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Journal of Sound and Vibration

Journal of Sound and Vibration 331 (2012) 4101–4114

0022-46

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/jsvi

Control of modulated vibration using an enhanced adaptive filteringalgorithm based on model-based approach

Byeongil Kim, Gregory N. Washington, Rajendra Singh n

Smart Vehicle Concepts Center, Department of Mechanical Engineering, The Ohio State University, 201 West 19th Avenue, Columbus, OH 43210, USA

a r t i c l e i n f o

Article history:

Received 2 May 2011

Received in revised form

2 March 2012

Accepted 9 April 2012

Handling editor: J. Lammode control (MPSMC) in the adaptive filtering system. Several amplitude and

Available online 12 May 2012

0X/$ - see front matter & 2012 Elsevier Ltd.

x.doi.org/10.1016/j.jsv.2012.04.007

esponding author. Tel.: þ614 292 9044; fax:

ail address: [email protected] (R. Singh).

a b s t r a c t

Conventional adaptive filtering algorithms, typically limited to the control of single or

multiple sinusoids, are not appropriate to control modulated vibrations, especially in

the presence of rich side band structures. To overcome this deficiency, a new control

algorithm is proposed that introduces a feedback loop with the model predictive sliding

frequency modulation cases are first computationally studied, and conventional and

proposed methods are comparatively evaluated in terms of estimation error, perfor-

mance in time and frequency domains, stability, and uncertainty in the reference signal.

To experimentally validate the proposed algorithm, an active strut (with longitudinal

vibrations) is constructed. Overall, the proposed adaptive algorithm yields superior

reductions at the main frequencies and at side bands; also, good attenuation is found on

a broadband basis.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Side band structures are commonly observed in the measured vibro-acoustic spectra of many mechanical elements androtating systems such as gears, bearings, fans, and motors [1–5]. Though the modulated spectral contents are often utilizedto develop algorithms for fault detection and preventive maintenance and to trouble shoot practical noise and vibration (orsound quality) problems, several aspects of the problem are still not well understood. For instance, amplitude modulation(AM) or frequency modulation (FM) theories do not fully describe the measured signatures, though these concepts areuseful in understanding the existence of frequencies [6]. Also, attempts to actively suppress amplitude or frequencymodulated signals have been partially successful [7–9].

In particular, rotorcraft cabin noise continues to pose challenges as significant structure-borne noise from the gearboxis transmitted to the cabin [10–14]. The spectral content is usually rich in terms of gear mesh frequencies and their sidebands [1–2,5,10]. Fig. 1 illustrates a typical casing acceleration spectrum for a unity gear pair. Several researchers havesuggested active noise or vibration control methods for rotorcraft application. Millott et al. [11] proposed a system withinertial actuators and noise reduction at the first gear mesh frequency was only examined for several operating conditions.Gembler et al. [12], Maier et al. [13], and Hoffmann et al. [14] have developed improved active strut designs and found thatvibration magnitudes at the fundamental gear mesh frequencies were considerably reduced, but higher gear meshharmonics and side bands remained unchanged or even amplified after the control.

All rights reserved.

þ614 292 3163.

www.elsevier.com/locate/jsvi

www.elsevier.com/locate/jsvi

dx.doi.org/10.1016/j.jsv.2012.04.007

dx.doi.org/10.1016/j.jsv.2012.04.007

dx.doi.org/10.1016/j.jsv.2012.04.007

mailto:[email protected]

dx.doi.org/10.1016/j.jsv.2012.04.007

Fig. 1. Typical gearbox casing acceleration spectrum measured on a spur gear test facility. Here the reference for dB is 1.0 g rms. The gear mesh frequency

(fm) is 2800 Hz and 7 i100 Hz side bands are seen at 7 jfm where i¼1,2,y and j¼1,y,4.

Fig. 2. Adaptive filtering system with least mean squares (LMS) and filtered-X LMS (FX-LMS) (denoted by ) algorithms.

B. Kim et al. / Journal of Sound and Vibration 331 (2012) 4101–41144102

Several researchers have investigated the feasibility of controlling modulated vibration or noise signals. Guan et al. [7]proposed an active shaft vibration control system in a gearbox structure but considered only the first two gear meshfrequencies. Mucci and Singh [8] examined the feasibility of attenuating modulated motions of a simply supported beam.Asnani et al. [9] compared two control algorithms to suppress amplitude and frequency modulated sounds in a duct overthe plane wave regime. In such limited studies, side bands could only be reduced up to certain levels. Therefore, furtherinvestigation is necessary to develop a new or improved control method that can concurrently address modulatedvibrations on both narrow and broad band bases. This is the chief goal of this article; computational and experimentalstudies are conducted to validate the proposed algorithm.

