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applied sciences Article

Modified LMS Strategies Using Internal Model Control for Active Noise and Vibration Control Systems Byeongil Kim 1 and Jong-Yun Yoon 2, * 1 2

*

School of Mechanical Engineering, Yeungnam University, (Dae-dong) 280 Daehak-ro, Gyeongsan-si 38541, Gyeongsangbuk-do, Korea; [email protected] Department of Mechatronics Engineering, Incheon National University, (Songdo-dong) 119 Academy-ro, Yeonsu-gu, Incheon 22012, Korea Correspondence: [email protected]; Tel.: +82-32-835-8682

Received: 26 April 2018; Accepted: 19 June 2018; Published: 20 June 2018







Abstract: Traditional adaptive filtering algorithms are non-recursive systems that cannot use a time-variant reference input in real time and are not appropriate for control signals with uncertainties and unanticipated conditions. The main purpose of this research is to design novel adaptive digital filtering algorithms based on internal model control (IMC). The new methods consist of a process model for the target plant so as to estimate its dynamic behavior for active vibration and noise attenuation schemes in order to improve the stability, robustness, and tracking performance. On the basis of on the existing least mean squares, the methods are combined with an internal model controller, or the whole adaptive filtering system could become a feedback control system structure based on IMC. The performances were validated in numerical simulations with various conditions that could have happened in realistic applications, and the results were compared with the original algorithms. This study shows that the active noise and vibration systems that are applied to vehicles, mechanical systems, and other targets are enhanced through improving the performance of conventional adaptive filtering algorithms and by using internal model control effectively. Keywords: active vibration and noise mitigation; adaptive filtering system; filtered-X LMS; internal model control; least mean squares; smart structures

1. Introduction Various components, devices, and systems that have been created with ferroelectric substances have been broadly employed for energy transformation and precise positioning devices. Currently, a number of research efforts have been dedicated to active vibration and noise mitigation by smart materials and structures. Active control applications apply force actively with various devices, such as piezoelectric, electromagnetic, and magnetorheological actuators, in order to mitigate the vibration and unexpected noise of given systems. For active control systems on noise, vibration, and harshness (NVH), adaptive digital filtering systems with the least mean square (LMS) algorithm have been used. Previous research on adaptive algorithms could deal with signals that contain relatively simple frequency spectra without unanticipated situations, but the realistic conditions would be affected by various factors, such as unexpected disturbance, uncertainties, and noise. In addition, even after the amplitude of a governing frequency component is considerably attenuated, the left over spectral elements still induce severe vibration and noise. Therefore, more investigations on enhanced adaptive filtering algorithms are required for better performance of the active control systems (such as tracking, robustness, stability, and adaptability).

Appl. Sci. 2018, 8, 1007; doi:10.3390/app8061007

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There are large volumes of research on enhanced LMS strategies. For example, hybrid structures have been proposed, which combine a feedback control with the filtered-X least mean square (FX-LMS) algorithm [1,2]. A different feedback control scheme is introduced with internal model control (IMC) [3]. A modified FX-LMS method with adaptive step-size is developed for improving the rate of convergence [4,5]. An LMS algorithm with variable steps is applied so as to determine the hysteresis parameters and a creep model for stability and faster convergence [6]. It has shown that a changeable coefficient for convergence could also improve the performance [7]. Because the LMS can be unstable under unpredictable noise, a novel LMS strategy is investigated, where reference signals are calculated in advance in order to guarantee convergence and stability, particularly in case where the signal magnitude exceeds a defined threshold [8]. An LMS algorithm against saturation is introduced so as to enhance the performance, by restraining the controller output with a large disturbance [9]. An additional filter is used to decrease disturbances in the primary path, which were induced from an uncorrelated noise of the secondary path [10]. A new LMS algorithm with phase compensation is suggested so as to deal with signals with DC (direct current) components [11]. There have also been efforts devoted to using Iin order MC to improve the traditional LMS method. Wang and Gan investigated an internal model feedback control system based on the FX-LMS algorithm and applied it to stochastic analyses of band-limited white noise [12]. Shafiq and Riyaz developed an adaptive IMC scheme based on adaptive finite impulse response (FIR) filters [13]. Zhu et al. proposed an internal-model–P (proportional) cascade control system, with all of the models in the control system constructed by LMS filters for online identification [14]. The main purpose of this study is to suggest a LMS algorithm based on state-of-the-art internal model control (IMC) for smart structures. The fundamental element is the IMC, which integrates the model as part of the controller. This algorithm enables effective mitigation of unwanted disturbances with adaptive filtering systems, and can handle noise and unexpected signals with systems containing uncertainties, using a feedback loop in the active control system. Strong vibration and noise cancelling in a broad frequency range is possible, and this technique can be applied to structural vibration controls in mechanical systems and can be used for structure-borne noise attenuation inside vehicles. The process model and the feedback loop in the control structure can effectively control the multi-spectral components of a target signal with the assistance of IMC, based on the traditional LMS algorithm. In addition, the nonlinearities, model uncertainties, and other unpredicted conditions can be managed by the feedback loop. The objective of this research is to propose two enhanced adaptive algorithms based on IMC, with the perspective of improving the LMS algorithm for managing the stability of the active control system and multi-spectral signal control, while maintaining its inherent structure. The specific objectives are as follows: (1) examine the concept of the internal model least mean squares method 1 (IM-LMS 1); (2) examine the concept of the internal model least mean squares method 2 (IM-LMS 2); and (3) conduct computational studies to validate the new methods for various conditions and to compare them with the traditional LMS method. 2. Adaptive Digital Filtering System Commonly, digital filters are used for modeling the response properties of given inputs on acoustic and structural systems. The FIR filter is widely used for modeling system responses on account of its intrinsic stability, easy application, and simplicity. Adaptive digital filters also contain a notable capability for analysis and control of adaptive structures. Adaptive digital signal processing is used to adjust the coefficients of digital filters so as to generate a wanted signal to a given system. Adaptive filters update their filter coefficients with a reference signal and error, based on a repetitive algorithm, and are commonly used in active control applications such as noise and vibration attenuation, inverse modeling, and system identification. Among a number of adaptive filtering algorithms, the LMS method is the most prevalent feedforward control algorithm because of its relative simplicity [15]. It adjusts a digital filter by

