From Volterra series through block-oriented approach to Volterra series

Transcription

1 From Volterra series through block-oriented approach to Volterra series Pawel Wachel and Przemyslaw Sliwinski Department of Control Systems and Mechatronics Wroclaw University of Technology European Research Network on System Identification (ERNSI) Varberg 215 P. Wachel, P. Sliwinski

2 Outline Introduction Volterra series and block oriented models Nonparametric kernel methods basic concepts Nonparametric kernel regression in application to system identification Mixed (parametric nonparametric) system identification Aggregative modeling dictionary approach Application of aggregative approach to Volterra series modeling of LNL system P. Wachel, P. Sliwinski 2/33

3 Volterra vs. block oriented approach Classic Volterra approach Volterra series Classic identification methods Curse of dimensionality problems Block oriented alternatives U n m Z n Y n U n p i i W n m V n q j j Z n Y n U n i i W n Z n m Y n m i i U n W n Y n Z n Z n Z n U n p i i W n m Y n U n m W n i i Y n P. Wachel, P. Sliwinski 3/33

4 General block oriented identification setup Input signal i.i.d. random sequence Nonlinearity Nonparametric prior knowledge Dynamic subsystem(s) Asymptotic stability assumed P. Wachel, P. Sliwinski 4/33

5 Nonparametric kernel density estimation Let us consider the following estimation problem: Given the set of i.i.d. measurements U 1,U 2,,U N with the unknown probability density function f U find (estimate) f U in some point u from the set U 1,U 2,,U N P. Wachel, P. Sliwinski 5/33

6 Nonparametric kernel density estimation The main idea behind the considered estimate is based on the observation that for any u and relatively small constant : h f U u 1 h uh/2 uh/2 f U vdv. P. Wachel, P. Sliwinski 6/33

7 Nonparametric kernel density estimation Let now K be a box kernel function of the form Kv 1 for v 1 2 ; 1 2 for v 1 2 ; 1 2 How the kernel works? Kv K vu h v v P. Wachel, P. Sliwinski 7/33

8 Nonparametric density estimation Using the kernel function we obtain further f U u 1 h uh/2 uh/2 f U vdv 1 h f U vk vu h 1 h E K dv U 1u h P. Wachel, P. Sliwinski 8/33

9 Nonparametric density estimation Finally, on the basis of f U u 1 h E K U 1u h we obtain the following Kernel Density Estimate (Rosenblatt, Parzen) f U u 1 N Nh i1 K U iu h (which is a sum of scaled and translated kernel functions) P. Wachel, P. Sliwinski 9/33

10 Nonparametric kernel identification Consider a static nonlinear system described by the formula Z n Y n mu n Z n U n m Y n m u P. Wachel, P. Sliwinski 1/33

11 Nonparametric kernel identification Consider the following local least squares quality function Goal: find c Q c;u Y i c 2 K such that m N i1 U iu h c argmin cr Q c;u u u h 2 u h 2 u P. Wachel, P. Sliwinski 11/33

12 Nonparametric kernel identification N Q c;u Y i c 2 K i1 U iu h Q c;u is a (non-negative) quadratic function of variable. c Q c;u c c Solution: Nadaraya Watson kernel estimate c m u N Y i K i1 N K i1 U i u h U i u h P. Wachel, P. Sliwinski 12/33

13 Hammerstein system nonlinearity estimation Z n U n m W n i i Y n W n mu n, Y n i i W ni Z n u N Y i K i1 N K i1 U i u h U i u h u mu, as N in probability P. Wachel, P. Sliwinski 13/33

14 Wiener system Gaussian input assumed U n i i W n Z n m Y n W n l Y n mw n. l U nl Z n, Main concept: Replace input measurements with output ones in the kernel estimate. y m 1 y as N in probability. y N U i K i1 N K i1 Y i y h Y i y h P. Wachel, P. Sliwinski 14/33

15 Orthogonal algorithms and wavelets 2nd order Daubechies 3rd order Daubechies P. Wachel, P. Sliwinski 15/33

16 Orthogonal wavelet regression estimate Orthogonal estimate of the nonlinearity K u N Y i K u,u i i1 N K u, U i i1 NOTE SIMILARITY! u N Y i K i1 N K i1 U i u h U i u h where n 2 K u,v Knu Knv nn 1 and Kn is a family of orthogonal functions. P. Wachel, P. Sliwinski 16/33

17 Application to predistortion Doherty amplifier Predistortion problem: Estimate an inverse of the system s nonlinearity m 1 and linearize system by replacing input U n with m 1 U n. Doherty amplifier Scheme Hammerstein model Peak amplifier (class C) OFDMA U n m W n i i Z n Y n Splitter Combiner Carrier amplifier (class AB) P. Wachel, P. Sliwinski 17/33

18 Predistortion practical example 1 Estimate of the system nonlinearity Estimate of the inverse of the nonlinearity Effect of linearization P. Wachel, P. Sliwinski 18/33