2. Problem formulation

Adaptive digital filtering systems with recursive algorithms are typically utilized to actively control noise and vibration[15]. The least mean squares (LMS) method, as shown in Fig. 2, is the most popular feedforward algorithm because of itssimpler formulation [16]. Here, Pd(z), f[k], d[k], and y[k] describe the disturbance plant, filter output, disturbance orunwanted signal, and filtered output through the secondary path dynamics, respectively; integer k is the time step. Theoutput of the ‘‘adaptive filter’’ f[k] converges to the output of the ‘‘disturbance plant’’ d[k], which is an unwanted signal.From the least mean squares (LMS) algorithm theory, coefficients of the adaptive filter are adjusted at each samplinginstant based on the error between f[k] and d[k], and the algorithm tries to minimize that error. The filter coefficient vectorw½k�, the reference vector u[k], and the error signal e[k] are related as follows where p is the number of filter coefficientsand m is a parameter which regulates the stability and convergence rate.

w½kþ1� ¼ w½k�þmUu½k�Ue½k�,

u½k� ¼ u½k� u½k�1� � � � u½k�pþ1�h iT

: (1a2b)

Since it is virtually impossible to ignore the secondary path dynamics (denoted by Ps(z)) in a practical situation, the LMSalgorithm may destabilize the entire system, or it might just converge to a wrong solution [17]. The filtered-X least meansquares (FX-LMS) algorithm overcomes this deficiency by employing a filter (denoted by PsðzÞ in Fig. 2) and bycompensating for phase and time delays [17]. The modified formulation includes the filtered reference vector ~u½k� as

w½kþ1� ¼ w½k�þmU ~u½k�Ue½k�: (2)

Table 1Summary of simulation studies and typical vibration reduction achieved.

Case Signal type Insertion loss DL (dB) with model predictive sliding mode — least mean squares

Narrowband Broadband

S1 Amplitude modulated (AM) 30–60 22–25

S2 Frequency modulated (FM) 23–27 15–20

S3 Product of AM and FM 15–25 10–15

S4 Gaussian noise not applicable 30

Table 2Summary of experimental studies and typical vibration reductions.

Case Type of modulation Frequency range (Hz) Insertion loss DL (dB) with model predictive sliding mode — least mean squares

Narrowband Broadband

E1a Amplitude modulated (AM) 1–100 6–17 7–9

E1b 1–100 (Narrow gap) 10–12 6–8

E1c 500–1500 3–15 3–6

E2a Frequency modulated (FM) 1–100 3–18

E2b 1–100 (Narrow gap) 5–10 5–7

E2c 500–1500 2–10 2–7

E3 Product of AM and FM 1–100 2–8 1

B. Kim et al. / Journal of Sound and Vibration 331 (2012) 4101–4114 4103

There are (at least) two ways to control multi-spectral contents with the conventional LMS algorithm. First, utilize a multi-channel LMS scheme containing the same number of LMS blocks in parallel as the number of spectral contents to betreated [18]. Second, employ a very large number of filter coefficients that could improve tracking [19]. Nevertheless, suchsolutions (with significant computational burden) may not adequately manage many complex signals in practicalapplications, including modulated disturbances. Mucci and Singh [8] employed the FX-LMS to minimize the beamvibration at a location where an accelerometer was placed to sense errors. However, some spectral components orunintended modulations (as caused by the experimental system nonlinearities) could not be controlled. Most significantattenuation was observed at the carrier frequency, while the residual error at side bands was the same or greater inmagnitude than at the carrier frequency. Asnani et al. [9] overcame some of the deficiencies by utilizing the narrowbandadaptive noise equalizer (ANE) [20–22] in the cancellation mode; significant success was found as the ANE couldcompletely cancel side bands as well as carrier frequencies. Further, the ANE permitted spectral shaping of the errorspectrum over individual frequency components and thus some residual noise could be retained, though dominant sidebands were drastically reduced. However, the ANE needs a priori knowledge of the carrier frequency and side bands.

Specific objectives of this article include the following: (1) develop a new control algorithm that combines model-basedand nonlinear control methods to enhance the conventional LMS algorithm while retaining its inherent advantages; (2)comparatively evaluate the proposed and conventional algorithms for AM and FM signals (as well as a product of both) andGaussian noise as listed in Table 1; (3) construct an active strut and conduct experimental studies using both algorithms,and measure vibration residuals for several modulated vibration signals as listed in Table 2. The stability issues and effectsof mistuned reference signals or disturbance model will be also discussed.

