EP2436003B1 - Method and device for narrow-band noise suppression in a vehicle passenger compartment - Google Patents

Description

The present invention relates to a method and a device for rejecting noise in a passenger compartment of a vehicle, in particular an automobile, by active control. It has applications in the industrial field of motor vehicles, this term being taken in the broad sense including including light vehicles, heavy, road, rail, boats, barges, submarines, and that of electroacoustic equipment as per example the car radios to which such a function can be added.

Some acoustic noises occurring in a passenger compartment of a vehicle may have a broad spectrum, others may instead be approximately single-frequency. This is particularly the case of the noise generated by the rotation of the motor shaft, known as a buzz that results in a noise whose spectrum is composed of lines whose frequencies are proportional to the frequency of rotation of the motor. 'motor shaft with a fundamental and harmonics.

These frequencies are variable according to the rotational speed of the motor shaft, nevertheless they can be known precisely thanks to the information from the tachometer generally integrated into the vehicle.

It has already been proposed to reduce or eliminate these noises by active acoustic means. One can mention in this regard an assessment of the state of the art in the field of active control applied to motor vehicles, made by Elliot in December 2008 in an article entitled "A review of active noise and vibration control in road vehicles" (ISVR technical memorandum No. 981 - University of Southampton ).

There are two main active acoustic control structures. Firstly, a so-called feedforward or precompensation structure. This structure requires a loudspeaker, an error microphone at which it seeks to cancel the noise and a controller receiving a reference signal, correlated with the signal to be canceled, producing a correction signal sent on the speaker. This structure is schematically represented on the Figure 1 of the state of the art. This structure has given rise to a series of algorithms based on the least square square (LMS) method: Fx-LMS, FR-LMS, whose goal is to minimize the least squares signal. from the error microphone, by exploiting the reference signal.

Still in the case of a structure called "feedforward", we can cite the article by Sano and All. titled "NV Countermeasure Technology for a Cylinder - On-Demand Engine-Development of an Active Booming Noise Control Applied Adaptive Notch Filter (SAE 2004). The authors present an algorithm based on an adaptive notch filter, the noise attenuation frequency being known. The device is based on an algorithm whose structure is of the "feedforward" type, named FR-SAN, which is an adaptation of the FR-LMS algorithm in the case where the noise to be attenuated is of the single-frequency type. When implementing this algorithm, the problems arising when the transfer function of the passenger compartment varies, for example as a function of the number of passengers, are not taken into account. Moreover, with this algorithm, it is not possible to know otherwise than experimentally the behavior of the control system at frequencies other than those to which it acts.

Second, a structure called "feedback" or feedback. This structure is schematically represented on the Figure 2 of the state of the art. This structure does not require a reference signal, unlike the so-called "feedforward" structure. It is then in a conventional feedback structure and all the tools of the conventional automatic (including measurement of robustness, stability analysis, performance) can be used. In particular, a robustness analysis of the looped system with respect to the variation of the transfer function of the passenger compartment can be performed. One can also study the frequency behavior of the system, not only at the disturbance rejection frequency, but also at other frequencies.

The document AMARA, KABAMBA, ULSOY: "Adaptive sinusoidal disturbance rejection in linear discrete-time systems-Part II: Experiments "discloses a finite impulse response filter and the implementation of a correction control law comprising applying a Youla parameter to a central corrector and such that only the Youla parameter has frequency dependent coefficients noise to be attenuated in said correction control law, the central corrector having fixed coefficients, the Youla parameter being in the form of a finite impulse response filter.

The document EP 0 578 212 describes an infinite impulse response filter, but the coefficients used are not used to calculate the Youla parameter.

The present invention is classified in this second type of structure, called "feedback". It relates more particularly to an active process in real time, by feedback, attenuation of a narrow-band noise, essentially single-frequency at at least one determined frequency, in a passenger compartment of a vehicle by emission of a sound by at least one a transducer, typically a loudspeaker, controlled with a signal u (t) or U (t) in a monovariable or multivariable case respectively, generated by a programmable computer, as a function of an acoustic measurement signal y (t) or Y (t) depending on the case, made by at least one acoustic sensor, typically a microphone, the use of a sensor corresponding to a single-variable case and the use of several sensors corresponding to a multivariable case, and in a first phase of design, the electroacoustic behavior of the assembly formed by the passenger compartment, the transducer, and the sensor being modeled by an electroacoustic model in the form of an electronic transfer function. which is determined and calculated, a correction control law then being determined and calculated from a global model of the system in which the correction control law is applied to the electroacoustic transfer function whose output additionally receives a signal of noise to attenuate p (t) to give the signal y (t) or Y (t) in said design phase, said correction control law making it possible to produce the signal u (t) or U (t) as a function of acoustic measurements y (t) or Y (t), and in a second phase of use, said calculated correction control law being used in the computer to produce the signal u (t) or U (t) then sent to the transducer according to the signal y (t) or Y (t) received from the sensor for attenuation of said noise.

• on récupère la fréquence courante du bruit à atténuer,
• on fait calculer au calculateur la loi de commande de correction, comprenant le correcteur central avec le paramètre de Youla, en utilisant, pour le paramètre de Youla les coefficients mémorisés d'une fréquence déterminée correspondant à la fréquence courante du bruit à atténuer.

According to the invention, a correction control law is implemented comprising the application of a Youla parameter to a central corrector and such that only the Youla parameter has coefficients dependent on the frequency of the noise to be attenuated in said correction control law, the central corrector having fixed coefficients, the Youla parameter being in the form of an infinite impulse response filter and after determining and calculating the correction control law, it is stored in a memory of the calculator at minus said variable coefficients, preferably in a table as a function of the determined noise frequency / frequencies p (t) used in the design phase and in the use phase, in real time:

• we recover the current frequency of the noise to be attenuated,
• the correction control law, comprising the central corrector with the Youla parameter, is made to the computer, using, for the Youla parameter, the coefficients stored at a determined frequency corresponding to the current frequency of the noise to be attenuated.

In other words, a correction control law is implemented comprising a fixed coefficient part called a central corrector and a variable coefficient part as a function of the frequency of the noise to be attenuated, which is here a parameter of Youla, the part the variable coefficient corrector being an infinite impulse response filter and after determining and calculating the correction control law, at least one of said variable coefficients is stored in a computer memory, preferably in a table according to the frequency (s); noise determinations p (t) used in the design phase and in the utilization phase, in real time: the current frequency of the noise to be attenuated is recovered and the calculator is computed with the correction control law, including the corrector fixed coefficient coefficient with the variable coefficient part using for the variable coefficient part, the coefficients nts stored at a determined frequency corresponding to the current frequency of the noise to be attenuated. Thus, in the context of the invention, a central corrector with fixed coefficients is used for the attenuation of the noise at at least one determined frequency, to which is added a block with variable coefficients which is a parameter of Youla in the form of a Q block of Youla.
the monovariable case can be called SISO ("single input single outpout": an input, an output) and the multivariable case can be called MIMO ("multi-inputs-multi-outputs": multiple inputs, multiple outputs).

The term signal in the context of the invention relates both to analog signals such as the electrical signal coming out of the microphone itself as digital signals such as the output signal of the block Q (q ^{- 1} ) Youla. In addition, it is understood that the terms transducer and sensor are used in a generic and functional manner and that in practice interface electronics are associated therewith such as in particular analog-digital or digital-to-analog converters, one or more filters. anti-aliasing, amplifiers (for the speaker (s) and microphone). The term signal also covers single-ended cases (a sensor and thus a single input of acoustic measurements) and multivariable (several sensors and therefore several inputs of acoustic measurements) and regardless of the number of loudspeakers. Thus, the invention can be applied both to a monovariable case (a single microphone, that is to say a single location where the noise will be attenuated in the cabin), as multivariable cases (several microphones, that is to say, as many locations where the noise will be attenuated). It is also understood that the invention is applicable both to attenuation of a noise which is at a particular fixed frequency over time (for example noise of a truck refrigeration compressor) that a noise whose frequency can evolve over time and in this case, in the design phase, it is preferable to determine and calculate Youla parameters, block Q (q ^{- 1} ) , for several determined frequencies in order to take during the phase of using the result of the calculation of the Youla parameter for a determined frequency which corresponds (equal to or close to, which in fact corresponds best or is interpolated otherwise) to the current frequency of the noise to be attenuated. It is understood that the more the frequency mesh will be fine, the more likely it is to find a Youla parameter calculation result with a determined frequency which corresponds to the frequency of the current noise to be attenuated. Indeed, we will see that in the correction control law, only the Youla parameter is variable (its coefficients in practice) as a function of the frequency of the noise, unlike the coefficients of the central corrector which remain fixed, independent of the frequency noise.

It may be noted that in a very different domain, the parameterization of Youla has already been used for sinusoidal disturbance rejection: it is the vibration control of an active suspension. The corresponding article is: "Adaptive narrow disturbance applied to an active suspension- an internai model approach" (Automatica 2005), whose authors are ID Landau , et al. In this last device, the Youla parameter is in the form of a finite impulse response filter (transfer function with a single polynomial without denominator) whereas in the present invention we will see that this parameter of Youla is in the form of an infinite impulse response filter (transfer function with a numerator and denominator). In addition, in this paper, Youla parameter coefficients are computed using an adaptive device, that is, disturbance frequency information is not known unlike the present invention where this frequency is known from measurements, including a revolution counter, and where the Youla parameter coefficients are stored in tables for their use in real time. The devices and method of the invention allow a much greater robustness of the control law. In the case of the invention, this corresponds to an insensitivity of the control law to the parametric variations of the electroacoustic model, that is to say to the variations of the configuration of the passenger compartment, which, of an industrial point of view, is a capital element.

We can also mention the article Mark A.Mcever, 44th AIAA / ASME / ASCE / AHS Structures, structural dynamics and material conference, 7-10 april 2003, "Adaptive Control for interior noise control in rocket fairings" . Here again, the Youla parameter is a finite impulse response (FIR) filter, which poses problems for the robustness of the system, the algorithm is adaptive and is not dedicated to the rejection of a particular frequency.

Finally, in the field of vibration control of a motor vehicle, mention may also be made of the article: "Active control of engine-induced vibration in automotive vehicles using disturbance observe gain scheduling" in Bohn's control engineering practice 12 (2004) 1029-1039 et al. The control law presented uses a state observer whose several elements are variable as a function of the frequency to be rejected, which leads to the control law having a number of variant parameters much larger than the optimal number. On the other hand, the present invention ensures that the number of variant parameters of the control law is minimal.

• la phase de conception est effectuée dans un calculateur programmable,
• le paramètre de Youla est déterminé et calculé par discrétisation d'une cellule du second ordre continue,
• dans le deuxième temps de la phase de conception, on détermine et calcule les polynômes Ro(q ^-1) et So(q ^-1) du correcteur central de manière à ce que ledit correcteur central seul garantisse des marges de gain et de phase, sans avoir d'objectif de rejet de perturbation,
• dans le cas monovariable, dans la phase de conception :
  • a) - dans un premier temps, on utilise un modèle électroacoustique linéaire, le modèle électroacoustique étant sous forme d'une fonction de transfert électroacoustique rationnelle discrète, et on détermine et calcule ledit modèle électroacoustique par excitation acoustique de l'habitacle par le(s) transducteur(s) et mesures acoustiques par le capteur puis application d'un procédé d'identification de système linéaire avec les mesures et le modèle,
  • b) - dans un deuxième temps, on met en oeuvre un correcteur central appliqué au modèle électroacoustique déterminé et calculé, le correcteur central étant sous forme d'un correcteur RS de deux blocs 1/So(q ^-1) et Ro(q ^-1), dans le correcteur central, le bloc 1/So(q ^-1) produisant le signal u(t) et recevant en entrée le signal de sortie inversé du bloc Ro(q ^-1), ledit bloc Ro(q ^-1) recevant en entrée le signal y(t) correspondant à la sommation du bruit p(t) et de la sortie de la fonction de transfert électroacoustique du modèle électroacoustique, et on détermine et calcule le correcteur central,
  • c) - dans un troisième temps, on adjoint un paramètre de Youla, qui est donc un bloc de transfert à coefficients variables, au correcteur central pour former la loi de commande de correction, le paramètre de Youla étant sous forme d'un bloc Q(q^-1), filtre à réponse impulsionnelle infinie, avec $Q\left(q^{-1}\right)=\frac{\beta \left(q^{-1}\right)}{\alpha \left(q^{-1}\right)}$
    
    adjoint au correcteur central RS, ledit bloc Q(q^-1) de Youla recevant une estimation du bruit obtenue par calcul à partir des signaux u(t) et y(t) et en fonction de la fonction de transfert électroacoustique et le signal en sortie dudit bloc Q(q ^{- 1}) de Youla étant soustrait au signal inversé de Ro(q ^-1) envoyé à l'entrée du bloc 1/So(q ^-1) du correcteur central RS, et on détermine et calcule le paramètre de Youla, donc le bloc de transfert à coefficients variables, dans la loi de commande de correction comportant le correcteur central auquel est associé le paramètre de Youla pour au moins une fréquence de bruit p(t) dont au moins la fréquence déterminée du bruit à atténuer,
  et dans la phase d'utilisation, en temps réel :
  • on récupère la fréquence courante du bruit à atténuer,
  • on fait calculer au calculateur la loi de commande de correction, comprenant le correcteur RS avec le paramètre de Youla, en utilisant, pour le paramètre de Youla les coefficients qui ont été calculés pour une fréquence de bruit correspondant à la fréquence courante du bruit à atténuer, les coefficients de Ro(q ^-1) et So(q ^-1) étant fixes,
• dans la phase de conception (cas monovariable), on effectue les opérations suivantes :
  • a) - dans le premier temps, on excite acoustiquement l'habitacle en appliquant au(x) transducteur(s) un signal d'excitation dont la densité spectrale est sensiblement uniforme sur une bande de fréquence utile,
  • b) - dans le deuxième temps, on détermine et calcule les polynômes Ro(q ^-1) et So(q ^-1) du correcteur central de manière à ce que ledit correcteur central soit équivalent à un correcteur calculé par placement des pôles de la boucle fermée dans l'application du correcteur central à la fonction de transfert électroacoustique, n pôles de la boucle fermée étant placés sur les n pôles de la fonction de transfert du système électroacoustique,
  • c) - dans le troisième temps, on détermine et calcule les numérateur et dénominateur du bloc Q(q ^{- 1}) de Youla au sein de la loi de commande de correction pour au moins une fréquence de bruit p(t) dont au moins la fréquence déterminée du bruit à atténuer, ceci en fonction d'un critère d'atténuation, le bloc Q(q ^-1 ) étant exprimé sous forme d'un rapport β(q^-1)/α(q^-1), afin d'obtenir des valeurs de coefficients des polynômes α(q ^-1) et β(q ^-1) pour la/chacune des fréquences, , le calcul de β(q^-1) et α(q^-1) se faisant par l'obtention d'une fonction de transfert discrète Hs(q^-1)/α(q^-1) résultant de la discrétisation d'une cellule du second ordre continu, le polynôme β(q^-1) se calculant par la résolution d'une équation de Bézout,
  et dans la phase d'utilisation, en temps réel, on effectue les opérations suivantes :
  • on fait calculer au calculateur la loi de commande de correction, correcteur central à coefficients fixes avec paramètre de Youla à coefficients variables, pour produire le signal u(t) envoyé au/aux transducteurs, en fonction des mesures acoustiques y(t) et en utilisant pour le bloc Q(q ^{- 1}) de Youla les valeurs des coefficients des polynômes α(q ^-1) et β(q ^-1) déterminées et calculées pour une fréquence déterminée correspondant à la fréquence courante,
• le calcul de l'estimation du bruit est obtenu par application du numérateur de la fonction de transfert électroacoustique à u(t) et soustraction du résultat à l'application de y(t) au dénominateur de la fonction de transfert électroacoustique,
• on utilise pour le modèle électroacoustique une fonction de transfert électroacoustique de la forme : $$\frac{y\left(t\right)}{u\left(t\right)}=\frac{q^{-d}B\left(q^{-1}\right)}{A\left(q^{-1}\right)}$$
  
  où d est le nombre de périodes d'échantillonnage de retard du système, B et A sont des polynômes en q ^-1 de la forme : $$B\left(q^{-1}\right)={\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">b</figure-callout>}}_0+{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">b</figure-callout>}}_1\cdot q^{-1}+\cdots b_{\mathit{nb}}\cdot q^{-\mathit{nb}}$$
  
  $$A\left(q^{-1}\right)=1+a_1\cdot q^{-1}+\cdots a_{\mathit{na}}\cdot q^{-\mathit{na}}$$
  
