Identification of FIR Wiener systems with unknown, non-invertible, polynomial non-linearities

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1 INT. J. CONTROL, 2003, VOL. 76, NO. 5, Identification of FIR Wiener systems with unknown, non-invertible, polynomial non-linearities SETH L. LACY{* and DENNIS S. BERNSTEIN{ Wiener systems consist of a linear dynamic system whose output is measured through a static non-linearity. In this paper we study the identification of single-input single-output Wiener systems with finite impulse response dynamics and polynomial output non-linearities. Using multi-index notation, we solve a least squares problem to estimate products of the coefficients of the non-linearity and the impulse response of the linear system. We then consider four methods for extracting the coefficients of the non-linearity and impulse response: direct algebraic solution, singular value decomposition, multi-dimensional singular value decomposition and prediction error optimization. Received August Revised June Accepted June * Author for correspondence. seth.lacy@kirtland.af.mil { Air Force Research Laboratory, Space Vehicles Directorate, Kirtland AFB, NM 877, USA. { Aerospace Engineering Department, University of Michigan, Ann Arbor, MI 4809, USA.. Introduction Non-linear system identification remains one of the most challenging and potentially useful problem areas in system theory. Numerous approaches have been developed for this problem, including black box and grey box techniques (Bayard and Eslami 984, Pajunen 985, Hunter and Korenberg 986, Korenberg and Hunter 986, Hasiewicz 987, Chen and Fassois 992, 997, Greblicki 992, 994, 997, 998, Westwick and Kearney 992, Wigren 994, Westwick and Verhaegen 996, Bai 998, Lovera et al. 2000, Van Pelt and Bernstein 2000, Lacy and Bernstein 200, Lacy et al. 200, Nelles 200). The grey box case includes the identification of block-structured models, such as the Hammerstein model (linear system with input non-linearity) and Weiner model (linear system with output non-linearity) (Haber and Keviczky 999 a, b). This paper is concerned with identifying Wiener systems under more general assumptions than have been previously considered. Many methods for Wiener system identification require the non-linearity to be known, invertible, differentiable, odd or require specially designed input sequences. In particular, the Wiener identification problem has been considered in Brillinger (970), Pajunen (985), Hasiewicz (987), Greblicki (992, 994, 997, 998), Westwick and Kearney (992), Wigren (994), Westwick and Verhaegen (996), Bai (998), and Lovera et al. (2000) under the assumption that the non-linearity is unknown but one-to-one. This assumption simplifies the problem considerably since the inverse system can be viewed as a Hammerstein system in the input to the nonlinearity is measured. If the non-linearity is known but not one-to-one, then identification is possible by first generating a candidate set of signals at the output of the linear system (Bayard and Eslami 984, Lacy et al. 200). These methods are applicable even if the output non-linearity is a step function in which case the output assumes at most two distinct values (Lacy et al. 200). If the input sequence can be chosen freely, the frequency content of the input sequence can be selected such that the effect of the non-linearity can be derived from the frequency content of the output (Pintelon and Schoukens 200). In the present paper we consider Wiener system identification in which the output non-linearity is both unknown and not necessarily one-to-one. In this case the goal