Learning based Robust Control: Guaranteeing Stability while - - PowerPoint PPT Presentation

Learning‐based Robust Control: Guaranteeing Stability while Improving Performance

Felix Berkenkamp and Angela P. Schoellig

IROS 2014 – Machine Learning in Planning and Control of Robot Motion Workshop (14. Sep.)

Why is model‐based control not always sufficient?

• Model inaccuracies limit achievable performance!
• Surface Material
• Topography
• Unknown dynamics

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Robust Robust control control

• Specify prior uncertainty

in model

• Guarantee stability and

performance for all possible models

Onl Online ne le learning

• Learn from online data
• Improve the model

True model Nominal model Set of possible models True model Nominal model Online learning

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How these problems have been tackled so far

Felix Berkenkamp

The missing link

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Robust Control Online learning Learning- based Robust Control

Models uncertainty

  

Guarantees stability

  

Improves

• nline

  

True model Nominal model Set of possible models Online learning True model Nominal model Set of possible models

Felix Berkenkamp

• S. Schaal and C. G. Atkeson, “Learning control in robotics,”

IEEE Robotics & Automation Magazine, vol. 17, no. 2, pp. 20–29, 2010.

Model for our approach

• Stabilization of an operating point
• Robust Control: Guaranteed stability / performance
• Gaussian Process: Online learning

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Framework: linear Robust Control

• Extended linear plant
• Output
• Uncertainty
• Disturbance
• Error signal

Goal: Find K such that

• Stab

ability ility: stable for all allowed

• Perform

Performance: minimized

Convex optimization problem

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Gaussian Process (GP) learning

• Objective: learn the model error with uncertainties from input/output data
• Assumption: Correlation of data given by a kernel function

Similar inputs will lead to similar outputs

• Hyperparameters (noise and scaling): learned from data

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Gaussian Process learning

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Gaussian Process in our context

• Model:
• GP inputs:
• GP outputs:

True model Nominal model Set of possible models

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Example: simple linear/affine system

• True model:
• Without prior knowledge, use GP to learn system dynamics

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Parameters estimated using GP

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Bringing GPs and Robust Control together

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This generalizes to non-linear, non-scalar, MIMO systems using the same process for element- wise uncertainty

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The complete process

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True model Nominal model Set of possible models True model Nominal model Set of possible models

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0.2 time [s] x position [m] step response 2 4 6 8 10 12 14

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0.2 time [s] y position [m] 800 1000 desired 800 1000 desired 2 4 6 8 10 12 14

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0.2 time [s] x position [m] step response 2 4 6 8 10 12 14

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0.2 time [s] y position [m] 800 1000 2000 desired 800 1000 2000 desired 2 4 6 8 10 12 14

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0.2 time [s] x position [m] step response 2 4 6 8 10 12 14

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0.2 time [s] y position [m] 800 1000 2000 3000 desired 800 1000 2000 3000 desired

Experimental results: https://youtu.be/YqhLnCm0KXY?list=PLC12E387419CEAFF2

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Summary

• Combined

Combined Gaussian Process lear arni ning ng with Linear Robust Robust Cont Control rol

• Enables contro

controlle ller perform performance to to im improve onlin

• nline while

providing stab stability ility guaran guarantees

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True model Nominal model Set of possible models True model Nominal model Set of possible models

Felix Berkenkamp

Thank you!

befelix@ethz.ch Felix Berkenkamp