Design of Norm-Optimal Iterative Learning Controllers: The Effect of - PowerPoint PPT Presentation

Design of Norm-Optimal Iterative Learning Controllers: The Effect of an Iteration-Domain Kalman Filter for Disturbance Estimation Nicolas Degen, Autonomous System Lab, ETH Zürich Angela P. Schoellig, University of Toronto Institute of Aerospace Studies 16 December 2014 53rd Conference on Decision and Control, Los Angeles

Quadrocopter Tracking Performance Problem : Unsatisfactory tracking performance Solution : Iterative Learning Control with Kalman Filter (K-ILC) desired trajectory updated reference ILC System trajectory Input Update Disturbance Estimator tracking error A. P. Schoellig, F. L. Mueller, and R. D’Andrea, “Optimization- F. L. Mueller, A. P. Schoellig, and R. D’Andrea, “Iterative learning of feed-forward based iterative learning for precise quadrocopter trajectory corrections for high-performance tracking,” in Proc. of the IEEE/RSJ International tracking,” Autonomous Robots, vol. 33, no. 1-2, pp. 103–127, 2012. Conference on Intelligent Robots and Systems (IROS), 2012, pp. 3276–3281. Nicolas Degen and Angela P Schoellig 2

Goal: Analytic Comparison of ILC Algorithms Compare QILC and K-ILC What are the differences? QILC K-ILC • Q uadratic cost criterion ILC • K alman-Filter - Enhanced ILC desired trajectory desired output updated control reference ILC input System trajectory Input Update System F Input Update tracking error Disturbance Estimator tracking error J. H. Lee, K. S. Lee, and W. C. Kim, “Model- A. P. Schoellig, F. L. Mueller, and R. D’Andrea, based iterative learning control with a quadratic “Optimization- based iterative learning for precise criterion for time-varying linear systems,” quadrocopter trajectory tracking,” Autonomous Automatica, vol. 36, pp. 641–657, 2000. Robots, vol. 33, no. 1-2, pp. 103–127, 2012. 3

Outline of the Presentation 1. Detailed Presentation of K-ILC Algorithm 2. Comparison with Standard QILC 3. Simulation Example 
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Lifted-Domain Representation desired output Lifted vector notation for j-th Iteration: control u j = [ u j [1] , u j [2] , ..., u j [ N ]] T ILC input Input Update System F Equivalent for all other signals Disturbance Estimator tracking error Nominal System Model: Linear, Discrete, Iteration-Constant {z | {z } F Time-constant linear system for illustration 2 3 4 CB 5 0 0 0 6 7 ... ... 6 7 CAB 0 6 7 y j = u j 6 7 . ... ... 6 7 . . 0 Measured tracking error: 4 5 CA N − 1 B . . . CAB CB e j = y j − y desired | {z } Nicolas Degen and Angela P Schoellig 5

Disturbance Estimation of K-ILC Algorithm desired output Linearised system F around desired trajectory: control ILC y j = Fu j input Input Update System F Disturbance Estimator tracking error System Model Including Modelled Disturbance as Stochastic Process : d j +1 = d j + ω j stochastic disturbance representing modelling errors y j = Fu j + d j + µ j modelled system output ω j ∼ N (0 , E j ) , µ j ∼ N (0 , H j ) random variable distributions d 0 ∼ N (0 , P 0 ) Kalman filter equations: = P j + E j S j iteration-varying = S j ( S j + H j +1 ) − 1 K j K j Kalman gain = ( I − K j ) S j . P j Nicolas Degen and Angela P Schoellig 6

Input Update of K-ILC Algorithm A Error prediction of next iteration: } + ˆ e j +1 = Fu j +1 − y d d j +1 ¯ | {z nominal model error Kalman filter used through ˆ = ˆ d j + K j ( y d − Fu j − ˆ d j ) d j +1 Estimation of Disturbance: B Updated input as solution of convex optimisation of cost function: { J j +1 ( u 0 u j +1 = argmin j +1 ) } e T J j +1 = ¯ j +1 W e ¯ e j +1 u 0 j +1 2 C Nicolas Degen and Angela P Schoellig 7

