The structure of optimal distributed controllers
What you get for free, and what you can impose
Bassam Bamieh
Mechanical Engineering, UCSB
, DISC, June ’09
slide 1/7
The Setting
The Plant has spatially distributed dynamics
The controller also has spatially distributed dynamics
, DISC, June ’09
slide 2/7
The Setting
The Plant has spatially distributed dynamics
The controller also has spatially distributed dynamics
For a given plant structure,
what’s the inherent structure of the Centralized Controller?
If we want to constrain the controller’s architecture,
what type of constraints lead to tractable problems?
, DISC, June ’09
slide 2/7
The Approach
W E WILL TAKE A BROAD VIEW OF spatially distributed dynamics
Systems described by Partial Differential Equations (PDEs)
Continuous Space
Dynamical systems over lattices and graphs
Discrete Space
, DISC, June ’09
slide 3/7
The Approach
W E WILL TAKE A BROAD VIEW OF spatially distributed dynamics
Systems described by Partial Differential Equations (PDEs)
Continuous Space
Dynamical systems over lattices and graphs
Discrete Space
Look for “interesting” special structures
Special structure ⇥
, DISC, June ’09
More detailed results
Insight
slide 3/7
The Approach
W E WILL TAKE A BROAD VIEW OF spatially distributed dynamics
Systems described by Partial Differential Equations (PDEs)
Continuous Space
Dynamical systems over lattices and graphs
Discrete Space
Look for “interesting” special structures
Structure
generality
Result's utility
, DISC, June ’09
slide 3/7
The Approach
W E WILL TAKE A BROAD VIEW OF spatially distributed dynamics
Systems described by Partial Differential Equations (PDEs)
Continuous Space
Dynamical systems over lattices and graphs
Discrete Space
Look for “interesting” special structures
Structure
generality
Result's utility
, DISC, June ’09
slide 3/7
Part I
What you get for free:
The inherent structure of the centralized controller for
spatially distributed plants
, DISC, June ’09
slide 4/7
Outline
Examples
Vehicular Platoons
Heat Equation with Distributed Control
Spatially-Invariant Plants
Optimal Controllers are Inherently Spatially Invariant
Optimal Centralized Controllers are Inherently Localized
Spatially-Varying Plants
Localized Plants over Arbitrary Networks
Notions of Distance and Spatial Decay
Central LQR Controllers are Inherently Localized
, DISC, June ’09
slide 5/7
Vehicular Platoons
w
⇥
wo
1
w1
⇥
w2
⇥
⇥
...
...
L⇥
x
1
L⇥
xo
L⇥
x1
x2
Objective: Design a controller for each vehicle to:
• Maintain constant small slot length L.
• Reject the effect of disturbances {wi} (wind gusts, road conditions, etc...)
Vehicular Platoons
w
⇥
wo
1
w1
⇥
w2
⇥
⇥
...
...
L⇥
x
1
L⇥
xo
L⇥
x1
x2
Objective: Design a controller for each vehicle to:
• Maintain constant small slot length L.
• Reject the effect of disturbances {wi} (wind gusts, road conditions, etc...)
Warning: Designs based on two vehicle models may lack “string stability”,
i.e. disturbances get amplified as they propagate through the platoon.
Vehicular Platoons
w
⇥
wo
1
w1
⇥
w2
⇥
⇥
...
...
L⇥
x
1
L⇥
xo
L⇥
x1
x2
Objective: Design a controller for each vehicle to:
• Maintain constant small slot length L.
• Reject the effect of disturbances {wi} (wind gusts, road conditions, etc...)
Warning: Designs based on two vehicle models may lack “string stability”,
i.e. disturbances get amplified as they propagate through the platoon.
Problem Structure:
• Actuators: each vehicle’s throttle input.
• Sensors: position and velocity of each vehicle.
9
Vehicular Platoons Set-up
xi: i’th vehicle’s position.
w
⇥
...
x̃i := xi
x̃1,i := x̃i
x̃2,i := x̃˙ i
xi
1
L
1
⇥
L⇥
C
x
1
wo
w1
L⇥
xo
w2
⇥
⇥
..
L⇥
x1
x2
Vehicular Platoons (Optimal LQR)
Centralized LQR design (Melzer & Kuo ’70, Athans & Levine ’66)
⇤
x̃˙
ṽ˙
⌅
⌅⇤ ⌅
⇤ ⌅
0 I
x̃
0
+
w,
0 0
ṽ
I
⌃ ⇥⇧
⇥
J =
q1 (x̃k x̃k 1 )2 + q2 ṽ2k + u2k
=
⇤
0
Feedback gains are
“localized”:
k
#!!
"!!
+,)-
!
!"!!
!#!!
!$!!
&!
%!
&!
$!
%!
$!
#!
#!
"!
*()
Inherent Localization:
, DISC, June ’09
"!
!
!
'()
Bamieh et. al, TAC ’02, Motee et. al. ’07
slide 33/76
Vehicular Platoons Set-up
xi: i’th vehicle’s position.
w
⇥
...
x̃i := xi
xi
L
1
1
⇥
L⇥
C
x
x̃1,i := x̃i
x̃2,i := x̃˙ i
1
wo
w1
L⇥
xo
w2
⇥
⇥
..
L⇥
x1
x2
Structure of generalized plant:
H =
H11 H12
H21 H22
⇥
⇤
⇥
⌥
⌥
= ⌥
⌥ ⇥
⇧
...
...
0
⇥
ho
h1
The generalized plant has a Toeplitz structure!
0
...
...
⌅
⌃
z
w
H
u
y
⇥
C
z = F(H, C) w
10
Optimal Controller for Vehicular Platoon
Example: Centralized H2 optimal controller gains for a 50 vehicle platoon
(From: Shu and Bamieh ’96)
#!!
"!!
+,)-
!
!"!!
!#!!
!$!!
&!
%!
&!
$!
%!
$!
#!
#!
"!
*()
Remarks:
"!
!
!
'()
Figure 1: Position error feedback gains for a 50 vehicle platoon
• For large platoons, optimal controller is approximately Toeplitz
• Optimal centralized controller has some inherent decentralization (“localization”)
Controller gains decay away from the diagonal
Optimal Controller for Vehicular Platoon
Example: Centralized H2 optimal controller gains for a 50 vehicle platoon
(From: Shu and Bamieh ’96)
#!!
