PREDICTION of the future is considered to be a difficult

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1 876 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 46, NO. 5, OCTOBER 1999 Polynomial Predictive Filtering in Control Instrumentation: A Review Sami Väliviita, Member, IEEE, Seppo J. Ovaska, Senior Member, IEEE, and Olli Vainio, Senior Member, IEEE Abstract Additional delay is an unavoidable drawback of conventional filters used frequently in industrial electronics. This delay is particularly harmful if the filtered primary signal is to be used for time-critical feedback or synchronization purposes. Therefore, predictive signal processing methods can offer significant advantages for these real-time applications. Polynomial predictive filters are specified without explicit passbands and stopbands, and they are behaving delaylessly or predictively for smoothly varying signal components. The degree of smoothness of the incoming signal sets the requirements for the applied filtering scheme and its parameters. Smoothness of a signal is a fuzzy and application-specific concept: the degree of smoothness depends on the ratio of the bandwidth of the primary signal and the applied sampling rate, as well as the noise component. In this paper, we review the most important polynomial predictive filtering methods and algorithms, their design and implementation techniques, and a collection of successful applications. Index Terms Measurement, prediction methods, smoothing methods. I. INTRODUCTION PREDICTION of the future is considered to be a difficult task. However, in certain fields of science and engineering, prediction has proved to be feasible, and many predictive algorithms have been successfully implemented for various real-time applications. The existing prediction techniques can be divided into two main categories: stochastic and deterministic signal prediction methods. Prediction of signals with stochastic nature is usually performed by using some well-established method of linear prediction [1]. The task is to predict future signal values by a linear system, the coefficients of which are optimized to minimize the expected squared prediction error. Computation of the filter coefficients requires knowledge of the autocorrelation function of the signal to be predicted. If the signal statistics change over time, the filter coefficients should be computed on-line. Stochastic methods based on nonparametric signal models, e.g., the least-mean-square (LMS) adaptive algorithm [2], do not require explicit knowledge of the autocorrelation function of the signal. Instead, the LMS-algorithm-based adaptive filters Manuscript received May 12, 1998; revised April 28, Abstract published on the Internet June 18, S. Väliviita was with the Institute of Intelligent Power Electronics, Helsinki University of Technology, FIN Espoo, Finland. He is now with Tellabs Oy, FIN Espoo, Finland ( sami.valiviita@tellabs.fi). S. J. Ovaska is with the Institute of Intelligent Power Electronics, Helsinki University of Technology, FIN Espoo, Finland. O. Vainio is with the Signal Processing Laboratory, Tampere University of Technology, FIN Tampere, Finland. Publisher Item Identifier S (99) perform data-driven approximation, and the filter coefficients are updated using a finite number of past input samples. Consequently, adaptive filters are attractive when the signal characteristics are either unknown, or fixed but satisfactory behavior cannot be achieved with time-invariant filters. Neural-network-based prediction methods [3] and fuzzy systems [4] belong to the class of deterministic methods. Their usefulness is based on their capability to associate pairs of multi-input multi-output patterns, and to generalize this, possibly nonlinear association. However, their design process is highly application specific, and they are often computationally intensive. Successful applications of neural networks and fuzzy systems in motor drives are discussed in [5]. Finally, deterministic prediction methods based on time-domain signal models, polynomials [6] or sinusoids [7], are highly applicable when the signal to be predicted can be approximated with sufficient accuracy using such a general model. When an appropriate prediction method is to be chosen for a particular application, the selection is typically based on the following issues: characteristics of the primary signal to be predicted and the corrupting noise component: deterministic or stochastic, wide-band or narrow-band signal, additive or multiplicative noise, and low or high signal-to-noise ratio (SNR); implementation environment, i.e., available computing power and accuracy (the alternatives include applicationspecific integrated circuits or digital signal processors, and fixed-point or floating-point arithmetic); availability of algorithms and computer-aided design and implementation tools. In control instrumentation, the feedback signals are delayed due to two main reasons. One source of inherent delay is the data acquisition and signal processing circuitry. For instance, analog-to-digital conversion always delays the measured signal to some extent. Second, most of the traditional filtering algorithms further delay the signal. Therefore, predictive filters with efficient noise attenuation would be highly beneficial, because we are often dealing with real-time applications where the system has a strictly limited time to react to its reference and feedback signals. Closed-loop control can even become unstable if feedback signals are delayed. Fortunately, there exist numerous signals with highly deterministic behavior, and this a priori knowledge can be utilized in the design of predictive algorithms. A predictive filter is defined as an algorithm that estimates future values of the primary signal, while it simultaneously attenuates the noise components /99$ IEEE

