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Regularized & Distributionally RobustData-Enabled Predictive ControlFlorian DorflerETH Zurich Peking University Seminar
Acknowledgements
Jeremy Coulson
John Lygeros
Linbin Huang
Ivan Markovsky
Paul Beuchat
Further:Ezzat Elokda,Daniele Alpago,Jianzhe (Trevor) Zhen,Saverio Bolognani,Andrea Favato,Paolo Carlet, &IfA DeePC team
1/28
Perspectives on model-based control
x+ = Ax + Buy = Cx + Du
controller
system
model
system→ models useful forsystem analysis, design,estimation, . . . control→ modeling from firstprinciples & system ID
recurring themes• modeling & system ID
are very expensive• models not always
useful for control• need for end-to-end
automation solutions
From experiment design to closed-loop control!
Hakan Hjalmarsson∗
Department of Signals, Sensors and Systems, Royal Institute of Technology, S-100 44 Stockholm, Sweden
1. Introduction
Ever increasing productivity demands and environmental
standards necessitate more and more advanced control meth-
ods to be employed in industry. However, such methods usu-
ally require a model of the process and modeling and system
identification are expensive. Quoting (Ogunnaike, 1996):
“It is also widely recognized, however, that obtaining the
process model is the single most time consuming task in the
application of model-based control.”
In Hussain (1999) it is reported that three quarters of the
total costs associated with advanced control projects can
be attributed to modeling. It is estimated that models exist
for far less than one percent of all processes in regulatory
control. According to Desborough and Miller (2001), one of
the few instances when the cost of dynamic modeling can
be justified is for the commissioning of model predictive
controllers.
It has also been recognized that models for control pose
special considerations. Again quoting (Ogunnaike, 1996):
“There is abundant evidence in industrial practice that
when modeling for control is not based on criteria related
to the actual end use, the results can sometimes be quite
disappointing.”
Hence, efficient modeling and system identification tech-
niques suited for industrial use and tailored for control de-
sign applications have become important enablers for indus-
trial advances. The Panel for Future Directions in Control,
(Murray, Aström, Boyd, Brockett, & Stein, 2003), has iden-
tified automatic synthesis of control algorithms, with inte-
grated validation and verification as one of the major future
challenges in control. Quoting (Murray et al., 2003):
“Researchers need to develop much more powerful design
tools that automate the entire control design process from
model development to hardware-in-the-loop simulation.”
2/28
Control in a data-rich world• ever-growing trend in CS & applications:
data-driven control by-passing models• canonical problem: black/gray-box
system control based on I/O samples
Q: Why give up physical modeling &reliable model-based algorithms ?
data-driven
control
u2
u1 y1
y2
Data-driven control is viable alternative when• models are too complex to be useful
(e.g., fluids, wind farms, & building automation)
• first-principle models are not conceivable(e.g., human-in-the-loop, biology, & perception)
• modeling & system ID is too cumbersomee.g., robotics, drives, & electronics applications
Central promise: Itis often easier to learncontrol policies directlyfrom data, rather thanlearning a model.Example: PID [Astrom, ’73]
3/28
Abstraction reveals pros & consindirect (model-based) data-driven control
minimize control cost(x, u
)
subject to(x, u
)satisfy state-space model
where x estimated from(u, y)
& model
where model identified from(ud, yd
)data
→ nested multi-level optimization problem
}outeroptimization
}middle opt.
}inner opt.
separation &certaintyequivalence(→ LQG case)}no separation(→ ID-4-control)
direct (black-box) data-driven control
minimize control cost(u, y)
subject to(u, y)
consistent with(ud, yd
)data
→ trade-offsmodular vs. end-2-end
suboptimal (?) vs. optimalconvex vs. non-convex (?)
Additionally: all above should be min-max or E(·) accounting for uncertainty . . .4/28
A direct approach: dictionary + MPC1© trajectory dictionary learning• motion primitives / basis functions• theory: Koopman & Liouville
practice: (E)DMD & particles
2© MPC optimizing over dictionary span
→ huge theory vs. practice gap→ back to basics: impulse response
y4y2
y1y3 y5
y6
y7
u2 = u3 = · · · = 0
u1 = 1
x0 =0
dynamic matrix control(Shell, 1970s): predictivecontrol from raw data
yfuture(t) =[y1 y2 y3 . . .