  les b_i et a_i étant des scalaires, et q ^-1 étant l'opérateur retard d'une période d'échantillonnage, et le calcul de l'estimation du bruit est obtenu par application de la fonction q ^{- d}B(q ^-1 ) à u(t) et soustraction du résultat à l'application de y(t) à la fonction A(q ^-1),
• pour le temps b), on détermine et calcule les polynômes Ro(q ^-1) et So(q ^-1) du correcteur central par une méthode de placement des pôles, n pôles dominants de la boucle fermée munie du correcteur central étant choisis égaux aux n pôles de la fonction de transfert électroacoustique et que m pôles auxiliaires sont des pôles situés en haute fréquence,
• dans la phase de conception :
  • a) - dans un premier temps, on utilise un modèle électroacoustique linéaire, le modèle électroacoustique étant sous forme de représentation d'état de blocs matriciels H, W, G et q^-1.I, G étant une matrice de transition, H étant une matrice d'entrée, W étant une matrice de sortie et I la matrice identité, ladite représentation d'état pouvant s'exprimer par une équation de récurrence : $$X\left(t+\mathit{Te}\right)=G\cdot X\left(t\right)+H\cdot U\left(t\right)$$
    
    $$Y\left(t\right)=W\cdot X\left(t\right)$$
    
    avec X(t) : vecteur d'état, U(t) : vecteur des entrées, Y(t) : vecteur des sorties,
    et on détermine et calcule ledit modèle électroacoustique par excitation acoustique de l'habitacle par les transducteurs et mesures acoustiques par les capteurs puis application d'un procédé d'identification de système linéaire avec les mesures et le modèle,
  • b) - dans un deuxième temps, on met en oeuvre un correcteur central appliqué au modèle électroacoustique déterminé et calculé, le correcteur central étant sous forme observateur d'état et retour d'état estimé qui exprime X̂ un vecteur d'état de l'observateur itérativement en fonction de Kf un gain de l'observateur, Kc un vecteur de retour sur l'état estimé, ainsi que du modèle électroacoustique précédemment déterminé et calculé, soit $$\widehat{X}\left(t+\mathit{Te}\right)=G\cdot \widehat{X}\left(t\right)+H\cdot U\left(t\right)+\mathit{Kf}\cdot \left(Y\left(t\right)-W\cdot \widehat{X}\left(t\right)\right)$$
    
    avec une commande U(t) = -Kc · X̂(t),
    et on détermine et calcule ledit correcteur central,
  • c) - dans un troisième temps, on adjoint un paramètre de Youla, qui est donc un bloc de transfert à coefficients variables, au correcteur central pour former la loi de commande de correction, le paramètre de Youla étant sous forme d'un bloc Q multivariables, de matrices d'état AQ, BQ, CQ, adjoint au correcteur central exprimé également sous forme de représentation d'état, bloc Q dont la sortie additionnée à la sortie du correcteur central produit un signal qui forme l'opposée de U(t) et dont l'entrée reçoit le signal Y(t) auquel est soustrait le signal W·X̂(t), et on détermine et calcule le paramètre de Youla, donc le bloc de transfert à coefficients variables, dans la loi de commande de correction comportant le correcteur central auquel est associé le paramètre de Youla pour au moins une fréquence de bruit P(t) dont au moins la fréquence déterminée du bruit à atténuer, le calcul des coefficients des matrices AQ, BQ, CQ se faisant par l'obtention de fonctions de transfert discrètes Hsi(q^-1)/αi(q^-1) résultant de la discrétisation de cellules du second ordre continues et par un placement de pôles ainsi que la résolution d'équations de rejet asymptotique,
  et dans la phase d'utilisation, en temps réel :
  • on récupère la fréquence courante du bruit à atténuer,
  • on fait calculer au calculateur la loi de commande de correction, comprenant le correcteur central à coefficients fixes avec le paramètre de Youla à coefficients variables, en utilisant, pour le paramètre de Youla les coefficients qui ont été calculés pour une fréquence de bruit correspondant à la fréquence courante du bruit à atténuer,
• dans la phase de conception (cas multivariables), on effectue les opérations suivantes :
  • a) - dans le premier temps, on excite acoustiquement l'habitacle en appliquant aux transducteurs des signaux d'excitation dont la densité spectrale est sensiblement uniforme sur une bande de fréquence utile, les signaux d'excitation étant décorrélés entre-eux,
  • b) - dans le deuxième temps, on détermine et calcule le correcteur central de manière à ce qu'il soit équivalent à un correcteur avec observateur d'état et retour sur l'état calculé par placement des pôles dans l'application du correcteur central à la fonction de transfert électroacoustique, à cette fin on choisit un gain de l'observateur nul, soit Kf=0 (choix du gain de l'observateur égal à la matrice nulle), et un gain de retour d'état Kc choisi de façon à introduire des pôles haute fréquences dans la boucle afin d'assurer la robustesse de la loi de commande munie du paramètre de Youla, le calcul de Kc étant par exemple effectué par optimisation LQ (linéaire quadratique),
  • c) - dans le troisième temps, on détermine et calcule en considérant une représentation d'observateur d'état augmenté, les pôles du bloc Q de Youla au sein de la loi de commande de correction pour au moins une fréquence de bruit P(t) dont au moins la fréquence déterminée du bruit à atténuer en fonction d'un critère d'atténuation, afin d'obtenir des valeurs de coefficients du paramètre de Youla pour la/chacune des fréquences,
  et dans la phase d'utilisation, en temps réel, on effectue les opérations suivantes :
  • on fait calculer au calculateur la loi de commande de correction, correcteur central à coefficients fixes avec paramètre de Youla à coefficients variables, pour produire le signal U(t) envoyé au/aux transducteurs, en fonction des mesures acoustiques Y(t) et en utilisant pour le paramètre de Youla les valeurs des coefficients déterminées et calculées pour une fréquence déterminée correspondant à la fréquence courante,
• dans le deuxième temps, le calcul de Kc est effectué par optimisation LQ (linéaire quadratique),
• le procédé est adapté à un ensemble de fréquences déterminées de bruit à atténuer et on répète le temps c) pour chacune des fréquences déterminées et, en phase d'utilisation lorsque aucune des fréquences déterminées ne correspond à la fréquence courante du bruit à atténuer, on fait une interpolation à ladite fréquence courante pour les valeurs des coefficients du bloc Q de Youla à partir des valeurs de coefficients dudit bloc Q de Youla connus pour les fréquences déterminées,
• les signaux sont échantillonnés à une fréquence Fe et au temps a) on utilise une bande de fréquence utile du signal d'excitation qui est sensiblement [0, Fe/2],
• le signal d'excitation a une densité spectrale uniforme,
• avant la phase d'utilisation, on ajoute à la phase de conception un quatrième temps d) de vérification de la stabilité et de la robustesse du modèle du système électroacoustique et de la loi de commande de correction, correcteur central avec paramètre de Youla, obtenus précédemment aux temps a) à c) en faisant une simulation de la loi de commande de correction obtenue aux temps b) et c) appliqué au modèle électroacoustique obtenu au temps a) pour la/les fréquences déterminées et lorsque un critère prédéterminé de stabilité et/ou robustesse n'est pas respecté, on réitère au moins le temps c) en modifiant le critère d'atténuation,
• dans le quatrième temps d) de la phase de conception, lorsqu'un critère prédéterminé de stabilité et/ou robustesse n'est pas respecté, on réitère en outre le temps b) en modifiant les pôles auxiliaires de la boucle fermée
• la phase de conception est une phase préalable et elle est effectuée une fois, préalablement à la phase d'utilisation, avec mémorisation des résultats des déterminations et calculs pour utilisation dans la phase d'utilisation, (par exemple, dans le cas monovariable, mémorisation des coefficients des blocs R, S et Q pour la loi de commande de correction calculée ainsi que de la fonction de transfert électroacoustique calculée, pour le bloc Q des tables de coefficients pouvant être mises en oeuvre du fait de calculs pour plusieurs fréquences déterminées)
• on sélectionne le critère d'atténuation en fonction d'au moins un des deux éléments suivants : la profondeur (amplitude) de l'atténuation et la largeur de bande de l'atténuation,
• la fréquence courante du bruit à atténuer est récupérée à partir d'une mesure compte-tour d'un moteur du véhicule.

In various embodiments of the invention, the following means can be used alone or in any technically possible combination, are employed:

• the design phase is performed in a programmable computer,
• the Youla parameter is determined and calculated by discretization of a continuous second order cell,
• in the second stage of the design phase, the polynomials Ro ( q ^-1 ) and So ( q ^-1 ) of the central corrector are determined and calculated so that the central corrector alone guarantees gain and phase margins, without having a disturbance rejection objective,
• in the monovariable case, in the design phase:
  • a) - initially, a linear electroacoustic model is used, the electroacoustic model being in the form of a discrete rational electroacoustic transfer function, and said electroacoustic model is determined and calculated by acoustic excitation of the passenger compartment by the ) transducer (s) and acoustic measurements by the sensor then application of a linear system identification method with the measurements and the model,
  • b) - in a second step, we implement a central corrector applied to the determined and calculated electroacoustic model, the central corrector being in the form of a corrector RS of two blocks 1 / So ( q ^-1 ) and Ro ( q ^{- 1} ), in the central corrector, the block 1 / So ( q ^-1 ) producing the signal u (t) and receiving as input the inverted output signal of the block Ro ( q ^-1 ), said block Ro ( q ^-1 ) receiving as input the signal y (t) corresponding to the summation of the noise p (t) and the output of the electroacoustic transfer function of the electroacoustic model, and the central corrector is determined and calculated,
  • c) - in a third step, a Youla parameter, which is a transfer block with variable coefficients, is added to the central corrector to form the correction control law, the Youla parameter being in the form of a block Q (q ^-1 ), infinite impulse response filter, with $Q\left(q^{-1}\right)=\frac{\beta \left(q^{-1}\right)}{\alpha \left(q^{-1}\right)}$
    
    assistant to the central corrector RS, said Q (q ^-1 ) block of Youla receiving an estimate of the noise obtained by calculation from the signals u (t) and y (t) and as a function of the electroacoustic transfer function and the signal in output of said block Q (q ^{- 1} ) of Youla being subtracted from the inverted signal of Ro ( q ^-1 ) sent to the input of the block 1 / So ( q ^-1 ) of the central corrector RS, and the parameter is determined and calculated of Youla, therefore the transfer block with variable coefficients, in the correction control law comprising the central corrector which is associated with the Youla parameter for at least one noise frequency p (t) of which at least the determined frequency of the noise at mitigate,
  and in the use phase, in real time:
  • we recover the current frequency of the noise to be attenuated,
  • calculating the correction control law, comprising the corrector RS with the Youla parameter, using, for the Youla parameter, the coefficients that have been calculated for a noise frequency corresponding to the current frequency of the noise to be attenuated , the coefficients of Ro ( q ^-1 ) and So ( q ^-1 ) being fixed,
• in the design phase (monovariable case), the following operations are performed:
  • a) - in the first stage, the cabin is acoustically excited by applying to the transducer (s) an excitation signal whose spectral density is substantially uniform over a useful frequency band,
  • b) - in the second step, the polynomials Ro ( q ^-1 ) and So ( q ^-1 ) of the central corrector are determined and calculated so that said central corrector is equivalent to a corrector calculated by placing the poles of the closed loop in the application of the central corrector to the electroacoustic transfer function, n poles of the closed loop being placed on the n poles of the transfer function of the electroacoustic system,
  • c) - in the third step, one determines and calculates the numerator and denominator of the block Q (q ^{- 1} ) of Youla within the law of control of correction for at least a frequency of noise p (t) of which at least the determined frequency of the noise to be attenuated, this according to an attenuation criterion, the block Q (q ^{- 1} ) being expressed in the form of a ratio β (q ^-1 ) / α (q ^-1 ), in order to obtain coefficient values of the polynomials α (q ^-1 ) and β ( q ^-1 ) for the / each of the frequencies, the calculation of β (q ^-1 ) and α (q ^-1 ) being obtained by obtaining a discrete transfer function Hs (q ^-1 ) / α (q ^-1 ) resulting from the discretization of a cell of the second continuous order , the polynomial β (q ^-1 ) being calculated by solving a Bézout equation,
  and in the use phase, in real time, the following operations are performed:
  • the correction control law, a fixed coefficient central corrector with Youla parameter with variable coefficients, is made to calculate the computer to produce the signal u (t) sent to the transducer (s), as a function of the acoustic measurements y (t) and using for the Q (q ^{- 1} ) block of Youla the values of the coefficients of the polynomials α ( q ^-1 ) and β ( q ^-1 ) determined and calculated for a determined frequency corresponding to the current frequency,
• the calculation of the noise estimate is obtained by applying the numerator of the electroacoustic transfer function to u (t) and subtracting the result from the application of y (t) to the denominator of the electroacoustic transfer function,
• an electroacoustic transfer function of the form is used for the electroacoustic model: $$\frac{\mathrm{there}\left(t\right)}{u\left(t\right)}=\frac{q^{-d}B\left(q^{-1}\right)}{\mathrm{AT}\left(q^{-1}\right)}$$
  
  where d is the number of delay sampling periods of the system, B and A are polynomials in q ^-1 of the form: $$B\left(q^{-1}\right)={\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">b</figure-callout>}}_0+{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">b</figure-callout>}}_1\cdot q^{-1}+\cdots b_{\mathit{nb}}\cdot q^{-\mathit{nb}}$$
  
  $$\mathrm{AT}\left(q^{-1}\right)=1+{\mathrm{at}}_1\cdot q^{-1}+\cdots {\mathrm{at}}_{\mathit{n\; /\; A}}\cdot q^{-\mathit{n\; /\; A}}$$
  
  the b _i and a _i being scalars, and q ^-1 being the delay operator of a sampling period, and the calculation of the noise estimate is obtained by applying the q ^{- d function} B (q ^{- 1} ) to u (t) and subtraction of the result from the application of y (t) to the function A ( q ^-1 ),
• for time b), the polynomials Ro ( q ^-1 ) and So ( q ^-1 ) of the central corrector are determined and calculated by a method of placing the poles, n dominant poles of the closed loop provided with the central corrector being chosen equal. to the n poles of the electroacoustic transfer function and m auxiliary poles are poles located at high frequency,
• in the design phase:
  • a) - firstly, a linear electroacoustic model is used, the electroacoustic model being in the form of a state representation of matrix blocks H, W, G and q ^-1 .I, G being a transition matrix, H being an input matrix, W being an output matrix and I the identity matrix, said state representation being able to express itself by a recursion equation: $$X\left(t+\mathit{You}\right)=\mathrm{BOY\; WUT}\cdot X\left(t\right)+H\cdot U\left(t\right)$$
    
    $$Y\left(t\right)=W\cdot X\left(t\right)$$
    
    with X (t) : state vector, U (t) : vector of the inputs, Y (t) : vector of the outputs,
    and said electroacoustic model is determined and calculated by acoustic excitation of the passenger compartment by the transducers and acoustic measurements by the sensors, then application of a linear system identification method with the measurements and the model,
  • b) - in a second step, a central corrector is applied to the determined and calculated electroacoustic model, the central corrector being in state observer form and estimated state feedback which expresses X a state vector of the iteratively observes, according to Kf, a gain of the observer, Kc a vector of return on the estimated state, as well as the electroacoustic model previously determined and calculated, $$\widehat{X}\left(t+\mathit{You}\right)=\mathrm{BOY\; WUT}\cdot \widehat{X}\left(t\right)+H\cdot U\left(t\right)+\mathit{kf}\cdot \left(Y\left(t\right)-W\cdot \widehat{X}\left(t\right)\right)$$
    