2 VÄLIVIITA et al.: POLYNOMIAL PREDICTIVE FILTERING IN CONTROL INSTRUMENTATION 877 (a) (b) (c) Fig. 1. Three steps in the evolution from the classical Newton predictor to the efficient recursive linear smoothed Newton (RLSN) predictor. (a) Newton predictor [16]. (b) Linear smoothed Newton (LSN) predictor [17]. (c) RLSN predictor [18]. This paper is organized as follows. The use of the polynomial signal model as a basis of predictive filtering is motivated in Section II. Other prediction methods briefly described above can also be useful in control instrumentation, but they are beyond the scope of this review. In Section III, we present the theoretical foundation of polynomial prediction and different types of polynomial predictors. The design and implementation issues are discussed in Section IV. In Section V, successful applications are reviewed. The future trends and possibilities of polynomial predictive filtering techniques are considered in Section VI. Finally, Section VII concludes this paper. II. POLYNOMIAL-LIKE SIGNALS In this section, we discuss the viability of modeling signals encountered in motor drives with piecewise low-degree polynomials. However, the use of polynomial predictive filtering is not limited to this particular application domain. Signals encountered in motor drives often possess deterministic characteristics. A measured signal contains a primary component, as well as additional noise and disturbances. Primary signals can often be approximated either by low-degree polynomials or sinusoids. For instance, the substantial inertia of the motor and load prevents sudden stepwise changes of the angular velocity. Due to the inertia, practical velocity curves are always smooth. Smooth signals, in turn, can be modeled as piecewise low-degree polynomials with sufficient accuracy within narrow time windows. Smoothness of a signal is a fuzzy and application-specific concept [8]. The degree of smoothness depends on the ratio of the bandwidth of the primary signal and the applied sampling rate, as well as the noise content. If the primary signal is slowly changing compared to the sampling rate, it can be considered as a smooth signal. Application-specific smoothness index is defined in [9]. Due to the fuzzy smoothness concept, the parameters of polynomial predictive filters are selected experimentally. Polynomial modeling is regarded as one of the most powerful tools in signal processing [10]. The polynomial model for the velocity curve is particularly accurate in high-performance elevator drives where, in order to achieve smooth and comfortable movement of an elevator car, the desired velocity is a piecewise second-degree polynomial [11]. There are numerous practical applications where the angular velocity can be approximated by a low-degree polynomial within narrow time windows, as illustrated in the experimental sections of [12] [14]. Furthermore, if the angular velocity curve can be regarded as a piecewise low-degree polynomial, the angular acceleration and position are also polynomials, obtained by differentiating and integrating the velocity signal, respectively. In addition to angular velocity and acceleration, signals such as power, temperature, and the amplitude of magnetic flux can often be successfully approximated by low-degree polynomials. The angular velocity, as well as other signals encountered in motor drives, contains some noise. By using an optical pulse encoder and measuring the time between adjacent pulses, the angular velocity is obtained as a time-derivative approximation, where is the time difference between the encoder pulse appearance moments, and is the fixed rotation angle corresponding to one pulse. The angular velocity estimate obtained in this way contains several error sources [13]. At low velocities, a new velocity sample is not necessarily received during a fixed sampling period of the velocity controller, and a replica of the previous sample has to be used instead to

3 878 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 46, NO. 5, OCTOBER 1999 Fig. 2. Magnitude responses of the classical Newton predictor [16] (dash-dotted line), the LSN predictor (N =16)[17] (dashed line), and the RLSN predictor (N =16;c =0:1) [18] (solid line). The applied polynomial degree is M =1. substitute for the missing sample. The sampling frequency of the control loop is usually constant and, therefore, it is not synchronized to the availability moments of the velocity samples. Besides, the angular distance between two consecutive pulses is not constant, but has some random variation due to manufacturing tolerances. In digital computations, the time counter between adjacent pulses is discrete valued. Finally, the time-derivative approximation (backward difference) itself produces delayed velocity estimates. To tackle efficiently these error sources, a predictive method is needed. III. POLYNOMIAL PREDICTIVE FILTERING TECHNIQUES Polynomials are the simplest class of mathematical functions. A polynomial has one independent variable, and it produces an output value by forming a weighted sum of the positive, integer-valued powers of. An th-degree polynomial sequence is given by where the polynomial coefficients,, are unknown real constants. The common attractive feature of polynomial predictive filters is that they are independent of the actual polynomial coefficients. A. Newton-Type Polynomial Predictors The idea of extrapolating a function according to its derivatives has been well known in numerical analysis for a long time [15]. The classical Newton predictor forms an estimate (1) of a future value of a signal as a weighted sum of past samples, based on Newton s divided difference formula [16]. A -step-ahead Newton predictor for th-degree polynomials can be expressed as This transfer function realizes the sum of successive differences of an input signal as illustrated in Fig. 1(a). The Newton predictor would be attractive for extrapolating polynomial-like signals due to its low computational complexity. However, this type of polynomial predictive filtering is seldom applicable because wide-band noise is greatly amplified by the transfer function of (2), as illustrated in Fig. 2. Therefore, evolutionary improvements have been proposed. The linear smoothed Newton (LSN) predictor is based on the observation that most of the noise gain of the Newton predictor originates from the highest degree successive difference [17], which approximates the highest order nonzero derivative. Since the th derivative of an th-degree polynomial is constant, it can be low-pass filtered without delay. Thus, the noise gain originating from the highest degree difference is reduced. The transfer function of the LSN predictor becomes where is the transfer function of a low-pass filter, e.g., a moving averager of length. The structure of the LSN predictor and the corresponding magnitude response are (2) (3)

