Adaptation and Synchronization over a Network : stabilization - - PowerPoint PPT Presentation
Adaptation and Synchronization over a Network : stabilization without a reference model Travis E. Gibson (tgibson@mit.edu) Harvard Medical School Department of Pathology, Brigham and Womens Hospital 55 th Conference on Decision and Control
Travis E. Gibson (tgibson@mit.edu)
Harvard Medical School Department of Pathology, Brigham and Women’s Hospital
55th Conference on Decision and Control December 12-14, 2016
Adaptive Systems Network Consensus
Parameter Adaptation Error Generation System Consensus Error
◮ How do we achieve consensus and learning
without a reference model?
◮ Can synchronous inputs enhance adaptation?
2 / 22
◮ Synchronization can hurt learning
◮ Example of two unstable systems (builds on Narendra’s recent work)
◮ Synchronization and Learning in Networks
◮ Results Using Graph Theory
◮ Concrete connections to classic adaptive control (if time allows)
3 / 22
4 / 22
Consider two unstable systems [Narendra and Harshangi (2014)] Σ1 : ˙ x1(t) = a1(t)x1(t) + u1(t) Σ2 : ˙ x2(t) = a2(t)x2(t) + u2(t) Update laws ˙ a1(t) = −x1(t)e(t) a1(0) > 0 ˙ a2(t) = x2(t)e(t) a2(0) > 0 with e = x1 − x2. No Input u1 = 0 u2 = 0
t
5 10
x1, x2
5 10 15 20
x1 x2
t
5 10
a1, a2
1 2 3
a1 a2 5 / 22
Synchronous Input u1 = −e u2 = +e e = x1 − x2
t
5 10
x1, x2
×105 2 4 6 8 10 12
x1 x2
t
5 10
a1, a2
1 1.2 1.4 1.6 1.8 2
a1 a2
Desynchronous Input u1 = +e u2 = −e
t
5 10
x1, x2
2 4 6 8
x1 x2
t
5 10
a1, a2
2 4
a1 a2 6 / 22
Σ1 : ˙ x1(t) = a1(t)x1(t) + u1(t) Σ2 : ˙ x2(t) = a2(t)x2(t) + u2(t) ˙ a1(t) = −x1(t)e(t) ˙ a2(t) = x2(t)e(t) Theorem: Synchronous Inputs The dynamics above with synchronous inputs have a set of initial conditions with non-zero measure for which x1 and x2 tend to infinity while e ∈ L2 ∩ L∞ and e → 0 as t → ∞. Furthermore, this set of initial conditions that are unstable is also unbounded. Theorem: Desynchronous Inputs The dynamics above with desynchronous inputs are stable for all a1(0) = a2(0)
7 / 22
8 / 22
Graph : G(V, E) Vertex Set : V = {v1, v2, . . . , vn} Edge Set : (vi, vj) ∈ E ⊂ V × V v1 v2 v3 v4 Adjacency Matrix : [A]ij =
if (vj, vi) ∈ E
In-degree Laplacian : L(G) = D(G) − A(G) In-degree of Node i : [D]ii Consensus Problem Σi : ˙ xi = −
(xi − xj) Using Graph Notation Σ : ˙ x = −Lx, x = [x1, x2, . . . , xn]T
9 / 22
Σ : ˙ x = −Lx Theorem: (Olfati-Saber and Murray, 2004) For the dynamics above with G strongly connected it follows that limt→∞ x(t) = ζ1, for some finite ζ ∈ R. If G is also balanced then ζ = 1
n
n
i=1 xi(0), i.e. average consensus is reached.
strongly connected there is a walk between any two vertices in the network. balanced if the in-degree of each node is equal to its out-degree.
◮ Any strongly connected digraph can be balanced
(Marshall and Olkin, 1968).
◮ Distributed algorithms exist to balance a digraph
(Dominguez-Garcia and Hadjicostis, 2013).
10 / 22
Consider a set of n possibly unstable systems Σi ˙ xi(t) = aixi + θi(t)x Update Law ˙ θi = −xi
(xi − xj) Compact form Σ : ˙ x = Ax + diag(θ)x [A]ii = ai ˙ θ = −x ◦ Lx θ = [θ1, θ2, . . . , θn]T
11 / 22
˙ x = Ax + diag(θ)x ˙ θ = −x ◦ Lx Theorem For the dynamics above with G a strongly connected digraph, and all the ai + θi(0) not identical it follows that limt→∞ x(t) = 0.
◮ G is strongly connected =
⇒ λi(L) ∈ closed right-half plane of C.
