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{Network Flows, Queues, Combinatorial Optimization, ... }+

Dynamics

CCI Mini-Workshop on Theoretical Foundations of CPSApril 14, 2017

Ketan Savla([email protected])

University of Southern California

Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 1 / 11

Personal Perspective on Theoretical CPS

optimizationqueueing systems

network flowsgame theory

motion kinematicspsychologydynamicsphysics

Physical Sciences(physics, human behavior, dynamics)

Systems Sciences(operations research, network science, decision theory)

⇑Research Theme

Static Network Flowfig19

� �

cut Link flow capacity CeFeasible z:

ze ≤ Ce∑incoming ze =

∑outgoing ze′

Max flow min cut theorem

λ ≤ mincut

e∈cutCe

︸︷︷︸min cut capacity

⇐⇒ feasible z

Static Network Flowfig19

� �

cut Link flow capacity CeFeasible z:

ze ≤ Ce∑incoming ze =

∑outgoing ze′

Max flow min cut theorem

λ ≤ mincut

e∈cutCe

︸︷︷︸min cut capacity

⇐⇒ feasible z

Transmission Capacity of the Power Grid

Cmaxtrans

maxw∈[wl,wu]

µ Nonconvex

s.t. −c ≤ z(w, µ) ≤ c

Robustness Formulation

R(p) =

min�: k�k1=1,1T �=0

maxw,µ

µ Nonconvex

s.t. �c f(w, p + �µ) c

wl w wu

Simple Case: p contains one supply-demand pair(default in this talk) kpk1 = 1; � = ±p

maxw2[wl,wu]

µ Nonconvex

s.t. �c f(w, p + pµ) c

Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 3 / 13

Transmission Capacity Function & Equivalent Weight

Equivalent Weight

H(w, p) :=�pTL†(w)p

��1

Parallel: H(w) =P

Series: H(w) = (P

⇠ Thevenin e↵ective resistance

Rayleigh: rwH(w) � 0

Cmaxtrans = maxweq2[H(wl),H(wu)]

Ctranseq(weq)

Ctranseq(weq) = maxw: H(w)=weq

Ctrans(w)

Ctranseq(weq)

0 weqwleq

Cmaxtrans

(wl1, wl

(wu1 , wu

Combining Transmission Capacity Functions

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Cmaxtrans

maxw∈[wl,wu]

µ Nonconvex

s.t. −c ≤ z(w, µ) ≤ c

R(p) =

min�: k�k1=1,1T �=0

maxw,µ

µ Nonconvex

s.t. �c f(w, p + �µ) c

wl w wu

maxw2[wl,wu]

µ Nonconvex

Equivalent Weight

��1

Parallel: H(w) =P

Series: H(w) = (P

Ctranseq(weq)

Ctrans(w)

Ctranseq(weq)

0 weqwleq

Cmaxtrans

(wl1, wl

(wu1 , wu

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Cmaxtrans

maxw∈[wl,wu]

µ Nonconvex

s.t. −c ≤ z(w, µ) ≤ c

R(p) =

min�: k�k1=1,1T �=0

maxw,µ

µ Nonconvex

s.t. �c f(w, p + �µ) c

wl w wu

maxw2[wl,wu]

µ Nonconvex

Equivalent Weight

��1

Parallel: H(w) =P

Series: H(w) = (P

Ctranseq(weq)

Ctrans(w)

Ctranseq(weq)

0 weqwleq

Cmaxtrans

(wl1, wl

(wu1 , wu

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Cmaxtrans

maxw∈[wl,wu]

µ Nonconvex

s.t. −c ≤ z(w, µ) ≤ c

R(p) =

min�: k�k1=1,1T �=0

maxw,µ

µ Nonconvex

s.t. �c f(w, p + �µ) c

wl w wu

maxw2[wl,wu]

µ Nonconvex

Equivalent Weight

��1

Parallel: H(w) =P

Series: H(w) = (P

Ctranseq(weq)

Ctrans(w)

Ctranseq(weq)

0 weqwleq

Cmaxtrans

(wl1, wl

(wu1 , wu

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Series

1 2 3e1 e2

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Parallel

Ctranseq(weq)

0 weqwleq

Cmaxtrans

Link-reducible

Max Residual Capacity under Local Routing

δ∗ =min node residual capacity

:= minv

e∈E+v

(Ce − z∗e )

� �out(t)

Min node residual capacity= min{1.5 + 1.5, 0.75}= 0.75

δ = 0.6

Cez∗e

(1.5, 2)

(0.5, 2) (0.5, 1.25)

:= minv

e∈E+v

(Ce − z∗e )

� �out(t)

δ = 0.6

Cez∗e

(1.5, 2)

(0.5, 2) (0.5, 1.25)

:= minv

e∈E+v

(Ce − z∗e )

� �out(t)

δ = 0.6

static routing

Cez∗e

(1.5, 2)(1.5, 1.4)

