dynamics - ccicci.usc.edu/wp-content/uploads/2017/06/main-savla.pdffnetwork flows, queues,...
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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
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 2 / 11
Personal Perspective on Theoretical CPS
Physical Sciences(physics, human behavior, dynamics)
Systems Sciences(operations research, network science, decision theory)
optimizationqueueing systems
network flowsgame theory
+
motion kinematicspsychologydynamicsphysics
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 2 / 11
Personal Perspective on Theoretical CPS
⇑Research Theme
⇓
Physical Sciences(physics, human behavior, dynamics)
Systems Sciences(operations research, network science, decision theory)
optimizationqueueing systems
network flowsgame theory
+
motion kinematicspsychologydynamicsphysics
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 2 / 11
Personal Perspective on Theoretical CPS
⇑Research Theme
⇓
Physical Sciences(physics, human behavior, dynamics)
Systems Sciences(operations research, network science, decision theory)
optimizationqueueing systems
network flowsgame theory
+
motion kinematicspsychologydynamicsphysics
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 2 / 11
Personal Perspective on Theoretical CPS
⇑Research Theme
⇓
Physical Sciences(physics, human behavior, dynamics)
Systems Sciences(operations research, network science, decision theory)
optimizationqueueing systems
network flowsgame theory
+
motion kinematicspsychologydynamicsphysics
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 2 / 11
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
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 3 / 11
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
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 3 / 11
Transmission Capacity of the Power Grid
Cmaxtrans
maxw∈[wl,wu]
maxµ
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
pk
pi pj
�i
�j
�l
Simple Case: p contains one supply-demand pair(default in this talk) kpk1 = 1; � = ±p
maxw2[wl,wu]
maxµ
µ Nonconvex
s.t. �c f(w, p + pµ) c
pi pj
�i
�j
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
i wi
Series: H(w) = (P
i1wi
)�1
⇠ Thevenin e↵ective resistance
Rayleigh: rwH(w) � 0
pi pj
�i
�j
Cmaxtrans = maxweq2[H(wl),H(wu)]
Ctranseq(weq)
Ctranseq(weq) = maxw: H(w)=weq
Ctrans(w)
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
(wl1, wl
2)
(wu1 , wu
2 )
w1
w2
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 6 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 4 / 11
Transmission Capacity of the Power Grid
Cmaxtrans
maxw∈[wl,wu]
maxµ
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
pk
pi pj
�i
�j
�l
Simple Case: p contains one supply-demand pair(default in this talk) kpk1 = 1; � = ±p
maxw2[wl,wu]
maxµ
µ Nonconvex
s.t. �c f(w, p + pµ) c
pi pj
�i
�j
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
i wi
Series: H(w) = (P
i1wi
)�1
⇠ Thevenin e↵ective resistance
Rayleigh: rwH(w) � 0
pi pj
�i
�j
Cmaxtrans = maxweq2[H(wl),H(wu)]
Ctranseq(weq)
Ctranseq(weq) = maxw: H(w)=weq
Ctrans(w)
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
(wl1, wl
2)
(wu1 , wu
2 )
w1
w2
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 6 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 4 / 11
Transmission Capacity of the Power Grid
Cmaxtrans
maxw∈[wl,wu]
maxµ
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
pk
pi pj
�i
�j
�l
Simple Case: p contains one supply-demand pair(default in this talk) kpk1 = 1; � = ±p
maxw2[wl,wu]
maxµ
µ Nonconvex
s.t. �c f(w, p + pµ) c
pi pj
�i
�j
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
i wi
Series: H(w) = (P
i1wi
)�1
⇠ Thevenin e↵ective resistance
Rayleigh: rwH(w) � 0
