable eigenvalues for i6=j. Then, if (Fi, Gi)
are stabilizable pairs, the system is stabilizable. In particular,
node–stabilizability and stabilizability are equivalent.
The proof of Claim 2 follows from the Popov criterion:
(F, G)is stabilizable iff rank [λI −F|G] = nfor all
unstable eigenvalues. Consider an unstable eigenvalue λ, say
of the first block F1, and let ˜
F1=blockdiag{F2, . . . , Fr},
˜
G1= [G⊤
2. . . G⊤
r]⊤. We must have
rank λI −F10G1
0λI −˜
F1˜
G1=n
The condition is true, since [λI −˜
F1]has full rank because
λis an eigenvalue of F1only and [λI −F1|G1]has full rank
in view of the stabilizability of (F1, G1).
Theorem 1: Under Assumption 2, the following condi-
tions are equivalent:
•the system is stabilizable;
•the system is node–stabilizable;
•the system can be stabilized by means of a network–
decentralized control.
Proof: By Claim 2, if the subsystems do not share
unstable eigenvalues, node–stabilizability is equivalent to
stabilizability. If the system is decentralized–stabilizable,
then it is stabilizable. Therefore, we just need to show that
(node) stabilizability implies decentralized stabilizability.
Assume that the system is node–stabilizable. We consider
the first input u1and we rewrite the system so that the p
subsystems in N1are each KL-transformed (with respect to
input u1) and grouped in the first columns of A and B, as
below:
[A||B] =
S1R1. . . 0 0 0 V1X
0U1. . . 0 0 0 0 X
.
.
..
.
.....
.
..
.
..
.
..
.
..
.
.
0 0 ...SpRp0VpX
0 0 . . . 0Up0 0 X
0 0 . . . 0 0 Λ 0 ˜
B
Matrix Λcontains the subsystems not in N1and Xare
entries we can neglect. By rearranging the blocks we get:
[A||B] =
S1. . . 0R1. . . 0 0 V1X
.
.
.....
.
..
.
.....
.
..
.
..
.
..
.
.
0. . . Sp0. . . Rp0VpX
0. . . 0U1. . . 0 0 0 ˜
B1
.
.
.....
.
..
.
.....
.
..
.
..
.
..
.
.
0. . . 00. . . Up0 0 ˜
Bp
0. . . 0 0 . . . 0 Λ 0 ˜
B
Again by Claim 2, we can stabilize the blocks S1. . . Spby
means of the first input: we feed back the substates associated
with these blocks, xS
1(t), . . . , xS
p(t), so that
S1. . . 0
.
.
.....
.
.
0. . . Sp
+
V1
.
.
.
Vp
˜
K1. . . ˜
Kp= Φ1
is stable. We achieve the following block–triangular form
[ˆ
A||B] = Φ1XΓ11 Γ12
¯
0Φ20 Γ22 .(4)
We will not feed back the substates xS
1(t), . . . , xS
p(t)any-
more, so that a) the term denoted as ¯
0in (4) is preserved,
b) Φ1remains untouched. The procedure is iterated by
considering the remaining part:
[Φ2||Γ22] =
U1. . . 0 0 ˜
B1
.
.
.....
.
..
.
..
.
.
0. . . Up0˜
Bp
0. . . 0 Λ ˜
B
Note that this subsystem
•includes the inputs we still have to exploit, u2, . . . , uM;
•has a block–diagonal structure and meets Assumption 2;
•is node–stabilizable, hence stabilizable.
Therefore the problem of its stabilization is exactly as the
one we started with. By exploiting u2and feeding back
components of the state which are not among those of
xS
1(t), . . . , xS
p(t), we reach a new triangular form and so
on. At each step, we deal with the last part [Φk||Γkk ]of a
system of the form
[ˆ
A|| ˆ
B]k=
Φ1X . . . X Γ11 Γ12 . . . Γ1k
0 Φ2. . . X 0 Γ22 . . . Γ2k
.
.
..
.
.....
.
..
.
..
.
.....
.
.
0 0 . . . Φk0 0 . . . Γkk
The assumed node–stabilizability assures that the proce-
dure terminates successfully, because the unstable modes of
the residual system [Φk||Γkk ]are unreachable by the inputs
u1...uk−1, therefore they can necessarily be stabilized by
some of the remaining agents.