"!!
+,)-
!
!"!!
!#!!
!$!!
&!
%!
&!
$!
%!
$!
#!
#!
"!
*()
Remarks:
"!
!
!
'()
Figure 1: Position error feedback gains for a 50 vehicle platoon
• For large platoons, optimal controller is approximately Toeplitz
• Optimal centralized controller has some inherent decentralization (“localization”)
Controller gains decay away from the diagonal
Q: Do the above 2 results occur in all “such” problems?
11
Simple Example; Distributed LQR Control of Heat Equation
⇤
⇤2
⇥(x, t) = c 2 ⇥(x, t) + u(x, t)
⇤t
⇤x
⇥
d ˆ
⇥( , t) =
dt
c
2
ˆ , t) + û( , t)
⇥(
Solve the LQR problem with Q = qI, R = I. The corresponding ARE family:
2c
2
p̂( )
p̂( )2 + q = 0,
and the positive solution is:
p̂( ) =
c
2
+
c2
4
+ q.
Remark: In general P̂ ( ) an irrational function of , even if Â( ), B̂( ) are rational.
i.e. PDE systems have optimal feedbacks which are not PDE operators.
Let {k(x)} be the inverse Fourier transform of the function { p̂( )}.
26
Then optimal (temporally static) feedback
u(x, t) =
k(x
R
) ⇥( , t) d
!"#$%
!
Remark: The “spread” of {k(x)} indicates information required from distant sensors.
27
Distributed LQR Control of Heat Equation (Cont.)
Important Observation: {k(x)} is “localized”. It decays exponentially!!
k̂( ) = c
2
c2
4
+ q.
Distributed LQR Control of Heat Equation (Cont.)
Important Observation: {k(x)} is “localized”. It decays exponentially!!
⌥
2
k̂(⇥) = c⇥
c2⇥4 + q.
This can be analytically extended by:
⌥
2
k̂e(s) = cs
c2s4 + q,
which is analytic in the strip
⇧
⇧ ⇤ ⌅1 ⌃
2 q 4
s ⇤ C ; Im{s} <
.
2 c2
Therefore: ⌅M such that
|k(x)| ⇥ M e
|x|
,
for any
⇧ ⇤ ⌅1
2 q 4
<
.
2
2 c
⇥
⇥
⇥
⇥
⇥
⇥
⇥1
q 4
c2
⇥
⇥
⇥
⇥
⇥
Distributed LQR Control of Heat Equation (Cont.)
Important Observation: {k(x)} is “localized”. It decays exponentially!!
⌥
2
k̂(⇥) = c⇥
c2⇥4 + q.
This can be analytically extended by:
⌥
2
k̂e(s) = cs
c2s4 + q,
which is analytic in the strip
⇧
⇧ ⇤ ⌅1 ⌃
2 q 4
s ⇤ C ; Im{s} <
.
2 c2
Therefore: ⌅M such that
|k(x)| ⇥ M e
|x|
,
for any
⇥
⇥
⇥
⇥
⇥
⇥
⇥1
q 4
c2
⇥
⇥
⇥
⇥
⇥
⇧ ⇤ ⌅1
2 q 4
<
.
2
2 c
This results is true in general: under mild conditions
Solutions of AREs always inverse transform to exponentially decaying convolution
kernels
28
Outline
Examples
Vehicular Platoons
Heat Equation with Distributed Control
Spatially-Invariant Plants
Optimal Controllers are Inherently Spatially Invariant
Optimal Centralized Controllers are Inherently Localized
Spatially-Varying Plants
Localized Plants over Arbitrary Networks
Notions of Distance and Spatial Decay
Central LQR Controllers are Inherently Localized
, DISC, June ’09
slide 5/7
Distributed Systems with Special Structure
• General Infinite-dimensional Systems Theory
– Well posedness issues (semi-group theory)
– Constructive (convergent) approximation techniques
Distributed Systems with Special Structure
• General Infinite-dimensional Systems Theory
– Well posedness issues (semi-group theory)
– Constructive (convergent) approximation techniques
T HEME : Make infinite-dimensional problems look like finite-dimensional ones
Distributed Systems with Special Structure
• General Infinite-dimensional Systems Theory
– Well posedness issues (semi-group theory)
– Constructive (convergent) approximation techniques
T HEME : Make infinite-dimensional problems look like finite-dimensional ones
• Special Structure
– Distributed control and measurement (now more feasible)
– Regular (lattice) arrangement of devices
Together =
Spatial Invariance
Distributed Systems with Special Structure
• General Infinite-dimensional Systems Theory
– Well posedness issues (semi-group theory)
– Constructive (convergent) approximation techniques
T HEME : Make infinite-dimensional problems look like finite-dimensional ones
• Special Structure
– Distributed control and measurement (now more feasible)
– Regular (lattice) arrangement of devices
Together =
–
–
–
–
Spatial Invariance
Control of “Vehicular Strings”, (Melzer & Kuo, 71)
Discretized PDEs, (Brockett, Willems, Krishnaprasd, El-Sayed, ’74, ’81)
“Systems over rings”, (Kamen, Khargonekar, Sontag, Tannenbaum, ...)
Systems with “Dynamical Symmetry”, (Fagniani & Willems)
More recently:
– Controller architecture and localization, (Bamieh, Paganini, Dahleh)
– LMI techniques, localization, (D’Andrea, Dullerud, Lall)
6
System Representations
All signals are spatio-temporal, e.g. u(x, t), ⌅(x, t), y(x, t), etc.
