applied sciences Article
Enhanced Adaptive Filtering Algorithm Based on Sliding Mode Control for Active Vibration Rejection of Smart Beam Structures Byeongil Kim 1 and Jong-yun Yoon 2, * 1 2
*
School of Mechanical Engineering, Yeungnam University, 280, (Dae-dong) Daehak-ro, Gyeongsan-si, Gyeongsangbuk-do 38541, 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
Academic Editor: César M. A. Vasques Received: 3 July 2017; Accepted: 19 July 2017; Published: 24 July 2017
Abstract: This article investigates vibration rejection for a continuous smart structure using piezoelectric bimorph patches. Such a structure has inherent nonlinearities, such as hysteresis and creep, and the whole system may experience unexpected disturbances, uncertainties, and noise from external sources. Thus, it is very important to design the active control scheme carefully with adaptive filtering systems to deal with these conditions. An advanced adaptive filtering algorithm was developed based on the conventional least mean squares (LMS) method and sliding mode control for the active vibration rejection system. The sliding mode controller is applied to the standard LMS algorithm to overcome problems with misadjustment and excess error in an optimal manner. A numerical analysis and laboratory experiment show that the technique can significantly attenuate the vibration of the smart structure at different levels and broadband frequency spectra. In addition, unidentified impedance is chosen to change the distribution of the mass, and the robustness and the adaptivity of the proposed approach are verified. The experimental results show that the method can isolate impulse-type vibrations of at least 2.8 dB, even with the adjusted mass arrangement. Keywords: active vibration rejection; adaptive filtering system; filtered-X LMS; sliding mode control; least mean squares algorithm; smart structures
1. Introduction The primary structural components that experience external forces are usually modeled as continuous cantilevered beams in engineering structures such as vehicles, aircrafts, buildings, and bridges. Such structures commonly have dynamic behavior, which cannot be modeled perfectly and occur from unexpected external sources. These influences could induce undesirable vibrations, resulting in fatigue and likely severe failure. Smart structures with active control methods are one possible solution for isolating these detrimental vibrations, but the control system itself and mistuned control parameters can cause instabilities. Therefore, it is important to design control methods that are inherently insensitive to parameter changes to attenuate structural vibrations. Lead zirconate titanate (PZT) patches have a number of real-world applications for active control for reducing noise, vibration, and harshness (NVH). Some automotive parts have thin walls and can transmit unexpected noise and vibration from many sources. Thus, there have been many efforts to apply PZT patches to automotive parts such as the body panels [1], muffler system [2], and powertrain oil pan [3]. PZT patches are also used for mechanical and aerodynamic NVH reduction applications, such as aircraft fin-tips [4], rotorcraft blades [5], and aeroelastic flutter [6]. PZT patches are also used in active fault detections and energy harvesting devices. Appl. Sci. 2017, 7, 750; doi:10.3390/app7070750
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Piezoelectric shunt circuits can control vibrations from the actuator arms of hard disk drives [7], and they can be used for detecting or fixing cracked beams [8,9]. A hybrid-type switching control method has been used in order to isolate unwanted vibration as well as to improve the energy efficiency [10]. Active vibration isolation techniques have been implemented with piezoelectric actuators and sensors for the functionally graded material beams [11]. A smart structure with shunted piezoelectric patches was constructed, and topology optimization was performed for maximizing the electromechanical coupling factor in terms of the damping efficiency, piezo patch geometries, and patch locations [12]. A visualization technique was proposed to identify and reduce the vibrational energy of a 2D plate using a constrained damping layer [13]. To achieve satisfactory vibration attenuation, the structures and vibrational responses should be analyzed. An analytical solution was introduced for a piezoelectric bimorph cantilever beam with suitable boundary conditions [14]. The Airy stress function can also be used to study other different piezoelectric cantilevers, including multilayer piezoelectric cantilevers under different loads. The active vibration control of parametrically excited systems with resonance was examined to avoid the parametric instability, and an active control system was introduced, which includes a beam subjected to an axial harmonic load [15]. An enhanced method was proposed to stabilize the system via velocity feedback and pole placement by relocating the transition curves. Finally, generally layered smart beams were observed with a one-dimensional finite element method [16]. This state space model is based on first-order shear deformation beam theory with Hermite shape functions and the effect of external electric loads. Appropriate control algorithms must be designed for individual systems to achieve desirable performance. Piezoelectric shunt circuits have been applied extensively for the semi-active control of vibrating structures, but a single piezoelectric shunt circuit can deal with only one vibration mode at a time [17]. Many methods have been explored to manage more than one mode, such as multiple piezoelectric shunt circuits [18], synchronized switch voltage damping [19], and periodic piezoelectric shunts [20]. However, these techniques are still passive methods, for which adjustment is needed in advance. Thus, a number of active control schemes have been examined. These schemes include positive position feedback [21,22], acceleration feedback [23,24], full-state feedback [25], dynamic hysteresis compensators [26], pole placement control [27], and a hybrid of variable structure control and Lyapunov-based control [28]. Decoupled models were proposed to manage many degrees of freedom (DOF), and a backstepping method was introduced to design a robust controller with sliding mode and adaptive strategies [29]. To avoid the high computational burden of full-order controllers, a direct minimization method was introduced to design reduced-order H-infinity controllers and genetic algorithms [30]. The conventional approaches for active vibration control are often too complicated and thus difficult to implement. Furthermore, the control targets have to be determined, which limits the tracking, convergence, and stability performance. If a correct mathematical model of flexible structures can be derived, the structures can be regulated actively to isolate the structural vibration. However, several control schemes show performance degradation when the model is not precise enough and the structural dynamics varies with time due to structural modifications. To solve these issues, a simple and robust control algorithm should be developed without any correlation to the original structural dynamics and with low computational burden. This study uses an enhanced active vibration control method attenuate the vibration of a nonlinear elastic smart structure. A new control scheme called as the sliding mode least mean squares (SM-LMS) algorithm is proposed and validated both numerically and experimentally [31]. The control algorithm combines the optimal control characteristics of the least mean squares algorithm with the robustness of sliding mode control (SMC). It has been shown that SM-LMS can be successfully used for the displacement control of piezoelectric actuators [32]. However, no study has applied SM-LMS to the active vibration control of flexible structures. To validate the scheme, the controller was implemented on a smart structure constructed with an aluminum beam with two PZT actuators attached to form a
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bimorph structure and one PZT sensor. For the actual tests, we would not know the exact model of the system due to unknown changes in the operating environment, and the plant may vary over time. This changes the control parameters needed to achieve the best tracking performance. Thus, experiments of two different cases were prepared: (1) control for a baseline structure; and (2) control for a structure with additional dynamics that cannot be predicted. Additional impedance is added to the baseline structure to simulate a sudden change in the plant. The scope is limited to a single-input single-output system, and the first four peaks of the disturbance plant (the natural modes) are considered. Specific objectives of this research are described below. (1) (2)
(3) (4)
Introduce the characteristics of the novel nonlinear control methodology (sliding mode least mean squares) and examine the performance on active vibration control through numerical simulations. Identify the dynamic model of the active beam in the frequency domain with an impact hammer test. Estimate parameters of the linear beam model through a frequency domain identification method. Conduct numerical validations with a conventional approach for attenuating vibration level of the structure under impulsive excitations and observe limitations. Conduct experimental validations and examine the performance of the controller for the baseline structure and the modified structures in order to show the robustness of the new control technique.
2. Adaptive Filtering Systems for Active Control Conventionally, digital filters have been applied for modeling dynamic responses to any type of given inputs in engineering structures and acoustic platforms. They apply numerical processes in the discrete time domain for a certain signal to obtain a desired signal shape, and they are broadly used in electronics, computer science, and mechanical engineering. The finite impulse response (FIR) filter is extensively used to represent system responses because of its inherent simplicity, stability, and ease of implementation. In addition, digital filters notably applicable in adaptive structural design and control. Adaptive digital signal processing is applied to update digital filter coefficients automatically in order to create a desired signal from a reference signal and error based on recursive algorithms. Digital filters are thus commonly used in active noise and vibration isolation, system identification, and inverse modeling. An adaptive digital filtering system was designed for active control of the unwanted vibration of continuous structures with a new adaptive filtering algorithm. 2.1. Least Mean Squares Method The least mean squares (LMS) algorithm is the most widely used feedforward control algorithm for adaptive digital filtering systems, since it is relatively simple and easy to implement. The stochastic gradient of the squared error signal is the core of this algorithm [33], and it iteratively regulates the coefficients of transversal filters. These filters use the following two values for the updating formulation: (1) a reference input u[k] containing frequency components to be dealt with; and (2) the measured error value e[k] at the current instant, which is the difference between the filter output y[k ] and the disturbance (desired value) d[k] to be tracked. The algorithm adjusts the adaptive digital filter to determine a filter output y[k ] that is as close as possible to the disturbance value d[k]. A formulation of the LMS algorithm for updating these coefficients can be derived by minimizing the mean squared error with a pre-defined cost function. N
y[k] =
∑ w j [ k ] · u[ k − j ]
(1)
j =0
The most attractive feature of the LMS method is that it does not require average correlation functions from gradient estimates and matrix inversions [34]. It starts from the steepest descent method, and the output of the FIR filter is calculated by multiplying a reference vector u by a vector
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of filter coefficients or weights w. In Equation (1), u[k − j] is the reference input signal at time index 7, 750 4 of 23 th k − j,Appl. andSci. w2017, j [ k ] is the j filter coefficient at time index k. The desired signal d [ k ] is the output of the disturbance plant to beortracked and [k]the is reference the deviation the filter u[error k j ] eis of filter coefficients weightsand In Equation (1),the input between signal at time w . reduced, output y[k ] and the desired signal dth [k]. index k j , and w j [k ] is the j filter coefficient at time index k . The desired signal d[k ] is the
output of the disturbance plant to be tracked reduced, and the error e[ k ] is the deviation e[k] = dand [k] − y[k]
(2)
between the filter output y[k ] and the desired signal d[k ] .
A cost function is chosen to minimize the error. The mean squared error ξ is typically selected as e[ k ] d [ k ] y[ k ] (2) a cost function since it is a quadratic function and thus guarantees a unique minimum, which is a very cost function chosen to minimize error. The mean squared error is typically selected importantAadvantage foris the digital filteringthe system. as a cost function since it is a quadratic function and thus guarantees a unique minimum, which is a very important advantage for the digital filtering ξ = Esystem. [ e [ k ]2 ] 2
E[e[kEquations ]] The cost function is expanded by substituting (1) and (2) into Equation (3):
(3)
(3)
The cost function is expanded by substituting Equations (1) and (2) into Equation (3):
ξ = E[d[k]2 ] − 2w[k] T P + w[k ] T Rw[k], 2 T T E[d [k ] ] 2w[k ] P w[k ] Rw[k ] , P = E[u[k] · d[k ]], P E[u[ k ] d [ k ]] ,
R = E [u[ k ] · u[ k ] T ]
(4) (4)
(5) (5)
(6)
(6)
T
R Ethe [u[kreference ] u[k ] ] input and the desired signal, and R is the P is a cross correlation vector between autocorrelation matrix of the reference signal.the The steepest descent concept used and to develop correlation vector between reference input and the desiredissignal, P is a cross R is an algorithm that adjusts matrix the vector filter weights w. The reason whyconcept this method is applied here is the autocorrelation of theofreference signal. The steepest descent is used to develop an that italgorithm is very straightforward and has beenweights validated in many applications. that adjusts the vector of filter reason why this method is applied here is w . The that it is very straightforward and has been validated in many applications.