3. Development of new control algorithm

3.1. Conceptual considerations

The proposed algorithm is illustrated in Fig. 3. A horizontal line (dotted) distinguishes the dynamic system from thecontrol system. The dynamic system shows the disturbance plant Pd(z), an input u[k], and the plant output d[k], which is anunwanted signal. The remaining part indicates the overall control system. It integrates the LMS algorithm with the model-based and nonlinear control schemes. Here, n[k] is the sensor noise, q[k] is the unexpected perturbation, and uncertainty insystem models or signals is quantified by time-varying Ps(z). The primary difference from the conventional adaptivefiltering is the location of the feedback loop. The LMS algorithm still tries to make the output of the adaptive filter convergeto the output of the disturbance plant, except that the error is once again controlled by the feedback loop for bettertracking. The model predictive sliding mode control (MPSMC) method can be designed based on Pd(z) after it has beenidentified, although a very rough model would be obtained. The control input c[k] from the feedback loop, as derived inSection 3.2, is utilized to create a modified filter output m[k]; it is defined as follows:

c½k� ¼ f ½k��m½k�: (3)

Fig. 3. Proposed model predictive sliding mode — least mean squares (MPSM-LMS) control algorithm.


Then, m[k] is employed as an input to Ps(z). Note that a filtered reference ~u½k�, as observed in Fig. 2, is not utilized sincea delay originating from Ps(z) could be managed by the feedback loop without ~u½k�. Here, the tracking error e[k] is definedas follows.

e½k� ¼ d½k��y½k�: (4)

The potential advantages of adding a feedback loop are discussed below.

Tracking performance: The transfer function Gef(z) between e[k] and f[k] determines the tracking performance.

Gef ðzÞ ¼EðzÞ

FðzÞ¼

PsðzÞ

1þPsðzÞPlðzÞ: (5)

When 9Gef(z)9-0, we expect a good tracking performance over a wide frequency range. Thus, any misadjustmentinduced by the LMS could be treated by adjusting the controller Pl(z) in the feedback loop.Rejection of perturbation: When the adaptive control system is subjected to a perturbation q[k], the feedback loop couldmanage this by minimizing the sensitivity of e[k] to q[k]. The error transfer functions EQ(z)/Q(z) and EF(z)/F(z) can becalculated, assuming that other inputs are zero.

EQ ðzÞ

Q ðzÞ¼

1

1þPsðzÞPlðzÞ, (6)

EF ðzÞ

FðzÞ¼

PsðzÞ


Assume 9PsðzÞPlðzÞ9c1, the effect of the perturbation could be suppressed since 9EQ ðzÞ=Q ðzÞ9{1. Conversely, 9EF(z)/F(z)9-91/Pl(z)9 and thus the performance should not be affected by a variation in Ps(z).Low sensitivity to time-varying secondary path dynamics: Define the estimated transfer function of the actual secondarypath dynamics Ps(z) as PsðzÞ, and the estimated error is

DPsðzÞ ¼ PsðzÞ�PsðzÞ: (8)

A change in Gef(z) is derived from Eqs. (5) to (8).

DGef ðzÞ ¼PsðzÞþDPsðzÞ

1þfPsðzÞþDPsðzÞgPlðzÞ�

PsðzÞ


From Eqs. (5) and (9), the sensitivity F(z) is defined as follows:

DGef ðzÞ

Gef ðzÞ¼

DPsðzÞ

PsðzÞFðzÞ, (10a)

FðzÞ ¼1

1þPlðzÞPsðzÞ: (10b)

When 9PlðzÞPsðzÞ9c1, 9F zð Þ9{1 and thus the effect of DPs(z) on DGef(z) is reduced comparing with the feedforward onlysystem. Thus, a more precise control could be achieved by using the feedback system.Low sensitivity to sensor noise: The sensitivity to sensor noise is related to the transfer function Gen(z) that is defined as:

GenðzÞ ¼ENðzÞ

NðzÞ¼�

PsðzÞPlðzÞ



When 9Gen(z)9 is minimized with the help of Pl(z), the effect of the sensor noise on the system performance shoulddiminish.

The proposed conceptual considerations employ successive error reduction methods. The LMS algorithm and thefeedback controller are sequentially applied at each time instant [k]. Both could be essentially designed independentlywith each other. Finally, the use of a feedback control system should theoretically reduce sensitivity to externalperturbations, sensor noise, and variations in system parameters.

3.2. Model predictive sliding mode control (MPSMC)

The sliding mode control (SMC) belongs to a class of nonlinear control techniques [23]. Discontinuous control inputs drivethe target system states between two sides of a discontinuous sliding surface and eventually reach the sliding mode, whichdefines the expected system characteristics. Since this type of controller is often more robust and insensitive to original plantdynamics and other unexpected situations, it is preferred for higher order complex or nonlinear systems [24].