    with a command U ( t ) = -Kc · X ( t ),
    and said central corrector is determined and calculated,
  • c) - in a third step, a Youla parameter, which is a transfer block with variable coefficients, is added to the central corrector to form the correction control law, the Youla parameter being in the form of a block Q multivariable, of state matrices AQ, BQ, CQ, assistant to the central corrector also expressed in the form of a state representation, block Q whose output added to the output of the central corrector produces a signal which forms the opposite of U ( t) and whose input receives the signal Y (t) from which the signal W · X (t) is subtracted, and the Youla parameter, and hence the variable coefficient transfer block, is determined and calculated in the control law correction device comprising the central corrector with which the Youla parameter is associated with at least one noise frequency P (t) of which at least the determined frequency of the noise to be attenuated, the calculation of the coefficients of the matrices AQ, BQ, CQ being carried out by obtaining function s of discrete transfer Hsi (q ^-1 ) / αi (q ^-1 ) resulting from the discretization of continuous second-order cells and by a placement of poles as well as the resolution of asymptotic rejection equations,
  and in the use phase, in real time:
  • we recover the current frequency of the noise to be attenuated,
  • the correction control law, comprising the fixed coefficient central corrector with the variable coefficient Youla parameter, is calculated from the calculator using, for the Youla parameter, the coefficients which have been calculated for a noise frequency corresponding to the current frequency of the noise to be attenuated,
• in the design phase (multivariable cases), the following operations are performed:
  • a) - in the first stage, the cabin is acoustically excited by applying to the transducers excitation signals whose spectral density is substantially uniform over a useful frequency band, the excitation signals being decorrelated between them,
  • b) - in the second step, the central corrector is determined and calculated so that it is equivalent to a corrector with state observer and returns to the state calculated by placing the poles in the application of the central corrector to the electroacoustic transfer function, for this purpose a gain of the null observer is chosen, ie Kf = 0 (choice of the gain of the observer equal to the null matrix), and a gain of state return Kc chosen from way to introduce high frequency poles into the loop in order to ensure the robustness of the control law provided with the Youla parameter, the calculation of Kc being for example carried out by optimization LQ (quadratic linear),
  • c) - in the third step, it determines and calculates, considering an augmented state observer representation, the poles of the Q block of Youla within the correction control law for at least one noise frequency P (t ) of which at least the determined frequency of the noise to be attenuated according to an attenuation criterion, in order to obtain coefficient values of the Youla parameter for the / each of the frequencies,
  and in the use phase, in real time, the following operations are performed:
  • the correction control law, a fixed coefficient central corrector with Youla parameter with variable coefficients, is calculated in the computer to produce the signal U (t) sent to the transducer (s), as a function of the acoustic measurements Y (t), and using for the Youla parameter the values of the coefficients determined and calculated for a determined frequency corresponding to the current frequency,
• in the second step, the computation of Kc is carried out by optimization LQ (linear quadratic),
• the method is adapted to a set of determined frequencies of noise to be attenuated and the time c) is repeated for each of the determined frequencies and, in the use phase when none of the determined frequencies corresponds to the current frequency of the noise to be attenuated, interpolates at said current frequency for the Youla Q block coefficient values from the Youla Q block coefficients known for the determined frequencies,
• the signals are sampled at a frequency Fe and at time a) a useful frequency band of the excitation signal is used which is substantially [0, Fe / 2],
• the excitation signal has a uniform spectral density,
• before the phase of use, a fourth time d) of verification of the stability and the robustness of the model of the electroacoustic system and the correction control law, central corrector with parameter of Youla, obtained in the design phase are added previously at times a) to c) by simulating the correction control law obtained at times b) and c) applied to the electroacoustic model obtained at time a) for the determined frequency / frequencies and when a predetermined criterion of stability and / or robustness is not respected, we reiterate at least the time c) by modifying the attenuation criterion,
• in the fourth step d) of the design phase, when a predetermined criterion of stability and / or robustness is not respected, the time b) is further reiterated by modifying the auxiliary poles of the closed loop
• the design phase is a preliminary phase and is carried out once, prior to the use phase, with the results of the determinations and calculations for use in the use phase are stored (for example, in the single-variable case, storage coefficients of the blocks R, S and Q for the calculated correction control law as well as the calculated electroacoustic transfer function, for the Q block of the coefficient tables that can be implemented due to calculations for several determined frequencies)
• the attenuation criterion is selected as a function of at least one of the following two elements: the depth (amplitude) of the attenuation and the bandwidth of the attenuation,
• the current frequency of the noise to be attenuated is recovered from a rev-counter measurement of a vehicle engine.

More generally, the invention also relates to a device specially adapted for implementing the method of the invention for attenuation of narrow-band noise, essentially single-frequency at at least one determined frequency, the device comprising at least one transducer , typically a loudspeaker, controlled with a signal generated by a programmable computer, as a function of a signal of acoustic measurements made by at least one acoustic sensor, typically a microphone, a correction control law having been determined and calculated in a first design phase, said calculated correction control law being used in a second phase of use in the computer to produce a signal sent to the transducer according to the signal received from the sensor for attenuation of said noise, and which device comprises means implementation in the calculator of a correction control law c omportant the application of a Youla parameter to a central corrector, only the Youla parameter having coefficients dependent on the frequency of the noise to attenuate in said correction control law the central corrector having fixed coefficients and a memory of the calculator stores at least said variable coefficients, preferably in a table as a function of the determined noise frequency (s) p (t) used in the design phase.

An aspect of the invention also relates to an instruction medium for directly or indirectly controlling the computer so that it operates according to the method of the invention and in particular in real time in the use phase.

• la Figure 1 de l'état de la technique qui est une représentation schématique d'une structure dite « feedforward » ou à précompensation d'un système d'atténuation de bruit,
• la Figure 2 de l'état de la technique qui est une représentation schématique d'une structure dite « feedback » ou à contre-réaction d'un système d'atténuation de bruit,
• la Figure 3 de l'état de la technique qui est une représentation du schéma de principe d'un système à bouclage électroacoustique avec loi de commande pour habitacle de véhicule,
• la Figure 4 qui est une représentation schématique du temps de la stimulation du système réel électroacoustique de l'habitacle du véhicule destiné à déterminer et calculer le modèle électroacoustique qui sera utilisé,
• la Figure 5 qui est une représentation d'un système bouclé sur le modèle électroacoustique avec correcteur du type RST, dit correcteur central, avec T=0 et dans le cas monovariable,
• la Figure 6 qui est un exemple de fonction de sensibilité directe et qui montre que par application du théorème de Bode-Freudenberg-Looze, les deux aires, au dessous et au dessus de l'axe 0 dB, sont égales,
• la Figure 7 qui est une représentation d'un cas monovariable de loi de commande de correction appliquée au modèle électroacoustique et comportant un correcteur central de type RS auquel on a adjoint un paramètre de Youla,
• la Figure 8 qui représente le schéma complet d'une loi de commande de correction avec un correcteur central de type RS auquel on a adjoint un paramètre de Youla et calculé en temps réel en phase d'utilisation pour atténuation de bruit dans l'habitacle,
• la Figure 9 qui est une représentation d'un schéma du transfert sur un système 2 haut-parleurs et deux microphones, donc dans le cas multivariables,
• la Figure 10 qui est une représentation sous forme de schéma bloc du système à commander, c'est à dire le modèle électroacoustique de l'habitacle, dans le cas multivariables,
• la Figure 11 qui est une représentation sous forme de schéma bloc du correcteur central, dans le cas multivariables,
• la Figure 12 qui est une représentation sous forme de schéma bloc du correcteur central appliqué au modèle électroacoustique de l'habitacle, dans le cas multivariables,
• la Figure 13 qui est une représentation sous forme de schéma bloc de la loi de commande de correction, correcteur central + paramètre de Youla appliqué au modèle électroacoustique de l'habitacle, dans le cas multivariables, et
• la Figure 14 qui est une représentation sous forme de schéma bloc de la loi de commande de correction, correcteur central + paramètre de Youla tel qu'utilisé en temps réel pour atténuation du bruit, dans le cas multivariables.

The present invention, without being limited thereby, will now be exemplified with the following description in relation to:

• the Figure 1 of the state of the art which is a schematic representation of a so-called feedforward or pre-compensation structure of a noise attenuation system,
• the Figure 2 of the state of the art which is a schematic representation of a so-called "feedback" or feedback structure of a noise attenuation system,
• the Figure 3 of the state of the art which is a representation of the schematic diagram of an electroacoustic loopback system with control law for a vehicle cabin,
• the Figure 4 which is a schematic representation of the time of the stimulation of the real electroacoustic system of the vehicle cabin intended to determine and calculate the electroacoustic model that will be used,
• the Figure 5 which is a representation of a looped system on the electroacoustic model with corrector of the RST type, called central corrector, with T = 0 and in the monovariable case,
• the Figure 6 which is an example of a direct sensitivity function and which shows that by applying the Bode-Freudenberg-Looze theorem, the two areas, below and above the 0 dB axis, are equal,
• the Figure 7 which is a representation of a monovariable case of correction control law applied to the electroacoustic model and comprising a central corrector of RS type to which a parameter of Youla has been added,
• the Figure 8 which represents the complete diagram of a correction control law with a RS type central corrector to which a Youla parameter has been added and calculated in real time during the use phase for noise attenuation in the passenger compartment,
• the Figure 9 which is a representation of a diagram of the transfer on a system 2 loudspeakers and two microphones, so in the multivariable case,
• the Figure 10 which is a representation in block diagram form of the system to be controlled, ie the electroacoustic model of the passenger compartment, in the multivariable case,
• the Figure 11 which is a block diagram representation of the central corrector, in the multivariable case,
• the Figure 12 which is a block diagram representation of the central corrector applied to the electroacoustic model of the passenger compartment, in the multivariable case,
• the Figure 13 which is a representation in block diagram form of the correction control law, central corrector + Youla parameter applied to the electroacoustic model of the passenger compartment, in the multivariable case, and
• the Figure 14 which is a block diagram representation of the correction control law, central corrector + Youla parameter as used in real time for noise attenuation, in the multivariable case.

We will now explain in detail the principles underlying the operation of the device of the invention for active control of noise in the passenger compartment, this device, under control of a programmable computer, consisting of a microphone and a microphone. one or more speakers connected to each other and integrated into the vehicle. The loudspeakers are controlled by a control law which generates control signals from the signal received from the microphone. The control law and the methodology for regulating this control law will therefore be described in detail. In order to simplify the explanations, in a first part we will be interested in the simpler single-variable case (only one microphone) then in a second part in the multivariable case (several microphones).

In its generality, the schematic diagram with control law and establishment of an electroacoustic loopback in the vehicle is presented Figure 3 .

Basically, the device of the invention (and the method that is implemented therein) comprises means for rejecting a monofrequency disturbance (noise), the frequency of which is assumed to be known thanks to external information such as, for example, the speed rotation of the engine of the vehicle given by a tachometer ...

In order to synthesize a control law, one must have a model of the real system consisting of the electroacoustic and acoustic elements of the passenger compartment including the / speakers (transducers), microphones (sensor), associated electronic elements (amplifiers, converters ...). This model called the electroacoustic model must be in the form of a rational transfer function, that is to say behaving as an infinite, discrete impulse response filter.

It should be noted that the computer being digital, analog-digital and digital-to-analog converters are used in particular for sampling the analog signals. The calculator therefore processes sampled signals of period Te (in second) and frequency Fe = 1ATe in (Hertz).

It is advantageous to carry out a linear approximation of the real system consisting of the electroacoustic and acoustic elements of the passenger compartment taking into account the level of the signals in play. It is even possible, in advanced implementation variants, to use means intended to avoid non-linear phenomena of saturation or other (by example compression / expansion of signals, anti-spectrum overlap frequency filters ...).

It should also be taken into account that the equations governing the real behavior of the passenger compartment are partial differential equations, ie the transfer function representing exactly the real system is of infinite dimension (parameter model distributed). Therefore, in order to put the invention into practice, it is necessary to find a compromise for defining the electroacoustic model and the order of the transfer function of said model is chosen with a sufficiently small dimension so as not to result in a large volume of calculations. but large enough to correctly approximate the model. It follows from this constraint that oversampling is to be avoided. For example, for a maximum disturbing noise frequency of 120 Hz, a sampling frequency of 500 Hz can be chosen. One of the advantages of choosing a moderate sampling frequency lies in a reduction in the charging load. calculation of the on-board computer. It should be noted that since the speaker amplifier has a much higher sampling frequency (or even operates with analog components), it is desirable to place between the computer output and the speaker input. a low-pass filter operating at the frequency of the loudspeaker amplifier, the cut-off frequency of said filter being constant, in order to reduce the harmonic distortions due to the transition between different sampling period signals.

In the context of the present invention, it has been chosen a particular form of electroacoustic model that will now be presented. However, it is understood that other forms of electroacoustic models can be used in the context of the invention and in particular in the case where the determinations and calculations of the attenuation system applied to this electroacoustic model would not give a satisfactory solution (see further the implementation of an optional time of verification of the stability and robustness of the model of the electroacoustic system and the RS correction system with Youla parameter during the design phase).

• d est le nombre de périodes d'échantillonnage de retard du système,
• B et A étant des polynômes en q ^-1, q ^-1 étant l'opérateur retard d'une période d'échantillonnage. En particulier, on a : $$B\left(q^{-1}\right)={\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">b</figure-callout>}}_0+{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">b</figure-callout>}}_1\cdot q^{-1}+\cdots b_{\mathit{nb}}\cdot q^{-\mathit{nb}}$$
  
  $$A\left(q^{-1}\right)=1+a_1\cdot q^{-1}+\cdots a_{\mathit{na}}\cdot q^{-\mathit{na}}$$
  
  les bi et ai étant des scalaires.

We can express the transfer function of the electroacoustic model which describes the behavior of the real electroacoustic system between the points u (t) and y (t) of the system in the absence of any loopback. If we put q ^-1 the delay operator of a sampling period, the transfer function sought, in the absence of any looping and noise (the noise that is to be attenuated is not present), is of the form: $$\frac{\mathrm{there}\left(t\right)}{u\left(t\right)}=\frac{q^{-d}B\left(q^{-1}\right)}{\mathrm{AT}\left(q^{-1}\right)}$$

• d is the number of delay sampling periods of the system,
• B and A being polynomials at q ^-1 , where q ^-1 is the delay operator of a sampling period. In particular, we have: $$B\left(q^{-1}\right)={\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">b</figure-callout>}}_0+{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">b</figure-callout>}}_1\cdot q^{-1}+\cdots b_{\mathit{nb}}\cdot q^{-\mathit{nb}}$$
  
  $$\mathrm{AT}\left(q^{-1}\right)=1+{\mathrm{at}}_1\cdot q^{-1}+\cdots {\mathrm{at}}_{\mathit{n\; /\; A}}\cdot q^{-\mathit{n\; /\; A}}$$
  
  the bi and ai being scalars.

The identification is performed by stimulating the real system with a signal u (t) whose spectral density is substantially uniform, over the frequency range [0, Fe / 2], Fe / 2 being the Nyquist frequency. It is understood that the frequency / frequencies of the noise that one seeks to attenuate must also be included in the same range and Fe is therefore chosen according to the highest frequency of the noise to be attenuated. Such a stimulation excitation signal can be produced for example by an SBPA (pseudo-random binary sequence). This stimulation, shown schematically Figure 4 , is performed in the absence of disturbing outside noise. All the test data u (t) and y (t) during the test time on the real system (cockpit with its electroacoustic components) are recorded in order to be exploited in the preferential setting of an offline processing.

The algorithms for identifying linear systems that can be used are numerous. In order to have an overview of the usable methodologies, we can refer for example to ID Landau: "Systems Control" (2002) ). After obtaining the rational transfer function, the identification must be validated to ensure that the electroacoustic model obtained is correct. Various validation methods exist depending on the assumptions made about the disturbing noise affecting the model (for example, the whiteness test of the prediction error). In order to increase the reliability of the model obtained, it is also possible to validate the model obtained by comparisons between simulation results on the model obtained and the real system subjected to single-frequency excitation (comparison on the amplitude and the phase of the signals) on a frequency range corresponding to the range of interest for the rejection of disturbances.

Preferably, this identification operation with stimulation is performed for all occupancy configurations of the passenger compartment of the actual model. This occupation may correspond to placements of passengers, accessories (additional seats for example), change of acoustic or electronic equipment, or any other condition that may change the electroacoustic behavior of the passenger compartment. Thus, it is desirable to make identifications for all occupancy configurations of the passenger compartment of the vehicle because the multiple models obtained in fact have differences in gain and phase for each frequency.

After obtaining the transfer function of the electroacoustic model and following its validation using the appropriate tools indicated, we will now synthesize the control law allowing the rejection of a variable frequency disturbance.

The characterization of the level of rejection of the acoustic disturbance which acts on the cabin is done by means of the direct sensitivity function of the looped system noted Syp.

$$S\left(q^{-1}\right)=1+s_1\cdot q^{-1}+\cdots s_{\mathit{ns}}\cdot q^{-\mathit{ns}}$$

Suppose that the control law is of the RST type, that is to say a law composed of three blocks with here T = 0, and R, S being polynomials such that: $$R\left(q^{-1}\right)={\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">r</figure-callout>}}_0+{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">r</figure-callout>}}_1\cdot q^{-1}+\cdots r_{\mathit{nr}}\cdot q^{-\mathit{nr}}$$

$$S\left(q^{-1}\right)=1+{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">s</figure-callout>}}_1\cdot q^{-1}+\cdots s_{\mathit{ns}}\cdot q^{-\mathit{ns}}$$

The order law with: $$u\left(t\right)=\frac{R\left(q^{-1}\right)}{S\left(q^{-1}\right)}\cdot \mathrm{there}\left(t\right)$$

est la fonction de transfert du modèle électroacoustique décrite plus haut. Dans ce bloc diagramme, p(t) est l'équivalent de la perturbation acoustique que l'on a déportée en sortie du système, sans perte de généralité.The RST corrector is the most general form of implementation of a monovariable corrector. We can then schematize the system looped by the block diagram of the Figure 5 in which $\frac{q^{-d}B\left(q^{-1}\right)}{\mathrm{AT}\left(q^{-1}\right)}$

is the transfer function of the electroacoustic model described more high. In this block diagram, p (t) is the equivalent of the acoustic disturbance that was deported at the output of the system, without loss of generality.

The direct sensitivity function Syp can be defined as the transfer function between the perturbation signal p (t) and the microphone signal y (t). This transfer function describes the behavior of the closed loop concerning the rejection of acoustic disturbance.

In particular, obtaining this function makes it possible to know at any frequency the quality of disturbance rejection.

We can show that this transfer function is written: $$S_{\mathit{yp}}=\frac{\mathrm{AT}\left(q^{-1}\right)S\left(q^{-1}\right)}{\mathrm{AT}\left(q^{-1}\right)S\left(q^{-1}\right)+q^{-d}B\left(q^{-1}\right)R\left(q^{-1}\right)}$$

The object of the control law being to allow the rejection of disturbance at a fpert frequency, it is necessary that at said frequency the Syp module is low, in practice much below 0 dB.

