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A UNIFIED ADAPTIVE ACTIVE CONTROL MECHANISM FOR NOISE CANCELLATION AND VIBRATION SUPPRESSION
Mechanical Systems and Signal Processing (1996) 10(6), 667–682
A UNIFIED ADAPTIVE ACTIVE CONTROL MECHANISM FOR NOISE CANCELLATION AND VIBRATION SUPPRESSION M. O. T Department of Automatic Control and Systems Engineering, The University of Sheffield, Sheffield, U.K.
M. A. H Department of Computer Science, University of Dhaka, Dhaka, Bangladesh (Received August 1994, accepted February 1996) This paper presents the design and implementation of an active control mechanism for noise cancellation and vibration suppression within an adaptive control framework. A control mechanism is designed within a feedforward control structure on the basis of optimum cancellation at an observation point. The design relations are formulated such that to allow on-line design and implementation and thus result in a self-tuning control algorithm. The algorithm is implemented on an integrated digital signal processing (DSP) and transputer system. Simulation results verifying the performance of the algorithm are presented and discussed. 7 1996 Academic Press Limited
1. INTRODUCTION
Noise and vibration have long been recognised as sources of environmental pollution, having numerous adverse effects on human life. Many attempts have been made in the past to devise methods of tackling the problems arising from unwanted noise and vibration. Traditional methods of noise cancellation and vibration suppression utilise passive control techniques which consist of mounting layers of passive material on or around the source. Investigations have shown that these methods are efficient at high frequencies but expensive and bulky at low frequencies [1, 2]. Active noise/vibration control consists of artificially generating cancelling sources to interfere destructively with the unwanted source and thus result in a reduction in the level of the noise/vibration (disturbances) at desired locations. Active noise/vibration control is not a new concept. It is based on principles that were initially proposed by Lueg in the early 1930s for noise cancellation [3]. Since then a considerable amount of research has been devoted to the development of methodologies for the design and realisation of active noise/vibration control systems in various applications. Not much work has been done in investigating the problems of noise and vibration together [4]. In many practical applications, the problems of noise and vibration are found to be closely related. For example, a close coherence is observed in numerous situations between the two due to secondary effects [5]. Many sources of noise, for instance, vibrate continuously while in operation and the vibration is found to be coherent with the acoustic 667 0888–3270/96/060667 + 16 $25.00/0
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waves they emit. On other occasions, it is often noticed in buildings, for instance, that noise due to chattering of windows or motion of articles is caused through ground vibrations by passing trains and/or vehicles. In these cases a control solution aimed at reducing, for example, the level of vibration will result in significant reduction in the level of noise and, to some extent, vice versa [6]. Moreover, the control of noise and vibration by active means is based on the same design principle. These form the basis of design unification in this work by adopting a systems approach for the development of an active control strategy for both noise cancellation and vibration suppression. Thus, the method proposed in this work can be utilised to tackle the problem of noise and vibration either in isolation or together. The results presented in this paper, however, correspond to the former. Active noise/vibration control is realised by detecting and processing the noise/vibration by a suitable electronic controller so that when superimposed on the disturbances, cancellation occurs. Due to the broadband nature of these disturbances, it is required that the control mechanism realises suitable frequency-dependent characteristics so that cancellation over a broad range of frequencies is achieved [1, 5]. In practice, the spectral contents of these disturbances as well as the characteristics of system components are in general subject to variation, giving rise to time-varying phenomena. This implies that the control mechanism is further required to be intelligent enough to track these variations, so that the desired level of performance is achieved and maintained. Through his experiments of reducing transformer noise, Conover was the first to realise the need for a ‘black box’ controller that would adjust the cancelling signal in accordance with information gathered at a remote distance from the transformer [7]. Later it has been realised by numerous authors that an essential requirement for an active noise control (ANC) system to be practically successful is to have an adaptive capability [1, 8–13]. Implementing an adaptive control algorithm within the ANC system will allow the system to adjust the controller characteristics in accordance with changes in the noise pattern of the source. Chaplin and his associates [9, 14–16] have reported some success with an adaptive scheme based on a trial and error waveform generation. This method relies on the source noise being periodic. The method has been tested with rotary machines such as a motor vehicle, fan, etc. where the noise spectrum is harmonically related to the fundamental (engine firing) frequency. The engine firing frequency provides a clock signal to the algorithm. A noticeable amount of theoretical and practical investigations have subsequently been reported in this area [10–12, 17–26]. Among these, the scheme developed by Eriksson and his co-workers relies on the development of a model of the source. The scheme reported by Nelson, Elliott and co-workers is based on the minimisation (in the least square sense) of the sound level at discrete locations in the medium. The control scheme reported here, however, is based on optimum cancellation of noise/vibration. This paper presents an investigation into the development of an active control mechanism for noise cancellation and vibration suppression within an adaptive control framework. An ANC system is designed on the basis of optimum cancellation of broadband noise at an observation point in a linear 3-D propagation medium. The controller design relations are formulated such that to allow on-line design and implementation and, thus, yield a self-tuning control algorithm. The algorithm is implemented on an integrated DSP-transputer system and its performance assessed within a simulated three-dimensional free-field medium. A flexible beam system in transverse vibration is considered for vibration suppression. Such a system has an infinite number of resonance modes although in most cases, when considering a vibration problem, the lower modes are the dominant ones requiring attention. In considering a vibro-acoustic problem, however, it is likely that higher modes
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will also be contributing significantly to the noise level. In these circumstances a consideration of these modes up to a practically reasonable number will be required. First-order central finite difference (FD) methods are used to study the behaviour of the beam and develop a suitable test and verification platform. The unwanted vibrations in the structure are assumed to be caused by a single point disturbance of broadband nature. These are detected and suitable suppression signals generated via a point actuator to yield vibration suppression over a broad frequency range along the beam. The self-tuning control algorithm developed within the ANC system is implemented on the integrated DSP-transputer system and its performance assessed.
2. ACTIVE NOISE CONTROL
A schematic diagram of the geometrical arrangement of a feedforward ANC structure is shown in Fig. 1(a). An unwanted (primary) point source emits broadband noise into the propagation medium. This is detected by a detector located at a distance re relative to the primary source, processed by a controller of suitable transfer characteristics and fed to a cancelling (secondary) point source located at a distance d relative to the primary source and a distance rf relative to the detector. The secondary signal thus generated is superimposed on the primary signal to achieve cancellation of the noise at and in the vicinity of an observation point located at distances rg and rh relative to the primary and secondary sources respectively. A frequency-domain equivalent block diagram of the ANC structure is shown in Fig. 1(b), where E, F, G and H are transfer functions of the acoustic paths through the distances re , rf , rg and rh respectively. M, M0 , C and L are transfer characteristics of the detector, the observer, the controller and the secondary source respectively. UD and UC are the primary and secondary signals at the source locations whereas YOD and YOC are the corresponding signals at the observation point. UM is the detected signal and YO is the observed signal. The block diagram in Fig. 1(b) can be thought of either in the continuous frequency (s) domain or the discrete frequency (z) domain. Therefore, unless specified, the analysis and design developed in this paper apply to both the continuous-time and the discrete-time domains. The control structure in Fig. 1 corresponds to that proposed by Lueg in his patent where the time delay is implemented by the physical separation between the primary and secondary sources [3]. A significant amount of consideration has subsequently been given to this strucuture in various applications. Conver [7], Hesselman [27] and Ross [28] have employed this structure in the cancellation of transformer noise. Ross has considered the design, Roure has analysed the stability and Eriksson, et al. have considered the implementation of this structure in the cancellation of the one-dimensional duct noise [13, 21–23, 29]. Nelson et al. have analysed the performance of this structure in the cancellation of enclosed sound fields [30–32]. Tokhi and Leitch have considered the design, stability, performance and implementation of this structure in the cancellation of noise in three-dimensional propagation medium [1–2, 5, 26, 33–34]. The detected and observed signals, namely UM and YO in Fig. 1(b), can each be thought of as composed of the sum of two signals: one due to the primary source and one due to the secondary source. Using the block diagram in Fig. 1(b), these can be expressed as UM = MEUD + MFUC YO = MO GUD + MO HUC
(1)
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Figure 1. Active noise control structure. (a) Schematic diagram; (b) block diagram.