The design process of the SMC has two steps. First, a sliding surface s is defined as s¼Gx, where x is a state vector and Gis a row vector which determines the dynamics of a sliding mode. Next, a discontinuous control input is designed to satisfythe conditions for the existence and accessibility of the sliding mode. A typical form of the control input for a linear systemis us¼�M sign(s) where M is a positive gain sufficiently large enough to guarantee s_so0 so that the state trajectory willreach the sliding surface. A conceptual diagram for the initiation of a sliding mode is shown in Fig. 4 (left) where thesystem trajectory in the state space defined by (x, _x), reaches the sliding surface defined as s¼0 and moves towards theorigin along the surface.

Nevertheless, a true sliding mode cannot be achieved since the discontinuous actions may induce chattering or actuatormalfunctioning. Moreover, the controller is likely to be saturated in the case of discrete sliding mode control since thesystem states are forced to reach the sliding surface within one sampling instant [23]. In order to overcome such problems,the model predictive control (MPC) could be introduced to approach the sliding surface in an optimal manner. The modelpredictive control has superior reference tracking capability due to predicted inputs and outputs up to a certain timeinstant (the receding horizon) by a pre-defined process model [25]. Consequently, the sliding mode control is combinedwith the model predictive control to establish a model-based nonlinear control method; it has been designated as themodel predictive sliding mode control (MPSMC) by Neelakantan [26]. The MPSMC enforces the system states to approachthe sliding mode in a more continuous manner unlike with the discontinuous approach followed by the SMC, thuspreventing the chattering phenomenon. Fig. 4 illustrates the convergence characteristics of both methods. Note that thestate trajectory with MPSMC is approaching (but not exactly reaching) the sliding surface asymptotically, since an idealsliding mode cannot be achieved with the modified continuous input.

Dynamic system equations are defined next where x[k], us[k], and h[k] indicate the state, input, and time-varyingdisturbance vectors, respectively.

x½kþ1� ¼Ax½k�þBus½k�þDh½k�: (12)

Here, A is the state matrix, B is the input matrix, and D is the disturbance matrix, respectively. An error vector e[k] isdefined below where d[k] combines the cumulative effects of nonlinearities, disturbances, model uncertainties, andreference inputs.

e½k� ¼ x½k��r½k�, (13)

e½kþ1� ¼ Ae½k�þBus½k�þd½k�, (14)

Fig. 4. Convergence profiles of two control techniques. Key: , sliding mode control (SMC); , model predictive sliding mode control

(MPSMC).


d½k� ¼Ar½k�þDh½k��r½kþ1�: (15)

Next, the augmented state equations are defined by combining the above formulation with an input update. Define a newstate vector z[k] that consists of e[k] and us[k�1].

z½k� ¼e½k�

us½k�1�

" #, (16)

e½kþ1�

us½k�

" #¼

A B

0 I

� �e½k�

us½k�1�

" #þ

B

I

� �Dus½k�þ

I

0

� �d½k�, (17)

z½kþ1� ¼Cz½k�þUDus½k�þHd½k�, (18a)

C¼A B

0 I

� �, U¼

B

I

� �, H¼

I

0

� �: (18b2d)

In the new state equation, Dus[k]¼us[k]�us[k�1] is now used as the control input. Next, a sliding surface s[k] isdefined in the augmented state plane where P determines the system characteristics in the sliding mode as

s½k� ¼ Pz½k�: (19)

The following relationship is obtained by combining Eqs. (18a) and (19) at the [kþ1] time step.

s½kþ1� ¼ PðCz½k�þUDus½k�þHd½k�Þ: (20)

Combining Eq. (20) from [kþ1] to the [kþN] time step, up to as many steps as the receding horizon N, the future slidingmode s(kþ1:kþN) is predicted as follows, where Dus(k:kþN�1), and d(k:kþN�1) indicate the future input vector anddisturbance vector, respectively.

sðkþ1:kþNÞ ¼ Cz½k�þUDusðk:kþN�1Þ þHdðk:kþN�1Þ, (21a)

sðkþ1:kþNÞ ¼ s½kþ1� � � � s½kþN�h iT

, Dusðk:kþN�1Þ ¼ Dus½k� � � � Dus½kþN�1�h iT

,

dðk:kþN�1Þ ¼ d½k� � � � d½kþN�1�h iT

: (21b)

The C, U, and H matrices are defined as follows.

C¼ P

CC2

^

CN

26664

37775, (21c)

U¼ P

U 0 � � � 0

CU U � � � 0

^ ^ & 0

CN�1U CN�2U � � � U

26664

37775, (21d)

H¼ P

H 0 � � � 0

CH H � � � 0

^ ^ & 0

CN�1H CN�2H � � � H

26664

37775: (21e)

The scalar cost function J is defined next, and it is minimized to obtain an optimized Dus where l is a weighting factor forthe square norm of Dus(:).

minDusð:Þ

J¼ :sð:Þ:2þl:Dusð:Þ:

2: (22)

Next, derive Dus[k] where e[1] is a vector used to select the initial value of Dus(:) so that it can be utilized at the very nextstep.