]·
ufuture(t)
ufuture(t− 1)ufuture(t− 2)
...
today : arbitrary, finite, & corrupted data, . . . stochastic & nonlinear ?
5/28
ContentsI. Data-Enabled Predictive Control (DeePC): Basic Idea
J. Coulson, J. Lygeros, and F. Dorfler. Data-Enabled Predictive Control: In theShallows of the DeePC. [arxiv.org/abs/1811.05890].
II. From Heuristics & Numerical Promises to TheoremsJ. Coulson, J. Lygeros, and F. Dorfler. Distributionally Robust Chance ConstrainedData-enabled Predictive Control. [https://arxiv.org/abs/2006.01702].
I. Markovsky and F. Dorfler. Identifiability in the Behavioral Setting. [link]
III. Application: End-to-End Automation in Energy & Robotics
L. Huang, J. Coulson, J. Lygeros, and F. Dorfler. Decentralized Data-EnabledPredictive Control for Power System Oscillation Damping.[arxiv.org/abs/1911.12151].
E. Elokda, J. Coulson, P. Beuchat, J. Lygeros, and F. Dorfler. Data-EnabledPredictive Control for Quadcopters. [link].
[click here] for related publications
Previewcomplex 4-area power system:large (n=208), few sensors (8),nonlinear, noisy, stiff, inputconstraints, & decentralizedcontrol specifications
control objective: oscillationdamping without model(grid has many owners, models areproprietary, operation in flux, . . . )
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!&( seek a method that worksreliably, can be efficientlyimplemented, & certifiable→ automating ourselves
6/28
Reality check: magic or hoax ?surely, nobody would put apply such a shaky data-driven method• on the world’s most complex engineered system (the electric grid),• using the world’s biggest actuators (Gigawatt-sized HVDC links),• and subject to real-time, safety, & stability constraints . . . right?
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at least someone believes that DeePC is practically useful . . .7/28
Behavioral view on LTI systemsDefinition: A discrete-time dynamicalsystem is a 3-tuple (Z≥0,W,B) where
(i) Z≥0 is the discrete-time axis,
(ii) W is a signal space, and
(iii) B ⊆ WZ≥0 is the behavior.
B is the set ofall trajectories
Definition: The dynamical system (Z≥0,W,B) is(i) linear if W is a vector space & B is a subspace of WZ≥0
(ii) and time-invariant if B ⊆ σB, where σwt = wt+1.
LTI system = shift-invariant subspace of trajectory space
y
u
8/28
LTI systems and matrix time seriesfoundation of state-space subspace system ID & signal recovery algorithms
u(t)
t
u4
u2
u1 u3
u5u6
u7
y(t)
t
y4
y2
y1
y3
y5
y6
y7
(u(t), y(t)
)satisfy recursive
difference equationb0ut+b1ut+1+. . .+bnut+n+
a0yt+a1yt+1+. . .+anyt+n = 0
(ARX / kernel representation)
⇐under assumptions
⇒
[ 0 b0 a0 b1 a1 ... bn an 0 ] in left nullspaceof trajectory matrix (collected data)
HT
(ud
yd
)=
(ud1,1
yd1,1
) (ud1,2
yd1,2
) (ud1,3
yd1,3
)...
(ud2,1
yd2,1
) (ud2,2
yd2,2
) (ud2,3
yd2,3
)...
......
......
(udT,1
ydT,1
) (udT,2
ydT,2
) (udT,3
ydT,3
)...
where ydt,i is tth sample from ith experiment
9/28
Fundamental Lemma [Willems et al. ’05], [Markovsky & Dorfler ’20]
u(t)
t
u4
u2
u1 u3
u5u6
u7
y(t)
t
y4
y2
y1
y3
y5
y6
y7
Given: data(udiydi
)∈ Rm+p & LTI complexity parameters
{lag `order n
set of all T -length trajectories ={
(u, y) ∈ R(m+p)T : ∃x ∈ Rn s.t.
x+ = Ax + Bu , y = Cx + Du}
︸ ︷︷ ︸ ︸ ︷︷ ︸parametric state-space model non-parametric model from raw data
colspan
(ud
1,1
yd1,1
) (ud
1,2
yd1,2
) (ud
1,3
yd1,3
)...