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Among number Appl. Sci. 2018, 8,a1007

of adaptive filtering algorithms, the LMS method is the most prevalent 3 of 16 feedforward control algorithm because of its relative simplicity [15]. It adjusts a digital filter by the reference input uk, with frequency components to be dealt with; the measured error ek at the current the reference , with frequency components to bethe dealt with;value the measured error ek at the current moment; the input gap ofukthe adaptive filter output yk and desired dk. This algorithm makes the moment; the gap of the adaptive filter output y and the desired value d . This algorithm makes k output of the filter close to the plant output, which is the desired signal.k Updating the equation the for output of the filter close to the plant output, which is the desired signal. Updating the equation for the the filter coefficients of the LMS can be acquired with the minimization of the mean squared error filter coefficients of the LMS can be acquired with the minimization of the mean squared error chosen chosen by a cost function [16]. by a cost function [16]. wkk++1 = =w + μµuukk ·⋅ eekk (1) w wk + (1)

Here, w and µ adjusts thethe stability andand the the speed of the wkk stands stands for forthe theweight weightvector vectorofofthe thefilter filter and μ adjusts stability speed of process. Figure 1 represents the diagram of a common digital filter system. It can be noticed that the the process. Figure 1 represents the diagram of a common digital filter system. It can be noticed that filter coefficients cannotcannot track the actual the steepest the weight are the filter coefficients track the surface actual of surface of thedescent, steepestsince descent, since changes the weight calculated bycalculated gradient estimates withestimates simplification factor would rather changes are by gradient with assumptions. simplificationThis assumptions. Thisinduce factora would crude inducetracking a ratherperformance. crude tracking performance.

Figure least mean mean square square (FX-LMS) (FX-LMS) algorithm. algorithm. Figure 1. 1. Digital Digital filtering filtering system system with with filtered-X filtered-X least

The LMS algorithm is suitable for cases where there are no dynamics in the secondary path of The LMS algorithm is suitable for cases where there are no dynamics in the secondary path of the control system. Nevertheless, in most practical situations, some influences could exist from the the control system. Nevertheless, in most practical situations, some influences could exist from the sensors, actuators, amplifiers, filters, digital converters, and so on. The algorithm would not sensors, actuators, amplifiers, filters, digital converters, and so on. The algorithm would not function function appropriately with such effects. The most usual type of the secondary path dynamics is a appropriately with such effects. The most usual type of the secondary path dynamics is a time delay time delay or a phase shift, and the overall system seems to converge to a trivial value or become or a phase shift, and the overall system seems to converge to a trivial value or become unstable, unstable, because the* signal yk* and the filter output yk are distinct. because the signal yk and the filter output yk are distinct. The FX-LMS is a modification of the above LMS method that incorporates the dynamics of the The FX-LMS is a modification of the above LMS method that incorporates the dynamics of the secondary path, which have been obtained beforehand [17]. This algorithm consists of a filtered secondary path, which have been obtained beforehand [17]. This algorithm consists of a filtered reference and it is broadly known that FX-LMS is more stable when the dynamics of the secondary reference and it is broadly known that FX-LMS is more stable when the dynamics of the secondary path cannot be ignored. The below equation defines the FX-LMS. path cannot be ignored. The below equation defines the FX-LMS. ˆ k +1 = w ˆ k + μ u k * ⋅ ek w (2) ˆ k +1 = w ˆ k + µuk ∗ · ek w (2) The significant property of the Pc*(z) filter is that its response with impulse contains the The significant of the Pc *the (z) filter is thatofitsthe response withpath impulse the there identical identical quantity ofproperty time delay with dynamics secondary Pc(z).contains However, are quantity ofdefects time delay with the dynamics the secondary path Pc(ID) (z). However, are noticeable noticeable of the FX-LMS, which of a system identification procedurethere is needed so as to * (z) defects of the a systemAn identification (ID)ID procedure is possible needed soalso, as tobut obtain the Pc the obtain the Pc*FX-LMS, (z) offlinewhich in advance. online system might be it causes offline advance. An online system IDismight be possible also, but it causes the overall control system overallincontrol system complex and it difficult to implement. complex and it is difficult to implement. 3. Internal Model Control 3. Internal Model Control IMC uses the plant model as a component of the controller [18]. The IMC has a number of IMCcompared uses the with plantthe model as a component of the controller [18]. The has number of benefits traditional PI (proportional-integral) control. FirstIMC of all, thea closed-loop benefits compared with the traditional PI (proportional-integral) control. First of all, the closed-loop stability is guaranteed merely by selecting a stable controller if an accurate plant model can be stability is guaranteed merely by selecting a stable controller if an accurate plant model can be

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obtained. Furthermore, the controller design is relatively straightforward and closed-loop obtained. Furthermore, thesuch controller design is relatively straightforward performance properties, as the rising time, settling time, and and timeclosed-loop constant, performance are directly properties, such as the rising time, settling time, and time constant, are directly correlated with the correlated with the control parameters. This makes online tuning of the controller very convenient. control parameters. makes online of the controller very convenient. The IMC structure is a The IMC structure isThis a modification oftuning a classical feedback structure, with the process model applied modification of a classical feedback structure, with the process model applied definitely as an internal definitely as an internal part of the controller [19]. The general structure of an IMC system is shown part of the2.controller [19]. The general structure of an IMC system is shown in Figure 2. in Figure

Figure 2. Internal model control (IMC) system. Figure 2. Internal model control (IMC) system.

The purpose of designing an IMC controller is to force the output to respond to a set point Thewith purpose of designing IMC controller to force output to respond to a set point change desired mannersan and counter the is effects of the disturbances. If the controller gainchange is the with desired manners and counter the effects of disturbances. If the controller gain is the inverse of the inverse of the model gain, and the controller is tuned to assure stability, the process output y will model gain, and the controller is tuned to assure stability, the process output y will eventually reach eventually reach and maintain the set point ys in the absence of new disturbances. Unfortunately, a and maintain set point ys in the ifabsence of new a model is never model is neverthe perfect. In addition, the model has disturbances. any dynamics Unfortunately, at all, rather than just a gain, no perfect. In addition, if the model has any dynamics at all, rather than just a gain, no controller can controller can perfectly invert the process model. A controller can, however, come very close to perfectly the process model. A controller can,come however, very to inverting the process invertinginvert the process model. Just how close it can is thecome subject ofclose the IMC design problem, in model. Just how close it can come is the subject of the IMC design problem, in order to expect perfect order to expect perfect models. When no zeros are near the imaginary axis or in the right half of the models. no zeros aremodel near the in the right half If of the the form s-plane, the inverse s-plane, When the inverse of the is imaginary stable and axis not or overly oscillatory. of the process of is the model stable and overly If polynomials the form of the is defined ascontroller Equation can (3), defined as is Equation (3), not where N(s)oscillatory. and D(s) are of s,process the internal model where N(s)easily and D(s) are polynomials of s, the internal model controller can be chosen easily as follows. be chosen as follows. N((ss)) N = pp((ss)) = D((ss)) D D (s) Dλs ( s+ ) 1)r N ( s )( q ( s ) = q ( s ) f ( s ) = r q(s) = qe(s) f (s) =

