Convergence analysis of the Filtered-U LMS algorithm for active noise control in case perfect cancellation is not possible☆
Introduction
In active control (AC) applications, such as in active noise control, active vibration control, active structural acoustical control, etc., finite-impulse response (FIR) adaptive filters are widely used. Usually, the Filtered-X LMS (FxLMS) algorithm [23] is used to adapt the FIR filter coefficients of the controller. However, the infinite-impulse response (IIR) adaptive filter is preferred in case the optimal controller has one or more poles close to the unit circle, which yields a very long impulse response of the optimal controller. In case, the plant to be controlled contains intrinsic feedback (in active noise control applications called acoustical feedback), i.e. the control signal influences the input to the controller, the optimal controller might contain unstable poles. Then, it is even impossible to use an FIR adaptive filter to do optimal control without reducing the bandwidth of the controller. For this reason, the Filtered-U LMS (FuLMS) algorithm, which adapts the coefficients of IIR filters for AC applications, has been proposed [4], [5]. The algorithm was based on the LMS algorithm for adaptive IIR filtering proposed by Feintuch [6]. However, due to the non-quadratic nature of the cost function to be minimized in case of updating an IIR filter, convergence to a local minimum might occur, as illustrated in [8] (see also [17]).
The convergence of the algorithm proposed by Feintuch, which is a pseudo linear regression (PLR), is analyzed by various approaches, see [12] for stochastic approaches and [16] for a deterministic approach using the small-gain theorem. A well-known approach to derive conditions for global convergence of PLRs was proposed by Ljung in [9], [10] and is referred to as the ordinary differential equation (ODE) method. Using this method, the main condition for global convergence of PLRs is that a certain transfer function should be strictly positive real (SPR). In [22] the ODE method has been applied to derive conditions for global convergence of the FuLMS algorithm (see also [13] and for the multiple-channel case see [14]). To use the ODE method it was assumed that there exists a controller which yields perfect cancellation. This assumption is equivalent to the assumption that the system is in the model set, which is assumed in the analysis of recursive identification methods using the ODE method [12]. However, in most AC applications, perfect cancellation is not achievable due to pure delays and non-minimum phase zeros in the system, which are often present in real situations.
The main contribution of this paper is to show that the assumption requiring perfect cancellation is not necessary to analyze the convergence of the FuLMS algorithm via the ODE method. The relaxation of the assumptions, made in the convergence analysis in [13], [14], [22], supports the practical evidence known by the practical engineer, that the FuLMS algorithm can even converge when canceling the primary noise is not perfectly possible. This new global convergence result is based on the analysis of the causal Wiener filter, which yields optimal performance in the mean-square error sense (in case of no intrinsic feedback). The main step in the derivation is to show that the residual disturbance obtained by the causal Wiener filter is stochastically independent of the regression vector, which is necessary to apply the ODE method. Using this new convergence result and the internal model control (IMC) principle, convergence of the FuLMS algorithm for feedback AC systems could also be investigated analytically.
Though the FuLMS algorithm converges to the global minimum under certain conditions presented in this paper, its convergence rate can be very slow. The structure of the causal Wiener filter suggests methods of preconditioning to increase the convergence speed of the FuLMS algorithm. For FxLMS it can be shown that preconditioning increases the convergence rate by prewhitening the reference signal and using the inner–outer factorization of the secondary path transfer function [2], [3], [15] (see also [24], [18] for comparable algorithms to improve the convergence rate of FxLMS). Similar preconditioning can be used to increase the convergence rate of the FuLMS algorithm. Therefore, the second contribution of this paper is the presentation of a preconditioned FuLMS algorithm, which can increase the convergence rate significantly as demonstrated by simulation examples.
The paper is organized as follows. Section 2 explains the notation and some mathematical preliminaries. Section 3 describes the AC problem, the derivation of the optimal controller, including the causal Wiener filter and the optimal performance. These results are well known, but give insight on the conditions for which perfect cancellation is not achievable, and on the characteristics of the remaining residual signal. Section 4 recapitulates briefly the existing conditions for global convergence of the FuLMS algorithm [22]. Section 5 shows that the assumption of perfect cancellation is not necessary and presents the new theorem for global convergence of the FuLMS algorithm. Section 6 shows how the new theorem can be used to prove global convergence of the FuLMS algorithm for feedback AC systems, in which no reference signal is available. Section 7 presents the preconditioned FuLMS algorithm which can increase the convergence rate significantly as is demonstrated in Section 8 by two simulation examples. Finally, Section 9 concludes the paper.