(ud
2,1
yd2,1
) (ud
2,2
yd2,2
) (ud
2,3
yd2,3
)...
......
......
(udT,1
ydT,1
) (udT,2
ydT,2
) (udT,3
ydT,3
)...
if and only if the trajectory matrix has rank m · T + n for all T > `
10/28
set of all T -length trajectories ={
(u, y) ∈ R(m+p)T : ∃x ∈ Rn s.t.
x+ = Ax + Bu , y = Cx + Du}
︸ ︷︷ ︸ ︸ ︷︷ ︸parametric state-space model non-parametric model from raw data
colspan
(ud
1,1
yd1,1
) (ud
1,2
yd1,2
) (ud
1,3
yd1,3
)...
(ud
2,1
yd2,1
) (ud
2,2
yd2,2
) (ud
2,3
yd2,3
)...
......
......
(udT,1
ydT,1
) (udT,2
ydT,2
) (udT,3
ydT,3
)...
all trajectories constructible from finitely many previous trajectories
• can also use other matrix data structures: (mosaic) Hankel, Page, . . .
• novelty (?) : motion primitives, (E)DMD, dictionary learning, subspacesystem id, . . . all implicitly rely on this equivalence→ c.f. “fundamental”
• standing on the shoulders of giants:classic Willems’ result was only “if” &required further assumptions: Hankel,persistency of excitation, controllability
11/28
Control from matrix time series data
We are all writing merely the dramatic corollaries . . .
implicit & stochastic
→ Markovsky & ourselves
explicit & deterministic
→ Groningen: Persis, Camlibel, . . .
→ lots of recent momentum (∼ 1 ArXiv / week) with contributions byScherer, Allgower, Matni, Pappas, Fischer, Pasqualetti, Goulart, Mesbahi, . . .
→ more classic subspace predictive control (De Moor) literature12/28
Data-driven prediction [Markovsky & Rapisarda ’08]
Problem : predict future output y ∈ Rp·Tfuture based on• input signal u ∈ Rm·Tfuture
• past data col(ud, yd) ∈ BTdata
→ to predict forward
→ to form trajectory matrix
Solution: given (u1, . . . , uTfuture )→ compute g & (y1, . . . , yTfuture ) from
HTfuture
(ud
yd
)g =
ud1,1 ud
2,1 ud3,1 · · ·
......
......
ud1,Tfuture
ud2,Tfuture
ud3,Tfuture
· · ·
yd1,1 yd
2,1 yd3,1 · · ·
......
......
yd1,Tfuture
yd2,Tfuture
yd3,Tfuture
· · ·
g =
u1
...uTfuture
y1
...yTfuture
Issue: predicted output is not unique → need to set initial conditions !13/28
Data-driven prediction & estimationRefined problem : predict future output y ∈ Rp·Tfuture based on• initial trajectory col(uini, yini) ∈ R(m+p)·Tini
• input signal u ∈ Rm·Tfuture
• past data col(ud, yd) ∈ BTdata
→ to estimate initial xini
→ to predict forward
→ to form trajectory matrix
Solution: given u & col(uini, yini)→ compute g & y from
HTini
(ud
yd
)
HTfuture
(ud
yd
)
g =
ud1,1 ud
2,1 ud3,1 · · ·
.
.
....
.
.
....
ud1,Tini
ud2,Tini
ud3,Tini
· · ·yd
1,1 yd2,1 yd
3,1 · · ·...
.
.
....
.
.
.yd
1,Tiniyd
2,Tiniyd
3,Tini· · ·
ud1,Tini+1
ud2,Tini+1
ud3,Tini+1
· · ·...
.
.
....
.
.
