Robust Control of Heterogeneous Networks (e.g. congestion control for the Internet)

Transcription

1 Robust Control of Heterogeneous Networks (e.g. congestion control for the Internet) Glenn Vinnicombe University of Cambridge & Caltech 1/29

2 Introduction Is it possible to build (locally verifiable) stable and robust large scale feedback interconnections of linear dynamical systems? Yes, but only limited results so far (e.g. Internet congestion control). 2/29

3 Networks, Modules and Protocols We regard a network as being a collection of interconnected modules. Modules are elements of parameterized classes of linear, time-invariant, dynamical systems. Protocols are the rules by which interconnections between modules be made. Will present sufficient conditions for network stability for fairly general classes of asymptotically stable modules but fairly restricted interconnection protocols. 3/29

4 Modules and Protocols Σ f T Module: e.g. y(t) = u i (t T ) Protocol: e.g. f/b interconnection allowed if green s T < red s τ Σ g τ Module: e.g. τẏ(t) = u i (t) Note: this protocol still allows each red module to be connected to many green modules and each green module to be connected to many red modules. 4/29

5 Signal Flow Graphs (Mason 53) Σ f T i f T o f T f T Assume linearity, and that Laplace transforms have been taken everywhere, so f T o (s) = f T (s)f T i (s) 5/29

6 Example: Congestion Control 6/29

7 Network Paradigm y 1 = x 1 y 2 = x 1 + x 2 acks (p 2 ) src 2 x 2 dst 2 link 2 (p2) p 2 (y 2 ) x 1 (p 1 ) (p 1 + p 2 ) x 1 link 1 src 1 dst 1 p 1 (y 1 ) acks (p 1 + p 2 ) Important: Feedback delayed by a Round Trip Time 7/29

8 Dynamic fluid model (links) link 2 link 1 x 2 p 2 x 1 src 2 p 2 x 1 src 1 p 1 Flow through each link given by: y j (t) = r :r uses j x r (t τ jr ). τ jr is the propagation delay from source r to link j. Links set their prices according to: p j (t) = G ( {y j (τ) : τ t} ) (t) 8/29

9 Dynamic fluid model (sources) link 2 link 1 x 2 p 2 x 1 src 2 p 2 x 1 src 1 p 1 Aggregate price at a source given by: q r (t) = p j (t τ jr ) j:j used by r where τ jr is the delay from link j back to source r. T r = τ jr + τ jr (RTT) The sources set their rates according to: x r (t) = F ( {q r (τ) : τ t} ) (t) 9/29

10 Stability of Networks We shall consider networks where all connections are symmetric, and all cycles have even length. graph is bipartite. This structure is clearly guaranteed by the protocol described. 10/29

11 Nyquist stability criterion r (s) + Σ z(s) ē(s) G(s) K(s) represents the simultaneous equations: e(t) = r (t) z(t) and z(s) = K(s)G(s)ē(s) (i.e. z(t) = k g e) We write ȳ(s) = K(s)G(s) 1 + K(s)G(s) r (s) L(s) = G(s)K(s) (the Return Ratio) 11/29

12 If L(s) is stable, then Nyquist stability criterion I I I 1 R 1 R 1 R L(jω) L(jω) L(jω) L(s) 1 + L(s) is asymptotically stable L(s) 1 + L(s) is marginally stable L(s) 1 + L(s) is unstable 12/29

13 2 Nyquist diagram, k=1 k = 1 2 Closed-loop step response Imag Axis L( 2 j) L(jω) PSfrag replacements Closed-loop is marginally stable nts Real Axis Closed-loop poles are at the roots of s 3 + s 2 + 2s + 2 = 0, i.e. s = 1 and 0 ± j (since 1 + L(1.4142j) = 0 13/29

14 k = Nyquist diagram, k=0.8 2 Closed-loop step response Imag Axis PSfrag replacements Closed-loop is asymptotically stable Real Axis Closed-loop poles are at the roots of s 3 + s 2 + 2s = 0, i.e. s = j, j, /29

15 k = Nyquist diagram, k=1.2 2 Closed-loop step response Imag Axis PSfrag replacements Closed-loop is unstable Real Axis Closed-loop poles are at the roots of s 3 + s 2 + 2s = 0, i.e. s = j, j, /29

16 Breaking the loop g 4 f 2 g 3 f 1o (s) = f 1 (s)f 1i (s) etc g 2 f 1 g 1 Open the loop at the red systems, to get [ f1i f 2i ] = [ ] g g g }{{} g R }{{} R T [ f1o f 2o R ij = 1 if f i is connected to g j 16/29 ]

