NASA/TM-2002-211776
A Numerical Scheme for Ordinary Differential Equations Having Time Varying and Nonlinear Coefficients on the State Transition Matrix
Robert
E. Bartels
Langley
August
Research
2002
Center,
Hampton,
Virginia
Based
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NASA/TM-2002-211776
A Numerical Scheme for Ordinary Differential Equations Having Time Varying and Nonlinear Coefficients on the State Transition Matrix
Robert
E. Bartels
Langley
Research
Center,
National
Aeronautics
and
Hampton,
Space Administration Langley Research Center Hampton, Virginia 23681 2199
August
2002
Virginia
Based
Available from: NASA Center for AeroSpace 7121 Standard Drive Hanover, MD 21076-1320 (301) 621-0390
Information
(CASI)
National Technical Information 5285 Port Royal Road Springfield, VA 22161-2171 (703) 605-6000
Service (NTIS)
A numerical and
scheme nonlinear
for ordinary coefficients
differential equations having based on the state transition Robert
time varying matrix
E. Bartels
Abstract A variable order method of integrating initial value ordinary differential equations that is based on the state transition matrix has been developed. The method has been evaluated for linear time variant and nonlinear systems of equations. While it is more complex than most other methods, it produces exact solutions at arbitrary time step size when the time variation of the system can be modeled exactly by a polynomial. Solutions to several nonlinear problems exhibiting chaotic behavior have been computed. Accuracy of the method has been demonstrated by comparison with an exact solution and with solutions obtained by established methods.
Section 1.0. Introduction. Numerical methods to solve ordinary differential equations are at a state where efficient and accurate solutions are routine. Accuracy is often achieved by adapting time step size, which in some applications is inconvenient or impossible. It may not be possible to adapt time step size when the equations of dynamics are integrated while coupled to another solver (e.g. computational fluid dynamics desirable intervals.
(CFD) code). In other cases an arbitrarily large constant time step size may be such as when externally supplied data is applied to a system at constant time
One approach to address this dilemma has been the construction based on classical solutions. An exact numerical scheme based
of numerical schemes on the state transition
matrix (STM) is an example. One advantage of a scheme based on the STM is that it offers an exact solution of the system for arbitrary time step size. The ability to compute at arbitrary time step size is useful, for instance, when coupling a continuous system to a discretely sampled control system with discrete feed back. The system can be integrated with absolute accuracy between samples and discrete feed back with one or several steps, as required to capture the discrete nature of the problem. While numerical methods for time invariant systems based on the STM have been in use for many years 1, only recently has it been applied to time variant systems. 2 The present paper proposes a discrete form of the STM that has application nonlinear time varying (NLTV) Consider
to any linear time varying systems.
the simple first order differential 2 = A(t)x
where
X = (XI ,...,XN )T, Xn E R,
A e
R NxN
(LTV)
system,
and some
equation + B(t)u(t)
,Be
R NxN
(1) t > t o. Settingthe
dimensionof
the matrix B identical to matrix A is purely for convenience; the results are applicable to a matrix B of general form. In this paper the classical method of solving differential equations will be used to construct the numerical solution of equation (1). Between times
tm and tm+ h, where the solution
h is time step size, a discrete
approximation
will be performed
using
t m +h
x(tm + h) = _,(tm + h, t_)x(t_) + I _(tm + h,T)B(T)u(T)dT
(2)
t m
For any LTV system, series
the STM between
times tm and tm+ h is given by the Neumann
jr"
A(t_)jt
A(t 2)dt2dt _
(3) t "t'-f"
+h
tI A(tl)
tm
I
t2 A(t2)
I
tm
This result is called the Peano-Baker matrix A is constant, is this solution
A(_3)d_3d_2d_
1
-t.-...
tm
formula. Only in special equal to the more familiar
¢(t m +h,tm)-expL_.,t
Adt
cases, such as when the expression
=I+Ah+_(Ah)
2 +....
In all applications discussed in this paper, a discrete form of equation That solution may have some value if an approximation can be found accurate and robust.
