Data Driven Discrete Time Modeling of Continuous Time Nonlinear Systems. Problems, Challenges, Success Stories. Johan Schoukens

Transcription

1 1/51 Data Driven Discrete Time Modeling of Continuous Time Nonlinear Systems Problems, Challenges, Success Stories Johan Schoukens

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3 fyuq (,, )

4 4/51 System Identification Data Distance Model

5 5/51 System Identification Data Linear System Y = GU Distance Model

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7 Problem solved? Mission accomplished? 7/51

8 8/51 System Identification in a real world Linear Nonlinear Time-Varying

9 9/51 Linear SI versus Nonlinear SI Linear SI - mature field - well developed tools - inexpensive

10 10/51 Linear SI versus Nonlinear SI Linear SI Nonlinear SI - mature field - hot research topic - well developed tools - inexpensive - expensive

11 11/51 Linear SI versus Nonlinear SI Linear SI Nonlinear SI - mature field - hot research topic - well developed tools - inexpensive - expensive Preliminary Questions - Do we face a nonlinear identification problem? - Safe to use a linear system identification approach? - How much to gain with a nonlinear model?

12 12/51 Data Driven Discrete Time Modeling of Continuous Time Nonlinear Systems: Problems and Challenges Do we need a nonlinear model? Detection, qualification, quantification NL Linear identification in the presence of nonlinear distortions Nonlinear system identification

13 13/51 Detection, qualification, quantification NL Goal characterize nonlinear behaviour no increase of the measurement time little user interaction Result: a well informed decision: linear or nonlinear model?

14 14/51 Detection, qualification, quantification NL Goal Tool characterize nonlinear behaviour no increase of the measurement time little user interaction periodic excitations ut () = 1 -- å F A F k cos( 2 p kf 0 t + j k ) k = 1

15 15/51 Detection, Qualification, Quantification NL Basic Idea, =, ut () = 2coswt = e jwt e jwt w 1 U( w) w

16 16/51 Detection, Qualification, Quantification NL Basic Idea, =, ut () = 2coswt = e jwt e jwt w 1 U( w) f u 3 = ( e jwt e jwt )( e jwt e jwt )( e jwt e jwt )

17 17/51 Detection, Qualification, Quantification NL Basic Idea, =, ut () = 2coswt = e jwt e jwt w 1 U( w) f u 3 = ( e jwt e jwt )( e jwt e jwt )( e jwt e jwt ) Output: all possible combinations, 3 by 3, of the frequencies -1 and U( w) w

18 18/51 Detection, Qualification, Quantification NL Basic Idea input linear f f 0 + f f 0 even odd f f 0 = output f f f f 0

19 19/51 Detection, qualification, quantification of nonlinear distortions u ms 2 + ds+ k 1 y k 3 y 3

20 20/51 Detection, qualification, quantification of nonlinear distortions u ms 2 + ds+ k 1 y k 3 y 3 odd NL even NL noise

21 21/51 Detection, qualification, quantification of nonlinear distortions Acknowledgement: These measurements were done in a collaboration with Bart Peeters (LMS International, part of Siemens Product Lifecycle Management), Jean-Philippe Noël and Gaetan Kerschen (University of Liège) during the LMS Ground Vibration Testing Master Class held in September 2014 at the Saffraanberg military basis in Belgium

22 22/51 Detection, qualification, quantification of nonlinear distortions odd NL even NL noise

23 23/51 Detection, qualification, quantification of nonlinear distortions odd NL even NL noise

24 24/51 Data Driven Discrete Time Modeling of Continuous Time Nonlinear Systems: Problems and Challenges Do we need a nonlinear model? Detection, qualification, quantification NL Linear identification in the presence of nonlinear distortions Nonlinear system identification

25 25/51 Nonlinear system identification Data Distance Model

26 26/51 Nonlinear system identification Model Data - Continuous time - Discrete time - Unstructured - Structured - Coupled - Decoupled - Reflect future use - Input design Data Model Distance Cost - Model errors - Noise errors - Process noise Success stories

27 27/51 Model: Continuous time - Discrete time ZOH-Assumption Physical system What we like: explicit model What we get: implicit model d xt () = FCT ( xt ()ut, ) dt x DT ( k + 1) = F DT ( x DT ( k), u DT ( k) ) x DT ( k) = F DT( x DT ( k), u DT ( k) ) Goodwin et. al. (2013) : Truncated Taylor Series to get explicit model x DT ( k) = x CT ( kt s ) + OT ( s ) Assumption: NLSS is affine in the input

28 28/51 Model: Continuous time - Discrete time Alternative Assumption: Low Pass Assumption Bandlimited signal Low pass signal

29 29/51 Experimental verification on silverbox Output Spectrum Relative Error 30 db/decade

30 30/51 Nonlinear models - User desires Unstructured - Structured models A model should be more than a bunch of parameters Flexible: cover a wide range of nonlinear systems Multiple Input - Multiple Output Reliable Provides intuitive insight in system behaviour

31 31/51 Nonlinear Models unstructured structured x + = Ax+ Bu+ Fxu (, ) y= Cx+ Du+ Gxu (, )

32 32/51 Nonlinear Models unstructured structured x + = Ax + Bu + F( x, u) y = Cx + Du + G( x, u)

