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Rolf
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Stufen der Verkehrsoptimierung
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4 Einleitung Maximilian Merkert // Die intelligente Ampelkreuzung
System
DATA System
Model based control
Data
Model
Laboratory for Systems Theory and Automatic ControlOtto-von-Guericke University Magdeburg
wind
Fusing Predictive Control and Machine Learning Towards Safe Autonomous SystemsRolf Findeisen, J. Matscheck, M. Maiworm, J. Bethge, H.H. Nguyen, T. Zieger, H. Rewald, A. Savchenko
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Autonomous systems
Challenge: standard adaptive/disturbance rejection control/planning approaches often fail►Exploit / fuse machine learning approaches with control & estimation
• Should operate autonomously ►involves many tasks• Perception/estimation• Planning• Control• Communication• …
wind
• Autonomous systems are increasingly important• Robotics, autonomous cars, drones, …• Industrial production, …• Chemical/bio-chemical plants, …• Energy systems, …
• Should be able to adapt/react to even large changes in environment• Disturbances• Changes in operation modes• Tasks & objectives• …
H2O
H2
CO2
SynGas
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+
Machine learning for safe autonomous systems?
How can one fuse control and machine learning approaches to achieve• Safety and reliability ►satisfaction of constraints (collisions, ….)• Stability, performance, transparency, …• Exploit pre-knowledge (models, physical insight, …)
• Significant advancements in the field of machine learning/AI• Deep-networks, re-enforcement learning, Gaussian processes, …
• Mainly driven by• Availability of data for training due to digitalization• (Cloud) computing resources for learning• Big companies like Amazon, Google, …• Freely available software tools/environments (tensorflow, PyTorch, …)
• Widely used for• (Image-)classification, image processing• Learning customer demands and wishes• Autonomous driving• …
So far only limited use in control/planning• Challenging to provide guarantees/safety certificates for machine learning approaches• Real-time learning
System
DATA
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Mainly used “tool”: model predictive control
1. Obtain state
2. Predict system behavior and optimize input
3. Apply optimal input signal
Predictive control = repeated optimal control
prediction horizon
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Model predictive control?
• direct consideration of constraints ►safety • nonlinear systems with multiple inputs ►flexibility• use of preview information and model knowledge ►trustability, adaptation • possibility to “optimize” a cost ►performance • many stability & robustness results available ►stability & robustness• efficient embedded optimization strategies
3. Apply optimal input
2. Calculate optimal input
1. Obtain system state
s.t.
How can we handle highly uncertain systems / systems with limited model knowledge?
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Pure learning-based control
+ Exploits available data + No physical model needed- Lot’s of data needed- Physical insight?- Retraining required- Guarantees?
System+ Uncertainty
DATA
Pure model-based control
+ Exploits physical knowledge+ Provides physical insight+ Exploits model and data+ Guarantees possible- challenging to handle large changes
System+ Uncertainty
MPC
Data
Dynamical Model
Fusing learning and control
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Pure learning-based control
System+ Uncertainty
DATA
Learning supported model-based control
+ Exploits physical knowledge+ Provides physical insight+ Exploits model and data+ Guarantees possible+ Allows adaptation
System+ Uncertainty
MPC
Data
Dynamical Model
Fusing learning and control
SystemMPC
references, disturbances
Here: focus on learning supported control (MPC) with safety and stability “guarantees“
SystemMPC
model
SystemMPC
controller
+ Exploits available data + No physical model needed- Lots of data needed- Physical insight?- Retraining required- Guarantees?