Ideally, it would be desirable for Syp to be as low as possible at all frequencies, however this objective is not attainable because of the Bode-Freudenberg-Looze theorem which shows that if the closed-loop system is asymptotically stable and is also stable in open loop, we have: $${\displaystyle {\int }_0^{\mathrm{0.5.}\mathit{Fe}}\log \left|S_{\mathit{yp}}\left(e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="imgf0004.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathit{\pi f}.\mathit{Fe}}\right)\right|\mathit{df}=0}$$

This means that the sum of the areas between the sensitivity module curve and the 0 dB axis taken with their sign is zero. This implies that the attenuation of the perturbation in a certain frequency zone will necessarily lead to the amplification of disturbances in other frequency zones.

An example of a direct sensitivity function is shown in FIG. Figure 6 and the two areas, below and above the 0 dB axis, are equal.

We saw above that the denominator of Syp is written A ( q ^-1 ) S ( q ^-1 ) + q ^-d B ( q ^-1 ) R ( q ^-1 ) which is a polynomial in q ^-1 . The roots of this polynomial constitute the poles of the closed loop.

The computation of the coefficients of the polynomials R ( q ^-1 ) and S ( q ^-1 ) can in particular be done by a technique of placement of poles. There are also other calculation techniques for synthesizing a linear corrector, but preferably the pole placement technique is used here. It amounts to calculating the coefficients of R and S by specifying the poles of the closed loop which are the roots of the polynomial P, ie: $$P\left(q^{-1}\right)=\mathrm{AT}\left(q^{-1}\right)S\left(q^{-1}\right)+q^{-d}B\left(q^{-1}\right)R\left(q^{-1}\right).$$

After choosing these poles, we express P and solve equation (2) which is a Bézout equation. The detail of the resolution of Bézout's equation can be found, for example, in the work of I.D. Landau cited above, on pages 151 and 152. It goes through the resolution of a Sylvester system. In addition, there are computational routines associated with Matlab® and Scilab® software that enable this resolution. The choice of poles can be made according to various strategies. One of these strategies is explained below.

The cancellation of the effect of the disturbances p (t) on the output is obtained at frequencies where $$\mathrm{AT}\left(e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="imgf0004.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathit{\pi f}/\mathit{Fe}}\right)S\left(e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="imgf0004.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathit{\pi f}/\mathit{Fe}}\right)=0$$

Si on a $\begin{array}{ll}{h}_1& =-2\cos \left(2\pi .\mathit{fpert}/\mathit{Fe}\right)\\ h_2& =1\end{array},$

on introduit une paire de zéros complexes non amortis à la fréquence fpert.Also, in order to compute a corrector rejecting a perturbation at the frequency Fpert, we first specify a part of S, by imposing in equation (2) that S factorizes by Hs second order polynomial for a single-frequency perturbation. That is to say : $$\mathit{Hs}=1+{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">h</figure-callout>}}_1\cdot q^{-1}+{\mathrm{<figure-callout\; id="2"\; label="h"\; filenames="imgf0004.png"\; state="\{\{state\}\}">h</figure-callout>}}_2\cdot q^{-2}$$

If we have $\begin{array}{ll}{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">h</figure-callout>}}_1& =-2\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="imgf0004.png"\; state="\{\{state\}\}">cos</figure-callout>}\left(2\pi .\mathit{fpert}/\mathit{<figure-callout\; id="2"\; label="Fe\; h"\; filenames="imgf0004.png"\; state="\{\{state\}\}">Fe\; </figure-callout>},\right)& \\ h_{}{}_2& =1\end{array},$

a pair of unamortized complex zeros are introduced at the fpert frequency.

If h ₂ ≠ 1 we can introduce a pair of complex zeros with non-zero damping in S, damping chosen according to the desired attenuation at a certain frequency.

The Bézout equation to solve is then: $$S\text{'}\left(q^{-1}\right)\cdot \mathit{Hs}\left(q^{-1}\right).\mathrm{AT}\left(q^{-1}\right)+B\left(q^{-1}\right)R\left(q^{-1}\right)=P\left(q^{-1}\right)$$

et ce pour chaque fréquence à rejeter. On voit que cela amène un gros volume de calcul s'il fallait implémenter, notamment en temps réel, la résolution de cette équation. Par ailleurs, tous les coefficients S et R du correcteur sont appelés à varier lors d'un changement de fréquence. Cela aboutit à un algorithme très lourd et qui nécessite une puissance de calcul considérable. Ainsi, même si cette solution de correcteur RS simple est applicable, on préfère mettre en oeuvre une autre solution qui évite ce problème et qui minimise le nombre de coefficients de la loi de commande de correction variant avec la fréquence de la perturbation à rejeter.In practice, the frequency of the noise to be rejected is variable over time, depending in particular on the speed of rotation of the motor shaft of the vehicle, also the block Hs must vary according to said frequency. It follows then that we also have to solve a Bézout equation of the form: $$S\text{'}\left(q^{-1}\right)\cdot \mathit{Hs}\left(q^{-1}\right).\mathrm{AT}\left(q^{-1}\right)+B\left(q^{-1}\right)R\left(q^{-1}\right)=P\left(q^{-1}\right)$$

and this for each frequency to reject. We see that this brings a large volume of calculation if it was necessary to implement, in particular in real time, the resolution of this equation. Moreover, all the coefficients S and R of the corrector are called to vary during a change of frequency. This results in a very heavy algorithm that requires considerable computing power. Thus, even if this simple RS correction solution is applicable, it is preferred to implement another solution which avoids this problem and which minimizes the number of correction control law coefficients varying with the frequency of the disturbance to be rejected.

Thus, to overcome this problem, we propose in the following a solution based on the Youla-Kucera parameterization concept applied to an RS-type corrector.

Such a monovariable system driven by an RS-type corrector to which the Youla parameter has been added is schematized on the Figure 7 .

Such a corrector is based on a so-called central corrector RS consisting of the blocks Ro ( q ^-1 ) and So ( q ^-1 ) . Ro and So being polynomials in q ^-1

The Youla parameter is the block $Q\left(q^{-1}\right)=\frac{\beta \left(q^{-1}\right)}{\alpha \left(q^{-1}\right)}.$

β and α being polynomials in q ^-1 .

As we have seen, the blocks q ^{- d} B ( q ^-1 ) and A ( q ^-1 ) are the numerator and denominator of the transfer function of the electroacoustic system to be controlled.

It can be shown that the entire corrector thus produced and represented Figure 7 is equivalent to an RS-type corrector whose R and S blocks are equal to: $$\begin{array}{l}R\left(q^{-1}\right)=\mathit{ro}\left(q^{-1}\right)\cdot \alpha \left(q^{-1}\right)+\mathrm{AT}\left(q^{-1}\right)\cdot \beta \left(q^{-1}\right)\\ S\left(q^{-1}\right)=\mathit{So}\left(q^{-1}\right)\cdot \alpha \left(q^{-1}\right)-q^{-d}B\left(q^{-1}\right)\cdot \beta \left(q^{-1}\right)\end{array}$$

Suppose now that a central corrector has been created and that it stabilizes the system.

Without parameterization of Youla the characteristic polynomial of the system, Po, as seen above, is written: $$\mathit{Po}\left(q^{-1}\right)=\mathrm{AT}\left(q^{-1}\right).\mathit{So}\left(q^{-1}\right)+q^{-d}B\left(q^{-1}\right).\mathit{ro}\left(q^{-1}\right)$$

$$P\left(q^{-1}\right)=\mathit{Po}\left(q^{-1}\right).\alpha \left(q^{-1}\right)$$

On voit donc que les pôles de Q (zéros de a) viennent s'adjoindre aux pôles de la boucle fermée équipée seulement du correcteur central dont le polynôme caractéristique est Po.By providing the central corrector with the Youla parameter, the characteristic polynomial of the system is written: $$P\left(q^{-1}\right)=\mathrm{AT}\left(q^{-1}\right).\left(\mathit{So}\left(q^{-1}\right).\alpha \left(q^{-1}\right)-q^{-d}B\left(q^{-1}\right).\beta \left(q^{-1}\right)+q^{-d}B\left(q^{-1}\right).(\mathit{ro}\left(q^{-1}\right).\alpha \left(q^{-1}\right)+\mathrm{AT}\left(q^{-1}\right).\beta \left(q^{-1}\right)\right)$$

$$P\left(q^{-1}\right)=\mathit{Po}\left(q^{-1}\right).\alpha \left(q^{-1}\right)$$

We see, therefore, that the poles of Q (zeros of a) come to join the poles of the closed loop equipped only with the central corrector whose characteristic polynomial is Po.

afin de spécifier le bloc S avec un bloc de préspécification Hs, c'est-à-dire : $$S\prime \left(q^{-1}\right).\mathit{Hs}\left(q^{-1}\right)=\mathit{So}\left(q^{-1}\right).\alpha \left(q^{-1}\right)-q^{-d}B\left(q^{-1}\right)\beta \left(q^{-1}\right)$$

Soit : $$S\prime \left(q^{-1}\right).\mathit{Hs}\left(q^{-1}\right)+q^{-d}B\left(q^{-1}\right)\beta \left(q^{-1}\right)=\mathit{So}\left(q^{-1}\right).\alpha \left(q^{-1}\right)$$

qui est également une équation de Bézout, permettant notamment de trouver β si α et Hs sont définis.Moreover, we can use the equation: $$S\left(q^{-1}\right)=\mathit{So}\left(q^{-1}\right).\alpha \left(q^{-1}\right)-q^{-d}B\left(q^{-1}\right)\beta \left(q^{-1}\right)$$

in order to specify the block S with a prespecification block Hs, that is to say: $$S\text{'}\left(q^{-1}\right).\mathit{Hs}\left(q^{-1}\right)=\mathit{So}\left(q^{-1}\right).\alpha \left(q^{-1}\right)-q^{-d}B\left(q^{-1}\right)\beta \left(q^{-1}\right)$$

Is : $$S\text{'}\left(q^{-1}\right).\mathit{Hs}\left(q^{-1}\right)+q^{-d}B\left(q^{-1}\right)\beta \left(q^{-1}\right)=\mathit{So}\left(q^{-1}\right).\alpha \left(q^{-1}\right)$$

which is also a Bézout equation, allowing in particular to find β if α and Hs are defined.

Let Sypo be the direct sensitivity function of the looped system with the central corrector without Youla parameter.

The function of direct sensitivity of the looped system with corrector provided with the Youla parameter, is written: $$S_{\mathit{yp}}=S_{\mathit{ypo}}-\frac{q^{-d}B\left(q^{-1}\right)}{P\left(q^{-1}\right)}Q\left(q^{-1}\right)$$

Thus, from a looped system comprising a central corrector which is not intended to reject a sinusoidal disturbance at a particular frequency fpert, it is possible to add to the central corrector the parameter of Youla which will modify the sensitivity function Syp, while by keeping the poles of the closed loop provided with the central corrector, to which will add the poles of Q. It is thus possible to create a notch in Syp at the frequency fpert.

résulte de la discrétisation d'un bloc continu du second ordre par la méthode de Tustin avec « prewarping » : $$\frac{\frac{s^2}{{\left(2\pi .\mathit{fpert}\right)}^2}+\frac{2\cdot {\varsigma }_1.s}{\left(2\pi .\mathit{fpert}\right)}+1}{\frac{s^2}{{\left(2\pi .\mathit{fpert}\right)}^2}+\frac{2\cdot {\varsigma }_2.s}{\left(2\pi .\mathit{fpert}\right)}+1}$$

For this, we calculate Hs and α such that the transfer function $\frac{\mathit{Hs}\left(q^{-1}\right)}{\alpha \left(q^{-1}\right)}$

results from the discretization of a continuous block of the second order by the method of Tustin with "prewarping": $$\frac{\frac{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}^2}{{\left(2\pi .\mathit{<figure-callout\; id="2"\; label="fpert"\; filenames="imgf0004.png"\; state="\{\{state\}\}">fpert</figure-callout>}\right)}^2}+\frac{2\cdot {\mathrm{<figure-callout\; id="1"\; label="\varsigma "\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">\varsigma </figure-callout>}}_1.\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}{\left(2\pi .\mathit{fpert}\right)}+1}{\frac{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}^2}{{\left(2\pi .\mathit{<figure-callout\; id="2"\; label="fpert"\; filenames="imgf0004.png"\; state="\{\{state\}\}">fpert</figure-callout>}\right)}^2}+\frac{2\cdot {\mathrm{<figure-callout\; id="2"\; label="\varsigma "\; filenames="imgf0004.png"\; state="\{\{state\}\}">\varsigma </figure-callout>}}_2.\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}{\left(2\pi .\mathit{<figure-callout\; id="1"\; label="fpert\; +"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">fpert\; </figure-callout>},\right)}+1}$$

,

sont des coefficients d'amortissement d'une cellule du second ordre.Hs and α are polynomials in q ^-1 of degree 2 and

,

are damping coefficients of a second-order cell.

Furthermore, the discretization operation of the continuous transfer function (in s) can be performed by means of calculation routines which can be found, for example, in computer software dedicated to the automatic. In the case of Matlab®, this is the "c2d" function.

It can be shown that the attenuation M at the frequency fpert is given by the relation: $$M=20\mathrm{<figure-callout\; id="1"\; label="log\; \varsigma "\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">log\; </figure-callout>}\left(\frac{{\varsigma }_{}}{}\right)\left(\frac{{}_1}{{\mathrm{<figure-callout\; id="2"\; label="\varsigma "\; filenames="imgf0004.png"\; state="\{\{state\}\}">\varsigma </figure-callout>}}_2}\right)\mathrm{with}{\mathrm{<figure-callout\; id="1"\; label="\varsigma "\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">\varsigma </figure-callout>}}_1<{\mathrm{<figure-callout\; id="2"\; label="\varsigma "\; filenames="imgf0004.png"\; state="\{\{state\}\}">\varsigma </figure-callout>}}_2$$

In addition, it is necessary ${\varsigma }_{}$${}_1<1$

on montre que l'encoche sur la fonction de sensibilité Syp est d'autant plus large que

est grand. Mais plus cette encoche est large, plus |Syp| se trouve déformée aux fréquences autres que fpert (conséquence du théorème de Bode Freudenberg Looze). Aussi on détermine un compromis par le choix de

,

afin de créer une atténuation suffisamment large autour de fpert sans provoquer une remontée trop importante de |Syp| aux autres fréquences. Des valeurs typiques des facteurs d'amortissement sont : ${\varsigma }_1=\mathrm{0,01}$

${\varsigma }_2=\mathrm{0,1,}$

Ces valeurs peuvent constituer un point de départ pour un affinage.Moreover, for an equal ratio of $\frac{{\mathrm{<figure-callout\; id="1"\; label="\varsigma "\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">\varsigma </figure-callout>}}_1}{{\mathrm{<figure-callout\; id="2"\; label="\varsigma "\; filenames="imgf0004.png"\; state="\{\{state\}\}">\varsigma </figure-callout>}}_2}$

we show that the notch on the sensitivity function Syp is even wider than

is tall. But the wider this notch, the more | Syp | is deformed at frequencies other than fpert (consequence of the Bode Freudenberg Looze theorem). Also a compromise is determined by the choice of

,

in order to create a sufficiently wide attenuation around fpert without causing too much rise of | Syp | at other frequencies. Typical values of damping factors are: ${\varsigma }_1=0.01$

${\mathrm{<figure-callout\; id="2"\; label="\varsigma "\; filenames="imgf0004.png"\; state="\{\{state\}\}">\varsigma </figure-callout>}}_2=0.1$

These values can be a starting point for refining.

We can then calculate β by solving the Bézout equation (10).

It is shown that this choice of Hs and α creates a notch in the sensitivity function Syp while having an almost negligible effect at the other frequencies with respect to Sypo, even if the Bode Freudenberg Looze theorem applies, which brings up of the Syp module with respect to Sypo at frequencies other than fpert.

This rise of Syp can decrease the robustness of the closed loop measurable by the module margin (distance at point -1 of the frequency locus of the open loop corrected in the Nyquist plane) equal to the inverse of the maximum of | Syp | in the frequency range [0; Fe / 2].

de plus β est d'ordre 1 $$\beta \left(q^{-1}\right)={\beta }_1.q^{-1}+{\beta }_2.q^{-2}$$

The main advantage of using the Youla parameterization is that α is of order 2: $$\alpha \left(q^{-1}\right)=1+{\mathrm{<figure-callout\; id="1"\; label="\alpha "\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_1.q^{-1}+{\mathrm{<figure-callout\; id="2"\; label="\alpha "\; filenames="imgf0004.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_2.q^{-2}$$

more β is order 1 $$\beta \left(q^{-1}\right)={\mathrm{<figure-callout\; id="1"\; label="\beta "\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">\beta </figure-callout>}}_1.q^{-1}+{\mathrm{<figure-callout\; id="2"\; label="\beta "\; filenames="imgf0004.png"\; state="\{\{state\}\}">\beta </figure-callout>}}_2.q^{-2}$$

Thus, with the proposed RS type corrector system to which is added the Youla parameter, the number of variant parameters as a function of the frequency of the disturbing noise to be rejected in the control law is only 4. The calculation of these parameters parameters as a function of the frequency f of the disturbance to be rejected can be performed offline, previously, by solving the Bézout equation (10), during the design phase of the control law, the parameters that can be stored in tables on the on-board programmable computer in the vehicle and called, in real time, according to the frequency to be rejected.

