[PPT] - Approximate MPC based on machine learning and probabilistic PowerPoint Presentation

SLIDE 1

Approximate MPC based on machine learning and probabilistic verification

Sergio Lucia Technische Universität Berlin Einstein Center Digital Future www.iot.tu-berlin.de

SLIDE 2

Motivation

Solving NMPC problems in real time is still challenging:

For very fast systems
On low-cost embedded hardware

Even more challenging in the case of robust st NMPC

Go

Goal: : Development of an approach that simultaneously

Obtains approximate optimal robust solutions
Has small memory footprint
Can be rapidly evaluated on an embedded device

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SLIDE 3

Explicit MPC in the linear case

The MPC control law for LTI systems is a piecewise se-af affine ne funct unction

n
Depends only on the current state (and possibly parameters)
Can be offline precomputed and stored

Each region is described by a polyhedron:

[A. Bemporad, M Morari, V. Dua, E.N. Pistikopoulos, 2002]

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SLIDE 4

Reducing complexity of explicit MPC

Optimal representations:
Merging of regions with same feedback law
Lattice representation
Suboptimal approximations:
Trade-off between complexity reduction and performance
Simplicial partitions
Neural networks

[T. A. Johannsen, A. Bemporad, F. Borrelli, C. Jones, M. Morari, M. Kvanisca and others]

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SLIDE 5

Using neural networks to approximate MPC

It is an old idea: already done in 1995 for nonlinear MPC

What is new?

Common practice until recent successes in deep learning was:

Because of unive

versa sal approxi ximation theorem: use only 1 layer What are the possible adva vantages s of deep learning for approximating complex MPC laws?

[T. Parisini and R. Zoppoli, 1995, Akesson and Toivonen, 2006]

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SLIDE 6

Deep neural networks (DNN)

Neural network with L hidden laye yers and M neurons per layer

Affine transformation
Activation function
tanh:
ReLU:

gl(fl) = tanh(fl) = efl − e−fl efl − e−fl

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SLIDE 7

Why deep (and not shallow)?

Number of linear regions represented by a ReLU network of depth L and width M

Exp

xponential growth of regions w.r.t. depth

Greater expressiveness with same

amount of weights

[Montufar et al., 2014]

7

SLIDE 8

Proposed approach

2: Offline training of the deep neural network 1: Generate training samples by solving many MPC problems

math

˙ x = f(x, u)

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3: High performance implementation

n low-cost embedded hardware

Place here your preferred (overcomplicated) robust st NMPC Method

(x0, u∗

0)

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8

SLIDE 9

Increasingly popular

In many cases including strategies to have some guarantees:

Chen et al., ACC 2018 (Projection at the output to achieve guarantees)
Hertneck et al., IEEE Control System Letters 2018 (Hoeffdings inequality)
Zhang, Bujarbaruah and Borrelli, ACC 2019 (Statistical validation)
Drgona et al., Applied Energy 2018 (application on building control)
Karg and Lucia, ECC 2018, NMPC 2018, ECC 2019 (applications, validation)

9

SLIDE 10

It works well in practice

B

[Lucia, Andersson, Brandt, Diehl and Engell. JPC 2014]

10

SLIDE 11

An industrial polymerization reactor

̇ 𝑛# = ̇ 𝑛#,& ̇ 𝑛' = ̇ 𝑛',& − 𝑙*+𝑛',* − 𝑞+𝑙*-𝑛'#.𝑛' 𝑛/01 ̇ 𝑛2 = 𝑙*+𝑛',* + 𝑞+𝑙*-𝑛'#.𝑛' 𝑛/01 ̇ 𝑈* = 1 𝑑7,*𝑛/01 ̇ 𝑛&𝑑7,& 𝑈& − 𝑈* + Δ𝐼*𝑙*+𝑛',* − 𝑙:𝐵 𝑈* − 𝑈

< −

̇ 𝑛'#.𝑑7,* 𝑈* − 𝑈=: ̇ 𝑈

< = 1/(𝑑7,<𝑛<) 𝑙:𝐵 𝑈* − 𝑈 < − 𝑙:𝐵 𝑈 < − 𝑈A

̇ 𝑈A = 1 𝑑7,#𝑛A,:# ̇ 𝑛A,:#𝑑7# 𝑈A

BC − 𝑈A + 𝑙:𝐵 𝑈 < − 𝑈A

̇ 𝑈=: = 1 𝑑7,*𝑛'#. ̇ 𝑛'#.𝑑7,# 𝑈* − 𝑈=: − 𝛽 𝑈=: − 𝑈

'#. + 𝑞+𝑙*-𝑛'𝑛'#.ΔH*

𝑛/01 ̇ 𝑈

'#. =

1 𝑑7,#𝑛'#.,:# ̇ 𝑛'#.,:#𝑑7,F 𝑈

'#. BC

− 𝑈

'#. − 𝛽 𝑈 '#. − 𝑈=:

