Design of Norm-Optimal Iterative Learning Controllers: The Effect of an Iteration-Domain Kalman Filter for Disturbance Estimation

Transcription

1 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 rd Conference on Decision and Control, Los Angeles

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

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

4 Outline of the Presentation 1. Detailed Presentation of K-ILC Algorithm 2. Comparison with Standard QILC 3. Simulation Example 4

5 Lifted-Domain Representation Lifted vector notation for j-th Iteration: u j =[u j [1],u j [2],...,u j [N]] T Equivalent for all other signals desired output ILC Input Update Disturbance Estimator control input System F Nominal System Model: Linear, Discrete, Iteration-Constant y j = {z F 2 3 {z } CB CA B... CB CAB 4 CAB CA N 1 B... CAB CB {z } u j Time-constant linear system for illustration Measured : e j = y j y desired 5

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

7 Input Update of K-ILC Algorithm A Error prediction of next iteration: ē j+1 = Fu j+1 y d {z } + ˆd j+1 nominal model error Kalman filter used through Estimation of Disturbance: ˆd j+1 = ˆd j + K j (y d Fu j ˆdj ) B Updated input as solution of convex optimisation of cost function: u j+1 = argmin u 0 j+1 2C {J j+1 (u 0 j+1)} J j+1 =ē T j+1w e ē j+1 7

8 Video of ILC in Action 8

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

10 Comparison of Input Update QILC A Error prediction K-ILC ē j+1 = F u j+1 + e j u j+1 = u j+1 u j ē j+1 = Fu j+1 y d {z } + ˆd j+1 nominal model error B Input update cost function noise filtering J j+1 =ē T j+1w e ē j+1 + u T j+1w u u j+1 J j+1 =ē T j+1w e ē j+1 10

11 Parameters Defining the Algorithms QILC Parameters K-ILC d j+1 = d f +! j y j = Fu j + d j + µ j, J j+1 =ē T j+1w e ē j+1 + u T j+1w u u j+1! j N (0,E j ),µ j N (0,H j ) d N 0 N (0,P 0 ) noise filtering S j = P j + E j K j = S j (S j + H j+1 ) 1 P j =(I K j )S j. 2 Weighting Matrices 3 Covariance Matrices 11

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

13 Mass-Spring-Damper Simulation Example 1.2 QILC equivalent of converged K-ILC robust, but converging slowly kek QILC equivalent of initial K-ILC converging fast, but not robust once converged noise 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 13

14 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 desired output ILC Input Update desired trajectory System kek2 0.8 K-ILC QILC QILC-c QILC-i Disturbance Estimator Iteration count

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