.ud
1,Tini+Tfutureud
2,Tini+Tfutureud
3,Tini+Tfuture· · ·
yd1,Tini+1
yd2,Tini+1
yd3,Tini+1
· · ·...
.
.
....
.
.
.yd
1,Tini+Tfutureyd
2,Tini+Tfutureyd
3,Tini+Tfuture· · ·
g =
uiniyiniuy
⇒ observability condition: if Tini ≥ lag of system, then y is unique 14/28
Output Model Predictive ControlThe canonical receding-horizon MPC optimization problem :
minimizeu, x, y
Tfuture−1∑
k=0
‖yk − rt+k‖2Q + ‖uk‖2R
subject to xk+1 = Axk +Buk, ∀k ∈ {0, . . . , Tfuture − 1},yk = Cxk +Duk, ∀k ∈ {0, . . . , Tfuture − 1},xk+1 = Axk +Buk, ∀k ∈ {−Tini − 1, . . . ,−1},yk = Cxk +Duk, ∀k ∈ {−Tini − 1, . . . ,−1},uk ∈ U , ∀k ∈ {0, . . . , Tfuture − 1},yk ∈ Y, ∀k ∈ {0, . . . , Tfuture − 1}
quadratic cost withR � 0, Q � 0 & ref. r
model for predictionover k ∈ [0, Tfuture − 1]
model for estimation(many variations)
hard operational orsafety constraints
For a deterministic LTI plant and an exact model of the plant,MPC is the gold standard of control : safe, optimal, tracking, . . .
15/28
Data-Enabled Predictive ControlDeePC uses Hankel matrix for receding-horizon prediction / estimation:
minimizeg, u, y
Tfuture−1∑
k=0
‖yk − rt+k‖2Q + ‖uk‖2R
subject to H(ud
yd
)g =
uiniyiniuy
,
uk ∈ U , ∀k ∈ {0, . . . , Tfuture − 1},yk ∈ Y, ∀k ∈ {0, . . . , Tfuture − 1}
quadratic cost withR � 0, Q � 0 & ref. r
non-parametricmodel for predictionand estimation
hard operational orsafety constraints
• trajectory matrix H(ud
yd
)=
HTini
(udyd
)
HTfuture
(udyd
)
from past data
• past Tini ≥ lag samples (uini, yini) for xini estimation
collected offline(could be adapted online)
updated online16/28
Consistency for LTI SystemsTheorem: Consider DeePC & MPC optimization problems. If therank condition holds (= rich data), then the feasible sets coincide.
Corollary: closed-loop behaviors under DeePC and MPC coincide.
Aerial robotics case study :Thus, most of MPC carries overto DeePC . . . in the nominal casec.f. stability certificate [Berberich et al. ’19]
Beyond LTI: what about noise,corrupted data, & nonlinearities ?
. . . playing certainty-equivalencefails → need robustified approach
17/28
Noisy real-time measurements
minimizeg, u, y
Tfuture−1∑
k=0
‖yk − rt+k‖2Q + ‖uk‖2R + λy‖σini‖p
subject to H(ud
yd
)g =
uiniyiniuy
+
0σini00
,
uk ∈ U , ∀k ∈ {0, . . . , Tfuture − 1},yk ∈ Y, ∀k ∈ {0, . . . , Tfuture − 1}
Solution : add `p-slackσini to ensure feasibility→ receding-horizonleast-square filter→ for λy � 1: constraintis slack only if infeasible
c.f. sensitivity analysisover randomized sims
100
102
104
106
106
108
1010
Co
st
Cost
100
102
104
106
0
5
10
15
20
Du
ratio
n v
iola
tio
ns (
s)
Constraint Violations
18/28
Trajectory matrix corrupted by noise
minimizeg, u, y
Tfuture−1∑
k=0
‖yk − rt+k‖2Q + ‖uk‖2R + λg‖g‖1
subject to H(ud
yd
)g =
uiniyiniuy
,
uk ∈ U , ∀k ∈ {0, . . . , Tfuture − 1},yk ∈ Y, ∀k ∈ {0, . . . , Tfuture − 1}
Solution : add a`1-penalty on g
intuition: `1 sparsely selects{trajectory matrix columns}= {motion primitives}∼ low-order basis
c.f. sensitivity analysisover randomized sims
0 200 400 600 8000
1
2
3
4
5
6
7
Co
st
107 Cost
0 200 400 600 8000
5
10
15
20
Du
ratio
n v
iola
tio
ns (
s)
Constraint Violations
19/28
Towards nonlinear systemsIdea : lift nonlinear system to large/∞-dimensional bi-/linear system→ Carleman, Volterra, Fliess, Koopman, Sturm-Liouville methods→ nonlinear dynamics can be approximated by LTI on finite horizon
regularization singles out relevant features / basis functions in data
case study :DeePC+ σini slack+ ‖g‖1 regularizer+ more columns
in H(ud
yd
)
-1.5
1
-1
0.5-0.2
-0.5
00
0
0.2-0.5 0.4
0.5
0.6-1
1
1.5
2
flukeorsolid ?