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Existing works MPC and learning with guarantees (incomplete!)Learning references/disturbances and predictive control• GPs as reference/disturbance model in MPC [Klenske ´16,Ortmann ´17], ...• NN as reference model in MPC [Ling´18], ...Exploiting Gaussian process models in robust/stochastic MPC• Variance in cost function [Ažman ´08, Cao´17, Murray-Smith ´03,Yang ´15], ...• Constraint tightening/chance constraints [Grancharova ´08, Ostafew´16,Hewing ´19, Likar ´07, Wang ´16, Soloperto 18]Learning models in predictive control with guarantess• Learning dynamic models via Gaussian processes with Guarantees [Berkenkamp ´16, ´17], …• Deep learning for dynamic models in MPC [Terzi ´19], …• Support vector machine to learn part of the model and input-to-state stability guaranteed [Yoo ´17], ...• Invariant safe sets & layer control frame [Akametalu´14, Gilluala´12, Koller´18, Fisac´18, Wabersich´18, Aswani´ 13], ...• Lipschitz constant for constructing upper and lower bounds (LACKI) [Limon ´17]Learning MPC controllers• Explicit learning of an linear MPC controller [Chen ´18], [Karg ’18, ´19], …• Learning an Approximate MPC with Guarantees [Hertneck ´18], ...• Safe and Fast Tracking on a Robot Manipulator [Nubert ´20], ...Verification of learned controllers• Use Robust Control Theory approach [Wang ´19, Jin ´18], ...• Exploit the characteristics of specific type of nonlinear activation function [Ivanov ´18], ...and many, many more!
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Fusing Predictive Control and Machine Learning Towards Safe Autonomous Systems
MPC and learned references & disturbances• Support controller with learned reference/disturbance previews• Use reachable sets ► Safety / constraint satisfaction and repeated feasibility
Key message: fusion of model based predictive control and learning verypromising for the control of autonomous systems with guarantees
MPC subject to learned system dynamics• Learning input-output models with Gaussian processes• Guarantee safety and feasibility under learned switching dynamics► Enforcing stability despite learning and multiple modes
Learned baseline controllers• Neural network-based learning of a baseline controller► Validation of closed loop stability of learned controller
SystemMPC
SystemMPC
SystemMPC
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Challenges• Complete ablation despite uncertainties• Avoid thermal overheating of spinal cord• Avoid injury of spinal cord/sensible areas
Robot supported intervention
Remove tumor by heating via electrodes
• Precise needle positioning & stabilization• Precise control of applied forces
Motivation - robot supported spine radio frequency tumor ablation
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Cont
rol T
asksForce controlConstrained path following
control
Many sub-tasks • Tracking• Registration• Human-Robot-Interface• Minimize tooltip errors• Collision free & optimal path planning• Optimal robot configurations
Towards precise & safe robot supported intervention
Force limitation
& compensation
Access path constraints
Predictive force feedback control
Exp. Design Optimal Control
Therapyplanning
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Predictive force feedback path following control
Predictive force feedback path following control
Under suitable conditions one can guarantee feasibility & stability & safety
Basic idea• add controllable dynamics by a virtual system for reference evolution• cost penalizes path-following error
[Faulwasser & Findeisen`09, '12, ‘16; Matschek et.al. 2017]
Cost functional path following error
constraints on forces
error with force &position components
reference speed = additional degree
of freedom
• small error• real-time capable
(22nd order model,3ms sampling time)
Experimental validation
path
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Cont
rol T
asksForce controlConstrained path following
controlLearning / adaptation
Many sub-tasks • Tracking• Registration• Human-Robot-Interface• Minimize tooltip errors• Collision free & optimal path planning• Optimal robot configurations
Towards precise & safe robot supported intervention
Movementcompensation
Force limitation
& compensation
Access path constraints
Combine predictive control & learning. Safety and performance guarantees?
Exp. Design Optimal Control
Therapyplanning
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Basic idea: identify reference model for prediction
Gaussian processes
Patient movement MPC
Measure-ments Reference /
DisturbancePrediction
Robot
Tool position
Approach should• Be able to handle noisy measurements• Loss of information• Allows inclusion of prior knowledge• Self-adapting• Safety-guarantees
Past observation
Future prediction
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Gaussian processes-based learning
Advantages of Gaussian Processes• Non-parametric, data-based modelling
o No explicit physical model neededo Adapts to current situation (e.g. via online data update)
• Robust against overfitting despite noisy measurements• Provides estimate of uncertainty • Inclusion of prior knowledge via mean and covariance functions
Gaussian Process
• Generalisation of Gaussian distribution to function space• Mean function• Covariance function
GP as reference generator:Use posterior mean as reference
Past observation Future prediction
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GP learning for movement compensation
GP setup• 6 independent GPs (position, orientation of goal structure)
• GP regressor is time • GP prior mean is constant with known parameter
prior data
posterior
Prior knowledge: Selection of covariance - motion includes quasi-periodic effects
Squared exponentialcovariance function
Periodiccovariance function
Learning
MPC
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Learning and MPC based movement compensation - results
Learning supported MPC for motion control allows precise tracking
Closed loop maximum errors 0.3 mm, 0.03°
Can we provide safety and stability guarantees despite learning?• Enforce reasonable/feasible references?• Stability of the closed loop?