The Figure 8 represents the complete schema of the correction control law (central corrector RS + parameter of Youla Q).

To achieve the synthesis of the corrector, it is preferable to use an electroacoustic model that can be described as median, that is to say, a model corresponding to an intermediate level of occupancy of the passenger among the models. electroacoustics corresponding to different occupancy configurations of the passenger compartment.

For the synthesis of the central corrector, it is preferably sought that it guarantees maximum margins without a particular objective of disturbance rejection. This can be achieved, for example, by a pole placement technique, and, if necessary, one can consult the book of ID Landau already cited for this, in particular, the whole of chapter 3. More precisely, it can be seen proceed as explained later.

We choose to perform the placement of the poles of the closed loop by placing n dominant poles of the closed loop on the n poles of the system to control either the roots of A ^{( q -1 )} , n being the degree of the polynomial A. There is no prespecification of the block So because one does not seek disturbance rejection by means of the central corrector alone. By performing this operation, the central corrector does not reject the disturbances p (t), but ensures maximum robustness.

A certain number of "high frequency" auxiliary poles whose value is between 0.05 and 0.5 in the complex plane are also placed (in the case where there is no oversampling). Recall that a sampled system is stable if all its poles are strictly included in the unit circle in the complex plane. These auxiliary poles have the role of increasing the robustness of the control law, when adding the Youla parameter.

d'inconnue So et R'o.After having thus chosen the poles of the closed loop, that is to say the roots of Po ( q ^-1 ), we express Po ( q ^-1 ), which is a polynomial in ( q ^-1 ) of degree n + m. We then solve the Bézout equation using the aforementioned routines: $$\mathit{So}\left(q^{-1}\right).\mathrm{AT}\left(q^{-1}\right)+q^{-d}B\left(q^{-1}\right).R\text{'}o\left(q^{-1}\right)=\mathit{Po}\left(q^{-1}\right)$$

of unknown So and R'o.

The central corrector was thus determined and calculated.

The coefficients of the Youla Q parameter (that is to say α and β) which are the only variant polynomials of the control law as a function of the frequency of the perturbation to be rejected are then calculated.

,

de l'équation (12), de telle sorte à régler la profondeur de l'atténuation de Syp à la dite fréquence, ainsi que la largeur de l'encoche (largeur de bande) à la fréquence fpert dans Syp, tout en ménageant une robustesse suffisante mesurable par la marge de module décrite plus haut (maximum de Syp). On peut se fixer par exemple pour objectif une marge de module de 0,7, ce qui correspond à un niveau de robustesse important de la boucle fermée, robustesse qui garantira la stabilité du système de contrôle actif lors des variations de configuration d'habitacle.For each of the frequencies fpert of the perturbation to be rejected, the damping factors are selected

,

of equation (12), so as to adjust the depth of attenuation of Syp at said frequency, as well as the width of the notch (bandwidth) at the frequency fpert in Syp, while providing a sufficient robustness measurable by the module margin described above (maximum Syp). It is possible, for example, to set a target for a module margin of 0.7, which corresponds to a high level of robustness of the closed loop, a robustness that will guarantee the stability of the active control system during variations in passenger compartment configuration.

It is known that a servo is all the more robust that the poles of the closed loop are close to the system to be controlled. This condition is achieved entirely through the choice of placement of poles during the synthesis of the central corrector.

The polynomials Hs ( q ^-1 ) and α ( q ^-1 ) are calculated as explained above by discretizing a second order cell and the Bézout equation (10) is solved to determine β ( q ^-1 ). .

Preferably, this calculation resulting in the determination of α ( q ^-1 ) and β ( q ^-1 ) as a function of fpert is performed over the entire frequency range where it is intended to perform a disturbance rejection. For example, α and β can be calculated for frequencies varying from 2 Hz to 2 Hz over a range from 30 to 120 Hz.

In addition to the electroacoustic model (s) and the RS central corrector model obtained, the set of coefficients of the polynomials α ( q ^-1 ) and β ( q ^-1 ) as a function of fpert is stored in memory, a table for these last, the calculator. The tables make it possible to find the data that will be used in real time according to the current conditions, in particular the current frequency of the noise to be attenuated and possibly a current configuration of occupancy of the passenger compartment.

,

(profondeur et largeur fréquentielle du rejet). Si cela n'est toujours pas suffisant, on peut alors essayer de prendre pour modèle électracoustique un autre modèle parmi ceux obtenus pour les diverses configuration d'habitacle ou, alors, jouer sur l'emplacement des pôles auxiliaires de la boucle fermée (pôles haute fréquence).The correction control law (corrector RS + Youla parameter) is then synthesized. It is possible, in an optional phase of the design phase, to verify that it has a stability and a correct level of robustness (module margin> 0.5) with a simulation of the looped system and disturbance rejection over the entire range of Frequency for all occupancy configurations of the passenger compartment using the electroacoustic models identified in the various configurations. If this is not the case, we come back to the design of the control law by playing on the coefficients

,

(depth and frequency width of the rejection). If this is still not enough, we can then try to take as another electro acoustic model among those obtained for the various cockpit configurations or, then, to play on the location of the auxiliary poles of the closed loop (high poles frequency).

These previous design and synthesis times require significant calculations and are therefore preferably performed offline. Once this synthesis is carried out, the models obtained in real time can be applied to the computer to obtain the attenuation of the noise in the passenger compartment.

When the calculator works in real time as shown Figure 8 , the stored data, in particular the coefficients of the polynomials α (q ^-1 ) and β ( q ^-1 ) for the Youla parameter, are called as a function of the information on the current frequency of the noise to be rejected originating, for example, indirectly , a tachometer measurement on the motor shaft. For current frequency values that do not correspond directly to the frequencies of the table entries (current frequency between two frequencies of calculation of the values of the table), it is possible to estimate the coefficients of the polynomials α (q ^-1 ) and β ( q ^-1 ) by interpolating between coefficients calculated for two or more known frequency values. In the latter case, it is preferable that the frequency mesh is not too great between the frequencies used for the calculations of the coefficients, a mesh of 2 Hz in 2 Hz is generally suitable.

To summarize the previous example, it can be considered that the invention relates to a real-time active method, by feedback, attenuation of a narrow-band noise, essentially single-frequency at at least a predetermined frequency, in a passenger compartment of a vehicle by emission of a sound by at least one transducer, typically a loudspeaker, controlled with a signal u (t) generated by a programmable computer, depending of a signal of acoustic measurements y (t) made by an acoustic sensor, typically a microphone, in a first phase of design, the electroacoustic behavior of the assembly formed by the passenger compartment, the transducer, and the sensor being modeled by an electroacoustic model in the form of an electroacoustic transfer function which is determined and calculated, a correction control law then being determined and calculated from a global model of the system in which the correction control law is applied to the electroacoustic transfer function whose output additionally receives a noise signal p (t) to give the signal y (t) in said pha design, said correction control law making it possible to produce the signal u (t) as a function of the acoustic measurements y (t), and in a second phase of use, said calculated correction control law being used in the calculator to produce the signal u (t) then sent to the transducer as a function of the signal y (t) received from the sensor for attenuation of said noise.

• a) - dans un premier temps, on utilise comme modèle électroacoustique une fonction de transfert électroacoustique rationnelle discrète et on détermine et calcule ladite fonction de transfert électroacoustique par excitation acoustique de l'habitacle par le transducteur et mesures acoustiques par le capteur puis application d'un procédé d'identification de système linéaire avec les mesures et le modèle de la fonction de transfert,
• b) - dans un deuxième temps, on met en oeuvre une loi de commande de correction comportant un correcteur RS dit central de deux blocs 1/So(q ^-1) et Ro(q ^-1), dans le correcteur central, le bloc 1/So(q ^-1) produisant le signal u(t) et recevant en entrée le signal de sortie inversé du bloc Ro(q ^-1), ledit bloc Ro(q ^-1) recevant en entrée le signal y(t) correspondant à la sommation du bruit p(t) et de la sortie de la fonction de transfert électroacoustique du modèle électroacoustique, et on détermine et calcule le correcteur central,
• c) - dans un troisième temps, on introduit un paramètre de Youla dans la loi de commande de correction sous forme d'un bloc Q(q ^{- 1}) de Youla adjoint au correcteur central RS, ledit bloc Q(q ^-1 ) de Youla recevant une estimation du bruit obtenue par calcul à partir des signaux u(t) et y(t) et en fonction de la fonction de transfert électroacoustique et le signal en sortie dudit bloc Q(q ^{- 1}) de Youla étant soustrait au signal inversé de Ro(q ^-1) envoyé à l'entrée du bloc 1/So(q ^-1) du correcteur central RS, et on détermine et calcule le bloc Q(q ^{- 1}) de Youla dans la loi de commande de correction comportant le correcteur central auquel est associé le paramètre de Youla pour au moins une fréquence de bruit p(t) dont au moins la fréquence déterminée du bruit à atténuer,

et dans la phase d'utilisation, en temps réel :

• on détermine la fréquence courante du bruit à atténuer,
• on fait calculer au calculateur la loi de commande de correction, comprenant le correcteur RS avec le paramètre de Youla, en utilisant celle qui a été calculée pour une fréquence déterminée correspondant à la fréquence courante du bruit à atténuer.

More specifically, in the design phase:

• a) - firstly, a discrete rational electroacoustic transfer function is used as an electroacoustic model, and said electroacoustic transfer function is determined and calculated by acoustic excitation of the passenger compartment by the transducer and acoustic measurements by the sensor then application of a linear system identification method with the measurements and the model of the transfer function,
• b) - in a second step, a correction control law is implemented comprising a central RS corrector of two blocks 1 / So ( q ^-1 ) and Ro ( q ^-1 ), in the central corrector, the block 1 / So ( q ^-1 ) producing the signal u (t) and receiving as input the inverted output signal of the block Ro ( q ^-1 ), said block Ro ( q ^-1 ) receiving as input the signal y (t) corresponding to the summation of the noise p (t) and the output of the electroacoustic transfer function of the electroacoustic model, and the central corrector is determined and calculated,
• c) - in a third step, a Youla parameter is introduced into the correction control law in the form of a block Q (q ^{- 1} ) of Youla associated with the central corrector RS, said block Q (q ^{- 1} ) of Youla receiving an estimate of the noise obtained by calculation from the signals u (t) and y (t) and as a function of the electroacoustic transfer function and the output signal of said block Q (q ^{- 1} ) of Youla being subtracted from the signal inverted of Ro ( q ^-1 ) sent to the input of the block 1 / So ( q ^-1 ) of the central corrector RS, and the block Q (q ^{- 1} ) of Youla is determined and calculated in the correction control law having the central corrector associated with the Youla parameter for at least one noise frequency p (t), of which at least the determined frequency of the noise to be attenuated,

and in the use phase, in real time:

• the current frequency of the noise to be attenuated is determined,
• the correction control law, comprising the corrector RS with the Youla parameter, is calculated by the computer using the one calculated for a determined frequency corresponding to the current frequency of the noise to be attenuated.

So far we have presented a simple implementation with a cockpit with a single microphone and a speaker, or a group of speakers, all excited by the same signal.

However, it turns out that the noise reduction / silence that can be obtained by an active control method is spatially very localized. In the article "A review of active noise and vibration control in road vehicles" already cited, Eliott indicates that the zone of silence around the error microphone does not exceed one tenth of the wavelength of the noise to be rejected. about 110 cm for a noise of 30 Hz, 55 cm for a noise of 60 Hz, 28 cm for a noise of 120 Hz at room temperature.

So we see that it is not possible to obtain a uniform noise reduction in a cabin of a car somewhat spacious with a single microphone and it is therefore necessary to multiply the number of error microphones and distribute them in the passenger compartment to increase the space where there is noise reduction.

In what follows, in order to generalize the explanations, we will consider the case where the cabin is equipped with several microphones and several speakers (or groups of speakers). This generalization makes it possible to understand the applications that are more specific to specific loudspeaker (s) and microphone (s) numbers.

A first solution is to use the previously established control scheme for a single microphone to make a loudspeaker-microphone loop one by one. This solution, however, may give very poor results, or even instability. Indeed, a given loudspeaker of a modeled system will have an influence on all the microphones of the cabin, even those which are not of its own modeled system.

We therefore propose another more global solution from the point of view of the automatic. Here, with several microphones, there is a multivariable problem, that is to say with several inputs and several coupled outputs.

For example, there is shown on the Figure 9 a diagram of the electroacoustic transfer on a 2 * 2 system (2 loudspeakers, 2 microphones). In this example, the microphone 1 is sensitive to the acoustic effects of the speaker 1 (HP1) and the speaker 2 (HP2). Similarly, the microphone 2 is sensitive to the acoustic effects of speaker 2 (HP2) and speaker 1 (HP1). This exemplary system can be modeled by the following transfer function matrix: $$\left[\begin{array}{c}\mathrm{there}1\left(t\right)\\ \mathrm{there}2\left(t\right)\end{array}\right]=\left[\begin{array}{cc}\mathrm{The}11& \mathrm{The}12\\ \mathrm{The}21& \mathrm{The}22\end{array}\right]\cdot \left[\begin{array}{c}u1\left(t\right)\\ u2\left(t\right)\end{array}\right]$$

Still, always in the case 2 * 2 $$\left[\begin{array}{c}\mathrm{there}1\left(t\right)\\ \mathrm{there}2\left(t\right)\end{array}\right]=\left[\begin{array}{cc}\frac{B11\left(q^{-1}\right)}{\mathrm{AT}11\left(q^{-1}\right)}& \frac{B12\left(q^{-1}\right)}{\mathrm{AT}12\left(q^{-1}\right)}\\ \frac{B21\left(q^{-1}\right)}{\mathrm{AT}21\left(q^{-1}\right)}& \frac{B22\left(q^{-1}\right)}{\mathrm{AT}22\left(q^{-1}\right)}\end{array}\right]\cdot \left[\begin{array}{c}u1\left(t\right)\\ u2\left(t\right)\end{array}\right]$$

The representation of a multivariable system by transfer function is actually impractical, it prefers the representation of state, which is a universal representation of linear systems (multivariable or not).

• nu : le nombre d'entrées du système (soit le nombre de haut parleurs ou groupes de haut-parleurs reliés ensemble) ;
• ny : le nombre de sorties du système (soit le nombre de microphones) ;
• n : l'ordre du système.

Is

• nu: the number of system inputs (the number of speakers or groups of speakers connected together);
• ny: the number of outputs of the system (ie the number of microphones);
• n: the order of the system.

In what follows we consider that nu = ny to simplify the explanations but this is not restrictive, the following may also apply to the case n> ny.

avec :

• X : vecteur d'état du système de taille (n*1)
• U : vecteur des entrées du système de taille (nu*1)
• Y: vecteur des sorties de taille (ny*1)

et:

• G une matrice dite matrice d'évolution de taille (n*n)
• H la matrice d'entrée du système de taille (n*nu)
• W la matrice de sortie du système de taille (ny*n).

The state representation of the electroacoustic system (of the passenger compartment) can be written in the form of a so-called equation of state equation of recurrence: $$\begin{array}{l}X\left(t+\mathit{You}\right)=\mathrm{BOY\; WUT}\cdot X\left(t\right)+H\cdot U\left(t\right)\\ Y\left(t\right)=W\cdot X\left(t\right)\end{array}$$

with:

• X : size system state vector (n * 1)
• U : vector size system inputs (nu * 1)
• Y: size output vector (ny * 1)

and:

• G a matrix called size evolution matrix (n * n)
• H the input matrix of the size system (n * naked)
• W the output matrix of the size system (ny * n).

The coefficients of matrices G, H, W define the multivariable linear system.

It is specified that X (t) corresponds to the vector X at time t and X (t + Te) corresponds to the vector X at time t + Te (ie a sampling period after X (t)).

The correction control law is based on this state representation, so, as for the monovariable case, is it necessary to determine and calculate the model of the electroacoustic system to be controlled (electroacoustic model of the passenger compartment), that is to say say the coefficients of matrices G, H, W.

dans p₁...p_ny, p_i étant la perturbation sur la sortie i.On the Figure 10 we have a block diagram of the electroacoustic model of the passenger compartment in the multivariable case where I corresponds to the identity matrix and corresponds to formula (18). By analogy with the monovariable case, P (t) is the vector of disturbances on the outputs, ie: $$P\left(t\right)=\left(\begin{array}{c}{p}_1\left(t\right)\\ ⋮\\ p_{\mathit{ny}}\left(t\right)\end{array}\right)$$

in p ₁ ... p _ny , where p _i is the disturbance on the output i.