The secondary source signal UC , on the other hand, can be expressed, through the controller path, as UC = CLUM
(2)
For complete cancellation of the noise to be achieved at the observation point the condition YO = 0
(3)
must be satisfied. This is equivalent to the minimum variance design criterion in a stochastic environment [35–37]. This requires the primary and secondary signals at the observation point to be equal in amplitudes and have a phase difference of 180° relative to each other. Thus, synthesising the controller within the block diagram of Fig. 1(b) on the basis of this objective yields C=
G ML(FG − EH)
(4)
This is the required controller transfer function for optimum cancellation of broadband noise at the observation point. In designing the controller, causality is ensured by making the number of zeros in C either less than or equal to the number of its poles. This is achieved by using a suitable realisation structure for the controller transfer characteristics. Note in equation (4) that, for given secondary source and detector, the controller characteristics are determined by the geometric arrangement of system components. This incorporates the location of the detector and the observer with respect to the primary and
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secondary sources. Among these, there exist arrangements that will lead to FG − EH, in the denominator of equation (4), approaching zero and thus requiring the controller to have an impractically large gain. Thus, an analysis of the system on this basis yielding the locus of detection and observation points for which the controller will be required to have an impractically large gain for optimum cancellation is important at a design stage. Moreover, note in Fig. 1(b) that the secondary signals reaching the detector form a positive feedback loop that can cause the system to become unstable for certain geometrical arrangements of system components. Therefore, further analysis of the system from a stability point of view leading to a robust design of the system is important at a design stage. Tokhi and Leitch have analysed and studied both these issues for ANC systems in 3-D non-dispersive propagation media [5, 33]. 2.1. - In practice, the characteristics of sources of noise vary due to operating conditions leading to time-varying spectra. Moreover, the characteristics of transducers, sensors and other electronic equipment used are subject to variation due to environmental effects, ageing, etc. To design an ANC system so that the controller characteristics are updated in accordance with these changes in the system, a self-tuning control strategy, allowing on-line design and implementation of the controller, can be utilised such that the required performance is achieved and maintained. Self-tuning control is distinguished as a class of adaptive control mechanisms [35–37]. It essentially consists of the processes of identification and control, both implemented on-line. The identification process is mainly concerned with on-line modeling of the plant being controlled and, thus, should incorporate a suitable system identification algorithm. The control process, on the other hand, is concerned with the design and implementation of the controller using the plant model and, thus, should incorporate a suitable controller design criterion. Consider the system in Fig. 1 as a single-input single-output system with the detected signal, UM , as input and the observed signal, YO , as output. Moreover, owing to the state of the secondary source, let the system behaviour be characterised by two sub-systems, namely, when the secondary source is off, with an equivalent transfer function denoted by Q0 , and when the secondary source is on, with an equivalent transfer function denoted by Q1 . Using the block diagram of Fig. 1(b), these can be obtained as Q=
Mo G , ME
Q1 =
$
Mo G ML(FG − EH) 1− ME G
%
(5)
Note that in obtaining Q0 and Q1 in equation (5) the controller block in Fig. 1 is replaced simply by a switch; prior to an estimation/measurement process the controller transfer function is not known. Note further that, the primary source is on throughout the process of estimating/measuring Q0 and Q1 . Manipulating equation (5) and using equation (4) yields an equivalent design relation for the controller in terms of Q0 and Q1 as
$
C= 1−
%
Q1 Q0
−1
(6)
This is the required controller design rule given in terms of transfer characteristics Q0 and Q1 which can be measured/estimated on-line. An on-line design and implementation of the controller can thus be achieved as
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(a) obtain Q0 and Q1 using a suitable system identification algorithm, for example a recursive least squares (RLS) parameter estimation algorithm; (b) use equation (6) to calculate the controller transfer function; and (c) implement the controller on a digital processor. Moreover, to monitor system performance and update the controller characteristics upon changes in the system, a supervisory-level control can be realised. The supervisor can be designed to monitor system performance on the basis of a pre-specified quantitative measure of cancellation as an index of performance, so that if the cancellation achieved is within the specified range then the algorithm implementation is to remain at the control level, step (c) above. However, if the cancellation is outside the specified range then self-tuning is to be reinitiated at the identification level, step (a) above. This results in a self-tuning ANC mechanism as depicted in Fig. 2. The ‘plant’ in Fig. 2 designates the system in Fig. 1 between the detection and observation points, in which during the identification process the controller is replaced by a switch. During the control process, however, the estimated controller transfer function is implemented to replace the switch. In implementing the self-tuning control algorithm described above, several issues of practical importance need to be given careful consideration. These include properties of the disturbance signal, robustness of the estimation and control, system stability and processor-related issues such as word length, speed and computational power [2]. In employing the minimum variance design criterion, a problem commonly encountered is that of instability of system, specially, when non-minimum phase models are involved [35–37]. A model is non-minimum phase if it has one or more zeros outside the stability region. The process of minimum variance design involves cancellation of open-loop plant zeros. Thus, in controlling a non-minimum phase plant, the estimated plant model will also be non-minimum phase. This will require the cancellation of the plant zeros with the estimated model zeros to be exact. However, due to finite processor word lengths and
Figure 2. Self-tuning controller.