(0.5, 2) (0.5, 1.25)

:= minv

e∈E+v

(Ce − z∗e )

� �out(t)

δ = 0.6

Cez∗e

(1.5, 2)

(0.5, 2) (0.5, 1.25)

:= minv

e∈E+v

(Ce − z∗e )

� �out(t)

δ = 0.6

aggressive routing

Cez∗e

(1.5, 2)(0.5, 1.4)

(0.5, 2)(1.5, 2)

(0.5, 1.25)

:= minv

e∈E+v

(Ce − z∗e )

� �out(t)

δ = 0.6

aggressive routing

Cez∗e

(1.5, 2)(0.5, 1.4)

(0.5, 2)(1.5, 2)

(0.5, 1.25)

:= minv

e∈E+v

(Ce − z∗e )

� �out(t)

δ = 0.6

downstream cascades

Cez∗e

(1.5, 2)(0.5, 1.4)

(0.5, 2)(1.5, 2)

(0.5, 1.25)

General Dynamical Network Flowfig12

zi Rik

Mass Conservation

x = λ+R(x)T z(x)− z(x)

= λ+(RT (x)− I

xi : mass on link i

λi : external inflow into link i

R(x) : sub-stochastic routing matrix

zi(x) : outflow from link i

Flow Constraints:z(x) ∈ Z(x)

Positive Systems: for all i,xi → 0+ =⇒ zi(x)→ 0+

Objectives

Stability

Robustness

Optimal control

zi Rik

Mass Conservation

= λ+(RT (x)− I

xi : mass on link i

Objectives

Stability

Robustness

Optimal control

zi Rik

Mass Conservation

= λ+(RT (x)− I

xi : mass on link i

Objectives

Stability

Robustness

Optimal control

Stability of Simple Nonlinear Modelfig 56

zi(xi) fig3

zkRki = Rhi; Rii = 0∂∂xj

Rki(x) ≥ 0 i 6= j

Decentralized control

x = λ+(RT (x)− I

)z(x) =: g(x)

∂gi(x)∂xj

≥ 0∑

i∂gi(x)∂xj

24⇤ ⇤ ⇤⇤ ⇤ ⇤⇤ ⇤ ⇤

`1 Contraction Principle

If [∂gi/xj ]ij is compartmental ∀x,

then ddt‖x(t)− x(t)‖1 ≤ 0 ∀ x(0), x(0)

zi(xi) fig3

Rki(x) ≥ 0 i 6= j

x = λ+(RT (x)− I

)z(x) =: g(x)

∂gi(x)∂xj

∑i∂gi(x)∂xj

24⇤ ⇤ ⇤⇤ ⇤ ⇤⇤ ⇤ ⇤

then ddt‖x(t)− x(t)‖1 ≤ 0 ∀ x(0), x(0)

zi(xi) fig3

Rki(x) ≥ 0 i 6= j

x = λ+(RT (x)− I

)z(x) =: g(x)

∂gi(x)∂xj

≥ 0∑

i∂gi(x)∂xj

24⇤ ⇤ ⇤⇤ ⇤ ⇤⇤ ⇤ ⇤

then ddt‖x(t)− x(t)‖1 ≤ 0 ∀ x(0), x(0)

zi(xi) fig3

Rki(x) ≥ 0 i 6= j

x = λ+(RT (x)− I

)z(x) =: g(x)

∂gi(x)∂xj

≥ 0∑

i∂gi(x)∂xj

24⇤ ⇤ ⇤⇤ ⇤ ⇤⇤ ⇤ ⇤

then ddt‖x(t)− x(t)‖1 ≤ 0 ∀ x(0), x(0)

State-dependent Queue: Human-in-the-loop Systems

x =b− xτ

, τ > 0

Server dynamics

b(t) =

⇢1 server busy at t0 server idle at t

x(t) 2 [0, 1]: server state

x =b � x

⌧, x(0) = x0

⌧ > 0: sensitivity of the server

t1 t2 t3

small ⌧ big ⌧

Ketan Savla (LIDS, MIT) Dynamical Queues September 14, 2010 5 / 13

Server dynamics

b(t) =

x =b � x

⌧, x(0) = x0

t1 t2 t3

small ⌧ big ⌧

Objectives

Stability

�maxu = max{� | lim supt!1 queue length(t) < 1 8x0 a.s.}

�max = supu �maxu ; stability () � �max

u maximally stabilizing () �maxu = �max

Dynamical version of a D/G/1 queue

Trivial bounds: (wS(1))�1 �max (wSmin)�1

Homogeneous tasks: one-task equilibrium

fW = �w; u(t) ⌘ ON

xi xi+1

ti ti + S(xi) ti + 1�

R(x,�low)

R(x,�med)

R�x, �

xxmed,1 xmed,2xlow

xeq(�): set of one-task eqm states

� := max{� | xeq(�) 6= ;}x := xeq(�)

