Data processing: generic control systems or specific application – Generic control system – apparatus or process – Optimization or adaptive control
Reexamination Certificate
1998-04-28
2001-11-06
Gordon, Paul P. (Department: 2121)
Data processing: generic control systems or specific application
Generic control system, apparatus or process
Optimization or adaptive control
C700S040000, C700S042000, C701S045000
Reexamination Certificate
active
06314327
ABSTRACT:
TECHNICAL FIELD
The present invention herein resides in the art of predicting a function value from a set of past derivative values. More particularly, the present invention relates to integrators that provide high accuracy for polynomials and oscillatory functions. Specifically, the present invention relates to a predictor-corrector integrator that provides improved accuracy and larger stability regions over known polynomial-based integrators.
BACKGROUND ART
In certain applications it is very beneficial to be able to predict the future velocity of a moving object. For example, such information could be employed to calculate velocities in relation to automobile air-bag use. This information could then be employed to more quickly deploy air-bags and/or to control their rate of deployment. Other instances of where prediction of future velocity values may be implemented is in controlling the operation of the moving object, in regulating the spacing of moving vehicles on an automated highway system and the like.
Past integrators and related methods for predicting velocity from accelerometer samples have been polynomial based and have been documented as not providing accurate predictions. A skilled artisan is aware that a general explicit linear multi-step integration method can be written as
f
(
t
)
=
∑
n
=
1
N
a
n
(
0
)
f
(
t
-
nT
)
+
T
∑
n
=
1
N
a
n
(
1
)
f
′
(
t
-
nT
)
,
(
1
)
where T is the uniform sampling interval. Traditional methods usually assume knowledge of only one function value, and that is generally chosen as the most recent value, to give a formula of the form
f
(
t
)
=
a
1
0
f
(
t
-
T
)
+
T
∑
n
=
0
N
-
1
a
n
1
f
′
(
t
-
nT
)
.
(
2
)
Typically, samples of acceleration are integrated to obtain a velocity value. Thus, samples of f′(t) represent acceleration samples and f(t−T) represents a corresponding velocity value.
When the coefficients in formulas similar to Equation (2) are computed so that the formula is accurate for the highest degree polynomials possible, then one obtains the well-known Adams-Bashforth (AB) method. For example, N=1 gives the Euler formula
ƒ(
t
)=ƒ(
t−T
)
+T
ƒ′(
t−T
), (3)
or with N=2 one obtains the AB two-step method,
f
(
t
)
=
f
(
t
-
T
)
+
T
[
3
2
f
′
(
t
-
T
)
-
1
2
f
′
(
t
-
2
T
)
]
(
4
)
The three-step AB formula is
f
(
t
)
=
f
(
t
-
T
)
+
T
(
23
12
f
′
(
t
-
T
)
-
16
12
f
′
(
t
-
2
T
)
+
5
12
f
′
(
t
-
3
T
)
)
(
5
)
valid for polynomials of the third degree. Such formulas can be derived by using Lambert's equations and by using matrix methods to find the coefficients in the formula. These AB methods are explicit methods, since they are based on past samples of the function and derivative.
Predictor-corrector integration methods employ both a predictor, which is an explicit type of formula as given above, as well as a corrector, which is an implicit formula that includes a current value of the derivative, i.e., f′(t). The Adams-Moulton (AM) family of formulas is the usual companion to the AB family of equations described above and is similarly polynomial-based. The coefficients in the corrector formula,
f
(
t
)
=
b
1
(
0
)
f
(
t
-
T
)
+
T
∑
n
=
0
N
-
1
b
n
(
1
)
f
′
(
t
-
nT
)
(
6
)
are chosen so that the formula is accurate for polynomials of the highest degree possible. The AM-one (trapezoidal) and AM-two step formulas are given, respectively, by
ƒ(
t
)=ƒ(
t−T
)
+T[
1/2ƒ′(
t
)+1/2ƒ′(
t−T
)] (7)
ƒ(
t
)=ƒ(
t−T
)
+T[
5/12ƒ′(
t
)+8/12ƒ′(
t−T
)−1/12ƒ′(
t−
2
T
)] (8)
Any method utilizing the above equations for predicting velocity or for determining any appropriate function from knowledge of its derivative's values are based upon the polynomial and have not been established as providing the most accurate predictions thereof.
DISCLOSURE OF INVENTION
Based on the foregoing, it is a first aspect of the present invention to provide a method for predicting future velocity or function values utilizing acceleration or derivative values related thereto.
Another aspect of the present invention, as set forth above, is to use a sensor associated with an object for predicting the velocity or the function values of the object.
Still another aspect of the present invention, as set forth above, is to provide a processor with a prediction formula, which when provided with predetermined variables such as a sampling rate and a frequency value, generates prediction coefficients for the prediction formula, which in turn predicts the velocity or other function value of the object.
A further aspect of the present invention, as set forth above, is to generate the prediction coefficients by either Gaussian elimination or by computing eigenvectors.
Yet a further aspect of the present invention, as set forth above, is to correct the predicted future velocity or function value by including a current sample of the acceleration or derivative value in a prediction-correction formula.
The foregoing and other aspects of the present invention, which shall become apparent as the detailed description proceeds, are achieved by a method for calculating a future function value associated with an object from derivative values associated with the object, to then control operation of the object, comprising the steps of: using a sensor coupled to the object for generating derivative values; predetermining a sampling rate T of the derivative values; determining a highest frequency W in the sampled signal; acquiring at least one function value from the sensor; determining a value for &tgr; by multiplying 2*T*W, wherein &tgr; is a Nyquist value, and wherein &tgr; must always be less than one; computing prediction coefficients for use in a prediction formula, wherein the prediction coefficients are found by minimizing the integral
∫
-
2
π
TW
2
π
TW
&LeftBracketingBar;
1
-
a
k
(
0
)
ⅇ
-
ⅈ
k
ω
-
∑
n
=
1
N
a
n
(
1
)
ⅈ
ωⅇ
-
ⅈ
n
ω
&RightBracketingBar;
2
ⅆ
ω
wherein said prediction formula is in the form of
f
(
t
)
=
a
k
(
0
)
f
(
t
-
kT
)
+
T
∑
n
=
1
N
a
n
(
1
)
f
′
(
t
-
nT
)
,
to generate a future function value; and utilizing the future function value to control operation of the object.
In accordance with other aspects of the present invention, as set forth above, the method can generate corrected-predicted values by incorporating current derivative values of the monitored object. The method may also utilize a look-up table to store prediction coefficients based on preselected criteria.
BRIEF DESCRIPTION OF THE DRAWING
The drawing is a schematic representation of a system which incorporates the concepts of the present invention.
REFERENCES:
patent: 5394322 (1995-02-01), Hansen
patent: 5502658 (1996-03-01), Relin
patent: 5513109 (1996-04-01), Fujishima
patent: 5546307 (1996-08-01), Mazur et al.
patent: 5559697 (1996-09-01), Wang
patent: 5583771 (1996-12-01), Lynch et al.
patent: 5602736 (1997-02-01), Toya et al.
patent: 5608628 (1997-03-01), Drexler et al.
Mugler Dale H.
Wu Yan
Gordon Paul P.
Renner Kenner Greive Bobak Taylor & Weber
The University of Akron
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