(3) (3)

(4) (4) N ( s )(λ s + 1) In this form, the filter f (s) has two parameters, λ and r. The relative order r is the order of the denominator minus thefilter order ofhas the two numerator and it λ is and selected to be large enough forthe q(s),order in order to In this form, the f(s) parameters, r. The relative order r is of the be proper. The filter time constant λ is selected in order to avoid excessive noise amplification. The main denominator minus the order of the numerator and it is selected to be large enough for q(s), in order advantage is that simply by choosing a stable internal model controller. to be proper. Theclosed-loop filter time stability constantisλassured is selected in order to avoid excessive noise amplification. In addition, it has systematic and relatively straightforward methods, with advantages such The main advantage is that closed-loop stability is assureddesign simply by choosing a stable internal as internal stability, and performance robustness. The structure also intuitive, model controller. Instability addition,robustness, it has systematic and relatively straightforward designismethods, with and the closed-loop characteristics directly related to the controller parameters, advantages such as performance internal stability, stability are robustness, and performance robustness. The making online tuning of the IMC very convenient. structure is also intuitive, and the closed-loop performance characteristics are directly related to the Consider the following model for an arbitrary plant with second-order linear system dynamics controller parameters, making online tuning of the IMCT(s) very convenient. P(s) and time-varying delay term D(s),for as follows: Consider the following model an arbitrary plant T(s) with second-order linear system dynamics P(s) and time-varying delay term D(s), as follows: aωn 2 T (s) = P(s) · D (s) = 2 · e−τs (5) aωn sn 2+ ωn 2 −τ s s + 2ζω T ( s ) = P ( s ) ⋅ D( s ) = 2 e ⋅ (5) 2 + 2ζωfrequency, n s + ωn Here, a, ω n , ζ, and τ are the proportional gain,s natural damping ratio, and time delay, respectively. As IMC is a linear controller that requires a linear system model, the exponential function Here, a, ωn, ζ , and τ are the proportional gain, natural frequency, damping ratio, and time in the model needs to be linearized. For this purpose, (0, 1) the Padé approximation is applied so as to delay, respectively. As IMC is a linear controller that requires a linear system model, the exponential function in the model needs to be linearized. For this purpose, (0, 1) the Padé

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convert the exponential function into a rational function, such that the following, and then the actuator model, becomes linearized, as follows: e−τs ≈

1 e (s) =D τs + 1

1 aωn 2 e(s) = Y (s) · D e (s) = T · 2 2 V s + 2ζωn s + ωn τs + 1

(6)

(7)

With this model, an internal model controller can be designed. By taking the reciprocal of the plant model, the controller transfer function can be written as follows: C (s) =

s2 + 2ζωn s + ωn 2 τs + 1 · 2 aωn (λs + 1) p

(8)

where the term (λs + 1)p in the denominator is introduced in order to make the transfer function proper (i.e., the order of the denominator is not lower than that of the numerator). Therefore, p needs to be determined by the order of the numerator in the controller transfer function. In this case, p = 3 is used. The time constant λ in the new term depends on the modeling errors and the allowable noise amplification by the controller. Although IMC uses the plant model in the controller design, good performance cannot be expected unless the exact model of the plant is known. Moreover, if there are discrepancies between the plant model and the actual plant dynamics, IMC can render the controlled system unstable. The main limitation of the IMC stems from the fact that they are basically linear control methodologies, while the plant is nonlinear. 4. Enhanced LMS Methods with IMC This part discusses the investigation of the two enhanced LMS methods, with a supporting model-based control scheme. The key topic is the use of IMC. Two different structures of adaptive digital filtering systems have been proposed, based on IMC for active noise and vibration attenuation. These systems promote stability performance and enable the multi-spectral control of disturbances in order to manage noise signals and a number of frequency elements in a certain signal simultaneously. 4.1. Novel Adaptive Filtering Algorithm Based on the IMC 1: Non-Recursive Method 4.1.1. Conceptual Explanation As mentioned, the LMS algorithm could cause instability of the whole control system, or an incorrect solution may be obtained after convergence, since neglecting the secondary path dynamics Pc is essentially unavoidable in realistic situations. The FX-LMS algorithm overcomes this inadequacy by using a reference filter for the input and by compensating for time and phase delays that are induced by several auxiliary parts that are included in the filtering system (sensors, converters, amplifiers, etc.). However, the model of the secondary path dynamics must also be identified offline in advance, in order to obtain the coefficients of the filter, which adds to the effort that is needed to implement the control system. Thus, the IMC can be substituted for the filter after the reference signal in the FX-LMS system, as shown in Figure 3. The IMC system is inserted before the secondary path dynamics, so that the effect of Pc can be compensated without destabilizing the adaptive filtering system. If we roughly know the dynamics of the secondary path, there is no need to identify it offline beforehand, and the control system can be implemented immediately. The key point for this novel method is that the filtered-X LMS algorithm is not needed for the system with secondary path dynamics and the offline process for identification is not required if we have ‘even’ a rough model of the secondary path.

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Figure 3. Schematic of a new digital filtering system with LMS, based on the internal model control Figure 3. Schematic of a new digital filtering system with LMS, based on the internal model control (IM-LMS 1). (IM-LMS 1).

The shaded region of Figure 3 indicates the IMC system for the secondary path dynamics Pc. The shaded region of Figure indicates based the IMC for themodel secondary path Pc .  c . Let The internal model controller Q is3designed on system an estimated thedynamics value right P ˆ c . Let the value right before The internal model controller Q is designed based on an estimated model P before the internal model controller Q be tk. Then, the following equation holds. the internal model controller Q be tk . Then, the following equation holds.