Spatially distributed inputs, states, and outputs
• State space description
d
dt ⌅(x, t)
y(x, t)
= A ⌅(x, t) + B u(x, t)
= C ⌅(x, t) + D u(x, t)
A, B, C, D translation invariant operators
⇥
spatially invariant system
• Spatio-temporal impulse response h(x, t)
y(x, t) =
h(x
⇥, t
⇤ ) u(⇥, ⇤ ) d⇤ d⇥,
• Transfer function description
Y ( , ⇧) = H( , ⇧) U ( , ⇧)
2
Spatio-temporal Impulse Response
Spatio-temporal impulse response h(x, t)
y(x, t) =
h(x
,t
⇥ ) u( , ⇥ ) d⇥ d ,
Interpretation
h(x, t): effect of input on output a distance x away and time t later
Example: Constant maximum speed of effects
7
Example: Distributed Control of the Heat Equation
u
⇥
y
2
1
u
1
yo
⇥
uo
y1
⇥
u1
⇥
y2
u2
⇥
ui: input to heating elements.
yi: signal from temperature sensor.
Dynamics are given by:
⇥
.. ⇥
..
.. ⇥
..
⌃
⇧
⇧ ...
⇧ y 1 ⌃
H 1,0
... ⌃
⌃⇧ u 1 ⌃
⌃
⇧
⇧
⌃ ⇧ uo ⌃
⇧ yo ⌃ = ⇧
H0, 1 H0,0 H0,1
⌃
⌃⇧
⇧
⌃
⇧
⌅
⇤
⇤ ...
⇤ y1 ⌅
u1 ⌅
H1,0
...
..
..
..
..
Each Hi,j is an infinite-dimensional SISO system.
Fact: Dynamics are spatially invariant
H is Toeplitz
The input-output relation can be written as a convolution over the actuator/sensor
index:
yi =
⇥
⌥
H̄(i
j)
uj ,
j= ⇥
7
The limit of large actuator sensor array:
⇤⇥
⇤ 2⇥
(x, t) = c 2 (x, t) + u(x, t)
⇤t
⇤x
⇥x =
⇥
Hx
u d ,
⇥
8
Spatial Invariance of Dynamics
Indexing of actuator and sensor signals:
ui(t) := u(i1,...,in)(t),
yi(t) := y(i1,...,in)(t).
i := (i1, . . . , in) a spatial multi-index,
i ⇥ G := G1
...
Gn.
Spatial Invariance of Dynamics
Indexing of actuator and sensor signals:
ui(t) := u(i1,...,in)(t),
yi(t) := y(i1,...,in)(t).
i := (i1, . . . , in) a spatial multi-index,
i ⇤ G := G1
Linear input-output relations:
yi =
X
j⇤G
Hi,j uj ,
⇥
...
Gn.
A general linear system from u to y :
y(i1,...,in) =
X
j1 ⇤G1
...
X
jn ⇤Gn
H(i1,...,in),(j1,...,jn) u(j1,...,jn),
Spatial Invariance of Dynamics
Indexing of actuator and sensor signals:
ui(t) := u(i1,...,in)(t),
yi(t) := y(i1,...,in)(t).
i ⌃ G := G1 ⇥ . . . ⇥ Gn.
i := (i1, . . . , in) a spatial multi-index,
Linear input-output relations:
yi =
X
Hi,j uj ,
j⌃G
⇧
A general linear system from u to y :
y(i1,...,in) =
X
...
j1 ⌃G1
X
jn ⌃Gn
H(i1,...,in),(j1,...,jn) u(j1,...,jn),
Spatial Invariance:
Assumption 1: Set of spatial indices = commutative group
G := G1 ⇥ . . . ⇥ Gn,
each Gi a commutative group.
Remark: “spatial shifting” of signals
Compare with: Time shift by ⇥ (S⇥ u)(t) := u(t
(S u)i := ui
Assumption 2: Spatial invariance
⌥
⌃ G,
HS =S H
⇤⌅
⇧
⇥)
Commute with spatial shifts
S
1
HS = H
12
Examples of Spatial Invariance
Generally: Spatial invariance easily ascertained from basic physical symmetry!
• Vehicular platoons: signals index over Z.
Examples of Spatial Invariance
Generally: Spatial invariance easily ascertained from basic physical symmetry!
• Vehicular platoons: signals index over Z.
• Channel flow: Signals indexed over {0, 1}
y(l,i) =
⇥
j= ⇥
H(l
0,i j)
u(0,j) +
⇥
Z:
H(l
1,i j)
u(1,j),
j= ⇥
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
l = 0, 1.
Examples of Spatial Invariance
Generally: Spatial invariance easily ascertained from basic physical symmetry!
• Vehicular platoons: signals index over Z.
• Channel flow: Signals indexed over {0, 1}
⇥
y(l,i) =
H(l
0,i j)
j= ⇥
⇥
u(0,j) +
Z:
H(l
1,i j)
l = 0, 1.
u(1,j),
j= ⇥
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
Remark: The input-output mapping of a spatially invariant system can be rewritten:
yi =
Ḡi
j
uj ,
j⇤G
⇥
y(i1,...,in) =
...
j1 ⇤G1
Ḡ(i1
j1 ,...,in jn )
u(j1,...,jn).
jn ⇤Gn
A spatial convolution
13
Symmetry in Dynamical Systems and Control Design
• Many-body systems always have some inherent dynamical symmetries:
e.g. equations of motion are invariant to certain coordinate transformations
• Question: Given an unstable dynamical system with a certain symmetry,
is it possible to stabilize it with a controller that has the same symmetry?
(i.e. without “breaking the symmetry”)
• Answer: Yes! (Fagnani & Willems ’93)
Symmetry in Dynamical Systems and Control Design
• Many-body systems always have some inherent dynamical symmetries:
e.g. equations of motion are invariant to certain coordinate transformations
• Question: Given an unstable dynamical system with a certain symmetry,
is it possible to stabilize it with a controller that has the same symmetry?
(i.e. without “breaking the symmetry”)
• Answer: Yes! (Fagnani & Willems ’93)
Remark: Spatial invariance is a dynamical symmetry
This answer applies to optimal design as well
i.e.
For best achievable performance, need only consider spatially-invariant controllers
14
The Standard Problem of Optimal and Robust Control
z
w
The standard problem:
H
Signal norms:
w
p
p
:=
i G
u
y
⇥
R
⇥
|wi(t)|pdt =
w
i G
p
p
C
z = F(H, C) w
The Standard Problem of Optimal and Robust Control
z
w
The standard problem:
H
Signal norms:
w
p
p
:=
i⇥G
⇥
R
⇥
|wi(t)|pdt =
w
p
p
p i
:=
sup
w⇥LP
z
w
p
.
p
The H2 norm:
F(G, C)
2
H2
=
z
2
2
=
zi
2
L2 ,
i⇥G
with impulsive disturbance input wi(t) =
C
z = F(H, C) w
i⇥G
Induced system norms:
F(G, C)
u
y
(i) (t).