is to simultaneously identify both the linear system dynamics and the non-linearity despite the noninvertibility of the output non-linearity. To do this we assume that the non-linearity can be represented as a finite sum of polynomials. We use multi-index notation (Evans 998, Dunkl and Xu 200) to expand this polynomial and write the output as a linear-inparameters sum of known terms with unknown coefficients. These coefficients consist of products of the system parameters. We then present several methods for extracting the system parameters. This approach of expanding the polynomial output non-linearity requires that the linear system output depend only on past inputs, that is, the linear dynamics are assumed to be FIR. This paper is organized as follows. In } 2 we define the problem and list the assumptions. In } 3 we introduce notation used throughout the paper. In } 3. we present an algebraic solution. In } 3.2 we present a solution based on a singular value decomposition. In } 3.3 we present a solution based on a multi-dimensional singular value decomposition (Andersson and Bro 2000). In } 3.4 we present an optimality approach based on a prediction-error cost function. In } 4 we apply all of these methods to an example to illustrate their implementation and compare their effectiveness. International Journal of Control ISSN print/issn online # 2003 Taylor & Francis Ltd DOI: 0.080/

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3 Identification of FIR Wiener systems Problem description Here we study the identification of a single-input, single-output, linear time-invariant system with finite impulse response whose output is measured through a static non-linearity. This system, which is represented in figure, is modelled by yðkþ ðh 0 uðkþþh uðk Þþþh m uðk mþ Xm zðkþ NðyðkÞÞ Figure. h i uðk iþ ðþ ð2þ u is the input to the system, y is the unmeasured output of the linear system, h i are Markov parameters, and z is the measured output of the non-linearity. We assume that the non-linear function N : R! R is a polynomial of the form NðyðkÞÞ Xp Wiener system. c i yðkþ i ð3þ If N is not a polynomial, then (3) can be regarded as an approximation. We assume that the order m of the FIR dynamics and the degree p of the polynomial N are known. The non-linearity N is otherwise unknown and not necessarily one-to-one. In the practical situation m and p are not known. Also, it may be difficult to obtain satisfactory results if these parameters are underestimated. However, if upper bounds on these parameters are known, then the bounds can be used in () and (3) at the expense of increasing the computational complexity. Alternatively, the values for p and N can be incremented until satisfactory performance is achieved, at the expense of increasing the computational load at each increment. The identification problem is to estimate the coefficients h i and c i using measurements of u and z. We adopt a two-stage approach. First, we solve a least squares problem to obtain an estimate ^h of a vector h whose entries are the unknown parameters and products of the unknown parameters. Next, we present several techniques that use ^h to estimate the individual unknown parameters. In addition, we minimize a prediction error cost function to further refine the parameter estimates and compare to the direct approaches. 3. Wiener identification Using (3), we rewrite equation (2) as zðkþ Xp Xp c i yðkþ i Xp c i X ðkþ T h c i X m j0! h vðkþ Xp h j uðk jþ! i! vðkþ c jj h ð4þ h ½h 0 h h m Š T 2 R mþ ð5þ ½ 2 mþ Š T 2 N mþ 0 ð6þ vðkþ ½uðkÞ uðk Þ uðk mþš T 2 R mþ ð7þ h ½c jj h Š jj p 2 R Dmþ P ð8þ ðkþ! vðkþ 2 R Dmþ P ð9þ jj p and 2 N0 mþ is a multi-index whose order is m þ, N 0 is the set of positive integers and zero. A multi-index is a vector whose components are nonnegative integers (see Evans 998, Dunkl and Xu 200). We define jj þ 2 þþ mþ Xmþ i i! ð!