4 VÄLIVIITA et al.: POLYNOMIAL PREDICTIVE FILTERING IN CONTROL INSTRUMENTATION 879 Fig. 3. Magnitude responses of one-step-ahead predictive FIR filters of Heinonen and Neuvo [6] of lengths N = 8, 16, 32, and 64 (dotted, dash-dotted, dashed, and solid lines, respectively) for the first-degree polynomial model. illustrated in Figs. 1(b) and 2, respectively. Here, the wideband noise is not essentially amplified, but it is not attenuated either. The polynomial coefficients of practical signals are usually time varying. Therefore, the length of the moving averager is a tradeoff between the noise gain and prediction error in transient conditions, when the polynomial degree is rapidly varying, i.e., when the assumption on the viability of the specific polynomial model is temporarily violated. The recursive linear smoothed Newton (RLSN) predictor enhances the LSN predictor structure in two ways. First, a feedback loop is added to smoothen the incoming signal. Second, all the successive differences are filtered [18], as illustrated in Fig. 1(c). The recursive extension replaces the current signal value with a weighted average of the current input sample and its estimate which is available as the delayed output of the predictor, i.e., the prediction result that has been delayed by sampling periods. The current signal value is weighted by, and the delayed prediction is weighted by. Unlike the th successive difference, the successive differences of the degrees, cannot be simply low-pass filtered because they are varying over time. This problem can be tackled by noticing that differentiating an th-degree polynomial results in a polynomial of the degree. Since a predictor which smooths the polynomial signals is available, it is possible to apply delayed prediction values as smoothed estimates of these time-varying derivatives. The recursive transfer function of the RLSN predictor becomes (4), as shown at the bottom of the page. A magnitude response of the RLSN predictor is depicted in Fig. 2. Obviously, the noise attenuation capabilities of the RLSN predictor are considerably improved from the LSN predictor. It should be noted that the design of a polynomial predictive differentiator for th-degree polynomials is accomplished conveniently by cascading a scaled backward difference operator and an th-degree RLSN predictor [19]; is the applied sampling period. The polynomial degree of Newton predictors must be decided upon at the design phase. Even if the signal only occasionally contains significant higher order derivatives, and would in most cases be adequately predicted with a lower degree polynomial approximation, the predictor has to be designed to predict a fixed polynomial degree at all times. However, the inclusion of higher degree successive differences increases the noise sensitivity of the system and should, therefore, be avoided. All polynomial predictors and differentiators share the same characteristic: as the degree of the polynomial model is increased, the noise sensitivity increases. Therefore, in real-world applications, mainly polynomial degrees one and two are applicable. These polynomial degrees are adequate for modeling practical feedback signals because the sampling rates of control systems are commonly 10 bandwidth. 50 times the signal (4)

5 880 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 46, NO. 5, OCTOBER 1999 To overcome the problem discussed above, the adaptive recursive linear smoothed Newton (ARLSN) predictor was proposed in [20]. There, the highest degree of successive differences considered in prediction is changed dynamically according to the characteristics of the input signal. A derivative detector periodically monitors the magnitudes of successive differences and gives a weight to each difference, dictating how much effect each consecutive branch has when computing the output of the predictor. B. Finite-Duration-Impulse Response (FIR) Polynomial Predictors FIR polynomial predictors can be considered as an extension of data smoothing filters, as discussed in [21]. When the formulas for the FIR-filter-based polynomial predictors are derived, it is explicitly assumed that the input signal can be modeled as a polynomial of degree, as in (1), by fitting an th-degree polynomial model through latest samples of the input signal. Future estimates of the input signal are produced by extrapolating. The extra degrees of freedom, obtained when the filter length is greater than the minimum allowable length, can be spent on minimizing the whitenoise gain [defined in the Appendix in (A4)] of the filter, as in the case of the polynomial predictive FIR filter of Heinonen and Neuvo (H N) [6]. It turns out that the coefficients of that filter are not dependent on the input signal, but only on the values assigned to, and. Estimates of the future values are calculated by applying FIR filtering where, are the optimized FIR filter coefficients. The values assigned to the forward prediction step are not limited to integers here. The derivation of the H N FIR predictor coefficients, based on the polynomial constraints of the form of (5), is given in detail in the Appendix. This versatile technique can also be used to obtain closed-form formulas for predictive polynomial differentiators [22] and sinusoid predictors [23], among other applications. For instance, by requiring where denotes the time derivative of, we can define the optimization constraints for a predictive polynomial FIR differentiator. Further, additional constraints can be set, e.g., by requiring zero response