pi pj
�i
�j
Cmaxtrans = maxweq2[H(wl),H(wu)]
Ctranseq(weq)
Ctranseq(weq) = maxw: H(w)=weq
Ctrans(w)
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
(wl1, wl
2)
(wu1 , wu
2 )
w1
w2
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 6 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 4 / 11
Transmission Capacity of the Power Grid
Cmaxtrans
maxw∈[wl,wu]
maxµ
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
pk
pi pj
�i
�j
�l
Simple Case: p contains one supply-demand pair(default in this talk) kpk1 = 1; � = ±p
maxw2[wl,wu]
maxµ
µ Nonconvex
s.t. �c f(w, p + pµ) c
pi pj
�i
�j
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
i wi
Series: H(w) = (P
i1wi
)�1
⇠ Thevenin e↵ective resistance
Rayleigh: rwH(w) � 0
pi pj
�i
�j
Cmaxtrans = maxweq2[H(wl),H(wu)]
Ctranseq(weq)
Ctranseq(weq) = maxw: H(w)=weq
Ctrans(w)
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
(wl1, wl
2)
(wu1 , wu
2 )
w1
w2
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 6 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Combining Transmission Capacity Functions
Series
1 2 3e1 e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Parallel
1 2
e1
e2
Ctranseq(weq)
0 weqwleq
wueq
Cmaxtrans
Link-reducible
1
2
3
4
e1
e2
e3
e4
e5
weq
Ctranseq(weq) {e 4,5}{e 2,4,5}{e 1,3}{e 1,2,3,4,5}
1
2
3
4
e1 e3
e2 e6
1 4
e7
e8
1 4e9
Ketan Savla (CEE, USC) Robustness of DC Power Networks March 7, 2017 7 / 13
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 4 / 11
Max Residual Capacity under Local Routing
δ∗ =min node residual capacity
:= minv
∑
e∈E+v
(Ce − z∗e )
fig16
v
� �out(t)
E+v
Min node residual capacity= min{1.5 + 1.5, 0.75}= 0.75
δ = 0.6
v1
v2
2
Cez∗e
(1.5, 2)
(0.5, 2) (0.5, 1.25)
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 5 / 11
Max Residual Capacity under Local Routing
δ∗ =min node residual capacity
:= minv
∑
e∈E+v
(Ce − z∗e )
fig16
v
� �out(t)
E+v
Min node residual capacity= min{1.5 + 1.5, 0.75}= 0.75
δ = 0.6
v1
v2
2
Cez∗e
(1.5, 2)
(0.5, 2) (0.5, 1.25)
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 5 / 11
Max Residual Capacity under Local Routing
δ∗ =min node residual capacity
:= minv
∑
e∈E+v
(Ce − z∗e )
fig16
v
� �out(t)
E+v
Min node residual capacity= min{1.5 + 1.5, 0.75}= 0.75
δ = 0.6
static routing
v1
v2
2
Cez∗e
(1.5, 2)(1.5, 1.4)
(0.5, 2) (0.5, 1.25)
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 5 / 11
Max Residual Capacity under Local Routing
δ∗ =min node residual capacity
:= minv
∑
e∈E+v
(Ce − z∗e )
fig16
v
� �out(t)
E+v
Min node residual capacity= min{1.5 + 1.5, 0.75}= 0.75
δ = 0.6
v1
v2
2
Cez∗e
(1.5, 2)
(0.5, 2) (0.5, 1.25)
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 5 / 11
Max Residual Capacity under Local Routing
δ∗ =min node residual capacity
:= minv
∑
e∈E+v
(Ce − z∗e )
fig16
v
� �out(t)
E+v
Min node residual capacity= min{1.5 + 1.5, 0.75}= 0.75
δ = 0.6
aggressive routing
v1
v2
2
Cez∗e
(1.5, 2)(0.5, 1.4)
(0.5, 2)(1.5, 2)
(0.5, 1.25)
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 5 / 11
Max Residual Capacity under Local Routing
δ∗ =min node residual capacity
:= minv
∑
e∈E+v
(Ce − z∗e )
fig16
v
� �out(t)
E+v
Min node residual capacity= min{1.5 + 1.5, 0.75}= 0.75
δ = 0.6
aggressive routing
v1
v2
2
Cez∗e
(1.5, 2)(0.5, 1.4)
(0.5, 2)(1.5, 2)
(0.5, 1.25)
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 5 / 11
Max Residual Capacity under Local Routing
δ∗ =min node residual capacity
:= minv
∑
e∈E+v
(Ce − z∗e )
fig16
v
� �out(t)
E+v
Min node residual capacity= min{1.5 + 1.5, 0.75}= 0.75
δ = 0.6
downstream cascades
v1
v2
2
Cez∗e
(1.5, 2)(0.5, 1.4)
(0.5, 2)(1.5, 2)
(0.5, 1.25)
4 = 1
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 5 / 11
General Dynamical Network Flowfig12
ixi k
j
zj
zi Rik
Mass Conservation
x = λ+R(x)T z(x)− z(x)