19 Average Derivative Estimation Wiener system Fact (chain rule): Let m be a differentiable function, then d dx mx Conclusion: Derivative can extract factor in the argument of m. d dx mx embedded Z n U n p i i W n m Y n p Y n m i VECTOR i U ni Z n Y n m T NOTATION U n Z n P. Wachel, P. Sliwinski 19/33

20 Identification from structured input data Consider the following MISO Hammerstein system: m i i U n W n Y n Z n U n R D Fact: Convergence rate of the nonparametric estimators strongly depends on the number of system inputs. Assumption: Input data are structured i.e. grouped on unknown d dimensional manifold ( d D). P. Wachel, P. Sliwinski 2/33

21 Identification from structured input data P. Wachel, P. Sliwinski 21/33

22 AGGREGATIVE MODELING P. Wachel, P. Sliwinski 22/33

23 Dictionary modeling The a priori knowledge about the system is collected in a dictionary in a form of the system components, models, preestimates etc. Prior identification results Orthogonal families Volterra series components Heuristic models P. Wachel, P. Sliwinski 23/33

24 Aggregative modeling The class of considered SISO systems can be described by the general discrete-time input-output equation Assumptions: X n Y n mx n,x n1,x n2,,x np Z n, is a stationary sequence of inpendent random variables. mx is any bounded function, such that mx L. Z n is a zero-mean i.i.d. random sequence with finite variance. There is a dictionary of maps, S m i : R p1 L,L,i 1,,D, D 2 collected by a user to model the system. P. Wachel, P. Sliwinski 24/33

25 Aggregative modeling The algorithm combines all the dictionary entries into a single one, referred to as the aggregated empirical model. where mx; D i1 i m i x argmin Q, subject to 1 1, and where Q 1 N p N mx i ; Y i 2. ip1 P. Wachel, P. Sliwinski 25/33

26 Convergence For a wide class of nonlinear systems and aggregative (dictionary) model, it holds that EQ Q C NlnD N p where and where Q EmX n ; Y n 2 argmin A Q P. Wachel, P. Sliwinski 26/33

27 l 1 constrained vs. standard LS modeling.4.35 MSE N const. D N lnd N Apply standard approach Apply l 1 constrained approach D P. Wachel, P. Sliwinski 27/33

28 Why it works? l 2 ball for D=3 l 1 ball for D=3 V V P. Wachel, P. Sliwinski 28/33

29 Why it works? Volume of a unit l 1 ball vs. space dimension (D) Volume of a unit l 2 ball vs. space dimension (D) Volume of a unit l 1 ball vs. space dimension (D) Volume of a unit l 2 ball D D D P. Wachel, P. Sliwinski 29/33

30 Experiment settings Wiener-Hammerstein system: U n p i i W n m V n q j j Z n Y n Nonlinearity: mw w 2. Impulse responses: 1 1 i j Input signal: Uniform ( 3, 3) Noise signal: SNR=1 Dictionary: Unique Volterra series components (L =4, P=2) P. Wachel, P. Sliwinski 3/33

31 Experiments results 1.4 Diagonal of the 2nd order Volterra kernel by AGGREST 1.5 Second order Volterra kernel by AGGREST h(i, j) FFT of measured validation output for N = 124 and SNR = 1 5 Re(FFT(System output)) 4 Re(FFT(Model output)) 3 Re(FFT(Z)) 2 Impulse h 2 (x 1, x 2 ) x 1 x 2 Outpusts of AGGREST model and test system System output Model output 4 Re(FFT) P. Wachel, P. Sliwinski 31/33

32 LS Method vs Aggregative modeling Dictionary weights evaluated by the LS method and aggregative approach. LS METHOD AGGREGATIVE APPROACH 1E E E-4 1E-4 1E-7 1E-7 1E-1 1E-1 P. Wachel, P. Sliwinski 32/33

33 Bibliography Greblicki W. and Pawlak M. Nonparametric system identification. Cambridge: Cambridge University Press, 28. Greblicki W. and Pawlak M. "Identification of discrete Hammerstein systems using kernel regression estimates." Automatic Control, IEEE Transactions on 31.1 (1986): Greblicki W. "Nonparametric identification of Wiener systems."information Theory, IEEE Transactions on 38.5 (1992): Wachel P. Sliwinski P. and Hasiewicz Z. "Nonparametric identification of MISO Hammerstein system from structured data." Journal of Systems Science and Systems Engineering 24.1 (215): Hasiewicz Z. and Sliwinski P. "Identification of non-linear characteristics of a class of blockoriented non-linear systems via Daubechies wavelet-based models." International Journal of Systems Science (22): Hasiewicz Z. and Mzyk G. "Combined parametric-nonparametric identification of Hammerstein systems." Automatic Control, IEEE Transactions on 49.8 (24): Sliwinski P. Nonlinear system identification by Haar wavelets. Vol. 21. Springer Science & Business Media, 212. Mzyk, G. Combined Parametric-Nonparametric Identification of Block-Oriented Systems. Berlin: Springer, 214. Wachel P. and Mzyk G. "Direct identification of the linear block in Wiener system." International Journal of Adaptive Control and Signal Processing (215). P. Wachel, P. Sliwinski 33/33

34 Thank you P. Wachel, P. Sliwinski 34/33