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computational errors the cancellation is not often exact, leading to an unstable closed-loop system. Note also in the controller design rules, equation (6), that such a situation will result in a non-minimum phase and an unstable controller with the unstable poles approximately cancelling the corresponding zeros that are outside the stability region. Thus, to avoid this problem of instability either the estimated models can be made minimum-phase by reflecting those zeros that are outside the stability region into the stability region and using the resulting minimum-phase models to design the controller, or once the controller transfer function has been obtained, the poles and zeros that are outside the stability region can be reflected into the stability region. In this manner, a factor (1 − pz−1) corresponding to a pole/zero at z = p, in the complex z-plane, that is outside the stability region can be reflected into the stability region by replacing the factor with (p − z−1). The supervisory-level control described above is used as a performance monitor. In addition to monitoring system performance, it can also be facilitated with further levels of intelligence. For example, monitoring system stability and avoiding problems due to non-minimum phase models as described above, verifying controller characteristics on the basis of practical realisation to make sure that impractically large controller gains are not required as discussed earlier, system behaviour in a transient period, model structure validation, etc. 2.2. To assess the performance of the self-tuning ANC algorithm a simulation environment characterising the structure in Fig. 1 was constructed using practical measurements of noise propagation in a 3-D medium. The propagation medium is assumed to be non-dispersive (linear), where a sound wave in propagating through a distance r is attenuated in amplitude inversely with r and delayed in phase directly with the product of r and the frequency of the wave. The experimental set-up for construction of the simulation environment consisted of loudspeaker and a microphone at a set distance in front of it. The loudspeaker was driven by a pseudo-random binary sequence (PRBS) signal from a signal generator and the frequency response between the loudspeaker input and the microphone output was measured. The measured data thus obtained is the equivalent transfer characteristics of the loudspeaker L, microphone M, the associated drive and pre-amplifiers and the acoustic path between the loudspeaker and the microphone. Noting in equations (4) and (5) that the transfer characteristics of the acoustic paths in Fig. 1, conceptually, appear in cascade with the characteristics of the detector and the secondary source, the measured frequency response can be used to obtain the frequency response of the acoustic paths in Fig. 1(a) for known distances and hence calculate signal propagation through the system within the block diagram in Fig. 1(b). In this manner, a simulation environment characterising the system within the propagation medium is constructed. A fast Fourier transform algorithm is utilised to transform the signals concerned from the time domain into the frequency domain and vice versa as required. The self-tuning ANC algorithm was implemented on an integrated DSP-transputer system incorporating an i860 DSP device and a T805 transputer using DSP and parallel processing (PP) methods to allow efficient and relatively accurate implementation. The DSP device with its extensive signal processing power is 64-bit processor and is capable of 80 million floating-point operations per second (Mflops). The transputer, on the other hand, with its powerful communication facility, is a 32-bit processor capable of 4 Mflops. The ANSI C toolset is used as a common software for program development and implementation.
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Figure 3. Performance of the self-tuning ANC system. (a) Power spectral density of the noise; (b) attenuation in spectral density.
To assess the performance of the algorithm a broadband (0–512 Hz) PRBS signal was used as the unwanted primary noise. Figure 3(a) shows the autopower spectral density of the noise at the observation point before cancellation as GDo (v) and after cancellation as Goo (v). Figure 3(b) shows the difference GDo (v) − Goo (v), i.e. the actual attenuation in the spectral density of the noise. It is noted that an average broadband cancellation level of 20–25 dB below 150 Hz and more than 40 dB cancellation above 150 Hz is achieved. As compared with previous implementations, using DSP devices only, this significant level of cancellation over a broad range of frequencies is due to the computing power achieved by the utilisation of DSP and PP methods.
3. ACTIVE VIBRATION CONTROL
Active vibration control (AVC) is realised in a similar manner as ANC. The design of an AVC system depends upon the complexity of the structure under consideration and the nature of the disturbance process. A flexible beam system is considered in transverse vibration in a fixed-free form. Such a system has an infinite number of resonance modes although in most cases the lower modes are the dominant ones requiring attention. A schematic diagram of the AVC system, incorporating a cantilever beam, is shown in Fig. 4. The unwanted vibrations in the structure are assumed to be caused by a single point disturbance force (UD ) of broadband nature. These are detected by a point detector and processed by a controller to generate suitable suppression signals (UC ) via a point actuator to yield vibration suppression over a broad frequency range at an observation point along
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Figure 4. Schematic diagram of the AVC structure.
the beam. A frequency domain equivalent block diagram of the AVC system in Fig. 4 will give rise to that of the ANC system in Fig. 1(b), with a similar interpretation of the transfer functions and signals involved. In this manner, the required controller transfer function for optimum vibration suppression at the observation point is, therefore, given as in equation (4) with the corresponding equivalent relation suitable for on-line design and implementation as in equation (6). Therefore, a similar formulation of the self-turning control algorithm developed for noise cancellation applies to vibration suppression in Fig. 4, yielding a self-tuning AVC algorthm. To meet the observability and controlability requirements, the system components in Fig. 4 are suitably placed relative to one another. For the disturbance signal to be measured properly, for instance, the detector is required to be placed close to the primary source to detect the dynamic modes of interest of the system. Moreover, to achieve vibration suppression over a broad range of frequencies, the actuator should be located such that to excite all the dynamic modes of interest of the beam. Integral to these requirements is, additionally, to ensure that the arrangement results in a practically realisable controller and stable system, as discussed earlier within ANC. 3.1. A schematic diagram of the cantilever beam system is shown in Fig. 5, where l represents the length of the beam, U(x, t) represents an applied force at a distance x from the fixed (clamped) end of the beam at time t and y(x, t) is the deflection of the beam from its stationary (unmoved) position at the point where the force has been applied. The motion of the beam in transverse vibration is governed by the well-known fourth-order partial differential equation (PDE) [38] m2
1 4y(x, t) 1 2y(x, t) 1 + = U(x, t) 1x 4 1t 2 m
Figure 5. The cantilever beam flexure.
(7)
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where m is a beam constant given by m2 = EI/rA, with r, A, I and E representing the mass density, cross-sectional area, moment of inertia of the beam and the Young’s modulus respectively, and m is the mass of the beam. The corresponding boundary conditions at the fixed and free ends of the beam are given by y(0, t) = 0
and
1y(0, t) =0 1x
1 2y(l, t) =0 1x 2
and
1 3y(l, t) =0 1x 3
(8)
To obtain a solution to the PDE, describing the beam motion, the partial derivative terms in equation (7) and the boundary conditions in equation (8) are approximated using first-order central FD approximations. This involves a discretisation of the beam into a finite number of equal-length sections (segments), each of length Dx, and considering the beam motion (deflection) for the end of each section at equally spaced time steps of duration Dt (see Appendix A). 3.2. In these simulation studies, an aluminium type cantilever beam of length l = 0.635 m, m = 0.03745 kg and m = 1.3511 with Dt = 0.3 ms was considered. Investigations were carried out to determine a suitable number of segments for the beam so that reasonable accuracy is achieved by the simulation algorithm. This was determined as 19 sections. To investigate the performance of the self-tuning AVC algorithm in broadband vibration suppression, the beam simulation algorithm, as a test and evaluation platform, was implemented on the integrated DSP-transputer system. The investigations here were focused onto a broad frequency range covering almost all the resonance modes of the beam. To assess the performance of the algorithm a fixed (finite-duration) disturbance was used as the unwanted primary disturbance (UD ). While the frequency range of excitation used was 0–3000 Hz, analysis of the performance of the system was focused to 0–800 Hz which gave a sufficiently clear insight into the behaviour of the system.
Figure 6. Performance of the self-tuning AVC system. (a) Power densities before and after cancellation; (b) attenuation in spectral density.