µ ∇ξ 2
w[k + 1] = w[k] −
w[ k 1] w[ k ]
2
(7) (7)
Next, Equation (7) is partially differentiated with respect to w[k ], and then the gradient of the cost Next, Equation (7) is partially differentiated with respect to w[ k ] , and then the gradient of the function ∇ ξ can be derived: cost function can be derived: ∇ξ = 2Rw[k] − 2P (8) descent 2Rw[k ] algorithm 2P From Equations (7) and (8), the steepest is obtained:
(8)
From Equations (7) and (8), the steepest descent algorithm is obtained:
w[k + 1] = w[k] + µ(P − Rw[k]) w[k 1] w[k ] (P Rw[k ])
(9)
(9)
where µ is a parameter thatthat regulates the speedand andthe thestability stability of the adaptation where is a parameter regulates theconvergence convergence speed of the adaptation procedure. Figure 1 shows a block diagram of a typical adaptive digital filtering system. procedure. Figure 1 shows a block diagram of a typical adaptive digital filtering system.
Figure 1. Typical adaptive digital filtering system.
Figure 1. Typical adaptive digital filtering system.
The gradient of the cost function is calculated in every iteration with an average of squared
The gradient the cost times. function ∇ξ is calculated every iteration withthe ancross average of squared errors at recentofsampling However, it is usuallyinimpossible to predict correlation errors at recent sampling times. However, it is usually impossible to predict the cross correlation vector
vector P and the correlation matrix R at every sample time. Thus, to simplify the updating equation for the filter coefficients, the LMS algorithm uses instant estimations for P , R , and the 2
cost function [35]. Therefore, e[ k ]
is chosen as an estimate of , and the correlation matrix and
cross correlation vector are assumed to be: Appl. Sci. 2017, 7, 750
2 ˆ e[k ] ,
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(10)
P and the correlation matrix R at every sample time. Thus, to simplify the updating equation for the T (11) ˆ instant R u[k ] u[estimations k] filter coefficients, the LMS algorithm uses for P, R, and the cost function ξ [35]. 2 Therefore, e[k ] is chosen as an estimate of ξ, and the correlation matrix and cross correlation vector (12) are assumed to be: Pˆ u[ k ] d [ k ] ξˆ = e[k]2 , (10) ˆ of the cost function is obtained from Equations (8) and Next, the estimation of the gradient ˆ = u[ k ] · u[ k ] T R (11) (10)–(12): ˆ u[ k ] · d [ k ] (12) ˆ 2u[ k ] u[ kP]T = ˆ [ k ] 2u[ k ] d [ k ] w ˆ ξ of the cost function is obtained from Equations(13) Next, the estimation of the gradient ∇ (8) 2u[ k ]( d [ k ] y[ k ]) 2u[ k ] e[ k ] and (10)–(12): ˆξ= ∇ 2u[into k ] · uEquation [k]T · w ˆ [k] (7), − 2uthe [k] ·final d[k] form of the LMS algorithm is Finally, substituting Equation (13) (13) = −2u[k](d[k] − y[k]) = −2u[k] · e[k] derived.
ˆ [k ] (7), ˆ [k into Finally, substituting Equation (13) w 1] Equation w u[k ]the e[kfinal ] form of the LMS algorithm is derived. (14)
ˆ[k[ k+] 1cannot Notably, the filter coefficients w ˆw ]=w ˆ [ktrace ] + µuthe [k] ·actual e[k] steepest descent surface because (14) the changes of the weights are based on estimated or simplified gradients, which produce noisy Notably, the filter coefficients w ˆ [k] cannot trace the actual steepest descent surface because the tracking. changes of the weights are based on estimated or simplified gradients, which produce noisy tracking. 2.2. Filtered-X Least Mean Squares Method 2.2. Filtered-X Least Mean Squares Method In most practical applications, there are usually effects from secondary path dynamics due to In most practical applications, are usually effects secondary path others. dynamics due to AD/DA converters, filters, actuators,there sensors, amplifiers, andfrom circuit noise, among However, AD/DA converters, filters, actuators, sensors, amplifiers, and circuit noise, among others. However, we assume that the whole control system includes no such dynamics and that the algorithm will not we assume that the whole includes no such dynamics and that algorithm not behave properly with suchcontrol effects.system The most common types of secondary paththe dynamics arewill phase behave properly with such effects. The most common types of secondary path dynamics are phase shifts or time delays. The whole control structure become unstable or converges to an inappropriate shifts or time The whole control structure become unstable converges to an f [k ] or y[kinappropriate ] subtracted solution underdelays. these effects, which changes the filter output and the signal solution under these effects, which changes the filter output f [k] and the signal y[k ] subtracted from from the desired signal d[k ] . the desired signal d[k]. The filtered-X LMS (FX-LMS) algorithm is a modified version of the traditional LMS algorithm The filtered-X LMS (FX-LMS) algorithm is a modified version of the traditional LMS algorithm that manages the secondary path dynamics, which are identified in advance [36]. This algorithm that manages the secondary path dynamics, which are identified in advance [36]. This algorithm requires a filtered reference signal u [k ] with a numerical model of the secondary path dynamics e [k] with a numerical model of the secondary path dynamics Pˆs (z). requires a filtered reference signal u ˆ P z y[k ]is. ( ) . The signal is applied in the LMS algorithm obtain the adaptive filter The signal is applied in the originaloriginal LMS algorithm to obtaintothe adaptive filter output y[output k]. FX-LMS s more stable LMS than method secondary path dynamics cannot be dynamics disregarded. Figurebe2 FX-LMS is than morethe stable thewhen LMSthe method when the secondary path cannot shows a schematic of the FX-LMS algorithm. disregarded. Figure 2 shows a schematic of the FX-LMS algorithm.
Figure Figure 2. 2. Filtered-X Filtered-X least least mean mean square square algorithm. algorithm.
The formulation for the FX-LMS algorithm is almost the same as that of the LMS method except e [k ] is used in the updating equation of the filter weights: that the filtered reference input u
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w ˆ [ k + 1] = w ˆ [k ] + µe u[ k ] · e [ k ]
(15)
The most important property of the filter Pˆ s (z) is that its impulse response contains the same amount of time lag as the secondary path dynamics Ps (z). However, a significant defect of FX-LMS is that an offline system identification procedure is essential to obtain Pˆ s (z) beforehand. Although online system identification might be possible, it makes the overall controller complex and difficult to implement. 2.3. Limitations Many studies have modified the LMS algorithm to address defects such as misadjustment and excessive error with respect to the reference inputs. Hybrid structures were proposed by combining the LMS method with feedback control [37,38]. The vibration isolation was improved at main frequencies and the speed of the convergence was increased, but the error over a broadband frequency range could not be managed. An alternative feedback structure was suggested with internal model control. This method achieves a better transient response and faster convergence, but an incorrect secondary path model seems to degrade the total control performance [39]. An LMS algorithm with adaptive step size was investigated to increase the convergence rate [40,41]. Similarly, a variable-step LMS scheme was applied along with a creep model to accomplish rapid convergence and stability and to identify the hysteresis parameters [42]. A variable convergence coefficient corresponding with an input signal can improve the performance, but it may result in noisy step sizes with unwanted disturbances and thus influence the mitigation at other frequencies [43]. Another LMS scheme prevents instability with impulsive noise, and the reference is pre-defined to assure stability and convergence, particularly when the magnitude of the signal is larger than a certain threshold [44]. An anti-saturation LMS algorithm was introduced to overcome the controller output with large disturbance, which showed improved performance. A different cost function is needed to obtain an enhanced recursive formulation [45]. An extra filter has been used to suppress disturbances in the primary path resulting from uncorrelated noise in the secondary path. This method accomplishes faster convergence, but additional filters increase the complexity and noise in the control system [46]. A new LMS algorithm with phase compensation was introduced to deal with signals that include DC components. The traditional LMS may not be effective for such signals, and when a high pass filter is used to eliminate the DC components, it can produce a phase variation [47]. 3. Enhanced LMS Algorithm 3.1. Sliding Mode Control SMC is one of the most popular nonlinear control methodologies in use today. The strength of this controller is that it is robust, insensitive to plant dynamics, and able to decouple higher-order systems, regardless of their nonlinearity and uncertainty [48]. Once a sliding surface is defined, the state trajectory moves between both sides of the surface by high frequency switching control with discontinuous inputs, and it eventually arrives on the pre-defined sliding surface. The motion of a system is not associated with its original dynamics, and it depends on only the sliding mode equation. The main advantage of SMC lies in its robustness and insensitivity to noise, since the system prescribes the dynamics associated with desirable performance characteristics and the system response is insensitive to disturbances or uncertainties in the system parameters [49]. The SMC design procedure includes the following two parts: (1) a discontinuity surface called the sliding surface is chosen to have dynamics related to desirable performance properties; and (2) a high-frequency switching control input is designed to drive the initial system states to the sliding surface. Because the discontinuous input leads the system states towards the sliding surface, the states eventually reach the surface and move along it, and this condition is referred to as “sliding mode.” The sliding surface is defined for linear systems:
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states eventually reach the surface and move along it, and this condition is referred to as “sliding Appl. Sci. 2017, 7, 750 7 of 23 mode.” The sliding surface is defined for linear systems:
s Gs x , (16) s = Gs x, (16) where x is a state vector, and Gs is a row vector that decides the dynamics of the sliding mode. where x is a state vector, and G is a row vector that decides the dynamics of the sliding mode. The discontinuous inputs u+s ( x , t ) s and− u s ( x , t ) are then chosen to fulfill the requirements for the The discontinuous inputs us (x, t) and us (x, t) are then chosen to fulfill the requirements for the existence existence and the reachability of the sliding mode. A common model of the control input for linear and the reachability of the sliding mode. A common model of the control input for linear system is: system is:
− Msign us us= Msign (s) , (s),
(17) (17)
.
00soso where positivegain gainthat thatisisfairly fairlylarge largetotoinsure insuress ss that state trajectory will reach where M M isisaapositive < that thethe state trajectory will reach the the sliding surface. sliding surface. Figure Figure3 3shows showsconceptual conceptualfigures figuresofofananideal idealsliding slidingmode modeand anda areal realsliding slidingmode. mode.The Thesystem system . x )x)arrives trajectory the state space defined bybys s= 0 0and trajectoryinin the state space( x(,x, arrivesatatthe thesliding slidingsurface surface defined andslides slidestowards towards theorigin originalong alongthe thesurface. surface.ToToaccomplish accomplishideal idealsliding slidingmode modecontrol, control,discontinuous discontinuouscontrol controlinputs inputs the should be able to switch at a fairly high frequency (theoretically infinite), as shown in the figure. should be able to switch at a fairly high frequency (theoretically infinite), as shown in the figure. However,most mostmechanical mechanicalactuators actuatorscannot cannotrealistically realisticallyreplicate replicatethis thiscondition, condition,which whichresults resultsinin However, chatteringproblems problemsthat thatcould couldpossibly possiblymake makethe theoverall overallsystem systemunstable unstablewhen whencombined combinedwith with chattering unmodeled dynamics. unmodeled dynamics.
(a)
(b)
Figure 3. Phase trajectory in sliding mode: (a) ideal sliding mode; and (b) real sliding mode. Figure 3. Phase trajectory in sliding mode: (a) ideal sliding mode; and (b) real sliding mode.
