Recent trends in vibration testing have moved toward low acceleration profiles and simple—but dynamically poor—payloads. Poor payloads exhibit large undamped resonances that occur within the range of test frequencies. Resonant points on the plate can show acceleration levels at least 100 times greater than the control level. To address these needs, it is important to understand recent developments in vibration control systems such as low-level testing and adaptive processing.
Low-Level Testing
Typical vibration tests used by computer, computer peripherals, and other component manufacturers, performed at acceleration levels of less than 1 g, often start from low displacement levels and low frequencies. Generally, vibration test systems have a wide, but fixed, dynamic range (0.1 to 100 g), most of which is underutilized during these tests.
The acceleration return signals from the test package are much smaller than the maximum input signal the vibration controller can accept. The vibration controller converts this small input signal to a digital signal using an analog-to-digital converter (ADC). With a small input signal, only a limited range of the ADC is used.
To improve the dynamic range and allow the vibration controller to measure signals with respect to 1 g using the whole range of the ADC, it is necessary to use a programmable gain amplifier (PGA) (Figure 1). The PGA amplifies the small input signal to a level near the maximum level acceptable to the ADC.
The gain of the PGA can be programmed automatically by the vibration controller to deal with varying signals specified by the user or returned by the test package. In this way, the full quantization range of the ADC is used, making the full dynamic range of the controller available at 1 g or 100 g, depending on the test requirements. The capability to control dynamically poor payloads with large undamped resonances at low test levels is significantly improved.
Adaptive Processing
The subject of adaptive filters has been a research topic since the 1960s. Subsequently, these filters are used in many areas, specifically in communications data transmission such as echo cancellation, speech coding, and channel equalization.
An adaptive filter is time-varying and nonlinear since it adapts differently to two dissimilar input signals. The self-adjustment property of adaptive filters makes them suitable for use in closed-loop systems.
In designing a vibration controller, a fundamental issue is to select an algorithm that suits the signal-processing application of interest. As the shakers are nonlinear and the UUT can vary in shape and size and have unknown dynamics, an adaptive algorithm would be most appropriate to perform the control operation.
Closed-Loop Adaptive Filtering
An adaptive system consists of a shaker and an adaptive (programmable) filter. The filter must be adjusted to pass, without corruption, the desired components of the input signal and to remove the undesired components.
In a typical adaptive system, the input signal is fed through the adaptive filter to yield an output. In turn, this output is compared with a desired (or reference) signal to produce an error signal. The error signal then is used to update the weights of the programmable filter to minimize the error signal. This results in the filter output that approximates the reference signal.
Various cost functions may minimize the error signal to reach an optimum solution (see sidebar). The choice of the cost function depends on the adaptation algorithm.
In designing an adaptive filter, a number of issues must be considered:
Choice of programmable filter, either finite impulse response (FIR) or infinite impulse response (IIR).
The adaptive filter structure.
The adaptation algorithm.
Choice of Programmable Filter
A major drawback with IIR filters is the possible instability of the filter. One way to overcome this instability is to use nonrecursive or FIR filters which are all-zero filters with no poles. In the case of vibration testing, FIR filters are suitable.
Adaptive Filter Structure
There are four basic classes of adaptive filtering structures: prediction, system modelling, inverse modelling, and interference cancelling (see sidebar). Each structure suits a certain type of application, and it is essential to select the correct implementation to match the system requirements. In vibration testing, it is important that the shaker output follows the signal you specified at the input, so inverse modelling is most suitable.
Adaptation Algorithm
In designing a vibration controller, select an algorithm that suits the application of interest and that correctly adjusts the characteristics of the adaptive filter. The algorithm of interest must meet these criteria:
The adaptive filter must have a long-impulse response to produce a FIR filter with a sharp cutoff frequency.
The algorithm must adapt quickly as long-duration adaptations can cause fatigue in the UUT mounted on the shaker.
The algorithm must be stable and robust.
The algorithm must withstand noise and external disturbances.