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The self-tuning algorithm was realised with the primary and secondary sources located at grid points 15 and 19 respectively, the detector at grid point 15 and the observer at grid point 11 along the beam. Figure 6(a) shows the performance of the self-tuning control algorithm, where GDo (v) and Goo (v) represent the autopower spectral densities of the vibration signal at the observation point before and after cancellation respectively. Figure 6(b) shows the difference GDo (v)–Goo (v), i.e. the attenuation in the spectral density. It is noted that an average cancellation level of more than 25 dB is achieved over the full broad frequency range of the disturbance. The sharp spikes/dips noted in Fig. 6(b) occur at the resonance frequencies of the beam. It is noted in Fig. 6 that the resonance behaviour of the uncontrolled beam is dominantly governed by the first mode of vibration of the beam. This is due to the nature of the excitation used, which has an energy of dominant level
Figure 7. Performance of the self-tuning AVC system. (a) Beam fluctuation before cancellation; (b) beam fluctuation after cancellation.
. . . .
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Figure 8.
Average autopower spectral density level along the beam length.
around the first mode of vibration of the beam and insufficient excitation energy around higher modes. This phenomenon is more clearly evidenced in the resonance behaviour of the controlled beam where, with the beam excited by both the primary and secondary signals, there is sufficient energy put into the system to further excite higher resonance modes of the beam. By using an excitation, for example, of a PRBS nature the uncontrolled beam would show sufficient evidence of resonance behaviour and hence a better control of the system at higher resonance modes of the beam will be achieved. The corresponding time domain descriptions of the beam fluctuation before and after cancellation are shown in Fig. 7. It is noted that the level of unwanted disturbance is significantly reduced throughout the length of the beam. This is further evidenced in the corresponding frequency-domain description in Fig. 8 given in terms of the frequency-averaged signal power at each point throughout the length of the beam.
4. CONCLUSION
The design and implementation of a self-tuning control algorithm for ANC and AVC have been presented, discussed and verified through simulation experiments. An active control mechanism for broadband cancellation of noise and vibration has been developed within an adaptive control framework. The algorithm, thus developed and implemented on an integrated DSP-transputer system, incorporates on-line design and implementation of the controller in real-time. Moreover, a supervisory-level control has been incorporated within the control mechanism which allows on-line monitoring of system performance and controller adaptation. The performance of the algorithm has been verified in the cancellation of broadband noise in a free-field medium and in the suppression of broadband vibration in a flexible beam structure. A significant amount of cancellation has been achieved over the full frequency range of the disturbance in each case. These demonstrate the significance of the self-tuning control mechanism for broadband noise cancellation and vibration suppression.
ACKNOWLEDGEMENT
Dr Hossain acknowledges the support of a research fellowship of the Association of Commonwealth Universities.
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APPENDIX A: FINITE DIFFERENCE SIMULATION OF THE BEAM
Let y(x, t) be denoted by yi,j representing the beam deflection at point i at time step j (grid point i, j). Let y(x + vDx, t + wDt) be denoted by yi + v,j + w , where v and w are non-negative integer numbers. Using a first-order central FD method the partial derivatives 12y/1t 2 and 14y/1x 4 can be approximated as 1 2y(x, t) yi,j + 1 − 2yi,j + yi,j − 1 = 1t 2 (Dt)2 1 4y(x, t) yi + 2,j − 4yi + 1,j + 6yi,j − 4yi − 1,j + yi − 2,j = 1x 4 (Dx)4
(A1)
Substituting for 1 2y/1t 2 and 1 4y/1x 4 from equation (A1) into equation (7) and simplifying yields yi,j + 1 = 2yi,j − yi,j − 1 − l 2{yi + 2,j − 4yi + 1,j + 6yi,j − 4yi − 1,j + yi − 2,j } +
(Dt)2 U(x, t) m
(A2)
where, l 2 = ((Dt)2/(Dx)4)m 2. Equation (A2) gives the deflection of point i along the beam at time step j + 1 in terms of the deflections of the point at time steps j and j − 1 and deflections of points i − 1, i − 2, i + 1 and i + 2 at time step j. Note that in evaluating the deflection at the grid point i = 1 the fictitious deflection y−1,j will be required. Similarly, in evaluating the deflection at the free end of the beam, i = n (n representing the total number of sections along the beam), the fictitious deflections yn + 1,j and yn + 2,j will be required. To obtain these, the boundary conditions in equation (8) are used. In a similar manner as above, the boundary conditions in equation (8) can be expressed in terms of the FD approximations as y0,j = 0,
y1,j − y−1,j =0 2Dx
yn + 1,j − 2yn,j + yn − 1,j = 0, (Dx)2
681
yn + 2,j − 2yn + 1,j + 2yn − 1,j − yn − 2,j =0 2(Dx)3
(A3)
Solving equation (A3) for the deflections y−1,j and y0,j at the fixed end and yn + 1,j and yn + 2,j at the free end yields y0,j = 0
and
y−1,j = y1,j
yn + 1,j = 2yn,j − yn − 1,j
and yn + 2,j = 2yn + 1,j − 2yn − 1,j + yn − 2,j
(A4)
Equations (A2) and (A4) give the complete set of relations necessary for the construction of the simulation algorithm. Substituting the discretised boundary conditions for the fixed and free ends from equations (A4) into equation (A2) yield the beam deflection at the grid points along the beam as
60
y1,j + 1 = −y1,j − 1 − l 2
7−
1
7
2 y − 4y2,j + y3,j + f l 2 1,j
6
0
y2,j + 1 = −y2,j − 1 − l 2 −4y1,j + 6 −
1
1
0
2 y +f l 2 n,j
2 y − 4y3,j + y4,j + f l 2 2,j
. . .
6
yn,j + 1 = −yn,j − 1 − l 2 2yn − 2,j − 4yn − 1,j + 2 −
1 7
where, f = ((Dt)2/m)U(x, t). The above equations can be written in a matrix form as Yj + 1 = −Yj − 1 − l 2SYj + (Dt)2U(x, t)
1 m
(A5)
where,
K y1,j + 1 L Gy G 2,j + 1 Yj + 1 = G . G , Yj = G .. G k yn,j + 1 l
Ky1,jL Gy G G .2,jG , Yj − 1 = G .. G kyn,jl
K y1,j − 1 L Gy G G 2,j.− 1 G , G .. G k yn,j − 1 l
and S is a stiffness matrix, given (for n = 20, say) as
K G G G G S=G G G G G k
a −4 1 −4 b −4 1 −4 b 0 1 −4
0 1 −4 b
0 0 1 −4
0 0 0 1
··· ··· ··· ··· ···
··· ··· ··· ··· ···
··· ··· 1 0 0
··· ··· −4 1 0
··· ··· ··· ··· ···
··· ··· ··· ··· ···
··· ··· ··· ···
··· ··· ··· ···
0 0 0 0
··· ··· ··· ··· ··· ··· b −4 1 −4 c −2 2 −4 d
L G G G G G G G G G l
. . . .
682 2
where, a = 7 − 7/l , b = 6 − 2/l2, c = 5−2/l2 and d = 2 − 2/l2. Equation (A5) is the required relation for the simulation algorithm, characterising the behaviour of the flexible beam system, which can be implemented easily on a digital processor.
