Correct-by-construction techniques, such as control barrier functions (CBFs), can be used to guarantee closed-loop safety by acting as a supervisor of an existing legacy controller. However, supervisory-control intervention typically compromises the performance of the closed-loop system. On the other hand, machine learning has been used to synthesize controllers that inherit good properties from a training dataset, though safety is typically not guaranteed due to the difficulty of analyzing the associated learning structure. In this paper, supervised learning is combined with CBFs to synthesize controllers that enjoy good performance with provable safety. A training set is generated by trajectory optimization that incorporates the CBF constraint for an interesting range of initial conditions of the truck model. A control policy is obtained via supervised learning that maps a feature representing the initial conditions to a parameterized desired trajectory. The learning-based controller is used as the performance controller and a CBF-based supervisory controller guarantees safety. A case study of lane keeping (LK) for articulated trucks shows that the controller trained by supervised learning inherits the good performance of the training set and rarely requires intervention by the CBF supervisor.

References

References
1.
Ames
,
A. D.
,
Grizzle
,
J. W.
, and
Tabuada
,
P.
,
2014
, “
Control Barrier Function Based Quadratic Programs With Application to Adaptive Cruise Control
,”
53rd IEEE Conference on Decision and Control
(
CDC
),
Los Angeles, CA
,
Dec. 15–17
, pp.
6271
6278
.
2.
Xu
,
X.
,
Grizzle
,
J. W.
,
Tabuada
,
P.
, and
Ames
,
A. D.
,
2016
, “
Correctness Guarantees for the Composition of Lane Keeping and Adaptive Cruise Control
,” preprint arXiv: 1609.06807.
3.
Hsu
,
S.-C.
,
Xu
,
X.
, and
Ames
,
A. D.
,
2015
, “
Control Barrier Function Based Quadratic Programs With Application to Bipedal Robotic Walking
,”
American Control Conference
(
ACC
),
Chicago, IL
,
July 1–3
, pp.
4542
4548
.
4.
Chen
,
Y.
,
Peng
,
H.
, and
Grizzle
,
J.
,
2017
, “
Obstacle Avoidance for Low-Speed Autonomous Vehicles With Barrier Function
,”
IEEE Trans. Control Syst. Technol.
,
26
(
1
), pp.
194
206
.
5.
Ames
,
A. D.
,
Xu
,
X.
,
Grizzle
,
J. W.
, and
Tabuada
,
P.
,
2016
, “
Control Barrier Function Based Quadratic Programs With Application to Automotive Safety Systems
,” preprint arXiv: 1609.06408.
6.
Gomi
,
H.
, and
Kawato
,
M.
,
1993
, “
Neural Network Control for a Closed-Loop System Using Feedback-Error-Learning
,”
Neural Networks
,
6
(
7
), pp.
933
946
.
7.
Da
,
X.
,
Hartley
,
R.
, and
Grizzle
,
J. W.
,
2017
, “
Supervised Learning for Stabilizing Underactuated Bipedal Robot Locomotion, With Outdoor Experiments on the Wave Field
,”
IEEE International Conference on Robotics and Automation
(
ICRA
),
Singapore
,
May 29—June 3
, pp.
3476
3483
.
8.
Bojarski
,
M.
,
Del Testa
,
D.
,
Dworakowski
,
D.
,
Firner
,
B.
,
Flepp
,
B.
,
Goyal
,
P.
,
Jackel
,
L. D.
,
Monfort
,
M.
,
Muller
,
U.
,
Zhang
,
J.
,
Zhang
,
X.
,
Zhao
,
J.
, and
Zieba
,
K.
,
2016
, “
End to End Learning for Self-Driving Cars
,” preprint arXiv: 1604.07316.
9.
Sallab
,
A. E.
,
Abdou
,
M.
,
Perot
,
E.
, and
Yogamani
,
S.
,
2016
, “
End-to-End Deep Reinforcement Learning for Lane Keeping Assist
,” preprint arXiv: 1612.04340.
10.
Oh
,
S.-Y.
,
Lee
,
J.-H.
, and
Choi
,
D.-H.
,
2000
, “
A New Reinforcement Learning Vehicle Control Architecture for Vision-Based Road Following
,”
IEEE Trans. Veh. Technol.
,
49
(3), pp.
997
1005
.
11.
Bertsekas
,
D.
,
1995
,
Dynamic Programming and Optimal Control
,
1
,
Athena Scientific
,
Belmont, MA
.
12.
Gillula
,
J. H.
, and
Tomlin
,
C. J.
,
2012
, “
Guaranteed Safe Online Learning Via Reachability: Tracking a Ground Target Using a Quadrotor
,”
IEEE International Conference on Robotics and Automation
(
ICRA
),
St. Paul, MN
,
May 14–18
, pp.