The SMC should also be considered in the discrete time domain since controllers are usually The SMC alsodigital be considered in the timediscrete domain time sincesliding controllers arecontrol usually implemented onshould modern computers. Fordiscrete traditional mode implemented on modern computers. Fornext traditional discrete time[50], sliding (DSMC), (DSMC), the sliding modedigital is forced in the very sampling moment the mode point control of which may the sliding mode is forced in the very next sampling moment [50], the point of which may cause cause controller saturation. The control system will then not work as expected and can even controller saturation. The control system will then not work as expected and can even destabilize the destabilize the target structure. Therefore, there is a need for a novel control technique that can solve target structure. Therefore, there is a need for a novel control technique that can solve these issues. these issues. 3.2. Sliding Mode LMS (SM-LMS) 3.2. Sliding Mode LMS (SM-LMS) A different structure of the adaptive filtering system is introduced using SMC for active vibration A different structure of the adaptive filtering system is introduced using SMC for active isolation. This new system prevents misadjustment from improper initial states and excessive error vibration isolation. This new system prevents misadjustment from improper initial states and problems with respect to the amplitude of the desired values for the sake of the SMC. The sliding mode excessive error problems with respect to the amplitude of the desired values for the sake of the SMC. control is improved by the LMS in terms of smooth convergence and stability. With these properties, The sliding mode control is improved by the LMS in terms of smooth convergence and stability. With it is expected that the technique will enable multi-spectral control of disturbances to manage noise these properties, it is expected that the technique will enable multi-spectral control of disturbances to signals and a number of frequency elements simultaneously with optimized tracking schemes. manage noise signals and a number of frequency elements simultaneously with optimized tracking The SMC technique is combined with the LMS to track or control x (t) due to its intrinsic schemes. benefits of insensitivity to the plant dynamics, robustness, and ability to decouple systems with The SMC technique is combined with the LMS to track or control x(t ) due to its intrinsic high dimension [49]. The proposed SM-LMS technique is illustrated in Figure 4. benefits of insensitivity to the plant dynamics, robustness, and ability to decouple systems with high dimension [49]. The proposed SM-LMS technique is illustrated in Figure 4.
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f [k] in Figure 2 is defined as follows:
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N
f [k ] in Figure 2 is defined as follows:
∑ w j [ k ] · u [ k − j ], j =0 f [k ] w [k ] u[k j ] , f [k] =
(18)
N
(18)
j
j 0
Figure 4. Proposed slidingmode modeleast least mean mean squares (SM-LMS) method. Figure 4. Proposed sliding squarescontrol control (SM-LMS) method.
where w j [ k ] indicates the jth filter coefficient at sampling index [k ] . e[k ] is defined with the
where w j [k ] indicates the jth filter coefficient at sampling index [k]. e[k ] is defined with the assumption that Ps ( z ) can be disregarded: assumption that Ps (z) can be disregarded: e[k ] d [k ] f [k ]
(19)
e[k] = d[k] − f [k]
Next, the sliding mode s[k ] in the error state plane is designated as follows. Here, α determines
(19)
the slope of the sliding surface. Next, the sliding mode s[k] in the error state plane is designated as follows. Here, α determines s[k ] e[k ] e[k 1] the slope of the sliding surface. (20) s [ k ] = α · e [ k ] + e [ k − 1] (20) Using a similar procedure to the derivation of the LMS algorithm, the cost function for the
E the SM-LMS is now defined as Equation (21), where of means expected value. Notefunction that the ξcost Using a similar procedure to the derivation LMSthealgorithm, the cost for the 2 function is applied in the LMS algorithm. E [ e [ k ] ] SM-LMS is now defined as Equation (21), where E means the expected value. Note that the cost function ξ = E[e[k]2 ] is applied in the LMS algorithm. E[ s[ k ]2 ] (21) A unique minimum value is guaranteed ξ = since E[s[k]2 ] is a quadratic function. By expanding (21) using Equations (4) and (20), the following results can be obtained:
A unique minimum value is TguaranteedT since ξ is a quadratic function. By expanding ξ using 2 2 E[d [k ] ] 2w[k ] P1 [k ] w[k ] R1 [k ] w[k ] Equations (4) and (20), the following results can be obtained: α2
n
2
T
T
E[d [k 1] ] 2w[k 1] P2 [k ] w[k 1] R 2 [ko ] w[k 1] 2
T
T
(22)
T E[d[k2] ] −E[2w +[kw[1] k]T P· R [k] · wT[k] 1 [ k] w [k1] w[k ] P4 [k ] w[k ] R3 [ko ] w[k 1] 2 d[k[]k ]d[k· P1]] n 3 + E[d[k − 1] ] − 2w[k − 1] T · P2 [k] + w[k − 1] T · R2 [k] · w[k − 1] (22) n o The vector P indicates the expected cross correlation between the reference and unwanted T T T + 2α [d[kmatrix ] · d[k −R 1]]is− wexpected [k − 1] ·autocorrelation P3 [k] − w[k] of · Pthe + w [ k ] · R3 [ k ] · w [ k − 1 ] signals, andEthe the 4 [ k ]reference:
ξ=
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The vector P indicates the expected cross correlation between the reference and unwanted signals, and the matrix R is the expected autocorrelation of the reference:
P1 [k] = E[u[k ] · d[k]]
(23)
P2 [k] = E[u[k − 1] · d[k]]
(24)
P3 [k] = E[u[k − 1] · d[k − 1]]
(25)
P4 [k] = E[u[k] · d[k − 1]]
(26)
R1 [ k ] = E [ u [ k ] · u [ k ] T ]
(27)
R2 [ k ] = E [ u [ k − 1 ] · u [ k ] T ]
(28)
R3 [ k ] = E [ u [ k ] · u [ k − 1 ] T ]
(29)
The next step is using the steepest descent scheme to obtain an algorithm for regulating the weights w[k ]. w[k + 1] = w[k] − 0.5µ∇ξ (30) Equation (21) is partially differentiated with respect to w[k] in order to derive the gradient of the cost function ∇ξ: ∇ξ = −2α2 · P1 [k] + 2α2 · R1 [k] · w[k] (31) − 2α · P4 [k] + 2α · R3 [k] · w[k − 1] For this control algorithm, the gradient ∇ξ along with P and R, have to be determined in every time instant k, as mentioned in the previous section. This is a clumsy process, and to simplify the calculating operations, instantaneous estimates of ξ, P, and R should be obtained at a specific sampling instant in the LMS algorithm [34]. Thus estimations of these values indicated by the hat symbol (“ˆ”) are defined for time step k: ξˆ = s[k ]2 (32) Pˆ 1 [k] = u[k] · d[k ]
(33)
Pˆ 4 [k] = u[k ] · d[k − 1]
(34)
ˆ 1 [ k ] = u[ k ] · u[ k ] R
T
ˆ 3 [ k ] = u[ k ] · u[ k − 1] T R
(35) (36)
After substituting Equations (34) and (35) for P and R in Equation (30), a new estimate for ∇ξˆ is derived: ∇ξˆ = −2α2 · u[k] · e[k] − 2α · u[k] · e[k − 1] (37) Upon combining Equations (29) and (36), w[k] is updated as follows: n o w ˆ [ k + 1] = w ˆ [ k ] + µ · u[ k ] · α2 · e [ k ] + α · e [ k − 1]
(38)
As shown in Figure 5, the inessential maneuvers of error states are decreased by following a pre-defined sliding surface. Therefore, less input effort or energy would be needed, and faster convergence can be achieved with the introduction of sliding mode control.
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Figure 5. Error surface illustration for: (a) the conventional LMS method; and (b) SM-LMS. Figure 5. Error surface illustration for: (a) the conventional LMS method; and (b) SM-LMS.
ˆ [k ] with the optimized solutions of SM-LMS when the We now discuss the convergence of w We now discuss the2 convergence of w ˆ [k ] with the optimized solutions of SM-LMS when the ˆ [k ]] cost function E[ s[ k ] ] is minimized. The expected value of the filter weight vector E[w cost function ξ = E[s[k]2 ] is minimized. The* expected value of the filter weight vector E[w ˆ [k]] would ˆ after a specific number of iterations when the gradient would accomplish the optimal accomplish the optimal solution solution w ˆ ∗* after awspecific number of iterations when the gradient ∇ becomes ˆ can be derived from Equation (38): becomes zero ideally. w zero ideally. Thus, w ˆ ∗ can Thus, be derived from Equation (38): 0 2 2 P1 [ k ] 2 2 R 1 [ k ] w * 2 P4 [ k ] 2 R 3 [ k ] w * , (39) ∇ = 0 = −2α2 · P1 [k] + 2α2 · R1 [k] · w∗ − 2α · P4 [k] + 2α · R3 [k] · w∗ , (39) w * ( R 1 [ k ] R 3 [ k ]) 1 ( (40) −1 P1 [ k ] P4 [ k ]) w∗ = ( α · R (40) 1 [ k ] + R3 [ k ]) ( α · P1 [ k ] + P4 [ k ]) Using the expectation operator on both sides of Equation (37) gives the following equation: Using the expectation operator on both sides of Equation (37) gives the following equation: 2
ˆ [ k 1]] E[ w ˆ [ k ]] E[u[ k ] e[ k ]] E[u[ k ] e[ k 1]] E[ w E [w ˆ [k + 1]] = E[w ˆ [k]] + µα2 E[u[k] · e[k]] + µαE[u[k ] · e[k − 1]] 2 ˆ [ k ]] ˆ [Pk4]] [ k ]) [ k+ ] µαP4 [k ] ( I = (RI1 − µα 2ER[ w 1 [ k ]) · E [w
(41) (41)
2 P [ k ] − µαR [ k ] · E [w[ k − 1]] 2 3 [ k 1]] 1 R [ k ] E[ w ] P+[ kµα 1
3
The The principal principal axis axis coordinate coordinate should should be be changed changed to to solve solve this this equation. equation. Hence, Hence, the the translated translated ∗ * variable ˆwˆ − ˆˆ ,, and and Equation Equation (40) (40) becomes: variable is is defined defined as as VV=w becomes: w w 2 EE[[V ( I (I R 12[R k ]) [ VE[[kV]][k]] [ k ]3E[k[ V V[[kk +1]] 1]] = − µα k]) −RµαR ] E[[kV[1]] k − 1]] 3 1 [E
(42) (42)
The coordinates are rotated again with V QV ' to produce the following. The coordinates are rotated again with V = QV0 to produce the following.