þð 2!Þð mþ!þ Ymþ i! i vðkþ v ðkþ v 2 ðkþ 2 v mþ ðkþ mþ h h h 2 2 h mþ mþ Ymþ i h i i Ymþ i v i ðkþ i ð0þ ðþ ð2þ ð3þ The number of multi-indices of order m þ of fixed absolute value i is given by Ci mþ m þ i m þ i ðm þ iþ! ð4þ i m m!i! and the number of multi-indices of order m þ of absolute value less than or equal to p is given by D mþ p Xp C mþ i Xp ðm þ iþ! m!i! ð5þ We need to define an order relation for multi-indices of the same order. Let, 2 N0 mþ be multi-indices. If jj > jj then >.Ifand have the same order,

4 502 S. L. Lacy and D. S. Bernstein jj jj, then we choose the standard dictionary ordering. The notation 2 3 ½ f ðþš ½ f ½ðÞŠ jj ½ f ðþš jj p. ð6þ ½ f ðþš jjp denotes the column vector whose components are f evaluated at every multi-index such that jj p. The components are ordered according to the above ordering scheme. This vector has D mþ p components. Thus h and ðkþ have D mþ p components. To estimate h we rewrite (4) as z F T h ð7þ z ½zðm þ Þ zð ÞŠ T 2 R m ð8þ F ½ðmþÞ ð Þ 2R Dmþ p m ð9þ We assume FF T is non-singular, which is a persistency of excitation condition that requires Dp mþ þ m. Then we calculate the least squares estimate ^h of h given by To estimate the remaining components of ^c and ^h, we arrange the components of h into the matrix AðhÞ 2R mþdmþ p, and AðhÞ h T ½c jjþ h Š jj<p Then we calculate the singular value decomposition to obtain the estimates Að^hÞ USV T ^h Sð; ÞVð; :Þ ð22þ ð23þ ð24þ ð25þ ^ Uð; :Þ ð26þ the scalar 6 0 selects the normalization constraint. Finally, we extract the non-linearity coefficients ^c i from ^. Specifically, ^c is given directly by ^ðþ, while the remaining coefficients are calculated using least squares estimation and ^h Multi-dimensional SVD First, we define the tensors A 0 2 R, A j 2 j i Rnþ ^h ðff T Þ z ð20þ A 0 ðhþ c 0 ð27þ Next we develop several methods for obtaining estimates ^c and ^h based on ^h. Note that an arbitrary scaling and its reciprocal can be applied to the linear system and the output non-linearity. We remove this ambiguity by introducing a normalization constraint, thereby selecting a single system from a class of equivalent systems. We can normalize ^c and ^h by setting ^c i a, ^h i a, kck a, khk a or various other constraints. A ðhþ c h h A A 2 ðhþ c 2 h h 2 ih A2 A 3 ðhþ c 3 h h h 3 ih A3 A p ðhþ c p p i h p i h A p ð28þ ð29þ ð30þ ð3þ 3.. Direct solve We have m þ p þ 2 unknown parameters in h and c, one normalization constraint and D mþ p equations in terms of ^h. We can normalize as above, then choose m þ p þ independent equations. These m þ p þ equations must also be independent of the constraint equation, which constitutes equation number m þ p þ 2. A symbolic manipulator such as Mathematica can be used to invert these non-linear equations and obtain estimates ^h and ^c of h and c SVD To begin, ^c 0 can be estimated directly by h An c =n n h ð32þ We use a multi-dimensional singular value decomposition (Andersson and Bro 2000) to obtain the estimate ^h Ai of h Ai. To do this we note that ½h A h A2 h Ap Šhd T ð33þ d ½c c =2 2 c =p p Š T ð34þ Hence we compute the singular value decomposition ½ ^h Ai ^h Ap ŠUSV T ð35þ ^c 0 ^hðþ ð2þ to obtain the estimates