A UNIFIED ADAPTIVE ACTIVE CONTROL MECHANISM FOR NOISE CANCELLATION AND VIBRATION SUPPRESSION M. O. T Department of Automatic Control and Systems Engineering, The University of Sheffield, Sheffield, U.K.
M. A. H Department of Computer Science, University of Dhaka, Dhaka, Bangladesh (Received August 1994, accepted February 1996) This paper presents the design and implementation of an active control mechanism for noise cancellation and vibration suppression within an adaptive control framework. A control mechanism is designed within a feedforward control structure on the basis of optimum cancellation at an observation point. The design relations are formulated such that to allow on-line design and implementation and thus result in a self-tuning control algorithm. The algorithm is implemented on an integrated digital signal processing (DSP) and transputer system. Simulation results verifying the performance of the algorithm are presented and discussed. 7 1996 Academic Press Limited
1. INTRODUCTION
Noise and vibration have long been recognised as sources of environmental pollution, having numerous adverse effects on human life. Many attempts have been made in the past to devise methods of tackling the problems arising from unwanted noise and vibration. Traditional methods of noise cancellation and vibration suppression utilise passive control techniques which consist of mounting layers of passive material on or around the source. Investigations have shown that these methods are efficient at high frequencies but expensive and bulky at low frequencies [1, 2]. Active noise/vibration control consists of artificially generating cancelling sources to interfere destructively with the unwanted source and thus result in a reduction in the level of the noise/vibration (disturbances) at desired locations. Active noise/vibration control is not a new concept. It is based on principles that were initially proposed by Lueg in the early 1930s for noise cancellation [3]. Since then a considerable amount of research has been devoted to the development of methodologies for the design and realisation of active noise/vibration control systems in various applications. Not much work has been done in investigating the problems of noise and vibration together [4]. In many practical applications, the problems of noise and vibration are found to be closely related. For example, a close coherence is observed in numerous situations between the two due to secondary effects [5]. Many sources of noise, for instance, vibrate continuously while in operation and the vibration is found to be coherent with the acoustic 667 0888–3270/96/060667 + 16 $25.00/0
7 1996 Academic Press Limited
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. . . .
waves they emit. On other occasions, it is often noticed in buildings, for instance, that noise due to chattering of windows or motion of articles is caused through ground vibrations by passing trains and/or vehicles. In these cases a control solution aimed at reducing, for example, the level of vibration will result in significant reduction in the level of noise and, to some extent, vice versa [6]. Moreover, the control of noise and vibration by active means is based on the same design principle. These form the basis of design unification in this work by adopting a systems approach for the development of an active control strategy for both noise cancellation and vibration suppression. Thus, the method proposed in this work can be utilised to tackle the problem of noise and vibration either in isolation or together. The results presented in this paper, however, correspond to the former. Active noise/vibration control is realised by detecting and processing the noise/vibration by a suitable electronic controller so that when superimposed on the disturbances, cancellation occurs. Due to the broadband nature of these disturbances, it is required that the control mechanism realises suitable frequency-dependent characteristics so that cancellation over a broad range of frequencies is achieved [1, 5]. In practice, the spectral contents of these disturbances as well as the characteristics of system components are in general subject to variation, giving rise to time-varying phenomena. This implies that the control mechanism is further required to be intelligent enough to track these variations, so that the desired level of performance is achieved and maintained. Through his experiments of reducing transformer noise, Conover was the first to realise the need for a ‘black box’ controller that would adjust the cancelling signal in accordance with information gathered at a remote distance from the transformer [7]. Later it has been realised by numerous authors that an essential requirement for an active noise control (ANC) system to be practically successful is to have an adaptive capability [1, 8–13]. Implementing an adaptive control algorithm within the ANC system will allow the system to adjust the controller characteristics in accordance with changes in the noise pattern of the source. Chaplin and his associates [9, 14–16] have reported some success with an adaptive scheme based on a trial and error waveform generation. This method relies on the source noise being periodic. The method has been tested with rotary machines such as a motor vehicle, fan, etc. where the noise spectrum is harmonically related to the fundamental (engine firing) frequency. The engine firing frequency provides a clock signal to the algorithm. A noticeable amount of theoretical and practical investigations have subsequently been reported in this area [10–12, 17–26]. Among these, the scheme developed by Eriksson and his co-workers relies on the development of a model of the source. The scheme reported by Nelson, Elliott and co-workers is based on the minimisation (in the least square sense) of the sound level at discrete locations in the medium. The control scheme reported here, however, is based on optimum cancellation of noise/vibration. This paper presents an investigation into the development of an active control mechanism for noise cancellation and vibration suppression within an adaptive control framework. An ANC system is designed on the basis of optimum cancellation of broadband noise at an observation point in a linear 3-D propagation medium. The controller design relations are formulated such that to allow on-line design and implementation and, thus, yield a self-tuning control algorithm. The algorithm is implemented on an integrated DSP-transputer system and its performance assessed within a simulated three-dimensional free-field medium. A flexible beam system in transverse vibration is considered for vibration suppression. Such a system has an infinite number of resonance modes although in most cases, when considering a vibration problem, the lower modes are the dominant ones requiring attention. In considering a vibro-acoustic problem, however, it is likely that higher modes
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will also be contributing significantly to the noise level. In these circumstances a consideration of these modes up to a practically reasonable number will be required. First-order central finite difference (FD) methods are used to study the behaviour of the beam and develop a suitable test and verification platform. The unwanted vibrations in the structure are assumed to be caused by a single point disturbance of broadband nature. These are detected and suitable suppression signals generated via a point actuator to yield vibration suppression over a broad frequency range along the beam. The self-tuning control algorithm developed within the ANC system is implemented on the integrated DSP-transputer system and its performance assessed.
2. ACTIVE NOISE CONTROL
A schematic diagram of the geometrical arrangement of a feedforward ANC structure is shown in Fig. 1(a). An unwanted (primary) point source emits broadband noise into the propagation medium. This is detected by a detector located at a distance re relative to the primary source, processed by a controller of suitable transfer characteristics and fed to a cancelling (secondary) point source located at a distance d relative to the primary source and a distance rf relative to the detector. The secondary signal thus generated is superimposed on the primary signal to achieve cancellation of the noise at and in the vicinity of an observation point located at distances rg and rh relative to the primary and secondary sources respectively. A frequency-domain equivalent block diagram of the ANC structure is shown in Fig. 1(b), where E, F, G and H are transfer functions of the acoustic paths through the distances re , rf , rg and rh respectively. M, M0 , C and L are transfer characteristics of the detector, the observer, the controller and the secondary source respectively. UD and UC are the primary and secondary signals at the source locations whereas YOD and YOC are the corresponding signals at the observation point. UM is the detected signal and YO is the observed signal. The block diagram in Fig. 1(b) can be thought of either in the continuous frequency (s) domain or the discrete frequency (z) domain. Therefore, unless specified, the analysis and design developed in this paper apply to both the continuous-time and the discrete-time domains. The control structure in Fig. 1 corresponds to that proposed by Lueg in his patent where the time delay is implemented by the physical separation between the primary and secondary sources [3]. A significant amount of consideration has subsequently been given to this strucuture in various applications. Conver [7], Hesselman [27] and Ross [28] have employed this structure in the cancellation of transformer noise. Ross has considered the design, Roure has analysed the stability and Eriksson, et al. have considered the implementation of this structure in the cancellation of the one-dimensional duct noise [13, 21–23, 29]. Nelson et al. have analysed the performance of this structure in the cancellation of enclosed sound fields [30–32]. Tokhi and Leitch have considered the design, stability, performance and implementation of this structure in the cancellation of noise in three-dimensional propagation medium [1–2, 5, 26, 33–34]. The detected and observed signals, namely UM and YO in Fig. 1(b), can each be thought of as composed of the sum of two signals: one due to the primary source and one due to the secondary source. Using the block diagram in Fig. 1(b), these can be expressed as UM = MEUD + MFUC YO = MO GUD + MO HUC
(1)
670
. . . .
