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Automatica 36 (2000) 1659}1664

Brief Paper

Adaptive controller using "lter banks to reject multi-sinusoidal disturbance夽 P. Micheau *, P. Coirault GAUS, Mechanical Department, Universite& de Sherbrooke, 2500 Boul. Universite& , Sherbrooke, Que& bec, Canada J1K 2R1 E! SIP, Universite& de Poitiers, 40 Av. du Recteur Pineau, 86022 Poitiers, France Received 10 February 1998; revised 18 June 1999; received in "nal form 10 February 2000

Abstract The purpose of this paper is to apply "lter banks to the control problem involving the rejection of multi-sinusoidal disturbance from output of slowly time-varying stable systems. The use of "lter banks allows to distribute the control e!ort in many independent adaptive controllers, each of them taking care of a sinusoidal component of the disturbance. By varying the "lter banks speci"cations, the method handles the trade-o! between a time-behavior controller, with interesting settling time and a tonal-behavior controller showing the properties of simpli"ed control, reduced computational time and on-line system identi"cation with the system output noise via the feedback loop. Numerical examples are presented to illustrate this trade-o!. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Adaptive control; Filter banks; Rejection; Sinusoidal signals; Convergence analysis

1. Introduction The motivation for this work was to design an adaptive feedback controller for active control of pulsed #ow. In terms of control, the objective is to reject a multisinusoidal disturbance from output of stable system. In the "eld of active control of sound, Nelson and Elliot (1992) used an adaptive feedforward approach to control the anti-noise with high precision. However, this feedforward approach needs a measured reference signal. Without a reference signal, the problem is to design a feedback loop that is able to give the same attenuation as an adaptive feedforward approach. The feedback loop achieves desirable rejection of sinusoid despite the presence of modeling uncertainty in the plant model and slowly time-varying sinusoidal disturbance. But, in the case of slowly time-varying system, the feedback controller must be adapted to achieve both stability and optimal reduction. 夽 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor K.W. Lim under the direction of Editor F.L. Lewis. * Corresponding author Tel.: #1-819-821-8000; fax: #1-819-8217163. E-mail address: [email protected] (P. Micheau).

In the "eld of feedback theory, a well-known Bode's theorem suggests that reduction in one frequency band must be traded o! against increase of sensitivity at other frequencies. Indeed, to obtain a perfect attenuation at one frequency, without undesirable property at other frequencies, the feedback controller must be narrow band. To deal with this control problem, di!erent controllers applied to real-world systems have been proposed. Sievers and Flotow (1992) have presented a global comparison of control methods for narrow band disturbance rejection such as higher harmonic control, tonal control, repetitive control, learning control, LMS adaptive feedforward "ltering, classical and modern control. The more complex design involving a LQ-based method requires an accurate model of the plant dynamics. On the other hand, the tonal controller is the simplest compensator because it ignores the underlying dynamics, except for the plant gain and the phase at the disturbance frequencies. Micheau, Coirault, Hardouin and Tartarin (1996) experimented with a controller working in a narrow band to reject the preponderant harmonic of a pulsed #ow. Good experimental results have motivated extension of the approach to the case of multiple sinusoids. For this purpose, the authors investigated "lter banks, which allow analysis and synthesis in many narrow bands. This approach is an adaptation for a control

0005-1098/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 0 7 2 - 8

1660

P. Micheau, P. Coirault / Automatica 36 (2000) 1659}1664

problem of the subbands echo canceller, using frequency techniques proposed by Gilloire and Vetterli (1988), developed by Perez and Amano (1990) and Tang, Camache and Flores (1995). Micheau, Coirault, Renault and Tartarin (1995) have successfully implemented this new controller in real situations. The objective of this paper is to show that an adaptive controller using "lter banks is between a time-behavior controller and a tonal-behavior controller. The paper is organized in three sections. In Section 2, the problem of the rejection of a multi-sinusoidal disturbance from a stable system as well as the signal processing tools are presented. Section 3 introduces the direct adaptive feedback controller. The on-line adaptation needs the identi"cation of the system and disturbance models in each subband, which is performed using a complex generalized recursive estimator with forgetting factor. However, without extra-signal or dead zone algorithm, this adaptive controller raises the crucial problem of estimator convergence. Finally, Section 4 presents a numerical simulation showing the trade-o! between interesting settling time, which needs `time-behavior controllera, and estimator convergence, which needs tonal-behavior controller.