5 Identification of FIR Wiener systems 503 ^h Sð; ÞUð:; Þ ^d Vð:; Þ ð37þ ^c i ^d i i 6 0 selects the normalization constraint. ð36þ 2 pe 2i!j! 2 ^h 2i! X j! 0 ^h X ^h jjj! j ^h e j ðzðkþ ^zðkþþ vðkþ vðkþ ð46þ ^h Xp! vðkþ^c jj j ^h e j A 3.4. Prediction error cost function Consider the prediction error cost function J pe ð^c; ^hþ kz ^zk ð39þ ^z is the response of the estimated system, that is! ^zðkþ Xp X m ^c i ^h j uðk jþ ð40þ j0 2 ^h ^h j 2 Xp! ^c jj i vðkþ " ð j ij Þ ^h e i e j ðzðkþ ^zðkþþ ^h e i ð47þ Hence J 2 peð^c; ^hþ X zðkþ Xp The derivatives of @ ^h i! vðkþ^c jj ^h A 2 ð4þ X jj! vðkþ ^h i! X ^h! vðkþ ð42þ Xp! vðkþ^c jj i ^h e i ð43þ e i is the ith column of I mþ. Hence 2 i ðzðkþ ^zðkþþ X i!! vðkþ ^h A and ^h 2i! 2 ^h i 2 X 2 Xp ðzðkþ ^zðkþþ@ vðkþ ðzðkþ ^zðkþþ 0! ^c jj i ^h e i Xp ð44þ! vðkþ^c jj i ^h e ia vðkþ ðzðkþ ^zðkþþ Xp #! vðkþ^c jj j ^h e j ( ij if i j 0 else ð48þ ð49þ Using the above expressions, we implement a gradient-based optimization algorithm to minimize J pe and obtain estimates ^c and ^h. Assuming c 6 0, we normalize by letting ^c, and thus remove it from the optimization problem. 4. Example Let m, p 3, , h ½2 Š T, c ½ 20 0 Š T, and NðyÞ 20 þ y þ 0y y 3, which is shown in figure 2. Thus yðkþ h 0 uðkþþh uðk Þ zðkþ c 0 þ c h uðk Þþc h 0 uðkþþc 2 h 2 uðk Þ 2 þ 2c 2 h 0 h uðkþuðk Þþc 2 h 2 0uðkÞ 2 þ c 3 h 3 uðk Þ 3 þ 3c 3 h h 2 0uðkÞuðk Þ 2 þ 3c 3 h 2 0h uðkþ 2 uðk Þþc 3 h 3 0uðkÞ 3 þ wðkþ ð50þ ð45þ ðkþ T h þ wðkþ ð5þ The second derivatives are given by

6 504 S. L. Lacy and D. S. Bernstein ^c 0 ^hðþ; ^c ; ^c 2 ^hð4þ ^hð2þ 2 ð59þ ^c 3 ^hð7þ ^hð2þ 3 ; ^h 0 hð2þ; ^h ^hð3þ ð60þ Figure 3(a) shows ^h for 00 simulations, each having a different realization of the input and noise sequences. Figure 3(b) shows the identified non-linearity. h ½c jj h Š jj p ðkþ ½c 0 c h c h 0 c 2 h 2 c 2 h 0 h c 2 h 2 0 c 3 h 3 c 3 h 0 h 2 c 3 h 2 0h c 3 h 3 0 Š! vðkþ jj p ½ uðk Þ uðkþ uðk Þ 2 2uðkÞuðk Þ uðkþ 2 uðk Þ 3 ð52þ 3uðkÞuðk Þ 2 3uðkÞ 2 uðk Þ uðkþ 3 Š T ð53þ and wðkþ is a realization of a zero-mean Gaussian process such that the signal to noise ratio S=N Then (7) is replaced by z ½zð2Þ zð ÞŠ T F ½ð2Þ ð ÞŠ Figure 2. kz wk kwk NðyÞ. z F T h þ w 0 w ½wðm þ Þ wð ÞŠ T ½wð2Þ wð ÞŠ T ð54þ ð55þ ð56þ ð57þ ð58þ We assume FF T is non-singular, and estimate the parameter vector using (20). 4.. Direct solve We have m þ p þ 2 6 unknown parameters, one normalization constraint, and D mþ p 0 equations. We choose to normalize by setting ^c, and solve m þ p þ 5 equations. Using ^hðþ, ^hð2þ, ^hð3þ, ^hð4þ, ^hð7þ and ignoring the rest of ^h we obtain 4.2. SVD Here we arrange the components of ^h into a matrix that is also an outer product of h and. Then we compute the singular value decomposition of this matrix to find ^h and ^. Finally, we extract ^c from ^. First, ^c 0 can be estimated directly as in (2). To find the remaining parameters, we arrange the components of ^h as AðhÞ h T ½c jjþ h Š jj<p h T c hð3þ hð2þ c 2 h hð5þ hð4þ c 2 h 0 hð6þ hð5þ c 3 h 2 ½h 0 h Š ð6þ hð8þ hð7þ 6 4 c 3 h 0 h hð9þ hð8þ 7 5 c 3 h 2 0 hð0þ hð9þ We calculate the singular value decomposition of Að^hÞ as in (24). Then we obtain ^h and ^ from (25) and (26). Next, we extract the non-linearity coefficients c i from ^. ^c is given directly by ^ðþ, and we calculate the remaining c i using least squares estimation. In this case the normalization constraint is selected by the choice of the scalar. We choose to normalize by setting Uð; Þ such that ^c ^ðþ. In figure 3(c) we plot ^h for 00 simulations. In figure 3(d) we plot the identified non-linearity Multi-dimensional SVD We arrange the elements of h into several matrices as in (27) (3), corresponding to c i and the ith power of h A 0 c 0 hðþ " # " # hð3þ A c h c h 0 h hð2þ h A " # h 0 A 2 c 2 h h c 2 ½h 0 h Š h " # hð6þ hð5þ h A2 h A2 hð5þ hð4þ ð62þ ð63þ ð64þ