for a specific frequency component while satisfying the constraints of (5), given in the Appendix [24]. The prediction band of FIR polynomial predictors, i.e., the range of frequencies with negative group delay is narrow due to the utilization of a low-degree polynomial model. The narrow prediction band is not a problem, however, because most of the energy of low-degree polynomial signals lie near the zero frequency. The disadvantages of FIR predictors, in turn, include the considerable gain peak in the passband, (5) (6) which can disturb closed-loop control, and unpractically high stopband gain for low. Note that the gain peak is a natural consequence of prediction: without the gain peak, there is no prediction. The magnitude responses of H N FIR predictors with different lengths are depicted in Fig. 3. It can be observed that the stopband attenuation is enhanced and the passband gain peak is reduced as the filter length is increased. To alleviate the problem of considerable passband gain peak and low stopband attenuation of the H N FIR predictors, the use of application-specific low-pass prefilters of the infiniteimpulse response (IIR) type has been suggested. The idea is to fix the coefficients of the IIR prefilter, and then optimize the coefficients of the FIR postfilter so that the cascade, as a whole, satisfies the requirement of exact prediction. Two variants of this method have been suggested. The first one minimizes the noise gain of the cascade [25], and the second one explicitly splits the frequency axis into a passband and stopband [26]. C. Augmented FIR Predictors A polynomial predictive filter structure where the predictive FIR filter of Heinonen and Neuvo is augmented with feedback was introduced in [27]. The augmented structure is advantageous compared to the H N FIR predictor in terms of attainable magnitude response shape and lower computational complexity. The passband gain peak can be reduced and the stopband attenuation can be enhanced while the filter order remains low. The augmented FIR predictor is designed to retain the steady-state predictive property of the H N FIR predictor, while allowing flexible shaping of the magnitude response. The structure of the augmented FIR predictor is illustrated in Fig. 4. The original H N FIR predictor is inside the dashed box. The output of the filter is a future estimate of the input signal, and the discrete prediction step is. When the output of the predictor is delayed by time steps, the result is an estimate of the current input sample. This kind of delayed prediction is denoted in Fig. 4 by. By taking a weighted average of the actual input value and its delayed prediction, we obtain a smoothed unbiased estimate of the present input sample. When the delayed prediction is further delayed, we can construct a secondary delay line. The final output is computed similarly as in the H N FIR predictor, except that smoothed estimates are used instead of pure input samples. A similar strategy can be applied to form efficient recursive differentiators [28]. In polynomial differentiation, however, the output of the filter is the derivative of the polynomial-like input signal. Thus, before using it as a feedback signal it must be integrated accordingly. The task remains to determine the appropriate feedback coefficients to shape the desired magnitude response. In [27], the feedback coefficients were optimized by minimizing the weighted white-noise gain and the maximum gain of the magnitude response using the simplex algorithm [29]. Recently, Harju and Ovaska optimized the feedback coefficients of the augmented FIR predictor using a multidirectional search algorithm [30] and a genetic algorithm [31]. These more advanced techniques further improve the magnitude response

6 VÄLIVIITA et al.: POLYNOMIAL PREDICTIVE FILTERING IN CONTROL INSTRUMENTATION 881 Fig. 4. Augmented FIR polynomial predictive filter [27]. Fig. 5. Magnitude responses of the cascade of the recursive differentiator [28] for second-degree polynomial model and polynomial estimator (solid line), and the FIR differentiator [22] of length 103 (dash-dotted line). In the small window, the magnitude response of the ideal differentiator is plotted with the dotted line. shape of the augmented FIR predictor because the desired magnitude response can be defined more arbitrarily. Due to the binary encoding of the filter coefficients in the genetic algorithm approach, quantized feedback coefficient values can be produced directly. Comprehensive empirical experiments suggested that it is difficult to optimize an augmented FIR predictor with a large value of. Therefore, the use of additional polynomial estimators (PE s) was proposed in [27]. PE s are otherwise similar to the augmented FIR predictors, but their output is instead of, i.e., they estimate the current value of the polynomial-like input signal. A polynomial predictive composite filter can then be formed by cascading one polynomial predictor and one or more PE s. This kind of filter structure is needed when a short augmented FIR predictor alone cannot provide the desired shape for the magnitude response. In [28], for instance, a cascade of a recursive differentiator and one PE was optimized. The order of the cascade system was five. For comparison, an FIR differentiator [22] of length 103 was required to obtain the same white-noise gain. The corresponding magnitude responses are illustrated in Fig. 5. It can be observed that the slope band gain peak of the cascade differentiator is almost negligible while the gain peak of the FIR differentiator is considerable. The upper band attenuation of the cascade differentiator could be further enhanced by cascading more than one PE with the same IIR structure. Here, the slope band is defined as the range of frequencies in which the magnitude response curve of a band-limited differentiator is monotonically rising, while the frequencies above the slope band belong to the upper band. The predictive differentiation band is defined as the range of frequencies in which the magnitude response follows exactly the ideal differentiator.

