A New Control Theory for Dynamic Data Driven Systems
A New Control Theory for Dynamic Data Driven Systems Nikolai Matni Computing and Mathematical Sciences
Joint work with Yuh-Shyang Wang, James Anderson & John C. Doyle
New application areas
1
New application areas
2
Reflex Distributed
Fast (offline computation)
Rigid
Unstable real dynamics
Planning Centralized
Slow (online computation)
Flexible
Stable virtual dynamics 3
Designing reflex layers for large-scale cyber-physical systems
4
inspired from the second-order dynamics of a power network X, NO. XX, SEPTEMBER 2016 or mechanical our companionregulation paper [16] for a Powersystem grid:– see frequency detailed discussion.
5]. (14) nto tic mal ent the ed larized ) - as arly caexpenameters ves
11
11 10 20 9 18 8 16 7 14 6 12
✓i
✓j
˙i ✓
˙j ✓
5 10 48 36 2 4 1 2 0 00 0
42
84
126
16 8
20 10
(a) topology (b) Interactionbetween between neighboring (a) Interconnected Interconnected topology (b) Interaction neighborTopology Subsystem interaction subsystems ing subsystems 2
6
10
14
18
e, each on example interaction graph. 3. Simulation Example oror aFig. 3. Simulation Fig.
5
Centralized and dense control Controller
6
Sparse, distributed & localized control w/ delayed communications C
C
C
C C
C
C C C
C
C
C
C
C C
C
C
C
C
7
new challenges info exchange constraints
NP-hard problems
6
new challenges info exchange constraints
comms channels
7
new challenges info exchange constraints A
comms channels infrastructure co-design
S
S A
A A
S
S
S
A
A S
S A
S
10
new challenges info exchange constraints
~125 million homes powered
comms channels
~100 billion neurons, ~100k miles of axons
infrastructure co-design large (huge) scale
billions of nodes billions of devices 9
A unified theory for system design robustness info exchange scale to efficiency + comms in the loop + millions, billions, resilience co-design trillions of nodes
11
10k
time (s)
100
Distributed
Centralized
state of the art Localized (serial)
system level synthesis
1
Localized (parallel)
.01
100
1000 # States
10k
100k 10
DDDAS + SLS for Automated Control System Design and Redesign
14
Sys ID
Controller Synthesis
Controller
Experts Needed One Shot ID/Design 15
2
task. The dynamic interaction between two neighboring subsystems is shown in Fig. 3(b), in which state x i ,1 can affect state x j ,2 if i and j are neighbors. This interaction model is inspired from the second-order dynamics of a power network or mechanical system – see our companion paper [16] for a detailed discussion.
The analytic solution to (13) is given by the vectorial softthresholding operator [22]: vec(M 2 ) kin+j ,1d = (1 −
λj / ⇢ )+ · ||(M k1 + 1 + ⇤k ) in j , d ||H 2 20
vec(M k1 + 1 + ⇤k ) in j , d
(14)
18 16 14
in which (c) + = max(0, c) for any scalar c. Thus usingCONTROL, these analytic expressions implement the 2016 EE TRANSACTIONS ON AUTOMATIC VOL. XX, NO.to XX, SEPTEMBER iterate updates (11a) and (11b), the actuator regularized LLQR optimization problem (8) can be solved nearly as 11 quickly as the state-feedback problem, as the most expen10 Consider the LLQGsive problem in [25]. (a) Interconnected topology (b) Interaction between neighborstep is proposed to compute the update equation parameters 9 ing subsystems bal optimization problem (F oaj , d , Fobis ) decomposed as defined in (12)into – once this 8is done, each j ,d 7 ✓ update can be implemented via a matrix multiplication or a i Example ✓j oblems, each subproblem is a convex quadratic Fig. 3. Simulation 6 soft-thresholding. 5
Sys ID
12 10
8
Controller Synthesis
Controller
11
6 4 2 0
0
2
4
6
8
10
12
14
16
18
20
cted on an affine set. In this case, the proximal The diagonal and off-diagonal entries of A are drawn affine function [20], [25], [30]. We only need 1 If we only ˙j 0.4, − 0.2][ considered subsystems in the d-outgoing set, the localized ˙[0.4, from a uniform distribution over✓ 0.8] and ✓ [− i constraint L would have no effect on the synthesized controllers, thus we s affine function once. The iteration in (30a) [0.2, 0.4], respectively. The open loop plant is unstable, with extend the region by 1 to incorporate “ boundary” subsystems that must have be carried out using multiple multiplicaa zero responsematrix to the disturbance at subsystem j . the spectral radius of A given by 1.2246 in the example that (a) Interconnected topology (b) Interaction between neighboring duced