2723
2730
.
13.
Akametalu
,
A. K.
,
Fisac
,
J. F.
,
Gillula
,
J. H.
,
Kaynama
,
S.
,
Zeilinger
,
M. N.
, and
Tomlin
,
C. J.
,
2014
, “
Reachability-Based Safe Learning With Gaussian Processes
,”
IEEE 53rd Annual Conference on Decision and Control
(
CDC
),
Los Angeles, CA
,
Dec. 15–17
, pp.
1424
1431
.
14.
Smit-Anseeuw
,
N.
,
Vasudevan
,
R.
, and
Remy
,
C. D.
,
2017
, “
Safe Online Learning Using Barrier Functions
,” Proceedings of Dynamic Walking, Mariehamn, Finland, June 4–9.
15.
Berkenkamp
,
F.
, and
Schoellig
,
A. P.
,
2015
, “
Safe and Robust Learning Control With Gaussian Processes
,”
European Control Conference
(
ECC
),
Linz, Austria
,
July 15–17
, pp.
2496
2501
.
16.
Isidori
,
A.
,
2013
,
Nonlinear Control Systems
,
Springer Science & Business Media
, London.
17.
Khalil
,
H. K.
,
1996
,
Nonlinear Systems
, Vol.
2
,
Prentice Hall
,
Upper Saddle River, NJ
, pp.
5
1
.
18.
Westervelt
,
E. R.
,
Grizzle
,
J. W.
,
Chevallereau
,
C.
,
Choi
,
J. H.
, and
Morris
,
B.
,
2007
,
Feedback Control of Dynamic Bipedal Robot Locomotion
,
CRC Press
,
Boca Raton, FL
.
19.
Shiriaev
,
A.
,
Perram
,
J. W.
, and
Canudas-de-Wit
,
C.
,
2005
, “
Constructive Tool for Orbital Stabilization of Underactuated Nonlinear Systems: Virtual Constraints Approach
,”
IEEE Trans. Autom. Control
,
50
(
8
), pp.
1164
1176
.
20.
Maggiore
,
M.
, and
Consolini
,
L.
,
2013
, “
Virtual Holonomic Constraints for Euler–Lagrange Systems
,”
IEEE Trans. Autom. Control
,
58
(
4
), pp.
1001
1008
.
21.
Grizzle
,
J. W.
,
Di Benedetto
,
M. D.
, and
Lamnabhi-Lagarrigue
,
F.
,
1994
, “
Necessary Conditions for Asymptotic Tracking in Nonlinear Systems
,”
IEEE Trans. Autom. Control
,
39
(
9
), pp.
1782
1794
.
22.
Miege
,
A. J.
, and
Cebon
,
D.
,
2005
, “
Optimal Roll Control of an Articulated Vehicle: Theory and Model Validation
,”
Veh. Syst. Dyn.
,
43
(
12
), pp.
867
884
.
23.
Wong
,
J. Y.
,
2008
,
Theory of Ground Vehicles
,
Wiley
,
Hoboken, NJ
.
24.
Xu
,
X.
,
Tabuada
,
P.
,
Grizzle
,
J. W.
, and
Ames
,
A. D.
,
2015
, “
Robustness of Control Barrier Functions for Safety Critical Control
,”
IFAC-PapersOnLine
,
48
(
27
), pp.
54
61
.
25.
Parrilo
,
P. A.
,
2003
, “
Semidefinite Programming Relaxations for Semialgebraic Problems
,”
Math. Program.
,
96
(
2
), pp.
293
320
.
26.
Bochnak
,
J.
,
Coste
,
M.
, and
Roy
,
M.-F.
,
2013
,
Real Algebraic Geometry
, Vol.
36
,
Springer Science & Business Media
,
New York
.
27.
Stengle
,
G.
,
1974
, “
A Nullstellensatz and a Positivstellensatz in Semialgebraic Geometry
,”
Math. Ann.
,
207
(
2
), pp.
87
97
.
28.
Papachristodoulou
,
A.
, and
Prajna
,
S.
,
2002
, “
On the Construction of Lyapunov Functions Using the Sum of Squares Decomposition
,”
41st IEEE Conference on Decision and Control
(
CDC
),
Las Vegas, NV
,
Dec. 10–13
, pp.
3482
3487
.
29.
Prajna
,
S.
, and
Jadbabaie
,
A.
,
2004
, “
Safety Verification of Hybrid Systems Using Barrier Certificates
,”
International Workshop on Hybrid Systems: Computation and Control
(
HSCC
),
Philadelphia, PA
,
Mar. 25–27
, pp.
477
492
.
30.
Prajna
,
S.
,
Papachristodoulou
,
A.
, and
Wu
,
F.
