This paper presents an approximate method for obtaining truncated balance realizations of systems represented by non-rational transfer functions, that is infinite dimensional systems. It is based on the approximation to the Hankel operator

Deabreu-Garcia, J. Alex

Hartley, Tom T.

English

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N9 2- 22-4
Approximate Truncated Balanced Realizations for .....
Infinite Dimensional Systems /F_- It//'
Tom T. Hartley
J. Alexis De Abreu - Garcia
Department of Electrical Engineering
The University of Akron
Akron, OH 44325-3904
216-972-5991
October 1990
Abstract
This paper presents an approximate method for obtaining truncated
balanced realizations of systems represented by non-rational
transfer functions, that is infinite dimensional systems. It is based
on an approximation to the Hankel operator.
Acknowledgement
This research was supported by the Advanced Control Technology
Branch of NASA Lewis Research Center via NASA Grant NAG 3-904.
Introduction
The benefits of obtaining accurate low order models are obvious to
the control system designer. In particular, if an accurate low order
model can be obtained, then a low order controller can be designed
which will hopefully maintain the designed closed loop robustness
properties when applied to the real system. Much effort has been
given to the model reduction problem, particularly in the finite
dimensional situation. While input-output methods were prevalent
in the 1960's and 1970's [1], the 1980's saw tremendous effort in
the state space approach to model reduction [2].
Infinite dimensional systems, on the other hand, have always been of
https://ntrs.nasa.gov/search.jsp?R=19920013250 2020-03-17T12:00:46+00:00Z
interest, but have not always been so easy to deal with. Part of this
problem comes from the extensive training that the typical control
engineer receives in finite dimensional systems via rational
polynomial transfer functions. The existence of a non-rational
transfer function, other than a time delay operator, is often found to
be surprizing. Furthermore, the origins of such a transfer function
are usually very mysterious. This is probably due to the fact that
control engineers typically do not understand spatial boundary value
problems. None-the-less, non-rational transfer functions for
describing partial differential equations have been in use since
Heaviside and the interested reader should consult the wonderful
book [3].
Many non-rational transfer functions arise from systems described
by wave equations. These will usually have several time-delay
operators occupying terms in both the numerator and denominator.
Historically, even in the 1950's, these were being approximated by
Pade approximations [4]. Here the individual time-delays are
replaced by an appropriate order Pade approximation. Then after
some tedious algebra, a rational approximation results. The
accuracy of the resulting approximation depends significantly on the
order of the Pade approximations used with the cost of improved
accuracy being increased system order.
Other than some exhausting nonlinear optimization techniques, little
significant progress in the reduction of infinite dimensional
systems occured until the late 1980's. This work stemmed from the
research in the robust control area and deals with balanced
realizations in state space and well as Hankel norms [5]. These
methods will be discussed at length in the next section. Several
examples will be included to demonstrate the utility of the
approximate method developed here.
Approximate Truncated Balanced Realizations
Truncated balanced realizations for finite dimensional systems are
well known and are readily implemented [2]. Many variations are
available and the reader is encouraged to consult the large amount of
literature for more information. Recently, Glover et al [5] have been
able to develop this technique for infinite dimensional systems. The
approach is similar to the finite dimensional situation in that it is
based on singular value decomposition. In the finite dimensional
case, the balanced realization comes from the leading terms in the
singular value decomposition of the necessarily finite rank system
Hankel matrix. Similarly, in the infinite dimensional case, the
balanced realization is based on an orthogonal expansion (singular
value decomposition) of the Hankel operator
x
(Fu)(t) = I h(t+s)u(s)ds
where h(t) is the system impulse response. The difficulty here is
that the Hankel operator effectively has infinite rank as nonzero
singular values are infinite in number. The real problem in the
single-input-single-output case is that this is an integral equation
requiring orthogonal expansion of its symmetric kernel, h(t+s). This
approach is referred to as that of Hilbert-Schmidt. A good
discussion of these matters can be found in [6]. This orthogonal
expansion for symmetric kernels is based on the eigenvalue-
eigenfunction problem
Un(t) = _.n I h(t+s)Un(S) ds
Vn(S)= ?_nI h(t+s)Vn(t)dt
where u.(t)and v.(s) are the orthogonal eigenfunctions, or Schmidt
pairs, corresponding to the eigenvalue _.n. Once these are found, the
Hankel kernel can then be written as
h(t+s) = Z Un(t) Vn(S)_ Z (Yn Un(t) Vn(S) •
n=l _kn n=l
The kernel can then be truncated after the least significant singular
value O'k to yield a rank k expansion. Once this is done, it is shown
