Nonlinear Active Noise Control Using NARX Model Structure Selection

Transcription

1 2009 Aerican Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 10-12, 2009 FrC13.6 Nonlinear Active Noise Control Using NARX Model Structure Selection R. Naoli and L. Piroddi, Meber, IEEE Abstract In this work a novel Nonlinear Active Noise Control (NANC) schee is roosed to deal with reference and error icrohone saturation issues. Polynoial NARX odels are used in the adative controller for enhanced odel flexibility. A suitable odel selection aroach is used off-line to find an accurate and coact odel structure for the adative controller. The controller araeters are successively udated with an aroriate adative algorith based on the error gradient and on the residual noise. Soe siulation exerients are rovided to show the effectiveness of the roosed aroach coared to existing alternatives. R I. INTRODUCTION ECENT algorithic and technologic advances in digital signal rocessing have fostered the research in the area of Active Noise Control (ANC), resulting in several successful alications and theoretical develoents [1]. Briefly, ANC exloits the rincile of destructive interference to counteract an offending noise traveling on a riary acoustic ath using a secondary acoustic source suitably governed by a controller. Both broad- and narrow-band, singleand ulti-channel control aroaches are docuented in the literature, and various odel identification and adative algoriths have been roosed in this context using different tye of filters [2]. Most of the studies resented in the literature are concerned with linear ANC, although there are several sources of nonlinearity that can affect the syste. For exale, the noise affecting the syste ay be a nonlinear deterinistic, ossibly chaotic, rocess [3]. The involved acoustic aths ay also dislay a nonlinear behavior. For exale, a coon source of nonlinearity is due to high sound ressure causing distortion and saturation in soe of the devices that ake u the control equient [4]. To effectively deal with these effects, the odels and algoriths eloyed in ANC ust be roerly extended and rearranged. The ter NANC (nonlinear ANC) collectively refers to adative nonlinear control schees that address these issues [5]. Various tyes of nonlinear filters have been studied in the NANC literature, such as truncated Volterra exansions [6], [7], radial basis functions [6], ulti-layer artificial neural networks (MLANN) [4], [8], functional link artificial neural Manuscrit received Seteber 15, R. Naoli is with Esion s.r.l, 23900, Lecco, Italy (e-ail: berto.naoli@gail.co) L. Piroddi is with the Diartiento di Elettronica e Inforazione, Politecnico di Milano, 20133, Milano, Italy (hone: ; fax: ; e-ail: iroddi@elet.olii.it). networks (FLANN) using trigonoetric functional exansions [9] or iecewise linear functional exansions [10], adative bilinear filters [11], and general function exansion nonlinear filters [5]. The first issue with these filters is the develoent of suitable adatation echaniss able to deal with nonlinear odels, since it is easy to show that classical ethods such as the Filtered-X Least Mean Squares (FXLMS) aroach [2] are not alicable to the nonlinear case without roer odifications. An even ore critical issue is the odel size which is tyically high in the nonlinear case. This greatly affects both the coutational load and the eory requireents of the adative schee. Also, if large over-araeterized odels are eloyed, in order to guarantee sufficient odel flexibility, several known unwanted robles ay arise, such as overfitting, araeter fluctuation and even odel instability [12], [13]. This aer exlores the use of a different nonlinear odel class, naely the olynoial Nonlinear AutoRegressive odels with exogenous variables (NARX), which is well known for its flexibility and reresentation caabilities [14], [15]. Also, the linear-in-the-araeters structure allows the use of sile identification algoriths of the Least Squares (LS) faily. What atters ost for the NANC alication is that several algoriths have been develoed for the identification of NARX odels which are greatly concerned with the selection of a suitable and coact odel structure (see, e.g., [16], [17], [13]). A NANC schee is here roosed based on the off-line structure identification of a NARX odel and the on-line adative tuning of its araeters. The case of saturation nonlinearities acting both at the reference and the error icrohone (see, e.g., [11], [18]), is here considered, as reresented in Figure 1. x(k) x (k) P C d(k) y(k) S y (k) e(k) ē(k) Fig. 1. Block diagra of a NANC syste in the resence of saturated reference and error signals. P and S denote the riary and secondary ath, resectively, and C the controller. II. NONLINEAR ACTIVE NOISE CONTROL The use of a nonlinear controller has been roved to outerfor linear ANC solutions if either the secondary ath /09/$ AACC 5616