where where
E[ V '[ k 1]] ( I 2 1 ) E[ V '[ k ]] 2 E[ V '[ k 1]] , E[V0[k + 1]] = ( I − µα2 Λ1 ) E[V0[k ]] − µαΛ2 E[V0[k − 1]],
(43) (43)
1 Q 1 R 1−Q1 Λ 1 = Q R1 Q
(44) (44)
=1Q 2Λ2 Q R− Q1 R3 Q 3
(45) (45)
Equation it from time steps k + 1k to and1 the higher-power terms 11, to Equation(42) (42)can canbe besolved solvedbybyextending extending it from time steps , and the higher-power 2 Λ ) and 2 of ( I − µα (− µαΛ ) are ignored. terms of ( I 1 1 ) and 2( 2 ) are ignored. 2 k k 2 1 ) Ek[ V '[1]] ( 2 ) E[ Vk '[0]] [ V0[k'[ k+11]] ((I I− EE[V ]] = µα Λ1 ) E[V0[1]] + (−µαΛ2 ) E[V0[0]] + · · ·
(46) (46)
From Equation (45), it is confirmed that the convergence of the SM-LMS is guaranteed only if From Equation (45), it is confirmed that the convergence of the SM-LMS is guaranteed only if the the following rules are satisfied: following rules are satisfied: 1 |1 2 1,max | 1 → 2 (47) 1 0 , 2 > µ > 0, (47) 1 − µα λ1,max < 1 → 1,max α2 λ1,max
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1 | 2,max | 1 →1 1 11 0 , → → → → 0,0, 00, , | | 1 | 1 | |2,max | 1 | | 1 2,max 2,max2,max 2,max
(48) (48) (48) (48) 11 (48) of 23 2,max 2,max 2,max2,max where 1,max and 2,max indicate the largest eigenvalues of R1 and R3 , respectively. Relatively fast R3R RRand RR3 3, ,respectively. Rand where thethe largest eigenvalues of ofRof , respectively. Relatively fast where and indicate the largest eigenvalues of and respectively. Relatively fast where and indicate the largest eigenvalues Relatively fast where and indicate largest eigenvalues Relatively fast 1,max 2,max and indicate 1,max 2,max 1 1 and 3 , respectively. 1,max 2,max 1,max 2,max 1 1of1 the convergence could be achieved with2,max the| µ > 0, (48) |− convergence could bebe achieved with thethe introduction of 2,max the mode control forfor the LMS convergence could be achieved with the introduction ofsliding the sliding mode control for the LMS convergence could be achieved with the introduction of the sliding mode control for the LMS convergence could achieved with introduction of the sliding mode control the LMS αλ algorithm. Nonetheless, the possible range of the convergence parameter is reduced compared algorithm. Nonetheless, thethe possible range of of the parameter isis reduced compared algorithm. Nonetheless, the possible range ofconvergence the convergence parameter reduced compared algorithm. Nonetheless, the possible range of the convergence parameter isisreduced compared algorithm. Nonetheless, possible range the convergence parameter reduced compared where λ1,max andLMS λ2,max indicate the largest eigenvalues of R R3 , respectively. fast 1 and with that of the algorithm, which could compromise the benefits of using theRelatively sliding mode with that of the algorithm, which could compromise the benefits of of using the sliding mode with that ofLMS the LMS algorithm, which could compromise the benefits of using using the sliding mode with that of the LMS algorithm, which could compromise the benefits of the sliding mode with that of the LMS algorithm, which could compromise the benefits using the sliding mode convergence could be achieved with the introduction of the sliding mode control for the LMS algorithm. control. control. control. control. control. Nonetheless, the possible range of the convergence parameter µ is reduced compared with that of the Appl. Sci. 2017, 7, 750
LMS algorithm, which could 3.3. Numerical Evaluation of thecompromise SM-LMS the benefits of using the sliding mode control. 3.3. Numerical Evaluation of the SM-LMS 3.3. Numerical Evaluation the SM-LMS 3.3. Numerical Evaluation ofof the SM-LMS 3.3. Numerical Evaluation of the SM-LMS A simple numerical validation was carried out to examine the signal tracking performance of 3.3. Numerical Evaluation of the SM-LMS A simple numerical validation was carried outout to examine thethe signal tracking performance of of of Asimple simple numerical validation was carried out toexamine examine the signal tracking performance of A numerical validation was carried out to the signal tracking performance A simple numerical validation was carried to examine signal tracking performance the proposed algorithm under the following conditions: (1) a second-order linear time-invariant (LTI) thethe proposed algorithm under the following conditions: (1) a second-order linear time-invariant (LTI) the proposed algorithm under the following conditions: (1) a second-order linear time-invariant (LTI) the proposed algorithm under the following conditions: (1) a second-order linear time-invariant (LTI) proposed algorithm under the following conditions: (1) a second-order linear time-invariant (LTI) A simple numerical validation was carried out to examine the signal tracking performance of system with a primary frequency of n 30 Hz and damping ratio of 0.2 ; (2) a finite impulse 30 30 30 Hz Hz Hz n n30 Hz system with awith primary frequency and damping of a; ;time-invariant finite impulse system primary frequency of and damping ratio of (2) finite impulse system with aaalgorithm primary frequency and damping ratio (2) aafinite impulse system with a primary frequency ofof and damping ratio of of ; (2) a finite impulse 0.2 ;(2) 0.2 0.2 0.2 the proposed underofthe following conditions: (1) aratio second-order linear (LTI) n n u ( t ) A sin( t ) response (FIR) filter with three coefficients; and (3) a single-frequency sine wave system with a primary frequency of ωand 30 damping ratio of ζ= n = u0.2; (tu)((2) tsin( (tt)asin( )Afinite sin( Asin( A tu)u(A impulse t)tt))) response (FIR) filter with three coefficients; (3) aHz single-frequency sine wave response (FIR) filter with three coefficients; and (3) single-frequency sine wave response (FIR) filter with three coefficients; and (3) aasingle-frequency sine wave response (FIR) filter with three coefficients; and (3) a and single-frequency sine wave with amplitude excitation frequency phase ( (t) 0=).AThe control A with 1 , three asingle-frequency 120 Hz , and nosine response (FIR)Afilter coefficients; and sin (ωt + φ) with amplitude excitation frequency and no (wave control with amplitude excitation frequency and no phase ). The The control with amplitude , , excitation frequency , , and no phase control with amplitude frequency , and nophase phase control A1A,A1 ,11excitation (3) 120 ( (0(u 120 Hz 120 Hz 120 ).0).The 00 ).The Hz, Hz parameters are A selected as 15 and ω . Figure 6a shows the(φtime-domain tracking results 0.3 with amplitude = 1, excitation frequency = 120 Hz, and no phase = 0). The control parameters 15 and 15 15 15 parameters areare selected as asas 6a6a shows thethe time-domain tracking results parameters are selected as and Figure 6ashows shows the time-domain tracking results parameters are selected and . .Figure 6a the time-domain tracking results parameters selected and . Figure shows time-domain tracking results 0.3 0.3 .Figure 0.3 0.3 are α = 15 and When µ = 0.3. shows the time-domain tracking results the two withselected the twoasalgorithms. theFigure signal6ahas sudden changes in the direction and with amount, the with thethe two algorithms. When thethe signal has sudden changes in in the and amount, thethethe with the two algorithms. When the signal has sudden changes indirection the direction and amount, the with the two algorithms. When the signal has sudden changes in the direction and amount, with two algorithms. When signal has sudden changes the direction and amount, algorithms. signal hashas sudden changes in the direction amount,algorithm, the conventional LMS conventionalWhen LMSthe algorithm worse performance than theand SM-LMS and higher conventional LMS algorithm has performance than thetheSM-LMS algorithm, and higher conventional LMS algorithm has worse performance than the SM-LMS algorithm, and higher conventional LMS algorithm has worse performance than the SM-LMS algorithm, and higher conventional LMS algorithm hasworse worse performance than SM-LMS algorithm, and higher algorithm worse performance thancompared the SM-LMS algorithm, anderror higher fluctuations frequency has fluctuations are observed to the estimation forfrequency both cases, as shownare in frequency fluctuations areare observed compared to to the error forfor both cases, as as shown in in in frequency fluctuations are observed compared toestimation the estimation error for both cases, as shown shown in frequency fluctuations are observed compared to the estimation error for both cases, as frequency fluctuations observed compared the estimation error both cases, shown observed the estimation errorthat for the bothfilter cases, as shown in Figure 6b.approach It is also important to Figure 6b.compared It is also to important to ensure weights at steady state the same or Figure 6b.6b. It6b. is also important to to ensure that the filter weights at at steady state approach thethe same or or or Figure 6b. also important toensure ensure that the filter weights atsteady steady state approach the same or Figure ItItalso isisalso important to that the filter weights at state approach the same Figure It is important ensure that the filter weights steady state approach same ensure that the at steady state approach or closeshow values forthe both algorithms, close values forfilter bothweights algorithms, as shown in Figure the 7a. same The results that sliding mode close values forfor both algorithms, as as shown in in Figure 7a.7a. The results show that the sliding mode close values for both algorithms, as shown shown in Figure Figure 7a. The results show that the sliding mode close values for both algorithms, as in 7a. The results show that the sliding mode close values both algorithms, shown Figure The results show that the sliding mode as shownonly in Figure 7a. The results show that sliding mode controlbetter only supplements the original control supplements the original LMSthe algorithm to achieve optimized convergence control only supplements thethe original LMS algorithm to to achieve better optimized convergence control only supplements the original LMS algorithm to achieve achieve better optimized convergence control only supplements the original LMS algorithm to better optimized convergence control only supplements original LMS algorithm achieve better optimized convergence LMS algorithm to achieve better optimized convergence changing its primary characteristics. without changing its primary characteristics. Figure 5b without shows the trajectory of the coefficient plane without changing itsits primary characteristics. Figure 5b5b shows thethe trajectory of of the plane without changing itsprimary primary characteristics. Figure 5bshows shows the trajectory ofcoefficient the coefficient plane without changing its characteristics. Figure 5b the trajectory of the coefficient plane without changing primary characteristics. Figure shows trajectory the coefficient plane Figure 5b shows the trajectory of the coefficient plane (the filter weights), and the path to the desired (the filter weights), and the path to the desired steady state is relatively straight and fast for the SM(the filter weights), and the path to theto steady state is relatively straight and fast for the SM(the filter weights), and the path todesired the desired steady state relatively straight and fast for the SM(the filter weights), and the path the desired steady state isisrelatively straight and fast for the SM(the filter weights), and the path to the desired steady state is relatively straight and fast for the SMsteady state is relatively for theinSM-LMS. Therefore, changes of the filter weights in LMS. Therefore, changesstraight of the and filterfast weights the SM-LMS case show superior control without LMS. Therefore, changes of of the weights in in the case show superior control without LMS. Therefore, changes offilter the filter weights inSM-LMS the SM-LMS case show superior control without LMS. Therefore, changes of the filter weights in the SM-LMS case show superior control without LMS. Therefore, changes the filter weights the SM-LMS case show superior control without the SM-LMS case showmovement. superior control without inappropriate system movement. inappropriate system inappropriate system movement. inappropriate system movement. inappropriate system movement. inappropriate system movement.
(a) (a)(a)(a) (a)
(b) (b)(b)(b) (b)
Figure 6. Control of a single sinusoid at 120 Hz in the time domain: (a) predicted with least mean Figure 6.6.Control Control single sinusoid at 120 in the time domain: (a) predicted with least mean Figure 6. Control of of a single sinusoid at at 120at inHz thethe time domain: (a)(a) predicted with least mean Figure Control ofsingle single sinusoid atHz 120 Hz in the time domain: (a)predicted predicted with least mean Figure 6. of aaasingle sinusoid 120 (a) with least mean Figure 6. Control aof sinusoid 120 Hz in time domain: predicted with least mean squares and sliding mode least mean squares (Key: , original signal; , filter output); squares and sliding mode least mean squares (Key: , original original signal; , filter filter output); squares and sliding mode least mean squares (Key: signal; output); squares and sliding mode least mean squares (Key: original signal; filter output); squares and sliding mode least mean squares (Key: ,,,original signal; ,, ,filter output); squares and sliding mode least mean squares (Key: , original signal; , filter output); and (b) predicted estimation error (Key: , with least mean squares; , with sliding and (b) predicted estimation error (Key: ,, with least mean squares; mode and (b) predicted estimation error (Key: , with least mean squares; with sliding and (b) predicted estimation error (Key: with least mean squares; ,, with with sliding and (b) predicted estimation error (Key: , ,with least mean squares; , ,sliding with sliding and (b) predicted estimation error (Key: with least mean squares; , with sliding mode least mean squares). least mean squares). mode least mean squares). mode least mean squares). mode least mean squares). mode least mean squares).