Among the many adaptation algorithms available, the most important family is the Least Mean Square (LMS) algorithm.1 The LMS algorithm is popular because of its simplicity and ease of implementation. As a result, it is used in many signal-processing applications where computational requirements must be minimized. If the input and desired responses of an adaptive system are available at each iteration, then this algorithm generally is the best choice for many different adaptive signal-processing applications.2
However, this algorithm has slow convergence properties; that is, a large number of time-consuming iterations are required before an error signal is driven to zero. This makes the LMS algorithm an unattractive solution for vibration testing.
Most adaptive-filter algorithms operating in the time domain require the implementation of a mathematical process called convolution. In this case, convolution integrates the products of two time-varying functions.
When many iterative calculations are required to achieve high accuracy, numerical integration can be a very slow process. This shortcoming makes any time-domain-based algorithm implementation unattractive for the long-impulse response adaptive-filter applications required for vibration testing.
Fortunately, it has been mathematically proven—and verified in practice—that a convolution of sequences defined in the time domain can be replaced by a simple multiplication of the same sequences in the frequency domain. This mathematical equivalency can be exploited to minimize the input-to-response time lag by merely using frequency-domain-based adaptive-filter implementations.
Figure 2 shows a block diagram of how such an implementation may be accomplished. The input signal is applied simultaneously to the shaker and to a delay element. The delayed input signal and the vibration response signal are converted separately into the frequency domain via fast Fourier transform (FFT) algorithms to provide the signal and control inputs for the adaptive filter. Then, the output of the filter is reconverted into the time domain via the N-point inverse FFT.
Since the adaptive-filter impulse response is the inverse to that of the shaker and payload, it compensates for any shaker and load-induced response signal aberrations. As a result, the combined output approximates that of the signal containing the applied reference spectrum as required for proper closed-loop system operation.
References
1. Cowan, C. and Grant, P., Adaptive Filters, Prentice Hall, 1985.
2. Widrow, B. and Stearns, S., Adaptive Signal Processing, Prentice Hall, 1985.
3. Oppenheim, A. and Schafer, R., Discrete-Time Signal Processing, Prentice Hall, 1989.
About the Author
Sarah Nobari is employed by Imperial College, London University, and has worked with Ling Dynamic Systems for two years in the area of adaptive signal processing. She has a B.Sc. degree in Information Systems Engineering from Imperial College, and is working towards an MPhil. Ling Dynamic Systems, Heath Works, Baldock Rd., Royston, Herts SG8 5BQ, UK. 44 1763 242424.
SIDEBAR
Definitions
Cost Function—A cost function is defined in terms of a parameter that is to be minimized (or maximized). Examples of this type of function are the variance or the modulus of the parameter to be optimized. The minimum/maximum is obtained by differentiating the cost function and setting it to zero. In the case of adaptive filtering, this parameter is the error signal between the desired signal and the output of the adaptive filter.
IIR Filters—IIR, or recursive, filters have an infinite-duration impulse response. The impulse response of a system defines the relationship between the system input and output in the time domain. With IIR filters, it is possible to create sharp cutoff filter characteristics with low filter complexity.
However, IIR filters have two major disadvantages:
No control is offered on the phase response of the filter.
The filter may not be stable unless restrictions are placed on the coefficients of the filter.
FIR Filters—FIR or nonrecursive filters are zero, except for a limited period of time when the coefficients of the filter are non-zero. The major drawback of this type of filter is the complexity of the filter when sharp cutoff frequency responses are required.
However, these types of filters always are stable and can provide a linear-phase response.
Four Classes of Adaptive Filtering—The four classes are major configurations of adaptive filtering, each suited to a particular application. In the diagram,
s = the input signal.
d = the desired signal (which can be the same as the input signal).
y = the adaptive processor output.
e = the error.
The first configuration is prediction that is applied in signal encoding and noise reduction (a). The second configuration is system modelling where the transfer function of an unknown system is identified. As shown in b, the unknown system is the shaker.
The third configuration is equalization or inverse modelling. This type of application generally is used to undo the effect of a transducer, a communications channel, or some other system, or to produce an inverse model of the shaker (c).
The final configuration is interference cancelling (d). In this case, the signal is corrupted by additive noise, n´, and a distorted but correlated version of the noise n is available. The goal is for the output y to resemble n so that the signal e closely follows the input signal s.
Copyright 1997 Nelson Publishing Inc.
June 1997
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