The procedure provides a control which might take advantage
of only a subset of the control agents. If this is an issue,
we can fully exploit the available arcs: we find a structured
feedback u=−¯
Kx, we solve the Lyapunov equation to find
aPfor the closed–loop system, and finally we derive a more
suitable Kwith this Pby solving
min kKk2: (A−BK)⊤P+P(A−BK)<0, K ∈ S(B⊤)
This convex optimization problem is always feasible and the
obtained control exploits all available links.
IV. CAS E O F SH ARE D UNS TAB LE EI GEN VALU ES
To consider the case of common unstable eigenvalues,
we first provide a general sufficient condition in terms of a
structured LMI [8]. We will show that this condition becomes
sufficient and necessary under additional assumptions.
Proposition 1: Consider system (2), with Ablock–
diagonal and Bblock–structured. If the following LMI
SA⊤+AS −2γBB⊤<0(5)
has a solution S > 0in the form
S=P−1=blockdiag{P−1
1, P −1
2, . . . , P −1
N},(6)
with Pkof the same dimensions of Ak, then the problem
admits a network–decentralized stabilizing feedback control.
Proof: The LMI is solvable if and only if
(A−γBB⊤P)⊤P+P(A−γBB⊤P)<0(7)
with P=S−1, [11]. The network–decentralized control
K=γB⊤Passures
(A−BK)⊤P+P(A−BK)<0.(8)
The LMI condition is sufficient, but not necessary, to guar-
antee network decentralized stabilizability [8].
Solvability conditions can be provided under additional
assumptions. The first assumption is motivated by Example
1, in which the subsystems share a single unstable eigenvalue
(λ= 0). In this case, if the system is stabilizable then
the LMI is feasible, hence the problem of decentralized
stabilization can be solved.
Proposition 2: Assume that all the matrices Aihave a
single unstable eigenvalue λ≥0of ascent 1(i.e. the largest
Jordan block associated with λhas dimension 1). Then the
following conditions are equivalent:
•the system is stabilizable;
•the system can be stabilized by means of a network–
decentralized control;
•the LMI (5) has a structured solution (6).
Proof: We prove that stabilizability implies that the
structured LMI is solvable. The remaining proofs are trivial.
We apply to the blocks Akseparate transformations Tk, so
that T−1
kAkTk=blockdiag{λIk,ˆ
AS
k}, where the stable part
ˆ
AS
khas the identity Ias Lyapunov matrix:
(ˆ
AS
k)⊤+ ( ˆ
AS
k) = −Qk<0
By rearranging all the blocks and joining all λIk, we can put
the system in the form
ˆ
A=λI 0
0ˆ
ΛS,ˆ
B=ˆ
Bλ
ˆ
BS
where ˆ
ΛS=blockdiag{ˆ
AS
1,ˆ
AS
2, . . . , ˆ
AS
N}is stable.
If the system is stabilizable, then ˆ
Bλhas full row rank, as
it can be immediately seen by means of the Popov criterion.
We consider the candidate block–diagonal matrix
ˆ
S=I0
0µI ,
where µ > 0has to be decided, and the feedback
u=−γ[ˆ
B⊤
λ0]x=−ˆ
Kx,
which is network–decentralized. Then
ˆ
S(ˆ
A−ˆ
Bˆ
K)⊤+ ( ˆ
A−ˆ
Bˆ
K)ˆ
S=2(λI −γˆ
Bλˆ
B⊤
λ)−γˆ
Bλˆ
B⊤
S
−γˆ
BSˆ
B⊤
λ
−µˆ
Q,
where ˆ
Q=blockdiag{Q1, Q2, . . . , QN}. Since ˆ
Bλˆ
B⊤
λ>0,
for γlarge enough the block 2(λI −γˆ
Bλˆ
B⊤
λ)is negative
definite. Let us fix such a γ. Since ˆ
Q > 0, we can
subsequently take µlarge enough to assure that
ˆ
S(ˆ
A−ˆ
Bˆ
K)⊤+ ( ˆ
A−ˆ
Bˆ
K)ˆ
S < 0.(9)
By using the backward transformations, we restore all the
blocks to the original position and thus we find a structured S
as desired. Then we take P=S−1, which is also structured,
to get (7). Thus (5) is satisfied with S > 0structured.
