Spectral Observer with Reduced Information Demand

Transcription

1 Spectral Observer with Reduced Iformatio Demad Gy. Orosz, L. Sujbert, G. Péceli Departmet of Measuremet ad Iformatio Systems, Budapest Uiversity of Techology ad Ecoomics Magyar tudósok krt. 2., -52 Budapest, ugary Phoe: , fax: , Keywords: adaptive sigal processig, spectral observer, wireless sesor etwork, etworked sigal processig. Itroductio The sig error observer algorithm itroduced i the paper is based o the relatioship betwee the least mea square (LMS) [] ad resoator based observer algorithms [2][3], but utilizes the sig error LMS algorithm [4] for estimatig the state variables of the observed system. Sice the algorithm uses the sigum of the error of the estimatio, sigificat reductio i the computatioal demad ad i the amout of required data ca be achieved. Possible applicatios of the observer are the Fourier decompositio of sigals ad adaptive cotrol. The advatage of this structure is that it requires reduced amout of data ad lower computatio demad tha the origial oe, hece the utilizatio of this algorithm reduces desig restrictios i systems with limited resources (e.g. badwidth of commuicatio chaels). A applicatio example is a active oise cotrol (ANC) system [8] that uses wireless sesor etwork (WSN) for oise sesig [6]. This is a straightforward field for the deploymet of this algorithm, sice ANC systems require lots of sesors ad relatively high samplig frequecy compared to the typical badwidth of the WSN's radio stadards (e.g. ZigBee), so data reductio plays importat role. The paper is structured as follows. Sectio 2 summarizes the operatio of the origial spectral observer ad itroduces the sig error observer. A method is also described that improves the properties of the sig error structure. I Sectio 3 the siged error observer based active oise cotroller is preseted. 2. Proposed approach 2.. Review of the traditioal resoator based observer structure The resoator based observer was desiged to follow the state variables of the so-called coceptual sigal model [2]. The sigal model is described as follows: x + = x ; x = [x i, ] T () c = [c i, ]; c i, = y = c x = L c i, xi, i= L j ω j i e i ω = e (2), i= L L (3) where x is the state vector of the sigal model at time step, y is its output (the iput of the observer), c represets the basis fuctios. To geerate a real sigal ω -i =-ω i shall be satisfied. This restrictio is ot ecessary, but advatageous i most cases. Obviously, i these cases the

2 correspodig state variables shall form complex cojugate pairs. The coceptual sigal model ca be cosidered as a summed output of resoators which ca geerate ay multisie with compoets up to the half of the samplig frequecy. The correspodig observer is (Fig. ): x ˆ + = + g (y c ) = + g (y y ) = + g e ; g = [g i, ] T = [r i c* i, ] T (4) y + e g, g 2, g N, z- z- z- x ˆ, x ˆ 2, x ˆ N, c, c 2, c N, Figure. Basic cofiguratio of the resoator based observer y g i, z- Q i (z) c i, i-th resoator chael ˆ x i, where{ =[ x ˆ i, ] T ; i= N; N=2L+} is the estimated state vector, {r k ; k= N} are free parameters to set the poles of the system, ad * deotes the complex cojugate operator. N is the umber of harmoic compoets. Due to the complex expoetials, the chaels of the observer ca be cosidered as time-ivariat systems with a pole o the uit circle. This is why they are called resoators. If the resoator poles are arraged uiformly o the uit circle, ad {r k = / N ; k= N} g = / N c ( deotes the cojugate traspose), the observer has fiite impulse respose, ad the observer correspods to the recursive discrete Fourier trasform (RDFT) [2]. If the aligmet of the resoators is ot uiform, the settlig is o loger deadbeat, but the system is still stable. Due to the formal correspodece, (4) ca be iterpreted as the state variables would be updated by the complex LMS algorithm, where the referece sigal is c. Usig this relatioship betwee the observer ad LMS [3] i the proposed ew observer structure the sig error LMS (SE-LMS) algorithm is used for updatig the state variable The siged error observer structure The proposed sig error structure ca be see i Fig. 2. The update procedure is the followig: x ˆ + = + g sg(e ); g = [g k, ] T =[α c* k, ] T = α c where e = (y y ) is the error of the estimatio. sg(x)= x /x, i.e. sg(x)=+ if x>, if x < ad sg(x) = if x =. It meas that ν = i Fig. 2 i the case of this simple sig error observer. This updatig requires oly the kowledge of the sig of the error, so it eeds less computatio, ad the amout of data required for the operatio is reduced. This is advatageous if it is implemeted i systems with costraied resources. α is used for settig the trasiet ad steady state behavior of the observer. (5)