2 is non-iniu hase or the riary ath is nonlinear [6], [7]. The cited works eloy a 2 nd order truncated Volterra series for the controller structure: L1 y(k) = w1 ( 1 ; k)x(k 1 ) 1 =0 L1 L1 w2 ( 1, 2 ; k) x(k 1 ) x(k 2 ) (1) 1 =0 2 = 1 where y(k) reresents the outut of the Volterra filter, and w 1 ( 1 ; k) and w 2 ( 1, 2 ; k) denote the (tie-varying) coefficients of the linear and quadratic kernel, resectively. Notice that, since odel (1) has a nonlinear Finite Iulse Resonse (FIR) structure, it adits only one fixedoint, so that it is not suitable for odeling nonlinear systes with a ore colex static behavior. Also, due to odel size exlosion, ractical alications are liited to odels of the 2 nd order such as (1). Even so, since FIR odels generally require a long window L of ast ters to rovide sufficient accuracy, this odel tyically has a large nuber of araeters to estiate (actually, it requires L 2 3L 1 ters). This greatly affects both the araeter estiation confidence and the robustness of the odel, in ters of its generalization caabilities. Finally, as already entioned, over-araeterization can be resonsible for several unwanted dynaic effects [12], [13]. On the ositive side, odel (1) is a linear-in-the-araeters odel, so that classical adatation algoriths used for FIR filters can still be alied with inor odifications. The so called Volterra FXLMS (VFXLMS) algorith is used for odel (1). This algorith is based on a ulti-channel structure for feedforward active noise control. The reference signal x(k) is searated in its linear and quadratic coonents, which are then fed to distinct controller blocks whose oututs are added to coose the overall control signal y(k). Assuing that the secondary ath is linear the FXLMS can be alied correctly as long as each coonent of x(k) is indeendently filtered. In a siilar fraework, [9] roosed the use of FLANN filters using trigonoetric bases, which can be udated using a Filtered-S Least Mean Squares (FSLMS) algorith which exloits the articular odel structure. An efficient filter-bank adatation schee in the frequency doain was also roosed for these nonlinear filters by the sae research grou [19]. Other tyes of eleentary functions have been used with FLANN filters, such as iecewiselinear exansions [10]. A filtered-x affine rojection algorith is also introduced in [10] for both the adative Volterra filter and the FLANN structures. Infinite-iulse resonse (IIR) tye filters such as the outut-error bilinear filter L y(k) = i=0 L ai (k)x(ki) bj (k)y(kj) L L ci,j (k)x(ki)y(kj) (2) i=0 have been suggested [11] as an efficient alternative to Volterra odels. A tye of FXLMS aroach is adoted also in this case, which assues a linear secondary ath and filters the gradient of the controller s outut with resect to its weights. Although in ters of L odels (1) and (2) have the sae nuber of araeters, it is shown in [11] that the bilinear odel aroach can achieve erforances coarable to Volterra filters by using a shorter filter length L, i.e. less araeters. Few solutions have been roosed in the literature so far for the ore general NANC roble with nonlinear secondary ath. In [4], [8], MLANN are used for this urose. More recently, [5] develoed an adative control algorith for general function exansion nonlinear filters that works for both linear and nonlinear secondary aths. This algorith is based on the so called virtual secondary ath filter, which is related to the gradient of the secondary ath. Notice, however, that the odel class eloyed in [5] has still a substantially FIR structure, since the controller s outut is actually constructed as a linear cobination of nonlinear functions of the inut reference signal. As