= λ+(RT (x)− I
)z(x)
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
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 6 / 11
General Dynamical Network Flowfig12
ixi k
j
zj
zi Rik
Mass Conservation
x = λ+R(x)T z(x)− z(x)
= λ+(RT (x)− I
)z(x)
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
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 6 / 11
General Dynamical Network Flowfig12
ixi k
j
zj
zi Rik
Mass Conservation
x = λ+R(x)T z(x)− z(x)
= λ+(RT (x)− I
)z(x)
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
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 6 / 11
Stability of Simple Nonlinear Modelfig 56
Ci
xi
zi(xi) fig3
k
i
jh
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
≤ 0
fig3
24⇤ ⇤ ⇤⇤ ⇤ ⇤⇤ ⇤ ⇤
35
P 0
� 0
`1 Contraction Principle
If [∂gi/xj ]ij is compartmental ∀x,
then ddt‖x(t)− x(t)‖1 ≤ 0 ∀ x(0), x(0)
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 7 / 11
Stability of Simple Nonlinear Modelfig 56
Ci
xi
zi(xi) fig3
k
i
jh
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
≤ 0
fig3
24⇤ ⇤ ⇤⇤ ⇤ ⇤⇤ ⇤ ⇤
35
P 0
� 0
`1 Contraction Principle
If [∂gi/xj ]ij is compartmental ∀x,
then ddt‖x(t)− x(t)‖1 ≤ 0 ∀ x(0), x(0)
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 7 / 11
Stability of Simple Nonlinear Modelfig 56
Ci
xi
zi(xi) fig3
k
i
jh
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
≤ 0
fig3
24⇤ ⇤ ⇤⇤ ⇤ ⇤⇤ ⇤ ⇤
35
P 0
� 0
`1 Contraction Principle
If [∂gi/xj ]ij is compartmental ∀x,
then ddt‖x(t)− x(t)‖1 ≤ 0 ∀ x(0), x(0)
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 7 / 11
Stability of Simple Nonlinear Modelfig 56
Ci
xi
zi(xi) fig3
k
i
jh
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
≤ 0
fig3
24⇤ ⇤ ⇤⇤ ⇤ ⇤⇤ ⇤ ⇤
35
P 0
� 0
`1 Contraction Principle
If [∂gi/xj ]ij is compartmental ∀x,
then ddt‖x(t)− x(t)‖1 ≤ 0 ∀ x(0), x(0)
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 7 / 11
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
0
0
1
1
t
t
x(t)
b(t)
t1 t2 t3
t1 t2 t3
small ⌧ big ⌧
Ketan Savla (LIDS, MIT) Dynamical Queues September 14, 2010 5 / 13
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
0
0
1
1
t
t
x(t)
b(t)
t1 t2 t3
t1 t2 t3
small ⌧ big ⌧
Ketan Savla (LIDS, MIT) Dynamical Queues September 14, 2010 5 / 13
Objectives
t
x(t)
x
S(x)
Smin
S(1)
1
u
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
Ketan Savla (LIDS, MIT) Dynamical Queues September 14, 2010 6 / 13
Homogeneous tasks: one-task equilibrium
fW = �w; u(t) ⌘ ON
0
1
t
x(t)
xi xi+1
ti ti + S(xi) ti + 1�
x0 1
S(x)
R(x,�low)
R(x,�med)
R�x, �
�
Smin
Smax
1/�
xxmed,1 xmed,2xlow
xeq(�): set of one-task eqm states
� := max{� | xeq(�) 6= ;}x := xeq(�)
Ketan Savla (LIDS, MIT) Dynamical Queues September 14, 2010 7 / 13
Threshold policy is throughputoptimal:
uT(t) =
{ON if x(t) ≤ x,OFF otherwise.
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 8 / 11
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
0
0
1
1
t
t
x(t)
b(t)
t1 t2 t3
t1 t2 t3
small ⌧ big ⌧
Ketan Savla (LIDS, MIT) Dynamical Queues September 14, 2010 5 / 13
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
0
0
1
1
t
t
x(t)
b(t)
t1 t2 t3
t1 t2 t3
small ⌧ big ⌧
Ketan Savla (LIDS, MIT) Dynamical Queues September 14, 2010 5 / 13
Objectives
t
x(t)
x
S(x)
Smin
S(1)
1
u
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
Ketan Savla (LIDS, MIT) Dynamical Queues September 14, 2010 6 / 13
Homogeneous tasks: one-task equilibrium
fW = �w; u(t) ⌘ ON
0
1
t
x(t)
xi xi+1
ti ti + S(xi) ti + 1�
x0 1
S(x)
R(x,�low)
R(x,�med)
R�x, �
�
Smin
Smax
1/�
xxmed,1 xmed,2xlow
xeq(�): set of one-task eqm states
� := max{� | xeq(�) 6= ;}x := xeq(�)
Ketan Savla (LIDS, MIT) Dynamical Queues September 14, 2010 7 / 13
Threshold policy is throughputoptimal:
uT(t) =
{ON if x(t) ≤ x,OFF otherwise.