3 y + e ν ν e Q (z) Q 2 (z) Q N (z) y g i, z- ˆ Q i (z) x i, c i, i-th resoator chael Figure 2. Basic cofiguratio of the resoator based sig error observer The steady state error of the observer ca be determied by adaptig the results i [4] for this structure: E a ()= k= e 2 k + x N α 2α 2 where E a is the absolute mea error. (6) implies if (system is i steady state), the average absolute error is bouded by Nα/2 that is proportioal to the covergece parameter α. The settlig time M of the observer ca be estimated by the recursive expasio of (5): M - M = α c j sg( e j ) +. (7) j= Takig the absolute value, assumig that the iitial state ˆx = we get: M - j= M - M - α c sg( e j ) = j= j= x ˆ = α c sg( e ) α c = Mα N (8) M From (7) with the assumptio that estimatio of the settlig time is: j M x (the observer is i steady state at time istat M) the (6) x M (9) α N (6) ad (9) pose cotradictory coditios for the observer. The followig sectio itroduces the improved versio of the observer which esures fairly fast covergece with small steady state error The improved siged error observer structure I order to resolve the above metioed cotradictory coditios a adaptive tuig of the covergece parameter is proposed: β = α ν = α e m ; e m = [e m e m- e m-v+ ] T () where β is the ew covergece parameter. ν = e m, e m is a vector cosistig of the last V values of the error sigal at the time istat m whe β is modified. deotes the absolute orm. It ca be called ormalized siged error spectral observer. The updatig algorithm is the followig:

4 x ˆ + = + α c e m sg(e ) ; g =α e m c If the value of the error sigal is high the ν is also high, so the state variables are updated more radically (with larger steps), thus the covergece is faster. If the estimatio error is low the estimated ad real value of x are ear to each other is updated with lower modificatios so decreasig the error of the observatio. These facts mea that the utilizatio of the orm of the error improves the behavior of the sig error observer. The frequetly the parameter ν is calculated the faster the covergece is. If V =, the origial observer is got back. The optimal value of α i () ad () ca be calculated for the case whe resoators are aliged uiformly ad β is updated i each period of y. Let's deote the k-th period of the sigal by k. These coditios mea that V=N, ad m=kn i (), so e m =e kn =[e kn e kn-n+ ], sice for uiformly aliged resoators the legth of oe period of the sigal is N. I these circumstaces the observer algorithm miimizes e m 2, thus makes the power (i.e. mea square) of oe period of the error sigal miimal if the optimal α is utilized: NZ () α opt = (2) N N where N NZ is the umber of ozero elemets of e m. I practice this result ca be used as a iitial value whe the refreshig of the covergece factor is take place with other period or resoators are placed uevely. 3. Results Oe of the practical applicatios of the spectral observer is adaptive cotrol whe referece sigal is periodic [7], sice i this case the structure provides excellet behavior. The structure i Fig. 3 realizes a adaptive cotroller usig the resoator based observer. The plat to be cotrolled is A(z). The observer cotrols the output of A(z) to follow the periodic referece sigal y. Note that the algorithm ca also be used for costat y, sice this is a special case of periodic sigals ω=. The updatig procedure is the followig: x ˆ i, + = x ˆ i, + A (z i ) α c* i, e m sg(e ); g =[g i, ] T = α [A (z i ) c* i, ] T (3) where z i = exp(j ω i ) ad A (z i ) is the iverse of the trasfer fuctio o the i-th resoator frequecy. A (z i ) compesates the effect of A(z) i the feedback path, so it esures the uit egative feedback required for the stability [7]. A(z) should be idetified i advace. sesor A(z) Q (z) y y + e ν ν e Q 2 (z) Q N (z) y Figure 3. Adaptive cotrol structure