The Standard Problem of Optimal and Robust Control
z
w
The standard problem:
H
Signal norms:
⇥w⇥pp :=
u
y
i⇥G
⇥
R
⇥
|wi(t)|pdt =
i⇥G
⇥w⇥pp
C
z = F(H, C) w
Induced system norms:
⇥F(G, C)⇥p
i
⇥z⇥p
:= sup
.
⇥w⇥
P
p
w⇥L
The H2 norm:
⇥F(G, C)⇥2H2 = ⇥z⇥22 =
i⇥G
⇥zi⇥2L2 ,
with impulsive disturbance input wi(t) =
Note: In the platoon problem:
(i) (t).
finite system norm
string stability.
15
Spatially-Invariant vs. Spatially-Varying Controllers
Question: Are spatially-varying controllers better than spatially-invariant ones?
Answer: If plant is spatially invariant, no!
Spatially-Invariant vs. Spatially-Varying Controllers
Question: Are spatially-varying controllers better than spatially-invariant ones?
Answer: If plant is spatially invariant, no!
LSI := The class of Linear Spatially-Invariant systems.
LSV := The class of Linear Spatially-Varying systems.
Compare the two problems:
si
:=
inf
stabilizing C
C ⇥ LSI
F(G, C)
p i
The best achievable performance
with spatially-invariant controllers
sv
:=
inf
stabilizing C
C ⇥ LSV
F(G, C)
p i
The best achievable performance
with spatially-varying controllers
Theorem 1. If the plant and performance objectives are spatially invariant, i.e. if
the generalized plant G is spatially invariant, then the best achievable performance
can be approached with a spatially invariant controller. More precisely
si
=
sv .
16
Spatially-Invariant vs. Spatially-Varying Controllers (Cont.)
Related Problem: Time-Varying vs. Time-Invariant Controllers
Fact: For time-invariant plants, time-varying controllers offer no advantage over timeinvariant ones!
for norm minimization problems
Proofs based on use of YJBK parameterization. Convert to
ti
:=
inf
stable Q
Q ⇥ LT I
⇤T1
T2QT3⇤
tv
:=
T1, T2, T3 determined by plant, therefore time invariant.
inf
stable Q
Q ⇥ LT V
⇤T1
T2QT3⇤ ,
Spatially-Invariant vs. Spatially-Varying Controllers (Cont.)
Related Problem: Time-Varying vs. Time-Invariant Controllers
Fact: For time-invariant plants, time-varying controllers offer no advantage over timeinvariant ones!
for norm minimization problems
Proofs based on use of YJBK parameterization. Convert to
ti
:=
inf
stable Q
Q ⇥ LT I
⇤T1
T2QT3⇤
tv
:=
inf
stable Q
Q ⇥ LT V
⇤T1
T2QT3⇤ ,
T1, T2, T3 determined by plant, therefore time invariant.
• The H case: (Feintuch & Francis, ’85), (Khargonekar, Poolla, & Tannenbaum,
’85). A consequence of Nehari’s theorem
• The ⇥1 case: (Shamma & Dahleh, ’91). Using an averaging technique
• Any induced ⇥p norm: (Chapellat & Dahleh, ’92). Generalization of the averaging
technique
17
Spatially-Invariant vs. Spatially-Varying Controllers (Cont.)
Idea of proof:
si
:=
inf
stable Q
Q ⇤ LSI
After YJBK parameterization:
⌅T1
T2QT3⌅
⇥
Let Q̄ achieve a performance level ¯ = ⌅T1
sv
T2Q̄T3⌅.
:=
inf
stable Q
Q ⇤ LSV
⌅T1
T2QT3⌅ ,
Spatially-Invariant vs. Spatially-Varying Controllers (Cont.)
Idea of proof:
si
:=
After YJBK parameterization:
inf
stable Q
Q ⌅ LSI
⌃T1
T2QT3⌃
⇥
Let Q̄ achieve a performance level ¯ = ⌃T1
sv
:=
inf
stable Q
Q ⌅ LSV
⌃T1
T2QT3⌃ ,
T2Q̄T3⌃.
Averaging Q̄:
• If G is finite: define
Qav :=
1
|G|
⇥
⇤G
1
Q̄⇥. ⇤ Qav is spatially invariant, i.e. ⇧⇥ ⌅ G, ⇥
1
Qav ⇥ = Qav
Then
⌅T1
T2Qav T3⌅
=
⇥
⌅T1
T2
1 X‚
‚
‚
|G| ⇤G
1 X
|G| ⇤G
1
`
T1
1
Q̄
!
‚
‚ 1 X
‚
T3⌅ = ‚
‚ |G|
´ ‚
‚
T2Q̄T3 ‚ = ⌅T1
1
⇤G
T2Q̄T3⌅
`
T1
‚
´ ‚
‚
T2Q̄T3 ‚
‚
18
• If G is infinite, take a sequence of finite subsets M1 ⇥ M2 ⇥ · · · , with
,
Then define:
Qn :=
Qn converges weak
1
|Mn|
1
⇥
Mn = G
n
Q̄ .
⇥Mn
to a spatially-invariant Qav with the required norm bound.
19
Implications of the Structure of Spatial Invariance
Poiseuille flow stabilization:
...
C1
u
⇥
y
1
1
⇥
⇥
⇥
⌅
⇤⇥
uo
C
C
Co
2
...
1
yo
y1
u1
⇥
⇥
u2
y2
⇥
Channel
ui =
Ci
j
yj
j
20
Implications of the Structure of Spatial Invariance
Poiseuille flow stabilization:
...
C1
⇥
⇥
⇥
⌅
⇤⇥
u
⇥
C
Co
1
y
1
C
2
...
1
uo
yo
u1
⇥
⇥
y1
u2
y2
⇥
Channel
ui =
Ci
j
yj
j
21
Implications of the Structure of Spatial Invariance (Cont.)