As for the monovariable case, we obtain the coefficients of the model of the electroacoustic system to be controlled by an identification procedure during the design phase, that is to say by stimulation of the real electroacoustic system with spectral density noise. substantially uniform, the naked speakers being excited by signals that are decorrelated between them.

Thus, the input data (microphones measurements) and outputs (signals for the loudspeakers) are stored in a computer and are used therein to obtain a state representation of said system, this time using algorithms dedicated identification multivariable systems. These algorithms are for example provided in toolboxes of software specialized in the field of automation such as for example Matlab®. One can also consult advantageously L. LJUNG's book "System identification-Theory for the user" Prentice Hall, Englewood Cliffs, NS, 1987, the algorithms presented in this work having given birth to a box tool dedicated to the identification in Matlab® software. It is the same for the validation algorithms of the model obtained from the electroacoustic system to be controlled.

Another possible embodiment consists in identifying the one-to-one transfer functions one by one with the monovariable identification tools, and stimulating the loudspeakers one to one, and then proceeding to an aggregation of the nu * ny models in one, multivariable. This aggregation can be done, for example by the least squares method of innovation, algorithm described in Ph de Larminat's book: "Automatique appliquée" Hermès 2007.

As for the monovariable case, it is desirable to carry out an identification for each of the cockpit configurations and to take as a model of the electroacoustic system which is conserved for the rest of the design phase a model that will be described as "median ".

Once an input-output model of the electroacoustic system has been obtained in the form of a state representation and this model has been validated, it is possible to proceed to the determination and calculation of the correction control law. It is therefore now necessary to synthesize a correction control law making it possible to reject at the level of each of the microphones an acoustic disturbance of frequency fpert, said fpert frequency being able to evolve over time.

To do this, we generalize the concept of central corrector and the concept of Youla parameterization of the monovariable case to the multivariable case.

où :

• X̂ est le vecteur d'état de l'observateur de taille (n*1)
• Kf est le gain de l'observateur de taille (n*ny)

The electroacoustic system is considered to be described by the state representation (18). We can show that the central corrector is in the multivariable case in a state observer form + return on the estimated state of the form: $$\widehat{X}\left(t+\mathit{You}\right)=\mathrm{BOY\; WUT}\cdot \widehat{X}\left(t\right)+H\cdot U\left(t\right)+\mathit{kf}\cdot \left(Y\left(t\right)-W\cdot \widehat{X}\left(t\right)\right)$$

or :

• X is the state vector of the size observer (n * 1)
• Kf is the gain of the size observer (n * ny)

et la commande s'écrit : $$U\left(t\right)=-\mathit{Kc}\cdot \widehat{X}\left(t\right)$$

Kc étant le vecteur de retour sur l'état estimé du système de taille (nu*n).
On peut consulter avantageusement à ce sujet l'ouvrage « Robustesse et commande optimale » (Alazard et al., éditions CEPADUES, 1999, aux pages 224 et 225 ).So we have : $$\widehat{X}\left(t+\mathit{You}\right)=\left(\mathrm{BOY\; WUT}-\mathit{kf}\cdot W\right)\cdot \widehat{X}\left(t\right)+H\cdot U\left(k\right)+\mathit{kf}\cdot \left(Y\left(t\right)\right)$$

and the command is written: $$U\left(t\right)=-\mathit{kc}\cdot \widehat{X}\left(t\right)$$

Kc being the vector of return on the estimated state of the size system (nu * n).
The work can be advantageously consulted on this subject "Robustness and optimal control" (Alazard et al., CEPADUES editions, 1999, pages 224 and 225 ).

• eig(G - Kf · W) sont nommés : pôles de filtrage et
• eig(G - H · Kc) sont nommés : pôles de commande

avec eig() désignant les valeurs propres.In correspondence with these formulas, on the Figure 11 , we have the block diagram of the central corrector and on the Figure 12 the block diagram of the central corrector applied to the electroacoustic model of the passenger compartment, always in the multivariable case. This last correction structure is conventional in automatic. Under a principle called "principle of "separation", the poles of the closed loop consist of the eigenvalues of G - Kf · W and the eigenvalues of G - H · Kc , namely: $$\mathit{eig}\left(\mathrm{BOY\; WUT}-\mathit{kf}\cdot W\right)\cup \mathit{eig}\left(\mathrm{BOY\; WUT}-H\cdot \mathit{kc}\right).$$

• eig (G - Kf · W) are named: filtering poles and
• eig (G - H · Kc ) are named: command poles

with eig () designating the eigenvalues.

Thus the placement of the poles of the closed loop provided with the central corrector can be done by choosing the coefficients of Kf and Kc which are the adjustment parameters of this control structure. The number of poles to be placed is 2 * n.

We thus choose as central corrector this observer set and estimated state return. In the monovariable case, it had been shown that if we placed n poles of the closed loop on the n poles of the electroacoustic system (ie the roots of the polynomial of A ( q ^-1 )), we obtained a central corrector not rejecting specifically the disturbances, but with maximum robustness.

In the multivariable case, it is also sought that the central corrector has maximum robustness, without particular objective of disturbance rejection. Also, the filtering poles are chosen equal to the poles of the system to be controlled. It is therefore necessary that Kf · W = 0. The most trivial solution is: $$\mathit{kf}=0_{\mathrm{not}*\mathit{ny}}$$

So the equation of the central corrector becomes simply: $$\widehat{X}\left(t+\mathit{You}\right)=\left(\mathrm{BOY\; WUT}\right)\cdot \widehat{X}\left(t\right)+H\cdot U\left(t\right)$$

There remain n other poles to place (the control poles eig (G - H · Kc )). Following what has been done for the monovariable corrector, these poles will be chosen as a set of high frequency poles intended to ensure the robustness of the control law. It should be noted that since we are multivariable, the number of coefficients of Kc (nu * n) is greater than the number of poles yet to be placed (n), so these degrees of freedom can be used for perform a clean structure placement (choice not only of eigenvalues but also eigenvectors of ( G - H · Kc ) .

Another way of proceeding to calculate Kc consists of an optimization LQ (quadratic linear) for which the literature is very abundant. For example, one can refer to the book "Robustness and optimal order" CEPADUES editions, 1999 on pages 69-79 . One can also perform for the calculation of the coefficients of the matrix Kc, what Ph de Larminat calls an optimization LQ of type B, that is to say based on a horizon Tc. The details of this type B LQ optimization can be found in the Ph. De Larminat: "Automatic applied", Hermès, 2007 . In particular, we will find associated with this work a calculation routine for the software Matlab®, allowing the calculation of the coefficients of Kc following the optimization LQ type B.

The central corrector being determined and calculated, we will now present how to determine and calculate the Youla parameter which is associated with the central corrector for realize the correction control law in the multivariable case. The objective is always to reject sinusoidal perturbations of known frequency fpert, here at the level of each microphone, making sure that only the coefficients of the Youla parameter vary as fpert varies.

It can be shown that the Youla parameter associates with the central corrector to form the correction control law as shown in FIG. Figure 13 . The justification of the schema of the Figure 13 can be found for example in the book: "Robustness and optimal control" published by CEPADUES in 1999, pages 224-225 .

In the correction control law as represented symbolically on the Figure 13 , Q, parameter of Youla, is itself a multivariable block whose state representation can be written in the following way: $$X_Q\left(t+\mathit{You}\right)={\mathrm{AT}}_Q{X}_Q\left(t\right)+B_Q\left(Y\left(t\right)-W\cdot \widehat{X}\left(t\right)\right)$$

X _Q being the state vector of the Youla parameter.

cette loi de commande correspond à un retour d'état de l'observateur associé à un retour d'état du paramètre de Youla.The control law of the central corrector provided with the Youla parameter is then written: $$U\left(t\right)=-K_{\mathrm{vs}}\cdot \widehat{X}\left(t\right)-{\mathrm{VS}}_Q\cdot X_Q\left(t\right)$$

this control law corresponds to a return of state of the observer associated with a return of state of the parameter of Youla.

We will now show how to determine the parameters of Q so as to ensure a rejection of disturbances of known frequency.

par discrétisation d'une cellule du second ordre continue et α constituait alors le dénominateur du paramètre de Youla et Hs était utilisé dans une équation de Bézout permettant de trouver β, numérateur du coefficient de Youla.In the monovariable case, a transfer function was calculated $\frac{\mathit{Hs}\left(q^{-1}\right)}{\alpha \left(q^{-1}\right)}$

by discretizing a continuous second-order cell and α was then the denominator of the Youla parameter and Hs was used in a Bézout equation to find β, the Youla coefficient numerator.

où :

• X̂ _2i est le vecteur d'état du modèle de la perturbation i (taille 2*1)
• Z _2i est la perturbation additive sur la sortie i (taille 1*1)

avec : $$G_{2i}=\left[\begin{array}{cc}-h{s}_{1i}& 1\\ -h{s}_{2\mathrm{<figure-callout\; id="0"\; label="i"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">i</figure-callout>}}& 0\end{array}\right]$$

et : $$W_{2i}=\left[\begin{array}{cc}1& 0\end{array}\right]$$

In the multivariable case, let's put on each output n ° i a model of non controllable perturbation:
For each output i, this uncontrollable perturbation model is written: $$\begin{array}{l}{\widehat{X}}_{2i}\left(t+\mathit{You}\right)={\mathrm{BOY\; WUT}}_{2i}{\widehat{X}}_{2i}\left(t\right)\\ Z_{2i}\left(t\right)=W_{2i}{\widehat{X}}_{2i}\left(t\right)\end{array}$$

or :

• X _{2 i} is the state vector of the perturbation model i (size 2 * 1)
• Z _{2 i} is the additive disturbance on the output i (size 1 * 1)

with: $${\mathrm{BOY\; WUT}}_{2i}=\left[\begin{array}{cc}-h{s}_{1i}& 1\\ -h{s}_{2\mathrm{<figure-callout\; id="0"\; label="i"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">i</figure-callout>}}& 0\end{array}\right]$$

and $$W_{2i}=\left[\begin{array}{cc}1& 0\end{array}\right]$$

résultant de la discrétisation d'une cellule continue du second ordre, identique à celle utilisée dans le cas monovariable : $$\frac{\frac{s^2}{{\left(2\pi .\mathit{fpert}\right)}^2}+\frac{2\cdot {\varsigma }_{1i}.s}{\left(2\pi .\mathit{fpert}\right)}+1}{\frac{s^2}{{\left(2\pi .\mathit{fpert}\right)}^2}+\frac{2\cdot {\varsigma }_{2i}.s}{\left(2\pi .\mathit{fpert}\right)}+1}$$

It should be noted that the choice of the form of G _{2 i} W _{2 i} is not unique. Here we have adopted a canonical representation of observability. hs _{1 i} and hs _{2 i} are deduced from the numerator of a transfer function $\frac{H{s}_i\left(q^{-1}\right)}{{\alpha }_i\left(q^{-1}\right)}$

resulting from the discretization of a continuous cell of the second order, identical to that used in the monovariable case: $$\frac{\frac{s^2}{{\left(2\pi .\mathit{<figure-callout\; id="2"\; label="fpert"\; filenames="imgf0004.png"\; state="\{\{state\}\}">fpert</figure-callout>}\right)}^2}+\frac{2\cdot {\varsigma }_{1i}.\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}{\left(2\pi .\mathit{fpert}\right)}+1}{\frac{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}^2}{{\left(2\pi .\mathit{<figure-callout\; id="2"\; label="fpert"\; filenames="imgf0004.png"\; state="\{\{state\}\}">fpert</figure-callout>}\right)}^2}+\frac{2\cdot {\varsigma }_{2i}.\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}{\left(2\pi .\mathit{<figure-callout\; id="1"\; label="fpert\; +"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">fpert\; </figure-callout>},\right)}+1}$$

With: $$\begin{array}{l}H{s}_i\left(q^{-1}\right)=h_{0i}+h_{1i}\cdot q^{-1}+h_{2i}\cdot q^{-2}\\ h{s}_{1i}=\frac{h_{1i}}{h_{0i}}\mathrm{and}h{s}_{2i}=\frac{h_{2i}}{h_{0i}}\end{array}$$

The discretization of the continuous transfer function can be done for example by means of the calculation routine "c2d" of the Matlab® software.

avec : $$U\left(t\right)=-\mathit{Kc}\cdot \widehat{X}\left(t\right)-K{c}_2\cdot {\widehat{X}}_2\left(t\right)$$

où :

• Kf ₂ est de taille (2*ny,ny)
• Kc ₂ est de taille (nu,2*ny)

et avec : $${\mathrm{<figure-callout\; id="2"\; label="G"\; filenames="imgf0004.png"\; state="\{\{state\}\}">G</figure-callout>}}_2=\left(\begin{array}{cccc}{G}_{21}& 0& \cdots & 0\\ 0& G_{22}& & 0\\ ⋮& & & ⋮\\ 0& \cdots & & {\mathrm{<figure-callout\; id="2"\; label="G"\; filenames="imgf0004.png"\; state="\{\{state\}\}">G</figure-callout>}}_{2\mathit{ny}}\end{array}\right)\mathrm{matrice\; de\; taille}\left(2\mathrm{ny}*2\mathrm{ny}\right)$$

$${\widehat{X}}_2\left(t\right)=\left(\begin{array}{c}{\widehat{X}}_{21}\left(t\right)\\ {\widehat{X}}_{22}\left(t\right)\\ ⋮\\ {\widehat{X}}_{22}\left(t\right)\end{array}\right)\mathrm{vecteur\; de\; taille}\left(2\mathrm{ny}*1\right)$$

Ce vecteur' étant le vecteur d'état du modèle non commandable $$W_{}$$$${}_2=\left(\begin{array}{cccc}{W}_{21}& 0& \cdots & 0\\ 0& W_{22}& & 0\\ ⋮& & & ⋮\\ 0& 0& \cdots & {\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="imgf0004.png"\; state="\{\{state\}\}">W</figure-callout>}}_{2\mathit{ny}}\end{array}\right)\mathrm{matrice\; de\; taille}\left(\mathrm{ny}*2\mathrm{ny}\right)$$

One can then write the equation of state of an observer increased of the models of perturbation on the outputs which is then: $$\begin{array}{l}\widehat{X}\left(t+\mathit{You}\right)=\mathrm{BOY\; WUT}\cdot \widehat{X}\left(t\right)+H\cdot U\left(t\right)\\ {\widehat{X}}_2\left(t+\mathit{You}\right)={\mathrm{<figure-callout\; id="2"\; label="BOY\; WUT"\; filenames="imgf0004.png"\; state="\{\{state\}\}">BOY\; WUT</figure-callout>}}_2\cdot {\widehat{X}}_2\left(t\right)+\mathrm{<figure-callout\; id="2"\; label="K\; f"\; filenames="imgf0004.png"\; state="\{\{state\}\}">K\; </figure-callout>}{f}_{}{}_2\cdot \left(Y-W\cdot \widehat{X}\left(t\right)-{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="imgf0004.png"\; state="\{\{state\}\}">W</figure-callout>}}_2\cdot {\widehat{X}}_2\left(t\right)\right)\end{array}$$

with: $$U\left(t\right)=-\mathit{kc}\cdot \widehat{X}\left(t\right)-K{\mathrm{vs}}_2\cdot {\widehat{X}}_2\left(t\right)$$

or :

• Kf ₂ is large (2 * ny, ny)
• Kc ₂ is large (nude, 2 * ny)

and with : $${\mathrm{<figure-callout\; id="2"\; label="BOY\; WUT"\; filenames="imgf0004.png"\; state="\{\{state\}\}">BOY\; WUT</figure-callout>}}_2=\left(\begin{array}{cccc}{\mathrm{BOY\; WUT}}_{21}& 0& \cdots & 0\\ 0& {\mathrm{BOY\; WUT}}_{22}& & 0\\ ⋮& & & ⋮\\ 0& \cdots & & {\mathrm{<figure-callout\; id="2"\; label="BOY\; WUT"\; filenames="imgf0004.png"\; state="\{\{state\}\}">BOY\; WUT</figure-callout>}}_{2\mathit{<figure-callout\; id="2"\; label="ny\; size\; matrix"\; filenames="imgf0004.png"\; state="\{\{state\}\}">ny\; </figure-callout>}}& \\ \end{array},\right)\mathrm{size\; matrix}\left(2\mathrm{ny}*2\mathrm{ny}\right)$$

$${\widehat{X}}_2\left(t\right)=\left(\begin{array}{c}{\widehat{X}}_{21}\left(t\right)\\ {\widehat{X}}_{22}\left(t\right)\\ ⋮\\ {\widehat{X}}_{22}\left(\mathrm{<figure-callout\; id="2"\; label="t\; size\; vector"\; filenames="imgf0004.png"\; state="\{\{state\}\}">t\; </figure-callout>},\right)& \\ \end{array},\right)\mathrm{size\; vector}\left(2\mathrm{ny}*1\right)$$

This vector 'being the state vector of the non-controllable model $$W_{}$$$${}_2=\left(\begin{array}{cccc}{W}_{21}& 0& \cdots & 0\\ 0& W_{22}& & 0\\ ⋮& & & ⋮\\ 0& 0& \cdots & {\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="imgf0004.png"\; state="\{\{state\}\}">W</figure-callout>}}_{2\mathit{ny}}\end{array}\right)\mathrm{size\; matrix}\left(\mathrm{ny}*2\mathrm{ny}\right)$$

The equation (29) of the observer is written again: $$\begin{array}{l}\widehat{X}\left(t+\mathit{You}\right)=\mathrm{BOY\; WUT}\cdot \widehat{X}\left(t\right)+H\cdot U\left(t\right)\\ {\widehat{X}}_2\left(t+\mathit{You}\right)=\left({\mathrm{BOY\; WUT}}_2-\mathrm{<figure-callout\; id="2"\; label="K\; f"\; filenames="imgf0004.png"\; state="\{\{state\}\}">K\; </figure-callout>}{f}_{}{}_2.{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="imgf0004.png"\; state="\{\{state\}\}">W</figure-callout>}}_2\right)\cdot {\widehat{X}}_2\left(t\right)+\mathrm{<figure-callout\; id="2"\; label="K\; f"\; filenames="imgf0004.png"\; state="\{\{state\}\}">K\; </figure-callout>}{f}_{}{}_2\cdot \left(Y-W\cdot {\widehat{X}}_2\left(t\right)\right)\end{array}$$

We must now choose the coefficients of Kf ₂ , so as to place the poles of this part of the observer increased.