such, it still suffers fro the odel size roble. III. NARX MODELS Besides the different reresentation caabilities of the odels used in the aroaches described in Section II, the ain liitation of those ethods lies in the fact that they do not rovide any for of structure otiization, so that largely over-araeterized odels are invariantly used, with all the robles that this can cause. In this section, we introduce a different class of nonlinear odels, naely the olynoial NARX odels, which include both Volterra and bilinear odels, and for which it is ossible to devise a odel selection schee that generally yields sufficiently coact odels to be efficiently used in a NANC syste. Recursive inut-outut odels are widely used in blackbox nonlinear odel identification for their flexibility and reresentation caabilities (see, e.g., [20] for a corehensive review). In articular, NARX odels [14], [15] have been extensively eloyed in alications. The deterinistic NARX odel is an inut-outut recursive odel where the current outut is given by: y(k) = f(y(k1),..., y(kn y ), x(k1),..., x(kn x )), (3) where f( ) is a generic nonlinear function, x( ) and y( ) denote the inut and outut signals, resectively, and n x and n y are the axiu inut and outut lags. Several tyes of functional exansions can be used to describe f( ), one of the ost coon being the olynoial functional exansion y(k) = l ny =0 =0 n 1 =1 n =1 nu c, (n 1,...,n ) y(kni ) x(knh ), (4) where l is the axiu degree of the olynoial exansion. Exression (4) is linear with resect to the 5617

3 araeters to be estiated, so that Least Squares (LS) tye algoriths can be used for odel identification. Paraeter identification is, however, the easiest art of the roble, odel selection being generally the ain concern, since the nuber of ters in the full exansion (4) grows raidly with l, n x and n y, and coarably few ters are generally needed for successful odeling. The odel size should be ket sall, for odel robustness and araeter estiation confidence, and to avoid the unwanted dynaical consequences of odel redundancy [12], [13]. One of the ost oular identification algoriths for NARX odels is the Forward Regression Orthogonal Estiator (FROE) [16], which is based on the rediction error iniization aroach and uses Orthogonal LS (OLS) to estiate the araeters, exloiting the orthogonalization to decoule the regressors. Briefly, at each iteration of the FROE the regressor that enhances ost the rediction erforance of the current odel is included in it. An alternative to the OLS aroach is given by the Fast Recursive Algorith (FRA) develoed in [17], which is also based on the PEM aroach, but solves the LS roble recursively over the odel order without requiring atrix decoosition and transforation. This algorith is notable for the reduced colexity and the iroved nuerical stability it achieves with resect to the FROE. This akes it a good candidate for use in a NANC schee. Several iortant drawbacks of the PEM aroach for NARX identification have been discussed in the literature (see, e.g., [12], [13]), and the siulation error iniization (SEM) aroach has been ut forward for iroved structure selection and odel robustness. On this line, [13] introduces the SEMP (SEM with Pruning) algorith, which oerates iteratively as the FROE, but rates candidate odels based on their siulation erforance. The iterative rocedure includes a runing rocedure to avoid odel redundancy. Notice that, since odel siulation can only be couted iteratively, this aroach is coutationally intensive, although araeter identification is still erfored with a LS aroach for silicity. Coutational colexity can be soewhat reduced using cluster selection [21]. In view of the NANC alication where the controller s araeters are continually adjusted, allowing for a artial coensation of an irecise initial odel estiation, it is conjectured that the PEM-based NARX identification aroaches could suffice for this tye of alication. Here, the SEMP ethod has been eloyed to rovide an accurate reference odel for coarison uroses. Reark that the envisaged odel selection techniques are batch algoriths, so that to erfor this task, a suitable set of data ust be collected for the odel selection rocedure to rocess off-line. IV. METHODOLOGY DESCRIPTION The use of otiized NARX odels