Abstract
Active reduction of disturbing noise requires the generation of appropriate control signals in order to drive the canceling sources. For this purpose we have to organize the signal processing of data provided by sensors. These data contain information about the disturbance as well as about the systems state. Because we assume that we are able to work with reference signals that are well correlated to the disturbing noise, the upcoming chapter is restricted to signal processing in feed-forward control systems. It starts with a mathematical preparation, because we will use some concepts of matrix calculus, see (Zurmühl in Matrizen und ihre technischen Anwendungen, Springer, Berlin, 1964) or (Strang in Introduction to linear algebra, Cambridge University Press, Wellesley, 1993). It is continued with an introduction to feed-forward control signal processing, before we discuss active control of both harmonic excitations and stochastic disturbances. Besides the topic of optimal control, attention is paid to adaptive control that enables the controller to track down non-stationary effects, e.g. the change of the operational speed of a turbo-machinery, or changes in the system response that can be caused by a change of environmental conditions such as ambient pressure, air temperature or humidity as well as by the warming-up of the electro-acoustic equipment. Adaptive feed-forward control of harmonic disturbances is discussed in frequency domain, whereas we will present four time domain approaches for adaptive feed-forward control of stochastic disturbances. However, this chapter is far away from presenting the basics of digital signal processing and digital filters as described in (Antoniou in Digital filters: analysis and design, McGraw-Hill, New York, 1979), (Bose in Digital filters—theory and applications, Elsevier, New York, 1985), (Diniz in Adaptive filtering—algorithms and practical implementations, Springer, New York, 2008), (Haykin in Adaptive filter theory, Prentice Hall, London, 1996), (Hess in Digitale Filter—Eine Einführung, Teubner, Stuttgart, 1989), (Johnson in Digitale Signalverarbeitung, Hanser, München in Cooperation with Prentice Hall International, London, 1991), and (Lücker in Grundlagen digitaler Filter—Eine Einführung in die Theorie linearer zeitdiskreter Systeme und Netzwerke, 1980) to review the theory of adaptive filtering as presented in (Farhang-Boroujeny in Adaptive filters—theory and applications, Wiley, New York, 1998), (Honig and Messerschmidt in Adaptive filters—structure, algorithms and applications, Kluwer Academic, Boston, 1984), (Sayed in Fundamentals of adaptive filtering, Wiley, Hoboken, 2003) and (Widrow and Stearns in Adaptive signal processing, Prentice Hall International, London, 1985) or to discuss DSP implementations, see (Akpan et al. in Active noise and vibration control literature survey: controller technologies. DREA-CR-1999-177, Contractor Report, Defence Research Establishment Atlantic, Dartmouth NS (CAN); MARTEC Ltd, Halifax NS (CAN); Sherbrooke Univ, Sherbrooke QUE (CAN), 1999) and (Chassaing in Digital signal processing and applications with the C6713 and C6416 DSK, Wiley, Canada, 2005). Its intension is to provide a basis for adaptive feed-forward control of low frequency interior noise that also includes algorithmic formulations of the control schemes. The content of the upcoming chapter is therefore oriented on (Kuo and Morgan in Active noise control systems—algorithms and DSP implementations, Wiley, Canada, 1996) and (Elliott in Signal processing for active noise control, Academic Press, London, 2001)—two fundamental books about signal processing for ANC that are (without detailed algorithmic formulations) summarized in (Kuo and Morgan in Proc. IEEE 87(6):943–973, 1999) and (Elliott in Tokhi and Veres (eds.) Active sound and vibration control, Institution of Electrical Engineers, London, pp. 57–72, 2002).
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Notes
- 1.
In some textbooks, e.g. (Elliott 2001), the stability criteria (6.26) is motivated by the argument that the iterative mapping (6.28) can also performed with a non-hermitian matrix A. The stability criteria for the in this case complex eigenvalues λ m is given by
$$\vert 1-\mu\lambda_m\vert < 1 \quad \mbox{i.e.}\ 0 < \mu< \frac{2\operatorname{Re} (\lambda_m )}{\vert \lambda _m\vert ^2} .$$As shown in the present subsection, it is not necessary to argue this way.
- 2.
Active cancellation of sound pressure at a point in a pure tone diffuse sound field is discussed in (Elliott et al. 1988), whereas (Elliott and Garcia-Bonito 1995) reports on active cancellation of pressure and pressure gradient in a diffuse sound field. Active control of stationary random sound fields is analyzed in (Nelson et al. 1990).
- 3.
A modified approach, based on input power, that includes damping effects, unfortunately hidden behind real parts of transfer impedances, was presented in (Elliott 2001).
- 4.
The small shift between the curves for global control of acoustic potential energy and active control of total power input results from the fact that the kinetic acoustic energy (that is implicitly taken into account by controlling the total power input) has a significant contribution below the first acoustic mode, see (Curtis et al. 1987).
- 5.
This also implies that the secondary source is acting non-causally.
- 6.
Applying (6.77) to Q p results in \(L_{Q_{p}}=150.4\) dB.
- 7.
As shown in (Elliott 2001) the matrix H H H is not positive definite for an under-determined system (L<M) and will have at least M−L eigenvalues that are equal to zero. Therefore the solution of the under-determined minimization problem is not unique and will not be considered in the present work.
- 8.
- 9.
An example for sound pressure approximation is the forward prediction method, see (Munn 2003) and (Moreau et al. 2008), in which the error signals at the remote locations p M are approximated by the error signals obtained from physical error sensors p C such as p M =Fp C , where F contains weighting coefficients determined by finite difference schemes.
- 10.
If it is a not possible to use acoustic transducers for active control, information about the acoustical state may be obtained from non-acoustic reference signals measured by accelerometers. The transmissibility is in such a situation determined by the ratio between the primary noise measured at the l-th monitor microphone and the primary disturbance measured at the l-th accelerometer such as \(T_{l_{M}l_{C}} =P_{pM{l_{m}}}/ A_{pC{l_{C}}}\). For more details, especially on the more advanced approach of radiation mode sensing, the reader is referred to (Fahy and Gardonio 2007).
- 11.
The Sherman-Morrison-Woodbury formula states that inverse matrix to the matrix A+OP H is given by
To apply this formula we have to use the identifications \({\mathbf{A}}\equiv\hat{\mathbf{H}}_{0}^{H} {\mathbf{W}}_{p} {\mathbf{H}}_{0}\), \({\mathbf{O}}\equiv\hat{\mathbf{H}}_{0}^{H} {\mathbf{W}}_{p}\), and P H≡Δ H. For more details see (Golub and Loan 1996).
- 12.
As shown in (Elliott 2001) a control system that uses 6 reference signals, 4 canceling sources and 128 coefficients in each filter would lead to a filter matrix with 3072×3072 elements. Assembling and inverting of such a filter matrix in every time step n is not the goal of fast signal processing.
- 13.
- 14.
The index j corresponds to the j-th coefficient of the impulse response of the lm-th secondary path, whereas m corresponds to the m-th controller output signal.
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Kletschkowski, T. (2012). Active Control of Tonal and Broadband Noise. In: Adaptive Feed-Forward Control of Low Frequency Interior Noise. Intelligent Systems, Control and Automation: Science and Engineering, vol 56. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2537-9_6
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