Video of ILC in Action Nicolas Degen and Angela P Schoellig 8

Goal: Analytic Comparison of ILC Algorithms Objective: Compare QILC and K-ILC QILC K-ILC • Q uadratic cost criterion ILC • K alman-Filter - Enhanced ILC • Deterministic system model • Modelling errors as stochastic disturbance • Separated disturbance estimation and input update desired output desired output control control ILC input input Input Update System F ILC Algorithm System F Disturbance Estimator tracking error tracking error J. H. Lee, K. S. Lee, and W. C. Kim, “Model- A. P. Schoellig, F. L. Mueller, and R. D’Andrea, based iterative learning control with a quadratic “Optimization- based iterative learning for precise criterion for time-varying linear systems,” quadrocopter trajectory tracking,” Autonomous Automatica, vol. 36, pp. 641–657, 2000. Robots, vol. 33, no. 1-2, pp. 103–127, 2012. 9

Comparison of Input Update QILC K-ILC A Error prediction e j +1 = F ∆ u j +1 + e j } + ˆ ¯ e j +1 = Fu j +1 − y d d j +1 ¯ | {z nominal model error ∆ u j +1 = u j +1 − u j ➤ B Input update cost function noise filtering e T J j +1 =¯ j +1 W e ¯ e j +1 e T J j +1 = ¯ j +1 W e ¯ e j +1 + ∆ u T j +1 W ∆ u ∆ u j +1 Nicolas Degen and Angela P Schoellig 10

Parameters Defining the Algorithms QILC K-ILC Parameters d j +1 = d f + ω j y j = Fu j + d j + µ j , ω j ∼ N (0 , E j ) , µ j ∼ N (0 , H j ) ∼ N e T J j +1 =¯ j +1 W e ¯ e j +1 d 0 ∼ N (0 , P 0 ) + ∆ u T j +1 W ∆ u ∆ u j +1 = P j + E j S j = S j ( S j + H j +1 ) − 1 K j noise filtering = ( I − K j ) S j . P j 2 Weighting Matrices 3 Covariance Matrices Nicolas Degen and Angela P Schoellig 11

Quadratic Norm Allows an Explicit Comparison u nom − P j QILC u j +1 = QILC Le j Explicit notation i =1 possible with quadratic norm and no constraints ! u nom − P j K − ILC u j +1 = K − ILC L j e j i =1 0 = QILC L = ( W ∆ u + F T W e F ) − 1 F T W e = F − 1 K I = = F − 1 K K − ILC L j = F − 1 K j For given iteration QILC can be made equivalent to K-ILC ➤ K-ILC optimises gain for every iteration Nicolas Degen and Angela P Schoellig 12

Mass-Spring-Damper Simulation Example 1.2 QILC equivalent of converged K-ILC robust, but converging slowly 0.8 QILC equivalent of initial K-ILC k e k 2 converging fast, but not robust once converged noise 0.4 0 0 10 20 30 Iteration count K-ILC designed for the problem QILC designed for the problem QILC equivalent of converged K-ILC QILC equivalent of initial K-ILC Nicolas Degen and Angela P Schoellig 13

Advantages of K-ILC Algorithm Implications of Kalman filter usage: 1. Separation between disturbance estimation and input update 2. Straightforward iteration-varying and optimal input update behaviour: • Fast initial convergence behaviour • Noise-resilient converged behaviour 1.2 K-ILC desired output QILC QILC-c 0.8 desired QILC-i ILC System k e k 2 trajectory Input Update 0.4 Disturbance tracking error Estimator 0 0 10 20 30 Iteration count 14

Thank you! Nicolas Degen, ETH Zurich Angela P Schoellig, University of Toronto 53rd Conference on Decision and Control, 2014 Los Angeles