Section discrete
2.0. Approximation of the A(t) matrix for an LTV analog to the continuous functions above.
X21 =x_(t
m-h)
,
u2_ = uo(t_ - h) The numerical polynomial
X2 =x_(t m)
,
Xl_ =x_(t m +h)
, u° = uo(tm) , u_, = uo(t_ + h)
solution
of equation
approximation
(1) can be obtained
of the matrix
system.
(3) will be used. that is efficient,
Consider
the
,...,
x2 =x_(t
,...,
u'/,= uo(t_ + rh)
by starting
m +rh)
with a term by term
A (t)
I
a_,l(r)=
c_,_,ir i-1 / hi-1 i=1
,
r=t-tm,r_(O,h)
(4)
for schemes of various order. For instance, one can define the O(A) scheme where A = I -1. The coefficients in equation (4) are constructed from discrete values of the terms
1
-I
of matrix A(t), that is c i n,z = ci n,z(an,l,...,
can be easily illustrate
evaluated
since A(t)
the use of equation
approximated
is known
an,z ) • These interpolation
for time steps at least up to tm+h. To
(4), the an,_ term of the continuous
by polynomial
expressions
polynomials
of orders
matrix A (t) can be
zero, one and two for _-_ (0, h) by the
following 0
O(0) scheme:
an,l (r) = c 1n,l
where
cl n,l = an,l
O(I)
an,l(r)=Cln,l+C2n,lr/h
where
Cln,l=an,l
scheme:
O(II) scheme:
2
an,_(r) =cln,_
0
,
1
0
C2n,l=an,l--an,l
/h 2
+C2n,IT/h+c3n,IT
where Cl n,l
_
In a similar manner, term by term. approximation method based
('ln,l0
,
C2
higher
--_
n,l
(.in,l
__(.in,
l
C3n,
order approximations
l
_
_:.o
(.in,l
of the matrix
.t_(.in,
l
A (t) can be constructed
The accuracy of this scheme is dependent on the accuracy of the of matrix A(t) since equation (3) is exact. Under certain circumstances on equation (3) will be an exact numerical scheme.
a
Section 2.1. Approximation of the STM for an LTV system. Using the results of section 2.0, an algorithm is constructed to compute successive terms of the Peano-Baker formula. Since polynomials of order A = I - 1 are used to approximate the matrix A(O term by term, the integration
of those
in a straight forward begins as follows:
The first two terms
manner.
(Crn,,)l = C,n,, /i
,
expressions
i=l,I
over
(tm ,tin + h) can be accomplished
are easily found.
The computation
n=l,N,
I=I,N
(5a)
n=l,N,
I=I,N
(5b)
I
_n,,(t_
+h,t_)=dn,,
+h_-'(G_,,)l
,
i=1
where
fin,z = 1 ifn
index ranging Equations and provide kernel
= 1 and fin,z = 0 otherwise.
from 1 to I. In equation
(5 a) and (5 b) compute the start of the STM.
of a recursion
that computes
The superscript
i in equations
(5a-b)
is an
(4) the same index i is used as an exponent.
the sum of the first two terms in the series The terms
(o-_,z)1 computed
the remaining
terms
in equation
equation
(3)
(5a) are the
of the STM.
Starting at the double integral (the third term in equation (3)) the following recursive algorithm computes the remaining integral terms and adds their contribution to the STM.
