ADAPTIVE NARROW-BAND DISTURBANCE REJECTION FOR STABLE PLANTS UNDER ROBUST STABILIZATION FRAMEWORK. Jwu-Sheng Hu and Himanshu Pota

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1 ADAPIVE NARROW-BAND DISURBANCE REJECION FOR SABE PANS UNDER ROBUS SABIIZAION FRAMEWORK Jwu-Sheng Hu an Himanshu Pota Department of Electrical an Control Engineering National Chiao ung University Hsinchu, aiwan, 3, ROC School of Electrical Engineering Australian Defence Force Acaemy Northcott Drive, Canberra AC 6, AUSRAIA Abstract: An aaptive control algorithm for rejecting narrow-ban noise is propose in this paper. his algorithm is evelope uner the framework of robust stabilization an the internal moel principle. he internal moel which contains the noise ynamic is upate using the estimate isturbance signal. he parameter aaptation is constraine to meet the conition of robust stabilization when the parameters are converge. Simulation examples are given to show the effectiveness of the control law uner isturbance with unknown frequencies. Copyright 5 IFAC Keywors: Aaptive Control, Aaptive Digital Filters, Disturbance Rejection, Robust Stability, H-infinity Optimization.. INRODUCION Internal Moel Principle (IMP has been use for controller esign to track perioic trajectories (omizuka et al, 989; Kempf et al, 993 or reject narrow-ban noises (Hu, 996. For applications such as the active noise cancellation, IMP base controller has the avantage that the information of the noise source signal is not require, as compare with feeforwar-type algorithms such as the Filter- X (Elliot an Nelson, 993; Kuo an Morgan, 996; Bjarnason, 995. herefore, when a coherent noise source is ifficult to measure or the cost of sensor installation is high, IMP base controller becomes a goo caniate for noise rejection. he IMP base controller works well if the isturbance frequencies (or the funamental perio of a harmonic noise are known. For applications like isk rive tracking (Moon et al, 998, the conition is satisfie when the servo controller is synchronize with the isk rotation. In active noise cancellation (ANC, however, it may not be easy to locate these frequencies precisely. For example, narrow-ban noise cause by rotational machinery may rift aroun some nominal frequencies ue to small variations of the machine s spee. Earlier research efforts to ientify the frequencies an ajust the control law on-line were reporte (Hu, 99 but only limite to harmonic cases. For igital implementation, if the funamental perio of the harmonic noise is not an integer-multiple of the sampling perio, one also has to consier using techniques such as the fractional elay filter for the internal moel (Hu an Yu,. Another approach (Elliot an Goowin. 984; Palaniswami an Goowin, 987; Feng an Palaniswami 99; Palaniswami, 993 is to consier the non-minimal representation of the plant which inclues the isturbance moel. he isturbance moel is then extracte (or factore from the ientifie plant parameters. An inirect aaptive controller is implemente base on the pole-placement algorithm. Although it is not require to know the plant s parameters, extracting the isturbance moel from the composite estimate polynomial may not be easy (Palaniswami 99, especially when the number of isturbance frequencies is high. here were other aaptive IMP control schemes (Datta an ei, 998, 999; Silva an Datta, 999; Muramatsu an Watanabe, 3; Watanabe an Muramatsu, 3 but were aime at ientification of plants instea of the isturbance moel. In this paper, an aaptive internal moel control algorithm is investigate uner the framework of robust stabilization. he plant is assume stable an