can greatly enhance NANC alications. The ain idea is to eloy odel selection in a first off-line stage to find a coact odel structure for the NARX controller and then adat its araeters on-line. With resect to other NANC aroaches, this draws ost of the coutational load away fro the adatation task, which is critical since it has to work on-line, and transfers it to the odel selection task, which can be erfored off-line. The reduced colexity of the on-line adatation is obviously beneficial in ters of hardware requireents and overall syste erforance, but it also allows a ore flexible use of the odel s degrees of freedo that is generally exected to enhance its accuracy and robustness. A. Model selection To estiate the NARX odel of the controller with any of the odel selection rocedures exlained in the revious section, we need to forulate the controller s odel identification roble as a linear regression. This would be trivial if the desired controller outut was available, since then we could use directly the linear regression (4). Unfortunately, the error is directly easurable at the outut of the secondary ath and not at the controller s outut. To estiate the desired controller s outut, a odel inversion technique ay be used. Notice that while the use of an inverse odel of the secondary ath is generally avoided in ANC systes to revent the insertion of additional delays in the control chain, here the role of the inverse odel is only functional to the off-line odel structure selection of the controller and is not eant to interfere with the on-line execution of the adative ANC algorith. The odel selection task is based on the schee deicted in Figure 2, which uses an inverse odel of the secondary ath to estiate y(k) fro y (k). x (k) C y (k) y(k) e y (k) S S^1 d(k) y (k) e(k) Fig. 2. Reconstruction of signal y(k) for use in the odel selection hase. S denotes the secondary ath, C the controller, S^1 the estiated inverse of the secondary ath, and y (k) the reconstructed target controller outut. A general inverse identification schee (see Figure 3) can be alied, where the inverse odel is fed with the error signal, while the secondary ath is excited with a suitable artificially generated signal y(k), uncorrelated with d(k). Details on the identification algorith deend on the articular odel structure chosen for the inverse. The FROE or FRA algoriths ight be used in the NARX case. Using this inverse odel to obtain the reconstructed target controller outut y (k), a dataset of {x ( ), y (k)} sales ay be used for the controller s odel selection urose, directly referring to the linear regression (4). Notice that odel inversion is not always alicable, so that the suggested odel identification schee could turn 5618

4 out to be a critical oint of the overall ethod. Recall, however, that accurate inversion is only required in the disturbance frequency range, which can be quite circuscribed, esecially in narrowband control alications. Observe also that, as will be seen in the siulation exerients section, the on-line adative algorith that tunes the controller weights can artially coensate errors due to aroxiate inversion. y(k) S d(k) y (k) 1 S^1 y^(k) Fig. 3. Model identification schee for the estiation of the inverse secondary ath odel. S denotes the secondary ath, S^1 the estiated inverse of the secondary ath. B. On-line adatation of the controller s araeters At the end of the odel selection hase, the controller structure is fixed and an initial araeterization is available. At this stage an adative algorith ust be setu for continuous udating of the odel araeters, to deal with unodeled dynaics and tie-variant henoena. The algorith discussed in the following is an adatation to the NARX case of that eloyed in [5]. The objective of the adative algorith is to iniize the instantaneous squared error using the steeest descent algorith: ϑ(k1) = ϑ(k) µ 2 (e(k) 2 ) T ϑ ϑ(k) T = ϑ(k) µ e(k). (5) ϑ ϑ(k) Assuing a deterinistic NARX structure for both the controller (see equations (3-4)) and the secondary ath: y (k) = f(y (k1),..., y (kn y ), y(k1),..., y(k y )) l ny' = =0=0 n 1 =1 n =1 y c, (n 1,...,n ) y (kni ) y(knh ), the gradient ter / ϑ can be iteratively couted by eans of the recursive nonlinear filter: ny y ϑ (k) = y (kj) y y ϑ (kj) y(kj) y ϑ(kj). (6) where