• Moving horizon of training data• Mean value used in controller• Variance: additional safety layer for warnings
GP error is very small
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Safety of MPC reference trackingRequirements on learned reference
Performance• Reference known over prediction horizon• Smooth, i.e. no overfitting of noisy signal
Safety• Reference known for prediction horizon• Trackable under constraints
Idea: Use Gaussian process with constrained learning algorithm
Learning of references should• Provide predictions
(extrapolation)• Take constraints (=safety)
into account • Filter noisy data
GP
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Posterior mean and variance
Gaussian processes-based learning
►Include constraints in hyperparameter optimization
Training &prediction via
hyperparameteroptimization and
conditionalprobabilities(Bayes rule)
Hyper-parameters
+
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Constrained hyperparameter optimization
With negative logarithmic marginal likelihood
Constrained Gaussian-processes learning
Reachable tube with input constraints & dynamics ►stability
State constraints►safety
Theorem (Matschek et al. ‘20): under reasonable conditions we can guarantee tracktability & safety & stability.
How do we guarantee safety in the states? ► consider reachable points are inside of safe set
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Example constraint learning and safety
• standard learned reference not trackable
• Constrained GP provides a trackable reference
• Constrained GP learning allows to model, filter and predict references • Can achieve safety & stability in the sense of perfect tracking
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MPC and learned references & disturbances• Support controller with learned reference/disturbance previews• Use reachable sets ► Safety / constraint satisfaction and repeated feasibility
Key message: fusion of model predictive control and learning verypromising for the control of safe autonomous systems with guarantees
MPC subject to learned system dynamics• Learning input-output models with Gaussian processes• Guarantee safety and feasibility under learned switching dynamics► Enforcing stability despite learning and multiple modes
Learned baseline controllers• Neural network-based learning of a baseline controller► Validation of closed loop stability of learned controller
SystemMPC
SystemMPC
SystemMPC
Fusing Predictive Control and Machine Learning Towards Safe Autonomous Systems
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Learning input-output dynamics with Gaussian-processes
SystemMPC• Gaussian process learns NARX prediction model
• Optimal control problem formulation: nominal formulation, does not exploit uncertainty of GP
Stability, inherent robustness/performance despite ”learning”?
Past observation Future prediction
Theorem (ISS of MPC with a Gaussian process model) [Maiworm et al. (2018,2020)]The closed loop system is input-to-state stable under reasonable assumptions
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Simple example
• Hard input constraints:
• Soft output constraints:
Maiworm ET AL 11
FIGURE 5 Comparison of the three MPC schemes for thecase of initial training dataDref. Thin lines represent individualsimulations, thick lines represent mean values.
FIGURE 6 Comparison of the three MPC schemes for the caseof initial training data Dcomb. Thin lines represent individualsimulations, thick lines represent mean values.
This lack of knowledge leads even to a violation of the con-straints. The rGP however performs significantly better due tothe added data points, especially at the beginning of operation.
These simulations suggest that one should in general pre-fer the Dref case over the other cases, which is convenient forthe used MPC scheme because knowledge at the reference isrequired anyway to determine the terminal cost and controller.
In the second set of simulations we investigate the influenceof di�erent thresholds for the data inclusion approach, i.e.,
di�erent values for the maximum prediction error Ñep and the
maximum prediction variance �2. To this end, Fig. 7 combines
the rGP results of the previous figures for the three trainingdata cases, together with the evolution of the prediction errore
p and the prediction variance �2. In particular the latter illus-
trates nicely the di�erence between the three cases. In the caseof D0, the variance is small at the beginning and increasesaround t = 8min when the system leaves the neighborhoodof the initial condition and moves towards the reference. Thesame holds, but the other way round, for the case withDref. Theinitial high variance is caused by its computation before thefirst data point is added to the training set. In the case of Dcomb,the variance is almost always small, except at t = 5min andaround t = 9min, probably because the combinations of therespective output values and the relatively large input valuesare not present in Dcomb (see the input evolution in Fig. 6).