20/28
Experimental snippet
21/28
Consistent observations acrosscase studies — more than a fluke
22/28
let’s try to put some theorybehind all of this . . .
Distributional robust formulation [Coulson et al. ’19]
• problem abstraction : minx∈X c(ξ, x
)= minx∈X EP
[c (ξ, x)
]
where ξ denotes measured data with empirical distribution P = δξ
⇒ poor out-of-sample performance of above sample-average solution x?
for real problem: EP
[c (ξ, x?)
]where P is the unknown distribution of ξ
• distributionally robust formulation −→ “minx∈X max E[c (ξ, x)
]”
where max accounts for all stochastic processes (linear or nonlinear)that could have generated the data . . . more precisely
infx∈X supQ∈Bε(P ) EQ
[c (ξ, x)
]
where Bε(P ) is an ε-Wasserstein ballcentered at empirical sample distribution P :
Bε(P ) =
{P : inf
Π
∫ ∥∥ξ − ξ∥∥pdΠ ≤ ε
}
ξ
ξ
P
P
Π 23/28
note: Wasserstein ball does notonly include LTI systems withadditive Gaussian noise but
“everything” (integrable)
• distributionally robust formulation:
infx∈X supQ∈Bε(P ) EQ
[c (ξ, x)
]
where Bε(P ) is an ε-Wasserstein ballcentered at empirical sample distribution P :
Bε(P ) =
{P : inf
Π
∫ ∥∥ξ − ξ∥∥pdΠ ≤ ε
}
ξ
ξ
P
P
Π
Theorem : Under minor technical conditions:inf
x∈X supQ∈Bε(P ) EQ[c (ξ, x)
]≡ minx∈X c
(ξ, x)
+ εLip(c) · ‖x‖?p
Cor : `∞-robustness in trajectory space⇐⇒ `1-regularization of DeePC
10-5
10-4
10-3
10-2
10-1
100
0
0.5
1
1.5
2
2.5
3
3.5
Cost
105
cost
ǫ
Proof builds on methods by Shafieezadeh, Esfahani, & Kuhn : problem tractableafter marginalization, for discrete worst case, & with many convex conjugates. 24/28
Further ingredients [Coulson et al. ’19], [Alpago et al. ’20]
• multiple i.i.d. experiments→ sampleaverage data matrix 1
N
∑Ni=1 Hi(y
d)
• measure concentration: Wassersteinball Bε(P ) includes true distribution P
with high confidence if ε ∼ 1/N1/ dim(ξ)
• old online measurements→ Kalmanfiltering with explicit g? as hidden state
N = 1N = 10
• distributionally robust probabilistic constraintssupQ∈Bε(P ) CVaRQ
1−α ⇔ averaging + regularization + tightening
CVaRP1−α(X)
P(X) ≤ 1 − αVarP1−α(X)
25/28
All together in action for nonlinear& stochastic quadcoptor setup
control objective+ regularization+ matrix predictor+ averaging+ CVaR constraints+ σini estimation slack
→ DeePC works muchbetter than it should ! 0 2 4 6 8 10
-1
-0.5
0
0.5
1
1.5
2
main catch : optimization problems become large (no-free-lunch)→ models are compressed, de-noised, & tidied-up representations
26/28
Comparison: DeePC vs. ID + MPCconsistent across all nonlinearcase studies : DeePC always wins
reason (?) : DeePC is robust, whereascertainty-equivalence control is basedon identified model with a bias error
Closed‐loop cost
Number of simulations
DeePCPEM‐MPC
realized closed-loop cost =∑k∥∥yk − rk
∥∥2Q +
∥∥uk∥∥2Rstochastic LTI comparison (no bias)
show certainty-equivalence vs. robustcontrol trade-offs (mean vs. median)
link : DeePC includes implicit sys IDthough 1© biased by control objective,2© data not projected on LTI class, &3© robustified through regularizations
N4SID
+ MPC
DeePC
Open-loop tracking error (% increase wrt optimal)
→ more to be understood . . . ArXiv paper coming 27/28