dimension, which significantly improves subsystems mputation time. : Consider the LLQR problem with actuator Fig. 3. Simulation example interaction graph. [22], which is the state feedback version of mn-wise separable part is identical to the LLQG swing equation for power network. Consider the swing dyhe update (30b) can be computed using matrix namic equation Suppose that we use (32) as the actuator norm. X ¨i + di ✓ ˙i = − [22] that the row-wise separable part can be mi ✓ ki j ( ✓i − ✓j ) + wi + ui (40) multiple unconstrained optimization problems, j 2Ni operators given by vectorial soft-thresholding ˙i , m i , di , wi , ui are the phase angle deviation, where ✓i , ✓ rs an efficient way to update (30a). frequency deviation, inertia, damping, external disturbance, MM, there exists other distributed algorithms and control action of the controllable load of bus i . The sed to solve the optimization problem (29) in coefficient ki j is the coupling term between buses i and d scalable way. For instance, if either g( r ) (·) or ˙i ]> be the state of bus i and use j . We let x i := [✓i ✓ gly convex, we can use the alternating minimizaA∆ t e ⇡ I + A∆ t to discretize the swing dynamics. Equation (AMA) [31] to simplify the ADMM algorithm. (40) can then be expressed in the form of (2) with ternatives include the over-relax ADMM [20] 0 0 ated version of ADMM and AMA proposed in 1 ∆t Ai i = , Ai j = ki j , − ki ∆ t 1 − di ∆ t 4 3 2 1 0
0
2
4
6
8
10
Experts Needed Ever Changing System 16
Need a theory for local, scalable and automated
Adaptive Control and Sys ID Fault & Change Detection
Controller (re)design Actuation/sensing (re)design 17
Need a theory for local, scalable and automated
Adaptive Control and Sys ID Fault & Change Detection
Controller (re)design Actuation/sensing (re)design 18
A System Level Approach
A System Level Approach Novel parameterizations of stabilizing controllers
A System Level Approach Novel parameterizations of stabilizing controllers
(Approximate) Locality = scalability
controlled output
measurement
disturbance
control input
22
controlled output
measurement
disturbance
control input
info sharing constraints 23
control input
measurement sns
act
sns
act
controller sns
24
sns
act
sns
act
controller sns
25
sns
act
comm delay
sns
act sns
26
sns
act
comm delay
sns
act sns
27
controlled output
measurement
disturbance
control input
info sharing constraints 28
System level approach design entire closed loop response 29
What we care about
System level approach design entire closed loop response 30
What we design
System level approach design entire closed loop response 31
What we design
System level approach design entire closed loop response
Key idea: maintain the notion of state
32
What we design
System level approach design entire closed loop response
state structure
33
34
35
36
37
state structure
38
parameterize & design entire system response 39
need to ensure achievability of 40
need to ensure that there exists . such that the above holds 41
42
43
in affine space
44
Theorem [implementation]
45
sys response structure
controller internal structure
46
controller simulates full system response
47
system response and controller internal structure 48
(Convex) System level synthesis any (convex) functional
SLC: any (convex) set
Convex SLCs: system performance, controller structure & architecture, locality, etc. Unified framework for custom controller synthesis 49
State-Feedback System Level Parameterization response
implementation
achievability
50
Localizability Sensing / actuation delay: 1 Comm speed: 2 (compared to plant speed)
51
Localizability
52
Localizability
State deviation 53
Localizability activate
propagate 54
Localizability
55
Localizability
56
Localizability
57
Localizability
58
Idealized centralized control Control
State Space
- Comm Speed: Inf
Time 59
Distributed LQR (Lamperski, Doyle 2013) Control Control State action
Space
Comm speed: 2 Comm speed: Inf ~Global optimal Global optimal
Time
60
LLQR Control State action
Space
Comm speed: 2 ~Global optimal Time
61
LLQR Control State action
Space Comm range: 48 3
Time
62
LLQR Control State action
Space
Space-time region Comm speed: 2 ~Global optimal
Time
63
inspired from the second-order dynamics of a power network X, NO. XX, SEPTEMBER 2016 or mechanical our companionregulation paper [16] for a Powersystem grid:– see frequency detailed discussion.