To further investigate the problem, we introduce the follow-
ing definitions.
Definition 3: The network is locally stabilizable if each
agent uican stabilize each of the subsystems in Ni.
Note that this by no means implies that the agent can stabilize
simultaneously more than one subsystem in Ni.
Definition 4: The system is structurally triangularizable if
there exist a) an ordering of the nodes and b) a selection and
ordering of the agents such that the resulting Bhas a block
triangular structure.
For instance, the system in Example 2 is structurally triangu-
larizable by ordering the nodes as 1,2,4,3and disregarding
the last two agents (i.e. selecting the first four). To present
our next result, we need to consider the case in which some
of the subsystems are open–loop stable.
Definition 5: Given the structured system (A, B), we de-
fine the extended system (A, Bext)as follows. For each node
iwhich is asymptotically stable, Bis extended by adding a
fictitious block with nicolumns, which has an identity matrix
corresponding to Aiand zero blocks elsewhere.
For instance, if in the system of Example 2 the second
subsystem is asymptotically stable, we have to extend Bas
Bext := B|[0I0 0 ]⊤
It is understood that this is a fictitious system which cannot
be considered in practice. The following theorem holds.
Theorem 2: If the extended system is triangularizable and
locally stabilizable, then (5)–(6) are feasible.
Proof: If the system can be triangularized, we may
assume Bext in the form
Bext =¯
B˜
B=
¯
B11 X . . . X ˜
B1
0¯
B22 . . . X ˜
B2
.
.
..
.
.....
.
..
.
.
0 0 . . . ¯
BNN ˜
BN
where X are elements we can neglect. By assumption, ∀i,
either (Ai,¯
Bii)is stabilizable or Aiis already asymptotically
stable (then ¯
Bii =I, a fictitious column). Therefore, by
adopting the control
K=¯
K
0,¯
K=blockdiag{K1, K2. . . , KN},
where Kiis such that Ai−¯
BiiKiis asymptotically stable, the
resulting closed loop matrix is block triangular. Any asymp-
totically stable block–triangular matrix admits a Lyapunov
matrix which is block diagonal and satisfies (8); this means
that (5)–(6) are feasible.
Theorem 2 has demanding assumptions. However, there
are interesting structural assumptions under which LMI
solvability is guaranteed, such as when each control agent is
associated with an arc of a proper graph (not a hypergraph),
thus affects at most two subsystems.
Definition 6: A network is connected if the corresponding
graph is connected; it is connected with the external envi-
ronment if, in addition, the input matrix Bhas at least one
block–column with a single non–zero block.
Corollary 1: Assume that the system is locally stabiliz-
able and each agent controls at most two subsystems. If
the system is connected with the external environment, then
(5)–(6) are feasible, hence a decentralized stabilizing control
exists.
Proof: It is immediate, because, given a node connected
to the external environment, we can find a spanning tree and
form the triangular structure of Bstarting from this node.
An interesting case is that of flow networks, in which we
have a family of identical subsystems and each network link
connects a pair of nodes in such a way that its action has
opposite effects (inflow and outflow).
Proposition 3: Given a connected network, assume that
all the diagonal blocks of matrix Aare equal, Ai=A1, for
i= 1, . . . , N , and that all non–zero blocks of Bare ±B1.
Assume that there are at most two non–zero blocks in each
block column of matrix Band that, if they are two, they have
opposite sign. Then the following conditions are equivalent:
•the system is stabilizable;
•the system is locally stabilizable;
•the structured LMI is solvable;
•either A1is stable or the network is connected with the
external environment.
Proof: Ignoring the trivial case of A1stable, we show
that stabilizability implies both external connection and local
stabilizability; the rest of the proof follows from Corollary 1.