Summary & conclusionsmain take-aways• matrix time series serves as predictive model• data-enabled predictive control (DeePC)
X consistent for deterministic LTI systemsX distributional robustness via regularizations
future work→ tighter certificates for nonlinear systems→ explicit policies & direct adaptive control→ online optimization & real-time iteration
-1.5
1
-1
0.5-0.2
-0.5
00
0
0.2-0.5 0.4
0.5
0.6-1
1
1.5
2
Why have thesepowerful ideasnot been mixedlong before ?
Willems ’07: “[MPC] has perhaps too little systemtheory and too much brute force computation in it.”
The other side often proclaims “behavioral systemstheory is beautiful but did not prove utterly useful.”
28/28
appendix:
end-to-end automationcase study in power systems
Power system case study
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uncontrolled flow (p.u.)
• complex 4-area power system: large (n = 208), few measurements (8),nonlinear, noisy, stiff, input constraints, & decentralized control
• control objective: damping of inter-area oscillations via HVDC link• real-time MPC & DeePC prohibitive→ choose T , Tini, & Tfuture wisely
Centralized control
0 5 10 15 20 25 30
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5
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Fig. 5. Time-domain responses of the four-area system with the practicalsetting. The DeePC (or PEM-MPC) is activated at t = 10s. —– withoutwide-area control; —– with PEM-MPC (s = 60); —– with DeePC (s = 60).
Closed‐loop cost
Num
ber o
f sim
ulations DeePC
PEM‐MPC
Fig. 6. Cost comparison of DeePC and PEM-MPC under the practical setting.
Fig.7 plots the closed-loop cost (from 10s to 30) of thesystem with different DeePC parameters, which shows that a)the closed-loop cost dramatically drops with the increase of theprediction horizon N and then remains within an acceptablerange (here we set k = N
2 ); b) the closed-loop cost dropswhen T is increased from 800 to 1000 and then remains nearlythe same if further increasing T ; c) the closed-loop cost dropswith the increase of Tini from 5 to 40 and remains basically thesame with a larger Tini; and d) the system may have undesiredclosed-loop cost with a relatively small (or with a relativelylarge) �g but presents anticipated performance in between,which coincides with the fact the regularization on g providesrobustness against noisy measurements. Note that setting a toolarge �g (e.g., �g > 104) makes (5) focuses on minimizingkgk2
2 and results in inferior input/output performance. Fig.7also indicates the robustness of the DeePC with regards to thechoices of parameters, that is, the system presents anticipatedperformance with proper regularization on g (�g generally hasa wide admissible range) and sufficiently large N , Tini and T .
IV. MIN-MAX DEEPC
The DeePC algorithm presented above acts as a centralizedwide-area control, which is not resilient to communication fail-ures and less reliable than decentralized approaches especiallywhen more VSC-HVDC stations are considered. To this end,
Closed
‐loop
cost
Closed
‐loop
cost
Closed
‐loop
cost
Closed
‐loop
cost
Fig. 7. Closed-loop cost of the system with different DeePC parameters.
we present a Min-Max DeePC algorithm which further enablesdecentralized wide-area control.