Uneven distribution of sensors and actuators
Consider the following geometry of sensors and actuators:
• Sensor
Actuator
What kind of spatial invariance do optimal controllers have?
22
Implications of the Structure of Spatial Invariance (Cont.)
Uneven distribution of sensors and actuators (Cont.)
Consider the following geometry of sensors and actuators:
• Sensor
Actuator
Each “cell” is a 1-input, 2-output system.
underlying group is Z
Z
23
Transform Methods
Consider the following PDE with distributed control:
⇥
⇤⇥
(x1, . . . , xn, t) = A ⇥x⇥ 1 ,..., ⇥x⇥n ⇥(x1, . . . , xn, t) + B
⇤t
⇥
y(x1, . . . , xn, t) = C ⇥x⇥ 1 ,..., ⇥x⇥n ⇥(x1, . . . , xn, t),
where A, B, C are matrices of polynomials in
⇥
⇥
⇥x1 ,..., ⇥xn
⇥
u(x1, . . . , xn, t)
⇥
⇥xi .
Consider also combined PDE difference equations such as:
⇥
⇤⇥
1
1
(x1, . . . , xm, k1, . . . , kn, t) = A ⇥x⇥ 1 ,..., ⇥x⇥n , z1 , . . . , zn ⇥(x1, . . . , xn, k1, . . . , kn, t)
⇤t
⇥
1
1
+ B ⇥x⇥ 1 ,..., ⇥x⇥n , z1 , . . . , zn u(x1, . . . , xn, k1, . . . , kn, t)
We only require that the spatial variables x, k, belong to a commutative group
Taking the Fourier transform:
ˆ , t) :=
⇥(
⇤
e
j< ,x>
⇥(x, t) dx,
G
24
The above system equations become:
d⇥ˆ
ˆ , t) + B ( ) û( , t)
( , t) = A ( ) ⇥(
dt
ˆ , t),
ŷ( , t) = C ( ) ⇥(
where
Ĝ, the dual group to G.
Remark: This can be thought of as a parameterized family of finite-dimensional
systems.
25
B LOCK D IAGONALIZATION
BY
F OURIER T RANSFORMS
The Fourier transform converts:
spatially-invariant operators on L2(G)
⇥
multiplication operators on L2(Ĝ)
In general:
group: G dual group: Ĝ Transform
R
R
Fourier Transform
Z
⇥D
Z-Transform
⇥D
Z
Fourier Series
Zn
Zn
Discrete Fourier Transform
and the transforms preserve L2 norms:
⇤f ⇤22 =
G
|f (x)|2dx =
Ĝ
|fˆ( )|2d = ⇤fˆ⇤22
B LOCK D IAGONALIZATION
BY
F OURIER T RANSFORMS
The Fourier transform converts:
spatially-invariant operators on L2(G)
⇥
multiplication operators on L2(Ĝ)
In general:
group: G dual group: Ĝ Transform
R
R
Fourier Transform
Z
⇤D
Z-Transform
⇤D
Z
Fourier Series
Zn
Zn
Discrete Fourier Transform
and the transforms preserve L2 norms:
⇤f ⇤22 =
G
|f (x)|2dx =
Ĝ
|fˆ( )|2d = ⇤fˆ⇤22
The system operation is then spatially decoupled or “block diagonalized”:
d ˆ
dt ⇥(
ˆ , t) + B̂( )û( , t)
, t) = Â( )⇥(
ˆ , t) + D̂( )û( , t)
ŷ( , t) = Ĉ( )⇥(
⇤
⇤t ⇥(x, t)
= A ⇥(x, t) + B u(x, t)
y(x, t) = C ⇥(x, t) + D u(x, t)
A distributed,
spatially-invariant system
⇥
11
A parameterized family
of finite-dimensional systems
T RANSFORM M ETHODS
In physical space
d
⇥n = An ⇤ ⇥n + Bn ⇤ un
dt
yn = Cn ⇤ ⇥n
After spatial Fourier trans. (FT)
d ˆ
ˆ ) + B̂( ) û( )
⇥( ) = Â( ) ⇥(
dt
ˆ )
ŷ( ) = Ĉ( ) ⇥(
T RANSFORM M ETHODS
In physical space
d
⇥n = An ⇤ ⇥n + Bn ⇤ un
dt
yn = Cn ⇤ ⇥n
After spatial Fourier trans. (FT)
d ˆ
ˆ ) + B̂( ) û( )
⇥( ) = Â( ) ⇥(
dt
ˆ )
ŷ( ) = Ĉ( ) ⇥(
IMPLICATIONS
• Dynamics are decoupled by FT
(The A, B, C operators are “diagonalized” )
T RANSFORM M ETHODS
After spatial Fourier trans. (FT)
In physical space
d
⇥n = An ⇤ ⇥n + Bn ⇤ un
dt
yn = Cn ⇤ ⇥n
d ˆ
ˆ ) + B̂( ) û( )
⇥( ) = Â( ) ⇥(
dt
ˆ )
ŷ( ) = Ĉ( ) ⇥(
IMPLICATIONS
• Dynamics are decoupled by FT
• Quadratic forms preserved by FT
(The A, B, C operators are “diagonalized” )
=
Quadratically optimal control
problems are equivalent for FT
T RANSFORM M ETHODS
After spatial Fourier trans. (FT)
In physical space
d
⇥n = An ⇤ ⇥n + Bn ⇤ un
dt
yn = Cn ⇤ ⇥n
d ˆ
ˆ ) + B̂( ) û( )
⇥( ) = Â( ) ⇥(
dt
ˆ )
ŷ( ) = Ĉ( ) ⇥(
IMPLICATIONS
• Dynamics are decoupled by FT
• Quadratic forms preserved by FT
(The A, B, C operators are “diagonalized” )
=
Quadratically optimal control
problems are equivalent for FT
• Yields a parametrized family of mutually independent problems
T RANSFORM M ETHODS
After spatial Fourier trans. (FT)
In physical space
d
⇥n = An ⇤ ⇥n + Bn ⇤ un
dt
yn = Cn ⇤ ⇥n
d ˆ
ˆ ) + B̂( ) û( )
⇥( ) = Â( ) ⇥(
dt
ˆ )
ŷ( ) = Ĉ( ) ⇥(
IMPLICATIONS
• Dynamics are decoupled by FT
• Quadratic forms preserved by FT
(The A, B, C operators are “diagonalized” )
=⇥
Quadratically optimal control
problems are equivalent for FT