(

)

 c
tk = yk − tk QPc − tk QP tk = yk − tk QPc − tk QPˆ c

(9) (9)

The last two terms on the right-hand side are as follows. The last two terms on the right-hand side are as follows.

tk QPc = yk

(10a) (10a)

tk QPc = yk

yky* ∗k →→tktQ = kQ = PP c c → tk =

(10b) (10b)

y∗k

(10c)

* QP k c

y → tk = y∗k QPc ˆ ˆ

Pˆ c tk QPc = · Pc = y∗k Pc Pc

c From Equation (10b–d), Equation (9) becomes yk* the following: P * tk QP c = y∗k

Pc

⋅ Pc = 

yk Pc ˆ 

Pc = yk − y∗k 1 − Pc From Equation (10b–d), EquationQP (9)c becomes the following:

(10c) (10d)

(10d) (11)

 c  the original filter output yk and the By arranging Equation (11) with yky* ,* the difference  between P * k * 1 = y − y − (11) IMC compensated filter output yk can be obtained as k k  follows. QPc

y∗k



Pc 

1 By arranging Equation (11) with yk*, the=difference between the original filter output yk and (12) the Pˆ c 1 yk 1 − + * Pc follows. QPc IMC compensated filter output yk can be obtained as If an exact model of the secondary path yk* dynamics 1 can be obtained, Pˆ c = Pc , and Equation (12) = becomes the following:  y (12) y∗k k 1 − P c + 11 = QPc = (13) Pc (λsQP yk + 1c)r  c =value yk an andexact yk * would close to each other after applying IMC if the λ can be selected Equation (12) If modelbe ofvery the secondary path dynamics can be the obtained, P Pc , and properly, and the system instability that is caused by the secondary path dynamics would be decreased. becomes the following:

yk* 1 = QPc = (λ s + 1) r yk

(13)

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4.1.2. Numerical Validation

yThe k and yk* would be very was closeimplemented to each otherinafter applying the IMC and if thewas value λ can be proposed technique numerical simulations validated by selected properly, and the system LMS instability that is caused by the secondary path dynamics would comparison with the conventional algorithm. An adaptive digital filter with a FIR filter structure be hasdecreased. been defined with two coefficients. The target disturbance plant is assumed to be a second order system with a natural frequency of ω n = 50 Hz and a damping ratio of ζ = 0.1. A sinusoidal input 4.1.2. Validation signalNumerical x(t) = A1 sin(ω 1 t + θ 1 ) with one frequency component at ω 1 /2π = 100 Hz is used as both the inputThe reference andtechnique the disturbance plant input (A 1, θ 1 = 0).simulations The µ valueand waswas chosen as 0.4 for proposed was implemented in1 = numerical validated by both cases. comparison with the conventional LMS algorithm. An adaptive digital filter with a FIR filter In thehas firstbeen case,defined supposewith thattwo the input filter is The not used an adaptive plant filtering system when structure coefficients. targetindisturbance is assumed to bethe a secondary path dynamics P (z) cannot be neglected. The secondary path is presumed to be a first-order s second order system with a natural frequency of ωn = 50 Hz and a damping ratio of ζ = 0.1. A system, with a time constant 0.01, as in Equation (14b), and the exact model is assumed to be available. sinusoidal input signal x(t) = A1sin(ω1t + θ1) with one frequency component at ω1/2π = 100 Hz is used The internal model controller Q(z) can be designed as Equation (14c), with the model of the target as both the input reference and the disturbance plant input (A1 = 1, θ1 = 0). The μ value was chosen as plant Pˆs (z) following the procedure in Section 3. The value of λ is chosen as 0.002. 0.4 for both cases. In the first case, suppose that the input filter is not 1 used in an adaptive filtering system when Ps (be z) = (14a) the secondary path dynamics Ps(z) cannot neglected. 0.01s + 1The secondary path is presumed to be a first-order system, with a time constant 0.01, as in Equation (14b), and the exact model is assumed to be available. The internal model controllerPˆsQ(z) can be1designed as Equation (14c), with the model of (14b) (z) = 0.01s + 1  the target plant P s ( z ) following the procedure in Section 3. The value of λ is chosen as 0.002. 0.01s + 1 Q(z) = (14c) 1 Ps ( z ) = 0.002s + 1 (14a) 0.01 s + 1desired signal with a sinusoidal input under Figure 4a compares the time domain tracking of the the control of the LMS and the internal model LMS method 1 (IM-LMS 1). Figure 4b shows the 1 becomes unstable and diverges without the estimation error. The conventional system LMS  s ( zwith )= P (14b) filtered reference signal, but IM-LMS 1 can manage 0.01sthis + 1 condition just as with the filtered-X LMS system, and the whole system can follow the reference signal with stability.

0.01s + 1 1 Q(Pz()z=) = s 0.002 0.01ss + +11

(14c) (15a)

Figure 4a compares the time domain tracking of 1 the desired signal with a sinusoidal input Pˆs (z) = (15b) under the control of the LMS and the internal model 0.05sLMS + 1 method 1 (IM-LMS 1). Figure 4b shows the estimation error. The conventional system with LMS becomes unstable and diverges without 0.05s + 1 Q(1z)can = manage this condition just as with the filtered-X (15c) the filtered reference signal, but IM-LMS 0.002s + 1 LMS system, and the whole system can follow the reference signal with stability.

(a)

(b) Figure 4. Cont.

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(c) Figure 4. Simulation results in time domain with a single sinusoid at 100 Hz. (a) Time domain tracking with LMS and internal model LMS 1 methods; (b) estimation error; (c) predicted spectra.

Ps (z) =

1 0.01s + 1

(15a)

1 Pˆs ( z ) = (c) (15b) 0.05s + 1 Figure 4. Simulation in timedomain domain with a single sinusoidsinusoid at 100 Hz. (a) Figure 4. Simulation resultsresults in time with a single atTime 100domain Hz. tracking (a) Time domain 0.05 s + 1 error; (c) predicted spectra. with LMS and internal model LMS 1 methods; (b) estimation tracking with LMS and internal model LMS (b) estimation error; (c) predicted(15c) spectra. Q ( z1)methods; = 0.002 s + 1 Assuming conditions as the case, case, FigureFigure 5a compares the time-domain tracking Assuming the thesame same conditions as previous the previous 5a compares the time-domain of the desired under the under controlthe of control IM-LMSof with an 1, exact model and with 11, IM-LMS tracking of thesignal desired signal with an exact modeluncertainty. and with P = (z) Figure 5b shows the estimation error. In the transient state, up to around 0.1 s, uncertainty s uncertainty. Figure 5b shows the estimation error. In the transient state, up to around in 0.1the s, s + 0.01 1 model induces more error than the exact model. However, after that point, the error converges to zero uncertainty in the model induces more error than the exact model. However, after that point, the in the converges steady state. model hasmistuned been usedplant in themodel controlhas system, error to Although zero in thea significantly steady state. mistuned Althoughplant a significantly been the IM-LMS 1 method made the system converge and showed an acceptable tracking performance. used in the control system, the IM-LMS 1 method made the system converge and showed an 1 ˆ This suggests that the new algorithm has more self-adaptability. P z = ( ) acceptable tracking performance. Thisssuggests that the new algorithm has more self-adaptability.