A. Basic FormulationWe extend the unknown LTI system in (1) by adding a
measurable disturbance vector wt 2 Rq to (1) as⇢
xt+1 = Axt + But + Ewt
yt = Cxt + Dut + Fwt, (9)
where E 2 Rn⇥q and F 2 Rp⇥q .To be specific, the unknown system is subjected to some ex-
ternal disturbances (wt) whose past trajectory can be measuredbut the future trajectory is unknown. Let wd be a disturbancetrajectory of length T (i.e., wd 2 RqT ) measured from theunknown system such that col(ud, wd) is persistently excitingof order Tini + N + n. Note that here wt is regarded as anuncontrollable input vector of the unknown system. Similarto ud and yd, we use wd to construct the Hankel matrixHTini+N (wd), which is further partitioned into two parts as
WP
WF
�:= HTini+N (wd) , (10)
where WP 2 RqTini⇥(T�Tini�N+1) and WF 2RqN⇥(T�Tini�N+1).
Then, similar to (4), col(uini, wini, yini, u, w, y) is a trajec-tory of the unknown system (9) if and only if there existsg 2 RT�Tini�N+1 such that
26666664
UP
WP
YP
UF
WF
YF
37777775
g =
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uini
wini
yini
uwy
37777775
, (11)
where wini 2 RqTini is the most recent measured disturbancetrajectory and w = col(w0, w1, ..., wN�1) 2 RqN is the futuredisturbance trajectory, which is unknown but assumed to bebounded as wt 2 [w, w].
= Prediction ErrorMethod (PEM)System ID + MPC
t < 10 s : open loopdata collection withwhite noise excitat.
t > 10 s : control
Performance: DeePC wins (clearly!)
Closed‐loop cost
Number of simulations
DeePCPEM‐MPC
realized closed-loop cost =∑
k ‖yk − rk‖2Q + ‖uk‖2R
DeePC hyper-parameter tuningClosed‐loop cost
Closed‐loop cost
Closed‐loop cost
Closed‐loop cost
Tfuture
regularizer λg• for distributional robustness≈ radius of Wasserstein ball• wide range of sweet spots
→ choose λg = 20
estimation horizon Tini
• for model complexity ≈ lag• Tini ≥ 50 is sufficient & low
computational complexity
→ choose Tini = 60
Closed‐loop cost
Closed‐loop cost
Closed‐loop cost
Closed‐loop cost
Tfuture
prediction horizon Tfuture
• nominal MPC is stable ifhorizon Tfuture long enough
→ choose Tfuture = 120 andapply first 60 input steps
data length T
• long enough for low-rankcondition but card(g) grows
→ choose T = 1500(data matrix ≈ square)
Computational cost
time (s)
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• T = 1500
• λg = 20
• Tini = 60
• Tfuture = 120 & applyfirst 60 input steps• sampling time = 0.02 s• solver (OSQP) time = 1 s
(on Intel Core i5 7200U)⇒ implementable
Comparison: Hankel & Page matrix
Control Horizon k Control Horizon k
Averaged Closed‐loop Cost
S0=1
� Hankel matrix
� Hankel matrix withSVD (σthreshhold = 1)
� Page matrix
� Page matrix withSVD (σthreshhold = 1)
• comparison baseline: Hankel and Page matrices of same size• perfomance : Page consistency beats Hankel matrix predictors• offline denoising via SVD threshholding works wonderfully for
Page though obviously not for Hankel (entries are constrained)• effects very pronounced for longer horizon (= open-loop time)• price-to-be-paid : Page matrix predictor requires more data
Decentralized implementation
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uncontrolled flow (p.u.)
• plug’n’play MPC: treat interconnection P3 as disturbance variable wwith past disturbance wini measurable & future wfuture ∈ W uncertain• for each controller augment trajectory matrix with disturbance data w• decentralized robust min-max DeePC: ming,u,y maxw∈W
Decentralized control performance
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• colors correspondto different hyper-parameter settings(not discernible)
• ambiguity setWis∞-ball (box)
• for computationalefficiencyW isdownsampled(piece-wise linear)
• solver time ≈ 2.6 s
⇒ implementable