17 Multivariable Nyquist Criterion return ratio [ f1o f 2o ] [ f1o f 2o ] is given by [ ] [ ] g f g f g }{{}}{{} g F(s) R }{{}}{{} G(s) R T Assume each f i (s), g j (s) stable (i.e. analytic and bounded in R(s) > 0) and continuous on R(s) = 0 (i.e. approximable by rationals). Then Closed-loop stable { λi ( F(jω)RG(jω)R T ) : ω R } do not encircle +1 17/29

18 Aside: a graph viewpoint? Stability can also be related to the non-singularity of I f f nf [ ] 0 R g R T 0. }{{}. 0 A g ng (as det( ) = det(i FRGR T )) A = A T is the adjacency matrix of an undirected bipartite graph. Some sort of weighted Laplacian (note: f i, g j complex)? Non-symmetric A (directed graphs)? Non-bipartite graphs? 18/29

19 Simple case (G real) Assume all g s real and positive, and let Q = RGR T. Lemma: Let Q = Q 0, f i C, then Proof: σ ( diag(f 1, f 2,..., f n )Q ) Co(0 {f i })ρ(q) σ ( diag(f 1, f 2,..., f n )Q ) = σ (Q 1/2 diag(f 1, f 2,..., f n )Q 1/2) {v Q 1/2 diag(f 1, f 2,..., f n )Q 1/2 v : v = 1} ρ(q){w diag(f 1, f 2,..., f n )w : w 1} (since Q 1/2 v Q 1/2 = ρ(q)) = ρ(q){ i w i 2 f i : w 2 i 1} 19/29

20 Bounding ρ(q) Let y = Rx, q = R T p for real and positive x, p. e.g. x = 1 ng, p = 1 nf y is vector of degree of in-degrees of each f, (d i ), and q is vector of in-degrees of each g, (e j ). Then can rescale, and bound spectral radius in terms of row sums, to get ({ }) σ (FRGR T y i q j ) Co f i g j : i, j, R ij 0 p i x e.g. j σ (F(jω)RGR T ) Co ({ }) f i (jω)d i g j e j : i, j, R ij 0 Hence, 1 Co ({ }) f i (jω)d i g j e j : i, j, R ij 0, ω > 0 stability Note: this is a local condition 20/29

21 Example I For the source law ẋ r (t) = k r ( wr price/sec {}}{ ) x(t T r )q r (t) and link law willingness to pay ( p j (t) = g j yj (t) ) sufficient condition for (local) stability is k r T r j used by r ( ) g j + g j y j π 2 r as conjectured by Johari & Tan [2000], (proof: Massoulie [2000] (for 1 on RHS), Vinnicombe [2000]). since all loops have a return ratio of the form k e st st + ɛ 21/29

22 Example I OK 22/29

23 (More) General case: Dynamic G For X C let S(X) = ( Co ( X )) 2, where X := {y : y 2 X}. Note that S(X) is a little larger that Co(0 X) /29

24 (More) General case: Dynamic G Lemma: (f i, g j C) σ ( diag{f i }R diag{g j }R T ) ρ(r T R) Co { f i S{g j : R ij 0} : i } etc σ (GRFR ) f = (1 + j, 2 2j), g = (1, 2 + j), R 1, R 22 = 0 24/29

25 A proposed scheme The sources each have a utility function U r (x r ) and set their rates according to: T r ẋ r (t) = κ B x r (t) ( 1 q ) r (t) U r. (x r (t)) Links set prices (marking probabilities) according to: p j (z j ) = ( zj C j ) B, β j ż j + z j = y j with β j < K min r :r uses j T r. capacity 25/29

26 A proposed scheme Equilibrium is unique maximum of U r (x r ) r j yj 0 p j (v) dv Kelly, Maulloo & Tan [1998] (and, in the absence of delays, stable for all k > 0). Source law linearizes to δx r = κ B Link law linearizes to x r q r 1 st r + α r δq r, α r > 0 δp j = B p j 1 δy y j β j s + 1 j 26/29

27 Stability Proof Including delay: f r (s) = κ B x r q r e str st r + α r, g j (s) = B p j y j 1 β j s + 1 Interconnection permitted only if β j KT r. Note that y = Rx, q = R T p at equilibrium σ ( FR T GR ) { { }} (jω) κ Co f r (jω)s g j (jω) : R r j 0 { { } } e jx 1 κ Co jx S jy + 1 : 0 y Kx : x > 0 Scheme is (locally) stable if 1 this set. 27/29

28 Example: K = closed-loop stability for κ < 1. Note: this provides a stability certificate for the scheme. 28/29

29 Robust stability: K = 2, δ = if κ < then closed-loop stability is maintained for all source/delay and link dynamics within 0.1 of nominal. 29/29