2 a nominal moel is known. he output isturbance is narrow-ban whose frequencies are unknown or changing slowly. he control objective is to reject the isturbance without affecting the close loop stability uner the moel uncertainty. Simulation results are shown to emonstrate the effectiveness of the control law... FEEDBACK CONROER BASED ON OUPU DISURBANCE OBSERVAION A feeback controller which utilizes the observation of the output isturbance can be shown as Figure.. he iea is simple: if the estimation of the plant ynamic is accurate enough an the controller K(z provies a proper inversion of the plant within the banwith of the isturbance, the influence of the isturbance to the output can be minimize. Other than the uncertainty of the plant ynamic which may egrae the accuracy of isturbance observation, non-minimal phase zeros of the nominal plant also limit the performance of rejection. u K(z - G(z - ˆ G ( z ( y( ˆ( k Figure. Feeback control system base on output isturbance observation ( ˆ G ( z represents the nominal plant For stable plants, Figure. can be viewe as an Youla Parameterization where a stable controller K(z guarantees the close loop stablility. If the plant is accurately ientifie (i.e., ˆ G (, the isturbance to the output satisfies the following equation: Y ( = D( G( K( D( (. Using FIR filter as the structure of K(z, Eq.(. becomes a stanar Filter-X aaptive control setting if a signal correlate to ( can be obtaine, i.e., y( G( q K( q ( (.a θ φ ( where N K( z = a az az a N z (.b θ = [ a a a N ], φ ( k = [ g ( g ( k g ( k N] (.c g ( q ( (. Normally, ( is replace by the observe isturbance ˆ( k (Figure.. his metho has foun to be quite useful in rejecting narrow-ban noises (Rafaely an Elliott, 996. However, uner the plant uncertainty, the H norm of K(z must be constraine in orer to guarantee robustness. Seconly, when the isturbance is rich in frequency, an the plant is non-minimal phase, the length of the FIR filter has to be long enough. his woul result in slow convergence. Consier an alternative representation of the control system in Figure (see Figure. where G(z is the nominal plant hereafter. w z ( K(z G Q( G(z - G( Q( u( Figure. A unity feeback configuration of the control system base on output isturbance observation ( G : plant uncertainty Suppose the isturbance moel is W (z. o esign Q(z which satisfies the robustness an isturbance rejection, we can solve the following problem: Min ( QG W subject to Q < Q RH Another way to esign the controller is to solve K(z irectly using the H controller synthesis methos. Since all stablizing controller can be represente as shown in Figure. if the plant is stable (Doyle an Francis, 99, we can obtain Q(z as: K( Q ( = (.3 K( G( In other wors, we can obtain a nominal controller base on the unertstaning of the plant an the isturbance. o enhance the performance using aaptive methos, an aitional aaptive FIR filter can be ae to the controller as shown in Figure.3. G w z ( G u( Q(z Φ(z G(z - y( y( G(z - PAA ˆ( k Figure.3 Output isturbance rejection with a nomial controller Q(z an an aaptive FIR filter Φ(z. (PAA: parameter aaptation algorithm From Figure.3, if Q(z is compute to satisfy the robustness constraint, the constraint for Φ(z to maintain robusness at steay state (i.e., when parameters converge is, Φ( z < (.4 It is obvious that aing the nominal controller Q(z provies a normalization across the frequency for the constraint in parameter aaptation. 3. CONSRAINED ADAPIVE CONRO AGORIHM When the FIR filter Φ(z is ajuste for each sampling perio, the output of the system in Figure.3 can be erive as, y(

3 y( [ G( q G ] Φ( k, q Q( q ˆ( G( q Φ( k, q ˆ ( z( y( ν ( (3. where ˆ ( ( k = Q q ˆ( an z( represents the uncertainty signal. Denote the preicte isturbance signal as, ζ ( k = y( y( z( (3. Our goal is to ajust Φ( k, so that y ( matches (. In other wors, given a solution of Φ( k, that satisfies the goal (i.e., when parameters converge, the ynamics of y ( shall be able to represent (. et the filter Φ( k, be Φ ( = f ( f( z f ( z, If [ A, B, C, D ] are the state space matrices of the nominal plant, Eq.(3. an (3. can be represente in state space as, η ( k = F ( k η ( k (3.3a ζ ( k = H ( η( z( (3.3b where I F( =, (3.3c Bφ ( A H ( = [ Dφ ( C ], (3.3 θ ( η ( =, (3.3e x( θ ( = [ f ( f( f ( ], (3.3f φ ( = [ ˆ ( ˆ ( ˆ k k ( k ] (3.3g x( is the state vector of the plant an I is an unity matrix. he technique of H suboptimal causal filtering (Sayyarrosari et al, 998 can be applie to the formulation above. We wish to fin an optimal estimator such that, sup ν, η M k= [ ζ ( ˆ( ζ k ] η Πη M k= z( γ (3.4 where ˆ ζ ( k is the posteriori preiction of ζ (, η the initial guess of the state vector an a positive efinite matrix. Eq.(3.4 is vali as long as the the uncertainty signal z( is boune. o show this, it is necessary that the H norm of the time varying filter Φ( k, is less than (Eq.