5]. (14) nto tic mal ent the ed larized ) - as arly caexpenameters ves
11
11 10 20 9 18 8 16 7 14 6 12
✓i
✓j
˙i ✓
˙j ✓
5 10 48 36 2 4 1 2 0 00 0
42
84
126
16 8
20 10
(a) topology (b) Interactionbetween between neighboring (a) Interconnected Interconnected topology (b) Interaction neighborTopology Subsystem interaction subsystems ing subsystems 2
6
10
14
18
e, each on example interaction graph. 3. Simulation Example oror aFig. 3. Simulation Fig.
64
10k
time (s)
100
Distributed
Centralized
state of the art Localized (serial)
system level synthesis
1
Localized (parallel)
.01
100
1000 # States
10k
100k 10
Need a theory for local, scalable and automated
Adaptive Control and Sys ID Fault & Change Detection
Controller (re)design Actuation/sensing (re)design 66
Need a theory for local, scalable and automated
Adaptive Control and Sys ID Fault & Change Detection
Controller (re)design Actuation/sensing (re)design 67
LLQR Control action
Space
Space-time region
Actuation Failure
Comm speed: 2 ~Global optimal
Time
68
LLQR LOCAL Fault Det Control & Redesign action
Space
Space-time region
Comm speed: 2 ~Global optimal
Time
69
LLQR Control action
Space
Space-time region
Comm speed: 2 ~Global optimal
Time
70
Just scratching the surface… Novel parameterization: • new approach to constrained optimal control • new perspective on closed loop sys ID/adaptive? Locality = scalability: • game changer for optimal controller synthesis • consequences for DDDAS? Goal: automated control (re)design for DDDAS 71
Nikolai Matni Select references •
Y.-S. Wang, N. Matni and J. C. Doyle, System level parameterizations, constraints and synthesis, IEEE ACC 2017.
•
Y.-S. Wang, N. Matni and J. C. Doyle, A system level approach to controller synthesis, IEEE TAC 2017, Submitted.
•
N. Matni and V. Chandrasekaran, Regularization for design, IEEE TAC 2016.
•
N. Matni and J. C. Doyle, A theory of dynamics, optimization and control in layered architectures, IEEE ACC 2016.
Joint work with Yuh-Shyang Wang, James Anderson & John C. Doyle
New application areas
1
New application areas
2
Reflex Distributed
Fast (offline computation)
Rigid
Unstable real dynamics
Planning Centralized
Slow (online computation)
Flexible
Stable virtual dynamics 3
Designing reflex layers for large-scale cyber-physical systems
4
inspired from the second-order dynamics of a power network X, NO. XX, SEPTEMBER 2016 or mechanical our companionregulation paper [16] for a Powersystem grid:– see frequency detailed discussion.
5]. (14) nto tic mal ent the ed larized ) - as arly caexpenameters ves
11
11 10 20 9 18 8 16 7 14 6 12
✓i
✓j
˙i ✓
˙j ✓
5 10 48 36 2 4 1 2 0 00 0
42
84
126
16 8
20 10
(a) topology (b) Interactionbetween between neighboring (a) Interconnected Interconnected topology (b) Interaction neighborTopology Subsystem interaction subsystems ing subsystems 2
6
10
14
18
e, each on example interaction graph. 3. Simulation Example oror aFig. 3. Simulation Fig.
5
Centralized and dense control Controller
6
Sparse, distributed & localized control w/ delayed communications C
C
C
C C
C
C C C
C
C
C
C
C C
C
C
C
C
7
new challenges info exchange constraints
NP-hard problems
6
new challenges info exchange constraints
comms channels
7
new challenges info exchange constraints A
comms channels infrastructure co-design
S
S A
A A
S
S
S
A
A S
S A
S
10
new challenges info exchange constraints
~125 million homes powered
comms channels
~100 billion neurons, ~100k miles of axons
infrastructure co-design large (huge) scale
billions of nodes billions of devices 9
A unified theory for system design robustness info exchange scale to efficiency + comms in the loop + millions, billions, resilience co-design trillions of nodes
11
10k
time (s)
100
Distributed
Centralized
state of the art Localized (serial)
system level synthesis
1
Localized (parallel)
.01
100
1000 # States
10k
100k 10
DDDAS + SLS for Automated Control System Design and Redesign
14
Sys ID
Controller Synthesis
Controller
Experts Needed One Shot ID/Design 15
2
task. The dynamic interaction between two neighboring subsystems is shown in Fig. 3(b), in which state x i ,1 can affect state x j ,2 if i and j are neighbors. This interaction model is inspired from the second-order dynamics of a power network or mechanical system – see our companion paper [16] for a detailed discussion.