For j = O, J p
N
trrP-k+l_,1/(p ( 0, the outer subscript
( )1 denotes
( )1 denotes
= 0 with the third term
values
values
computed
in
from the previous
iteration of equations (6). There are a total ofJ + 3 terms of the Neumann series computed by this algorithm, combined from both sets of formulae, equations (5) and (6). For instance ifJ = 0, the first 3 terms are computed, namely the first two obtained from the former equations and that withj = 0 from the latter recursive formulae. When implemented, several indexing details are important. If/= 1, the index p ranging over 1 to (I-1) is computed once forp is computed once forp = 0. When I = 1, this solution
= 1, and the sequence
is just the exponential
with indexp
over (I-1) to 0+1)(I-1)
series for a constant
matrix
(i.e. exp(Ah))
or alternately a zero order approximation of a time varying A(t) matrix. Solution of an LTV system at this level of approximation corresponds simply to applying the exponential series of the steady state STM updated at each time step. Furthermore, when the O(0) scheme is used (I = 1) and J = -1 (i.e. only the first two terms computed), this algorithm is the explicit backward Euler method. Since this scheme uses as its basis an exact solution, it possible with enough terms to compute solutions using large time step sizes, as long as the matrix A (t) is sufficiently well behaved so as to be accurately approximated by a polynomial expansion. In the cases computed so far, relatively few terms in the series have been required
to achieve
very good accuracy.
It would be desirable to obtain a bound on the size of the leading order truncated term following the last of the J + 3 terms used in the computation. Denote the size of the leading order truncated term by Ej+s. While it may be possible to obtain Ej+s for any time
varyingsystemit is likely tobeundulyrestrictiveif basedonthemostgeneralformof theequationsabove.Ratherconsiderthe L × n order linear time invariant (LTI) equation dnxl - 2cixl, dt _ where
)_i , 1 = 1,..., L are the coefficients
empirically truncation
1 = I,...,L
of a diagonalized
system.
by the author that the largest value of the highest order term in the series error, using J + 3 terms of the series, will have for bound J+2
E j+3 _< h
c_
for
(_ilaaX)
o_ = {1 + int( J + 1)}
(J + 2)!
where
It has been found
)_n_x = max[l)_l,...,l)_L]]
of the matrix
(7)
n
• In the examples
that follow,
the largest
A(t) at each time step will be used as an estimate
magnitude
entry
of )_nl_x • By doing this,
the assumption is that this will result in a sufficiently accurate estimate of the truncation error. Equation (7) does not of course represent the total error in the scheme, which would
also include
the error in the approximation
in equation
(4).
The solution of the last term in equation (2) is facilitated by the fact that values of the STM at successive time steps are already available, which can be used to numerically evaluate the integral term. For constant step size h, write _
= _(t,_ + h,t,_)B(t,_)u(t,_)
=
+
=
+ =
A recursion relations.
for computing The vector
the vectors
of integrals
+ h, _1,,,..., _mz+2 can easily be constructed
is evaluated
by defining
from these
I _ R N where
t.,+h
I _Ii_(tm + h,T)B(T)u(T)dT tm
or an approximation
of this integral h
I
0
i=1
ciG
/hi-ldG
(8)
where
I-1 is the order,
_"= T - t m and the vector
2i = 2i (_1m,...,_m I+2) approximation
,
(equation
_lm,...,_m I+2 E R x , 2 i E R x . To illustrate (8)) can be evaluated
O(0) scheme:
I -= h21
O(I) scheme:
I-=h(21
This completes modification
development
for orders
+_22) of the method
of the procedure
where
21 = _2
where
21 =_2
that the linear and nonlinear
the procedure,
the
,
22 =_1__ The next section
outlines
system.
of the STM for an NLTV
= _4(t,x)x
dependence
zero and one by
for an LTV system.
for an NLTV
Section 2.3. Approximation in which
Suppose
2i has the functional
system.
Consider
now the case
+ B(t)u(t)
(9)
parts of the matrix
A(t, x) can be expressed
as
_4(t,x) = A(t) + Y(x) where
an,,--1 = an,,(t, _ -h)+
_F
an,,--° = an,,(t_)+Z°, ,'",
f_-11
F
an,, = an,, (G + rh) + f£,,
where
fL-) = fn,,(X(t_--h)) , Z°, = fn,,(X(tm)) ,'", fL = fn,,(X(t_+ rh)) and the coefficients _4(t,x)
can be found
the construction The nonlinear example,
1
0
an,z, an,z,..,
are the time varying
using equation
(4) where
of these coefficients terms
is a crucial
an,z are again approximated
now
part.