Dus½k� ¼ �e½1�TðUTUþlIÞ�1U

T½Cz½k�þHdð:Þ�, (23a)

e½1�T ¼ 1 0 0 � � � 0� �T

: (23b)

Observe that all terms in Eq. (23a) are known except d(:) which predicts the disturbance vector. Since the exactdynamics of a real-life disturbance may not be previously known, a simplified assumption is made to overcome thissituation. What is available at present [k] is d at [k�1], which could be derived by organizing Eq. (14) with d[k�1] on the


left. This is repeated N times to estimate the future disturbance vector as:

d½k�1� ¼ e½k��Ae½k�1��Bus½k�1�, (24)

destð:Þ ¼ d½k�1� d½k�1� � � � d½k�1�

h iT: (25)

4. Comparative evaluation of two algorithms

A finite impulse response (FIR) digital filter with two coefficients is utilized for the simulation examples of Table 1. Thedisturbance plant is assumed to be given by a linear second order system with natural frequency on/2p¼800 Hz anddamping ratio x¼0.1. The model predictive sliding mode LMS algorithm is evaluated for the following multi-spectralsignals: amplitude modulated (AM) signal, frequency modulated (FM) signal, a product of AM and FM signals, andGaussian noise. The AM case (S1) is considered first where the signal xa(t) is defined as follows.

xaðtÞ ¼ A0að1þgrandÞcosðocatþycÞU½1þB0a cosðooatþyoÞ�, (26a)

F xaðtÞ½ � ¼ A0ap dðo�ocaÞþdðoþocaÞ� �

þ1

2A0aB0ap dðo�ocaþooaÞþdðoþoca�ooaÞþdðo�oca�ooaÞþdðoþocaþooaÞ

� �:

(26b)

Here, F[ ] represents the Fourier transform, and d is the Dirac delta function located at frequency o(rad/s). In Eq. (26a), xa(t)consists of a sinusoid at ooa/2p¼120 Hz which is modulated with a carrier signal at oca/2p¼1500 Hz where B0a¼0.5 isthe modulation depth. The amplitude is contaminated with grand (with variance¼0.1 and mean¼0). For the sake ofconvenience, A0a is normalized to be 1.0, and the phase angles yc and yo are ignored. A sinusoid at oca is employed as areference input. The LMS parameter m in Eq. (1) is selected as 0.5 and the sliding mode P, l, and N are determined asfollows: P¼ 500 1 0

� �, l¼1, and N¼5. The signal is also controlled with the conventional multi-channel LMS for the

sake of comparison; individual sinusoids at 1500 Hz and 7120 Hz are employed as reference inputs for LMS channels.Several parameters associated with both algorithms may affect the performance in the frequency domain. Since the targetof this paper is not to optimize or to select the best value of these parameters, consistent parameters are used (pre-tunedfor the proper performance such as non-divergent, without any spillover and illustrating reduced amplitudes after control)in all example cases to compare the performance of both methods. Fig. 5(a) compares the unwanted signal and the filteroutput in time domain for both algorithms. When the unwanted signal changes its direction, an extreme error is observedin Fig. 5(b) with the multi-channel LMS, but not with the MPSM-LMS. The peak to peak error for the MPSM-LMS is about10% of the one for the multi-channel LMS.

Fig. 6(a) compares the spectral contents. Although some reduction at the primary peaks is achieved with the multi-channel LMS, spillover is observed in the higher frequency region. Conversely, the controlled signal with the MPSM-LMSyields a higher reduction at all peaks, as well as on a broadband basis. Observe that the multi-channel LMS does not showmuch reduction at higher frequencies as this controller is forced to deal with the lower frequency region. (Note that this is

Fig. 5. Amplitude modulated (case S1) signals in time domain: (a) predicted with multi-channel LMS and model predictive sliding mode — least mean

squares (MPSM-LMS) algorithms. Key: , disturbance; , filter output; (b) predicted estimation error. Key: , with multi-channel LMS;

, with MPSM-LMS.

Fig. 6. Amplitude modulated (case S1) signals in frequency domain: (a) predicted spectra. Key: , disturbance; , controlled signal with

multi-channel LMS; , controlled signal with model predictive sliding mode – least mean squares (MPSM-LMS); (b) attenuation spectra (DL).

