The DXHS algorithm is equivalent to a passive internal model for sinusoidal signals

The DXHS algorithm was proposed as an active noise control method and is often referred to as an adaptive method. This paper shows that the DXHS algorithm is a dynamical system that is equivalent to a passive internal model for sinusoidal signals. It follows that the disturbance attenuation ability is a consequence of the internal model principle, although the DXHS algorithm is proposed based on the standard way of thinking for adaptive methods. Moreover, the closed-loop system stability is guaranteed for strictly passive secondary paths.


Introduction
Active noise control (ANC) is an important control application area [1].Many ANC methods have been proposed so far.As one of such methods, the Delayed-X Harmonics Synthesizer (DXHS) algorithm was proposed [2,3].
The DXHS algorithm assumes that the disturbance is a finite sum of fundamental wave with a known frequency and its harmonics.Roughly speaking, the DXHS algorithm assumes a periodic disturbance with a known period.While the frequency is known, the amplitudes and the phases of the component sinusoidal signals are unknown.Thus, the DXHS algorithm estimates the coefficients of the sine and cosine functions based on the error signals.Due to this nature, the DXHS algorithm is often referred to as an adaptive method.
This paper shows that the DXHS algorithm is equivalent to an internal model.It follows that the disturbance attenuation ability is a consequence of the internal model principle and that the essence of the DXHS algorithm is not adaptation.Moreover, the equivalent internal model is passive.It follows that the closed-loop system with the DXHS algorithm and the passive secondary path is stable.
This paper is organized as follows: In Section 2, the DXHS algorithm is introduced.The main results are shown in Section 3. In Section 4, the main results are confirmed by using numerical examples.
The notation is standard.N and R are the sets of natural and real numbers.

The DXHS algorithm
Figure 1 depicts the scheme of sinusoidal disturbance suppression due to the DXHS algorithm [2].Originally, the DXHS algorithm was proposed in the discrete time domain, we formulate the algorithm in the continuous time domain.Note that the continuous time algorithm was also proposed in [4] in a different form.
The DXHS algorithm assumes that the disturbance d(t) is injected at the output of the secondary path.The disturbance d(s) is defined by where ω > 0 and m ∈ N are the fundamental frequency and the number of harmonics, respectively, while α k ∈ R and β k ∈ R are coefficients.We assume that ω and m are known, while α k and β k are unknown.To suppress d(t), the DXHS algorithm applies the control input u(t) of the following form: where a k (t) ∈ R and b k (t) ∈ R are coefficients to be tuned.Then, the suppression error e(t) is defined by where μ k > 0 is the update gain.The update laws are derived based on the steepest descent direction on e(t) 2 .Note that the update laws ( 4) and ( 5) are independent from the transfer function of the secondary path.Due to this update scheme, the DXHS algorithm is often referred to as an adaptive method.

Main results
The DXHS algorithm is composed of m number of components, where the kth component is a dynamical system from e(t) to u k (t).The kth component system can be written in a state space form.In fact, defining Equations ( 4), (5), and ( 2) can be written by The total system of the DXHS algorithm is the parallel interconnection of all the components.System ( 7) and ( 8) imply that the kth component of the DXHS algorithm is a linear periodic time-varying system with the period of 2π kω .Due to the Floquet decomposition [5], linear periodic time-varying systems can be equivalently transformed to linear time invariant (LTI) systems with real coefficient matrices, where, in general, the transformation matrix has the period twice as long as the coefficient matrices of the original linear periodic time-varying system.
Here, the equivalence of linear time-varying (LTV) systems is defined as follows [5]: Definition 3.1: Let two linear time-varying systems are given by 1 : where Note that both LTI and linear periodic time-varying systems are special cases of LTV systems.
In spite of the Floquet decomposition, the following lemma shows that the transformation from ( 7) and ( 8) to an LTI system can be accomplished by a matrix with the same period as the coefficient matrices in (7) and (8): Theorem 3.2: System (7) and ( 8) is equivalent, in the sense of Definition 3.1, to the following LTI system: where ξ k (t) ∈ R 2 is the state and Moreover, the equivalence can be established by the following transformation: where Proof: Matrix R(kωt) is a unitary matrix and nonsingular for any t.Moreover, the following equation holds for any θ ∈ R: Then, Equations ( 18) and ( 7) yield Furthermore, Equation ( 8) leads to Equation (13).
Theorem 3.2 establishes that the DXHS algorithm is equivalent to an LTI system.Theorem 3.2 also gives the transformation matrix (15) that transforms system (7) and ( 8) to the LTI system (12) and (13).Note that the matrix in ( 15) is periodically time-varying and has the same period as the coefficient matrices of the system ( 7) and (8).Despite the fact that the DXHS algorithm updates the coefficients a k (t) and b k (t) online to estimate the real coefficients α k and β k , the equivalent LTI system ( 7) and ( 8) does not involve any estimation.On the other hand, since the LTI system ( 7) and ( 8) is equivalent to the DXHS algorithm, the former system also suppresses d(t).
Since the two systems are equivalent, the same results should be obtained.However, in implementing the DXHS algorithm, the sinusoidal functions must be computed, while the internal model does not require such computations.Therefore, the DXHS algorithm is more computationally demanding.Moreover, the accuracy of the sinusoidal function computations tends to deteriorate when t tends to be large.The implementation of the internal models are free from those difficulties.
The closed-loop systems due to the DXHS algorithm are often stable and effective in many applications [6].The reason can be understood by calculating the transfer function G k (s) of system ( 7) and (8).By the straightforward calculations, G k (s) is given by In addition to the above fact, G k (s) in (20) reveals that system ( 7) and ( 8) is passive, as is shown by Example 2.17 in [7].It follows that, if the DXHS algorithm is applied to a strictly passive linear time invariant system, the closed-loop system is asymptotically stable and the disturbance is perfectly attenuated when t tends to ∞.
Recently, some papers [8,9] are reported on adaptive MPC applied to ANC.In [8,9], the identified models include the poles corresponding to the sinusoidal disturbance.Therefore, this kind of ANC may also be related to internal models, although the relationship is not clear yet, since the control input is generated by MPC and not represented explicitly.