A local approximation
c_n,l = c_n,l(an°,l,...,an,l
step additional
explanation
by polynomial
expressions.
orders zero and one are given by
O(0) scheme:
an,l (z') = c 1n,l
O(I)
an,z (r) = cl n,z + c2 n,zr/ h
scheme:
0
where
0
c 1n,l = an,l + f£,l
where 0
0
cl n,, = an,, + f_,,
1
,
0
1
0
C2n,, = an,, -- an,, + fn,, -- f_,,
).
of
Because
is required. For
and z- _ (0, h) . The value of the solution therefore
the function
is f2,z = f_,z (y1), rest of the method
Section 3.0. computed. Example
x(t m + h) has not yet been
f2,z is not yet known.
where
21 = g(x°,...,x
follows
-_+_) is derived
that outlined
in the previous
Examples.
In this section
3.1. This example
computes j( =
t4x
This term is computed
solutions
the following ,
computed
and
by extrapolation,
from previous
solutions.
that The
section.
of LTV and NLTV
second
systems
are
order system
x(O) = O, 5c(0) = 1
or
¢=A4 where
computed
on t" 0 _
1. The exact solution
is given by the series
x(t) =
(6m - 5)(6m - 6)
k(t) =
(6m-5)(6m-6)
(6k-5)
t 6k-6
Figure 1 presents the error in the O(I), O(II), O(III) and O(IV) schemes compared with the second order implicit Euler method. Convergence to the exact solution is shown as the order of the approximation ofA (t) increases. The O(IV) scheme is exact to machine accuracy since the time variation of the matrix A (t) is exactly captured with a 4th order polynomial. This example illustrates one of the advantages sufficiently modeled with the order polynomial
of this method. If the time variation is used, the method is nearly insensitive to
the size of the time step. Using the O(IV) scheme in this example, any time step size is permissible since the exact solution is being computed. This aspect will be explored more fully in the next example.
__ _ ___ _ _______ .... e.... __,__
10° -
2nd ord. implicit Euler, 0(I) scheme, 0(11)scheme, 0(111) scheme, O(IV)scheme,
h=O.04 h=O.04 h=O.04 h=O.04 h=O.04
1 0 -2
% Errolr0-4
_A-" _c ,_
_ _ ._eoeeeO
_-J_--'.....
10 -° 10 -8 10 -1° 0
I
I
I
I
I
I
I
I
I
0.5 Figure
I
I
I
I
I
1
t
I
I
I
I
1.5
1. Percent
I
I
2
error.
Example 3.2. This example computes the response of a pitch/plunge apparatus initially oscillating as a sinusoid, due to a change in the torsion stiffness to ¼ its original value. The plunge
and initial torsion
N - m / tad. Ic/4
stiffness
are K h = 1.21 × 106 N
m andK_
The mass is m h = 26.64 kg, the mass pitch moment
= 0.086
actual
stiffness
kg
system
- m 2 ,
and the static offset is s_ = 0.378 kg - m.
set up to study transonic
flutter. 3 The functional
= 6.68 × 103
of inertia
is
These values
variation
are for an
of the torsion
is given by K_ = K_ (1 - f) + K_2 f
t_ < t < t_ +At
where f = @(1 - cos[_(t and t_ = 0.24 sec, modes
At = 0.01 sec. The natural
are 205.4 rad/sec
diagonalized,
The matrix
resulting
A(t)
and 299.3 rad/sec in the equation
has the values
- ta) / At]) frequencies
of the plunge
for t < t_. The second
and torsion
order system
is
--42189 0 0 -89580
fort_tl,
_ = 1-39298 L 10533
10533-50876
for t >_tl + At
and
/_=
,
b/ =
/
.
b/2
The generalized
variables
equations
are written
sufficient
number
equation
q_ and q2 roughly
in state variable
of terms
adjusted
correspond
form and solved
to pitch
and plunge.
using equations
at each time step to
ensure
These
(5) and (6) with a _
