Nonlinear System Identification Based on Reduced Complexity Volterra Models Guodong Jin1,a* and Libin Lu1,b

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1 6th Iteratioal Coferece o Electroics, Mechaics, Culture ad Medicie (EMCM 205) Noliear System Idetificatio Based o Reduced Complexity Volterra Models Guodog Ji,a* ad Libi Lu,b Xi a research istitute of high techology, Xi a, 70025, P. R. Chia a jiguodog_army@63.com, blulibi@26.com Keywords: Noliear system; Volterra series model; Noparametric model idetificatio; Radom multi-toe excitatio Abstract. Covetioal Volterra series model is hardly applied to egieerig practice due to its parametric complexity ad estimatio difficulty. o solve this problem, oliear system idetificatio usig reduced complexity Volterra models is proposed. Sice the oliear compoets ofte play a secodary role compared to the domiat, liear compoet of the system, they sped the most of idetificatio cost. So it is worth establishig a balace betwee idetificatio cost ad model accuracy by reducig the complexity of oliear compoets. Refer to the idea of oliear output frecy respose fuctio, covetioal Volterra model is simplified. Ad the a miimum mea square error criterio based method to idetify the simplified model is proposed. he distiguishig feature of this method is high accuracy, good robustess, ad sigificat reductio i the computatioal riremets compare to the idetificatio of covetioal Volterra models. he simulatio show that the proposed method is effective, ad the reduced complexity Volterra model is of good geeralizatio ability i geeral. So this oliear system idetificatio approach is quite applicable to egieerig practice. Itroductio he Volterra series model provides a ituitive ad relatively geeral framework for aalyzig the behavior of o-liear systems []. Most real world systems are oliear i ature so thatoliear models are ofte preferable for represetig systems uder study. his is evideced by the wild spread applicatios of Volterra models i fields icludig cotrol [2], idetificatio [3], damage detectio [4] ad assessmet [5]. he objective of preset study is to develop a efficiet ad practical approach for idetificatio of quadratic oliear systems. A simplified oparametric model idetificatio method is preseted. his method leads to sigificat reductios i both the computatioal riremets ad the mathematical tractability comparig to traditioal Volterra model. Simulatios show preset algorithm has excellet ability of model geeralizatio ad is still effective with output measuremet oises, eve i low sigal-oise-ratio (SNR) coditio. Prelimiaries For a weakly oliear system up to secod order Volterra series represetatio, the discrete time Volterra model ca be expressed as () y (k ) h (i,, i ) u (t i ) i i i i where h (,, ) deote the th-order Volterra kerel fuctio. I the frecy domai, Eq. trasforms to (2) Y ( f ) H f U f H 2 f g, g U f g U g k where Y( f ), U( f ) deote the Fourier trasforms of the output ad iput respose, respectively. H f,, f is the -dimesioal Fourier trasform of h (,, ), which called the th order GFRF: 206. he authors - Published by Atlatis Press 475

2 H f,, f h,, exp j2 f 2 f d d (3) he Eq.3 ca be writte more compactly with the liear ad quadratic operators L(.) ad Q(.) : Y = L(U) +Q(U) +ε (4) whereεdeotes the model error i frecy domai. For practical reaso, we cosider iput sigals havig egligible power at the frecies larger tha some upper boud frecy fm. Correspodigly the frecy rages of each order outputs are f [0, f M ] for liear part ad f, g [ f M, f M ] or quadratic part. herefore the valid frecy rage of quadratic kerels is [0, 2 f M ] with the assumptio of the GFRFs are symmetric without loss of geerality. A Simplified Idetificatio Method Cosider the iput ad output sigals are sampled sychroously with the samplig frecy fs. Whe the expected discrete frecy resolutio 0 f ad the maximum iput frecy fm = Mf0 are give, the least data legth N should be take as 4M to avoid aliasig. he the quadratic Volterra model i the discrete frecy domai may be writte as Y kf0 H kf0 U kf0 H2 mf0, k m f0 U k m f0 kf0 (5) m for k,m ( N /2, N/ 2]. Sice the higher order kerels ofte play a secodary role compared to the domiat, liear compoet of the system, it is reasoable to assume that all H,, f f i the hyper plae f f = f are the same value. Uder this assumptio, a coceptio of ivalet kerel is defied as H kf0 0,, H k f k f0 (6) 2M k k k herefore, the Eq.5 may be simplified to Y kf0 H kf0 U kf0 H2 kf0 U mf0 U k m f0 kf0 (7) m he ukow kerels i Eq.7 ca be calculated approximately from the LSM (Least-SquareMiimizatio) of the power of model error. Defie the least square cost fuctio 2 2 (8) J E E Y L U Q U Where E{.} deotes the expectatio operator. o miimize Eq.8 rires J J H H2 0 ( 9 ) Substitutig Eq.8 ito Eq.9, we obtai Z Φ H ( 0 ) Z * SUY kf0 SUUY gf0, k g f0 () H H kf0 H2 kf0 ( 2 ) * SUU kf0 SUUU gf0, k g f Φ SUUU gf0, k g f0 2SUU gf0 SUU k g f (3) g for k [0, 2M], g [(k +)/2, M] whe k is odd ad g [k/2,m] whe k is eve. I Eq.0-Eq.3, S UU, S UY, S UUU ad S UUY are the auto spectrum, cross spectrum, auto bi-spectrum ad cross bi-spectrum, respectively, ad [.] deotes the traspose operator. Eq.0 is derived by assumig the fourth order cumulat fuctio of iput sigal vaishes. he Least-Square solutio of Eq.0 is (ΦΦ) ΦZ. By usig the ivalet kerel, the size ofφφ is reduced from (M k /2 +) (M k /2 +) to 2 2, which implies the sigificat reductio of computatioal riremets ad store spaces i solvig Eq