2. Problem setting in the time+frequency domain 2.1. Problem setting in the time domain Assume the system is of "nite order and can thus be described by the following model: y(l)"h(q\)u(l)#d(l)#b(l),

(1)

d(l)"CX(l),

(2)

X(l#1)"AX(l),

(3)

b(l)"s(q\)v(l).

(4)

The discrete real signals u(l) and y(l) are, respectively, the sampled input and output. The stochastic noise b(l) is assumed to be a white Gaussian noise v(l) "ltered by a stable "lter s(q\). The sub-system (C, A) models the disturbance d(l) as non-stabilizable mode, unstable (on the unit circle) and non-commandable with A"diag(a ,2, a , aH, , aH ) 3C*"* where *\ 2 *\ a "exp(2p i< /F ), C"[c 2c cH2cH ]3C* L L C *\ *\ where c "A exp(iP ). The values A and P are, reL L L L L spectively, the amplitude and the phase of a sinusoid of frequency < . The objective is to reject the "rst L N sinusoids of d(l) from the system output (1).

and to select the N ones that include one disturbance frequency component to reject. Thus, for a sampling frequency F , the frequency domain from 0 to F /2 is C C split into M frequency subbands. Each subband, indexed p, is located into [(p!1)F /2M; (p#1)F /2M]. The C C number of subbands must be chosen such that no more than one sinusoidal disturbance is present in each subband. The analysis consists in transforming the output signal y(l) into undersampled complex signals > (k). With the N down sampling operator [ M] and w"exp(!p i/M), the analysis implemented as a modulated "lter bank is > (k)"[ M](g(q\)y(l)wNl) N

(5)

Output y(l) is translated by !pF /2M in the frequency C domain by multiplication with wNl. Using the prototype low-pass FIR "lter g(q\) with a cut-o! frequency at F /2M, undersampling at the rate F /M is possible withC C out aliasing. The analysis "lter bank used can be interpreted as a time}frequency tool where the number M speci"es the time}frequency sampling. With M"1, there is only one band, no decimation and no low-pass "lter; the analysis "lter bank does not provide any frequency analysis. With MPR, there is an in"nity of subbands with in"nitely short frequency support, and the decimation is in"nite; the analysis "lter bank computes frequency analysis like discrete Fourier transform (DFT). For an intermediate value of M, the analysis "lter bank works in the time}frequency domain, it computes a discrete short time Fourier transform (STFT). The STFT is one of many time}frequency representations mapping a time signal into a function of two variables, corresponding to time and frequency. The synthesis consists in generating the input signal according to the undersampled complex signals ; (k). N With the upsampling operator [!M] and Re to design the real part, the synthesis is written as a modulated "lter bank: u(l)"Re w\Nl(g(q\)[!M]; (k)). N N

(6)

Each complex signal ; (k) is oversampled because of the N interpolating low-pass FIR "lter g(q\), and is translated by #pF /2M in the frequency domain by multiplication C with w\Nl. The synthesis of the control u(l) constitutes the inverse operation of the analysis. Thus, it can be interpreted as an inverse discrete short time Fourier transform.