Key: , with multi-channel LMS; , with MPSM-LMS.


not a property of the LMS algorithm, and it can be easily checked via numerical examples based on the Fourier expansion.)The enhanced controller manages such problems by reducing the residuals of controlled signal at high frequencies and bydiminishing the spillover. Although the new algorithm utilizes only one sinusoid as a reference signal, it is still capable ofaddressing multiple frequency components. Also, the MPSM-LMS shows similar vibration reduction ability when B0a isvaried. The insertion loss DL¼Lu�Lw (in dB) between the uncontrolled (Lu) and the controlled (Lw) vibration magnitudes isexamined next. Fig. 6(b) compares DL values for both algorithms. Even when the variance of grand is varied, the same DL

value is obtained which suggests that the controller is unaffected by the randomness of disturbance. Note that the multi-channel LMS controller shows a significant reduction at 1380 Hz, even though significant spillover (with negative DL

values) occurs in several frequency regimes.The frequency modulated (FM) case (S2) is examined next where the phase of xf(t) is now contaminated with grand.

xf ðtÞ ¼ A0f cosðocf tþB0f sinðoof tþgrandÞþyf Þ, (27a)

F xf ðtÞ� �

¼ A0f J0ðB0f Þp dðo�ocf Þþdðoþocf Þ� �

þ1

2A0f

X1n ¼ 1

ð�1ÞnJnðB0f Þp dðo�ocfþnoof Þþdðoþocf�noof Þ�

þð�1Þndðo�ocf�noof Þþð�1Þndðoþocfþnoof Þ�: (27b)

Here, Jn is the Bessel function (first kind) of order n, and A0f, B0f (modulation depth) and yf are chosen to be 1.0, 0.5, and0. Other parameters and reference input are the same as utilized for the AM case. Similar trends are observed for bothmethods although now xf(t) yields a more complicated spectrum, as shown in Fig. 7(a). Fig. 7(b) compares the DL spectraand again they seem to be unaffected by the disturbance intensity. When B0f is increased, more side bands are observedover the same frequency bandwidth. Nevertheless, the MPSM-LMS yields similar reduction on both narrow and broadbandwidths.

Now consider a product of AM and the FM signals (case S3) as it may simulate some practical cases [6]. The compositesignal xp(t) is defined as

xpðtÞ ¼ xaðtÞUxf ðtÞ ¼ A0pð1þgrandÞfsinðocatþycÞU½1þB0a sinðooatþyoÞ�gUfsinðoofU½1þB0f sinðocf tÞ�Utþyf Þg, (28a)

F½xpðtÞ� ¼ F½xaðtÞ�nF½xf ðtÞ�: (28b)

Essentially, a sinusoid (at ooa/2p¼40 Hz) is modulated with a carrier (at oca/2p¼550 Hz) to define the AM signal and asinusoid (at oof/2p¼30 Hz) is modulated with a carrier (at ocf/2p¼350 Hz) to construct the FM signal. Again, the phases(yc, yo and yf) are ignored, and A0p, B0a, and B0f are normalized to be 1.0, 0.5, and 0.5, respectively. Fig. 8 compares bothuncontrolled and controlled signals. Since the spectral contents of xp(t) are represented by a convolution of two individualspectra, as given by Eq. (28b), a relatively complicated spectrum is observed. Observe that a reduction of about 10–20 dB isachieved at all peaks, including the fundamental frequencies and their side bands, although lower attenuation in xp(t) isseen than individual xa(t) or xf(t) cases.

The Gaussian noise disturbance with a mean of 0 and a variance from 0.1 to 1.0 is selected next (case S4). Comparison ofspectra in Fig. 9(a) shows that the MPSM-LMS is indeed superior. Fig. 9(b) shows the DL spectrum, and it remains virtuallyunchanged with an increase in the variance. The DL value over the frequency range of interest is about 30 dB with theMPSM-LMS, while the multi-channel LMS controller yields only a reduction of 4.5 dB.

Fig. 7. Frequency modulated (case S2) signals in frequency domain: (a) predicted spectra; (b) attenuation spectra (DL). Key as in Fig. 6.

Fig. 8. Predicted spectra for AM and FM product (case S3) signals in frequency domain. Key as in Fig. 6.


The filtered-X LMS of Fig. 2 requires a system identification procedure for filter PcðzÞwhich complicates the filter systemdesign. For the sake of argument, assume that PcðzÞ is not utilized when Pc(z) is present (case S5a). Now apply the MPSM-LMS and describe Pc(z) by a two step time delay (z�2) element. While the LMS induces an instability without the PcðzÞ term,the MPSM-LMS does not seem to require any identification process, as shown in Fig. 10(a). Next, assume that one of thereference signals (say at 1000 Hz) is somehow mistuned as a sinusoid at 1500 Hz (case S5b). The results of Fig. 10(b) showa large discrepancy for the LMS; conversely, the MPSM-LMS tracks it relatively well. Finally, an incorrect system model(case S5c) is assumed by arbitrarily selecting the following: on/2p¼300 Hz (instead of 800 Hz) and z¼0.3 (instead of 0.1).It is clear from Fig. 10(c) that the MPSM-LMS works well with an incorrect model, although the error increases slightly.Overall, the proposed controller appears to be self-adaptive in the presence of uncertainties and spurious noise.