3 Idetificatio Algorithm Usig Radom Multi-toe Excitatio For give discrete frecy resolutio f0 ad the maximum iput frecy Mf0, a radom multi-toe excitatio sigal could be desiged as follow M msi 2 0 m (4) u t A mf m0 where {A m }is the costat amplitudes, ad {φm}is the radom phases which are idepedet, uiformly distributed radom variables o [0, 2π ). It is proved that the fourth order cumulats of such radom multi-toe sigals fall faster to zero tha the third order cumulats [6]. hus we caassume that the iputs have a vaishig tri-spectrum, ad therefore Eq.0 is valid. he steps of the idetificatio algorithm ca be described as follow: Step : Choose the frecy resolutio f0ad the maximum iput frecy Mf0. Geerate M radom umbers idepedetly for {φm}. he costruct the iput sigal by Eq.4. Step 2: Determie the samplig frecy f s ad the record legth N follow the priciple f s=n f0 Mf0. Step 3: Apply the excitatio to the real process, ad sample the iput ad output sigals sychroously. Step 4: Estimate the spectrums i Eq.-Eq.3, ad the obtai the estimates of H ad H2 accordig to the Eq.0 solutio. Step 5: est the geeralizatio ability for the obtaied model. If the result reaches rired precisio, ed the algorithm, if ot, tur back to step. Simulatios I this sectio, simulatios are provide usig the system y= 5y 0.y2 + u + 0.u2. Iput sigal for testig the geeralizatio ability of model is selected as u(t) = 2si(20.25t) + 2si(5.34t), which is ot used i idetificatio. I order to demostrate the effects of the idetificatio algorithm preseted i this study, two groups of umerical simulatio are coducted. Oe compares the idetificatio performaces usig radom iputs with usig radom multi-toe iputs. Aother examies the effects of additive oise i output measuremet. I latter simulatios, performaces betwee the represeted method ad the previous method i [7] are compared. he results are give i Fig. -Fig. 3. Figure. Output of idetified model usig radom (left) ad radom multi-toe (right) iputs 477

4 Figure 2. Output of idetified model usig previous (left) ad preset (right) methods (SNR: 20dB) Figure 3. Output of idetified model usig previous (left) ad preset (right) methods (SNR: 0dB) I Fig., we compare the performaces of idetified model usig radom ad radom multi-toe iputs. As ca be see i this figure, it is difficult to obtai accurate estimatio usig radom iputs. For it is impossible to geerate radom iputs with perfect cut-off i its power spectrum, the accuracy of the estimatio usig radom iput is certaily affected. O the other had, usig radom multi-toe iputs recover a excellet performace. Fig. 2 ad Fig. 3 compare the previous method ad preset method with additive oise i output measuremet. It ca be see that the preset method has a better ability of agaist the oise i output, which profit from the use of higher order spectrum. However, additive oise still result i iaccuracies i model idetificatio, ad by usig more realizatios, the error i kerel estimatio may be decreased. Summary I this study, a simplified oparametric idetificatio algorithm for quadratic Volterra systems is proposed. By usig multi-toe iputs ad the higher order spectrum of iput-output data, the geeralizatio ability ad the robustess of obtaied model ca be guarateed. Moreover, compared to the previous methods, the preset algorithm rires fewer calculatios ad store spaces. Refereces [] V. Volterra: heory of Fuctioals ad of Itegral ad Itegro-differetial Equatios (Dover, New York 959). 478

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