• Yields a parametrized family of mutually independent problems
T RANSFER
FUNCTIONS
operator-valued transfer function
H(s) = C (sI
A)
1
B
spatio-temporal transfer function
⇥ 1
H(s, ) = Ĉ( ) sI Â( )
B̂( )
T RANSFORM M ETHODS
After spatial Fourier trans. (FT)
In physical space
d
⇥n = An ⇤ ⇥n + Bn ⇤ un
dt
yn = Cn ⇤ ⇥n
d ˆ
ˆ ) + B̂( ) û( )
⇥( ) = Â( ) ⇥(
dt
ˆ )
ŷ( ) = Ĉ( ) ⇥(
IMPLICATIONS
• Dynamics are decoupled by FT
• Quadratic forms preserved by FT
(The A, B, C operators are “diagonalized” )
=⇥
Quadratically optimal control
problems are equivalent for FT
• Yields a parametrized family of mutually independent problems
T RANSFER
FUNCTIONS
operator-valued transfer function
H(s) = C (sI
A)
1
spatio-temporal transfer function
⇥ 1
H(s, ) = Ĉ( ) sI Â( )
B̂( )
B
A multi-dimensional system with temporal, but not spatial causality
12
M. J OVANOVI Ć , UCSB
31
Optimal Control of Infinite Platoons
☞ G OOD
M AIN
APPROXIMATION OF LARGE BUT FINITE PLATOONS
IDEA : EXPLOIT SPATIAL INVARIANCE
⇧
LA:
⌥
⇤
☞
⌃
⇤
˙
⌅˙
˙n
⌅˙ n
⌅
⌅
=
NOT STABILIZABLE AT
=
0 1
0
0 1
0
⇥ = 0
e
0
T
1
0
j
⇥
⇥
n
⌅n
⇥
⌦SPATIAL Z
⌅
⇥
+
0
1
+
⇥
un ,
n⇤Z
- TRANSFORM
0
1
⇥
u , 0 ⇥ ⇥ < 2⇤
Parameterized ARE solutions yield “localized” operators!
Consider unbounded domains, i.e. G = R (or Z).
Theorem 2. Consider the parameterized family of Riccati equations:
A (⇥)P (⇥) + P (⇥)A(⇥)
P (⇥)B(⇥)R(⇥)B (⇥)P (⇥) + Q(⇥) = 0,
⇥ ⇥ Ĝ.
Under mild conditions:
there exists an analytic continuation P (s) of P (⇥) in a region
{|Im(s)| < },
> 0.
Convolution kernel resulting from Parameterized ARE has exponential decay.
That is, they have some degree of inherent decentralization (“localization”)!
Parameterized ARE solutions yield “localized” operators!
Consider unbounded domains, i.e. G = R (or Z).
Theorem 2. Consider the parameterized family of Riccati equations:
A (⇥)P (⇥) + P (⇥)A(⇥)
P (⇥)B(⇥)R(⇥)B (⇥)P (⇥) + Q(⇥) = 0,
⇥ ⇥ Ĝ.
Under mild conditions:
there exists an analytic continuation P (s) of P (⇥) in a region
{|Im(s)| < },
> 0.
Convolution kernel resulting from Parameterized ARE has exponential decay.
That is, they have some degree of inherent decentralization (“localization”)!
Comparison:
• Modal truncation: In the transform domain, ARE solutions decay algebraically.
• Spatial truncation: In the spatial domain, convolution kernel of ARE solution
decays exponentially.
Therefore: Use transform domain to design ⇤⇥. Approximate in the spatial domain!
29
D ISTRIBUTED A RCHITECTURE
OF
Q UADRATICALLY O PTIMAL C ONTROLLERS
E XAMPLE : one dimensional array of systems indexed in Z.
u
y
2
⇥
1
u
y0
⇥
cell # -1
n:
u0
1
⇥
cell #0
the state of the system in the n’th cell
y1
u1
⇥
cell # 1
y2
u2
⇥
cell # 2
total state: {. . . ,
1,
o,
1 , . . .}
D ISTRIBUTED A RCHITECTURE
OF
Q UADRATICALLY O PTIMAL C ONTROLLERS
E XAMPLE : one dimensional array of systems indexed in Z.
u
y
1
2
⇥
u
y0
⇥
⇥
cell # -1
n:
u0
1
cell #0
the state of the system in the n’th cell
d
dt
n
An
=
m
m
m
m
Cn
yn =
m
Bn
+
m
n
m
um
y1
u1
⇥
cell # 1
y2
u2
⇥
cell # 2
total state: {. . . ,
1,
o,
1 , . . .}
D ISTRIBUTED A RCHITECTURE
OF
Q UADRATICALLY O PTIMAL C ONTROLLERS
E XAMPLE : one dimensional array of systems indexed in Z.
u
y
1
2
⇥
u
y0
⇥
⇥
cell # -1
n:
u0
1
cell #0
n
An
=
m
m
m
um
m
m
Cn
yn =
Bn
+
m
n
m
d
dt
⇥
n
= An ⇥
yn = Cn ⇥
n
u1
⇥
cell # 1
y2
u2
⇥
cell # 2
total state: {. . . ,
the state of the system in the n’th cell
d
dt
y1
+ Bn ⇥ un
n
7
1,
o,
1 , . . .}
D ISTRIBUTED A RCHITECTURE
⇥
u
⇥
2
y
⇥
OF
⇥
1
u
⇥
Q UADRATICALLY O PTIMAL C ONTROLLERS
⇥
1
yo
⇥
uo
⇥
⇥
y1
⇥
u1
⇥
⇥
y2
⇥
u2
⇥
Observer based controller has the following structure:
Controller
Plant
d
dt
n
= An ⇥
yn = Cn ⇥
n
n
+ Bn ⇥ un
ui = Ki ⇥ ˆi
d ˆ
ˆ
n = An ⇥ n + Bn ⇥ un
dt
+ Ln ⇥ (yn ŷn)
D ISTRIBUTED A RCHITECTURE
⇥
u
⇥
2
y
⇥
OF
⇥
1
u
⇥
Q UADRATICALLY O PTIMAL C ONTROLLERS
⇥
1
yo
⇥
uo
⇥
⇥
⇥
y1
u1
⇥
⇥
y2
⇥
u2
⇥
Observer based controller has the following structure:
Controller
Plant
d
dt
n
= An ⇥
yn = Cn ⇥
n
n
+ Bn ⇥ un
ui = Ki ⇥ ˆi
d ˆ
ˆ
n = An ⇥ n + Bn ⇥ un
dt
+ Ln ⇥ (yn ŷn)
R EMARKS :
• Optimal Controller is “locally” finite dimensional.