7 882 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 46, NO. 5, OCTOBER 1999 TABLE I CHARACTERISTICS OF DIFFERENT TYPES OF POLYNOMIAL PREDICTIVE FILTERS AND DIFFERENTIATORS IV. DESIGN AND IMPLEMENTATION The design and implementation of polynomial predictive filters is discussed in this section. The implementation of H N FIR predictors, for instance, is considered by Campbell and Neuvo in [32]. They presented a filter structure with a recursive running-sum replacing the traditional delay line substructure. A fixed number of multiplications and additions is required per sample irrespective of the FIR filter length. Another computationally effective, recursive structure for generating arbitrary polynomial responses [33], was proposed for the implementation of predictive FIR differentiators [22], requiring multiplication and additions for the seconddegree polynomial model, regardless of the filter length. The length of the required delay line is, however. Thus, the drawback of H N FIR predictors and FIR differentiators, as well as RLSN-type predictors, remains that a long delay line is required to provide efficient noise attenuation, and to reduce the gain peak in the passband, or slope band. When the predictor or differentiator is to be implemented with application-specific integrated circuits (ASIC s) or field-programmable gate arrays (FPGA s), where strict cost effectiveness is required, it would be important to avoid such a long delay line. The roundoff noise properties of augmented FIR predictors were studied in [34]. The reported results show that a disturbing level of roundoff noise is produced, which necessitates the use of high-accuracy fixed-point or floatingpoint representations in internal computations. The roundoff noise problem was also observed in [13] and [20], in the implementation of RLSN-type predictors. Therefore, in [13], 32-b arithmetic was used in internal computations, and in the prototyping architecture of [20], 32-b floating-point arithmetic was applied. On the other hand, Harju [35] reported that polynomial predictive filters with a low number of bits per coefficient can be constructed while preserving the desired magnitude response characteristics and predictive properties. Consequently, all three approaches (FIR type, RLSN type, and augmented FIR type) have certain advantages and disadvantages, as summarized in Table I. First, the design process of RLSN predictors is very simple indeed, requiring only two parameters to be defined, and the amount of multiplications and additions required in the implementation is small. The H N FIR predictor coefficients, in turn, provide optimal whitenoise attenuation. Besides, there are closed-form formulas available for the straightforward design process for polynomial predictive filters and differentiators. Both filter structures share common problems, however: the gain peak in their passbands (or slope bands) and the need of relatively high filter order for practical noise attenuation. These problems can be tackled by the approach described in [27], which provides malleable magnitude responses and realizations with low computational complexity. Since RLSN predictors and augmented FIR predictors are IIR filters, they can encounter roundoff noise problems when low-precision fixed-point arithmetic is used. Further, the iterative design process of augmented FIR predictors is more time consuming than the straightforward design processes of RLSN and H N FIR predictors. However, various cost functions can be applied to meet the desired design criteria. Thus, augmented FIR predictors can provide flexible design of high-performance polynomial predictors and differentiators. V. APPLICATIONS Due to the underlying assumptions on which polynomial predictors are designed, they are only applicable to a class of slowly changing signals. Thus, the applied signal model limits the application domain of polynomial predictors but, on the other hand, computationally efficient filtering schemes

8 VÄLIVIITA et al.: POLYNOMIAL PREDICTIVE FILTERING IN CONTROL INSTRUMENTATION 883 Fig. 6. Missing velocity sample predictor [11]. can be constructed. The implementations of [13] and [20] suggest, however, that high-precision arithmetic is required, which sets minimum requirements for the implementation architecture. As a conclusion, applications where polynomial prediction excels are characterized by good compliance with the polynomial signal model and low or moderate levels of noise and disturbances. Since the bandwidth of polynomial predictors gets narrower when the noise gain is decreased, the use of polynomial prediction is a tradeoff between the fitness of the polynomial model and the overall noise-reduction requirements. That is, if considerable noise attenuation is required, the primary signal must be highly polynomial like in order to obtain satisfactory results. In control instrumentation applications, we often deal with a time-variant physical process, the rate of change of which is slow and, therefore, the polynomial signal model is highly applicable for typically encountered sampling rates. A. Velocity Measurement In [11], the use of the H N FIR predictor is suggested for enhancing the time resolution of velocity samples provided by an optical pulse encoder. The application is elevator control, where the velocity curve can be well approximated by a second-degree polynomial. With high and moderate elevator speeds, velocity estimates are produced at a sufficiently high rate, but when the car slows down, the time difference between successive velocity samples becomes larger and eventually exceeds the applied sampling period. Pulse