5 The utilizatio of this kid of sig error cotroller ca be advatageous whe the calculatio of the sig fuctio ad the orm of the error sigal ca be performed o the sesor. I this case either the referece sigal has to be kow at the sesor or the sesor has to measure directly the error sigal. The data to be trasmitted at each time istat for the cetral cotroller where the observer is implemeted is the sig of the error sigal that has oly three possible values {,,}, so sigificat reductio i the amout of trasmitted data ca be achieved. The orm of the error sigal has to be trasmitted oly i each V-th samplig istat. The prelimiary practical test of the sig error observer was carried out i a test applicatio [6] that is a wireless active oise cotrol (ANC) system [8]. The structure of the system ca be see i Fig 4. ANC systems are such cotrol systems i which the plat to be cotrolled A(z) i Fig. 3 is a acoustic plat, the iput of which is a loudspeaker, ad the referece sigal y is the oise to be suppressed. If the sigal y is predicted correctly, the remaiig oise e decreases, so reductio i the power of oise is achieved. The error sigal e evolves as the superpositio of the oise (y ) ad the iverse of the estimated oise ( y ) ad it is sesed by a microphoe. Sice superpositio of the soud waves is a sum of the sigals, so the subtractio ca be achieved with the multiplicatio of y by i the DSP. The microphoe is situated o a wireless sesor ode that samples the error sigal (i.e. remaiig oise), performs the calculatio of the sigum fuctio ad the orm of the error sigal ad seds the data through a gateway to a DSP that implemets the observer structure. The samplig frequecy of the error sigal is.8 kz. Noise source y y sesor radio commuicatio DSP sg(e ); e m gateway Figure 4. Cofiguratio of the wireless ANC system Due to the utilizatio of the ormalized sig error observer the amout of the data to be trasmitted from the sesor to the DSP was oe sixth tha that i the case of ormal observer. The reaso is that istead of the curret value of the sigal (that is coverted by a 8 bit AD) oly the sig of the error ad the absolute orm of the error i V=32 sample log itervals were trasmitted. Data reductio decreases the load of the radio etwork, so it makes possible either the expasio of the umber of sesors with the same samplig frequecy, or icreasig the samplig frequecy. Durig the test a exteral oise y with the fudametal frequecy of 95 z was radiated by a loudspeaker which had to be suppressed by the DSP. y cosisted of 5 harmoic compoets. Measuremet results of the realized system ca be see i Fig. 5. For each of the five harmoics at least 2 db suppressio was achieved i steady state, the settlig time of the error sigal was approximately sec. These are similar results as i the case whe the sesor trasmits the value with the 8 bit accuracy of the AD coverter. Naturally the ubiquitous exteral acoustic oise cotributes to the remaiig error.

6 -2 spectrum i steady state cotrol off cotrol o.2 trasiet fft(e ) [db] frequecy [z] e time [sec] Figure 5. Measuremet results of the improved sig error cotroller Coclusios ad future plas I this paper a sig error spectral observer structure was itroduced which ca be effectively used i sesor etworks where the badwidth of the commuicatio chael is limited. A method was also proposed for the adaptive tuig of the covergece factor i order to improve the trasiet ad steady state properties of the system. A structure that uses the ormalized sig error observer for cotrol applicatios was also preseted. The test applicatio was a wireless active oise cotrol system. Prelimiary results show that the algorithm ca be applied with success i such cotrol systems where the period of the tuig of the covergece factor ca be eglected compared to the trasiets of the system, or settlig time is out of iterest. I the future the expasio of the theory to MIMO systems ad their implemetatio ca be expected. Refereces [] Widrow, B., S.D. Stears, Adaptive Sigal Processig, Pretice all, Ic [2] Péceli, G., A commo structure for recursive discrete trasforms, IEEE Tras. Circuits Syst. Vol. CAS-33, pp.35-36, Oct [3] Widrow, B., P. Baudreghie, M. Vetterli, P. Titcheer, Fudametal relatios betwee the LMS algorithm ad the DFT, IEEE Tras. o Circuits ad Syst., Vol. 34, Jul pp [4] Gersho, A., Adaptive filterig with biary reiforcemet, IEEE Trasactios o Iformatio Theory, Vol. 3, Mar. 984, pp [5] Bucklew, J.A., Kurtz, T.G., Sethares, W.A., Weak covergece ad local stability properties of fixed step size recursive algorithms, IEEE Trasactios o Iformatio Theory, Vol. 39, May 993 pp [6] Orosz, Gy., L. Sujbert, G. Péceli, Testbed for Wireless Adaptive Sigal Processig Systems, Proceedigs of the IEEE Istrumetatio ad Measuremet Techology Coferece, Warsaw, Polad, May -3, 27 [7] Sujbert, L., G. Péceli, Periodic oise cacellatio usig resoator based cotroller, 997 It. Symp. o Active Cotrol of Soud ad Vibratio, ACTIVE 97, pp , Budapest, ugary, Aug [8] Kuo, S. M., D. R. Morga, Active Noise Cotrol: A Tutorial Review, i Proceedigs of the IEEE, vol. 87. No. 6., pp , Jue. 999.