Download Controllability Transition and Nonlocality in Network Control

Controllability Transition and
Nonlocality in Network Control
Jie Sun
Department of Mathematics and Computer Science
Clarkson University
Adilson E. Motter
Department of Physics and Astronomy
Northwestern University
Dec 6, 2013 (Network Frontier Workshop)
Formulation of the Classical Control Problem
(
ẋ = f (x, u, ⇠)
y = g(x, v, ⌘)
u
x
⇠
v
y
⌘
Common assumptions:
The form of f and g are known.
The states of u, v, and y can be observed relatively accurately.
The “controllability” question:
Is it always possible to construct a control signal u(t) that steers the system
from an arbitrary initial state to an arbitrary final state in finite time?
The “observability” question:
Is it always possible to uniquely determine x(t) from y(t)?
Nonlinear controllability is generally an open problem.
G.W. Haynes and H. Hermes, “Nonlinear Controllability via Lie Theory”, SIAM J. Control (1970).
H.J. Sussmann and V. Jurdjevic, “Controllability of Nonlinear Systems”, J. Diff. Eqs. (1973).
R. Hermann and A.J. Krener, “Nonlinear Controllability and Observability”, IEEE Trans. Aut. Contr. (1977).
Applications of Controlling Network Dynamics
genes
species
neurons
social interactions
weather
S.P. Cornelius and A.E. Motter, “Realistic Control of Network Dynamics”, Nat. Commun. (2013).
Y.-Y. Liu, J.-J. Slotine, and A.-L. Barabasi, “Controllability of Complex Networks”, Nature (2011).
Network Control in the Simplest Form
dx(t)
= Ax(t) + Bu(t)
dt
u
x
(
x(t) : states of all nodes at time t
u(t) : control inputs at time t
(
A : network adjacency matrix
B : control matrix
R.E. Kalman, “Mathematical Description of Linear Dynamical Systems”, J.S.I.A.M. Control (1963).
Example: a network with 3 nodes and 1 control input
a11
@a21
A
=
1
0
0
1
b11 0 0
B = @ 0 0 0A
b31 0 0
u1
x1 1
2
3
x2
x3
0
0
0
a32
1
a31
a23 A
0
0
1
b11 u1 (t)
Bu(t) = @ 0 A
b31 u1 (t)
Geometry of Control Trajectories
If the system is controllable, what do control trajectories look like?
x(0)
initial state
x(0)
vs
final state
x(1)
x(1)
Mathematical characterization of the control trajectories at a state:
Strictly Locally Stable (SLC)
"
"
x
Not SLC
(1)
x
(0)
x(1)
x(0)
Note: SLC is different from the notion of “locally controllable”.
How Many States are SLC?
Example: a “network” with 2 nodes and 1 control input
(
1
1
ẋ1 = x1 + u1 (t)
ẋ2 = x1
2
0.6
Whenever the final state is
below the initial state x(0),
any control trajectory has
to cross the line x1 = 0.
x(0)
x2
0.4
0.2
−0.5
The state x(0) is not SLC.
−0.25
0
x1
0.25
0.5
Almost all states are not SLC unless all node are directly controlled.
The origin is the only state that is SLC if A is nonsingular.
Nonlocality of Control Trajectories: Numerics
Goal: control the system from x(0) to x(1) in the time interval [t0 , t1 ].
Control trajectory that minimizes the control energy:
x(t) =
(
(t, t0 ) x(0) + W (t0 , t)W
1
(t0 , t1 )[ (t0 , t1 )x(1)
x(0) ]
— transition matrix
(t , t) = e
R t1
W (t0 , t1 ) = t0 (t0 , t)BB T T (t0 , t)dt — controllability Gramian
(t0 t)A
0
Numerical Results for Weighted ER Networks
L
x(1)
4
10
2
10
0
10
−2
10
at the origin
at other states
−4
10
−6
10
−5
10
−4
−3
−2
10
10
10
Target distance,
−1
10
n = 100,
(b)
Length of control trajectories, L
x(0)
Length of control trajectories, L
(a)
n = 100, p = 0.1, q = 25
= 10
2
p = 0.1
p = 0.2
5
10
3
10
1
10
−1
10
0
20
40
60
80
100
Number of control inputs, q
(Theory confirmed) Almost all states are not SLC, except for the origin.
Control trajectories become shorter as more nodes are controlled.
Control Trajectory vs Controllability Gramian
The (controllability) Gramian is a positive semi-definite matrix of size n.
The Gramian is strictly positive definite if and only if the system is controllable.