FIGURE 7 rGP simulation results for the di�erent train-ing data cases together with the prediction error e
p and theprediction variance �
2.
These findings suggest that �2 is particularly suited to
include points into D during exploration and that Ñep should
be preferred to refine prediction quality during interpolation.This is verified in Fig. 8 and Fig. 9, where we focus for the
► Proofable “stability and safety“ of the closed loop►“Bounded model plant mismatch“ leads to “bounded error “ in the output
Seborg et al. (1989)
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Uncertain environment – multiple possible modes
Challenge: Often “discrete uncertainties” due to the environment• Other systems / objects are unknown (grabbing)• Decisions of other participants are unknow a priory
F
Changes in material
properties
Uncertainsystem
behavior
Handling of such uncertain modes• Safety guarantees?• Limited performance decrease?
Trafficcrossing
Grasping ofDifferent Objects
Stufen der Verkehrsoptimierung
zentral gesteuertvolle Information
dezentral gesteuertzentrale (volle)
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dezentral gesteuertlokale Information
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4 Einleitung Maximilian Merkert // Die intelligente Ampelkreuzung
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Motivation: optimization of traffic flow
• Can we avoid congestion by intelligent coordination and control?• Influence of autonomous vehicles on the traffic flow• Local decision making
Macroscopic (“global“ perspective)
Microscopic(single vehicle planning and local decisions)
?
• Behavior of single (autonomous) vehicles?•What is the optimal path (path, stop, ...)
• How should drive when?• Global or local decisions?
Cooperation: S. Sorgatz, H. Rewald (VW), S. Sager, M. Markert, H. Duc OvGU
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Safety and performance despite uncertain behaviorsIdea• Keep all modes that are possible in the prediction, guarantee for all constraints = safety• Learn and optimize performance with respect to most probable mode• Remove impossible modes
min Cost(𝛴1, 𝛴2, 𝛴3)
𝛴1(t)∊ 𝓧1
𝛴2(t)∊ 𝓧1
𝛴3(t)∊ 𝓧1
STOP
used for performance optimization
t1 t2 t3
70%
20%
10%
80%
20%
Classification and estimation phase 1
Classification and estimation phase 2
Classification and estimation phase 3
used for performance optimization
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Overall MPC formulationCore idea: consider all modes, improve performance by learning
Differnt possibilitiesrobust constraint
satisfactionfor all modes
(safety)
Learned partimprove performance
Cost functionoptimize performance
STOP
Theorem (Bethge et al. 2018, safety and feasibility)If all nominal models satisfy tightened constraints, then the real system satisfies the constraints = safety and the problem is repeatedly feasible
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Toy simulation results: quadcopter
With learning and multiple modes• Constraint satisfaction guaranteed• Learning improves performance
No learning• Standard MPC violates the constraints• Multi-Mode MPC ensures
robust constraint satisfaction
MPC with one model
Multi-mode MPC
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MPC and learned references & disturbances• Support controller with learned reference/disturbance previews• Use reachable sets ► Safety / constraint satisfaction and repeated feasibility
Key message: fusion of model predictive control and learning verypromising for the control of safe autonomous systems with guarantees
MPC subject to learned system dynamics• Learning input-output models with Gaussian processes• Guarantee safety and feasibility under learned switching dynamics► Enforcing stability despite learning and multiple modes
Learned baseline controllers• Neural network-based learning of a baseline controller► Validation of closed loop stability of learned controller
SystemMPC
SystemMPC
SystemMPC
Fusing Predictive Control and Machine Learning Towards Safe Autonomous Systems
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Towards nominal stability certification of deep learning-based control
Objectives
• Learning from an existing baseline controller (mathematical description does not need to be known to)
• Decrease computational load
• Learn and adapt unknown controller
• Improve robustness, ...