Numerical example
In this section, the main results are confirmed by using a numerical example.We consider the case that the secondary path is an SISO linear time invariant system.Let the transfer function of the secondary path be given by System P(s) in ( 22) is strictly passive.In fact, the nyquist contour of P(s) is depicted in Figure 2 and contained in the open right half plane for any nonzero ω ∈ R, while P(∞) = 0 and P(0) = 0. Suppose that the disturbance is given by i.e. m = 1.Moreover, we assume that the update gain is μ 1 = 1 and the initial value of the coefficients are We compare the responses due to the DXHS algorithm and the internal model in Equations ( 12) and (13).Figures 3 and 4 show the responses of y and e, respectively.The responses of both the DXHS algorithm and the internal model perfectly coincide in terms of not only the steady state but also the transient responses.
These results confirm that the DXHS algorithm and the internal model are equivalent.When the secondary path is an LTI system and its transfer function is known, the error due to the DXHS algorithm can be easily predicted by using Theorem 3.2.Suppose that d(t) is given by Since 0.9 is not a multiple of ω = 1, this disturbance is not perfectly attenuated.The error is plotted in Figure 5.Moreover, if d(t) is given by the error is plotted also in Figure 5.Note that 3 is a multiple of ω = 1, but m = 1 in this example and d 2 is not perfectly attenuated, neither.The steady state amplitudes of the errors coincide with the gain of the sensitivity function In fact, |S(0.9j)|= 0.22 and |S(3j)| = 1.14 are consistent with the results in Figure 5.Note that the responses in Figure 5 are given by the closed-loop system due to not the internal model but the DXHS algorithm.Since 0.9 is close to 1, d 1 is attenuated to some extent owing to the continuity of the gain of S(s).

Conclusion
We have shown that the DXHS algorithm is equivalent to the passive internal model for sinusoidal signals.
Although the DXHS algorithm is often referred to as an adaptive method, it is equivalent to an LTI system which has no adaptive mechanism.On the other hand, the DXHS algorithm is extended so as to track the frequency of the disturbance [3].A future work is to clarify the relationship between this DXHS algorithm and internal models.

Figure 1 .
Figure 1.Block diagram of the DXHS Algorithm.

Figure 5 .
Figure 5.Comparison of e for the cases of w 1 = 0.9 and w 1 = 3.
all the coefficient matrices have the compatible sizes.Then, 1 and 2 are equivalent, if there exists a differentiable matrix valued function T(t) ∈ R n×n , which is nonsingular for any t, such that x 2 Japan, in 1999, and his M.E.and Ph.D. degrees from Tokyo Institute of Technology in 2001 and 2004, respectively.He was an Assistant Professor at the Department of Systems Science, Graduate School of Informatics, Kyoto University from 2005 to 2011, and an Associate Professor from 2011 to 2017.He was a Professor at Nagoya University from 2017 to 2022.He is currently a Professor at Kyoto University.His research interests include network system, multi-agent system, and systems biology.