◮ −L is Metzler =
⇒ ∃ Diagonal D > 0 s.t. −LTD − DL ≤ 0.
◮ Non-increasing function
n
[D]iiθi(t) = − t xTDLx dt +
n
[D]iiθi(0) = −1 2 t xT(DL + LTD)x dt +
n
[D]iiθi(0).
◮ L1 = 0 =
⇒ 1T(DL + LTD)1 = 0.
◮ ∃ κ λ2(DL + LTD) > 0 =
⇒
i[D]iiθi(t) ≤ −κ
i θi(0)
when x / ∈ span(1).
12 / 22
◮ Any connected digraph can be partitioned into disjoint subsets called
Strongly Connected Components (SCCs) where each subsets is a maximal strongly connected subgraph Graph G Condensed Graph GSCC
condensed nodes in red root
◮ For any connected G the corresponding GSCC is a Directed Acyclic
Graph (DAG)
◮ Every connected DAG contains a root node (not unique).
13 / 22
˙ x = Ax + diag(θ)x ˙ θ = −x ◦ Lx Theorem For the dynamics above with the adaptation occurring over a con- nected graph G such that a root can be chosen in GSCC that is a condensed node, then limt→∞ x(t) = 0
◮ The root is a strongly connected subgraph (thus stabilizes itself) ◮ All information flowing over G decimates from a stable SCC. ◮ Stability of each SCC then follows from the hierarchical structure of
the DAG. GSCC
14 / 22
This node can never stabilize if initially unstable
15 / 22
Bring everything together as a layered architecture
◮ The communication graph is G ◮ Ga is the adaptation graph and is constrained by the communication
in G
◮ Gs is the synchronization graph and is similarly constrained
G Ga Gs
(Doyle and Csete, 2011), (Alderson and Doyle, 2010)
16 / 22
Σ : ˙ x = Ax + diag(θ)x ˙ θ = −x ◦ Lax Ga =
t
5 10
xi
0.5 1 1.5
t
5 10
θi
1
17 / 22
Σ : ˙ x = Ax + Lsx + diag(θ)x ˙ θ = −Γx ◦ Lax Ga = Gs =
t
5 10
xi
0.5 1
t
5 10
θi
2
18 / 22
Borrowing from Narendra, Murray, and My Thesis, we have
◮ Found that synchronization can hurt learning. ◮ As always context is important ◮ What about other learning paradigms, i.e. Jadbabaie’s work or the
broader Machine Learning literature Bibliography
Alderson, D. L and J. C Doyle. 2010. Contrasting views of complexity and their implications for network-centric infrastructures, Systems, Man and Cybernetics, Part A: Systems and Humans, IEEE Transactions on 40, no. 4, 839–852. Dominguez-Garcia, A. D. and C. N. Hadjicostis. 2013. Distributed matrix scaling and application to average consensus in directed graphs, Automatic Control, IEEE Transactions on 58, no. 3, 667–681. Doyle, J. C. and M. Csete. 2011. Architecture, constraints, and behavior, Proceedings
Marshall, A. W and I. Olkin. 1968. Scaling of matrices to achieve specified row and column sums, Numerische Mathematik 12, no. 1, 83–90. Narendra, K. S. and P. Harshangi. 2014. Unstable systems stabilizing each other through adaptation, American Control Conference, pp. 7–12. Olfati-Saber, R. and R. M Murray. 2004. Consensus problems in networks of agents with switching topology and time-delays, Automatic Control, IEEE Transactions on 49, no. 9, 1520–1533.
19 / 22
r xm x e γ ℓ Reference Model Plant feedback gain learning rate
20 / 22
Classic Open-loop Reference Model (ORM) Adaptive (ℓ = 0)
◮ The reference model
does not adjust to any
reference: xm plant: x t
Closed-loop Reference Model (CRM) Adaptive
◮ The reference model
adjusts to rapidly reduce the model following error e = x − xm
reference: xm plant: x t
21 / 22
γ = 10 ℓ = 0
t
10 20 30
State
0.5 1 1.5 2 2.5
xm xp
t
10 20 30
Parameter
0.5 1
θ k
γ = 100 ℓ = −100
t
10 20 30
State
0.5 1 1.5 2 2.5
xo
m
xm xp
t
10 20 30
Parameter
0.5 1
θ k
γ = 100 ℓ = −1000
t
10 20 30
State
0.5 1 1.5 2 2.5
xo
m
xm xp
t
10 20 30
Parameter
0.5 1
θ k
22 / 22