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

Brubaker, Thomas

Lee, Gordon K. F.

Shults, James R.

English

NASA Technical Reports Server

LN9 1 - 2 2 3ii, 2
A FAST ALGORITHM FOR CONTROL AND ESTIMATION USING A POLYNOMIAL STATE-SPACE STRUCTURE
James R. Shults and Thomas Brubaker
Colorado State University
Gordon K. F. Lee
North Carolina State University
ABSTRACT
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 computation time required for
implementation. With recent advances in computer technology, it is becoming a viable
tool for use in the on-line environment. The following paper describes a distributed
Kalman filter implementation 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.
INTRODUCTION
The parameters of flexible structure systems are generally dynamic. They change
under varying load and environmental conditions. When there is a need to control such
dynamic systems, these parameters must be measured or estimated. These systems are
usually very complex and often more parameters are needed for control than can be
measured. The parameters which cannot be measured must therefore be estimated in some
manner.
With the rapid evolution of computers, the Kalman filter is becoming an excellent
tool for estimation of system parameters. Previously, this filter could only be used
in off-line applications such as filtering of laboratory data Brubaker. Now, it is
becoming useful in on-line environments for state estimation.
The Kalman filter is a time-varying digital filter which is based upon a model of
the system being studied. The filter uses signals from the system to adapt the model
and estimate system parameters. These parameters along with the measured signals can
then be used to control the system. A key issue in the use of parameter estimation is
the sensor distribution and use of appropriate sensor types. Along with this is the
data fusion issue from sensors to provide appropriate control information. Here we only
describe the estimation procedure with comments on the sensor issue. A block diagram
of the filter structure is shown in Fig. I for a single input system. For multiple
sensors, multiple filters could be deployed and data fusion done with filter outputs.
This work was supported by NASA Subcontract G-1926-I.
PRE-,C_ING PAGE BLANK NOT F_LMED
https://ntrs.nasa.gov/search.jsp?R=19910012999 2020-03-19T17:53:22+00:00Z
KALMAN FILTER THEORY
The Kalman filter is a time-varying recursive digital filter which estimates the
states of a system from one or more sensor signals. The filter operates on time domain
signals using linear least squares estimation that utilizes all of the past data from
the output signals. This estimation can provide separation of signal components. These
components can be used to determine the states of the system. The Kalman filter can
also improve noise reduction on the signals under consideration [Brubaker].
The first step in the design of a Kalman filter is the choice of a linear or
linear_zed signal model that describes the signal or serves as an approximation to the
signal. One of the most flexible of these models is a polynomial. The input signal to
the filter (output from the system), z(t), is represented by a polynomial of order m.
At time t = nT, where T is the sampling period of the system, the state vector for the
system is given by
z(n_ -
z
dz
dt
d mz
t-nT
(I)
Here, the system is being represented in canonical state-space form with the components
of the state vector being the derivatives of the polynomial model. When a system is to
be represented in a different state-space form, a linear transformation can be performed
to change the state vector to the desired form.
To use the polynomial model for the Kalman filter, the state vector must be
redefined using a Taylor Series representation for each element of z(nT). The resulting
state vector is
x(nT) -
z
x --
T m dz m
m! dtm
t-nT
(2)
The state of the system at t = (n+h)T can now be described in terms of the state at t=nT
by the relationship
x [ (n+h) T] -_ (h) x [nT] (3)
42
where @[h] is the following (m+l) X (m+l) state transition matrix:
I h .. h _
0 1 .. m.hm-i
(I,[h] - . . ..
0 0 .. 1
(4)
The filter is implemented by first producing forecasts of the estimate
xl (nT) - (_(1)x[ (n-1) T] (5)
and covariance matrix
S1(nT) - @(1)S[(n-1)T]@Z(1) + Q
at t = nT using the previous estimate, x[(n-l)T], and covariance matrix, S[(n-I)T].
These forecasts are obtained by using the state transition relationship given in (4).
The matrix Q is the covariance of the driving noise. It allows the designer to "fade"
the effects of past inputs. The covariance forecast is then used to calculate the
Kalman Gain Matrix,
K(nT) - Si (nT)Mr[g 2 + MSI (nT)M r]-i (7)
The term o z is the variance of the measurement noise. The matrix M is a row matrix