5. Experimental validation

An aluminum rod of length (L) 355 mm and diameter 16 mm is selected as a passive strut. The rod is installed verticallywhere x¼L is a free boundary. An electro-mechanical shaker excites the rod longitudinally at x¼0 as shown in Fig. 11. Apiezoelectric actuator of thickness 10.2 mm and diameter 50.8 mm (PCB 710M03) is then inserted as an active element byattaching it at x¼0.5L with mounting threads; this induces longitudinal forces along the strut. The frequency range of thisactuator is from 150 Hz to 5000 Hz and its sensitivity is 0.067 N/V with the maximum input voltage of 100 V. Onepiezoelectric accelerometer (PCB 352C68) is installed at xE0.5L as the control sensor, and the other accelerometer (PCBA353B66) is installed at x¼L for computing the metric. Amplifiers are required to drive both the piezoelectric actuator and

Fig. 9. Gaussian noise (case S4) signals in frequency domain: (a) predicted spectra; (b) attenuation spectra (DL). Key as in Fig. 6.

Fig. 10. Case S5 signals in time domain: (a) no filtered reference; (b) mistuned reference; (c) mistuned model. Key as in Fig. 5(a).


electrodynamic shaker. The dSPACE system is employed for real-time data acquisition, controller design, and implementa-tion; refer to Fig. 11 for the schematic. Output gain is adjusted in each case in order to achieve as much reduction invibration amplitude as the system would allow.

Fig. 11. Active strut experiment: (a) a single smart strut installed vertically and excited by an electromechanical shaker; (b) schematic of the proposed

control system. Key: DAC¼Digital-to-Analog Converter; ADC¼Analog-to-Digital Converter; , signal flow; , force flow.

Fig. 12. Comparison of measured accelerometer spectra for amplitude modulated signals (case E1c). Key: , disturbance; , controlled signal

with model predictive sliding mode — least mean squares (MPSM-LMS) algorithm.


Since the MPSM-LMS is a model-based algorithm, a state-space formulation is required which is very difficult toimplement since the rod must be described by a partial differential equation or a high order discretized system. Therefore,a simplified second order system model is assumed to illustrate and implement the proposed algorithm. An impulsehammer experiment is first conducted to measure the longitudinal impulse response or transfer function of the passive rod(without the shaker). Only the first longitudinal mode of the rod is of interest, though many modes are excited by theimpulse test. The damping ratio at first mode is obtained by applying the half power method [27]. An FIR digital filter withtwo coefficients is employed, and the model predictive sliding mode control (MPSMC) is designed in the feedback loopwhere on of the second order system is set equal to the first longitudinal natural frequency.

Consider the amplitude modulated signal (case E1a) where a sinusoid at ooa/2p¼20 Hz is modulated with oca/2p¼50 Hz, as described in Eq. (26a). Again, A0a and B0a are set to be 1.0 and 0.5, yc and yo are ignored, and B0a is assumedto be 0.5. Assume that only one frequency at 50 Hz is targeted and therefore employed as a reference signal. The primaryfrequency, its side bands, and the broadband basis are attenuated by 6–17 dB. The proposed control system concurrentlyattenuates many other spectral contents, though they are unknown. In addition, the modulation depth B0a is variedbetween 0.1 and 1.0 and the controller still shows similar DL characteristics. Note that the highest frequency peak is notreduced as much as the other two peaks. This could be due to the dominance of lower frequency contents in xa(t). Also,there may be a limit on the actuator force or control gain. More reduction is possible with a higher gain, though thecontroller can cause spillover on the broadband basis. Next, a sinusoid at ooa/2p¼2 Hz is modulated with the same carrier


signal in order to narrow the gaps between peaks (case E1b). Observe that the DL value at the first peak is reduced but DL

at the third peak is increased. Finally, the smart strut is excited in the higher frequency region with a sinusoid atooa/2p¼200 Hz modulated with a carrier at oca/2p¼1000 Hz (case E1c). The DL value at the primary frequency (1000 Hz)is now 15 dB, but the side bands (at 800 Hz and 1200 Hz) are attenuated by only 3–4 dB; the broadband reduction is now3–6 dB, as shown in Fig. 12.