2.2. Filter banks

2.3. Models in the time}frequency domain

To reject the "rst N sinusoids of d(l) we propose to split the whole frequency domain into many subbands,

Consider N selected subbands of index p, where each includes one sinusoid component to reject. The subbands

P. Micheau, P. Coirault / Automatica 36 (2000) 1659}1664

are selected such that they are not adjacent to each other. For a given subband indexed p, the model of system (1)}(4), where the ; (k) generated by the "lter bank N contributes to the input u(t) and where > (k) is obtained N from y(t), is described by the equations (7)}(10): > (k)"q\B! (q\); (k)#D (k)#B (k), N N N N N

(7)

B (k)"S (q\)< (k), N N N

(8)

D (k)D (q\)"0, N N

(9)

D (q\)"1!Q q\. N N

(10)

The downsampling and upsampling of M in (5) and (6) lead to the use of bu!ers of M samples, both for the acquisition of y(l) and for the reconstruction of u(l). This block processing saves computation time, but introduces in the system model (7) a delay d of the downsampled clock k. To reduce the number of parameters needed to identify the transfer function ! (q\), it is proposed to N replace it by ¹(q\)H (q\). The "rst term, ¹(q\), reN sults from the decimation and the low-pass "ltering. With an ideal FIR low-pass "lter g(q\) of size MM both at N the analysis and the synthesis, and with a decimation factor M, the term ¹(q\) converges towards a FIR of size 2M . Moreover, in practice, this term can include not E only signal processing operations, but also the complete chain of signal acquisition and reconstruction. The second term H (q\) describes the system behavior in the N subband p. For an ideal low-pass "lter g(q\), the frequency response of the model for the subband p, HI (e S+), must be equal to the frequency response of the N whole model hI (e S) in the subband p. Eq. (11) resumes this equality in the frequency domain, HI (e S+)"hI (e S>pN+) for u3]!p/M;#p/M[. (11) N For M"1, the subband model H is the discrete time N model; its impulse response follows from h(q\). This is a time-behavior model. For MPR, the subband model converges towards the frequency model HI (e S)PhI (e pN+); the subband model H can come down N to a complex coe$cient equal to the frequency response of h for u"#pp/M. This is a tonal-behavior model. Between these two extremes, the subband model could be approximated by a FIR "lter with N complex coe$& cients. We could call it a `time}frequency modela. For MP1 the FIR model needs many coe$cients N PR. & For MPR the FIR model needs few coe$cients N P1. Consequently, in open loop, the transient re& sponse to a step input ; (k) can be estimated to last N (d#2M #N )M¹ seconds. This clearly shows the E & C drawback between time and frequency: by increasing M to converge towards the frequency model, the transient time is also increased because MM PR and E N P1. &

1661

3. Adaptive feedback controller design and analysis The rejection of the N sinusoidal components of the disturbance d(t) from the output y(t) is equivalent to the rejection of the periodic component D (k) from > (k) for N N each subband indexed p. For that purpose, N independent adaptive feedback controllers are designed; each of them rejecting a D (k) component from > (k). For each N N subband, the corrector is implemented as an internal model controller according to Morari and Za"riou (1989): DK (k)"> (k)!q\B¹(q\)H(hK , q\); (k) N N N N

(12)

; (k)"!R DK (k). N N N

(13)

For every time k and every subband p, Eq. (12) estimates the periodic disturbance DK (k), which is equivalent to N observe one unstable pole of the disturbance model (9), (10). A perfect rejection is obtained when the complex number R compensates exactly the gain and the phase N shift of the time}frequency transfer function at the frequency of the sinusoidal control signal: R " N (QK \B¹(QK \)H(hK ,QK \))\. If the estimates hK and QK are N N N N N N exact, the corrector realizes a deadbeat control in (d#2M #N )M¹ seconds. This clearly shows the E & C trade-o! arising in the "lter bank approach for the control: by increasing M to converge towards the tonalbehavior controller, the settling time is also increased. Application of this regulator requires knowledge of the system characterized by h and the frequency of D (k) N N characterized by Q . The time variation of the physical N parameters justi"es the implementation of on-line identi"cation of h and Q . This is done by using two complex N N recursive least-squares estimators with forgetting factor: CRLS-j and CRLS-j . Forgetting factor 0(j(1 is mainly used to track time-varying parameters. This factor must be chosen as a trade-o! between tracking capability and noise sensitivity. Based on the theory of recursive identi"cation of Ljung and Soderstrom (1983), the CRLS-j is used to identify the FIR model para meters as follows: e (k)"D(QK (k!1), q\)(> (k)!hK (k!1)u (k)), N N N N N

