The increased power of small computers makes the use of parameter estimation methods attractive. Such methods have a number of uses in analytical chemistry. When valid models are available, many methods work well, but when models used in the estimation are in error, most methods fail. Methods based on the Kalman filter, a linear recursive estimator, may be modified to perform parameter estimation with erroneous models. Modifications to the filter involve allowing the filter to adapt the measurement model to theexperimental data through matching the theoretical and observed covoriance of the filter innovations sequence. The adaptive filtering methods that result have a number of applications in analytical chemistry.

Adaptive Kalman filtering

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A Savitzky-Golay (SG) filter, widely used in signal processing applications, is a finite-impulse-response low-pass filter obtained by a local polynomial regression on noisy observations in the least-squares sense. The problem addressed in this paper is one of optimal order (or filter length) selection of SG filter in the presence of non-Gaussian noise, such that the mean-squared-error (or risk) between the underlying clean signal and the SG filter estimate is minimized. Since mean-squared-error (MSE) depends on the unknown clean signal, direct minimization is impractical. We circumvent the problem within a risk-estimation framework, wherein, instead of minimizing the original MSE, an unbiased estimate of the MSE (which depends only on the noisy observations and noise statistics) is minimized in order to obtain the optimal order. The proposed method gives an unbiased estimate of the MSE considering SG filtering in the presence of additive noise following any distribution with finite first- and second-order statistics and independent of the signal. The SG filter's order and length are optimized by minimizing the unbiased estimate of MSE. The denoising performance of the optimal SG filter is demonstrated on real-world electrocardiogram (ECG) signals as well as signals from the WaveLab Toolbox under Gaussian, Laplacian, and Uniform noise conditions. The proposed denoising algorithm is superior to four benchmark algorithms in low-to-medium input signal-to-noise ratio (SNR) regions ( $-5$ dB to 12.5 dB) in terms of the SNR gain.

Adaptive Savitzky-Golay Filtering in Non-Gaussian Noise

One of the major problems associated with the control of flexible structures is the estimation of system states. Since the parameters of the structures are not constant under varying loads and conditions, conventional fixed parameter state estimators can not be used to effectively estimate the states of the system. One alternative is to use a state estimator which adapts to the condition of the system. One such estimator is the Kalman filter. This filter is a time varying recursive digital filter which is based upon a model of the system being measured. This filter adapts the model according to the output of the system. Previously, the Kalman filter has only been used in an off-line capacity due to the computational time required for implementation. With recent advances in computer technology, it is becoming a viable tool for use in the on-line environment. A distributed Kalman filter implementation is described for fast estimation of the state of a flexible arm. A key issue, is the sensor structure and initial work on a distributed sensor that could be used with the Kalman filter is presented.

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A fast algorithm for control and estimation using a polynomial state-space structure

The design of nonrecursive and recursive digital filters using linear least squares or linear minimum variance is described. This method of design requires a model, and since polynomial approximations are widely used in laboratory automation, a polynomial state model is first utilized. Then an exact scalar signal model is employed that results in a completely time-varying filter. For both models, the design leads to the recursive Kalman filter. The operation of the filters for noise reduction is first demonstrated for simulated data, and design forms are compared. For the exact scalar signal model, noise reduction and peak separation are demonstrated using simulated data and data obtained from a mass spectrometer. By using the correct model, excellent peak separation and noise reduction can be obtained.

F.N. Cornett

Papers

Linear digital filtering for laboratory automation