Figure 1. Active noise control structure. (a) Schematic diagram; (b) block diagram.
The secondary source signal UC , on the other hand, can be expressed, through the controller path, as UC = CLUM
(2)
For complete cancellation of the noise to be achieved at the observation point the condition YO = 0
(3)
must be satisfied. This is equivalent to the minimum variance design criterion in a stochastic environment [35–37]. This requires the primary and secondary signals at the observation point to be equal in amplitudes and have a phase difference of 180° relative to each other. Thus, synthesising the controller within the block diagram of Fig. 1(b) on the basis of this objective yields C=
G ML(FG − EH)
(4)
This is the required controller transfer function for optimum cancellation of broadband noise at the observation point. In designing the controller, causality is ensured by making the number of zeros in C either less than or equal to the number of its poles. This is achieved by using a suitable realisation structure for the controller transfer characteristics. Note in equation (4) that, for given secondary source and detector, the controller characteristics are determined by the geometric arrangement of system components. This incorporates the location of the detector and the observer with respect to the primary and
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secondary sources. Among these, there exist arrangements that will lead to FG − EH, in the denominator of equation (4), approaching zero and thus requiring the controller to have an impractically large gain. Thus, an analysis of the system on this basis yielding the locus of detection and observation points for which the controller will be required to have an impractically large gain for optimum cancellation is important at a design stage. Moreover, note in Fig. 1(b) that the secondary signals reaching the detector form a positive feedback loop that can cause the system to become unstable for certain geometrical arrangements of system components. Therefore, further analysis of the system from a stability point of view leading to a robust design of the system is important at a design stage. Tokhi and Leitch have analysed and studied both these issues for ANC systems in 3-D non-dispersive propagation media [5, 33]. 2.1. - In practice, the characteristics of sources of noise vary due to operating conditions leading to time-varying spectra. Moreover, the characteristics of transducers, sensors and other electronic equipment used are subject to variation due to environmental effects, ageing, etc. To design an ANC system so that the controller characteristics are updated in accordance with these changes in the system, a self-tuning control strategy, allowing on-line design and implementation of the controller, can be utilised such that the required performance is achieved and maintained. Self-tuning control is distinguished as a class of adaptive control mechanisms [35–37]. It essentially consists of the processes of identification and control, both implemented on-line. The identification process is mainly concerned with on-line modeling of the plant being controlled and, thus, should incorporate a suitable system identification algorithm. The control process, on the other hand, is concerned with the design and implementation of the controller using the plant model and, thus, should incorporate a suitable controller design criterion. Consider the system in Fig. 1 as a single-input single-output system with the detected signal, UM , as input and the observed signal, YO , as output. Moreover, owing to the state of the secondary source, let the system behaviour be characterised by two sub-systems, namely, when the secondary source is off, with an equivalent transfer function denoted by Q0 , and when the secondary source is on, with an equivalent transfer function denoted by Q1 . Using the block diagram of Fig. 1(b), these can be obtained as Q=
Mo G , ME
Q1 =
$
Mo G ML(FG − EH) 1− ME G
%
(5)
Note that in obtaining Q0 and Q1 in equation (5) the controller block in Fig. 1 is replaced simply by a switch; prior to an estimation/measurement process the controller transfer function is not known. Note further that, the primary source is on throughout the process of estimating/measuring Q0 and Q1 . Manipulating equation (5) and using equation (4) yields an equivalent design relation for the controller in terms of Q0 and Q1 as
$
C= 1−
%
Q1 Q0
−1
(6)
This is the required controller design rule given in terms of transfer characteristics Q0 and Q1 which can be measured/estimated on-line. An on-line design and implementation of the controller can thus be achieved as
672
. . . .
(a) obtain Q0 and Q1 using a suitable system identification algorithm, for example a recursive least squares (RLS) parameter estimation algorithm; (b) use equation (6) to calculate the controller transfer function; and (c) implement the controller on a digital processor. Moreover, to monitor system performance and update the controller characteristics upon changes in the system, a supervisory-level control can be realised. The supervisor can be designed to monitor system performance on the basis of a pre-specified quantitative measure of cancellation as an index of performance, so that if the cancellation achieved is within the specified range then the algorithm implementation is to remain at the control level, step (c) above. However, if the cancellation is outside the specified range then self-tuning is to be reinitiated at the identification level, step (a) above. This results in a self-tuning ANC mechanism as depicted in Fig. 2. The ‘plant’ in Fig. 2 designates the system in Fig. 1 between the detection and observation points, in which during the identification process the controller is replaced by a switch. During the control process, however, the estimated controller transfer function is implemented to replace the switch. In implementing the self-tuning control algorithm described above, several issues of practical importance need to be given careful consideration. These include properties of the disturbance signal, robustness of the estimation and control, system stability and processor-related issues such as word length, speed and computational power [2]. In employing the minimum variance design criterion, a problem commonly encountered is that of instability of system, specially, when non-minimum phase models are involved [35–37]. A model is non-minimum phase if it has one or more zeros outside the stability region. The process of minimum variance design involves cancellation of open-loop plant zeros. Thus, in controlling a non-minimum phase plant, the estimated plant model will also be non-minimum phase. This will require the cancellation of the plant zeros with the estimated model zeros to be exact. However, due to finite processor word lengths and
Figure 2. Self-tuning controller.