0.05s + 1

Q( z ) =

0.05s + 1 0.002 s + 1

(15a)

(15b)

(15c)

Assuming the same conditions as the previous case, Figure 5a compares the time-domain tracking of the desired signal under the control of IM-LMS 1, with an exact model and with uncertainty. Figure 5b shows the estimation error. In the transient state, up to around 0.1 s, uncertainty in the model induces more error than the exact model. However, after that point, the error converges to zero in the steady state. Although a significantly mistuned plant model has been used in the control system, the IM-LMS 1 method made the system converge and showed an acceptable tracking performance. (a) This suggests that the new algorithm (b) has more self-adaptability. Figure 5. Cont.

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(c) Figure Figure 5. 5. Simulation Simulation results results in in the the time time domain domain (single (single sinusoid sinusoid at at 100 100 Hz) Hz) with with LMS LMS and and internal internal model LMS 1 methods. (a) With exact model (top) and with uncertainty (bottom); (b) comparison model LMS 1 methods. (a) With exact model (top) and with uncertainty (bottom); (b) comparison of of estimation estimation error; error; (c) (c) predicted predicted spectra. spectra.

Next, to check the tracking performance of the algorithm, the IM-LMS 1 algorithm was Next, to check the tracking performance of the algorithm, the IM-LMS 1 algorithm was evaluated evaluated for the amplitude-modulated (AM) signal xa(t), which is defined as follows: for the amplitude-modulated (AM) signal xa (t), which is defined as follows: xa (t ) = A0 a (1 + γ rand ) cos(ω ca t + θ c ) ⋅ 1 + B0 a cos(ω oa t + θ o ) (16a) x a (t) = A0a (1 + γrand ) cos(ωca t + θc ) · [1 + B0a cos(ωoa t + θo )] (16a)

[

]

= A0 aπ [δ (ω − ωca ) + δ (ω + ωca )] FF[[xxa ((tt)] a )] = A0a π [ δ ( ω − ωca ) + δ ( ω + ωca )] 11 A B π [δ(ω − ω + ω ) + δ(ω + ω − ω ) (16b) ca oa ca oa 0a 0a (16b) ++ 2 A 0 a B0 aπ [δ (ω − ωca + ωoa ) + δ (ω + ωca − ωoa ) +2δ(ω − ωca − ωoa ) + δ(ω + ωca + ωoa )] δ /2π (ω −=ω12 − ωwhich (ωmodulated + ωca + ωwith ca Hz, oa ) + δis oa )] a carrier signal at ω ca /2π = xa (t) contains a sinusoid at + ω oa 150 Hz has a modulation of=B12 0.5.which For convenience, as 1.0, and the 0a = 0a is normalized contains a sinusoid atdepth ωoa/2π Hz, is modulatedAwith a carrier signal at ω ca/2π = xa(t)and phase θ c and θ o are ignored. ω caconvenience, is used as aAreference input. Equation (16b), 150 Hzangles and has a modulation depth A ofsinusoid B0a = 0.5.at For 0a is normalized as 1.0, and the F[ ] illustrates the Fourier transform, and δ indicates the Dirac delta function that is located at frequency phase angles θc and θo are ignored. A sinusoid at ωca is used as a reference input. Equation (16b), F[ ] ω. The LMS parameter µ is chosen as 0.5. For comparison, the same signal is also tracked by the illustrates the Fourier transform, and δ indicates the Dirac delta function that is located at frequency conventional FX-LMS. μ is chosen as 0.5. For comparison, the same signal is also tracked by the ω. The LMS parameter Figure 6a compares the time domain tracking of the desired signal with the filter output, for both conventional FX-LMS. FX-LMS and IM-LMS 1. Although works, as as the FX-LMS, with the no need a filtered Figure 6a compares the time IM-LMS domain 1tracking of well the desired signal with filterfor output, for reference, it has a much worse tracking performance. The estimation error of IM-LMS 1 shown both FX-LMS and IM-LMS 1. Although IM-LMS 1 works, as well as the FX-LMS, with no need forin a Figure 6b even grows over indicates that IM-LMS 1 is The merely helping error the LMS method to filtered reference, it has a time. muchThis worse tracking performance. estimation of IM-LMS 1 accomplish a more control andindicates is not suitable for controlling multi-spectral signals. shown in Figure 6bstabilized even grows over system time. This that IM-LMS 1 is merely helping the LMS

method to accomplish a more stabilized control system and is not suitable for controlling 4.2. Novel Adaptive Filtering Algorithm Based on the IMC 2: Recursive Method multi-spectral signals. 4.2.1. Conceptual Explanation 4.2. Novel Adaptive Filtering Algorithm Based on the IMC 2: Recursive Method IM-LMS 1 showed a better stability performance than the conventional LMS algorithm without filtered reference Explanation input signals, and no offline tuning of the secondary path dynamics was needed, 4.2.1. Conceptual which makes the proposed method convenient. However, the tracking performance of this adaptive IM-LMS 1 showed a better stability performance than the conventional LMS algorithm without filtering system is at the same level as the established method, so more efforts with a modified algorithm filtered reference input signals, and no offline tuning of the secondary path dynamics was needed, and system structures are needed. Mechanical systems that generate noise and vibration, such as which makes the proposed method convenient. However, the tracking performance of this adaptive vehicles, engineering structures, and electromechanical systems, have become more complicated, filtering system is at the same level as the established method, so more efforts with a modified and the signals contain very rich frequency spectra. The selection of a reference input is important algorithm and system structures are needed. Mechanical systems that generate noise and vibration, as it determines the tracking performance. An alternative technique based on IMC is thus needed to such as vehicles, engineering structures, and electromechanical systems, have become more effectively deal with signals that have a complex frequency spectra. complicated, and the signals contain very rich frequency spectra. The selection of a reference input is

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important as8,it1007 determines the tracking performance. An alternative technique based on IMC is10thus Appl. Sci. 2018, of 16 needed to effectively deal with signals that have a complex frequency spectra. Appl. Sci. 2018, 8, x FOR PEER REVIEW

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important as it determines the tracking performance. An alternative technique based on IMC is thus needed to effectively deal with signals that have a complex frequency spectra.