7 Identification of FIR Wiener systems 505 Figure 3. Simulation results.

8 506 S. L. Lacy and D. S. Bernstein A 3 c 3 h h h h A3 h A3 h A hð0þ h 3 0 h 2 0h h 2 c 0h h 0 h 2 3 hð9þ 6 4 h 2 0h h 0 h hð9þ h 0 h 2 h 3 4 hð8þ 3 hð9þ hð8þ hð8þ 7 5 hð7þ ð65þ We use a multi-dimensional singular value decomposition (Andersson and Bro 2000) to estimate the scaled Markov vector h Ai. Next, we arrange these results in a matrix and compute the singular value decomposition as in (35) to estimate ^h, ^d, and ^c, see (36) (38). In this case the normalization constraint is enforced by how we choose. We choose to normalize by setting V A ð; Þ resulting in ^c. In figure 3(e) we plot ^h for 00 simulations. In figure 3(f) we plot the identified non-linearity Prediction error cost function We minimize the standard prediction error cost function, J pe ð^c; ^hþ kz ^zk, initialized with one of the three algorithms discussed earlier. Using the derivatives given previously we obtain ^c and ^h. Although the results of the optimization are generally independent of the method used to obtain the initial estimate, the number of iterations needed by the optimization routine to converge usually depends on the accuracy of the initial estimate. We used lsqnonlin in the Matlab optimization toolbox to minimize the function. In figure 3(g) we plot ^h for 00 simulations. In figure 3(h) we plot the identified non-linearity. Due to noise effects, the value of the cost function evaluated at the optimal parameters is generally less than the cost function evaluated at the true parameters J pe ð^c ; ^h Þ < J pe ðc; hþ Discussion The four methods presented, namely direct algebraic solution, singular value decomposition, multi-dimensional singular value decomposition and predictionerror minimization, produced solutions of varying accuracy. Solving some of the equations for the unknown parameters generally produced poor quality estimates, as compared to the other three. Since the direct algebraic solution method uses only a fraction of the entries of ^h to compute the estimates, it is less robust than the other three methods. In addition, the direct algebraic solution method requires user interaction to select which equations to solve. The singular value decomposition approch produced estimates that were close to the ones obtained by the prediction error minimization, but relied only on a singular value decomposition and some subsequent least squares steps. This method is the simplest to implement both for the user and numerically. The multi-dimensional singular value decomposition approach produced estimates comparable to the first singular value decomposition approach, but the multidimensional singular value decomposition is more complex to implement than the two-dimensional singular value decomposition approach. The prediction-error minimization approach is the most complex to implement. For best results, it should be initialized using one of the previous methods. However, it produced the best estimates of both the linear dynamics and the output non-linearity. 5. Conclusion This paper presented four methods for identifying FIR Wiener systems with polynomial non-linearities. We presented three methods for simultaneous direct estimation of the non-linearity and linear dynamics, and a prediction error optimization method. Many authors have studied Wiener system identification under the assumption that the non-linearity is unknown but one-to-one (Brillinger 970, Pajunen 985, Hasiewicz 987, Greblicki 992, 994, 997, Westwick and Kearney 992, Wigren 994, Westwick and Verhaegen 996, Bai 998, Lovera et al. 2000). Other methods for Wiener system identification require the non-linearity to be known, invertible, monotonic, odd, even, or require the use of specially designed input sequences. In this paper we require the non-linearity to be polynomial and the linear dynamics to have finite impulse response. These two assumptions are practical in that many non-linearities can be approximated with polynomials, and that many systems with infinite impulse response can be approximated with finite impulse response dynamics. Future work will focus on extending the method in three directions: first, the identification of sandwich nonlinear systems, i.e. systems with both input and output non-linearities; second, the identification of IIR Wiener systems; and finally, identification of Wiener systems with non-polynomial output non-linearities. While the identification of sandwich non-linear systems using this approach seems tractable, overcoming the FIR and polynomial assumptions on the linear dynamics and output non-linearity appears to be more challenging. Acknowledgements Supported in part by NASA under GSRP NGT and AFOSR under grant F and laboratory research initiative 00VS7COR.

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