encoders with a higher resolution are available commercially, but would be prohibitively expensive. When a velocity sample is available, it is directly used as a present estimate, but in the absence of a fresh velocity sample, an estimate of the sample, provided by a polynomial predictor, is used instead, as illustrated in Fig. 6. A feedback mechanism is applied to keep the FIR delay line full. It is reported that up to two succeeding missing samples can be effectively replaced by their estimates, which increases the virtual sensing resolution of the pulse encoder threefold for the lowest elevator speeds. Other strategies for predicting missing samples are discussed in [36]. In the system described in [11], uncertainty of the time instant at which the incoming velocity samples were actually measured degrades the accuracy of the produced estimates. Fig. 7. Block diagram of the cascade of USN and RLSN predictors used for predictive synchronization and restoration of corrupted velocity samples [13]. Unevenly spaced stream of velocity samples, and the corresponding time instants, are stored in a shift register. The USN algorithm is used to convert the unevenly spaced stream of samples into an evenly spaced stream of samples. The RLSN predictor is applied in smoothing the output of the USN filter. The error due to this unsynchronized sampling rate is called jitter. To overcome this problem, knowledge of the time instants at which the velocity samples become available is included to improve the accuracy of the velocity estimates in [37] by using the unevenly sampling Newton (USN) algorithm [38]. A practical implementation strategy employing a shift register is described. As an advanced stage of development of the velocity estimation application, the smoothing of the evenly spaced sequence of velocity samples provided by the approach of [37] is discussed in [13]. It is noted that, by applying the USN algorithm, noise in the time intervals between successive velocity samples is transformed into wide-band noise that requires filtering. This noise originates from physical pulse lengths that deviate randomly from the nominal value and from amplitude and time instant quantization. Since the use of conventional low-pass filters would introduce undesired delay to the signal, they are not feasible for this purpose. Instead, the RLSN algorithm is applied in smoothing the output of the USN filter. The proposed velocity measurement principle is illustrated in Fig. 7. Significant reductions in estimation error are reported as compared to using the USN filter alone. It is argued that the smoothed velocity data is of equal quality as that produced by expensive, high-resolution pulse encoders. The use of an application-specific FIR polynomial predictive filter is suggested for the smoothing of a tachometer signal in [24]. Different types of electrical interference degrade the quality of the analog tachometer signal, including wideband ripple, impulsive noise, power line frequency and its harmonics, as well as narrow-band noise due to, e.g., mechanical system resonances. In [24], the polynomial prediction constraints are accompanied by further requirements that en-

9 884 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 46, NO. 5, OCTOBER 1999 Fig. 8. Velocity as a function of time. Typical velocity curve of an elevator car (dashed line: ideal velocity; solid line: velocity based on the pulse encoder output). sure the removal of disturbing noise at a certain frequency, corresponding to the power line frequency, by applying similar optimization method as with the design of the H N FIR predictor. As a result, a filter for simultaneous smoothing of the tachometer signal and removal of power-line artifacts is obtained. A class of predictive analog filters is introduced in [39]. The design of those predictors is done using model transfer functions of augmented FIR predictors designed in the discrete-time domain. -to- domain mapping is realized using the inverse bilinear transformation. Such analog filters can be implemented with active RC structures, using the state-variable structure for biquads and a single-op-amp structure for real poles and zeros. The application examples include, among others, polynomial prediction for sensor signal smoothing. B. Acceleration Measurement The detection of angular acceleration can provide improved performance, e.g., in the motion control of servo drives. Acceleration feedback has been reported to improve the transient performance of a flexible mechanical system [14] and to provide higher stiffness for the drive without requiring higher bandwidths of the velocity and position control loops [40]. Although the acceleration feedback could be used to improve the performance of motor drives, acceleration is seldom measured due to unsatisfactory results of most existing methods, which typically produce either inaccurate, noisy, or delayed acceleration estimates. Direct differentiation of the velocity signal obtained from a pulse encoder by the backward difference operator amplifies inaccuracies to a level that cannot be accepted in practical control applications. Thus, prior research based on differentiating the angular velocity usually assumes a lownoise velocity signal, high-performance pulse encoder, or long sampling interval [41], [42]. In practice, accurate encoders with high pulse rate are expensive, low-noise velocity signals are not typical in industrial environments, and long sampling intervals cause undesired delay. Digital differentiating filters can attenuate the noise efficiently [43] but, at the same time, cause additional delay which also degrades