control control only supplements the original LMS algorithm to achieve better optimized convergence only supplements the original LMS algorithm to achieve better optimized convergence withoutwithout changing its primary characteristics. Figure Figure 5b shows trajectory of the coefficient plane plane changing its primary characteristics. 5b the shows the trajectory of the coefficient the filter weights), and theand paththe topath the desired steady state is state relatively straightstraight and fastand for fast the for SM-the SM(the filter weights), to the desired steady is relatively LMS. Therefore, changes of the filter in the SM-LMS case show control control withoutwithout LMS. Therefore, of theweights filter weights in the SM-LMS casesuperior show superior Appl. Sci.changes 2017, 7, 750 nappropriate system movement. inappropriate system movement. Appl. Appl.Sci. Sci. 2017, 750 Sci.2017, 2017,7, 7,750 750 Appl. Sci. 2017, 7,7, 750 Appl. Sci. 2017, 7, 750
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(a) (a) (a) (a) (a)
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(b)
Figure 7. Comparison of least mean modeleast leastmean mean squares: (a) change in filter Figure 7. Comparison of least meansquares squaresand and sliding sliding mode squares: (a) change in filter Figure 7.Comparison Comparison ofleast least mean squares andsliding sliding mode least mean squares: (a) changein infilter filter Figure 7. Comparison of of least mean mean squares squares and and sliding mode mode least least mean mean squares: squares: (a) (a) change change in filter Figure 6. Control of a single sinusoid at 120 Hz in the time domain: (a) predicted with least mean Figure 6. Control of a single sinusoid at 120 Hz in the time domain: (a) predicted with least mean Figure 7. Comparison of least mean squares and sliding mode least mean squares: (a) change in filter coefficients (Key: , w0; , w1; , w2); and (b) comparison of trajectories in the coefficients (Key: , w0; , w1; , w2); and (b) comparison of trajectories in the coefficients (Key: w0; w1; , w2); (b) comparison of trajectories coefficients coefficients (Key: (Key: , w0; , w0; , w1; , w1; , w2); and and (b) (b) comparison comparison of of trajectories trajectories in in the squares squares and sliding mode least mean squares (Key: original signal; , filter output); and sliding mode least mean squares (Key: , original signal; , filter output); coefficients (Key: , w0; , w1; , w2); and (b) comparison of trajectories in the plane of filter coefficients (Key: , least mean squares; , sliding mode least mean plane of filter coefficients (Key: , least mean squares; , sliding mode least mean the of filter coefficients (Key: , least mean squares; , sliding mean squares). plane plane of of filter filter coefficients coefficients (Key: (Key: , , least least mean mean squares; squares; ,,sliding ,sliding sliding mode mode least least mean mean and (b) and predicted estimation error (Key: , with least mean squares; , with (b) predicted estimation error (Key: , with least mean squares; ,mode with least sliding plane of filter coefficients (Key: , least mean squares; sliding mode least mean squares). squares). squares). squares). mode least mean squares). mode least mean squares). squares). 4. Problem Formulation 4. Problem Problem Formulation 4. Formulation
4. 4.Problem Problem Formulation continuous structure was prepared for the validation of the active control 4. Problem Formulation AFormulation one-dimensional A one-dimensional one-dimensional continuous structure structure was was prepared prepared for for the the validation validation of of the the active active control control A continuous strategy. The structure is a thin, aluminum, cantilevered elastic beam with piezoelectric ceramic A Aone-dimensional one-dimensional continuous structure structurewas wasprepared preparedelastic for forthe the validation validation of ofthe theactive activecontrol control strategy. The structure structurecontinuous is thin, aluminum, aluminum, cantilevered elastic beam with piezoelectric piezoelectric ceramic A one-dimensional continuous structure was prepared for the validation of the active control strategy. The is aa thin, cantilevered beam with ceramic patches (PZT) attached to both sides. The specific type of PZT material used is PZT-5A and “31” strategy. strategy. The The structure structure is is a a thin, thin, aluminum, aluminum, cantilevered cantilevered elastic elastic beam beam with with piezoelectric piezoelectric ceramic ceramic patches (PZT) attached to both sides. The specific type of PZT material used is PZT-5A and “31” strategy. The structure is a thin, aluminum, cantilevered elastic beam with piezoelectric ceramic patches (PZT) attached to both sides. The specific type of PZT material used is PZT-5A and mode “31” mode has been used for the actuator. It is better for actuator applications for dynamic operating patches patches (PZT) (PZT) attached attached to tothe both both sides. sides. The The specific specific type type of ofPZT PZTapplications material material used used isis isdynamic PZT-5A PZT-5Aconditions, and and“31” “31” has been used for the actuator. It is better for actuator applications for dynamic operating patches (PZT) attached to both sides. The specific type of PZT material used PZT-5A and “31” mode has been used for actuator. It is better for actuator for operating conditions, while it is still good for utilized as sensor applications. The beam can be bent when the mode mode has has been used for the actuator. actuator. ItItItisis isbetter better for foractuator actuator applications for dynamic dynamic operating operating mode has been used the actuator. better for actuator applications for dynamic operating whileceramic it isbeen still good for utilized as sensor applications. The applications beam canbeam befor bent when the ceramic conditions, while it isfor stillthe good for utilized as sensor applications. The can be8 bent when the hasused deformations with applied voltage, which is called a “bimorph”. Figure shows the conditions, conditions, while while it it is is still still good good for for utilized utilized as as sensor sensor applications. applications. The The beam beam can can be be bent bent when when the conditions, while it is still good for utilized as sensor applications. The beam can be bent when the has deformations with applied voltage, which is called a “bimorph”. Figure 8 shows the concept ceramic has of deformations appliedstructures. voltage,The which is called a “bimorph”. Figure 8 shows concept different typeswith of bimorph red arrows show the polarization direction, and the ceramic ceramic has has deformations deformations with with applied applied voltage, voltage, which which isis isshow called called aaaproperties “bimorph”. “bimorph”. Figure Figure 888shows shows the the ceramic has deformations with applied voltage, which called “bimorph”. Figure shows the of different types oftypes bimorph structures. The red arrows the polarization direction, and green green arrows indicate the strain direction. Table 1The shows material of the PZT patch. concept of different of bimorph structures. redthe arrows show the polarization direction, and concept concept of ofdifferent different types of bimorph bimorph structures. structures. The red red arrows arrows show show the thepolarization polarization direction, direction, and concept of different types of bimorph structures. The red arrows show the polarization direction, and arrows indicate thetypes strain direction. Table 1 shows the material properties of the patch. green arrows indicate theof strain direction. Table 1The shows the material properties ofPZT the PZT patch.and green greenarrows arrowsindicate indicatethe thestrain straindirection. direction.Table Table111shows showsthe thematerial materialproperties propertiesof ofthe thePZT PZTpatch. patch. green arrows indicate the strain direction. Table shows the material properties of the PZT patch.
(a)
(b)
Figure 8. Bimorph concept: (a) parallel bimorph; and (b) serial bimorph.
(a)
(b)
Table(a) 1. Material properties of the piezoelectric ceramic patches (PZT) (a) (b) (b)patch. (a) (b)
Figure 8. Bimorph concept: (a) parallel bimorph; and (b) serial bimorph. Figure 8. Bimorph concept: (a) parallel bimorph; and (b) serial bimorph. Properties Value Value Figure Figure 8.8.8.Bimorph Bimorphconcept: concept:(a) (a) parallel parallelbimorph; bimorph;Properties and and(b) (b)serial serialbimorph. bimorph. Figure Bimorph concept: (a) parallel bimorph; and (b) serial bimorph. k Coupling Coefficient Thickness 0.5 mm 0.72 Table 1. Material properties of the piezoelectric ceramic patches (PZT) patch. 33 Table 1. Material properties of the piezoelectric ceramic patches (PZT) patch. Table Table1.1.1.Capacitance Material Materialproperties propertiesof ofthe the piezoelectric ceramic ceramic patches patches(PZT) (PZT)patch. patch. Table Material properties of the piezoelectric ceramic patches patch. k(PZT) Coupling Coefficient 162piezoelectric nF 0.35 Properties Value Properties 31 Value 12 6 Properties Value Properties d Piezo. Strain Coefficient Polarization Field Properties Properties Value Value Properties Properties Value Value Properties Value Properties V/m Value 3900.5 10 33 Coefficient k 2 10 Value Thickness mm m/V Coupling 0.72 33
12 Thickness mm Coupling Coefficient kk33k3333533 10 5 0.72 Coupling Coupling Coefficient Coefficient Thickness Thickness 0.5 0.5 mm 0.72 0.72 Coupling Coefficient Piezo. Strain Coefficient d31 0.5 Thickness 0.5 mm 0.72 Depolarization Field m/V Initial V/m 190 mm 10 Coupling Capacitance 162 nF 0.35 0.35 Capacitance 162 nF CouplingCoefficient Coefficient kk31 31 3 3 −162 12 Voltage Coefficient390g 33 Density Coupling Coefficient Coefficient Capacitance Capacitance 162 nF nF m/V Coupling 0.35 0.35 24.0 m/V 10 Coupling Coefficient Capacitance 162 nF 0.35 Polarization Fieldkkk 7800 kg/m Piezo. StrainPiezo. Coefficient d33 2 × 106 V/m 10 12 d33 ×390 Piezo. Strain Coefficient Polarization Field 313131 2 10 6 V/m5 m/V 10 − 12 3 Piezo. StrainPiezo. Coefficient d31 Initial Depolarization Field350 °C6665 × 10 V/m −190 ×10 12 12 Voltage Coefficient 12 Curie Temperature m/V 11.6 m/V 10 ddd333333g 31390 Piezo. Piezo.Strain Strain Coefficient Coefficient Polarization Polarization Field Field Piezo. Strain Coefficient Polarization Field m/V V/m 390 −10 10 22210 105 V/m 310 3 m/V V/m 390 10 12m/V Piezo. Voltage g33 Density × 10 m/V d24.0 Piezo.Coefficient Strain Coefficient Initial Depolarization Field V/mkg/m 190−310 m/V 5 10 7800 31 ◦C 12 12 12 555 Piezo. Voltage Coefficient g Curie Temperature 350 − 11.6 × 10 m/V 31 ddd Piezo. Piezo.Strain StrainCoefficient Coefficient InitialDepolarization DepolarizationField Field Piezo. Strain Coefficient Initial Depolarization Field m/V m/VInitial V/m V/m
190 19010 10 m/V 190 10 Piezo. Voltage Coefficient 31g313133 24.0 10 3 m/V 33 Piezo. Piezo.Voltage VoltageCoefficient Coefficient ggg 24.0 Piezo. Voltage Coefficient m/V 24.010 1033m/V m/V 24.0 10 Piezo. Voltage Coefficient g33333133 11.6 10 m/V 333 Piezo. Piezo.Voltage VoltageCoefficient Coefficient ggg31313111.6 Piezo. Voltage Coefficient m/V 11.610 10 m/V m/V 11.6 10
Density Density Density Density Curie Temperature Curie CurieTemperature Temperature Curie Temperature
55510 10 V/m 10 7800 kg/m3 333 7800 7800kg/m kg/m 7800 kg/m 350 °C 350 350°C °C 350 °C
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Figure 9 shows a schematic of the experimental setup. Serial bimorph PZT actuators are attached Appl. Sci. 2017, 7, 750 13 of 23 Figureof9 shows a schematic of the and experimental setup. Serial PZT actuators areisattached to the surface the aluminum beam, an additional smallbimorph piezoelectric ceramic attached to to the surface of the aluminum and ancontrol additional small piezoelectric ceramic is attached to one their one side as a strain sensor for thebeam, feedback system. PZT sensors are selected check Figure 9 shows a schematic of the experimental setup. Serial bimorph PZT actuators aretoattached side as a strain sensor for the feedback control system. PZT sensors are selected to check theirtip of feasibility for theofactive vibration control applications. Anpiezoelectric accelerometer is attached attothe to the surface the aluminum beam, and an additional small ceramic is attached one feasibility for the active vibration control applications. An accelerometer is attached at the tip of the the beam as aa strain performance indicator. A bench visesystem. and anPZT electrodynamic shakerto arecheck installed side as sensor for the feedback control sensors are selected their on a beam as a performance indicator. A bench vise and an electrodynamic shaker are installed on a feasibility for the active vibration control applications. accelerometer is attached at the tip of the vibration isolation table, and the whole beam structure isAn gripped with the vise, resulting clamped-free vibration isolation table, and the whole beam structure is gripped with the vise, resulting clampedbeam conditions. as a performance indicator. A bench viseprovides and an electrodynamic shaker are installed on a in boundary The electrodynamic shaker impulsive force to the vise, which results free boundary conditions. The electrodynamic shaker provides impulsive force to the vise, which vibration isolation table, and the whole beam structure is gripped with the vise, resulting clampedlateralresults vibration of the beam that hasbeam to bethat mitigated. mass can be attached the right in lateral vibration of the has to beAdditional mitigated. Additional mass can be to attached to side boundary conditions. The electrodynamic shaker provides impulsive force to the vise, which of thefree beam, as an impedance of thechange structure. the rightwhich side of works the beam, which works as change an impedance of the structure. results in lateral vibration of the beam that has to be mitigated. Additional mass can be attached to the right side of the beam, which works as an impedance change of the structure.