y ϑ (k) = ϑ(k), y ϑ(k) = y(k), and the aroxiations ϑ(k) y (kj) y (kj) y(kj) and ϑ(k) ϑ(kj) ϑ(k) y(kj) have been used, ϑ(kj) assuing that the ste size is sufficiently sall to yield slow convergence [2]. The derivatives of y (k) with resect to ters y (kj), j = 1,, n y, and y(kj), j = 1,, y, aearing in exression (6) can be couted as follows: y (kn ) = l ny n nu c, (n 1,,n,,n ) =0=0 n 1 =1 n j =n n =1 i j y(kn ) = l =0=0 y (kn i ) y(knh ), ny j=1n 1 =1 n j =n n nu c, (n 1,,n,,n ) n =1 y (kni ) y(kn h ). h j Finally, the derivative y ϑ (k) of y(k) with resect to a generic controller weight ϑ (k) = c, (n 1,...,n ; k) can be couted recursively using the following nonlinear dynaic filter: y ϑ (k) = y(kn i) h= 1 [yϑ (kn j ) l ny x(kn h) i j =0=0 n 1 =1 nu c, (n 1,...,n ) n =1 y(kn i )] x(knh ), (7) eloying once again the aroxiation y(kj) ϑ(k) y(kj) ϑ(kj). Recursions (6) and (7) are articularly costly for on-line ileentation, and soe silification is highly desired. A tyical aroxiation in ANC systes is Feintuch s assution [2], [11]. This consists in neglecting the recursion based on the old outut gradients, on the grounds that all derivatives of ast oututs with resect to current weights are zero. With this silification, exression (6) silifies to y y ϑ (k) = y(kj) y ϑ(kj), (8) and the dynaic filter (7) to y ϑ (k) = y(kn i) x(kn h) (9) h= 1 These ore anageable exressions have been used in the following with satisfactory results. The develoed adatation algorith is denoted Nonlinear Filtered-Gradient LMS (NFGLMS), since it involves filtering the gradient of the controller outut with resect to its weights, instead of the reference signal. The NFGLMS algorith is related to the FXLMS-based NANC algorith resented in [5], in the way it eloys the gradient inforation in the udating rocess, but the different assutions of the two ethods lead to different solutions. Moreover, in [5] the idea is to kee a structure siilar to the one of FXLMS, and the function exansion block is introduced to avoid the robles related to the use of the FXLMS algorith for nonlinear systes. On the contrary, the NFGLMS concentrates all nonlinearities in the adative filter. The roosed algorith can also be envisaged as an extension of the outut-error bilinear filter aroach [11], in ters of the odel structure and the filtering oerations concerning the secondary ath. 5619

5 C. Coents on the algorith s colexity and usage The colexity of the the on-line adative hase scales with the nuber of odel ters to be adated, that is generally reduced to a iniu thanks to the selection rocess, so that the on-line adatation and control effort is uch lower than with the Volterra and bilinear odel-based algoriths. As for odel selection, a full analysis of the coutational colexity of the FRA is rovided in [17]. Model selection gets increasingly intensive as the set of candidate ters grows larger, so that there is a trade-off between odel flexibility and coutational load. It is however iortant to note that such a stage is unavoidable for robust nonlinear identification to avoid the well-known drawbacks of over-araeterization. Besides, the active noise controller need not be idle during odel selection. On the contrary, the last couted odel structure should be eloyed for on-line adatation and control until a new odel structure is available, and odel selection should be eriodically reeated. The rocess could be started, e.g., using a sile linear FIR odel and FXLMS. V. SIMULATION EXPERIMENTS In the following tests a ulti-haronic reference signal x(k), su of three sine waves at the noralized frequencies of 0.02, 0.04, and 0.08, is eloyed, with a saling frequency of 8000 Hz. The riary and secondary aths are odeled by two linear FIR filters (taken fro [22]) with coefficients of the iulse resonse given by [ ] and [ ], resectively. The nonlinearity consists in the saturation of the reference and/or the error icrohone. Two levels of saturations are considered for the reference signal, naely weak and strong, obtained by setting the cliing threshold at 90% and 50% of the axiu signal value, resectively [11]. The error signal is either non saturated or weakly saturated with a 50% threshold [18]. Overall, three settings have been studied: WU) weakly saturated reference, unsaturated error, SU) strongly