which relates the measurable state variables to the actual measurements. For this
filter, the measurements are assumed to be components of the state-vector z(Nt) given
in (i). Therefore, M relates the estimate vector x(Nt) given in (2) to the measured
components of z(Nt). The Kalman Gain Matrix is then used to obtain the covariance
estimate,
S(nT) - [I - K(nT)M]S 1(nT) (8)
and the state-vector estimate,
x(nT) - x i (nT) + K(nT) [y(nT) - Mx i(nT)] (9)
In (9), the term y(nT) is the data measurement vector at t=Nt which consists of the
measured components of z(nT). Equations (i) through (9) are the basis for the Kalman
filter. A complete derivation for these equations can be found in many texts for
example, Liebelt and Meditch.
43
KALMAN FILTER DESIGN
With the choice of a polynomial model for the input signal, the design of a Kalman
filter involves the determination of a few key parameters. These parameters are set
according to the system and design specifications.
The first group of parameters which must be determined are associated directly
with the properties of the system. The first is the sampling period, T. This period
is usually set according to the nyquist rate,
fs - I__ k 2f m (I0)
T
where fm is the maximum frequency of the bandlimited input signal and fs is the
sampling frequency.
The second parameter is the variance of the sensor noise, o z . This parameter is
a property only of the sensors which are employed. Another term which is determined
from the sensors which are utilized is the Data Measurement Vector. This is the number
of terms in the state vector of equation I which can be directly measured.
The remaining filter parameters are determined using both design and system
specifications. The order of the polynomial model must be set according to the sampling
period and the angular velocity of the input signal oscillations. The calculation time
of the filter is larger for higher orders. The order must be small enough to allow the
calculation time to be smaller than the sampling period. On the other hand, the order
must be large enough to allow the filter to track the input signal well.
The key design parameter of the Kalman filter is the covariance matrix of the
driving noise, Q. For this paper, the driving noise is assumed to be uncorrelated white
noise. This simplifies the matrix Q to a diagonal matrix. After testing different
forms for the Q matrix, little difference was found. Therefore, the matrix Q was taken
to be the identity matrix times a constant, f.
The constant f is known as the fading factor. This term determines how the filter
will handle past data. When f = O, the filter is simply an expanding memory filter.
All past data is used evenly to calculate the present estimate. This will cause the
variance of the estimate and its covariance matrix to decrease to zero as time
increases, but deterministic errors will become large. If the fading factor is greater
than zero, more emphasis is placed on the present sample than on past samples. Thus,
the past samples are faded from the filter's memory. Larger values of f cause past
samples to fade more quickly. The fading of past samples causes the Kalman filter to
have a much smaller deterministic error, but the variance of the estimate and its
covariance approach the values at the input.
One way to minimize the deterministic error associated with a small fading factor
is to periodically reinitialize the Kalman filter. This requires establishing a
relationship between the frequency of initialization and the fading factor. The fading
factor should be the minimum value for which the deterministic error just prior to
reinitialization is within specifications. This value will cause the Kalman filter to
provide maximum noise reduction while meeting deterministic error specifications.
44
To reinitialize the Kalman filter, a nonrecursive filter is used. This filter
utilizes a window of the past samples to estimate the state vector and its covariance.
These parameters are then passed to the Kalman filter.
DESIGN EXAMPLE
The Kalman filter was tested in two separate operating conditions. The first was
with data acquired from the hub of the flexible arm at CSU. The second test was with
an eighth order model of the flexible arm. This test included the simulation of the
system with control in a closed-loop environment. A program has been developed which
allows the design and testing of these and other filter structures. The results of
these experiments are discussed in the following paragraphs.
The data acquired from the hub of the flexible arm was position information
sampled at a rate of i00 samples per second. A polynomial model of order 3 was required
to accurately represent the position of the hub. Since velocity information was not
available, a Data Measurement Vector of one was used. The variance of the sensor noise