𝑙*+ = 𝑙G𝑓

I =J *.K 𝑙L+ 1 − 𝑉 + 𝑙L-𝑉

𝑙*- = 𝑙G𝑓

I =J *.NO (𝑙L+ 1 − 𝑉 + 𝑙L-𝑉)

8 differential states 3 control inputs 2 uncertain parameters

11

SLIDE 12

Simulation results for multi-stage NMPC

Simple sc scenario tree

Extreme values of the uncertainty
Branch the tree only one stage
Economic cost function

1 2

x

2 2

x

3 2

x

4 2

x

5 2

x

6 2

x

7 2

x

8 2

x

9 2

x

1 3

x

2 3

x

3 3

x

4 3

x

5 3

x

6 3

x

7 3

x

8 3

x

9 3

x

1 4

x

2 4

x

3 4

x

4 4

x

5 4

x

6 4

x

7 4

x

8 4

x

9 4

x

1 2

u

1 2

u

1 2

u

4 2

u

5 2

u

6 2

u

7 2

u

8 2

u

9 2

u

1 3

u

1 3

u

1 3

u

4 3

u

5 3

u

6 3

u

7 3

u

8 3

u

9 3

u

… … … … … … … … …

1 1

u

1 1

d

2 1

d

2 1

u

3 1

d

3 1

u

4 1

u

4 1

d

5 1

d

5 1

u

6 1

d

6 1

u

7 1

u

7 1

d

8 1

d

8 1

u

9 1 d 9 1 u 1 1

x

2 1

x

3 1

x

4 1

x

5 1

x

6 1

x

7 1

x

8 1

x

9 1

x

1 29

x

2 29

x

3 29

x

4 29

x

5 29

x

6 29

x

7 29

x

8 29

x

9 29

x

1 30

x

2 30

x

3 30

x

4 30

x

5 30

x

6 30

x

7 30

x

8 30

x

9 30

x

1 29

u

1 29

u

1 29

u

4 29

u

5 29

u

6 29

u

7 29

u

8 29

u

9 29

u

1 2

d

2 2

d

3 2

d

4 2

d

5 2

d

6 2

d

7 2

d

8 2

d

9 2 d 1 3

d

2 3

d

3 3

d

4 3

d

5 3

d

6 3

d

7 3

d

8 3

d

9 3 d 1 29

d

2 29

d

3 29

d

4 29

d

5 29

d

6 29

d

7 29

d

8 29

d

9 29 d 1

u

1

d

2

d

2

u

3

d

3

u

4

u

4

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Simulations for different values of 𝑙 and 𝛦𝐼 (±30%)

Multi-stage NMPC

12

SLIDE 13

Proposed approach

2: Offline training of the deep neural network 1: Generate training samples by solving many MPC problems

math

˙ x = f(x, u)

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3: High performance implementation

n low-cost embedded hardware

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(x0, u∗

0)

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13

SLIDE 14

Performance of deep-learning based ms-NMPC

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 85 90 95

TR [°C]

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 80 100 120

Tadiab [°C]

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 4

mF [kg/h]

104 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Time [hours]

300 350 400

TM

IN [°C]

multi-stage deep network shallow network

0.5 1 1.5 2 2.5 85 90 95

TR [°C]

0.5 1 1.5 2 2.5 80 100 120

Tadiab [°C]

0.5 1 1.5 2 2.5 2 4

mF [kg/h]

104 0.5 1 1.5 2 2.5

Time [hours]

300 350 400

TM

IN [°C]

Exact vs. deep vs. shallow multi-stage NMPC Deep-learning based multi-stage NMPC

14

SLIDE 15

Performance of deep-learning based ms-NMPC

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 85 90 95

TR [°C]

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 80 100 120

Tadiab [°C]

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 4

mF [kg/h]

104 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Time [hours]

300 350 400

TM

IN [°C]

multi-stage deep network shallow network

0.5 1 1.5 2 2.5 85 90 95

TR [°C]