the signal in realtime applications. Dedicated acceleration sensors could provide moderate results. However, angular accelerometers with an unlimited rotation range [14] are not widely available. Besides, motor drive developers would prefer to use only one feedback transducer, an optical pulse encoder or a tachogenerator. Therefore, the acceleration signal should be derived from the encoder pulses that are already available for position or velocity measurement. Recently, three alternative approaches for acceleration measurement have been proposed by the authors of this paper. These approaches employ a cascade of the backward difference operator and an RLSN predictor [19], predictive FIR differentiator [22], and recursive differentiator [28]. In the following, an example is given to illustrate the capabilities of the proposed systems. The test signal is a simulated velocity curve of an elevator car, encountered in high-performance elevator drives [11]. The velocity signal

10 VÄLIVIITA et al.: POLYNOMIAL PREDICTIVE FILTERING IN CONTROL INSTRUMENTATION 885 Fig. 9. Acceleration signal produced by applying the scaled backward difference approximation to the velocity signal of Fig. 8. The applied sampling interval is 1 ms. derived directly from the encoder pulses using the practical specifications of [13] is illustrated in Fig. 8. If the acceleration was derived as a time-derivative approximation using the difference between two consecutive velocity samples, the resulting acceleration signal would be useless as such, as illustrated in Fig. 9. However, by filtering the velocity signal with the recursive differentiator [28] of Fig. 5, the resulting acceleration estimate is drastically improved without harmful delay, as depicted in Fig. 10. During steadystate acceleration, the acceleration estimate follows the true acceleration curve accurately, but after transient conditions, when the applied polynomial model is temporarily violated, the estimated acceleration overshoots. It should be noted that the recursive differentiator produces an acceleration curve with small low-frequency oscillations because the gain peak in its slope band is considerably lower than it is in the two other approaches [19], [22]. C. Compensation of Control Delay The applications for polynomial prediction in the field of motor drives are mainly related to motion control, which is a natural application domain due to the underlying assumptions on which the polynomial prediction is built. However, the polynomial prediction has been considered for other applications as well. For instance, computing time delay may have greatly degrading effects on control system performance. The effects of nonzero computing delay can be classified as delay or loss problems. The delay problem occurs when the timevarying computing delay is less than the constant sampling interval, and the loss problem exists when the computing delay is greater than the sampling interval. In [44], a nearly allpass predictor to be used as a predictive compensator of the computing delay is proposed. The presented add-on prediction scheme is practical when the primary output of the controller can be approximated by a piecewise low-degree polynomial. Other applications for polynomial prediction include estimation of primary power signal subject to time-varying Rayleigh fading radio channel [45]. Hybrids of a median filter and polynomial predictive filters, in turn, have been applied to medical signal processing [46] and restoration of audio signals [47]. VI. FUTURE CONSIDERATIONS Future research in the field of polynomial predictive filters and differentiators includes combining conventional prediction methods with complementing neural and fuzzy technologies. These approaches will become practical for embedded applications together with the development of high-performance and low-cost signal processors. A typical fuzzy controller [48] is depicted in Fig. 11. It can be observed that there are two inputs: the error between the feedback and reference samples and the first-order difference of that error. The problem with that approach is that the power of the additive noise in the error signal is amplified by the backward difference operator. If the error signal can be approximated as a low-degree polynomial, polynomial prediction schemes could improve the performance of the fuzzy system by reducing the noise problem.

11 886 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 46, NO. 5, OCTOBER 1999 Fig. 10. Acceleration as a function of time. Typical acceleration curve of an elevator car (dashed line: ideal acceleration; solid line: acceleration estimate obtained by applying the optimized cascade connection of recursive differentiator and a polynomial estimator [28] to the velocity signal of Fig. 8. Fig. 11. Fuzzy controller [48]. VII. CONCLUSION In this paper, we have reviewed polynomial predictive filtering algorithms and methods, their design and implementation approaches, as well as successful applications in the field control instrumentation. The time-domain-modelingbased polynomial predictive filters and differentiators have shown their indisputable advantages in several applications. Their attractiveness is a natural result of the striven possibility to develop virtually delayless filters with respect to the primary component of the measurement signal. Successful applications were found, in particular, in the field of motion control, where the inertia of masses prevents sudden movements that would violate the underlying polynomial assumptions. Other viable applications include signals such as temperature, power, and the amplitude of magnetic flux.