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 8 / 11
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
0
0
1
1
t
t
x(t)
b(t)
t1 t2 t3
t1 t2 t3
small ⌧ big ⌧
Ketan Savla (LIDS, MIT) Dynamical Queues September 14, 2010 5 / 13
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
0
0
1
1
t
t
x(t)
b(t)
t1 t2 t3
t1 t2 t3
small ⌧ big ⌧
Ketan Savla (LIDS, MIT) Dynamical Queues September 14, 2010 5 / 13
Objectives
t
x(t)
x
S(x)
Smin
S(1)
1
u
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
Ketan Savla (LIDS, MIT) Dynamical Queues September 14, 2010 6 / 13
Homogeneous tasks: one-task equilibrium
fW = �w; u(t) ⌘ ON
0
1
t
x(t)
xi xi+1
ti ti + S(xi) ti + 1�
x0 1
S(x)
R(x,�low)
R(x,�med)
R�x, �
�
Smin
Smax
1/�
xxmed,1 xmed,2xlow
xeq(�): set of one-task eqm states
� := max{� | xeq(�) 6= ;}x := xeq(�)
Ketan Savla (LIDS, MIT) Dynamical Queues September 14, 2010 7 / 13
Threshold policy is throughputoptimal:
uT(t) =
{ON if x(t) ≤ x,OFF otherwise.
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 8 / 11
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
0
0
1
1
t
t
x(t)
b(t)
t1 t2 t3
t1 t2 t3
small ⌧ big ⌧
Ketan Savla (LIDS, MIT) Dynamical Queues September 14, 2010 5 / 13
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
0
0
1
1
t
t
x(t)
b(t)
t1 t2 t3
t1 t2 t3
small ⌧ big ⌧
Ketan Savla (LIDS, MIT) Dynamical Queues September 14, 2010 5 / 13
Objectives
t
x(t)
x
S(x)
Smin
S(1)
1
u
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
Ketan Savla (LIDS, MIT) Dynamical Queues September 14, 2010 6 / 13
Homogeneous tasks: one-task equilibrium
fW = �w; u(t) ⌘ ON
0
1
t
x(t)
xi xi+1
ti ti + S(xi) ti + 1�
x0 1
S(x)
R(x,�low)
R(x,�med)
R�x, �
�
Smin
Smax
1/�
xxmed,1 xmed,2xlow
xeq(�): set of one-task eqm states
� := max{� | xeq(�) 6= ;}x := xeq(�)
Ketan Savla (LIDS, MIT) Dynamical Queues September 14, 2010 7 / 13
Threshold policy is throughputoptimal:
uT(t) =
{ON if x(t) ≤ x,OFF otherwise.
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 8 / 11
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
HTQ2
(a) (b)
1x
2x
3x
1y 2y
3y0 L
' HTQ1
HTQ2
HTQ1Release Control
Policy
HTQ2
1x
2x
3x
1y 2y
3y0 L
1x
2x
3x
1y 2y
3y0 L
1x
2x
3x
1y 2y
3y0 L
1x
2x
3x
1y 2y
3y0 L
0 1�ix ix 1�ix L
0 1�ix ix 1�ix LHTQ
VTQ
ix�
minv
iv
' 'D iy
1x
2x
3x
1y2y
3y 0 L
arrM\
M
\
in
out
depM
service rate =∑i(xi+1 − xi)m
S1 S2
1x 2x 3x
1x 2x 3x
0 1
L0
10
1x 2x 3x
1x 2x
1I 1B 2I 2B
1 2 3
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 9 / 11
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
HTQ2
(a) (b)
1x
2x
3x
1y 2y
3y0 L
' HTQ1
HTQ2
HTQ1Release Control
Policy
HTQ2
1x
2x
3x
1y 2y
3y0 L
1x
2x
3x
1y 2y
3y0 L
1x
2x
3x
1y 2y
3y0 L
1x
2x
3x
1y 2y
3y0 L
0 1�ix ix 1�ix L
0 1�ix ix 1�ix LHTQ
VTQ
ix�
minv
iv
' 'D iy
1x
2x
3x
1y2y
3y 0 L
arrM\
M
\
in
out
depM
service rate =∑i(xi+1 − xi)m
S1 S2
1x 2x 3x
1x 2x 3x
0 1
L0
10
1x 2x 3x
1x 2x
1I 1B 2I 2B
1 2 3
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 9 / 11
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
HTQ2
(a) (b)
1x
2x
3x
1y 2y
3y0 L
' HTQ1
HTQ2
HTQ1Release Control
Policy
HTQ2
1x
2x
3x
1y 2y
3y0 L
1x
2x
3x
1y 2y
3y0 L
1x
2x
3x
1y 2y
3y0 L
1x
2x
3x
1y 2y
3y0 L
0 1�ix ix 1�ix L
0 1�ix ix 1�ix LHTQ
VTQ
ix�
minv
iv
' 'D iy
1x
2x
3x
1y2y
3y 0 L
arrM\
M
\
in
out
depM
service rate =∑i(xi+1 − xi)m