The frequency modulated signal at oof/2p¼20 Hz with a carrier at ocf/2p¼50 Hz is considered next with A0f¼1.0, asdescribed in Eq. (27a) for case E2a. Again, only one frequency at 50 Hz is employed as a reference signal. Vibrationreductions from 3 dB to 18 dB are seen at the fundamental peaks, side bands, and on a broadband basis. Moreover, achange in B0f or variance of grand does not affect the DL levels. Next, a lower frequency (at oof/2p¼2 Hz) is utilized toconstruct the FM signal, where peaks are distributed over a very narrow frequency bandwidth (case E2b). Dense peaksaround 50 Hz are reduced by 5–10 dB, and the broadband vibration is attenuated by 4.8–7.4 dB. More attenuation with ahigher output gain is possible although it causes significant spillover in other regions. Finally, a sinusoid at oof/2p¼200 Hzis modulated with ocf/2p¼1000 Hz (case E3c). Fig. 13 shows that the controller can simultaneously achieve reductionsover narrow and broadband bases.

Case E3 utilizes a product of AM and FM signals as defined by Eq. (28a). The AM signal is a sinusoid at ooa/2p¼15 Hzmodulated with oca/2p¼60 Hz, and the FM signal is a sinusoid at oof/2p¼20 Hz with ocf/2p¼50 Hz. It is observed inFig. 14 that 2–8 dB reductions are achieved up to 70 Hz but none in the higher frequency region, possibly due to spillover

Fig. 13. Comparison of measured accelerometer spectra for frequency modulated signals (case E2c). Key as in Fig. 12.

Fig. 14. Comparison of measured accelerometer spectra for AM and FM product signals (case E3). Key as in Fig. 12.


with an inappropriately high output gain. Table 2 compares DL values for all experimental results. Narrowband valuesindicate DL levels at the fundamental frequencies and their side bands.

The simulation trends qualitatively match well with experimental results, though they differ quantitatively. Somepossible reasons for a quantitative disagreement are as follows: (1) the active strut structure is modeled only as a secondorder system with the first longitudinal mode, though it has many elastic modes. Further, this controller could be affectedby the flexural motion of the rod; (2) vibration modes (in the experiment) are also affected by the electrodynamic shakerand connection between the struts and actuator; (3) uncontrolled signals from the structure are contaminated with highfrequency noise from actuator, amplifier and sensor, and thus limiting the gains. A low pass filter helps this situation but itcomplicates the control system design.

6. Conclusion

This article has proposed a novel adaptive algorithm with the model predictive sliding mode controller (MPSMC) in thefeedback loop which compensates for the error from the original LMS algorithm (essentially a feedforward controlmethod). This concept is not just a simple addition of two methods; rather a novel structure of the adaptive filteringsystem is proposed. This new structure overcomes limitations of the conventional adaptive filtering system and itcombines the strengths of nonlinear control theory (manage uncertainties, disturbances, and nonlinearities), model-basedcontrol methods (enable effective multi-spectral control with a known or assumed system model), and feedback loop. Anactive strut experiment is constructed to show feasibility of the proposed method with some complicated signals that arefound in the components of automobiles, aircrafts, and other engineering structures. Unlike previous methods[7–9,11–14], it is possible to simultaneously attenuate the main frequencies, their side bands, and even the broadbandlevels even when the disturbance spectrum is partially known. Only a rough system model is needed and the controllerseems to be adaptable even if mistuned reference signals are employed. Finally, there is no need for a filter after thereference signal to control its stability in the presence of the secondary path dynamics, unlike the FX-LMS method [17].Nevertheless, three potential limitations of the proposed controller are as follows. First, the calculation required for thefeedback loop with the model predictive sliding mode control (MPSMC) is an additional burden since the proposedalgorithm employs two controllers in parallel. Although matrices C, H, and U are created from the optimization process(an off-line process), matrix and vector manipulations involved in the control algorithm(s) simply impose a calculationburden. Second, a number of pre-actions are required. For instance, matrices C, H, and U should be calculated off-linebased on the system model; the plant model (even though it is very simple) must be pre-identified for designing thecontroller. The output gain for the piezo actuator control input should be manually adjusted, and this could result inspillover or an under-controlled system when the gain is not properly adjusted. Thus, it may be difficult to quickly applythe algorithm to a new problem. Finally, the multiple ordered systems (with higher number of states) are somewhatdifficult to implement for a real-life problem because all states may not be available. Thus far only a single-input andsingle-output control system method has been applied to examine the longitudinal motions of a rod (while modeling onlyone elastic mode). Future work should therefore focus on multi-mode problems that would employ a multiple-input andmultiple-output control system. In addition, the proposed algorithm should be extended to 2D and 3D structures as well asto practical gearboxes [11–14] and other rotating devices.

Acknowledgments

We are grateful to the member organizations of the Smart Vehicle Concepts Center (www.SmartVehicleCenter.org) andthe National Science Foundation Industry/University Cooperative Research Centers program (www.nsf.gov/eng/iip/iucrc)for supporting this work.

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