12 VÄLIVIITA et al.: POLYNOMIAL PREDICTIVE FILTERING IN CONTROL INSTRUMENTATION 887 APPENDIX In the case of the polynomial predictive FIR filter of Heinonen and Neuvo [6], estimates of future values of a signal are calculated by applying FIR filtering. Since the input signal is modeled as an th-degree polynomial, we can substitute the expression of (1) for the input signal in (5), yielding The Lagrange function can be constructed for the H N FIR predictor as (A7) equations, corre- Equation (A1) can be separated into sponding to exponents as (A1) By setting the derivatives with respect to equal to zero leads to (A8) (A2) This equation is solved for where. The polynomial coefficient of the obtained equations can then be removed by algebraic manipulation, and obtain (A3) Equation (A3) expresses the constraints that the filter coefficients must satisfy. However, if, additional constraints can be set. In the case of the FIR filter, the formula for noise gain (NG) is (A4) assuming that the additive noise has a flat power spectrum [49]. The H N FIR predictor is thus designed such that, after satisfying the constraints of (A3), the remaining degrees of freedom are spent on minimizing the NG of (A4). This can be accomplished by utilizing the method of Lagrange multipliers [29]. The Lagrange function has the form (A5) where is the function to be minimized, are the parameters of are the constraints to be satisfied, and are the Lagrange multipliers. The global minimum of the Lagrange function is obtained analytically by setting the partial derivatives with respect to all of the arguments equal to zero. The resulting set of linear equations can then be solved for the values to minimize. In the case of the H N FIR predictor, the noise gain of (A4) is considered as the function to be minimized, and equations of the form of (A3) are the constraint functions (A6) (A9) which is substituted into (A6). The resulting equations are then solved for, which are finally backsubstituted into (A9). The resulting equations in closed form for, and are available in [6]. For instance, with, the filter coefficients become where. REFERENCES (A10) [1] J. Makhoul, Linear prediction: A tutorial review, Proc. IEEE, vol. 63, pp , Apr [2] P. S. R. Diniz, Adaptive Filtering: Algorithms and Practical Implementation. Norwell, MA: Kluwer, [3] S. Haykin, Neural Networks: A Comprehensive Foundation, New York: Macmillan, [4] B. Kosko, Fuzzy Engineering, Upper Saddle River, NJ: Prentice-Hall, [5] B. K. Bose, Ed., Power Electronics and Variable Frequency Drives: Technology and Applications, New York: IEEE Press, [6] P. Heinonen and Y. Neuvo, FIR-median hybrid filters with predictive FIR substructures, IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, pp , June [7] P. Händel, Predictive digital filtering of sinusoidal signals, IEEE Trans. Signal Processing, vol. 46, pp , Feb [8] L. A. Zadeh, Information granulation and its centrality in human and machine intelligence, in Proc. IEEE Int. Conf. Systems, Man, and Cybernetics, Orlando, FL, Oct. 1997, pp [9] S. Väliviita, X. Z. Gao, and S. J. Ovaska, Polynomial predictive filters: Complementing technique to fuzzy filtering, in Proc. IEEE Int. Conf. Systems, Man, and Cybernetics, San Diego, CA, Oct. 1998, pp [10] C. S. Williams, Designing Digital Filters, Englewood Cliffs, NJ: Prentice-Hall, [11] S. J. Ovaska, Improving the velocity sensing resolution of pulse encoders by FIR prediction, IEEE Trans. Instrum. Meas., vol. 40, pp , June [12] Y. Dote, K. Saijo, and S. Nishizuka, Adaptive digital filter with variable structure applied to processing signals from tachometer and torque meter, in Proc. Int. Power Electronics Conf., Tokyo, Japan, 1983, pp [13] J. Pasanen, O. Vainio, and S. J. Ovaska, Predictive synchronization and restoration of corrupted velocity samples, Measurement, vol. 13, pp , July [14] I. Godler, A. Akahane, K. Ohnishi, and T. Yamashita, A novel rotary acceleration sensor, IEEE Contr. Syst. Mag., vol. 15, pp , Feb

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Yli-Hankala, DC-level detection of burst-suppression EEG, Methods Inf. Med., vol. 33, pp , [47] P. T. Harju, S. J. Ovaska, and V. Välimäki, Delayless signal smoothing using a median and predictive filter hybrid, in Proc. 3rd Int. Conf. Signal Processing, Beijing, China, Oct. 1996, pp [48] T. Gupta, R. R. Boudreaux, R. M. Nelms, and J. Y. Hung, Implementation of a fuzzy controller for DC-DC converters using an inexpensive 8-b microcontroller, IEEE Trans. Ind. Electron., vol. 44, pp , Oct [49] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, Sami Väliviita (S 97 M 99) received the M.Sc. (Tech.) and the D.Sc. (Tech.) degrees in electrical engineering from Helsinki University of Technology, Espoo, Finland, in 1996 and 1998, respectively. From 1996 to 1998, he was with Helsinki University of Technology as a Research Scientist. Since then, he has been with Tellabs Oy, Espoo, Finland, as a Design Engineer. His current research interests include telecommunications and signal processing applications, in particular, predictive filtering and transmission technology. Dr. Väliviita is the Technical Chair of the 1999 IEEE Midnight-Sun Workshop on Soft Computing Methods in Industrial Applications. Seppo J. Ovaska (M 85 SM 91), for a photograph and biography, see this issue, p Olli Vainio (S 84 M 88 SM 94) received the Diploma Engineer and Doctor of Technology degrees in electrical engineering from Tampere University of Technology, Tampere, Finland, in 1984 and 1988, respectively. He has held research and teaching positions with Tampere University of Technology and the Academy of Finland. During , he was a Visiting Scholar at the University of California, Santa Barbara, working on high-speed integrated circuit design. He is currently a Professor of Computer Systems Engineering, Tampere University of Technology. He is also Docent of Microelectronics, Lappeenranta University of Technology. His research concerns DSP algorithms and their implementations.