(14)

hK (k)"hK (k!1)#P (k)t (k)He (k), N N N N N

(15)

P (k)\"j P (k!1)\#t (k)Ht (k) N N N N

(16)

where u (k)"q\B¹(q\)[; (k)2; (k!N #1)], N N N & t (k)"D(QK (k!1), q\)u (k) N N N and hK (k)"[hK (k),2,hK & (k)]. N N N, \

1662

P. Micheau, P. Coirault / Automatica 36 (2000) 1659}1664

3.1. Adaptive feedback controller

With the persistent excitation condition, E[t(k)H t(k)]'0, if the measurement noise is not correlated with the vector t(k), the estimated parameter hK converges to h. The decorrelation condition (19) is obtained when

3.2. Discussion about the estimator convergence

d#v(¹)#v(D)#v(S)'d(D)#d(S)

Estimator convergence can be subdivided into two phases: convergence at the beginning (convergence far from the convergence point) and convergence close to the convergence point (asymptotic convergence). In the present case, with a starting point such that the closed loop is stable, there is no problem with the convergence at the beginning because the deterministic transient dynamics provides an appropriate excitation. This is the self-tuning property. However, in steady state, the measured noise provides the persistent excitation, due to the closed loop. In this case, the use of an adaptive feedback corrector can raise a major problem: the correlation between measured noise and the command can force the estimator to converge towards the inverse of the corrector. Usually, to ensure algorithm stability, the estimator is frozen, or a random extra-signal is superimposed to the control. This part investigates analytically the convergence behavior in the case of stochastic noise. For the proposed adaptive controller, the simulation and the experimentation show that the main problem is the convergence of the CRLS-j . Therefore, we limit the convergence discussion to this estimator and we consider that QK "Q anywhere. The objective is to show estimator convergence far from the beginning; in this case, the disturbance is perfectly rejected from the system output. To simplify the expressions, the index p is omitted. With QK "Q and ¹(q\)H(h, q\)"!(q\), the a priori error is

where v(¹) is the lowest order of the polynomial ¹(q\) and d(D) is the highest order of the polynomial D(q\). According to Eq. (10), v(D) is equal to zero and d(D) is equal to one. The analysis of the stochastic noise implies that v(S)"1. In fact, the "rst terms of ¹ could be neglected such that v(¹)"3. Therefore, condition (20) is d#3'd(S), and it clearly depends on the noise autocorrelation, via S(q\), and on the system delay d"3. For a window g of length N "M M, the analysis of the E E stochastic noise implies that d(S)"M !1 only when E MPR. To conclude, if a window of length N 46M is E used, with a good frequency selectivity, the estimator can converge for M su$ciently large.

The estimator CRLS-j is used to estimate Q using N the disturbance model D (k) de"ned by Eq. (10). More N details on the CRLS-j are given in Micheau et al. (1995).

(20)

4. Simulation

(17)

Simulation results illustrate the trade-o! between the controller dynamics and the estimator convergence as a function of M. The deterministic disturbance comprises ¸"4 sinusoidal components of zero phase which are classi"ed according to their amplitude. For the two sinusoidal components which are to be controlled, the amplitudes A "A "1.0 and the frequencies < "147 Hz, < "303 Hz are employed. For the two sinusoidal components which are not to be controlled, the amplitudes A "A "0.1 and the frequencies < "69 Hz, < "382 Hz are employed (Fig. 1). The frequency sampling is F "1000 Hz. To simulate a comC plex system, we use the transfer function

where

Lihat lebih banyak...
Brief Paper

Adaptive controller using "lter banks to reject multi-sinusoidal disturbance夽 P. Micheau *, P. Coirault GAUS, Mechanical Department, Universite& de Sherbrooke, 2500 Boul. Universite& , Sherbrooke, Que& bec, Canada J1K 2R1 E! SIP, Universite& de Poitiers, 40 Av. du Recteur Pineau, 86022 Poitiers, France Received 10 February 1998; revised 18 June 1999; received in "nal form 10 February 2000