By choosing the two roots of the denominators α _i ( q ^-1 ) as poles, we generalize to the multivariable case what we did in monovariable.

More precisely, one chooses: eig ( G _{2 i} - Kf _{2 i} · W _{2 i} ) equal to the roots of polynomials α _i ( q ^-1 ) mentioned above, these polynomials resulting as we said above from the discretization of a continuous cell of the second order.

The calculation of Kf _{2 i} as a function of G _{2 i} , W _{2 i} , α _i ( q ^-1 ) and is a classical pole placement. To do this, one can for example use the Matlab® software routine dedicated to this operation whose name is "PLACE".

Under this last condition the matrix Kf ₂ is diagonal in blocks, ie: $$\mathrm{<figure-callout\; id="2"\; label="K\; f"\; filenames="imgf0004.png"\; state="\{\{state\}\}">K\; </figure-callout>}{f}_{}{}_2=\left(\begin{array}{cccc}K{f}_{21}& 0& \cdots & 0\\ 0& K{f}_{22}& & \\ & & & ⋮\\ 0& 0& \cdots & \mathrm{<figure-callout\; id="2"\; label="K\; f"\; filenames="imgf0004.png"\; state="\{\{state\}\}">K\; </figure-callout>}{f}_{}{}_{2\mathit{ny}}\end{array}\right)$$

It remains to choose Kc ₂ of size (nu * 2ny). This choice is not free if we want to obtain an asymptotic rejection of the output disturbances.

avec : $$\begin{array}{l}\mathit{Ta}\cdot G_2-G\cdot \mathit{Ta}-H\cdot \mathit{Ga}=0\\ W\cdot \mathit{Ta}-W_2=0\end{array}$$

Kc _{2 must} satisfy the so-called asymptotic rejection equations which are: $$K{\mathrm{vs}}_2=\mathit{ga}+\mathit{kc}\cdot \mathit{Your}$$

with: $$\begin{array}{l}\mathit{Your}\cdot {\mathrm{BOY\; WUT}}_2-\mathrm{BOY\; WUT}\cdot \mathit{Your}-H\cdot \mathit{ga}=0\\ W\cdot \mathit{Your}-{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="imgf0004.png"\; state="\{\{state\}\}">W</figure-callout>}}_2=0\end{array}$$

The justification of equations (36) and (37) can be found in ph. de Larminat: "Automatic Applied" hermès 2007 on pages 202 205. The resolution of equations (37) leads to the resolution of a Sylvester system. It should be noted that a calculation routine for the Matlab® software solving the asymptotic rejection equations is provided with the aforementioned work.

By comparing the equations (24) and (25) with the equation (34), one then realizes that this state-augmented structure of the observer is none other than the central corrector as it has been defined, provided with the parameter of Youla with, taking again the notations of (24) and (25): $$\begin{array}{l}{\mathrm{AT}}_Q={\mathrm{BOY\; WUT}}_2-\mathrm{<figure-callout\; id="2"\; label="K\; f"\; filenames="imgf0004.png"\; state="\{\{state\}\}">K\; </figure-callout>}{f}_{}{}_2\cdot W_2\\ B_Q=\mathrm{<figure-callout\; id="2"\; label="K\; f"\; filenames="imgf0004.png"\; state="\{\{state\}\}">K\; </figure-callout>}{f}_{}{}_2\\ {\mathrm{VS}}_Q=K{\mathrm{vs}}_2\end{array}$$

It should be noted that these equations are valid because we have chosen Kf = 0.

Thus, for each perturbation frequency, the coefficients of A _Q , B _Q and C _Q can be calculated during the adjustment of the correction control law and put into tables in order, in use phase, to be called as a function of fpert on the real time calculator. The Figure 14 gives the scheme of application of the correction control law in the phase of use in real time in the programmable computer.

The Youla Q block can be implemented as a transfer matrix to minimize the number of variant coefficients in this block. Such an operation can be carried out for example by means of the "ss2tf" routine of the Matlab® software.

,

des cellules du second ordre continues, influençant les largeurs fréquentielles et profondeur des rejets des perturbations à la fréquence fpert.As we have seen, the adjustment parameters of the correction control law reside in the choice of control poles (by Kc parameters) which have an influence on the robustness of the control law. For each frequency, the choice of damping factors is available

,

continuous second-order cells, influencing the frequency widths and depth of the disturbance rejections at the fpert frequency.

It should be noted that the robustness of the servocontrol can be evaluated by calculating the infinite norm of the transfer matrix between P (t) and Y (t) (generalization of the monovariable case). Since the calculation of the infinite norm of a transfer matrix is done by calculating the singular values of said transfer matrix, one can also use the Matlab® software and in particular the "SIGMA" function of the "control toolbox".

These adjustment possibilities generalize the possibilities of setting the mono variable case.

• Obtention d'un modèle électroacoustique de l'habitacle de véhicule qui est linéaire, multivariables, sous forme de représentation d'état, calculé par identification,
• Synthèse d'un correcteur central sous forme observateur d'état et retour d'état estimé avec choix de Kf=0,
• Choix des coefficients de Kc correspondant à des pôles haute fréquence pour assurer la robustesse la loi de commande (éventuellement par optimisation LQ et notamment optimisation LQ de type B),
• Choix des facteurs d'amortissement
  
  ,
  
  pour un maillage de fréquences de perturbation à rejeter, maillage effectué en particulier dans le cas où plusieurs fréquences courantes de bruit à atténuer peuvent être rencontrées au cours du temps ou que la fréquence du bruit varie au cours du temps (comme pour le cas monovariable, une interpolation des paramètres variables en fonction de la fréquence peut être effectuée dans la phase d'utilisation),
• Calcul les coefficients du paramètre de Youla qui sont mis dans des tables du calculateur pour utilisation dans la phase d'utilisation en temps réel.

To summarize, the correction control law (central corrector + Youla parameter) intended to be applied to an electroacoustic vehicle cabin model, in the multivariable case, is obtained by following the following steps:

• Obtaining an electroacoustic model of the vehicle cabin that is linear, multivariable, in the form of a state representation, calculated by identification,
• Synthesis of a central corrector in state observer form and estimated state return with choice of Kf = 0,
• Choice of the coefficients of Kc corresponding to high frequency poles to ensure the robustness of the control law (possibly by LQ optimization and in particular LQ optimization type B),
• Choice of damping factors
  
  ,
  
  for a mesh of disturbance frequencies to be rejected, meshwork carried out in particular in the case where several common frequencies of noise to be attenuated may be encountered over time or the frequency of the noise varies over time (as in the case of the single-variable case, an interpolation of the variable parameters as a function of the frequency can be carried out in the use phase),
• Calculates the Youla parameter coefficients that are put in the calculator tables for use in the real-time usage phase.

égaux pour une fréquence de perturbation donnée.It should be noted that a reduction in the number of coefficients to be placed in the tables can be performed by choosing all the

equal for a given disturbance frequency.

The invention thus implements a central corrector with a Youla parameter which is in the form of an infinite impulse response filter with at least one input and at least one output, which number depends on the chosen embodiments (single variable, multivariable, number of sensors and transducers ...).

In the implementation examples presented above, the case of a rejection of a single frequency was considered in order to simplify the explanations. However, the invention is applicable to the rejection of several frequencies at once and we will now describe such a case.

Indeed, whether in the single-variable case or in the multivariable case, it is possible to reject more than one frequency simultaneously. This leads to introducing a second or even a third notch in the sensitivity function Syp. However, as a result, given the Bode Freudenberg Looze theorem, the realization of one or more additional notches in the sensitivity function necessarily leads to a rise of | Syp | at other frequencies resulting in a decrease in robustness.

• la fréquence courante que l'on nomme ici fpert pour reprendre les notations utilisées jusqu'à présent, et
• une seconde fréquence reliée à fpert que l'on notera η.fpert, η étant pas nécessairement entier, η peut être constant sans nécessairement être entier, mais il peut également être fonction de fpert, la seule condition étant que la fonction η(fpert) soit continue .

In the following we will assume that we reject two frequencies, but this is not limiting and is given by way of example. These two frequencies are:

• the current frequency that is called here fpert to use the notation used so far, and
• a second frequency connected to fpert that one will note η.fpert, η being not necessarily integer, η can be constant without necessarily being integer, but it can also be function of fpert, the only condition being that the function η (fpert) be continuous.

dont les inconnues sont toujours S'(q ^-1) et β(q ^-1), mais cette fois-ci Hs et α sont tels que la fonction de transfert $\frac{\mathit{Hs}\left(q^{-1}\right)}{\alpha \left(q^{-1}\right)}$

résulte de la discrétisation d'un bloc continu par la méthode de Tustin constitué d'un produit de deux cellules du second ordre continues : $$\frac{\frac{s^2}{{\left(2\pi .\mathit{fpert}\right)}^2}+\frac{2\cdot {\varsigma }_{11}\cdot s}{\left(2\pi .\mathit{fpert}\right)}+1}{\frac{s^2}{{\left(2\pi .\mathit{fpert}\right)}^2}+\frac{2\cdot {\varsigma }_{21}\cdot s}{\left(2\pi .\mathit{fpert}\right)}+1}.\frac{\frac{s^2}{{\left(2\pi .\eta .\mathit{fpert}\right)}^2}+\frac{2\cdot {\varsigma }_{12}\cdot s}{\left(2\pi .\eta .\mathit{fpert}\right)}+1}{\frac{s^2}{{\left(2\pi .\eta .\mathit{fpert}\right)}^2}+\frac{2\cdot {\varsigma }_{22}\cdot s}{\left(2\pi .\eta .\mathit{fpert}\right)}+1}$$

In the monovariable case, we always have the Bézout equation (10): $$S\text{'}\left(q^{-1}\right).\mathit{Hs}\left(q^{-1}\right)+q^{-d}B\left(q^{-1}\right)\beta \left(q^{-1}\right)=\mathit{So}\left(q^{-1}\right).\alpha \left(q^{-1}\right)$$

whose unknowns are always S ' ( q ^-1 ) and β ( q ^-1 ), but this time Hs and α are such that the transfer function $\frac{\mathit{Hs}\left(q^{-1}\right)}{\alpha \left(q^{-1}\right)}$

results from the discretization of a continuous block by the Tustin method consisting of a product of two continuous second-order cells: $$\frac{\frac{s^2}{{\left(2\pi .\mathit{<figure-callout\; id="2"\; label="fpert"\; filenames="imgf0004.png"\; state="\{\{state\}\}">fpert</figure-callout>}\right)}^2}+\frac{2\cdot {\varsigma }_{11}\cdot s}{\left(2\pi .\mathit{fpert}\right)}+1}{\frac{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}^2}{{\left(2\pi .\mathit{<figure-callout\; id="2"\; label="fpert"\; filenames="imgf0004.png"\; state="\{\{state\}\}">fpert</figure-callout>}\right)}^2}+\frac{2\cdot {\varsigma }_{21}\cdot s}{\left(2\pi .\mathit{<figure-callout\; id="1"\; label="fpert\; +"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">fpert\; </figure-callout>},\right)}+1}.\frac{\frac{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}^2}{{\left(2\pi .\eta .\mathit{<figure-callout\; id="2"\; label="fpert"\; filenames="imgf0004.png"\; state="\{\{state\}\}">fpert</figure-callout>}\right)}^2}+\frac{2\cdot {\varsigma }_{12}\cdot s}{\left(2\pi .\eta .\mathit{fpert}\right)}+1}{\frac{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}^2}{{\left(2\pi .\eta .\mathit{<figure-callout\; id="2"\; label="fpert"\; filenames="imgf0004.png"\; state="\{\{state\}\}">fpert</figure-callout>}\right)}^2}+\frac{2\cdot {\varsigma }_{22}\cdot s}{\left(2\pi .\eta .\mathit{<figure-callout\; id="1"\; label="fpert\; +"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">fpert\; </figure-callout>},\right)}+1}$$

sont des facteurs d'amortissement permettant tout comme dans le cas du rejet monofréquentiel de régler la largeur et la profondeur de l'encoche d'atténuation dans la courbe représentative du module de Syp, α(q ^-1) est un polynôme d'ordre 4 et β(q ^-1) un polynôme d'ordre 3. Le nombre de coefficient variables dans la loi de commande est donc plus élevé : il y a 4 coefficients supplémentaires à faire varier en fonction de fpert. Hs and α are here polynomials in q ^-1 of degree 4 and

are damping factors allowing just as in the case of single-frequency rejection to adjust the width and depth of the attenuation notch in the representative curve of the Syp module, α ( q ^-1 ) is a command polynomial 4 and β ( q ^-1 ) a polynomial of order 3. The number of variable coefficients in the control law is therefore higher: there are 4 additional coefficients to be varied according to fpert.

et l'on a également $$W_{2i}=\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]$$

In the multivariable case, the matrix G _{2 i} of equation (27) is now of dimension 4 * 4, ie: $${\mathrm{BOY\; WUT}}_{2i}=\left[\begin{array}{cccc}-h{s}_{1\mathrm{<figure-callout\; id="1"\; label="i"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">i</figure-callout>}}& 1& 0& 0\\ -h{s}_{2\mathrm{<figure-callout\; id="0"\; label="i"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">i</figure-callout>}}& 0& 1& 0\\ -h{s}_{3\mathrm{<figure-callout\; id="0"\; label="i"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">i</figure-callout>}}& 0& 0& 1\\ -h{s}_{4\mathrm{<figure-callout\; id="0"\; label="i"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">i</figure-callout>}}& 0& 0& 0\end{array}\right]$$

and we also $$W_{2i}=\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]$$

résultant de la discrétisation d'un produit de deux cellules du second ordre continues identiques à celles utilisées dans le cas monovariable, soit : $$\frac{\frac{s^2}{{\left(2\pi .\mathit{fpert}\right)}^2}+\frac{2\cdot {\varsigma }_{11}\cdot s}{\left(2\pi .\mathit{fpert}\right)}+1}{\frac{s^2}{{\left(2\pi .\mathit{fpert}\right)}^2}+\frac{2\cdot {\varsigma }_{21}\cdot s}{\left(2\pi .\mathit{fpert}\right)}+1}.\frac{\frac{s^2}{{\left(2\pi .\eta .\mathit{fpert}\right)}^2}+\frac{2\cdot {\varsigma }_{12}\cdot s}{\left(2\pi .\eta .\mathit{fpert}\right)}+1}{\frac{s^2}{{\left(2\pi .\eta .\mathit{fpert}\right)}^2}+\frac{2\cdot {\varsigma }_{22}\cdot s}{\left(2\pi .\eta .\mathit{fpert}\right)}+1}$$

avec : $$H_{\mathit{si}}\left(q^{-1}\right)=h_{0i}+h_{1i}\cdot q^{-1}+h_{2i}\left(q^{-1}\right)+h_{3i}\left(q^{-1}\right)+h_{4i}\left(q^{-1}\right)$$

et : $$h{s}_{1i}=\frac{h_{1i}}{h_{0i}}$$

$$h{s}_{2i}=\frac{h_{2i}}{h_{0i}}$$

$$h{s}_{3i}=\frac{h_{3i}}{h_{0i}}$$

$$h{s}_{4i}=\frac{h_{4i}}{h_{0i}}$$

Il résulte que maintenant que :