Figure 9. Schematic of experimental setup. Figure 9. Schematic of experimental setup. Figure 9. Schematic of experimental setup.
A data acquisition and control system (dSPACE 1104) is used to implement real-time control, A data system (dSPACE is used to implement real-time control, and and it acquisition is connectedand to control a desktop computer with1104) MATLAB/Simulink installed. The designed A data acquisition and control system (dSPACE 1104) is used to implement real-time control, it is connected desktop computer with module MATLAB/Simulink The designed controllers controllersto area uploaded to the dSPACE after compilinginstalled. the MATLAB files, and the control are and it is connected to a desktop computer with MATLAB/Simulink installed. The designed system prepared as module follows: (1) signals from the sensor and the accelerometer acquired by uploaded to isthe dSPACE after compiling thePZT MATLAB files, and the controlare system is prepared controllers are uploaded to the dSPACE module after compiling the MATLAB files, and the control ADC channels and used for the control; and (2) the control input calculated in the dSPACE module as follows: signals from the PZT sensorfrom and the thePZT accelerometer areaccelerometer acquired by are ADC channels system(1) is prepared as follows: (1) signals sensor and the acquired by and is produced from and a DAC channel and provided to the PZT actuators after amplification. Figure 10from used ADC for the control; (2) the control input calculated in the dSPACE module is produced channels and used for the control; and (2) the control input calculated in the dSPACE module shows the main components of the experimental setup. a DAC channel and to theand PZT actuators after Figure 10 shows the10main is produced from provided a DAC channel provided to the PZTamplification. actuators after amplification. Figure shows the components ofsetup. the experimental setup. components ofmain the experimental
Figure 10. Experimental setup. Figure10. 10. Experimental Experimental setup. Figure setup. from the impact test, and the The following frequency response function was obtained relationship between the input and output (the transfer function) can be predicted from the results. The following frequency response function was obtained from the impact test, and the The following objective isfrequency to investigate the controller performance withfrom this baseline structure andthe a modified The response function obtained thebe impact test,from and relationship between the input and output (thewas transfer function) can predicted the relationship results. structure with additional mass, which dramatically changes the structural dynamics and transfer between inputisand output (the function) can be predicted fromstructure the results. objective is The the objective to investigate thetransfer controller performance with this baseline and The a modified function. Figure 11 compares the frequency response functions of the baseline and the modified to investigate the controller with this baseline and adynamics modified structure structure with additional performance mass, which dramatically changesstructure the structural and transfer with structures, and obvious changes in natural frequencies can be observed. function. Figure 11 compares the changes frequencythe response functions of theand baseline andfunction. the modified additional mass, which dramatically structural dynamics transfer Figure 11 structures, and obvious changesfunctions in natural frequencies can be observed. compares the frequency response of the baseline and the modified structures, and obvious
changes in natural frequencies can be observed.
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(a) (a)
(b) (b)
Figure 11. Frequency response function of beam structure: (a) two different configurations; and (b)
Figure 11. Frequency Frequency response response function function of of beam beam structure: structure: (a) (a) two two different different configurations; configurations; and and (b) (b) comparison of frequency response function. comparison of frequency response function.
5. System Identification and Notch Filter Design
5. and Notch Notch Filter Filter Design Design 5. System System Identification Identification and
To design an appropriate controller, the structural dynamics of the system should be fully To design an appropriate controller, the dynamics of system be characterized. A frequency-domain system identification method was applied to derive theshould model of To design an appropriate controller, the structural structural dynamics of the the system should be fully fully the system. Figure 12 shows the frequency response function of the structural model to be identified. characterized. A frequency-domain system identification method was applied to derive the model characterized. A frequency-domain system identification method was applied to derive the model of of damping ratio each the peakfrequency can be estimated by function the quadrature picking method. It identified. is a the system. Figure 12 shows response of structural model to theThe system. Figure 12of shows the frequency response function of the thepeak structural model tobe be identified. classical method for determining the damping at resonance, half power (−3 dB) pointsmethod. of each The peak can can be estimated estimated by the the and quadrature peak picking The damping damping ratio ratio of of each each peak be by quadrature peak picking method. It It is is aa resonance magnitude should be identified.
classical resonance, and and half half power power ((−3 classical method method for for determining determining the the damping damping at at resonance, −3 dB) dB) points points of of each each resonance resonance magnitude magnitude should should be be identified. identified.
Figure 12. Frequency response function of the structural model to be identified.
For a particular mode, the damping ratio m can be found from Equation (49). Figure 13 illustrates the concept this method. Figure 12.of Frequency response function function of the structural model to be identified. Figure 12. Frequency response
b a
H (d )
, H (ratio ) H ) be found from Equation (49). (49) mthe damping m( For particularmode, mode, 13 For aa particular the damping ratio ζa m can bebcan found2from Equation (49). Figure 13 Figure illustrates 2 d illustrates the of this method. the concept ofconcept this method.
The accuracy of this method depends on the frequency resolution used for the measurement, H (d ) (ω )| For lightly damped since it determines how accurately the a a peak magnitude can be measured. d ,, H H ( ()ωb )| = | H√ mm = ωb b − ω (49) (toωa )measure | H( a )| = | Hbthe structures, high-resolutionζanalysis is required peak 2exactly. 22ω d d 2 Consequently, repeated measurements at each resonance frequency are normally required to achieve sufficient accuracy. The obtained resonances aremethod shown independs Table 2. onon The accuracy accuracy ofthis this method depends frequency resolution formeasurement, the measurement, of thethe frequency resolution usedused for the since
since it determines how accurately peak magnitude can be measured. lightlystructures, damped it determines how accurately the peakthe magnitude can be measured. For lightlyFor damped structures, high-resolution analysistoismeasure requiredthe to peak measure the peak exactly. Consequently, repeated high-resolution analysis is required exactly. Consequently, repeated measurements measurements at each resonance normally to achieveaccuracy. sufficient accuracy. The at each resonance frequency arefrequency normallyare required to required achieve sufficient The obtained obtained resonances resonances are shownare in shown Table 2.in Table 2.
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Figure 13. Quadrature peak picking method.
Figure 13. Quadrature peak picking method. Table 2. Natural frequency and damping ratio of each peak.
Table 2. Natural frequency and damping of each peak. Figure 13. Quadrature peak pickingratio method. Peak Number 1 2 3 4 5 6 7 8 Frequency (Hz) 14.0 30.0 112.0 539.0 896.07 1330.08 Table 2. Natural damping ratio of peak.6 Peak Number 1 2 frequency 3 and 281.0 4 5 each 833.0 Damping Ratio 0.0138 0.0222 0.0204 0.0074 0.0177 0.0058 0.0047 0.0053 Frequency (Hz) 14.0 30.0 112.0 Peak Number 1 2 3 4281.0 5539.0 6833.0 7 896.0 8 1330.0 Damping Ratio 0.0138 0.0222 0.0204 0.0074 0.0177 0.0058 0.0047 0.0053 A transfer function can be identified for the object structure based on the frequency response Frequency (Hz) 14.0 30.0 112.0 281.0 539.0 833.0 896.0 1330.0 data. The least square estimation of the linear system parameters is presented as a complex-domain Damping Ratio 0.0138 0.0222 0.0204 0.0074 0.0177 0.0058 0.0047 0.0053 formulation [51]. Figure 14 shows a general model of the frequency-domain identification of linear A transfer function can bebeidentified for the objectstructure structure based the frequency response A transfer function identified the object on on the ) , where systems. The system is can represented by itsfortransfer function H ( sbased Laplace s frequency j in theresponse data.data. The The leastleast square estimation of the linear system parameters is presented as a complex-domain square estimation of the linear system parameters is presented as a complex-domain domain. H ( s) has a rational form that is extended by a delay term e jtd : formulation Figure showsaageneral general model model of identification of linear formulation [51].[51]. Figure 1414 shows ofthe thefrequency-domain frequency-domain identification of linear m its transfer m 1 H ( s ) systems. system is represented by function , where in the Laplace s j systems. The The system is represented bynmits transfer function H ( s ) , where s = jω in the Laplace domain. s nm 1 s n1 s n0 jt ( s) thatn isbyextended term ed : e jtd : (50) − jωt H ( s) has a rational form by a delay n 1 H (s)domain. has a rational form that isHextended a delay term e d n s d n 1 s d1 s d 0 d
m
m 1
nma1 sFourier 1+series, jt n nm snmm s+nby sm− ·n·1 s· +nwith n 0 1 s+ The excitation signal isH represented amplitudes U i at angular 0 e complex (50) (50) H (s) =( s) n n m−1n n1 −1 · e− jωtd ddnn−11ssis Y s+ )U dd01. sBoth sd n s+ + ·d·(1· + d0the measured input and output H frequencies . The response ofdnthe system d
i
i
i
i
nu and complex amplitudes are iscorrupted by by noise termsseries, , respectively (errors ncomplex U i inatvariables The excitation signal aby Fourier with angularUi at y The excitation signal isrepresented represented a Fourier series, withamplitudes complex amplitudes model), which are usually assumed to be Gaussian, uncorrelated between the input and output, andinput i . The Yi H ( frequencies of the system Both and output angular frequencies ωi .response The response of theissystem isi )U Yii . = H (the ωi )measured Ui . Bothinput the measured uncorrelated between different frequency points. Based on the above assumptions, a maximumand complex output complex amplitudes are by corrupted by noise terms nu and ny , (errors respectively (errors in nu and amplitudes are corrupted noise terms in variables n , respectively likelihood estimate of the system parameters is developed in realy terms. variables model), which are usually assumed to be Gaussian, uncorrelated between the input model), which are usually assumed to be Gaussian, uncorrelated between the input and output, and and output, and uncorrelated between different frequency Based on the above assumptions, uncorrelated between different frequency points. Based points. on the above assumptions, a maximumlikelihood estimate ofestimate the system developed in terms. in real terms. a maximum-likelihood of parameters the systemisparameters is real developed
Figure 14. Frequency-domain identification of linear systems.