D ISTRIBUTED A RCHITECTURE
⇥
u
⇥
2
y
⇥
OF
⇥
1
u
⇥
Q UADRATICALLY O PTIMAL C ONTROLLERS
⇥
1
yo
⇥
uo
⇥
⇥
⇥
y1
u1
⇥
⇥
y2
⇥
u2
⇥
Observer based controller has the following structure:
Controller
Plant
d
dt
n
= An ⇥
yn = Cn ⇥
n
n
+ Bn ⇥ un
ui = Ki ⇥ ˆi
d ˆ
ˆ
n = An ⇥ n + Bn ⇥ un
dt
+ Ln ⇥ (yn ŷn)
R EMARKS :
• Optimal Controller is “locally” finite dimensional.
• The gains {Ki}, {Li} are localized (exponentially decaying) ⇥ “spatial truncation”
D ISTRIBUTED A RCHITECTURE
⇥
u
⇥
2
y
⇥
OF
⇥
1
u
⇥
Q UADRATICALLY O PTIMAL C ONTROLLERS
⇥
1
yo
⇥
uo
⇥
⇥
⇥
y1
u1
⇥
⇥
y2
⇥
u2
⇥
Observer based controller has the following structure:
Controller
Plant
d
dt
n
= An ⇥
yn = Cn ⇥
n
n
+ Bn ⇥ un
ui = Ki ⇥ ˆi
d ˆ
ˆ
n = An ⇥ n + Bn ⇥ un
dt
+ Ln ⇥ (yn ŷn)
R EMARKS :
• Optimal Controller is “locally” finite dimensional.
• The gains {Ki}, {Li} are localized (exponentially decaying) ⇥ “spatial truncation”
• After truncation, local controller need only receive information from neighboring
subsystems.
D ISTRIBUTED A RCHITECTURE
⇥
u
⇥
2
y
⇥
OF
⇥
1
u
Q UADRATICALLY O PTIMAL C ONTROLLERS
⇥
1
⇥
yo
uo
⇥
⇥
⇥
⇥
y1
u1
⇥
⇥
y2
⇥
u2
⇥
Observer based controller has the following structure:
Controller
Plant
d
dt
n
= An ⇥
yn = Cn ⇥
n
n
+ Bn ⇥ un
ui = Ki ⇥ ˆi
d ˆ
ˆ
n = An ⇥ n + Bn ⇥ un
dt
+ Ln ⇥ (yn ŷn)
R EMARKS :
• Optimal Controller is “locally” finite dimensional.
• The gains {Ki}, {Li} are localized (exponentially decaying) ⇥ “spatial truncation”
• After truncation, local controller need only receive information from neighboring
subsystems.
• Quadratically optimal controllers are inherently distributed and semi-decentralized
(localized)
8
Outline
Examples
Vehicular Platoons
Heat Equation with Distributed Control
Spatially-Invariant Plants
Optimal Controllers are Inherently Spatially Invariant
Optimal Centralized Controllers are Inherently Localized
Spatially-Varying Plants
Localized Plants over Arbitrary Networks
Notions of Distance and Spatial Decay
Central LQR Controllers are Inherently Localized
PhD Thesis of Nader Motee
Motee & Jadbabaie, Optimal control of spatially distributed systems
, DISC, June ’09
slide 5/7
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Part II
What you can impose
Architectural constraints that lead to convex
optimal control problems
, DISC, June ’09
slide 6/7
Outline
Controller Constraints that Lead to Convex Problems
The YJBK Parameterization
Funnel Causality
, DISC, June ’09
slide 7/7
Controller Architecture
Centralized vs. Decentralized control: An old and difficult problem
5
CENTRALIZED :
⇥
G0
⇥
⇤
G1
⇥
⇤
⌅
G2
⇥
BEST PERFORMANCE
⇤
⌅
EXCESSIVE COMMUNICATION
⌅
K
FULLY DECENTRALIZED :
⇥
G0
⇥
⇤
G1
⇥
⇤
⌅
G2
WORST PERFORMANCE
⇤
⌅
K0
⇥
NO COMMUNICATION
⌅
K1
K2
LOCALIZED :
⇥
G0
⇥
⇤
⇥
⇤
⌅
⇥
G1
K0
⇥
MANY POSSIBLE ARCHITECTURES
⇤
⌅
⇥
G2
K1
⌅
⇥
K2
⇥
6
Reasoning with the YJBK Parameterization
Let G be a stable MIMO plant
All stabilizing controllers (Internal Model Control)
K = Q(I + GQ)
1
Q stable
If G and Q belong to a CLASS closed under
additions, multiplications, inversions
Then
Q ⌅ CLASS ⇤ K ⌅ CLASS
, DISC, June ’09
slide 8/8
Reasoning with the YJBK Parameterization
Let G be a stable MIMO plant
All stabilizing controllers (Internal Model Control)
K = Q(I + GQ)
1
Q stable
If G and Q belong to a CLASS closed under
additions, multiplications, inversions
Then
Q ⌅ CLASS ⇤ K ⌅ CLASS
Optimal design becomes
Q
inf
stable, Q⇥CLASS
⇧H
UQV⇧
Convex CLASS ⇥ Convex problem
, DISC, June ’09
slide 8/8
Reasoning with the YJBK Parameterization
Let G be a stable MIMO plant
All stabilizing controllers (Internal Model Control)
K = Q(I + GQ)
1
Q stable
If G and Q belong to a CLASS closed under
additions, multiplications, inversions
Then
Q ⌅ CLASS ⇤ K ⌅ CLASS
Optimal design becomes
Q
inf
stable, Q⇥CLASS
⇧H
UQV⇧
Convex CLASS ⇥ Convex problem
If G is unstable, use a factorization G = NM
All stabilizing controllers
K = (Y + MQ)(X + NQ)
, DISC, June ’09
1
1,
XM
YN = I
Q stable
slide 8/8
Spatio-temporal Impulse Response
Spatio-temporal impulse response h(x, t)
y(x, t) =
h(x
,t
⇥ ) u( , ⇥ ) d⇥ d ,
Interpretation
h(x, t): effect of input on output a distance x away and time t later
Example: Constant maximum speed of effects
7
Funnel Causality
Def: A system is funnel-causal if impulse response h(., .) satisfies
h(x, t) = 0 for
where
f (.)