The analytic solution to (13) is given by the vectorial softthresholding operator [22]: vec(M 2 ) kin+j ,1d = (1 −
λj / ⇢ )+ · ||(M k1 + 1 + ⇤k ) in j , d ||H 2 20
vec(M k1 + 1 + ⇤k ) in j , d
(14)
18 16 14
in which (c) + = max(0, c) for any scalar c. Thus usingCONTROL, these analytic expressions implement the 2016 EE TRANSACTIONS ON AUTOMATIC VOL. XX, NO.to XX, SEPTEMBER iterate updates (11a) and (11b), the actuator regularized LLQR optimization problem (8) can be solved nearly as 11 quickly as the state-feedback problem, as the most expen10 Consider the LLQGsive problem in [25]. (a) Interconnected topology (b) Interaction between neighborstep is proposed to compute the update equation parameters 9 ing subsystems bal optimization problem (F oaj , d , Fobis ) decomposed as defined in (12)into – once this 8is done, each j ,d 7 ✓ update can be implemented via a matrix multiplication or a i Example ✓j oblems, each subproblem is a convex quadratic Fig. 3. Simulation 6 soft-thresholding. 5
Sys ID
12 10
8
Controller Synthesis
Controller
11
6 4 2 0
0
2
4
6
8
10
12
14
16
18
20
cted on an affine set. In this case, the proximal The diagonal and off-diagonal entries of A are drawn affine function [20], [25], [30]. We only need 1 If we only ˙j 0.4, − 0.2][ considered subsystems in the d-outgoing set, the localized ˙[0.4, from a uniform distribution over✓ 0.8] and ✓ [− i constraint L would have no effect on the synthesized controllers, thus we s affine function once. The iteration in (30a) [0.2, 0.4], respectively. The open loop plant is unstable, with extend the region by 1 to incorporate “ boundary” subsystems that must have be carried out using multiple multiplicaa zero responsematrix to the disturbance at subsystem j . the spectral radius of A given by 1.2246 in the example that (a) Interconnected topology (b) Interaction between neighboring duced dimension, which significantly improves subsystems mputation time. : Consider the LLQR problem with actuator Fig. 3. Simulation example interaction graph. [22], which is the state feedback version of mn-wise separable part is identical to the LLQG swing equation for power network. Consider the swing dyhe update (30b) can be computed using matrix namic equation Suppose that we use (32) as the actuator norm. X ¨i + di ✓ ˙i = − [22] that the row-wise separable part can be mi ✓ ki j ( ✓i − ✓j ) + wi + ui (40) multiple unconstrained optimization problems, j 2Ni operators given by vectorial soft-thresholding ˙i , m i , di , wi , ui are the phase angle deviation, where ✓i , ✓ rs an efficient way to update (30a). frequency deviation, inertia, damping, external disturbance, MM, there exists other distributed algorithms and control action of the controllable load of bus i . The sed to solve the optimization problem (29) in coefficient ki j is the coupling term between buses i and d scalable way. For instance, if either g( r ) (·) or ˙i ]> be the state of bus i and use j . We let x i := [✓i ✓ gly convex, we can use the alternating minimizaA∆ t e ⇡ I + A∆ t to discretize the swing dynamics. Equation (AMA) [31] to simplify the ADMM algorithm. (40) can then be expressed in the form of (2) with ternatives include the over-relax ADMM [20] 0 0 ated version of ADMM and AMA proposed in 1 ∆t Ai i = , Ai j = ki j , − ki ∆ t 1 − di ∆ t 4 3 2 1 0
0
2
4
6
8
10
Experts Needed Ever Changing System 16
Need a theory for local, scalable and automated
Adaptive Control and Sys ID Fault & Change Detection
Controller (re)design Actuation/sensing (re)design 17
Need a theory for local, scalable and automated
Adaptive Control and Sys ID Fault & Change Detection
Controller (re)design Actuation/sensing (re)design 18
A System Level Approach
A System Level Approach Novel parameterizations of stabilizing controllers
A System Level Approach Novel parameterizations of stabilizing controllers
(Approximate) Locality = scalability
controlled output
measurement
disturbance
control input
22
controlled output
measurement
disturbance
control input
info sharing constraints 23
control input
measurement sns
act
sns
act
controller sns
24
sns
act
sns
act