Abstract The purpose of this paper is to apply "lter banks to the control problem involving the rejection of multi-sinusoidal disturbance from output of slowly time-varying stable systems. The use of "lter banks allows to distribute the control e!ort in many independent adaptive controllers, each of them taking care of a sinusoidal component of the disturbance. By varying the "lter banks speci"cations, the method handles the trade-o! between a time-behavior controller, with interesting settling time and a tonal-behavior controller showing the properties of simpli"ed control, reduced computational time and on-line system identi"cation with the system output noise via the feedback loop. Numerical examples are presented to illustrate this trade-o!. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Adaptive control; Filter banks; Rejection; Sinusoidal signals; Convergence analysis

1. Introduction The motivation for this work was to design an adaptive feedback controller for active control of pulsed #ow. In terms of control, the objective is to reject a multisinusoidal disturbance from output of stable system. In the "eld of active control of sound, Nelson and Elliot (1992) used an adaptive feedforward approach to control the anti-noise with high precision. However, this feedforward approach needs a measured reference signal. Without a reference signal, the problem is to design a feedback loop that is able to give the same attenuation as an adaptive feedforward approach. The feedback loop achieves desirable rejection of sinusoid despite the presence of modeling uncertainty in the plant model and slowly time-varying sinusoidal disturbance. But, in the case of slowly time-varying system, the feedback controller must be adapted to achieve both stability and optimal reduction. 夽 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor K.W. Lim under the direction of Editor F.L. Lewis. * Corresponding author Tel.: #1-819-821-8000; fax: #1-819-8217163. E-mail address: [email protected] (P. Micheau).

In the "eld of feedback theory, a well-known Bode's theorem suggests that reduction in one frequency band must be traded o! against increase of sensitivity at other frequencies. Indeed, to obtain a perfect attenuation at one frequency, without undesirable property at other frequencies, the feedback controller must be narrow band. To deal with this control problem, di!erent controllers applied to real-world systems have been proposed. Sievers and Flotow (1992) have presented a global comparison of control methods for narrow band disturbance rejection such as higher harmonic control, tonal control, repetitive control, learning control, LMS adaptive feedforward "ltering, classical and modern control. The more complex design involving a LQ-based method requires an accurate model of the plant dynamics. On the other hand, the tonal controller is the simplest compensator because it ignores the underlying dynamics, except for the plant gain and the phase at the disturbance frequencies. Micheau, Coirault, Hardouin and Tartarin (1996) experimented with a controller working in a narrow band to reject the preponderant harmonic of a pulsed #ow. Good experimental results have motivated extension of the approach to the case of multiple sinusoids. For this purpose, the authors investigated "lter banks, which allow analysis and synthesis in many narrow bands. This approach is an adaptation for a control

0005-1098/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 0 7 2 - 8

1660

P. Micheau, P. Coirault / Automatica 36 (2000) 1659}1664

problem of the subbands echo canceller, using frequency techniques proposed by Gilloire and Vetterli (1988), developed by Perez and Amano (1990) and Tang, Camache and Flores (1995). Micheau, Coirault, Renault and Tartarin (1995) have successfully implemented this new controller in real situations. The objective of this paper is to show that an adaptive controller using "lter banks is between a time-behavior controller and a tonal-behavior controller. The paper is organized in three sections. In Section 2, the problem of the rejection of a multi-sinusoidal disturbance from a stable system as well as the signal processing tools are presented. Section 3 introduces the direct adaptive feedback controller. The on-line adaptation needs the identi"cation of the system and disturbance models in each subband, which is performed using a complex generalized recursive estimator with forgetting factor. However, without extra-signal or dead zone algorithm, this adaptive controller raises the crucial problem of estimator convergence. Finally, Section 4 presents a numerical simulation showing the trade-o! between interesting settling time, which needs `time-behavior controllera, and estimator convergence, which needs tonal-behavior controller.