• Kf ₂ est de taille (4*ny,ny)
• Kc ₂ est de taille (nu,4*ny)

avec G ₂ conforme à l'équation (31) mais de taille (4ny*4ny)It should be noted that the choice of the form of G _{2 i} W _{2 i} is not unique. Here we have adopted a canonical representation of observability.
where hs _{1 i} hs _{2 i} hs _{3 i} hs _{4 i} are the numerator coefficients of a transfer function. $\frac{\mathit{Hsi}\left(q^{-1}\right)}{\mathit{\alpha i}\left(q^{-1}\right)}$

resulting from the discretization of a product of two continuous second order cells identical to those used in the monovariable case, namely: $$\frac{\frac{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}^2}{{\left(2\pi .\mathit{<figure-callout\; id="2"\; label="fpert"\; filenames="imgf0004.png"\; state="\{\{state\}\}">fpert</figure-callout>}\right)}^2}+\frac{2\cdot {\varsigma }_{11}\cdot s}{\left(2\pi .\mathit{fpert}\right)}+1}{\frac{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}^2}{{\left(2\pi .\mathit{<figure-callout\; id="2"\; label="fpert"\; filenames="imgf0004.png"\; state="\{\{state\}\}">fpert</figure-callout>}\right)}^2}+\frac{2\cdot {\varsigma }_{21}\cdot s}{\left(2\pi .\mathit{<figure-callout\; id="1"\; label="fpert\; +"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">fpert\; </figure-callout>},\right)}+1}.\frac{\frac{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}^2}{{\left(2\pi .\eta .\mathit{<figure-callout\; id="2"\; label="fpert"\; filenames="imgf0004.png"\; state="\{\{state\}\}">fpert</figure-callout>}\right)}^2}+\frac{2\cdot {\varsigma }_{12}\cdot s}{\left(2\pi .\eta .\mathit{fpert}\right)}+1}{\frac{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}^2}{{\left(2\pi .\eta .\mathit{<figure-callout\; id="2"\; label="fpert"\; filenames="imgf0004.png"\; state="\{\{state\}\}">fpert</figure-callout>}\right)}^2}+\frac{2\cdot {\varsigma }_{22}\cdot s}{\left(2\pi .\eta .\mathit{fpert}\right)}+1}$$

with: $$H_{\mathit{if}}\left(q^{-1}\right)=h_{0i}+h_{1i}\cdot q^{-1}+h_{2i}\left(q^{-1}\right)+h_{3i}\left(q^{-1}\right)+h_{4i}\left(q^{-1}\right)$$

and $$h{s}_{1i}=\frac{h_{1i}}{h_{0i}}$$

$$h{s}_{2i}=\frac{h_{2i}}{h_{0i}}$$

$$h{s}_{3i}=\frac{h_{3i}}{h_{0i}}$$

$$h{s}_{4i}=\frac{h_{4i}}{h_{0i}}$$

It follows that now that:

• Kf ₂ is large (4 * ny, ny)
• Kc ₂ is big (naked, 4 * ny)

with G ₂ according to equation (31) but of size (4ny * 4ny)

The vector X ₂ (t) is this time of size (4ny * 1) and the matrix W ₂ is this time of size (ny * 4ny). The asymptotic rejection equations (36) and (37) are unchanged. The resolution of such a multivariable system is similar to the case of the rejection of a single previously detailed frequency.

What has been described for a number of frequencies simultaneously rejected equal to two may be optionally extended to a higher number of frequencies, however, as said above, the increase in the number of rejected frequencies causes a loss of robustness quickly becoming prohibitive .

It will be understood that the principle underlying the invention, a central corrector to which a Youla parameter is added, can be applied in practice for noise attenuation in other ways than the one detailed above. In particular, the type of electroacoustic model may be different, the methods for determining and / or synthesizing the central corrector and Youla parameter may also be different and it is useful to refer to the literature indicated for the practical implementation of these other modalities.

Claims (15)

1. A real-time active method for attenuating, through feedback, a narrow-band noise, essentially mono-frequency at at least one determined frequency, in a vehicle passenger compartment, by emitting a sound through at least one transducer, typically a loud-speaker, controlled by a signal u(t) or U(t) according to the case respectively, generated by a programmable calculator, as a function of a signal of acoustic measurements y(t) or Y(t) according to the case, performed by at least one acoustic sensor, typically a microphone, wherein the use of one sensor corresponds to a monovariable case, and the use of several sensors corresponds to a multivariable case, and the method comprising following steps:

   in a first phase of design, the electroacoustic response of the unit formed by the passenger compartment, the transducer and the sensor, is modeled by an electroacoustic model as an electroacoustic transfer function that is determined and calculated, a control law being then determined and calculated from a global model of the system in which the control law is applied to the electroacoustic transfer function whose output additionally receives a noise signal to be attenuated p(t) to give the signal y(t) or Y(t) in said design phase, said control law making it possible to produce the signal u(t) or U(t) as a function of the acoustic measurements y(t) or Y(t), and

   in a second phase of use, said calculated control law is used in the calculator to produce the signal u(t) or U(t) then sent to the transducer as a function of the signal y(t) or Y(t) received from the sensor for attenuating said noise,

   characterized in that a control law is implemented, which comprises the application of a Youla parameter to a central controller and which is such that only the Youla parameter has coefficients that depend on the frequency of the noise to be attenuated in said control law, the central controller having fixed coefficients, the Youla parameter being in the form of an infinite impulse response filter, and in that, after determination and calculation of the control law, at least said variable coefficients are stored into a memory of the calculator, preferably in a table as a function the determined noise frequency(ies) p(t) used in the design phase, and in that, in the use phase, in real time:

   - the current frequency of the noise to be attenuated is collected,

   - the calculator is caused to calculate the control law, comprising the central controller with the Youla parameter, using as the Youla parameter the memorized coefficients of a determined frequency corresponding to the current frequency of the noise to be attenuated.
2. A method according to claim 1, characterized in that, in the monovariable case, in the design phase:

   a) - in a first time, a linear electroacoustic model is used, the electroacoustic model being in the form of a discrete rational electroacoustic transfer function, and said electroacoustic model is determined and calculated by acoustic excitation of the passenger compartment by the transducer(s) and acoustic measurements by the sensor, then application of a linear system identification process with the measures and the model,

   b) - in a second time, a central controller is implemented, which is applied to the determined and calculated electroacoustic model, the central controller being in the form of a RS controller of two blocks 1/So(q ^-1) and , Ro(q ^-1), in the central controller, the block 1/So(q ^-1) producing the signal u(t) and receiving as an input the inverted output signal of the block Ro(q ^-1), said block Ro(q ^-1) receiving as an input the signal y(t) corresponding to the sum of the noise p(t) and of the output of the electroacoustic transfer function of the electroacoustic model, and the central controller is determined and calculated,

   c) - in a third time, a Youla parameter is adjoined to the central controller to form the control law, the Youla parameter being in the form of a block Q(q ^-1), a infinite impulse response filter, with $Q\left(q^{-1}\right)=\frac{\beta \left(q^{-1}\right)}{\alpha \left(q^{-1}\right)}$

   

   adjoined to the central RS controller, said Youla block Q(q ^-1) receiving a noise estimation obtained by calculation from the signals u(t) and y (t) and as a function of the electroacoustic transfer function and the output signal of said Youla block Q(q ^-1) being subtracted from the inverted signal of Ro(q ^-1) sent to the input of the block 1/So(q ^-1) of the central RS controller, and the Youla parameter in the control law comprising the central controller with which is associated the Youla parameter, is determined and calculated for at least one noise frequency p(t), including at least the determined frequency of the noise to be attenuated,
   and in that, in the use phase, in real time:

   - the current frequency of the noise to be attenuated is collected,

   - the calculator is caused to calculate the control law, comprising the RS controller with the Youla parameter, using as the Youla parameter the coefficients that have been calculated for a noise frequency corresponding to the current frequency of the noise to be attenuated, the coefficients of Ro(q ^-1) and So(q ^-1) being fixed coefficients.
3. A method according to claim 2, characterized in that, in the design phase, the following operations are performed:

   a) - in a first time, the passenger compartment is acoustically excited by applying to the transducer an excitation signal whose spectral density is substantially uniform over an effective band of frequencies,

   b) - in a second time, the polynomials Ro(q ^-1) and So(q ^-1) of the central controller are determined and calculated, so that said central controller is equivalent to a controller calculated by poles placement of the closed loop in the application of the central controller to the electroacoustic transfer function, n poles of the closed loop being placed onto the n poles of the transfer function of the electroacoustic system,

   c) - in a third time, the numerator and denominator of the Youla block Q(q ^-1) in the control law are determined and calculated for at least one noise frequency p(t), including at least the determined frequency of the noise to be attenuated, as a function of a criterion of attenuation, the block Q(q ^-1) being expressed in the form of a ratio, β(q^-1)/α(q^-1), so as to obtain coefficient values of the polynomials α(q ^-1) and β(q ^-1) for the/each frequency, the calculation of β(q ^-1) and α(q ^-1) being performed by obtaining a discrete transfer function Hs(q^-1)/α(q^-1) resulting from the discretization of a continuous transfer function of the second order, the polynomial β(q ^-1) being calculated by solving a Bezout equation,

   and in that, in the use phase, in real time, the following operations are performed:

   - the calculator is caused to calculate the control law, fixed-coefficient central controller with variable-coefficient Youla parameter, to produce the signal u(t) sent to the transducer, as a function of the acoustic measurements y(t) and using for the Youla block Q(q ^-1) the coefficient values of the polynomials α(q ^-1 ) and β(q^-1) determined and calculated for a determined frequency corresponding to the current frequency.
4. A method according to claim 2 or 3, characterized in that, for the electroacoustic model is used an electroacoustic transfer function of the form: $$\frac{y\left(t\right)}{u\left(t\right)}=\frac{q^{-d}B\left(q^{-1}\right)}{A\left(q^{-1}\right)}$$

   

   where d is the number of delay sampling periods, B and A are polynomials in q ^-1 of the form: $$B\left(q^{-1}\right)=b_0+b_1\cdot q^{-1}+\Lambda b_{\mathit{nb}}\cdot q^{-\mathit{nb}}$$

   

   $$A\left(q^{-1}\right)=1+a_1\cdot q^{-1}+\Lambda a_{\mathit{na}}\cdot q^{-\mathit{na}}$$

   

   where b_i and a_i are scalar quantities, and q ^-1 is the delay operator of a sampling period, and in that the calculation of the noise estimation is obtained by applying the function q ^{- d}B(q^-1) to u(t) and subtracting the result from the application of y(t) to the function A(q^-1).
5. A method according to claim 2, 3 or 4, characterized in that, for the time b), the polynomials Ro(q^-1 ) and So(q^-1 ) of the central controller are determined and calculated by a method of poles placement of the closed loop, n dominant poles of the closed loop provided with the central controller being chosen equal to the n poles of the electroacoustic transfer function and m auxiliary poles being poles located in high frequency.
6. A method according to claim 1, characterized in that, in the multivariable case in the design phase:

   a) - in a first time, a linear electroacoustic model is used, wherein the electroacoustic model is in the form of a state representation of matrix blocks H, W, G and q^-1.I, G being a evolution matrix, H being an input matrix, W being an output matrix, and I being the identity matrix, wherein said state representation can be expressed by a recurrence equation: $$X\left(t+\mathit{Te}\right)=G\cdot X\left(t\right)+H\cdot U\left(t\right)$$

   

   $$Y\left(t\right)=W\cdot X\left(t\right)$$

   

   with X(t): state vector, U(t): input vector; Y(t): output vector,
   and said electroacoustic model is determined and calculated by acoustic excitation of the passenger compartment by the transducers and acoustic measurements by the sensors, then application of a linear system identification process with the measures and the model,

   b) - in a second time, a central controller applied to the determined and calculated model is implemented, the central controller being in the form of a state observer and feedback of estimated state, that iteratively expresses, X, a state vector of the observer, as a function of Kf, a gain of the observer, Kc a vector of feedback on the estimated state, as well as the previously determined and calculated electroacoustic model, i.e.: $$\widehat{X}\left(t+\mathit{Te}\right)=G\cdot \widehat{X}\left(t\right)+H\cdot U\left(t\right)+\mathit{Kf}\cdot \left(Y\left(t\right)-W\cdot \widehat{X}\left(t\right)\right)$$

   

   with a control U(t) = -Kc·X̂(t)
   and said central controller is determined and calculated,

   c) - in a third time, a Youla parameter is adjoined to the central controller to form the control law, the Youla parameter being in the form of a multivariable block Q, of state matrices AQ, BQ, CQ, adjoined to the central controller also expressed in the form of a state representation, block Q whose output added to the output of the central controller produces a signal that forms the opposite of U(t), and whose input receives the signal Y(t) from which is subtracted the signal W · X̂(t), and the Youla parameter in the control law comprising the central controller with which is associated the Youla parameter is determined and calculated for at least one noise frequency p(t), including at least the determined frequency of the noise to be attenuated, the calculation of the coefficients of the matrices AQ, BQ, CQ being performed by obtaining discrete transfer functions Hsi(q^-1)/αi(q^-1) resulting from the discretization of continuous transfer functions of the second order and by placing poles, as well as solving equations of asymptotic rejection,

   and in that, in the use phase, in real time:

   - the current frequency of the noise to be attenuated is collected,

   - the calculator is caused to calculate the control law, comprising the fixed-coefficient central controller with the variable-coefficient Youla parameter, using as the Youla parameter the coefficients that have been calculated for a noise frequency corresponding to the current frequency of the noise to be attenuated.
7. A method according to claim 6, characterized in that, in the design phase, the following operations are performed:

   a) - in a first time, the passenger compartment is acoustically excited by applying to the transducers excitation signals whose spectral density is substantially uniform over an effective band of frequencies, the excitation signals being de-correlated from each other,

   b) - in a second time, the central controller is determined and calculated so that it is equivalent to a controller with a state observer and a feedback on the calculated state by poles placement in the application of the central controller to the electroacoustic transfer function, wherein, for that purpose, a null observer gain is chosen, i.e. Kf=0, and a gain of state feedback Kc is chosen so as to ensure the robustness of the control law provided with the Youla parameter, by means of a LQ optimization,

   c) - in a third time, considering a representation of increased state observer, the coefficients of the Youla block Q in the control law are determined and calculated for at least one noise frequency P(t) including at least the determined frequency of the noise to be attenuated, as a function of criterion of attenuation, so as to obtain coefficient values of the Youla parameter for the/each frequency,

   and in that, in the use phase, in real time, the following operations are performed:

   - the calculator is caused to calculate the control law, fixed-coefficient central controller with variable-coefficient Youla parameter, to produce the signal U(t) sent to the transducers, as a function of the acoustic measurements Y(t) and using for the Youla parameter the coefficient values determined and calculated for a determined frequency corresponding to the current frequency.
8. A method according to any one of claims 2 to 7, characterized in that the method is adapted to a set of determined frequencies of noise to be attenuated, and the time c) is repeated for each of the determined frequencies, and in that, in the use phase, when no one of the determined frequencies corresponds to the current frequency of the noise to be attenuated, an interpolation is made at said current frequency for the coefficient values of the Youla block Q, based on the coefficient values of said Youla block Q that are known for the determined frequencies.
9. A method according to any one of claims 2 to 8, characterized in that the signals are sampled at a frequency Fe and, at the time a), the effective band of frequencies used for the excitation signal is substantially equal to [0, Fe/2].
10. A method according to any one of claims 2 to 9, characterized in that, before the use phase, a fourth time d) is added to the design phase, for verifying the stability and the robustness of the electroacoustic system model and of the control law, central controller with Youla parameter, previously obtained at the times a) to c), by making a simulation of the control law obtained at times b) and c), applied to the electroacoustic model obtained at the time a), for the determined frequency(ies), and when a predetermined criterion of stability and/or robustness is not respected, at least the time c) is reiterated by modifying the criterion of attenuation.
11. A method according to any one of the preceding claims, characterized in that the design phase is a preliminary phase and it is performed once, preliminary to the use phase, with memorization of the determination and calculation results for use in the use phase.
12. A method according to any one of the preceding claims, characterized in that the current frequency of the noise to be attenuated is collected from a measurement of a motor revolution counter of the vehicle.
13. A method according to any one of the preceding claims, characterized in that the noise is at one determined frequency fpert.
14. A method according to any one claims 1 to 12, characterized in that the noise is at two determined frequencies, with a first frequency fpert, and a second frequency η.fpert, η being either constant or varying continuously with fpert.
15. A device specially adapted for the implementation of the method according to any one of the preceding claims, to attenuate a narrow-band noise, essentially mono-frequency at at least one determined frequency, the device comprising at least one programmable calculator, one acoustic sensor and one transducer, typically a loudspeaker, controlled with a signal generated by the programmable calculator, as a function of a signal of acoustic measurements performed by said acoustic sensor, typically a microphone, a control law having been determined and calculated in a first phase of design, said calculated control law being used, in a second phase of use, in the calculator, to produce a signal sent to the transducer, as a function of the signal received from the sensor for attenuation of said noise, characterized in that it comprises means for implementing, in the calculator, a control law comprising the application of Youla parameter to a central controller, wherein only one variable-coefficient transfer block corresponds to the Youla parameter having coefficients that depend on the frequency of the noise to be attenuated in said control law, the central controller having fixed coefficients, the Youla parameter being in the form of an infinite impulse response filter, and a memory of the calculator stores at least said variable coefficients, preferably in a table as a function of the determined noise frequency(ies) p(t) used in the design phase.

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