S1 S2
1x 2x 3x
1x 2x 3x
0 1
L0
10
1x 2x 3x
1x 2x
1I 1B 2I 2B
1 2 3
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 9 / 11
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
HTQ2
(a) (b)
1x
2x
3x
1y 2y
3y0 L
' HTQ1
HTQ2
HTQ1Release Control
Policy
HTQ2
1x
2x
3x
1y 2y
3y0 L
1x
2x
3x
1y 2y
3y0 L
1x
2x
3x
1y 2y
3y0 L
1x
2x
3x
1y 2y
3y0 L
0 1�ix ix 1�ix L
0 1�ix ix 1�ix LHTQ
VTQ
ix�
minv
iv
' 'D iy
1x
2x
3x
1y2y
3y 0 L
arrM\
M
\
in
out
depM
service rate =∑i(xi+1 − xi)m
S1 S2
1x 2x 3x
1x 2x 3x
0 1
L0
10
1x 2x 3x
1x 2x
1I 1B 2I 2B
1 2 3
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 9 / 11
TSP under Differential Constraints
n = λ− n
TSP(n)
Euclidean: TSP(n) = Θ(n1/2)
Dubins, Diff Drive, ...TSP(n) = Θ(n2/3)
fig5
Phase 1 Phase 2 Phase 3
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 10 / 11
TSP under Differential Constraints
n = λ− n
TSP(n)
Euclidean: TSP(n) = Θ(n1/2)
Dubins, Diff Drive, ...TSP(n) = Θ(n2/3)
fig5
Phase 1
Phase 2 Phase 3
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 10 / 11
TSP under Differential Constraints
n = λ− n
TSP(n)
Euclidean: TSP(n) = Θ(n1/2)
Dubins, Diff Drive, ...TSP(n) = Θ(n2/3)
fig5
Phase 1 Phase 2
Phase 3
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 10 / 11
TSP under Differential Constraints
n = λ− n
TSP(n)
Euclidean: TSP(n) = Θ(n1/2)
Dubins, Diff Drive, ...TSP(n) = Θ(n2/3)
fig5
Phase 1 Phase 2 Phase 3
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 10 / 11
Current and Future Work
fig16
Systems Science
Physical Science
operations research
network science
decision theory
⇑
Research
Theme
⇓
dynamics
human behavior
physics
Current and Future Work
{Cascades over Networks} +{Physics-based Failure and Recovery}{Game Theory} +{Geometric Strategy Space}{Compressed Sensing} +{Constrained Measurement Matrix}
Emphasis on fundamental performance limits
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 11 / 11
Current and Future Work
fig16
Systems Science
Physical Science
operations research
network science
decision theory
⇑
Research
Theme
⇓
dynamics
human behavior
physics
Current and Future Work
{Cascades over Networks} +{Physics-based Failure and Recovery}
{Game Theory} +{Geometric Strategy Space}{Compressed Sensing} +{Constrained Measurement Matrix}
Emphasis on fundamental performance limits
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 11 / 11
Current and Future Work
fig16
Systems Science
Physical Science
operations research
network science
decision theory
⇑
Research
Theme
⇓
dynamics
human behavior
physics
Current and Future Work
{Cascades over Networks} +{Physics-based Failure and Recovery}{Game Theory} +{Geometric Strategy Space}
{Compressed Sensing} +{Constrained Measurement Matrix}
Emphasis on fundamental performance limits
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 11 / 11
Current and Future Work
fig16
Systems Science
Physical Science
operations research
network science
decision theory
⇑
Research
Theme
⇓
dynamics
human behavior
physics
Current and Future Work
{Cascades over Networks} +{Physics-based Failure and Recovery}{Game Theory} +{Geometric Strategy Space}{Compressed Sensing} +{Constrained Measurement Matrix}
Emphasis on fundamental performance limits
Ketan Savla (CEE, USC) Theoretical CPS April 14, 2017 11 / 11