Direct Adaptive Feedback Attenuation of Narrow-Band Disturbances
This chapter presents the basic algorithm for direct adaptive feedback attenuation of unknown and time-varying narrow-band disturbances. This algorithm implements the Internal Model Principle for disturbance attenuation using a Youla–Kučera parametrizatio
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Direct Adaptive Feedback Attenuation of Narrow-Band Disturbances
12.1 Introduction One of the basic problems in active vibration control and active noise control is the strong attenuation of single or multiple unknown and time-varying narrow-band disturbances1 without measuring them. In this context, an adaptive feedback approach, termed as adaptive regulation is now generally used. In contrast with the feedforward compensation approach [1–4], the feedback approach, does not require an additional measurement highly correlated with the disturbance. This is a significant advantage. Feedback approaches avoid also the possible destabilizing positive feedback coupling between the compensator system and the measurement of the disturbance which occurs often in feedforward compensation schemes (see [5] and Sect. 1.5) and require less parameters to adapt. A common assumption is that the disturbance is white noise or a Dirac impulse passed through a filter which characterizes the model of the disturbance.2 To be more specific, the disturbances considered can be defined as “finite band disturbances.” This includes single or multiple narrow-band disturbances or sinusoidal signals. For the purpose of this chapter, the disturbances are considered to be unknown and timevarying, in other words, their model has time-varying coefficients. This motivates the use of an adaptive regulation approach since the objective is to attenuate unknown disturbances without measuring them. A popular methodology for this regulation problem in the case when the model of the disturbance is known, is to design a controller that incorporates the model of the disturbance (internal model principle—IMP). This technique has been introduced in Sect. 7.2.2 and has its roots described in the paper [6]. Additional references for the present context are [7–9]. The main problem, using the IMP principle, is that complete rejection of the disturbances is attempted (asymptotically) and this 1 Called
tonal disturbances. the chapter, it is assumed that the number of narrow-band disturbances is known (it can be estimated from data if necessary) but not their frequency characteristics.
2 Throughout
© Springer International Publishing Switzerland 2017 I.D. Landau et al., Adaptive and Robust Active Vibration Control, Advances in Industrial Control, DOI 10.1007/978-3-319-41450-8_12
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12 Direct Adaptive Feedback Attenuation of Narrow-Band Disturbances
may have a strong influence upon the sensitivity functions outside the frequency band in which attenuation is achieved. As long as rejection of a single narrow-band disturbance is considered [7, 9], the influence upon the output sensitivity functions does in general not pose problems. Nevertheless, if low damped complex zeros are located near the disturbance frequency, even in a single narrow-band disturbance context, the influence over Syp (z−1 ) represents a major challenge [10]. The IMP principle will be combined with a Youla–Kuˇcera parametrization of the controller (see Sect. 7.2.3) which will allow to dev
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