This document presents a summary of the paper "System Theory over Random Networks: Control and Estimation" by Marzieh Nabi Abdolyousefi and Mehran Mesbahi. The paper explores system-theoretic properties like controllability and observability of systems operating over random networks. It examines control and estimation of diffusion-like protocols over random graphs and decentralized control and estimation schemes involving randomness. The main results analyze controllability and stability properties of control systems defined over random networks.

1. System Theory over Random Networks:
Control and Estimation
Marzieh Nabi Abdolyousefi
Mehran Mesbahi
Aeronautics & Astronautics
University of Washington
System Theory over Random Networks: Control and Estimation, slide 1/14

2. some system properties are theoretically harder to deal
with on deterministic networks
Consider human-swarm interaction
System Theory over Random Networks: Control and Estimation, slide 2/14

3. explore system/ graph theoretic aspect of deterministic/
stochastic systems that operate over a random network ...
why?
Modeling Introduced by design
real systems are subjective to limited battery sources:
link failure large sensor networks:
unreliable communication sensing structural
limited bandwidth integrity
delays habitat monitoring
data loss firebugs
reactive sample rate: soil
moisture monitoring
System Theory over Random Networks: Control and Estimation, slide 3/14

4. how easy is it to control or observe a random networked
system via a small subset of nodes or edges chosen ran-
domly or deterministically?
t
System Theory over Random Networks: Control and Estimation, slide 4/14

5. outline
controllability and observability of diffusion-like protocols over
random networks generated based on Erd¨s- Renyi distribution
o
control over random networks
estimation over random networks
System Theory over Random Networks: Control and Estimation, slide 5/14

6. control over static network
random diffusion compromised diffusion over a random network
System Theory over Random Networks: Control and Estimation, slide 6/14

7. model the diffusion-like potocol
x(t + 1) = A(G(wt ))x(t) + B(G(wt ))u(t)
y (t) = C (G(wt ))x(t), (1)
wt : the sequence of mutually independent random events
G is a realization of the random graph
A(G(wt )) is related to the diffusion-like protocol, e.g.,
A(G(wt )) = e −L(G(wt )
B(G(wt )): input matrix
C (G(wt )): output matrix
System Theory over Random Networks: Control and Estimation, slide 7/14

8. controllability Gramian over Random Networks ...
let
St = Bt Bt
and consider the event
Ωt = St + At St−1 At + . . . + (At . . . A2 )S1 (A2 . . . At )
the controlled diffusion is weakly controllable
(Bougerol 1993) if for some t ≥ 1,
P{det(Ωt ) = 0} = 0
or if and only if for some t ≥ 1,
P{rank (Bt , At Bt−1 , At At−1 Bt−2 , . . . , At . . . A2 B1 ) = n} = 0.
System Theory over Random Networks: Control and Estimation, slide 8/14

9. design linear quadratic regulator:
choosing a suitable control function u(t) in (1) s.th
every initial state x(t0 ) = x0 is returned to the reference signal
x =0
the performance index
T −1
T
E{ρ ||x(T , x0 )||2
S + ρt ||x(t, x0 )||2 },
Q
t=t0
ρ > 0, Q > 0, S ≥ 0
is minimized
System Theory over Random Networks: Control and Estimation, slide 9/14

10. design the controller ... continued
the motions of (1) are asymptotically stable in the mean
square if
E(||x(t, x0 )||2 ) < ∞ and
lim E(||x(t, x0 )||2 ) = 0 for all x0 .
t→∞
and asymptotic stability with probability 1 if
||x(t, x0 )|| < ∞, and
2
lim ||x(t, x0 )|| = 0.
t→∞
System Theory over Random Networks: Control and Estimation, slide 10/14

11. design the controller ... continued
let u(t) = Kt x(t) and Act = At − Bt Kt ; then
the motions of (1) are asymptotically stable in the mean
square if and only if
|λi [E(Act ⊗ Act )]| < 1 (i = 1, . . . , n2 ).
where the operator ⊗ is the Kronecker product
the optimal control law is constant and is given by
Kalman(1962)
K∗ = (E[B P∗ B])−1 E[B P∗ A],
P∗ = ρE(Act P∗ Act ) + Q
= ρ mat[E(Act ⊗ Act )vec(P∗ )] + Q
System Theory over Random Networks: Control and Estimation, slide 11/14

12. examine coordinated decentralized estimator design over
networks in different scenarios involving randomness
Coordinator
Coordinator
ith
local
1−p p p 1−p est.
ith
local Measurements
est.
p p p p
Measurements
Both schemes have some notion of randomness and local
computation in common ...
System Theory over Random Networks: Control and Estimation, slide 12/14

13. decentralized estimation ... continued
Main result
Theorem
consider the system (1). Then the estimation error
x(t) − x (t) is almost surely asymptotically stable.
ˆ
or equivalently, there is a real number γ > 0, such that almost
surely
1
lim log ||(At − Kt Ct ), . . . , (A1 − K1 C1 )|| ≤ −γ
t→∞ t
for any solution of the random Riccati equation
Proof.
random Riccati map is contractive
utilize a stochastic Lyapunov approach
System Theory over Random Networks: Control and Estimation, slide 13/14

14. recap and ongoing work
system-theoretic perspective on random networks
controllability, and observability over random networks
optimal properties of random networks
decentralized operators over random networks
some of the ongoing work
threshold phenomena in system theoretic properties of random
networks
rate of convergence
effect of different distribution and probability p
System Theory over Random Networks: Control and Estimation, slide 14/14