The time-domain measurements are transformed into the frequency domain, and the calculated Figure 14.14.Frequency-domain identificationofof linear systems. Figure Frequency-domain linear systems. i , i 1, , G are Umi and complex input and output amplitudes at selectedidentification angular frequencies
Ymi , respectively. The unknown parameters are the unknown coefficients of the numerator and the
time-domain measurementsare aretransformed transformed into domain, andand the calculated The The time-domain measurements intothe thefrequency frequency domain, the calculated denominator of the transfer functionat (Equation (49)), frequencies which are collected in, Gtheare vector and C. , i 1, complex input and output amplitudes selected angular mi complex input and output amplitudes at selected angular frequencies iωi , i = 1, · · · , GUare Umi and Ymi , respectively. unknownparameters parameters are are the of the numerator and the Ymi , respectively. TheThe unknown theunknown unknowncoefficients coefficients of the numerator and the denominator oftransfer the transfer function (Equation are collected the C. vector C. denominator of the function (Equation (49)), (49)), whichwhich are collected in the in vector Furthermore,
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Furthermore, the exact complex input and output amplitudes U and Y are also unknown. The basic equations can be written as follows for the exact values (Equation (50)) and for the noisy the exact complex input and observations (Equation (51)):output amplitudes U and Y are also unknown. The basic equations can be written as follows for the exact values (Equation (50)) and for the noisy observations (Equation (51)): Yi H (i , C)U i , i 1, 2, , G (51) Yi = H (ωi , C)Ui , i = 1, 2, · · · , G (51) Ymi H ( i , C ) (U mi nui ) n yi , i 1, 2, , G
(52)
Ymi = estimate H (ωi , C)of· (the Umiunknown − nui ) + nquantities ·· ,G (52) The maximum-likelihood is·equivalent to the minimization yi , i = 1, 2, of the following cost function: The maximum-likelihood estimate of the unknown quantities is equivalent to the minimization of the following cost function: G (U U )(U U ) G (Y Y )(Y Y ) i mi i mi i mi i J (U, Y, C) mi (53) , ! 2 2 G i 1 uiU − U ) i 1 G (Y 2−yi Y )(Y − Y ) (Umi − U2i )( mi i mi i mi i J (U, Y, C) = ∑ +∑ , (53) 2σui 2 2σyi 2 i =1 1 , G The upper Yi H (i , C)U i , i 1, 2,i= which is subject to the constraints bar means the complex
conjugate, and and standard output, respectively. Using yi are the which is subject toui the constraints Yi = H (ωi , Cdeviations )Ui , i = 1,of 2, ·the · · ,input G Theand upper bar means the complex conjugate, σui and σyi aare the standard deviations of the input and output, respectively. this above and methodology, transfer function with 9 zeros and 11 poles has been identified, Using which this has above methodology, 9 zeros and 11should poles has been identified, four four major peaks upatotransfer aroundfunction 500 Hz.with These four peaks be attenuated with which properhas control major peaks up to15 around 500 Hz. four peaks should be attenuated proper control methods. methods. Figure compares the These frequency response functions from thewith experiment and the system Figure 15 compares the frequency response functions from the experiment and the system identification, identification, and Equation (53) shows the formulation of the transfer function. This result seems to and Equation (53) shows the formulation theHz. transfer This result seems tofor fit checking with the fit with the experimental results for up toof 500 Figurefunction. 15 also shows the root locus experimental results for up 500 peak. Hz. Figure also of shows the root which poles are which poles are related totoeach Some15parts the locus golocus overfor thechecking right half plane, which related to each peak. Some parts of the locus go over the right half plane, which indicates instability. indicates instability.
(a)
(b)
Figure 15. Comparison of: (a) frequency response functions; and (b) root locus. Figure 15. Comparison of: (a) frequency response functions; and (b) root locus. 2 +s10s 6140 140s 5153000) s 2 )( s 2 )(10 s 2)(s10 106 s ( s 210 13530)( 6ss2+18250)( 418800)( s(ss2 + +13530 6s+18250 s2s+10s +418800 +5153000) .. (s) = 2 G( s) G 2 2 2 2 2 2 2 )(ss+7738)( 2.4s+s7738 8s+34360 s s+28s +477700 +3117000s)(2 s2120 +120s 11470000) 8)(s s + s26+s26s 3117000)( s )(28 s )( s + ( s 30)((ss+30 2.4 34360)( 477700)( 11470000)
(54) (54)
With the identified poles and zeros of the system, notch filters are designed to reduce the amplitude With the identified poles and zeros of the system, notch filters are designed to reduce the of each peak. A stop band filter (BSF) or a notch filter refers to a band-elimination or blocking filter amplitude of each peak. A stop band filter (BSF) or a notch filter refers to a band-elimination or that exhibits sharp attenuation characteristics in only a specific frequency band. Peak amplitudes blocking filter that exhibits sharp attenuation characteristics in only a specific frequency band. Peak can be reduced by adding zeros near poles to maintain the DC gain. Since two zeros are added for amplitudes can be reduced by adding zeros near poles to maintain the DC gain. Since two zeros are each peak, two poles should also be added, which is similar to replacing the damping ratio at each added for each peak, two poles should also be added, which is similar to replacing the damping ratio resonance frequency. These are the resonance frequencies and damping ratios of each peak. The filters at each resonance frequency. These are the resonance frequencies and damping ratios of each peak. are derived by maintaining ωn and changing the damping ratio ζ to 0.7. This is done to confirm that The filters are derived by maintaining n and changing the damping ratio to 0.7. This is done an actual solution exists and that enough control authority can be obtained from the actuators for the to confirm that an actual solution exists and that enough control authority can be obtained from the actual system. Table 3 shows the transfer functions of the notch filters based on Equation (54), in which actuators for the actual system. Table 3 shows the transfer functions of the notch filters based on the original damping ratio ζ z replaces the higher damping ratio ζ p at each resonance frequency so that Equation (54), in which the original damping ratio z replaces the higher damping ratio p at each the amplitudes at each peak can be attenuated. resonance frequency so that the amplitudes at each peak can be attenuated.
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s 2 2 z n s n 2 2 2 s 2s2 2+ 2ζ s n s+ z ω n ωn Gc (s) = 2 p n s + 2ζ p ωn s + ωn 2
Gc ( s )
(55) (55)
Table 3. Notch filter design.
Peak 1Peak 2
1 2
3
3
4 4 5
5
Resonance (Hz) Resonance 13.8478 (Hz) 13.8478
28.6550
28.6550
Table 3. NotchRatio filter design. Damping Damping 0.7000 Ratio 0.7000
0.7000
0.7000
109.0439
0.7000
280.1127 280.1127
0.7000 0.7000
109.0439
538.0233
538.0233
0.7000
0.7000
0.7000
Notch Filter s 2 2.4s 7570 Gc 5 2Notch Filter s 121.8s 7570 2 2s + 2.4s + 7570 Gc5 = s 2 8s 32420 Gc 4 2 s + 121.8s + 7570 2 + 8ss + s s252.1 32420 32420 Gc4 = 2 2 + 32420 s s 2+28252.1s s 469400 Gc 3 2 s + 28s + 469400 Gc3 = s 2 959.2s 469400 s + 959.2s + 469400 s 2 s2 26 s 3098000 + 26s + 3098000 GcG2 c2 =2 2 + 2464s 3098000 s s2 2464 s + 3098000 s + 120s + 11430000 Gc1 =s 2 2 120s 11430000 Gc1 2 s + 4733s + 11430000 s 4733s 11430000
As shown shown in in Figure Figure 16, 16, the the amplitudes amplitudes of of the the peaks peaks are are significantly significantly reduced reduced compared compared to to the the original system, and the root locus of the filtered system has been changed. The part in the right-hand original system, and the root locus of the filtered system has been changed. The part plane plane is is automatically automatically removed removed by by these these filters filterswith withthe theadded addedzeros zerosnear nearpoles. poles.
(a)
(b)
Figure 16. Notch filter control result: (a) frequency response functions; and (b) changed root locus. Figure 16. Notch filter control result: (a) frequency response functions; and (b) changed root locus.
The control results are also described in the time domain. Numerical simulations were The control results are also described inthe thesystem. time domain. Numerical were conducted conducted with impulse and step inputs to Figures 17 and 18simulations show the impulse response with impulse and step inputs to the system. Figures 17 and 18 show the impulse response andvibration the step and the step response of the structure. The notch filters show dramatically reduced response of the structure. The notch filters show dramatically reduced vibration amplitudes compared amplitudes compared to the case with no control. However, these filters will work on only the to the case with no assuming control. However, these filters will work on only the baseline structure, assuming baseline structure, that the identified model is accurate. that the identified model is accurate.
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(a) (a)
(b) (b)
Figure 17. Notch filter control result: (a) tracking error of impulse response; and (b) zoomed plot of Figure 17. Notch filter control result: (a) tracking error of impulse response; and (b) zoomed plot of Figure 17. Notch filter control result: (a) tracking error of impulse response; and (b) zoomed plot of (a). (a). (a).
(a) (a)
(b) (b)
Figure 18. Notch filter control result: (a) tracking error of step response; and (b) zoomed plot of (a). Figure Figure 18. Notch Notch filter filter control control result: (a) tracking error of step response; and (b) zoomed plot of (a).
In a real situation, the exact model of the system cannot be known due to changes in the In realsituation, situation, exact model ofsystem the system cannot be due known due toinchanges in the In aareal thethe exact model of the be known to changes operational operational environment, and the plant may vary cannot over time. To incorporate the factthe that the plant operational environment, and the plant may vary over time. To incorporate the fact that the plant environment, and plant may vary over To incorporate the fact that that an theadditional plant willmass change and will change and tothe highlight the efficacy of time. the controller, it is assumed (about will change and to highlight the efficacy of the controller, it is assumed that an additional mass (about to highlight the efficacy of the controller, is assumed that an additional mass a 20% increase) is attached to the originalitsystem, as mentioned in Section 4. (about a 20% increase) ais20% increase) isoriginal attachedsystem, to the original system, as mentioned in Section 4. attached to the as mentioned in Section 4. Since an added mass implies decreased resonance frequencies, a modified transfer function Since anadded addedmass mass implies decreased resonance frequencies, a modified transfer function Since an implies decreased resonance modified function would would exist with reduced resonance frequencies forfrequencies, each peak. aIn the nexttransfer section, the SM-LMS would exist with reduced resonance frequencies for each peak. Insection, the next section, the SM-LMS exist with reduced resonance frequencies for each peak. In the next the SM-LMS algorithm’s algorithm’s target-tracking performance is observed for vibration attenuation using the baseline and algorithm’s target-tracking performance is observed for vibration attenuation using the baseline and target-tracking modified beam performance structures. is observed for vibration attenuation using the baseline and modified modified beam structures. beam structures. 6. Experiments 6. 6. Experiments Experiments Several tests were performed with the cantilevered flexible beam. The tests are divided into two Several flexible beam. beam. The Several tests tests were were performed performed with with the the cantilevered cantilevered flexible The tests tests are are divided divided into into two two cases of vibration reduction control: one for the baseline structure of the beam, and one for a beam cases of vibration reduction control: one for the baseline structure of the beam, and one for a beam cases of vibration reduction control: one for the baseline structure of the beam, and one for a beam with variable mass. The LMS and SM-LMS algorithms are applied to compare the vibration reduction with variable with variable mass. mass. The The LMS LMS and and SM-LMS SM-LMS algorithms algorithms are are applied applied to to compare compare the the vibration vibration reduction reduction performance. The control parameters were tuned to obtain the best results. Figure 19 shows the performance. Thecontrol controlparameters parameters were tuned to obtain the results. best results. 19 shows the performance. The were tuned to obtain the best FigureFigure 19 shows the control control results for the baseline beam structure. The plot on the left indicates the strain change of the control results for the baseline beam structure. The plot on the left indicates the strain change of the results for the baseline beam structure. The plot on the left indicates the strain change of the aluminum aluminum beam in the time domain from the PZT sensor outputs. Note that the vibration stops in 0.2 aluminum in the time domain from the PZT sensor outputs. Note that the vibration stops 0.2 beam in thebeam time domain from the PZT sensor outputs. Note that the vibration stops in 0.2 to 0.3 sinwith to 0.3 s with LMS and SM-LMS, whereas it takes longer than 1 s to stop without a controller. The plot to 0.3 s with LMS and SM-LMS, whereas it takes longer than 1 s to stop without a controller. The plot LMS and SM-LMS, whereas it takes longer than 1 s to stop without a controller. The plot on the right on the right compares the frequency response plots between open-loop and closed-loop cases, and on the right frequency plots between closed-loop and compares thecompares frequencythe response plotsresponse between open-loop and open-loop closed-loopand cases, and some cases, amplitude some amplitude reductions can be seen for each peak frequency. some amplitude can peak be seen for each peak frequency. reductions can bereductions seen for each frequency.
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(a)
(b)
Figure 19. Experimental result for the baseline structure: (a) time domain; and (b) frequency domain.
Figure 19. Experimental result for the baseline structure: (a) time domain; and (b) frequency domain.
After 0.3 s, the vibration amplitudes with both control algorithms are very similar, but LMS shows0.3 worse isolation performance thancontrol SM-LMS. The frequency response shows a more After s, thevibration vibration amplitudes with both algorithms are very similar, but LMS shows obvious performance difference between the two methods, as shown in Table 4. worse vibration isolation performance than SM-LMS. The frequency response shows a more obvious performance difference between the two methods, as shown in Table 4. Table 4. Vibrational amplitudes of each case for the original beam structure.