33 33/51 Nonlinear Models unstructured structured x + = Ax + Bu + F( x, u) y = Cx + Du + G( x, u) NN, polynomials,... System identification: Best Linear Approximation

34 34/51 Nonlinear Models should be more than a bunch of numbers x + = Ax + Bu + F( x, u) y = Cx + Du + G( x, u) F 1 ( xu, ) = a 1 x 1 + a 2 x 2 + a 3 x 3 a 11 x a 12 x 1 x 2 + a 13 x 1 x 3 + a 22 x 2 + a 12 x 2 x 3 + a 33 x 3 a 111 x 1 3 a 222 x a 112 x 1 x 1 x + a x 1 x 1 x + a x 1 x 2 x + a x 1 x 3 x 3 + a 212 x 2 x 1 x 2 + a 232 x 2 x 3 x 2 + a 233 x 2 x 3 x 3 + ¼

35 35/51 Decoupling multivariate polynomials - increased physical insight - number of parameters grows linearly instead of combinatorially - exact - approximatively - noise weighting Dreesen, P., Ishteva, M., and Schoukens, J. (2015). Decoupling multivariate polynomials using first-order information and tensor decompositions. SIAM Journal Matrix Analysis and Applications, pp

36 36/51 Link with Neural Network theory - in NN are all g i are the same - NN are universal approximators, rate 1 n

37 37/51 Decoupling nonlinear state space models xk ( + 1) = Axk ( ) + Buk ( ) + Fxk ( ( ), uk ( )) yk ( ) = Cxk ( ) + Duk ( ) + Hxk ( ( ), uk ( )) 1) full multivariate description F 2) Semi-decoupling: all coupling terms eliminated å n j = 1 - each G i ( z 1, ¼, z n ) = g ij ( z j ) with i = 1, ¼, n F G - number of terms n 2 d proportional with degree d >< combinatiorial - can be calculated from the decoupling matrices 3) full decoupling each state depends nonlinearly only on one other state - is this (always) possible?

38 Example 38/51

39 39/51 Nonlinear system identification Model - Continuous time - Discrete time - Unstructured - Structured - Coupled - Decoupled Data Cost - Reflect future use - Input design - Model errors - Noise errors - Process noise Joint >< Marginal distribution Success stories

40 40/51 Data: reflect future use: wet clutch example x 1 Estimated domain Domain of interest x 2 Widanage, W.D., Stoev, J., Van Mulders, A., Schoukens, J., and Pinte, G. (2011). Nonlinear system-identification of the filling phase of a wet-clutch system. Control Engineering Practice,

41 Wet Clutch results 41/51

42 Not solved Not understood Experiment design for nonlinear systems Optimal input design What we know - power spectrum - amplitude distribution - global solution --> a combined optimization is needed Simple example (M. Gevers) Optimize joint distribution Fi = I 11 I 12 I 21 I 22 Gevers, M., Caenepeel, M., and Schoukens, J. (2012). Experiment design for the identification of a simple Wiener system. 51st IEEE Conference on Decision and Control. December 10-13, Maui, Hawaii, USA, pp /51

43 43/51 Optimal input design What we are not able to do - what is the global solution? - how to generate a random signal with an arbitrary joint distribution We look into sub-optimal solutions - Gaussian mixtures - Dictionary of excitation signals - Brute force optimization of a single realization

44 44/51 Nonlinear system identification Model - Continuous time - Discrete time - Unstructured - Structured - Coupled - Decoupled Data Cost - Reflect future use - Input design - Model errors - Noise errors - Process noise Success stories

45 45/51 Choice of the cost function in the presence of model errors Some thoughts on model errors Avoidable model errors e.g.: unmodelled dynamics in linear SI Why wouldn t you eliminate these? Non-avoidable model errors e.g.: NL system is outside the NL model class User defined criterion needed: how to shape the error? Shaping with the noise variance is no obvious choice

46 46/51 Cost: Model errors - Noise errors noise weighting >< shaping the model errors Typical result nonlinear identification Output Linear error NL error Noise floor The errors are not dominated by the noise The error weighting in the cost function becomes a user choice: e.g. small error in freq. band of interest Using the noise weighting can destroy the quality of the fit Schoukens, M., Marconato, A., Pintelon, R., Vandersteen, G., and Rolain, Y. (2015). Parametric identification of parallel Wiener Hammerstein systems. Automatica 51, pp

47 47/51 Cost: Process noise Wiener system example - noise wt () and signal ut () get mixed in f (). - negelecting process noise can lead to an arbitrary large bias - solving the full problem is much more involved tools needed to detect the presence of process noise Hagenblad, A., Ljung, L., Wills A. (2008). Maximum likelihood identification of Wiener models. Automatica, pp

48 48/51 Nonlinear system identification Model - Continuous time - Discrete time - Unstructured - Structured - Coupled - Decoupled Data Cost - Reflect future use - Input design Input domain? Optimal input design - Model errors - Noise errors - Process noise Success stories

49 49/51 Nonlinear identification of a battery Nonparametric distortion analysis

50 50/51 Parametric NLSS model results Time domain Frequency domain

51 Do we need a nonlinear model? Conclusions Linear identification in the presence of nonlinear distortions Nonlinear System Identification Model - Continuous time - Discrete time - Unstructured - Structured - Coupled - Decoupled Data - Reflect future use - Input design Input domain? Optimal input design Cost - Model errors - Noise errors - Process noise Success stories 51/51