is
t < f (x),
(1) non-negative
(2) f (0) = 0
(3) {f (x), x ⇥ 0} and {f (x), x
0} are concave
i.e. supp (h) is a “funnel shaped” region
8
Funnel Causality (Cont.)
Properties of funnel causal systems
Let Sf be a funnel shaped set
• supp (h1) ⇥ Sf & supp (h2) ⇥ Sf
⇤
supp (h1 + h2) ⇥ Sf
• supp (h1) ⇥ Sf & supp (h2) ⇥ Sf
⇤
supp (h1 h2) ⇥ Sf
• (I +h1)
1
exists & supp (h1) ⇥ Sf
⇤
supp (I + h1)
1
⇥
⇥ Sf
i.e.
The class of funnel-causal systems is closed under
Parallel, Serial, & Feedback
interconnections
9
A Class of Convex Problems
• Given a plant G with supp (G22)
S fg
• Let Sfk be a set such that Sfg Sfk
i.e. controller signals travel at least as fast as the plant’s
Solve
inf
⇥F(G; K)⇥,
K stabilizing
supp (K) Sfk
z
w
G
u
y
⇥
K
10
YJBK Parameterization and the Model Matching Problem
Lf := class of linear systems w/ impulse response supported in Sf
• Let G22 ⇤ Lfg
G22 = N M 1 and XM
• Let Sfg ⇥ Sfk
Y N = I with N, M, X, Y ⇤ Lfg and stable
• Then all stabilizing controllers K such that K ⇤ Lfk are given by
K = (Y + M Q)(X + N Q)
1
,
where Q is a stable system in Lfk .
• The problem becomes
⌅H
inf
Q stable
Q ⇤ Lfk
U QV ⌅,
A convex problem!
11
Coprime Factorizations
Bezout identity: Find K and L such that A + LC and A + BK stable
⌃
⌥ ⇤
⌅
A + BK B
⇥
A + LC
B L
M
:=
,
:= ⌦
X
Y
K
I ↵,
N
K
I
0
C
0
then G = N M
If
⇧
1
and XM
Y N = I,
• etAB, CetA and CetAB are funnel causal
• K and L are funnel causal (Easy!)
then all elements of Bezout identity are funnel-causal
A + BK B
C
0 ↵
⌦
K
0
⇥
K
A B
⇧
⌃
C
0
⇤
⌅
I 0
⌅
⇤⇥
⇥
12
Example: Wave Equations with Input
1-d wave equation, x
State space
:
representation
The semigroup
etA
1
=
2
⇧
R:
⇥t
⇧
1
2
⇥t2 (x, t) = c2 ⇥x2 (x, t) + u(x, t)
⌃
⇧
⌃⇧
⌃ ⇧ ⌃
0 I
0
1
=
+
u
2 2
c ⇥x 0
I
⌃2
⇧
⇤
⌅
1
.
=
I 0
1
Tct + T ct
c Rct
c⇥x2 Rct Tct + T
2
ct
⌃
G(x,t)
.
1
Rct := spatial convolution with rec( ct
x)
Tct := translation by ct
x
x=!ct
all components are funnel causal
e.g. the impulse response h(x, t) =
t
x=ct
1
2c
rec
⇥
1
ct x
.
13
Example: Wave Equations with Input (cont.)
:= spatial Fourier transform variable (“wave number”)
⇧
⌃ ⇧ ⌃
⇥
0
1
0
A + BK =
+
k1 k2
2 2
c
0
1
⇧
⌃
0
1
=
.
2 2
c
+ k1 k2
Set k1 = 0, then
⇥(A+BK) =
⌥⇤
R
1
k2 ±
2
4c2
k22
2
⌅
=
⇧
⌃ ⌥
3 1
k2, k2
(k2 +jR)
2 2
Similarly for A + LC. Therefore, choose e.g.
K=
0
⇥
1 ,
L=
⇧
1
0
⌃
.
14
Elements of the Bezout Identity are thus:
⇧
⌃
1
1
0
1
⇥
⌦
1 0
= ⌥ c2 2 0
,
X
Y
0
1 1
0
⇧
⌃
0
1 0
⇤
⌅
c2 2
1 1 ⌦
M
⌦
=
⌦.
N
1 1
⌥ 0
1
0 0
Equivalently
M =
s2 + c2 2
,
s2 + s + c2 2
N =
1
,
2
s + s + c2 2
X =
Y
=
s2 + 2s + c2 2+1
,
s2 + s + c2 2
c2 2
.
s2 + s + c2 2
15
How easily solvable are the resulting convex problems?
• In general, these convex problems are infinite dimensional
i.e. worse than standard half-plane causality
• In certain cases, problem similar in complexity to half-plane causality
e.g. H 2 with the causality structure below
(Voulgaris, Bianchini, Bamieh, SCL ’03)
16
Generalizations
• Quick generalizations:
– Several spatial dimensions
– Spatially-varying systems
funnel causality
non-decreasing speed with distance
– Use relative degree in place of time delay
• Quadratic Invariance (Rotkowitz, Lall )
• Arbitrary graphs (Rotkowitz, Cogill, Lall )
• How to solve the resulting convex problems
Related recent work:
• Anders Rantzer
17