Estimation and Rejection of Unknown Sinusoidal Disturbances Using a

Proceedings of the 2007 American Control ConferenceMarriott Marquis Hotel at Times SquareThC01.2 New York City, USA, July 11-13, 2007Estimation and Rejection of Unknown Sinusoidal Disturbances Using aGeneralized Adaptive Forced Balancing MethodElnaz Vahedforough, Bahram Shafai, and Stuart BealeAbstract— This paper presents a method for estimation and rejection of sinusoidal disturbances with unknown amplitudes and frequencies arising in feedback control systems. A method known as adaptive forced balancing has been used successfully to reject sinusoidal disturbances with known frequency and unknown amplitude in magnetic bearing control systems. The occurrence of such a disturbance is a result of a common prob- lem to all mechanical systems with rotating shaft known as mass unbalance. Because mass unbalance consists of a sinusoidal sensor disturbance within the bandwidth, conventional methods suffer from instability or fail to preserve desired bandwidth. AFB overcomes these problems and offers unique features which preserve stability and performance robustness. In this paper, we extend AFB with a frequency tracking capability. This is achieved by using two different adaptive observer structures. We show that the modified version of AFB is capab le of estimating both amplitude and frequency of the disturbance signal and demonstrate its rejection in the feedback loop.I. INTRODUCTIONThe problem of disturbance rejection has always been a critical issue in control systems. Among various forms of disturbances, sinusoidal signals occur frequently in rotational machinery. In such applications it is desirable to estimate unknown amplitude, frequency and phase of disturbance signals. The literature reports a vast variety of methods for this purpose. Hsu and Regalia improved the performanceof notch filters and introduced an adaptive IIR version ofit in [1], [2]. Later on Marino and Tomei introduced the adaptive observers to estimate the frequency of sinusoidal disturbances in [3]. Guo and Bodson considered frequency estimation and tracking of multiple sinusoidal components using magnitude/phase locked loop in [4]. Among other solutions, Kalman Filtering [5], power spectrum method and harmonic decomposition methods, were also reported by researchers in [6], [7], and [8]. In spite of the fact thatmany researchers partially solved the problem, combined estimation of unknown frequency and amplitude of sinu- soidal disturbance signal in different scenarios; particularly in feedback control systems, still remains a challenging task.A particularly important application that requires estimation and compensation of disturbances is in active magnetic bearing control systems. Analysis and design of such systems have been studied by many researchers in the past. For an overview of results in this direction, refer to [9]-[14] and the references therein.In this paper we consider the problem of rejecting sinu- soidal disturbance signal in active magnetic bearing control systems. Such a signal occurs as a result of vibration caused by mass unbalance: the condition of a rotor in which its principle axis of inertia is not coincident with its axis of geometry. In [15], this disturbance is modeled as a sinusoidal signal with known rotational frequency and unknown ampli- tude. A novel approach called Adaptive Forced Balancing (AFB) was used to estimate the amplitude of the disturbance signal and made its rejection possible. However, a more de- sirable solution for the disturbance accommodation problem is when both amplitude and frequency are not known. In most applications including magnetic bearing control systems, it is essential to estimate the frequency in addition to amplitude. This is due to the fact that, if rotational sensors are not available, it is required to estimate the frequency as well. The modified AFB proposed in this paper incorporates a frequency tracking algorithm, which enables to estimate both amplitude and frequency of the disturbance signal. AFB offers additional advantages, which makes it worthwhile to use in the control loops. One of the main characteristics of this method as compared to the previous filter based methods is that it operates external to the basic feedback loop. The proposed structure of AFB has negligible effect on frequency response of the designed closed loop system performance.In contrast with model based methods, it doesn’t needthe knowledge of plant dynamics and therefore is robustto modeling errors and plant parameter variations. In this paper we discuss the Adaptive Forced Balancing structurein