(.4. A sufficient conition to guarantee it is, θ ( θ ( = f (, k (3.5 i= et Γ be the subspace of θ( (Eq.(3.3e such that θ ( θ ( =. Combining the constraint of Eq.(3.5 an the objective function, we can erive the parameter upate law as (Sayyarrosari et al, 998, s s = s (3.6a = F k ηˆ( k K ( k ( ζ ( k H ( k ηˆ( k ( If s s >, = Pr{ s ˆ( η k }, else ˆ( η k = s s (3.6b ˆ ζ ( k k = H ( k ηˆ( k H ( k P ( k H ( k R H ( k ( ζ ( k H ( k ηˆ( k (3.6c K ( = F( P( H ( RH ( (3.6 R H ( = H ( P( H ( (3.6e P ( k = F( P( F(, P( = Π (3.6f where Pr{.} enotes the projection onto the space Γ. Remark : he value of ζ( is compute by ζ ( = y( y( where y ( can be obtaine by running a simulate nominal plant as y( q u(. But this is impossible if the nominal plant is proper but not strictly proper, i.e., D. he following list summarizes the proceure: At every k-th sampling instant, Step : compute the nominal plant output y( q u( an from the measurement of y( compute ζ ( = y( y( Step : compute the estimate isturbance ˆ ( = y( ζ (. Step 3: compute ˆ ( ( k = Q q ˆ( an form the vector φ (. Step 4: compute the parameter an state upate ˆ η ( k from Eq.(3.6a (3.6f Step 5: compute the control u( = θ ˆ( k φ(. 4. DESIGN EXAMPE AND SIMUAION o illustrate the aaptive control algorithm, we consier the plant of an active controlle heaset shown. he nominal plant represents the case when the heaset is in a normal position which covers the full ear an the actual plant is when it covers ear partially. able 4. shows the corresponing parameters. Actual plant Nominal plant Poles Zeros Poles Zeros.96.8 ±.8j.57 ±.5j ±.56j.96.6 ±.78j.58 ±.5j ±.56j Gain = Gain = able 4. Poles, zeros an gain of the nominal an actual plant 4. Design of the Central Controller he proceure escribe in (Stoorvogel, 99 is followe to esign the central controller K(z of Figure. an the corresponing Q(z is compute. Figure 4. shows the structure of the controller esign setting where w an w are uncertainty inputs, z the uncertainty output, u the input an y the measure output.

4 w w u W ( G ( z y comparison, another simulation is performe using the feeforwar-type of algorithm where the observe noise ˆ( k (Figure.3 is replace by a 6 Hz tone. Figure 4.4 shows that the moulation of the frequency remains at the output an has the tenency of growing. he length of the FIR filter in both cases is 6.. Figure 4. he robustness control setting to esign the central controller (G(z : the nominal plant; W(z : the weighting function Denote the uncertainty signal z as w = z an w = z, the I/O relationship is, Y ( ( ( W ( (4. U ( ( W ( he plant uncertainty is efine as G as shown in Figure.. herefore, we have, G( G ( ( W ( Or, G = ( W ( (4. G( As a result, by properly choosing the weighting function W(z such that jω G W ( e >, ω [, π ] (4.3 jω G( e we can scale the plant uncertainty to be less than to esign the robust controller as well as give the unity constraint of the aaptive FIR filter (Eq.(.4. he controller K(z can be solve by using Matlab an the controller is transforme to the internal moelbase control structure by Eq.( Simulation results Figure 4. shows the simulation result when the noise is a single tone at 6 Hz. he length of the FIR filter is 6. Figure 4.3 Simulation result for single tone noise with a slight frequency moulation Figure 4.4 Simulation result for the feeforwar type of algorithm without consiering the change of noise frequency astly, a noise ata measure from a V8 internal combustion (IC engine at ile spee is use for the isturbance. IC engine noise contains many narrowban features ue to the perioic nature of its operation. Figure 4.5 shows the noise an the noise reuction after applying the controller. he reuction is more clearly seen from the spectrum in Figure 4.6. he narrow-ban noises at lower frequencies are reuce at the expense of slight increases at higher frequencies. Figure 4. Simulation result of rejecting a tonal noise at 6 Hz o see the rejection capability when the noise s frequency is changing, the following noise is applie. ( = sin(πf ( k / f s where f( = 6 sin(πkf m / f s an f m = 3. It means the frequency swings between 59 an 6 Hz in a spee of 3 Hz. Figure 4.3 shows that the controller is able to aapt the changes. For Figure 4.5 the simulation result when the isturbance is an engine noise (ot: original noise; soli: reuce noise