(a)

(b)

(a)

(b)

(c) (c) signal. Figure amplitude-modulated (AM) Figure 6. 6. Control Control of of an an amplitude-modulated (AM) signal. (a) (a) Time-domain Time-domain tracking tracking with with LMS LMS and and internal model LMS 1 methods; (b) estimation error; (c) predicted spectra. internal model LMS 1 methods; (b) estimation error; (c) predicted spectra. Figure 6. Control of an amplitude-modulated (AM) signal. (a) Time-domain tracking with LMS and internal model LMS 1 methods; (b) estimation error; (c) predicted spectra.

Consequently, a recursive loop is proposed into the digital filter system. The schematic of the Consequently, a recursive loop is proposed into the digital filter system. The schematic of the IMC-based adaptive digital filtering system with ainto ‘recursive’ LMS algorithm is shown in Figure Consequently, a recursive loop is proposed the digital filter system. The schematic of the 7. IMC-based adaptive digital filtering system with a ‘recursive’ LMS algorithm is shown in Figure 7. IMC-based adaptive digital filtering system with a ‘recursive’ LMS algorithm is shown in Figure 7.

Figure 7. Schematic of a novel digital filter system with ‘recursive’ LMS algorithm, based on internal

Schematic of aa novel novel digital filter filter system with ‘recursive’ LMS algorithm, based on internal Figure 7. Schematic of model control (IM-LMS 2). digital model control (IM-LMS 2).

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The fundamental concept of this system is to create or estimate a reference signal xk * for the LMS algorithm in real time, with an estimated output of the disturbance plant dˆk . The error signal ek is the difference between the output of the disturbance plant dk and the output of the secondary path dynamics yk * . The estimated output of dˆk can be obtained by adding ek to the filtered output signal yk Pˆ c with the internal model of the secondary path dynamics Pc . Here, k indicates the sampling instant, and a time delay block z−∆ is placed as a result of the estimation process and the secondary path dynamics. In order to minimize the effect of this type of time delay, the internal model controller Q is again introduced and it is designed based on the secondary path dynamics Pˆ c . It is basically a single-channel system with one reference signal and one error sensor, and it can be extended to cover multi-channel signals. The filtered output signal yf_k is defined by multiplying yk with an internal model Pˆ c , which estimates the impulse response of the secondary path dynamics: y f _k = Pˆ c T yk

(17)

The estimated reference signal xk * in the recursive adaptive filtering system can then be obtained. x∗k =

h

dˆk−∆

dˆk−∆−1

· · · dˆk−∆− L+1

xk∗ = dˆk−∆ = (ek − yk Pˆ c )Qz−∆

i

(18) (19)

The output signal yk of the adaptive filter wk is calculated using Equation (20), where wk is the weight vector of the filter and L is the length of the adaptive filter. yk = wk T · x∗k wk T =

h

wk_0

wk_1

· · · wk_L−1

(20) iT

(21)

The weight update algorithm is the same as the conventional digital filter system with the LMS. The adaptive weight vector wk is updated with the following equation: wk+1 = wk + µx∗k ek

(22)

where µ is the step size. The error signal ek is obtained, and the output of the filter that has been affected by the secondary path dynamics yk * is defined as follows: ek = dk − y∗k

(23)

y∗k = Pc · yk

(24)

The traditional feedback ANC system with the filtered-X LMS algorithm has filter before the reference signal. However, in this novel system, internal model control is introduced and it is used for compensating unavoidable time delay in the feedback loop. This kind of time delay is actually from the secondary path dynamics and the estimation process. At least, the effect from the secondary path dynamics can be managed and it does not need offline identification process for the secondary path again. 4.2.2. Numerical Validation The internal model LMS method 2 (IM-LMS 2) was validated for two different kinds of multi-spectral signals, namely, AM and frequency-modulated (FM) signals. The same AM signal is used for evaluating the method. Figure 8 compares the tracking performance and estimation error of the conventional LMS and IM-LMS 2. In the transient region, up to around 0.05 s, the IM-LMS 2 method

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shows a much tracking performance, as shown in Figure 8a. However, in the method showsbetter a much better tracking performance, as shown in Figure 8a.steady-state However, region in the after 0.05 s, the IM-LMS 2 has a slower convergence than FX-LMS, for around 0.4 s (0.05 t ≤in0.45), method shows a much better tracking performance, as convergence shown in Figure 8a. However, the steady-state region after 0.05 s, the IM-LMS 2 has a slower than FX-LMS, for≤around 0.4 and finally, the estimation errors of both of the methods have similar peak-to-peak values, as shown in steady-state region after 0.05 s, the IM-LMS 2 has a slower convergence than FX-LMS, for around 0.4 s (0.05 ≤ t ≤ 0.45), and finally, the estimation errors of both of the methods have similar peak-to-peak Figure 8b. svalues, (0.05 ≤ as t ≤shown 0.45), and finally,8b. the estimation errors of both of the methods have similar peak-to-peak in Figure Since FX-LMS targeted to the component ω ca , the ω time-domain tracking values, as shown inisFigure 8b. only Since FX-LMS is targeted only to 150 theHz 150frequency Hz frequency component ca, the time-domain in steady state would beenhave superior that IM-LMS Thiscomponent is clear in is the spectrum Sincein FX-LMS ishave targeted onlybeen to to the 150of Hz frequency ωclear cafrequency , the time-domain tracking steady state would superior to that of2.IM-LMS 2. This in the frequency plot of Figure 9, which shows a difference of in magnitude of almost 30 dB at 150 Hz. This figure tracking steady state would have beenasuperior to of that IM-LMS 2.ofThis is clear inat the frequency spectruminplot of Figure 9, which shows difference inof magnitude almost 30 dB 150 Hz. This compares the predicted spectra of the uncontrolled AM signal and controlled AM signal with both of spectrum plot of Figure 9, which shows a difference of in magnitude of almost 30 dB at 150 Hz. This figure compares the predicted spectra of the uncontrolled AM signal and controlled AM signal with the methods. Although a broadband reduction cannot be achieved with either FX-LMS or IM-LMS 2, figure compares the predicted spectra of the uncontrolled AM signal and controlled AM signal with both of the methods. Although a broadband reduction cannot be achieved with either FX-LMS or they have shown significant reductions at primary peaks (ω , ω − ω , ω + ω ). The amplitude ca be ca achieved oa ca, ca both of the a broadband reduction cannot either or IM-LMS 2, methods. they haveAlthough shown significant reductions at primary peaks (ω ωwith ca − oa ω oa, ωcaFX-LMS + ωoa). The reduction levels of FX-LMS at the primary peaks are higher than those of IM-LMS 2, while FX-LMS IM-LMS 2, they have shown significant reductions at primary peaks (ω ca , ω ca − ω oa , ω ca + ω oa ). The amplitude reduction levels of FX-LMS at the primary peaks are higher than those of IM-LMS 2, shows a severe spillover at of higher frequency, fromfrequency, around Hz to 270than Hz. Thistoof indicates that amplitude reduction levels FX-LMS atatthe primary peaks180 are higher IM-LMS 2, while FX-LMS shows a severe spillover higher from around 180those Hz 270 Hz. This the conventional LMS algorithm with a feedforward control structure is insufficient to control the while FX-LMS shows a severe spillover at higher fromcontrol aroundstructure 180 Hz to 270 Hz. This indicates that the conventional LMS algorithm withfrequency, a feedforward is insufficient to multi-spectral signals, without full knowledge of the original signal. indicates that the conventional LMS algorithm a feedforward controlsignal. structure is insufficient to control the multi-spectral signals, without full with knowledge of the original