saturated reference, unsaturated error, SW) strongly saturated reference, weakly saturated error. The ste size, or convergence factor, µ is chosen in order to guarantee stability and to ensure, at the sae tie, a sufficiently raid convergence. Finally, erfect knowledge of the secondary ath is assued, i.e. S^(z) = S(z), and the adative coefficients of the controller are initialized to zero. A. Test 1: setting WU The first test concerns setting WU. An inverse FIR filter of length L inv = 8 is used for the reconstruction of signal y(k), that is estiated using LMS with a ste size µ inv = 0.1, while an adative NARX filter of order L = 8 including quadratic ters is identified with the FRA algorith for 8 iterations. Finally, the weight udate is erfored with NFGLMS with a convergence factor of µ = 0.1. Aarently the odel selection hase (see Figure 4) is caable of achieving significant noise reduction by itself (10 4 error alitude), but the NFGLMS (after a brief transient) can still irove the erforance by several orders of agnitude (10 8 error alitude after sales). The 8-araeter odel identified by the FRA algorith is reorted in Table I x Fig. 4. Setting WU, residual noise with the roosed NANC schee, after odel selection (blue line) and during weight adatation (red line). B. Test 2: setting SU Setting SU has been tested in the sae exeriental conditions. Although the error transient in the weight udate hase is quite different to that of case WU (see Figure 5), convergence is finally achieved (10 6 error alitude after sales). As for the controller s odel structure (see Table I), soe siilarities ay be noticed with the odel obtained with setting WU. In articular, the first (and ore iortant) selected ters are coon to both odels. Observe that these ters are linear, and ay actually be exlained as necessary for the descrition of the syste itself (the first and secondary aths, the noise), while saturation effects are odeled by the nonlinear ters x Fig. 5. Setting SU, residual noise with the roosed NANC schee, after odel selection (blue line) and during weight adatation (red line). C. Test 3: setting SW Setting SW was tested, using LMS for the identification of the inverse with µ inv = 0.1, and NFGLMS with µ = The odel identified by the FRA algorith is reorted in Table I. Both the odel inversion and the weight udate are 5620

6 affected by error saturation since they heavily rely on the inforation conveyed by the error signal. The residual behavior (see Figure 6) shows that significant noise reduction can be achieved even in this case and that the NFGLMS can irove the results x Fig. 6. Setting SW, residual noise with the roosed NANC schee, after odel selection (blue line) and during weight adatation (red line). TABLE I FILTER MODELS ESTIMATED IN THE 3 TESTS (IN THE OFF-LINE PHASE) Test 1 Test 2 Test 3 Regressors Par.s Regressors Par.s Regressors Par.s y(k5) y(k5) y(k5) u(k4) u(k4) u(k4) u(k1) u(k1) u(k1) y(k3)u(k8) u(k3)u(k4) u(k3)u(k4) y(k8)u(k3) y(k7)u(k4) y(k2)u(k4) y(k4)y(k6) u(k3) u(k3) u(k6)u(k8) y(k6)u(k7) y(k1)u(k3) y(k8)u(k1) y(k5)y(k8) y(k6)u(k7) VI. CONCLUSIONS The roble of the resence of nonlinearities in real ANC systes has been faced focusing on the effects of saturation of the reference and error icrohones. A novel NANC schee has been roosed that generalizes the Volterra and bilinear filter-based FXLMS ethods, and overcoes soe liitations of those ethods. The ain idea is to identify a coact NARX odel of the noise canceling controller, using well known odel selection algoriths, suitably readated to the ANC context. A filter gradient based LMS algorith is then eloyed for adative araeter tuning. Such algorith has also the caability to artially coensate for odel structure errors left out by the revious stage. The roosed ethod has been tested on soe realistic scenarios siulating saturation of the reference and error icrohones, and has rovided interesting results that outerfor reviously roosed techniques. Coared to these techniques, the resented one allows soe additional degrees of freedo concerning the odel selection schee itself and its integration with the araeter udate algorith, that ay be tailored to the syste s characteristics and exected erforance. REFERENCES [1] S. J. 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