2. Problem setting in the time+frequency domain 2.1. Problem setting in the time domain Assume the system is of "nite order and can thus be described by the following model: y(l)"h(q\)u(l)#d(l)#b(l),

(1)

d(l)"CX(l),

(2)

X(l#1)"AX(l),

(3)

b(l)"s(q\)v(l).

(4)

The discrete real signals u(l) and y(l) are, respectively, the sampled input and output. The stochastic noise b(l) is assumed to be a white Gaussian noise v(l) "ltered by a stable "lter s(q\). The sub-system (C, A) models the disturbance d(l) as non-stabilizable mode, unstable (on the unit circle) and non-commandable with A"diag(a ,2, a , aH, , aH ) 3C*"* where *\ 2 *\ a "exp(2p i< /F ), C"[c 2c cH2cH ]3C* L L C *\ *\ where c "A exp(iP ). The values A and P are, reL L L L L spectively, the amplitude and the phase of a sinusoid of frequency < . The objective is to reject the "rst L N sinusoids of d(l) from the system output (1).

and to select the N ones that include one disturbance frequency component to reject. Thus, for a sampling frequency F , the frequency domain from 0 to F /2 is C C split into M frequency subbands. Each subband, indexed p, is located into [(p!1)F /2M; (p#1)F /2M]. The C C number of subbands must be chosen such that no more than one sinusoidal disturbance is present in each subband. The analysis consists in transforming the output signal y(l) into undersampled complex signals > (k). With the N down sampling operator [ M] and w"exp(!p i/M), the analysis implemented as a modulated "lter bank is > (k)"[ M](g(q\)y(l)wNl) N

(5)

Output y(l) is translated by !pF /2M in the frequency C domain by multiplication with wNl. Using the prototype low-pass FIR "lter g(q\) with a cut-o! frequency at F /2M, undersampling at the rate F /M is possible withC C out aliasing. The analysis "lter bank used can be interpreted as a time}frequency tool where the number M speci"es the time}frequency sampling. With M"1, there is only one band, no decimation and no low-pass "lter; the analysis "lter bank does not provide any frequency analysis. With MPR, there is an in"nity of subbands with in"nitely short frequency support, and the decimation is in"nite; the analysis "lter bank computes frequency analysis like discrete Fourier transform (DFT). For an intermediate value of M, the analysis "lter bank works in the time}frequency domain, it computes a discrete short time Fourier transform (STFT). The STFT is one of many time}frequency representations mapping a time signal into a function of two variables, corresponding to time and frequency. The synthesis consists in generating the input signal according to the undersampled complex signals ; (k). N With the upsampling operator [!M] and Re to design the real part, the synthesis is written as a modulated "lter bank: u(l)"Re w\Nl(g(q\)[!M]; (k)). N N

(6)

Each complex signal ; (k) is oversampled because of the N interpolating low-pass FIR "lter g(q\), and is translated by #pF /2M in the frequency domain by multiplication C with w\Nl. The synthesis of the control u(l) constitutes the inverse operation of the analysis. Thus, it can be interpreted as an inverse discrete short time Fourier transform.