connection to magnetic bearing control systems. Then,we generalize the method by including a frequency tracker, which consists of a Proportional Integral Adaptive Observer, to the basic AFB structure. We demonstrate that estimation and rejection of unknown sinusoidal disturbances is possible by using the generalized AFB method. The simulation results show the fast convergence rate of frequency and amplitude estimations as well as disturbance cancelation at the output.II. ADAPTIVE FORCED BALANCING INCONTROL LOOPAdaptive Forced Balancing is designed to estimate the amplitude and possibly the frequency of the disturbance signal in order to be able to reject the effect of the disturbance in feedback control loop. This method is neither a model based nor resides in the inner control loop. It is an outer control loop which provides a synchronous, rotor position reference signal for the inner control loop. The design of AFB with the proposed set up does not degrade the stability margin of the closed loop control system. It also offers additional robustness in comparison to previous methods. Fig. 1 demonstrates the block diagram of the Adaptive Forced Balancing system in a control loop. The input ofthe AFB system is the control signal represented by u(t),1-4244-0989-6/07/$25.00 ©2007 IEEE. 3529ThC01.2and its output is r Ω(t) which represents the synchronous reference signal. We assume that the disturbance d Ω(t) has the following time domain representation:d Ω(t) = αd sin(ωx t) + βd cos(ωx t),(1)where α(t), β(t) are the Fourier Coefficients of the sinu - soidal disturbance. It is required to cancel out the effect of d Ω(t) by designing AFB system. To accomplish this taskFig. 2.The structure of Adaptive Forced Balancing systemwe need to provide a synchronous signal with the following structure:Butterworth and Chebychev filters. In fact the filter is capable of passing the DC components of demodulated signal. The r Ω(t) = αd (t) sin(ωx t) + βd (t) cos(ωx t).(2) cut off frequency is less than or equal to ωx /3 , while the If we represent the sensitivity function of the closed loop system as:P (s)C(s)T (s) = , (3)1 + P (s)C(s) then the system output can be written as:sampling frequency of filter should be greater than or equal to 100ωx .B. Fourier Co efficient ComputerThe Fourier Coefficient Calculator is the main part of our adaptive forced balancing system. The coefficients of sine Y (s) = T (s)r(s) + T (s) (r Ω(s) − d Ω(s)) ,(4) and cosine are calculated on line such that it cancels out the effect of disturbance. The adaptive laws which govern α(k), and the effect of the disturbance at the output is eliminatedβ(k) are generated by:as r Ω(t) → d Ω(t). This can only be realized if:α(k + 1) = α(k) + (1 + q(k)) p +1(k)n(k),(6) lim r Ω(t) → d Ω(t).t→∞(5) β(k + 1) = β(k) + (1 − q(k)) p −1(k)n(k),(7)III. THE STRUCTURE OF AFBAs it is shown in Fig. 2, AFB system is composed of three fundamental parts:1) Synchronous RMS Calculator 2) Fourier Coefficient Computer 3) Modulator(Signal Generator)A. Synchronous RMS CalculatorIn this part we calculate the ”energy” of the synchronous component of control signal. This energy serves as an error input to an algorithm which calculates the amplitude and phase of a synchronous correction signal. To obtain the norm signal of n u (t), the control signal u(t) is demodulated and Where α(k), β(k) are the discrete forms of the corre-sponding time varying Fourier Coefficients and are calculated in the sampling period of T s . The input to the system is represented as n(k) which is updated every T s seconds. The variables p(k) and q(k) get values equal to {+1, −1} according to the algorithm, shown in Fig. 4.In this process an ”enable” variable q(k) ∈ {−1, +1} holds one Fourier Coefficient constant while the other is getting adapted, and ”polarity” variables p +1(k), p −1(k) ∈ {−1, +1} insert the corresponding signs in (6), (7), respec- tively, so that the norm of error ǫ(k) decreases with time. The error ǫ(k) represents the amount of aviation from desired values of αd , βd , which is shown by:filtered with low pass Bessel filter. Then a complex mag ni- ǫ(k) = (α(k) − αd ) + j(β(k) − βd ).(8)tude calculation is performed as illustrated in Fig. 3. Here we used Bessel filter to overcome the overshoot problem ofIf the sampling frequency ωs is chosen to be small enough such that ωs ≪ ωx ; then [α(k) − α(k − 1)]/T s , and [β(k) − β(k −1)]/T s become sufficiently small. Hence we can make the following assumption:u(t) ≈ µ (ωx ) [(αr (t) − αd ) sin(ωx t + Θωx )+(9)Fig. 1.Adaptive Forced Balancing system in a control loopFig. 