was arbitrarily set at 10 .4 . The filter was reinitialized every i00 samples, I second,
and a fading factor of 10 .9 was used.
The outputs of the filter are shown in Figs. 2 through 5. Figure 2 demonstrates
the ability of the filter to estimate the position signal. Figure 3 displays the
estimate of the velocity. Figures 4 and 5 show the estimates of the second and third
derivatives of the position signal. No actual data was available for the second and
third derivatives. Therefore, the accuracy of these estimates could not be determined.
The second test of the filter involved an eighth order simulation of the flexible
arm with the filter outputs used as control feedback. The simulation provided the
position of the tip as input to the filter. A third order polynomial model was used for
the Kalman filter. The variance of the sensor noise was taken to be 10 .4 and the fading
factor was 5x10 -12. The feedback control was implemented as simple proportional position
only negative feedback. The feedback gain was set to 2.5
The results of this simulation are shown in Figs. 6 through 9. Figure 6 shows a
comparison of the position signal before and after the Kalman filter. Figures 7 through
9 show the first, second and third derivatives of position.
FUTURE WORK
Future work on the Kalman filter consists of implementation issues. A good design
method has been set forward, but many implementation issues have not been addressed.
The major implementation problem comes with the selection of hardware to run the Kalman
filter. This hardware must consist of a microprocessor which is fast enough to meet the
sampling rate, but inexpensive enough to make its use feasible. Another major problem
comes in the sensors which will provide state information for the Kalman filter. These
sensors must be selected for each application such that they provide accurate
information with a low noise level. A final problem which must be addressed is a method
to download the Kalman filter program to the hardware. This downloading could be done
by the design program with the appropriate interfaces. After these problems have been
resolved, the Kalman filter will provide a very effective state estimator in many
applications.
45
A NEW SENSOR
Within the past few months our group has designed and performed initial tests on
a distributed fiber optic sensor. Here, the fiber is connected to a flexible structure
over a one meter length. The fiber is excited with a milliwatt laser and the defraction
pattern out of the end is used to provide an estimate of displacement. Figures I0 and
II illustrate the results via change in the output pattern. For implementation a CCD
memory could be used to store the pattern and subsequently the information is driven
into a computer where basic pattern recognition techniques are used to generate good
estimates of displacement. A well designed system can also be used to estimate
velocity. Within the context of this paper, a two-dimensional Kalman filter can be used
to estimate parameters. Note that in Figs. I0 and ii, black and white images are shown.
In a physical system color will be used.
CONCLUSIONS
An investigation has been started into the usefulness of the Kalman filter as a
state estimator in an on-line environment. Previously, the filter has been used
strictly in an off-line capacity to do data analysis. With the advances in computing
speed, the filter is now becoming feasible as a real-time state estimator.
A Kalman filter based on a polynomial state-space model has been tested on
flexible structure data. This filter has proven to give excellent state-estimation and
noise reduction in such systems. The polynomial model provides for a very easy design
of the filter. Do to the ease of design, a program has been written to provide
assistance in the design process.
The Kalman filter design program provides a very straight forward design
methodology with an interactive graphics approach. This approach allows the designer
to see how well the filter works and the effects of changes in the design parameters.
When the program is combined with the appropriate hardware, a very effective state
estimation tool will become available for use in the real-time environment.
REFERENCES
I. Brubaker, T. A. e_tt. al. Linear Digital Filtering for Laboratory Automation,
Proceedings of the IEEE, Vol. 63, No. I0, October 1975.
2. Liebelt, P. B., An Introduction to Optimal Estimation, Addison-Wesley 1967.
3. Medich, J. S., Stochastic Optimal Linear Estimation and Control, McGraw-Hill 1969.
46 ¸
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Figure 10. Fiber-Optic Output With Zero Beam Displacements
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ORIGINAL PAGE IS
OF _ _ALrI'Y


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

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