2.2. Filter banks

2.3. Models in the time}frequency domain

To reject the "rst N sinusoids of d(l) we propose to split the whole frequency domain into many subbands,

Consider N selected subbands of index p, where each includes one sinusoid component to reject. The subbands

P. Micheau, P. Coirault / Automatica 36 (2000) 1659}1664

are selected such that they are not adjacent to each other. For a given subband indexed p, the model of system (1)}(4), where the ; (k) generated by the "lter bank N contributes to the input u(t) and where > (k) is obtained N from y(t), is described by the equations (7)}(10): > (k)"q\B! (q\); (k)#D (k)#B (k), N N N N N

(7)

B (k)"S (q\)< (k), N N N

(8)

D (k)D (q\)"0, N N

(9)

D (q\)"1!Q q\. N N

(10)

The downsampling and upsampling of M in (5) and (6) lead to the use of bu!ers of M samples, both for the acquisition of y(l) and for the reconstruction of u(l). This block processing saves computation time, but introduces in the system model (7) a delay d of the downsampled clock k. To reduce the number of parameters needed to identify the transfer function ! (q\), it is proposed to N replace it by ¹(q\)H (q\). The "rst term, ¹(q\), reN sults from the decimation and the low-pass "ltering. With an ideal FIR low-pass "lter g(q\) of size MM both at N the analysis and the synthesis, and with a decimation factor M, the term ¹(q\) converges towards a FIR of size 2M . Moreover, in practice, this term can include not E only signal processing operations, but also the complete chain of signal acquisition and reconstruction. The second term H (q\) describes the system behavior in the N subband p. For an ideal low-pass "lter g(q\), the frequency response of the model for the subband p, HI (e S+), must be equal to the frequency response of the N whole model hI (e S) in the subband p. Eq. (11) resumes this equality in the frequency domain, HI (e S+)"hI (e S>pN+) for u3]!p/M;#p/M[. (11) N For M"1, the subband model H is the discrete time N model; its impulse response follows from h(q\). This is a time-behavior model. For MPR, the subband model converges towards the frequency model HI (e S)PhI (e pN+); the subband model H can come down N to a complex coe$cient equal to the frequency response of h for u"#pp/M. This is a tonal-behavior model. Between these two extremes, the subband model could be approximated by a FIR "lter with N complex coe$& cients. We could call it a `time}frequency modela. For MP1 the FIR model needs many coe$cients N PR. & For MPR the FIR model needs few coe$cients N P1. Consequently, in open loop, the transient re& sponse to a step input ; (k) can be estimated to last N (d#2M #N )M¹ seconds. This clearly shows the E & C drawback between time and frequency: by increasing M to converge towards the frequency model, the transient time is also increased because MM PR and E N P1. &

1661

3. Adaptive feedback controller design and analysis The rejection of the N sinusoidal components of the disturbance d(t) from the output y(t) is equivalent to the rejection of the periodic component D (k) from > (k) for N N each subband indexed p. For that purpose, N independent adaptive feedback controllers are designed; each of them rejecting a D (k) component from > (k). For each N N subband, the corrector is implemented as an internal model controller according to Morari and Za"riou (1989): DK (k)"> (k)!q\B¹(q\)H(hK , q\); (k) N N N N

(12)

; (k)"!R DK (k). N N N

(13)

For every time k and every subband p, Eq. (12) estimates the periodic disturbance DK (k), which is equivalent to N observe one unstable pole of the disturbance model (9), (10). A perfect rejection is obtained when the complex number R compensates exactly the gain and the phase N shift of the time}frequency transfer function at the frequency of the sinusoidal control signal: R " N (QK \B¹(QK \)H(hK ,QK \))\. If the estimates hK and QK are N N N N N N exact, the corrector realizes a deadbeat control in (d#2M #N )M¹ seconds. This clearly shows the E & C trade-o! arising in the "lter bank approach for the control: by increasing M to converge towards the tonalbehavior controller, the settling time is also increased. Application of this regulator requires knowledge of the system characterized by h and the frequency of D (k) N N characterized by Q . The time variation of the physical N parameters justi"es the implementation of on-line identi"cation of h and Q . This is done by using two complex N N recursive least-squares estimators with forgetting factor: CRLS-j and CRLS-j . Forgetting factor 0(j(1 is mainly used to track time-varying parameters. This factor must be chosen as a trade-o! between tracking capability and noise sensitivity. Based on the theory of recursive identi"cation of Ljung and Soderstrom (1983), the CRLS-j is used to identify the FIR model para meters as follows: e (k)"D(QK (k!1), q\)(> (k)!hK (k!1)u (k)), N N N N N