The condition number of the controllability Gramian is defined as:
(W ) = kW k · kW
k )
(W ) = 1/(W )
reciprocal condition number
(b)
0
10
p = 0.1
p = 0.2
−4
10
5
−8
10
10
3
L 10
−12
10
−1
10
−16
10
1
10
0
−15
10
−10
10
−5
10
0
10
20
40
60
80 100
Number of control inputs, q
Reciprocal condition
number,
Reciprocal condition
number,
(a)
1
0
10
−5
10
−20
10
−35
10
−6
10
0
5 10 15
−12
10
−18
10
50
q = 30
q = 40
q = 50
150
250
Network size, n
Length of the control trajectories strongly correlates with the
reciprocal condition number of the controllability Gramian.
Reciprocal condition number decreases exponentially as the
number of nodes in the network increases.
Analytical Controllability Conditions
A linear time-invariant system is controllable if and only if the
controllability Gramian has full rank.
The system is controllable if and only if K has full row rank.
Kalman’s controllability matrix K = [B, AB, A2 B, . . . , An 1 B]
Network examples:
controllable
1
1
uncontrollable
a
a
2
2
b
3
c
0
1
4
B0
K=B
@0
0
1
0
a
0
0
0
0
ab
0
0
b
1
1
B0
0
B
K
=
C
@0
0 C
0 A
0
abc
1
c
3
0
a
b
c
4
0
0
0
0
controllable
a
2
1
0
0C
C
0A
0
1
b
3
1
c
4
0
1
············
B· · · · · · · · · · · ·C
C
K=B
@· · · · · · · · · · · ·A
············
C.-T. Lin, “Structural Controllability”, IEEE Trans. Aut. Contr. (1974). N.J. Cowan et al., PLoS ONE (2013).
Y.-Y. Liu, J.-J. Slotine, and A.-L. Barabasi, “Controllability of Complex Networks”, Nature (2011).
With self-loops, all networks are structurally controllable!
Numerical Network Controllability
Question: for a system that is (theoretically) controllable,
does the numerical control trajectory reach the final state?
ER networks, n = 100,
= 10
2
success rate, r
1
always succeeds when
all nodes are controlled
0.8
0.6
0.4
(numerical) network
controllability transition
0.2
0
0
always fails when only a
single node is controlled
20
40
60
80
100
number of control inputs, q
D. Achlioptas, R.M. D’Souza, and J. Spencer, “Explosive Percolation in Random Networks”, Science (2009).
J. Gomez-Gardenes, S. Gomez, A. Arenas, and Y. Moreno, “Explosive
Synchronization Transitions in Scale-Free Networks”, PRL (2011).
P. Ji, T.K.DM. Peron, P.J. Menck, F.A. Rodrigues, and J. Kurths,
“Cluster Explosive Synchronization in Complex Networks”, PRL (2013).
Numerical Network Controllability Transition
success rate, r
transition width, q/n
0.8
0.6
0.4
0.2
0
0
(c)
(b) 0.06
1
n = 100
n = 300
(d)
0.8
0.6
0.4
0
0
=6
=3
30 60 90 120 150 180
number of control inputs, q
0.3
q /n 0.2
c
0.1
50
0.02
20 40 60 80 100 120 140
number of control inputs, q
1
0.2
0.04
0
50
transition width, q/n
success rate, r
(a)
100
150
n
250
150 200 250
network size, n
300
0
10
0.3
q /n 0.2
c
−1
10
0.1
2.5 3.5 4.5 5.5
−2
10
2.5
3.5
4.5
5.5
power−law exponent,
Transition width decreases as the network size increases, and
increases as the network becomes more heterogeneous.
Analyzing the Controllability Transition
c
✏
/
⌘
numerical precision
radius of convergence
critical reciprocal condition
number at the transition point
0
10
= 10−6
slope = 0.98
−5
c
10
−10
10
−15
10
−20
10
−25
10
−20
−15
10
10
numerical precision,
−10
10
Conclusion
— Control trajectories are generally nonlocal (for both linear
and nonlinear systems) unless all nodes are controlled.
— Control by “linearization” generally fails unless the initial
or final state is the equilibrium.
— A theoretically controllable system may not be numerically
controllable, even for the simplest case (linear, time invariant).
— Numerical control of linear random network systems exhibit
sharp transition from failure to success as the number of
controlled nodes increases.
J. Sun and A. E. Motter,
“Controllability Transition and Nonlocality of Network Control”, PRL (2013).
Thank you!