0.5 1 1.5 2 2.5 80 100 120

Tadiab [°C]

0.5 1 1.5 2 2.5 2 4

mF [kg/h]

104 0.5 1 1.5 2 2.5

Time [hours]

300 350 400

TM

IN [°C]

Deep-learning based multi-stage NMPC

Average performance

ver random 100 batches

Exact vs. deep vs. shallow multi-stage NMPC

15

SLIDE 16

Two main advantages

En Enabl ble l low-cost st em

emb. i

. impl plementa tati tion

32 bit ARM Cortex M0+
48 MHz with 32 kB RAM
Approx. robust NMPC:
Memory footprint: 27 kB
Evaluation time on a uC: 37 ms
Trivial code-generation (uC, FPGA)

16

SLIDE 17

Two main advantages

En Enabl ble l large ge(r)-sc scale syst systems Problem with 5 uncertainties

243 scenarios
~115,000 variables and constraints

0.5 1 1.5 2 2.5 85 90 95

TR [°C]

0.5 1 1.5 2 2.5 2 4

mF [kg/h]

104 0.5 1 1.5 2 2.5

Time [hours]

300 350 400

TM

IN [°C]

0.5 1 1.5 2 2.5 80 100 120

Tadiab [°C]

En Enabl ble l low-cost st em

emb. i

. impl plementa tati tion

32 bit ARM Cortex M0+
48 MHz with 32 kB RAM
Approx. robust NMPC:
Memory footprint: 27 kB
Evaluation time on a uC: 37 ms
Trivial code-generation (uC, FPGA)

17

SLIDE 18

Mixed-integer case: Energy management system

You can learn a gl

globa bal opt ptimal solution

18

SLIDE 19

Mixed-integer case

states inputs

[Karg and Lucia, ECC 2018]

19

SLIDE 20

Robust NMPC in 1 microsecond

20

Inductor Vitroceramic glass Control Power electronics Pan

In Indu ducti tion h heati ting g is currently used in many industrial and domestic applications Control switching frequency and duty cycle. Satisfy constraints under uncertainty

SLIDE 21

Hardware-in-the-loop implementation

Advanced approximate optimization-based control in 1 μs (on an FPGA). Easy to optimize FPGA implementation

21

SLIDE 22

Moving horizon estimation for sensor fusion

Fusing visual information and inertial sensors

Common problem in autonomous driving, robotics
Usually many assumptions to simplify online optimization (or EKF)

22

EKF MHE 600 800 1000 1200 1400 1600 1800 time [s] −100 100 position [m] MHE. x y z EKF x y z EKF x y z

Fiedler et al., ECC 2020

SLIDE 23

Wait a minute... guarantees?

Compute the maximum approximation error Design a controller that is robust against and iterate

23

d = max

x0 |πNN(x0) − πMPC(x0)|

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d

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SLIDE 24

Wait a minute... guarantees?

Compute the maximum approximation error Design a controller that is robust against and iterate

Computing the maximum is often not possible
Probabilist

stic Validation

24

d = max

x0 |πNN(x0) − πMPC(x0)|

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d

<latexit sha1_base64="atqb6Tk79/5uoOmMsunvLDvg1rA=">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</latexit>

SLIDE 25

Wait a minute... guarantees?

Compute the maximum approximation error Design a controller that is robust against and iterate

Computing the maximum is often not possible
Probabilist

stic Validation

Hertneck et al., IEEE CSL 2018:
Based on Hoeffdings inequality and indicator (binary functions)
Karg and Lucia, arXiv:1806.10644, (2018), ECC 2019
Based on Hoeffdings inequality and temporal logic with finite-time simulations
Zhang et al., ACC 2019:
Based on prob. validation results (Tempo, Bai, Dabbene, 1997) to achieve primal

and dual guarantees

25

d = max

x0 |πNN(x0) − πMPC(x0)|

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d

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SLIDE 26

Probabilistic validation with performance indicators

A general (not necessarily binary) finite-time performance indicator Given a controller , a final simulation step and i.i.d samples

26

w(j) = {x(j)(0), ˆ x(j)(0), d(j)(0), . . . , d(j)(Nsim)}, j = 1, . . . , N,

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φ(w; Nsim, κ) = φ(x(0), ˆ x(0), κ(ˆ x(0)), d(0), x(1), κ(ˆ x(1)), d(1), . . . , x(Nsim)).