control the multi-spectral signals, without full knowledge of the original signal.

(a) (a)

(b) (b)

Figure 8. 8. Control Control of of an an amplitude-modulated amplitude-modulated (AM) (AM) signal. signal. (a) (a) Time-domain Time-domain tracking tracking with with LMS LMS and and Figure Figure Control of an amplitude-modulated internal8.model model LMS methods; (b) estimation estimation(AM) error.signal. (a) Time-domain tracking with LMS and internal LMS 22 methods; (b) error. internal model LMS 2 methods; (b) estimation error.

Figure 9. Predicted spectra for an amplitude-modulated (AM) signal with LMS and internal model Figure Predicted spectra for an amplitude-modulated (AM) signal with LMS and internal model LMS 2 9. methods. Figure 9. Predicted spectra for an amplitude-modulated (AM) signal with LMS and internal model LMS 2 methods. LMS 2 methods.

In addition, the IM-LMS 2 method shows considerable attenuation at around 50 Hz, which is In addition, the IM-LMS 2 method considerable attenuation at aroundsimilar 50 Hz,amplitude which is the natural frequency of the target plant,shows whereas the FX-LMS case demonstrates the natural frequency of the target whereas the FX-LMS case demonstrates similar amplitude levels around this frequency. This plant, condition comes from the benefit of the recursive structure of the levels around this frequency. This condition comes from the benefit of the recursive structure of the adaptive control system, where untargeted peaks could be directly handled by feeding the estimated adaptive control untargeted peaks be directly handled by feeding the estimated plant output back towhere the reference input of could the LMS algorithm. Therefore, the whole control d k system, plant output d k back to the reference input of the LMS algorithm. Therefore, the whole control

system identifies the frequency components other than ωca. In contrast, the conventional FX-LMS still recognizes only the ωca component of x(t). Overall, IM-LMS 2 uses recursive control techniques and exploits both the conventional LMS algorithm and IMC simultaneously. However, there is some trade-off in the tracking performance with the conventional FX-LMS. Appl. Sci. 2018, 8, 1007 13 of 16 The FM signal was investigated similarly, as shown in Equation (25a,b).

x f (t ) = A0 f cos(ωcf t + B0 f sin(ω of t + γ rand ) + θ f ) (25a) In addition, the IM-LMS 2 method shows considerable attenuation at around 50 Hz, which is the natural frequency of the target plant, whereas the FX-LMS case demonstrates similar amplitude (t )] frequency. ( B0 fcondition )π [δ (ω − comes (ω +the )] ωcf ) + δfrom ωcfbenefit F [ x fthis = A0 f J 0This levels around of the recursive structure of the ∞ 1 n adaptive control system, +where untargeted peaks could be directly handled the estimated A0 f (−1) J n ( B0 f )π [δ (ω − ωcf + nωof ) + δ (ωby + ωfeeding cf − nωof ) (25b) 2 reference plant output dˆk back to the input of the LMS algorithm. Therefore, the whole control n =1 system identifies the frequency other than nω ca . In contrast, the conventional FX-LMS + (−1) n δcomponents (ω − ωcf − nω of ) + ( −1) δ (ω + ωcf + nωof )] still recognizes only the ω ca component of x(t). Overall, IM-LMS 2 uses recursive control techniques Jn indicates Bessel function of order n, and the depths A0f,However, B0f, and θthere f are selected and exploits botha the conventional LMS algorithm andmodulation IMC simultaneously. is some as 1.0, 0.5, 0, respectively. The reference input and other parameters that were used are the trade-off in and the tracking performance with the conventional FX-LMS. sameThe as those for the AM case. FM signal was investigated similarly, as shown in Equation (25a,b). Figure 10 compares the tracking performance and estimation error of the conventional LMS ωcthe B0 f signal sin(ωo case ) + θ f ) for up to 0.45 s. In(25a) and IM-LMS 2 methods.xSimilar trends are the f (t) = A 0 f cos(to f t +AM ft+γ randobserved transient region, for up to around 0.05 s, the IM-LMS 2 method shows a much better tracking F [ x f (t)]but= a slower A0 f J0 ( Bconvergence δ(ωFX-LMS, + ωc f )] for around 0.4 s (0.05 ≤ t ≤ 0.45) in the 0 f ) π [ δ ( ω − ωthan c f ) +the performance ∞ n 1 steady-state region. However, point, with (−1)that Jn ( B (ω peak-to-peak − ωc f + nωo f )value + δ(ωof+the ωc festimation − nωo f ) error (25b) + 2 A0 f ∑ after 0 f ) π [ δthe n = 1 the IM-LMS 2 is about three ntimes better than that of FX-LMS. +(−1) δ(ω − ωc f − nωo f ) + (−1)n δ(ω + ωc f + nωo f )] Figure 11 compares the predicted spectral contents of the uncontrolled FM signal and Jn indicates a Bessel of order n, and the modulation depthsreduction A0f , B0f , and θ f arebeselected as controlled FM signal for function both of the methods. Although a broadband cannot achieved 1.0, 0.5, and 0, respectively. The reference input and other parameters that were used are the same as with either FX-LMS or IM-LMS 2, they show significant reductions at primary peaks (ωcf, ωcf − nωof, those forof). the AMamplitude case. ωcf + nω The reduction levels of IM-LMS 2 at primary peaks are higher than those of Figure 10 compares tracking performance and estimation of the frequency. conventional and FX-LMS, except for at ωcfthe , while FX-LMS shows a severe spillover error at a higher In LMS addition, IM-LMS 2 methods. Similar trends to the AM signal case areatobserved to 0.45 s. In the IM-LMS 2 method shows considerable attenuation aroundfor 50up Hz, which is the thetransient natural region, for of upthe to around 0.05 s, whereas the IM-LMS 2 method shows a much better trackinglevels performance but frequency target plant, FX-LMS demonstrates similar amplitude around this afrequency. slower convergence than the FX-LMS, for around 0.4 s (0.05 ≤ t ≤ 0.45) in the steady-state region. Thus, the tracking performance of IM-LMS 2 is improved when increasing the However, point, the peak-to-peak the IM-LMS 2 is about complexityafter andthat frequency components thatvalue wereoftothe be estimation controlled error in thewith output of the disturbance three than of FX-LMS. plant,times whichbetter should bethat tracked.