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computational errors the cancellation is not often exact, leading to an unstable closed-loop system. Note also in the controller design rules, equation (6), that such a situation will result in a non-minimum phase and an unstable controller with the unstable poles approximately cancelling the corresponding zeros that are outside the stability region. Thus, to avoid this problem of instability either the estimated models can be made minimum-phase by reflecting those zeros that are outside the stability region into the stability region and using the resulting minimum-phase models to design the controller, or once the controller transfer function has been obtained, the poles and zeros that are outside the stability region can be reflected into the stability region. In this manner, a factor (1 − pz−1) corresponding to a pole/zero at z = p, in the complex z-plane, that is outside the stability region can be reflected into the stability region by replacing the factor with (p − z−1). The supervisory-level control described above is used as a performance monitor. In addition to monitoring system performance, it can also be facilitated with further levels of intelligence. For example, monitoring system stability and avoiding problems due to non-minimum phase models as described above, verifying controller characteristics on the basis of practical realisation to make sure that impractically large controller gains are not required as discussed earlier, system behaviour in a transient period, model structure validation, etc. 2.2. To assess the performance of the self-tuning ANC algorithm a simulation environment characterising the structure in Fig. 1 was constructed using practical measurements of noise propagation in a 3-D medium. The propagation medium is assumed to be non-dispersive (linear), where a sound wave in propagating through a distance r is attenuated in amplitude inversely with r and delayed in phase directly with the product of r and the frequency of the wave. The experimental set-up for construction of the simulation environment consisted of loudspeaker and a microphone at a set distance in front of it. The loudspeaker was driven by a pseudo-random binary sequence (PRBS) signal from a signal generator and the frequency response between the loudspeaker input and the microphone output was measured. The measured data thus obtained is the equivalent transfer characteristics of the loudspeaker L, microphone M, the associated drive and pre-amplifiers and the acoustic path between the loudspeaker and the microphone. Noting in equations (4) and (5) that the transfer characteristics of the acoustic paths in Fig. 1, conceptually, appear in cascade with the characteristics of the detector and the secondary source, the measured frequency response can be used to obtain the frequency response of the acoustic paths in Fig. 1(a) for known distances and hence calculate signal propagation through the system within the block diagram in Fig. 1(b). In this manner, a simulation environment characterising the system within the propagation medium is constructed. A fast Fourier transform algorithm is utilised to transform the signals concerned from the time domain into the frequency domain and vice versa as required. The self-tuning ANC algorithm was implemented on an integrated DSP-transputer system incorporating an i860 DSP device and a T805 transputer using DSP and parallel processing (PP) methods to allow efficient and relatively accurate implementation. The DSP device with its extensive signal processing power is 64-bit processor and is capable of 80 million floating-point operations per second (Mflops). The transputer, on the other hand, with its powerful communication facility, is a 32-bit processor capable of 4 Mflops. The ANSI C toolset is used as a common software for program development and implementation.
674
. . . .
Figure 3. Performance of the self-tuning ANC system. (a) Power spectral density of the noise; (b) attenuation in spectral density.
To assess the performance of the algorithm a broadband (0–512 Hz) PRBS signal was used as the unwanted primary noise. Figure 3(a) shows the autopower spectral density of the noise at the observation point before cancellation as GDo (v) and after cancellation as Goo (v). Figure 3(b) shows the difference GDo (v) − Goo (v), i.e. the actual attenuation in the spectral density of the noise. It is noted that an average broadband cancellation level of 20–25 dB below 150 Hz and more than 40 dB cancellation above 150 Hz is achieved. As compared with previous implementations, using DSP devices only, this significant level of cancellation over a broad range of frequencies is due to the computing power achieved by the utilisation of DSP and PP methods.
3. ACTIVE VIBRATION CONTROL
Active vibration control (AVC) is realised in a similar manner as ANC. The design of an AVC system depends upon the complexity of the structure under consideration and the nature of the disturbance process. A flexible beam system is considered in transverse vibration in a fixed-free form. Such a system has an infinite number of resonance modes although in most cases the lower modes are the dominant ones requiring attention. A schematic diagram of the AVC system, incorporating a cantilever beam, is shown in Fig. 4. The unwanted vibrations in the structure are assumed to be caused by a single point disturbance force (UD ) of broadband nature. These are detected by a point detector and processed by a controller to generate suitable suppression signals (UC ) via a point actuator to yield vibration suppression over a broad frequency range at an observation point along
675
Figure 4. Schematic diagram of the AVC structure.
the beam. A frequency domain equivalent block diagram of the AVC system in Fig. 4 will give rise to that of the ANC system in Fig. 1(b), with a similar interpretation of the transfer functions and signals involved. In this manner, the required controller transfer function for optimum vibration suppression at the observation point is, therefore, given as in equation (4) with the corresponding equivalent relation suitable for on-line design and implementation as in equation (6). Therefore, a similar formulation of the self-turning control algorithm developed for noise cancellation applies to vibration suppression in Fig. 4, yielding a self-tuning AVC algorthm. To meet the observability and controlability requirements, the system components in Fig. 4 are suitably placed relative to one another. For the disturbance signal to be measured properly, for instance, the detector is required to be placed close to the primary source to detect the dynamic modes of interest of the system. Moreover, to achieve vibration suppression over a broad range of frequencies, the actuator should be located such that to excite all the dynamic modes of interest of the beam. Integral to these requirements is, additionally, to ensure that the arrangement results in a practically realisable controller and stable system, as discussed earlier within ANC. 3.1. A schematic diagram of the cantilever beam system is shown in Fig. 5, where l represents the length of the beam, U(x, t) represents an applied force at a distance x from the fixed (clamped) end of the beam at time t and y(x, t) is the deflection of the beam from its stationary (unmoved) position at the point where the force has been applied. The motion of the beam in transverse vibration is governed by the well-known fourth-order partial differential equation (PDE) [38] m2
1 4y(x, t) 1 2y(x, t) 1 + = U(x, t) 1x 4 1t 2 m
Figure 5. The cantilever beam flexure.
(7)
676
. . . .
where m is a beam constant given by m2 = EI/rA, with r, A, I and E representing the mass density, cross-sectional area, moment of inertia of the beam and the Young’s modulus respectively, and m is the mass of the beam. The corresponding boundary conditions at the fixed and free ends of the beam are given by y(0, t) = 0
and
1y(0, t) =0 1x
1 2y(l, t) =0 1x 2
and
1 3y(l, t) =0 1x 3
(8)
To obtain a solution to the PDE, describing the beam motion, the partial derivative terms in equation (7) and the boundary conditions in equation (8) are approximated using first-order central FD approximations. This involves a discretisation of the beam into a finite number of equal-length sections (segments), each of length Dx, and considering the beam motion (deflection) for the end of each section at equally spaced time steps of duration Dt (see Appendix A). 3.2. In these simulation studies, an aluminium type cantilever beam of length l = 0.635 m, m = 0.03745 kg and m = 1.3511 with Dt = 0.3 ms was considered. Investigations were carried out to determine a suitable number of segments for the beam so that reasonable accuracy is achieved by the simulation algorithm. This was determined as 19 sections. To investigate the performance of the self-tuning AVC algorithm in broadband vibration suppression, the beam simulation algorithm, as a test and evaluation platform, was implemented on the integrated DSP-transputer system. The investigations here were focused onto a broad frequency range covering almost all the resonance modes of the beam. To assess the performance of the algorithm a fixed (finite-duration) disturbance was used as the unwanted primary disturbance (UD ). While the frequency range of excitation used was 0–3000 Hz, analysis of the performance of the system was focused to 0–800 Hz which gave a sufficiently clear insight into the behaviour of the system.
Figure 6. Performance of the self-tuning AVC system. (a) Power densities before and after cancellation; (b) attenuation in spectral density.