• Data collected from measurements or simulations
• Baseline controller may or may not provide guarantees
Application scenario: learn behavior of human-based driver for autonomous driving
baseline controller
learned controller Σ!
plant Σ"
data
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• Plant Σ!:
• Baseline controller
Key idea• Exploit classical Lyapunov stability conditions• Specific type of deep neural network:
NAIS-Net (Non-Autonomous Input-output Stable Network)• Special type of residual neural network • Share weights of matrices within one block
Question: Can we provide performance and safety guarantees for nominal system with learning-based controller?
baseline controller
learnedcontroller Σ!
plant Σ"
data
Towards nominal stability certification of deep learning based controller
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Basic Idea: stability of the closed-loop system with NAIS-Net based controller
NAIS-Net basedcontroller with
finite number oflayer
plant Σ"
NAIS-Net basedcontroller with
infinite number oflayer
plant Σ"
Disturbance 𝜹(𝒙)
When NAIS-Net is suitably trained, the closed-loop system is equivalent to an auxiliary system with bounded disturbances
• Auxiliary system is asymptotically stable• Disturbance bound depends on # of hidden layers• The plant can be unstable
Theorem: If
Then the closed-loop system is practically asymptotically stable
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Simple simulation example: continuously stirred tank reactor
• Baseline Controller:MPC without provable asymptotic stability properties
• NAIS-Net based controller• 30 nodes in each layer• 512 layers
MPC controller
DNN controller
Learned controller is stable, even so that the baseline NMPC controller has no stability guarantees!
Open question: How can we integrate the stability conditions in the learning?
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Many possible applications• energy systems, batteries, …• cooperative robotics, medical robotics, …• autonomous vehicles, UAVs, aircrafts
Outlook:• Enforcing safety of NN-based controllers using the last layer• Embedded learning: can we perform the learning also online?• …
Fusion of predictive control and learning for autonomous systems• Allows to handle uncertainty and provides adaptation• One can learn many things: model, reference, cost, constraints• Stability and performance guarantees possible
Key message: fusion of model predictive control and learning verypromising for the control of autonomous systems with guarantees
wind
Summary:Fusing Predictive Control and Machine Learning Towards Safe Autonomous Systems
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Thanks
Janine Matschek Johanna Bethge
Hannes Rewald
Hoang Hai Nguyen
All collaborators:• S. Sorgatz (VW), S. Sager (OvGU), Limon group (U. Sevilla), Braatz group (MIT), Mesbah group (UC Berkley), Colin group
(EPFL), Wagener/Tautz group (FZ Jülich), Diehl group (U Freiburg), Borrelli group (UC Berkley), di Cairano group(MERL), Limo group (U Sevilla), S. Waldherr (KU Leuven), J. How group (MIT), M. Zeilinger (ETH)
• Siemens CRT, Bosch, Volkswagen, IAV, Airbus, Baker Hughes, ....
Michael Maiworm Former group members• Prof. Timm Faulwasser (TU Dortmund)• Prof. Sergio Lucia (TU Berlin)• Prof. Stefan Streif (TU Chemnitz)• Prof. Masako Kishida (NII Tokio)• Prof. Steffen Borchers (HTW Berlin)
Tim Zieger
Anton Savchenko
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Selected References• J. Matschek and R. Findeisen. Learning supported Model Predictive Control for Tracking of Periodic References. In Proceedings of Machine Learning
Research, volume 120, pages 511-520, 2020.• M. Maiworm, D. Limón, and R. Findeisen. Online learning-based model predictive control with Gaussian process models and stability guarantees.
International Journal of Robust and Nonlinear Control, 2020.• M. Maiworm, D. Limón, J. M. Manzano, and R. Findeisen. Stability of Gaussian process learning based output feedback model predictive control. In 6th IFAC
Conference on Nonlinear Model Predictive Control (NMPC), pages 551-557, Madison, USA, 2018.• J. Matschek, T. Gonschorek, M. Hanses, N. Elkmann, F. Ortmeier, and R. Findeisen. Learning references with Gaussian processes in model predictive control
applied to robot assisted surgery. In Proceedings of European Control Conference (ECC), pages 362-367, 2020.• J. Matschek and R. Findeisen. Learning supported Model Predictive Control for Tracking of Periodic References. In Proceedings of Machine Learning
Research, volume 120, pages 511-520, 2020.• J. Matschek, A. Himmel, K. Sundmacher, and R. Findeisen. Constrained Gaussian process learning for model predictive control. In Proceedings of 20th IFAC
World Congress Berlin, 2020, to appear.• J. Matschek, R. Jordanowa, and R. Findeisen. Direct Robotic Force Control with Learning Supported Model Predictive Control. 2020. Conference on Control
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