3.The RMS Calculator structure3530ωx 2 ωx 2 1/2ThC01.2IV. THE GENERALIZED STRUCTURE OF AFBWITH FREQUENCY TRACKER In previous section AFB is used to compensate the sinu- soidal disturbances while the fundamental frequency ωx was known. However if hardware (e.g., a tachometer, resolver, or another type of rotational sensor) is not available to generate these waveforms, a frequency tracking algorithm, operating in synergy with AFB, must be included to provide estimates of ωx . Therefore a Frequency Tracker system is needed to estimate the frequency while the amplitude is estimated by AFB system. Fig. 5 demonstrates the generalized structure of AFB which consists of AFB system along with frequency tracker. The estimated frequency signal enters the sine and cosine generators and then is injected to AFB to estimate the amplitude enabling the disturbance rejection in the next stage.A. Frequency TrackerConsidering the disturbance signal y = A sin(ωt + φ), with its state space representation model as:Fig. 4.The Fourier Coef ficient Calculator algorithmx ˙ 1 x ˙ 2=0 1 −α 0 x 1 x 2(14)d(t) = y(t) = H T x = x 1(βr (t) − βd ) cos(ωx t + Θωx )],where y(t) is the disturbance signal entering the system, α = ω2, with initial conditions x 1(0) = A sin(φ), and where µ(ω) = |(S(j ω)C(j ω)|, Θ(ω) =x 2(0) = A ω cos(φ). since the system described by (14)arg(S(j ω)C(j ω)) and S(j ω) is the sensitivity function given by S(j ω) = 1/1 + P (s)C(s). Therefore n(k) can be expressed as:has the observable canonical form, we can use the adaptive observer method to estimate the frequency of the signal. In the following sections we will first discuss the adaptive n(k) = n u (kT s ),= v{[h(kT s ) ∗ (u(kT s ) sin(2πkω)] + (10)observer procedure and then extend it to Proportional Integral Adaptive Observer.1) adaptive observer method: Adaptive Observer method first was introduced by Robert Carroll and then was mod-n(k) =v2[h(kT s ) ∗ (u(kT s ) cos(2πk )] } , ωsµ(ωx ) (α(k) − αd )2 + (β(k) − βd )2,(11)i fied by Narndra and Kudva for parameter estimation [16], [17]. Such an adaptive observer has been applied in many scenarios. In this paper we use it for frequency estimation in active magnetic bearing control system. Considering theand it can be simpli fied as:n(k) =1 2µ(ωx )|ǫ(k)|.(12)Therefore the adaptation of p(k) and q(k) enables us to control the coef ficients error such that:|ǫ(k + 1)| < |ǫ(k)|.C. ModulatorAs it is shown in Fig. 2, Modulator receives the output signal from FCC and modulates it to the following form:r Ω(t) = αr (t) sin(ωx t) + βr (t) cos(ωx t),(13)This provides the generation of the synchronous signal needed for disturbance cancelation.3531Fig. 5.The AFB Compensator with Frequency Trackerρ(s) = 2ψ(s) = 2 yˆ(t) = h z ThC01.2z˙1 = k1(y − yˆ) + z2 + n1 (y − yˆ) (20)z˙2 = k2(y − yˆ) + n2 (y − yˆ) −αˆyyˆ(t) = h T zwhere n1 and n2 are positive scalars. Using the secondorder filters:Fig. 6. The Frequency Tracker−ss + d1s + d2y(s), (21)system described by (14), the conventional adaptive observer−c1s − c2s + d1s + d2y(s), (22)structure assumes the following form: the above state space form can be summarized in followingz˙ = Kz + [k − aˆ(t)]x1 + v + wT where K is a stable matrix described by: (15)form:yˆ˙ (t) = k c y + k p(y − yˆ) + αˆ(t)ρ(t) + ψ(t)αˆ˙ = −Γρ(y − yˆ)(23)K = −k1−k21,where c1, c2, d1, d2 are positive scalars. The PI AdaptiveObserver offers a fast the convergence rate in comparisonto adaptive observers. The simulations in the next sectionsand h T = 1 0 and v, w are the auxiliary signals. The demonstrate its performance in frequency estimation.objective is to :lim aˆ(t) → a,t→∞B. Amplitude TrackerAs it was described before,α(t), β(t) are the Fourierwhere a represents 0 −αT . Using auxiliary filters: Coefficients which provide the amplitude for sine and cosinerespectively. Therefore the magnitude of the amplitude isv˙ = −λv − y, (16) calculated from (24) as it is shown in (25).w˙ = −λw −λ2y, (17) Aˆ(t) = α(t) sin(ωx t) + β(t) cos(ωx t) (24) the state space form of (15) can be transformed to the follow- |Aˆ(t)| = α(t)2 + β(t)2 (25)ing single input single output form, which was formalized in [3]. V. SIMULATION RESULTS FOR AFB WITH UNKNOWN FREQUENCYyˆ˙ = λ(y − yˆ) + λy + vαˆ + w (18) We simulated the control loop for the case when all pa-rameters of sinusoidal disturbance are unknown. ConsideringThe overall schematic diagram of frequency tracker system is shown in Fig. 6. In this system the disturbance signal passes through the low pass filters described by (16), (17). The generated auxiliary signal is used in identification block to estimate the frequency at the next level which is given by: the disturbance asd(t) = α sin(ωx t + φ) + β