5 Figure 4.6 the spectrum of the noise an noise reuction (ot: original noise; soli: reuce noise 5. CONCUSION An aaptive internal moel-base control algorithm is presente in this paper. his algorithm utilizes the robustness control technique to scale the plant uncertainty an constrains the aaptive filter to meet the robustness requirement. An observer-base aaptive law with the H optimal filtering criterion is propose. Simulation shows that this algorithm is able to reject narrow-ban isturbances with unknown or slow-varying frequencies. ACKNOWEDGEMEN his work is sponsore in part by UNSW@ADFA an the National Science Council of aiwan uner the grant number NSC-93-8-E-9-3. REFERENCES Anton Stoorvogel. he H Control Problem: A State Approach. Prentice Hall International, UK, 99. Bjarnason, E., Analysis of the filtere-x MS algorithm, IEEE rans. Speech an Auio Processing, vol. 3, no. 3, pp , 995. Datta, A. an ei Xing, he theory an esign of aaptive internal moel control schemes, Proceeings of the 998 American Control Conference, Volume: 6, 4-6 June 998, Pages: vol.6. Datta, A. an ei Xing, Aaptive internal moel control: H optimization for stable plants, IEEE ransactions on Automatic Control, Volume: 44, Issue:, Nov. 999, Pages:3 34. Doyle, John C., Francis, Bruce A. an annenbaum, Allen R., Feeback Control heory, Maxwell Macmillan, 99. Elliot, H. an Goowin, G.C., Aaptive implementation of the internal moel principle, in Proc. 3 r IEEE Conf. Decision Contr., 984, pp Elliot, S.J. an Nelson, P.A., Active noise control, IEEE Sig. Proc. Mag., vol., pp.-35, Oct Feng, G. an Palaniswami, M., A stable aaptive implementation of the internal moel principle, IEEE ransactions on Automatic Control, Vol. 37, No. 8, August 99, pp. -5. Hu, J. S.,"Variable Structure Digital Repetitive Controller, "In Proceeings of the American Control Conference, 99, pp Hu, J., Active noise cancellation in ucts using internal moel base control algorithms, IEEE ransactions on Control System echnology, vol. 4, No., March 996, pp Kempf, C.; Messner, W.C.; omizuka, M.; Horowitz, R., Comparison of four iscrete-time repetitive control algorithms, IEEE Control Systems Magazine, Volume: 3 6, Dec. 993, Page(s: Kuo, S.M. an Morgan, D.R., Active Noise Control Systems, John Wiley an Sons, 996. Moon, Jung-Ho, ee, Moon-Noh an Chung, Myung Jin, Repetitive control for the track-following servo system of an optical isk rive, IEEE ransactions on Control Systems echnology, Volume: 6 5, Sept. 998, Page(s: Muramatsu, E. an Watanabe, K., Aaptive internal moel control of MIMO systems, SICE 3 Annual Conference, Volume: 3, August 4-6, 3, Pages: Palaniswami, M. an Goowin, G.C., An aaptive implementation of the internal moel principle, in Proc. 7 th Amer. Contr. Conf., 987. Palaniswami, M., Aaptive internal moel for isturbance rejection an control, IEE Proceeings-D, Vol. 4, No., January 993, pp Rafaely, B. an Elliott, S.J., An aaptive an robust feeback controller for active control of soun an vibration, UKACC International Conference on Control '96 (Conf. Publ. No. 47, Volume:, Sept. 996, Pages: Sayyarrosari, B., How, J.P., Hassibi, B. an Carrier, A., An H -optimal alternative to the FxMS algorithm, Proceeings of the 998 American Control Conference, Volume:, 4-6 June 998, Pages: 6. Silva, G.J. an Datta, A., Aaptive internal moel control: the iscrete-time case, Proceeings of the 999 American Control Conference, Volume:, -4 June 999, vol., Pages: omozuka M.,. S. sao an K. K. Chew, "Analysis an Synthesis of iscrete-time Repetitive Controllers, " Journal of Dynamic Systems, Measurement an Control, vol., no. 3, , 989. Watanabe, K. an Muramatsu, E., Aaptive internal moel control of SISO systems, SICE 3 Annual Conference, Volume: 3, August 4-6, 3, Pages: Yu, S.-H. an Hu, J., Asymptotic Rejection of Perioic Disturbances with Fixe or Varying Perio: From Iterative Operator Inversion to Repetitive Control, ASME Journal of Dynamic Systems, Measurement An Control, VO. 3, NO. 3, pp 34-39, December,.

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