(14)

hK (k)"hK (k!1)#P (k)t (k)He (k), N N N N N

(15)

P (k)\"j P (k!1)\#t (k)Ht (k) N N N N

(16)

where u (k)"q\B¹(q\)[; (k)2; (k!N #1)], N N N & t (k)"D(QK (k!1), q\)u (k) N N N and hK (k)"[hK (k),2,hK & (k)]. N N N, \

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P. Micheau, P. Coirault / Automatica 36 (2000) 1659}1664

3.1. Adaptive feedback controller

With the persistent excitation condition, E[t(k)H t(k)]'0, if the measurement noise is not correlated with the vector t(k), the estimated parameter hK converges to h. The decorrelation condition (19) is obtained when

3.2. Discussion about the estimator convergence

d#v(¹)#v(D)#v(S)'d(D)#d(S)

Estimator convergence can be subdivided into two phases: convergence at the beginning (convergence far from the convergence point) and convergence close to the convergence point (asymptotic convergence). In the present case, with a starting point such that the closed loop is stable, there is no problem with the convergence at the beginning because the deterministic transient dynamics provides an appropriate excitation. This is the self-tuning property. However, in steady state, the measured noise provides the persistent excitation, due to the closed loop. In this case, the use of an adaptive feedback corrector can raise a major problem: the correlation between measured noise and the command can force the estimator to converge towards the inverse of the corrector. Usually, to ensure algorithm stability, the estimator is frozen, or a random extra-signal is superimposed to the control. This part investigates analytically the convergence behavior in the case of stochastic noise. For the proposed adaptive controller, the simulation and the experimentation show that the main problem is the convergence of the CRLS-j . Therefore, we limit the convergence discussion to this estimator and we consider that QK "Q anywhere. The objective is to show estimator convergence far from the beginning; in this case, the disturbance is perfectly rejected from the system output. To simplify the expressions, the index p is omitted. With QK "Q and ¹(q\)H(h, q\)"!(q\), the a priori error is

where v(¹) is the lowest order of the polynomial ¹(q\) and d(D) is the highest order of the polynomial D(q\). According to Eq. (10), v(D) is equal to zero and d(D) is equal to one. The analysis of the stochastic noise implies that v(S)"1. In fact, the "rst terms of ¹ could be neglected such that v(¹)"3. Therefore, condition (20) is d#3'd(S), and it clearly depends on the noise autocorrelation, via S(q\), and on the system delay d"3. For a window g of length N "M M, the analysis of the E E stochastic noise implies that d(S)"M !1 only when E MPR. To conclude, if a window of length N 46M is E used, with a good frequency selectivity, the estimator can converge for M su$ciently large.

The estimator CRLS-j is used to estimate Q using N the disturbance model D (k) de"ned by Eq. (10). More N details on the CRLS-j are given in Micheau et al. (1995).

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4. Simulation

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Simulation results illustrate the trade-o! between the controller dynamics and the estimator convergence as a function of M. The deterministic disturbance comprises ¸"4 sinusoidal components of zero phase which are classi"ed according to their amplitude. For the two sinusoidal components which are to be controlled, the amplitudes A "A "1.0 and the frequencies < "147 Hz, < "303 Hz are employed. For the two sinusoidal components which are not to be controlled, the amplitudes A "A "0.1 and the frequencies < "69 Hz, < "382 Hz are employed (Fig. 1). The frequency sampling is F "1000 Hz. To simulate a comC plex system, we use the transfer function

where

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