,
2004
, “
Nonlinear Control Synthesis by Sum of Squares Optimization: A Lyapunov-Based Approach
,” Fifth Asian Control Conference (
ASCC
), Melbourne, Australia, July 20–23 , pp.
157
165
.https://ieeexplore.ieee.org/document/1425952
31.
Goh
,
K. C.
,
Turan
,
L.
,
Safonov
,
M. G.
,
Papavassilopoulos
,
G. P.
, and
Ly
,
J. H.
,
1994
, “
Biaffine Matrix Inequality Properties and Computational Methods
,”
American Control Conference
(
ACC
),
Baltimore, MD
,
June 29–July 1
, pp.
850
855
.
32.
Chen
,
Y.
,
Peng
,
H.
, and
Grizzle
,
J. W.
,
2018
, “
Validating Noncooperative Control Designs Through a Lyapunov Approach
,”
IEEE Trans. Control Syst. Technol.
,
27
(2), pp.
527
539
.
33.
Betts
,
J. T.
,
2010
,
Practical Methods for Optimal Control and Estimation Using Nonlinear Programming
,
SIAM
,
Philadelphia, PA
.
34.
Hereid
,
A.
,
Cousineau
,
E. A.
,
Hubicki
,
C. M.
, and
Ames
,
A. D.
,
2016
, “
3D Dynamic Walking With Underactuated Humanoid Robots: A Direct Collocation Framework for Optimizing Hybrid Zero Dynamics
,”
IEEE International Conference on Robotics and Automation
(
ICRA
),
Stockholm, Sweden
,
May 16–21
, pp.
1447
1454
.
35.
Ames
,
A. D.
,
Xu
,
X.
,
Grizzle
,
J. W.
, and
Tabuada
,
P.
,
2017
, “
Control Barrier Function Based Quadratic Programs for Safety Critical Systems
,”
IEEE Trans. Autom. Control
,
62
(
8
), pp.
3861
3876
.
36.
Stoer
,
J.
, and
Bulirsch
,
R.
,
2013
,
Introduction to Numerical Analysis
, Vol.
12
,
Springer Science & Business Media
,
New York
.
37.
Hereid
,
A.
, and
Ames
,
A. D.
,
2017
, “
FROST*: Fast Robot Optimization and Simulation Toolkit
,” IEEE/RSJ International Conference on Intelligent Robots and Systems (
IROS
), Vancouver, BC, Canada, Sept. 24–28, pp. 719–726.
38.
Abadi
,
M.
,
Agarwal
,
A.
,
Barham
,
P.
,
Brevdo
,
E.
,
Chen
,
Z.
,
Citro
,
C.
,
Corrado
,
G. S.
,
Davis
,
A.
,
Dean
,
J.
,
Devin
,
M.
,
Ghemawat
,
S.
,
Goodfellow
,
I.
,
Harp
,
A.
,
Irving
,
G.
,
Isard
,
M.
,
Jia
,
Y.
,
Jozefowicz
,
R.
,
Kaiser
,
L.
,
Kudlur
,
M.
,
Levenberg
,
J.
,
Mane
,
D.
,
Monga
,
R.
,
Moore
,
S.
,
Murray
,
D.
,
Olah
,
C.
,
Schuster
,
M.
,
Shlens
,
J.
,
Steiner
,
B.
,
Sutskever
,
I.
,
Talwar
,
K.
,
Tucker
,
P.
,
Vanhoucke
,
V.
,
Vasudevan
,
V.
,
Viegas
,
F.
,
Vinyals
,
O.
,
Warden
,
P.
,
Wattenberg
,
M.
,
Wicke
,
M.
,
Yu
,
Y.
, and
Zheng
,
X.
,
2016
, “
Tensorflow: Large-Scale Machine Learning on Heterogeneous Distributed Systems
,” preprint arXiv: 1603.04467.
39.
Frazzoli
,
E.
,
Dahleh
,
M. A.
, and
Feron
,
E.
,
2002
, “
Real-Time Motion Planning for Agile Autonomous Vehicles
,”
J. Guid., Control, Dyn.
,
25
(
1
), pp.
116
129
.
40.
Tabuada
,
P.
,
2007
, “
Event-Triggered Real-Time Scheduling of Stabilizing Control Tasks
,”
IEEE Trans. Autom. Control
,
52
(
9
), pp.
1680
1685
.
41.
Da
,
X.
, and
Grizzle
,
J.
,
2017
, “
Combining Trajectory Optimization, Supervised Machine Learning, and Model Structure for Mitigating the Curse of Dimensionality in the Control of Bipedal Robots
,” preprint arXiv: 1711.02223.
42.
Arnol'd
,
V. I.
,
2013
,
Mathematical Methods of Classical Mechanics
, Vol.
60
,
Springer Science & Business Media
,
New York
.
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