Threshold policy is throughputoptimal:

uT(t) =

{ON if x(t) ≤ x,OFF otherwise.

x =b− xτ

, τ > 0

Server dynamics

b(t) =

x =b � x

⌧, x(0) = x0

t1 t2 t3

small ⌧ big ⌧

Server dynamics

b(t) =

x =b � x

⌧, x(0) = x0

t1 t2 t3

small ⌧ big ⌧

Objectives

Stability

fW = �w; u(t) ⌘ ON

xi xi+1

R(x,�low)

R(x,�med)

R�x, �

xxmed,1 xmed,2xlow

� := max{� | xeq(�) 6= ;}x := xeq(�)

uT(t) =

x =b− xτ

, τ > 0

Server dynamics

b(t) =

x =b � x

⌧, x(0) = x0

t1 t2 t3

small ⌧ big ⌧

Server dynamics

b(t) =

x =b � x

⌧, x(0) = x0

t1 t2 t3

small ⌧ big ⌧

Objectives

Stability

fW = �w; u(t) ⌘ ON

xi xi+1

R(x,�low)

R(x,�med)

R�x, �

xxmed,1 xmed,2xlow

� := max{� | xeq(�) 6= ;}x := xeq(�)

uT(t) =

x =b− xτ

, τ > 0

Server dynamics

b(t) =

x =b � x

⌧, x(0) = x0

t1 t2 t3

small ⌧ big ⌧

Server dynamics

b(t) =

x =b � x

⌧, x(0) = x0

t1 t2 t3

small ⌧ big ⌧

Objectives

Stability

fW = �w; u(t) ⌘ ON

xi xi+1

R(x,�low)

R(x,�med)

R�x, �

xxmed,1 xmed,2xlow

� := max{� | xeq(�) 6= ;}x := xeq(�)

uT(t) =

State-Dependent Queue: Horizontal Traffic Queue

fig8 fig9 fig8

σ(service) > 0 σ(service) > 0stability = N/A >

σ(service) = 0 σ(service) = 0

HTQ1 Release Control Policy

HTQ2HTQ1

(a) (b)

' HTQ1

HTQ1Release Control

Policy

0 1�ix ix 1�ix L

0 1�ix ix 1�ix LHTQ

' 'D iy

3y 0 L

service rate =∑i(xi+1 − xi)m

1x 2x 3x

1I 1B 2I 2B

fig8 fig9 fig8

HTQ2HTQ1

(a) (b)

' HTQ1

HTQ1Release Control

Policy

' 'D iy

3y 0 L

1x 2x 3x

1I 1B 2I 2B

fig8 fig9 fig8

HTQ2HTQ1

(a) (b)

' HTQ1

HTQ1Release Control

Policy

' 'D iy

3y 0 L

1x 2x 3x

1I 1B 2I 2B

fig8 fig9 fig8

HTQ2HTQ1

(a) (b)

' HTQ1

HTQ1Release Control

Policy

' 'D iy

3y 0 L

1x 2x 3x

1I 1B 2I 2B

TSP under Differential Constraints

n = λ− n

TSP(n)

Euclidean: TSP(n) = Θ(n1/2)

Dubins, Diff Drive, ...TSP(n) = Θ(n2/3)

Phase 1 Phase 2 Phase 3

n = λ− n

TSP(n)

Phase 1

Phase 2 Phase 3

n = λ− n

TSP(n)

Phase 1 Phase 2

Phase 3

n = λ− n

TSP(n)

Phase 1 Phase 2 Phase 3

Current and Future Work

Systems Science

Physical Science

operations research

network science

decision theory

Research

dynamics

human behavior

physics

{Cascades over Networks} +{Physics-based Failure and Recovery}{Game Theory} +{Geometric Strategy Space}{Compressed Sensing} +{Constrained Measurement Matrix}

Emphasis on fundamental performance limits

Systems Science

Physical Science

operations research

network science

decision theory

Research

dynamics

human behavior

physics

{Cascades over Networks} +{Physics-based Failure and Recovery}

{Game Theory} +{Geometric Strategy Space}{Compressed Sensing} +{Constrained Measurement Matrix}

Systems Science

Physical Science

operations research

network science

decision theory

Research

dynamics

human behavior

physics

{Cascades over Networks} +{Physics-based Failure and Recovery}{Game Theory} +{Geometric Strategy Space}

{Compressed Sensing} +{Constrained Measurement Matrix}

Systems Science

Physical Science

operations research

network science

decision theory

Research

dynamics

human behavior

physics

{Cascades over Networks} +{Physics-based Failure and Recovery}{Game Theory} +{Geometric Strategy Space}{Compressed Sensing} +{Constrained Measurement Matrix}

dynamics - ccicci.usc.edu/wp-content/uploads/2017/06/main-savla.pdffnetwork flows, queues,...

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