Section snippets
Notation and mathematical preliminaries
The definition operator is denoted as ≔. The imaginary unit is denoted by j, which is such that j2=−1. As in the convergence analysis presented in [22], all signals are assumed to be scalar-valued. Rational transfer functions are denoted as T(q−1) with q−1 the unit-delay operator, usually T(q−1) is shortly written as T. The set of all rational scalar transfer functions is denoted as RH and the set of all stable transfer functions in RH is denoted as RH∞. The complex-conjugate operator is
Problem formulation
The AC problem considered in this paper, is illustrated in Fig. 1. The primary path is denoted by P(q−1)∈RH∞, the secondary path by S(q−1)∈RH∞, the detector path by D(q−1)∈RH∞ and the intrinsic feedback by F(q−1)∈RH∞. The primary disturbance, denoted by , is a zero-mean white noise process with (δ(0)≔1 and δ(i)≔0 for i≠0). The primary signal is denoted by , and should be counteracted by the secondary signal, denoted by . The measurement noise is a
Existing convergence result of FuLMS
The FuLMS algorithm has been proposed by Eriksson et al. [5] to minimize (4) by an adaptive controller. However, no proof of convergence to the optimal controller has been given. In [22] conditions were derived for the FuLMS algorithm to converge to the optimal controller. In this section, we recall this convergence result for the FuLMS algorithm.
Let the controller C in (3) be replaced by the time-varying controllerNote, that
Convergence of FuLMS when perfect cancellation is not achievable
In Section 3 we inferred that the optimal performance was completely determined by the causal Wiener filter (the intrinsic feedback was compensated for by IMC). By exploiting the structure of the causal Wiener filter, global convergence of the FuLMS algorithm can be proven also for the case where perfect cancellation is not achievable, as shown in the proof of the next theorem. Theorem 4 (General convergence of FuLMS). Under the following conditions: The order N of the numerator and M of the denominator of
Convergence analysis of FuLMS for feedback systems
The FuLMS convergence result of Theorem 4 can also be used to investigate the convergence of FuLMS for feedback AC systems, in which the reference signal is replaced by the residual signal as input to the controller (see Fig. 3(a)). Because perfect cancellation is not achievable in feedback systems the original result of [22], i.e. Theorem 3, could not be used to analyze the convergence of FuLMS for feedback systems.
Assuming that there is no measurement noise, i.e. vm(n)=0, the feedback AC
The preconditioned FuLMS algorithm
By Theorem 4 we know under which conditions the FuLMS algorithm converges to the causal Wiener filter. However, the rate of convergence can be extremely slow, which is illustrated by the simulation examples in Section 8. For the FxLMS algorithm, in [2], [3] a prewhitening and the use of the inner–outer factorization of the secondary path transfer function were proposed in an FIR/frequency domain context to increase the convergence rate. Similar preconditioning can be applied to increase the
An illustrative example of the convergence of Theorem 4
In the first example, the AC system is given byNote, that D and S have non-minimum phase zeros at 1.1 and −1.1, respectively and P has no non-minimum phase zeros at 1.1 and −1.1, hence perfect cancellation is not achievable. The causal Wiener filter Wc (10) to which the FuLMS should converge and the factor to which the preconditioned FuLMS should converge are given by
Conclusions
The structure of the optimal controller and the remaining residual noise have been analyzed for AC applications. Insights from this analysis have been used to generalize the convergence results of the FuLMS algorithm which were derived in [22] for the case where perfect cancellation is not achievable. It is shown that the regression vector used in the FuLMS algorithm is always uncorrelated with the residual signal obtained by using the optimal controller (in case the secondary path model
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2018, Control Engineering PracticeCitation Excerpt :As a result, a more computationally efficient algorithm can be obtained in which the system can be accurately described with generally less parameters. Examples are found in Fraanje, Verhaegen, and Doelman (2003) and Landau, Alma, and Airimitoaie (2011), but these works use IIR filters with self-tuning poles which can lead to instability because the poles are not bounded to the open unit disk. An alternative is found in using IIR filters with fixed poles (Ninness & Gómez, 1998; Williamson & Zimmermann, 1996), also called generalized FIR filters (De Callafon, Zeng, & Kinney, 2010).
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This research has been conducted in the framework of the Knowledge Center ‘Sound and Vibration UT-TNO’, programme ‘Robust Active Control’, a joint initiative of TNO, Delft, The Netherlands and the University of Twente, Enschede, The Netherlands.
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Currently visiting Delft University of Technology.