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κ

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Nsim

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N

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SLIDE 27

Probabilistic validation with performance indicators

A general (not necessarily binary) finite-time performance indicator Given a controller , a final simulation step and i.i.d samples With probability no smaller than

is the maximum value of simulated among all N, after removing the largest elements

Provided that:

27

N ≥ 1 ✏ r − 1 + ln M + r 2(r − 1) ln M

!

.

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w(j) = {x(j)(0), ˆ x(j)(0), d(j)(0), . . . , d(j)(Nsim)}, j = 1, . . . , N,

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δ

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φ(w; Nsim, κ) = φ(x(0), ˆ x(0), κ(ˆ x(0)), d(0), x(1), κ(ˆ x(1)), d(1), . . . , x(Nsim)).

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κ

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Nsim

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N

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Prob{i(w) > φ

N(r)} ≤ ✏, i = 1, . . . , M,

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ψφ

N(r)

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φi(w)

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r

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SLIDE 28

Differences with previous works

Validation on general closed-loop performance guarantees
Not only binary functions
Not only error in the controller. Validation includes e.g. estimation errors
Discard the largest values to facilitate successful validations
Simultaneous design of several controllers (finite families)

More details in: Karg, Alamo and Lucia, Probabilistic performance validation of deep learning-based robust NMPC controllers. arXiv:1910.13906 (2019)

28

r

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SLIDE 29

Some further results

Probabilistically safe, embedded robust output-feedback NMPC Objective is to maximize thrust Two states can be measured, EKF to estimate Uncertain aerodinamic coefficients and wind parameters

29

Erhard and Strauch, 2012

SLIDE 30

Results

Embedded real-time implementation on an ARM-Cortex M3

96 kB memory footprint, 32 ms running time for DNN and 28 ms for EKF

30

SLIDE 31

Summary

1. Efficient approximation of the MPC control law using deep learning
Enables simple embedded implementation with very low memory footprint
Enables real-time robust NMPC of large complex systems
2. Statistical verification can be used to achieve guarantees
3. Recently good results for many different applications

31

SLIDE 32

Some material for discussion

What is better:
First approximate then solve (usual path)
First solve (as complex as you can) then approximate
What is more rigorous:
A priori guarantees (assumes knowledge of reality, including unc. description)
Probabilistic validation (assumes a reality simulator exists)
Many approaches use safety sets / backup controllers:
Nonlinear optimization running online -> one probably needs safety checks anyway…
Hierarchy: learn a new controller when something changes
Are finite-time guarantees acceptable? (even t is large?)

32

SLIDE 33

Open Invited Track at IFAC WC 2020 in Berlin

Together with Ali Mesbah (UC Berkeley)
Open Invited Track on „Machine Learning and MPC“
Use submission code a1d

a1d55 55

Deadline just extended to November 18th

33

SLIDE 34

Graceful performance degradation

0.5 1 1.5 2 2.5 3 85 90 95

TR [°C]

0.5 1 1.5 2 2.5 3 50 100 150

Tadiab [°C]

0.5 1 1.5 2 2.5 3 2 4

mF [kg/h]

104 0.5 1 1.5 2 2.5 3

Time [hours]

300 350 400

TM

IN [°C]

0.5 1 1.5 2 2.5 3 80 90 100

TR [°C]

0.5 1 1.5 2 2.5 3 50 100 150

Tadiab [°C]

0.5 1 1.5 2 2.5 3 2 4

mF [kg/h]

104 0.5 1 1.5 2 2.5 3

Time [hours]

300 350 400

TM

IN [°C]

Larger set of random initial conditions and uncertain parameters (±40%)

Exact multi-stage NMPC Deep-learning based multi-stage NMPC

34

SLIDE 35

Training

Samples generated solving multi-stage NMPC (CasADi + IPOPT) 100 batches of data (with random initial cond. and uncertain param.)

50 s sampling time
Total of 21050 samples
Training with Keras / Tensorflow

Output neural network Output multi-stage NMPC

35

SLIDE 36

Summary

Scheme to approximate complex

x model predictive ve controllers

Efficient approximation of the MPC control law using de

deep p learning

Two main advantages
Enable embedded implementation with very low memory footprint
Enable real-time robust NMPC of large complex systems
Some kind of sa

safety y net is necessary to have guarantees

(don‘t we always need this in reality, at least for the complex nonlinear case?)
Other problems: adaptation and RL, optimal training, optimal structure

36