Civil Engineering Reference
In-Depth Information
results in many applications. However, in real-time applications of control and prediction, recursive
methods are generally needed. Many recursive identification methods are limited to single input-single-
output systems. For multiple sensor input data, recursive methods are limited to a general recursive least
squares (RLS) form. Within this general form are several applicable adaptations, including RLS with a
forgetting factor (RLS-FF), gradient approaches, and the Kalman filter. (These adaptation methods may
even be combined as shown by Pappas et al., 1988.)
The Kalman filter is the most general of the multiple input RLS methods used to estimate time varying
parameters (Ljung, 1987). Much of the current estimation research has focused on using Kalman filtering
for state (parameter) estimation. It is an optimal estimator for time varying parameters, minimizing the
state error variance at each time. The Kalman filter is optimal in the sense of estimating a pure signal
from one contaminated with white or Gaussian noise. Similarly, the Weiner filter minimizes the sum of
the state error variance over time, and thus, it is mainly suited for stationary parameter estimation. If
appropriately applied, the Kalman filter has been demonstrated to yield better results than the many of
the other RLS based methods (Danai and Ulsoy, 1987).
When all the parameters of the pure signal and noise are known, the (optimal) Kalman filter can be
easily obtained. It has been shown that the exponential convergence of the Kalman Filter makes it robust
to modeling errors and faulty covariance statistics (Ydstie, 1985). However, the Kalman filter has been noted
to be difficult to apply because of the choosing of the appropriate noise intensities, when they are functions
of time. Poor choices of noise covariances often cause filter divergence to occur. Adaptive noise estimation
schemes can eliminate the necessity of choosing
a priori
noise covariances. However, these techniques have
also been found to be difficult to implement, and when the noise is non-Gaussian the schemes may fail
(Ydstie, 1985; Ulsoy, 1989). Previous solutions addressing this difficulty have included the use of parallel
Kalman filters with variable memory length using variable forgetting factors (Wojcik, 1987).
It is also important to note that in real-time applications the matrix inversion requirement of the
multiple input Kalman Filter is computationally intensive. This has lead many authors to avoid the
Kalman filter for multiple input/multiple output (MIMO) systems. An improved Kalman filter approach
for multiple inputs is that taken by Maybeck and Pogoda (1989) who used a weighted average of several
single input filters to determine a parameter.
The most widely used method for estimation of a model is a least squares fit. Here data are fit to a
model form to have the minimal sum of the squares of the difference between the actual data and the
predicted model. This technique is readily modified for real-time, recursive calculations and multiple
sources of input.
Ulsoy (1989) has identified two such special cases of RLS that attain fairly good results for time-variable
parameter estimation. Many RLS approaches rely on windowing techniques (or forgetting factors) to
improve estimation of transient behavior. In some cases, this may cause the asymptotic parameter error
variation to be large (Wiberg, 1987), but this is generally not the case. In general RLS methods can be
obtained as a special case of the Kalman filter. Therefore, RLS methods including optimal estimators may
be best for the task of estimating transient responses with multiple input sources. More details on the
RLS-FF and Kalman filter approaches are provided in texts on digital control and digital signal processing.
Other methods for multi-variable input for estimation include basis functions and neural networks.
The basis function (BF) method uses a series expansion in terms of a limited number of known functions
that form a basis (such as a discrete Fourier Transform). The coefficients of the functions can be found
by applying RLS methods (Ljung and Soderstrom, 1983). BF methods work best when the number of
functions is limited and when a parameter variation can be expressed in few terms. It has also been
demonstrated that basis functions are not always robust, and require some prior knowledge of the
parameter variation to satisfactorily represent the data set. Since the time varying response of the grinding
model coefficient,
K
P
, is not known
a priori
it is difficult to limit the type and number of functions
needed to span a basis. Therefore, BF methods are not employed in this research effort.
Attempts at relating measured variables to models have included the use of pattern matching techniques
embodied as neural networks (Rangwala and Dornfield, 1990; Purushothaman and Srinivasa,1994).
Neural network approaches require no pre-determined model structure, as knowledge and relational