(a)

(b)

Figure 10. 10. Control Control of of aa frequency-modulated frequency-modulated (FM) (FM) signal. signal. (a) (a) Time-domain Time-domain tracking tracking with with LMS LMS and and Figure internal model LMS 2 methods; (b) estimation error. internal model LMS 2 methods; (b) estimation error.

Figure 11 compares the predicted spectral contents of the uncontrolled FM signal and controlled FM signal for both of the methods. Although a broadband reduction cannot be achieved with either FX-LMS or IM-LMS 2, they show significant reductions at primary peaks (ω cf , ω cf − nω of , ω cf + nω of ). The amplitude reduction levels of IM-LMS 2 at primary peaks are higher than those of FX-LMS, except for at ω cf , while FX-LMS shows a severe spillover at a higher frequency. In addition, the IM-LMS 2 method shows considerable attenuation at around 50 Hz, which is the natural frequency of the

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target plant, whereas FX-LMS demonstrates similar amplitude levels around this frequency. Thus, the tracking performance of IM-LMS 2 is improved when increasing the complexity and frequency components that were to REVIEW be controlled in the output of the disturbance plant, which should be tracked. Appl. Sci. 2018, 8, x FOR PEER 14 of 16

Figure 11. Predicted spectra for a frequency-modulated (FM) signal with least mean squares and Figure 11. Predicted spectra for a frequency-modulated (FM) signal with least mean squares and internal model least mean squares 2 methods. internal model least mean squares 2 methods.

5. Conclusions and Future Work 5. Conclusions and Future Work Two enhanced LMS methods were introduced for an adaptive digital filtering system combined Two enhanced LMS methods were introduced for an adaptive digital filtering system combined with IMC, in order to take advantage of having the process model inside the control system with IMC, in order to take advantage of having the process model inside the control system structure. structure. Major contributions of this research comprise the following. Firstly, the internal model LMS Major contributions of this research comprise the following. Firstly, the internal model LMS method 1 (IM-LMS 1) is introduced, which is a novel model-based algorithm. Internal mode control method 1 (IM-LMS 1) is introduced, which is a novel model-based algorithm. Internal mode control is applied in the traditional LMS algorithm, for assuring convergence of the control system without is applied in the traditional LMS algorithm, for assuring convergence of the control system without filtered inputs. It also showed acceptable tracking performance with a significantly mistuned plant filtered inputs. It also showed acceptable tracking performance with a significantly mistuned plant model. Secondly, the internal model LMS method 2 (IM-LMS 2) is proposed. A new recursive digital model. Secondly, the internal model LMS method 2 (IM-LMS 2) is proposed. A new recursive digital filter system is proposed with an internal model controller. This control system enhances robustness, filter system is proposed with an internal model controller. This control system enhances robustness, stability, and the ability to manage multi-spectral signals. Moreover, the tracking performance of stability, and the ability to manage multi-spectral signals. Moreover, the tracking performance of IM-LMS 2 is improved with increasing complexity and frequency components to be controlled in IM-LMS 2 is improved with increasing complexity and frequency components to be controlled in the the output of the disturbance plant, which should be tracked. Thirdly, with numerical analyses, output of the disturbance plant, which should be tracked. Thirdly, with numerical analyses, their their performances are validated. Proposed algorithms are numerically verified in several cases with performances are validated. Proposed algorithms are numerically verified in several cases with single sine waves and multi-spectral signals. Time domain tracking performances and estimation single sine waves and multi-spectral signals. Time domain tracking performances and estimation errors in time and frequency domains are compared and their characteristics are discussed. errors in time and frequency domains are compared and their characteristics are discussed. This research will be verified with large number of applications for active noise and vibration This research will be verified with large number of applications for active noise and vibration control, including vibration mitigation, vehicle noise, and vibration attenuation. This research could control, including vibration mitigation, vehicle noise, and vibration attenuation. This research could be used to improve the active control applications in vehicles and mechanical structures with effective be used to improve the active control applications in vehicles and mechanical structures with management of an expansive range of multi-spectral signals. effective management of an expansive range of multi-spectral signals. Author Contributions: B.K. and J.-Y.Y. initiated and developed the ideas related to this research work; B.K. and Authordeveloped Contributions: and J.-Y.Y. initiated and formulations, developed the and ideascarried relatedout to this research work; B.K. and J.-Y.Y. novelB.K. methods, derived relevant performance analyses numerical analyses;novel B.K. wrote the paper draft under formulations, J.-Y.Y.’s guidance; B.K.out finalized the paper. J.-Y.Y. developed methods, derived relevant and and carried performance analyses and numerical analyses; B.K. the paper draft by under J.-Y.Y.’s guidance; and B.K.Research finalized Grant the paper. Acknowledgments: This wrote work was supported the 2016 Yeungnam University (216A580005).

Acknowledgments: was supported by theof 2016 Yeungnam University Research Grant (216A580005). Conflicts of Interest:This The work authors declare no conflict interest. Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature D(s) D(s) F[ ] Jn L N(s) Pc(z)

time-varying delay denominator Fourier transform Bessel function of order n length of filter polynomials of numerator secondary path dynamics

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Nomenclature D(s) D(s) F[ ] Jn L N(s) Pc (z) T(s) Q a dk ek k p(s) q(s) r uk wk z− ∆ xa (t) xf (t) yk ys δ ξ λ µ τ ωn ˆ *

time-varying delay denominator Fourier transform Bessel function of order n length of filter polynomials of numerator secondary path dynamics arbitrary plant internal model controller proportional gain desired value measured error sampling instant process model internal model controller relative order reference input weight vector of filter time delay block amplitude-modulated frequency-modulated adaptive filter output set point Dirac delta function damping ratio filter time constant convergence factor time delay natural frequency parameters of FX LMS after secondary path

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

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