controller sns
25
sns
act
comm delay
sns
act sns
26
sns
act
comm delay
sns
act sns
27
controlled output
measurement
disturbance
control input
info sharing constraints 28
System level approach design entire closed loop response 29
What we care about
System level approach design entire closed loop response 30
What we design
System level approach design entire closed loop response 31
What we design
System level approach design entire closed loop response
Key idea: maintain the notion of state
32
What we design
System level approach design entire closed loop response
state structure
33
34
35
36
37
state structure
38
parameterize & design entire system response 39
need to ensure achievability of 40
need to ensure that there exists . such that the above holds 41
42
43
in affine space
44
Theorem [implementation]
45
sys response structure
controller internal structure
46
controller simulates full system response
47
system response and controller internal structure 48
(Convex) System level synthesis any (convex) functional
SLC: any (convex) set
Convex SLCs: system performance, controller structure & architecture, locality, etc. Unified framework for custom controller synthesis 49
State-Feedback System Level Parameterization response
implementation
achievability
50
Localizability Sensing / actuation delay: 1 Comm speed: 2 (compared to plant speed)
51
Localizability
52
Localizability
State deviation 53
Localizability activate
propagate 54
Localizability
55
Localizability
56
Localizability
57
Localizability
58
Idealized centralized control Control
State Space
- Comm Speed: Inf
Time 59
Distributed LQR (Lamperski, Doyle 2013) Control Control State action
Space
Comm speed: 2 Comm speed: Inf ~Global optimal Global optimal
Time
60
LLQR Control State action
Space
Comm speed: 2 ~Global optimal Time
61
LLQR Control State action
Space Comm range: 48 3
Time
62
LLQR Control State action
Space
Space-time region Comm speed: 2 ~Global optimal
Time
63
inspired from the second-order dynamics of a power network X, NO. XX, SEPTEMBER 2016 or mechanical our companionregulation paper [16] for a Powersystem grid:– see frequency detailed discussion.
5]. (14) nto tic mal ent the ed larized ) - as arly caexpenameters ves
11
11 10 20 9 18 8 16 7 14 6 12
✓i
✓j
˙i ✓
˙j ✓
5 10 48 36 2 4 1 2 0 00 0
42
84
126
16 8
20 10
(a) topology (b) Interactionbetween between neighboring (a) Interconnected Interconnected topology (b) Interaction neighborTopology Subsystem interaction subsystems ing subsystems 2
6
10
14
18
e, each on example interaction graph. 3. Simulation Example oror aFig. 3. Simulation Fig.
64
10k
time (s)
100
Distributed
Centralized
state of the art Localized (serial)
system level synthesis
1
Localized (parallel)
.01
100
1000 # States
10k
100k 10
Need a theory for local, scalable and automated
Adaptive Control and Sys ID Fault & Change Detection
Controller (re)design Actuation/sensing (re)design 66
Need a theory for local, scalable and automated
Adaptive Control and Sys ID Fault & Change Detection
Controller (re)design Actuation/sensing (re)design 67
LLQR Control action
Space
Space-time region
Actuation Failure
Comm speed: 2 ~Global optimal
Time
68
LLQR LOCAL Fault Det Control & Redesign action
Space
Space-time region
Comm speed: 2 ~Global optimal
Time
69
LLQR Control action
Space
Space-time region
Comm speed: 2 ~Global optimal
Time
70
Just scratching the surface… Novel parameterization: • new approach to constrained optimal control • new perspective on closed loop sys ID/adaptive? Locality = scalability: • game changer for optimal controller synthesis • consequences for DDDAS? Goal: automated control (re)design for DDDAS 71
Nikolai Matni Select references •
Y.-S. Wang, N. Matni and J. C. Doyle, System level parameterizations, constraints and synthesis, IEEE ACC 2017.
•
Y.-S. Wang, N. Matni and J. C. Doyle, A system level approach to controller synthesis, IEEE TAC 2017, Submitted.
•
N. Matni and V. Chandrasekaran, Regularization for design, IEEE TAC 2016.
•
N. Matni and J. C. Doyle, A theory of dynamics, optimization and control in layered architectures, IEEE ACC 2016.