677
The self-tuning algorithm was realised with the primary and secondary sources located at grid points 15 and 19 respectively, the detector at grid point 15 and the observer at grid point 11 along the beam. Figure 6(a) shows the performance of the self-tuning control algorithm, where GDo (v) and Goo (v) represent the autopower spectral densities of the vibration signal at the observation point before and after cancellation respectively. Figure 6(b) shows the difference GDo (v)–Goo (v), i.e. the attenuation in the spectral density. It is noted that an average cancellation level of more than 25 dB is achieved over the full broad frequency range of the disturbance. The sharp spikes/dips noted in Fig. 6(b) occur at the resonance frequencies of the beam. It is noted in Fig. 6 that the resonance behaviour of the uncontrolled beam is dominantly governed by the first mode of vibration of the beam. This is due to the nature of the excitation used, which has an energy of dominant level
Figure 7. Performance of the self-tuning AVC system. (a) Beam fluctuation before cancellation; (b) beam fluctuation after cancellation.
. . . .
678
Figure 8.
Average autopower spectral density level along the beam length.
around the first mode of vibration of the beam and insufficient excitation energy around higher modes. This phenomenon is more clearly evidenced in the resonance behaviour of the controlled beam where, with the beam excited by both the primary and secondary signals, there is sufficient energy put into the system to further excite higher resonance modes of the beam. By using an excitation, for example, of a PRBS nature the uncontrolled beam would show sufficient evidence of resonance behaviour and hence a better control of the system at higher resonance modes of the beam will be achieved. The corresponding time domain descriptions of the beam fluctuation before and after cancellation are shown in Fig. 7. It is noted that the level of unwanted disturbance is significantly reduced throughout the length of the beam. This is further evidenced in the corresponding frequency-domain description in Fig. 8 given in terms of the frequency-averaged signal power at each point throughout the length of the beam.
4. CONCLUSION
The design and implementation of a self-tuning control algorithm for ANC and AVC have been presented, discussed and verified through simulation experiments. An active control mechanism for broadband cancellation of noise and vibration has been developed within an adaptive control framework. The algorithm, thus developed and implemented on an integrated DSP-transputer system, incorporates on-line design and implementation of the controller in real-time. Moreover, a supervisory-level control has been incorporated within the control mechanism which allows on-line monitoring of system performance and controller adaptation. The performance of the algorithm has been verified in the cancellation of broadband noise in a free-field medium and in the suppression of broadband vibration in a flexible beam structure. A significant amount of cancellation has been achieved over the full frequency range of the disturbance in each case. These demonstrate the significance of the self-tuning control mechanism for broadband noise cancellation and vibration suppression.
ACKNOWLEDGEMENT
Dr Hossain acknowledges the support of a research fellowship of the Association of Commonwealth Universities.
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APPENDIX A: FINITE DIFFERENCE SIMULATION OF THE BEAM
Let y(x, t) be denoted by yi,j representing the beam deflection at point i at time step j (grid point i, j). Let y(x + vDx, t + wDt) be denoted by yi + v,j + w , where v and w are non-negative integer numbers. Using a first-order central FD method the partial derivatives 12y/1t 2 and 14y/1x 4 can be approximated as 1 2y(x, t) yi,j + 1 − 2yi,j + yi,j − 1 = 1t 2 (Dt)2 1 4y(x, t) yi + 2,j − 4yi + 1,j + 6yi,j − 4yi − 1,j + yi − 2,j = 1x 4 (Dx)4
(A1)
Substituting for 1 2y/1t 2 and 1 4y/1x 4 from equation (A1) into equation (7) and simplifying yields yi,j + 1 = 2yi,j − yi,j − 1 − l 2{yi + 2,j − 4yi + 1,j + 6yi,j − 4yi − 1,j + yi − 2,j } +
(Dt)2 U(x, t) m
(A2)
where, l 2 = ((Dt)2/(Dx)4)m 2. Equation (A2) gives the deflection of point i along the beam at time step j + 1 in terms of the deflections of the point at time steps j and j − 1 and deflections of points i − 1, i − 2, i + 1 and i + 2 at time step j. Note that in evaluating the deflection at the grid point i = 1 the fictitious deflection y−1,j will be required. Similarly, in evaluating the deflection at the free end of the beam, i = n (n representing the total number of sections along the beam), the fictitious deflections yn + 1,j and yn + 2,j will be required. To obtain these, the boundary conditions in equation (8) are used. In a similar manner as above, the boundary conditions in equation (8) can be expressed in terms of the FD approximations as y0,j = 0,
y1,j − y−1,j =0 2Dx
yn + 1,j − 2yn,j + yn − 1,j = 0, (Dx)2
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yn + 2,j − 2yn + 1,j + 2yn − 1,j − yn − 2,j =0 2(Dx)3
(A3)
Solving equation (A3) for the deflections y−1,j and y0,j at the fixed end and yn + 1,j and yn + 2,j at the free end yields y0,j = 0
and
y−1,j = y1,j
yn + 1,j = 2yn,j − yn − 1,j
and yn + 2,j = 2yn + 1,j − 2yn − 1,j + yn − 2,j
(A4)
Equations (A2) and (A4) give the complete set of relations necessary for the construction of the simulation algorithm. Substituting the discretised boundary conditions for the fixed and free ends from equations (A4) into equation (A2) yield the beam deflection at the grid points along the beam as
60
y1,j + 1 = −y1,j − 1 − l 2
7−
1
7
2 y − 4y2,j + y3,j + f l 2 1,j
6
0
y2,j + 1 = −y2,j − 1 − l 2 −4y1,j + 6 −
1
1
0
2 y +f l 2 n,j
2 y − 4y3,j + y4,j + f l 2 2,j
. . .
6
yn,j + 1 = −yn,j − 1 − l 2 2yn − 2,j − 4yn − 1,j + 2 −
1 7
where, f = ((Dt)2/m)U(x, t). The above equations can be written in a matrix form as Yj + 1 = −Yj − 1 − l 2SYj + (Dt)2U(x, t)
1 m
(A5)
where,
K y1,j + 1 L Gy G 2,j + 1 Yj + 1 = G . G , Yj = G .. G k yn,j + 1 l
Ky1,jL Gy G G .2,jG , Yj − 1 = G .. G kyn,jl
K y1,j − 1 L Gy G G 2,j.− 1 G , G .. G k yn,j − 1 l
and S is a stiffness matrix, given (for n = 20, say) as
K G G G G S=G G G G G k
a −4 1 −4 b −4 1 −4 b 0 1 −4
0 1 −4 b
0 0 1 −4
0 0 0 1
··· ··· ··· ··· ···
··· ··· ··· ··· ···
··· ··· 1 0 0
··· ··· −4 1 0
··· ··· ··· ··· ···
··· ··· ··· ··· ···
··· ··· ··· ···
··· ··· ··· ···
0 0 0 0
··· ··· ··· ··· ··· ··· b −4 1 −4 c −2 2 −4 d
L G G G G G G G G G l
. . . .
682 2
where, a = 7 − 7/l , b = 6 − 2/l2, c = 5−2/l2 and d = 2 − 2/l2. Equation (A5) is the required relation for the simulation algorithm, characterising the behaviour of the flexible beam system, which can be implemented easily on a digital processor.