According to the Popov criterion, the system is stabilizable
iff, for all unstable eigenvalues λ,z⊤[λI −A|B] = 0 implies
z= 0. Consider now the vector z⊤= [z⊤
1z⊤
1. . . z⊤
1],
where z16= 0 is a left eigenvector associated with A1, hence
z⊤[λI −A] = 0. By contradiction, assume the network is not
connected with the external environment. Then, each non–
zero block column would have exactly two blocks of opposite
sign, so we would also have z⊤B= 0 and the system would
not be stabilizable.
Local stabilizability can be proved in a similar way.
Indeed, if the subsystems were not locally stabilizable, there
would be an unstable eigenvalue λsuch that z⊤
1[λI −
A1|B1] = 0,z16= 0, but defining again z⊤=
[z⊤
1z⊤
1. . . z⊤
1]6= 0 we would have z⊤[λI −A|B] = 0.
V. EX AMP LE
We revisit Example 1 and we assume that the nodes in
Fig. 1 have the following dynamics:
Ai="−αiβi0
αi−βi0
0 1 0#where βi= 0 for i∈ {1,2,4}.
In each node, the first two states are the reservoirs volumes,
while the third represents the local integrator. Constants αi
[min−1] and βi[min−1] depend on the size of the reservoirs
and the diameter of the connecting pipes; α1= 15,α2= 20,
α3= 16,α4= 16.7,α5= 14,β3= 12 and β5= 22.
The overall system has the same structure as model (2),
where A=blockdiag{A1, A2, A3, A4, A5},
B=
Bu−Bd0 0 0 0
0Bu−Bd0 0 −Bd
0 0 Bd−Bu0 0
0 0 0 BdBu0
0 0 0 0 −BuBd
,
Bd=010⊤, Bu=100⊤,
d=−[0 1 0 0 1 0 0 1 0 0 1 0 0 1 0]⊤and E=I. Since the
only unstable eigenvalue is λ= 0, which has ascent one and
is common to all the subsystems, we can apply the results
of Proposition 2 to regulate the system with a decentralized
control; Kis obtained through the direct solution of the
LMI (5)–(6), numerically solved using the MATLAB LMI
toolbox [15]. To enforce a certain speed of convergence, in
the LMI Acan be replaced with A+σI,σ > 0, so that
the closed loop eigenvalues have real part less than −σ. We
have used σ= 0.15.
In Fig. 3 and 4 the decentralized control is com-
pared with a centralized LQ control, with state weight-
ing matrix Iand input weighting matrix I/2;x(0) =
[−8.80 3.63 −9.15 −8.57 0.43 −8.06 6.36 6.35 4.44 −7.00 3.19 0.37
9.45 2.97 6.00]⊤. As expected, the equilibrium values of the
states equipped with integral control are smoothly recovered.
VI. CO N CL U DI NG RE MAR KS
We have proved that the problem of designing a network–
decentralized control to stabilize a system formed by a set
of independent subsystems is solvable when the subsystems
have no common unstable eigenvalues. In the general case,
we have provided a structured LMI condition which in
principle is only sufficient, but we have seen that such an
LMI is always feasible in particular cases, e.g. that of flow
0 5 10 15 20
10
5
0
5
10
Time![min]
Reservoir!volumes![m3]
Optimal!LQ!control!(Riccati)
x1x2x4x5x7x8x10 x11 x13 x14
0 5 10 15 20
10
5
0
5
10
Time![min]
Reservoir!volumes![m3]
Network!decentralized!control!(LMI)
x1x2x4x5x7x8x10 x11 x13 x14
Fig. 3: Reservoir volumes evolution: decentralized (top) and optimal (bot-
tom) control.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−10
−5
0
5
10
Time [min]
Reservoir volumes [m3]
Comparison between centralized (LQ) and decentralized strategy
x2 LQ x2 LMI x10 LQ x10 LMI x13 LQ x13 LMI
Fig. 4: Detailed simulation for reservoir volumes x2,x10 and x13 (zoomed
from Fig. 3).
networks and that of subsystems with a single common
unstable eigenvalue of ascent 1. Unfortunately, the general
question whether, under possibly common eigenvalues, sta-
bilization implies stabilizability in the sense of networks is
still unsolved and is left as a subject of further investigation.
Another interesting problem is how can we solve the LMI
efficiently: when BB⊤is large and sufficiently sparse, ideas
from chordal decomposition methods [1] could be promising.
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