Table 4. Vibrational amplitudes of each case Amplitude for the original beam structure. Control Peak Open-Loop With LMS With SM-LMS Amplitude 1st PeakControl (15 Hz) 0.0008709 (12.6 dB↓) 0.0005123 (17.2 dB↓) 0.0037290 2nd Peak (33 Hz) 0.0008719 (2.7 dB↓) 0.0005129 (7.4 dB↓) 0.0011960 Open-Loop With LMS With SM-LMS Peak 3rd Peak (117 Hz) 0.0048300 (8.9 dB↑) 0.0028410 (4.3 dB↑) 0.0017240 1st Peak (15 Hz) 0.0037290 0.0008709 (12.6 dB↓) 0.0005123 (17.2 dB↓) 4th Peak Peak (33 (312Hz) Hz) 0.0041200 0.0024240 (1.2 0.0027970 2nd 0.0011960 0.0008719 (2.7(3.4 dBdB↑) ↓) 0.0005129 (7.4 dBdB↓) ↓) 4 4 4 Overall RMS (0.8 dB↑) (3.9 3.3386 10 3.6430 10 2.1429 10 3rd Peak (117 Hz) 0.0017240 0.0048300 (8.9 dB↑) 0.0028410 (4.3 dBdB↓) ↑) 4th Peak (312 Hz) 0.0027970 0.0041200 (3.4 dB↑) 0.0024240 (1.2 dB↓) There is excellent attenuation of mode 1 and slight attenuation of mode 2 with LMS,−while higher Overall RMS 3.3386 × 10−4 3.6430 × 10−4 (0.8 dB↑) 2.1429 × 10 4 (3.9 dB↓) frequencies show slight amplifications. SM-LMS shows excellent attenuation of modes 1 and 2 and slight attenuation of mode 4, while mode 3 shows slight amplification. At every peak, the difference in attenuated amplitude between two1 methods is about 3 to 5 dBs. The amplitudes of while 1st, 2nd, There is excellent attenuation of the mode and slight attenuation of mode 2 with LMS, higher and 4thshow modalslight frequencies are reduced, and the 3rd modal frequency has some spill-over SMfrequencies amplifications. SM-LMS shows excellent attenuation of modeswith 1 and 2 and LMS. However, with LMS, the 3rd and 4th peaks have spillovers, and the amounts are higher than slight attenuation of mode 4, while mode 3 shows slight amplification. At every peak, the difference in in the amplitude case with SM-LMS. observing theisvibration in broadband, the SM-LMS attenuated betweenWhen the two methods about 3 level to 5 dBs. The amplitudes of 1st,shows 2nd, and better vibration isolation performance, thus reducing the overall RMS value. 4th modal frequencies are reduced, and the 3rd modal frequency has some spill-over with SM-LMS. Next, to incorporate the fact that the plant will change and to highlight the efficacy of the However, with LMS, the 3rd and 4th peaks have spillovers, and the amounts are higher than in the case controller, an additional mass is attached to the original system (increasing the total weight by about with SM-LMS. When observing in broadband, theFigure SM-LMS shows better mass vibration a 20%). A bracket is fastenedthe nearvibration the tip oflevel the beam, as shown in 11. Since an added isolation performance, thus reducing the overall RMS value. implies decreased resonance frequencies, a modified frequency response with reduced resonance Next, to incorporate the fact that the plant will change and to highlight the efficacy the cases. controller, frequencies for each peak is obtained to measure the performance of the controller in of these Note that mass the SM-LMS is notto related to the original dynamics andweight can make system an additional is attached the original systemstructural (increasing the total bythe about a 20%). have desired dynamics. Moreover, the parameters of the SM-LMS used for the baseline structures are A bracket is fastened near the tip of the beam, as shown in Figure 11. Since an added mass implies maintained for this case of control in order to observe the robustness of reduced the controller. decreased resonance frequencies, a modified frequency response with resonance frequencies Again, the time-domain responses caused by impulsive disturbances show significant reduction for each peak is obtained to measure the performance of the controller in these cases. Note that the visible, as shown in Figure 20. Furthermore, the amplitudes of each resonance peak and other SM-LMS is not related to the original structural dynamics and can make the system have desired frequencies are decreased, regardless of the significant change in the plant dynamics. Similarly, the dynamics. Moreover, the parameters of the SM-LMS used for the baseline structures are maintained vibration stops in 0.4 to 0.5 s with LMS and SM-LMS, whereas it takes longer than 1 s without a for this case of control to observe of the controller. controller. The plotin onorder the right comparesthe therobustness frequency response between the open-loop and closedAgain, the time-domain responses caused by impulsive disturbances show significant reduction loop cases, in which some amplitude reductions can be seen for each peak frequency.
visible, as shown in Figure 20. Furthermore, the amplitudes of each resonance peak and other frequencies are decreased, regardless of the significant change in the plant dynamics. Similarly, the vibration stops in 0.4 to 0.5 s with LMS and SM-LMS, whereas it takes longer than 1 s without a controller. The plot on the right compares the frequency response between the open-loop and closed-loop cases, in which some amplitude reductions can be seen for each peak frequency.
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(a)
(b)
Figure 20. Experimental result for the modified structure: (a) time domain; and (b) frequency domain. Figure 20. Experimental result for the modified structure: (a) time domain; and (b) frequency domain.
After 0.5 s, the vibration amplitudes for both control algorithms are very similar, but LMS shows After 0.5 s, the vibration amplitudes forSM-LMS. both control very similar, LMSobvious shows worse vibration isolation performance than Thealgorithms frequency are response shows but a more worse vibration isolation performance than SM-LMS. The frequency response shows a more obvious performance difference between the two methods, as indicated in Table 5. LMS shows excellent performance two methods, as modes. indicated in Table 5. LMS shows excellent attenuation ofdifference mode 1 butbetween has verythe little effect on higher SM-LMS shows excellent attenuation attenuation of mode 1 but has very little effect on higher modes. SM-LMS shows excellent attenuation of mode 1 and slight attenuation of modes 2 and 3, while mode 4 shows very slight amplification. At of mode 1 and attenuation of modes 2 and 3, while mode shows very slight amplification. At every peak, theslight difference in attenuated amplitude between the 4two methods is about 2 to 3 dBs. The every peak, the difference in attenuated amplitude between the two methods is about 2 to 3 dBs. The amplitudes of the 1st, 2nd, and 3rd modal frequencies are reduced, and the 4th modal frequency has amplitudes of the 1st,SM-LMS. 2nd, and 3rd modal with frequencies are2nd reduced, and the 4th modal frequency some spill-over with However, LMS, the and 4th peaks have spillovers, and has the some spill-over with SM-LMS. However, with LMS, the 2nd and 4th peaks have spillovers, and the amounts are higher than in the case with SM-LMS. When observing the vibration level in broadband, amounts are higher in the case with SM-LMS. When observing the vibration level inoverall broadband, the SM-LMS showsthan better vibration isolation performance, thus showing a reduced RMS the SM-LMS shows better vibration isolation performance, thus showing a reduced overall RMS value, value, although it is 1.1 dB lower than the baseline case. although it is 1.1 dB than the baseline case. As shown in lower the simulation results, the time-domain responses caused by impulsive disturbances are significantly reduced, and the amplitude of each resonance peak is nearly flattened. Table 5. Vibrational amplitudes of each case for the original beam structure. These results verify that the SM-LMS method can reduce the vibration of the cantilevered beam considerably, even though the structural dynamics is dramatically changed with the additional Control Amplitude impedance. Peak
Open-Loop
With LMS
With SM-LMS
1st Peak (15 Hz) 5. Vibrational 0.0022920 0.0007087 dB↓) beam0.0004849 Table amplitudes of each case for(10.2 the original structure.(13.5 dB↓) 2nd Peak (33 Hz) 0.0013280 0.0016330 (1.8 dB↑) 0.0011170 (1.5 dB↓) Amplitude Control 3rd Peak (117 Hz) 0.0045400 0.0037110 (1.8 dB↓) 0.0025390 (5.0 dB↓) 4th Peak (312 Hz) 0.0028750 0.0046440 dB↑) 0.0031780 dB↑) Peak Open-Loop With(4.2 LMS With(0.9 SM-LMS Overall RMS 3.0888 × 10−4 3.2784 × 10−4 (0.5 dB↑) 2.2431 × 10−4 (2.8 dB↓)
1st Peak (15 Hz) 0.0007087 (10.2 dB↓) 0.0004849 (13.5 dB↓) 0.0022920 2nd Peak (33 Hz) 0.0016330 (1.8 dB↑) 0.0011170 (1.5 dB↓) 0.0013280 As shown in the simulation results, the time-domain responses caused by 0.0025390 impulsive (5.0 disturbances 3rd Peak (117 Hz) 0.0037110 (1.8 dB↓) dB↓) 0.0045400 are significantly reduced, and the amplitude of each resonance peak is nearly flattened. These results 4th Peak (312 Hz) 0.0046440 (4.2 dB↑) 0.0031780 (0.9 dB↑) 0.0028750 4 4 4 verify that the SM-LMS method cantilevered beam even Overall RMS dB↑) 3.0888can 10reduce the vibration 3.2784 of 10 the(0.5 2.2431considerably, 10 (2.8 dB↓) though the structural dynamics is dramatically changed with the additional impedance. 7. Conclusions and Future Work 7. Conclusions and Future Work An enhanced LMS algorithm has been proposed and applied to an adaptive digital filtering An enhanced LMS algorithm has been proposed and applied to an adaptive digital filtering system combined with sliding mode control. The method benefits from the nonlinear control system combined with sliding mode control. The method benefits from the nonlinear control technique technique inside the control system structure for better adaptability and tracking performance. This inside the control system structure for better adaptability and tracking performance. This adaptive adaptive filtering system was used for the vibration reduction and control of a continuous smart filtering system was used for the vibration reduction and control of a continuous smart structure with structure with piezoelectric patches attached to it. The SM-LMS method allows for significant and piezoelectric patches attached to it. The SM-LMS method allows for significant and robust vibration robust vibration reduction for not only the original structure, but also for a beam with modified dynamics. Thus, it can be a good solution for reducing the vibration of structures with unknown dynamics.
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reduction for not only the original structure, but also for a beam with modified dynamics. Thus, it can be a good solution for reducing the vibration of structures with unknown dynamics. The contributions of this research are summarized as: First, the proposed control system has improved stability, robustness, and capability to manage signals with multiple spectra with increasing complexity and frequency of components to be controlled in the output of the disturbance plant. Numerical analyses demonstrated that the method has adequate tracking performance. Second, a smart structure with piezoelectric patches was constructed and its dynamic model has been identified with frequency domain identification method; third, a conventional control method has been validated numerically in the case of active vibration control and showed limitations. Third, the method was validated for the original structure and a modified structure. The controller had acceptable tracking performance in both cases. This work could be validated with several real-world applications in active vibration control, such as vehicle vibration reduction and vibration attenuation for engineering structures. In addition, a hybrid controller could be proposed to control two or more directions simultaneously for realistic applications by effectively managing an extensive range of multi-spectral signals. Self-sensing actuator could be implemented with one PZT material, which acts as sensor and actuator simultaneously. Intelligent control methods such as neural networks could also be more adaptive and robust to changeable system dynamics. Acknowledgments: This paper was supported by the National Research Foundation of Korea (NRF) grant funded by the government of Korea (Ministry of Science, ICT & Future Planning) (No. 2016R1C1B1010888) as well as by the 2017 Yeungnam University Research Grant (217A380020). Author Contributions: Byeongil Kim and Jong-yun Yoon initiated and developed the ideas related to this research work. Byeongil Kim and Jong-yun Yoon developed novel methods, derived relevant formulations, and carried out performance analyses and numerical analyses. Byeongil Kim wrote the paper draft under Jong-yun Yoon’s guidance and Byeongil Kim finalized the paper. Conflicts of Interest: The authors declare no conflict of interest.
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