cos(ωx t + φ),where,α = 20, β = 10, ω = 25, andφ represents an arbitrary phase shift. As shown in Fig. 5 the disturbanceαˆ˙ = −Γv(y − yˆ) (19) is acting on frequency tracker and AFB system. Frequency estimation simulations are carried out using the adaptive observer method and modified PI Adaptive Observer which2) PI adaptive observer method: Using the concept of adaptive observers discussed above we generalized the es- timation procedure by using Proportional Integral Adaptive Observer. The PI Adaptive Observer consists of the regular proportional term in addition with an Integral term which improves the convergence of estimated signal in both single and multi parameter cases. The following state space form represents the PI formulation for the purpose of frequency estimation.are illustrated in Fig. 7 and Fig. 8, respectively. As figuresdemonstrate, the PI adaptive observer improves the fast con- vergence of the signal. The parameters used in simulationsfor PI adaptive observer are k p = 5, k c = 10,Γ = 500,where the poles of the auxiliary filters are adjusted to 2.The estimated frequency is applied to AFB system where the amplitudesα, β are estimated and the resulting signalis used as reference signal to reject the disturbance at theoutput. Fig. 9 illustrates that the fourier coefficientsα(t), 3532ThC01.235The Estimated Frequency Using Adaptive Observer302520151050 051015202530Time(second)Fig. 7.3025The estimated frequency using Adaptive ObserverThe Estimated Frequency Using PI Adaptive Observer2520 Fig. 9. Estimated Fourier Coef ficientsThe Estimated Amplitude20 15151010550 051015Time(second)2025300 0510 15 Time(second)2025Fig. 8.The estimated frequency using PI Adaptive ObserverFig. 10.Estimated Amplitudeβ(t) converge to the amplitudes of each sinusoidal term. The effective amplitude is then calculated as shown in Fig. 10. The results of estimated reference signal and its rejectedd Ω(t) =nαd(2i+1) sin(ωx(2i+1))+βd(2i+1) cos(ωx(2i+1)),effect at the output are demonstrated in Fig. 11 and Fig. 12, respectively.VI. CONCLUSIONSIn this paper we designed an Adaptive Forced Balancing (AFB) system to estimate the amplitude and frequency of sinusoidal disturbances. We considered the application scenario of magnetic bearing control systems whereby a si- nusoidal signal characterizes the disturbance caused by mass unbalance. First we designed the AFB under the assumption of known rotational frequency and unknown amplitude. A frequency tracker system is introduced with the original AFB in a generalized structure to estimate both the amplitude and frequency of the sinusoidal disturbance signal. Simulation results show that the generalized AFB method proposed in this paper makes it possible to completely reject unknown sinusoidal disturbance in a closed loop control system with negligible effect on the stability and performance robustness. This system is designed to estimate the fundamental fre- quency. However in some applications when the fundamental frequency is considerably small, it is needed to extend it to higher harmonics to achieve better estimation results. In this case, we de fine:i=1which demonstrates the disturbance signal composed of (2n + 1) harmonics, where α, β are the estimated am- plitude set, obtained by Fourier Coef ficient Calculators of AFB. Therefore, the estimated values for higher harmonic frequencies can be constructed by multiple modules of the frequency tracker con figuration shown in Fig. 5. It should be pointed out that the above analysis also enables us to reject disturbances of pulse shape through Fourier series expansion. This will be elaborated in the future publication.R EFERENCES[1] L. Hsu, R. Omega, and G. Damm, ”A globally convergent frequencyestimator ”, IEEE Trans. Automat. Contr., vol. 44, pp. 698-713, 1999. [2] P. A. Regalia, ”An improved Lattice-based IIR notch filter ”, IEEETrans. Signal Processing , vol. 39, pp. 2124-2128, 1991.[3] R. Marino and P. Tomei, ”Global adaptive compensation of noiseswith unknown frequency ”, IEEE Trans. Decision and Control , vol. 5, pp. 4926-4927, 2000.[4] X. Guo and M. Bodson, ”Frequency estimation and tracking ofmultiple sinusoidal components ”, IEEE Conf. Decision and Control , vol. 5, pp. 5360-5365, 2003.[5] S. Bittanti and S. M. Savaresi, ”Frequency tracking via extendedKalman filter: parameter design ”, IEEE Trans. American Control Conference , vol. 4, pp. 2225-2229, 2002.[6] T. Lobos, T. Kozina, and H. J. Koglin, ”Power system harmonics esti-mation using linear least squares method and SVD ”, IEE Proceedings in Generation Tarnsmission Distributon , vol. 148, pp. 567-572, 2001.3533感谢您试用AnyBizSoft PDF to Word。