[5] that a state space system of rank k can be developed which
approximates the original system, with error related to _k÷l. There
an output normal realization is given and is repeated here;
Bi = oivi(O)
Ci = ui(O)
Aii =- 1 u_(O)
oTC i Cj% =B*Bj- *
") '3
oTi-o 7
Thus, once the singular values and the Schmidt pairs are found, it is
then a fairly simple task to obtain a truncated balanced realization.
It should also be noted that this reduces to the standard approach
for finite dimensional systems. Glover [5] then goes on to give the
optimal Hankel norm approximation, but that is not pursued at this
time.
The problem with the approach is that it is extremely difficult in
most cases to analytically obtain the required orthogonal expansion,
that is, the Schmidt pairs. Thus some approximate method for doing
this is required. The approach taken by Fredholm in the solution of
integral equations [6] is particularly appealing for performing this
task. Basically, the Fredholm approach is to divide the integration
limits for the Hankel operator into evenly spaced sections in s and t,
and then to add up the resulting sampled kernels as follows using
the rectangular rule for integration;
f
u(s) = X/ h(t+s)u(Odt
L
then for a symmetric kernel and a timestep T
u(O)- _T[h(O+O)u(O) + h(O+l)u(1) +---+ h(O+n)u(n)] = 0
u(1)-XT[h(l+O)u(O) + h(l+l)u(1) ÷... ÷ h(l+n)u(n)] =0
u(n)- _T[h(n+O)u(O) + h(n+l)u(1) +.-. + h(n+n)u(n)] = 0
Clearly this is still an eigenvalue problem, but rather than finding
Schmidt pairs which are continuous functions of time, the Schmidt
pairs will be eigenvectors representing approximate sampled
versions of the eigenfunctions. The above equation can be written in
matrix form as
1- ;kThoo -;kThol .... XThon
-_,Thl o l- ;kThl 1 .... ;_Thln
-_'l-'hnO -_.Thn l • •. 1- XThnn
F U o
lu!
U
=0
or
I- ;_T H]!ul = 0
where H is now the Hankel matrix formed by sampling the system
impulse response every T seconds. It should also be remembered
that H is symmetric. Thus finding its eigenvalues and eigenvectors
is equivalent to a singular value decomposition. Hence, the left and
right singular vectors of H will approximate sampled versions of the
Schmidt pairs of the orthogonal expansion of the continuous Hankel
kernel. Its singular values however are approximations of the
singular values obtained from the orthogonal expansion.
Furthermore it should be noted that the singular values of the Hankel
matrix must be multiplied by T in order to obtain the above
approximation. Since the relative magnitudes should not change, it
is then necessary to divide the matrices of right and left singular
vectors by SQRT(T). As T gets smaller and smaller, then the
approximate version converges to the exact version. The major
problem with the approach is in running out of computer memory to
form H as T is allowed to get small, however reasonable results can
be obtained for fairly large T. Some examples of this are now
presented for comparison with the actual solutions.
Example 1: Approximation of the unit triangle impulse response
[5,7]. This is included for comparison with the exact solution.
Using T=.02 yeilds a 50x50 Hankel matrix and the following second
order system using the output normal realization given above
H(s) = .9778s + 3.4477
s2 + 3.6918s + 7.0187
as compared with the actual solution taken from [5]
H(s) = .9561s + 3.6402
s2 + 4s + 7.6145
Comparison of the impulse responses is found in Figure 1.
To further demonstrate the process, the solution process for T=.2 is
now presented. The sampled Hankel matrix is
H
- 1 .8.6.4.2-q
.8.6.4.2 0 !
.6.4.2 00i
I
.4.2 0 0 0
_.20000"
The SVD is then performed on this, followed by the T-scaling to give
0.2_ = [.4036 .0647 .0247 .0145 .0109]
1.6321 1.1675 .8121 .5051 .2424
1.2105 -.2230 -1.0630 -1.2828 -.8422
•8050 -1.1675 -.8121 .7707 1.3174
.4435 -1.3053 .9079 .7516 -1.3078
.1618 -.7215 1.3140 -1.3930 .8867
v _ [ul -u2 u3 -u4 usl.
A second order approximation is chosen, then using the output
normal realization given above
C = [1.6321 1.1675]
A=!-1.3319 -2.2807]
L .3752 -.6815J
which is given in input-output form as
H(s) - 1. 1665s +. 8515
s2 + 2.0134s + 1.7634
It can be seen from Figure 2 that the response is not all that bad
considering the grossness of the approximation.
Example 2: Approximation of the unit block impulse response [7].
This was done using T=.I The resulting system is
H(s) = .8605s + 3.0822
s2 + 1.8523s + 3.4681
with corresponding response in Figure 3. Although it doesn't look
much like a block, the response is almost identical to that given by
[7]. The response from a tenth order approximation is also given and
looks pretty good.
Example 3: Approximation of the impulse response sin(t)/t This
system is perhaps even more difficult than the others since it is of
the infinite impulse response type. Using T=.05 the approximation
obtained was
H(s) = 1'0169s2 + 1.0532s + 1.0793
s3 + 1.2004s 2 + 1.1813s + .6198
Although the response is not bad, it did not appear to really be that
good, as can be seen from Figure 4. This is probably due to the fact
that this system has an infinite impulse response. The reduction of
such systems would appear to require a great deal of memory in
order to obtain an accurate approximation.
Example 4: Approximation of a simplified supersonic inlet [see 8 for
background]. Here the system to be approximated is
H(s) = 50 e -.015s
s + 2 + 50 e--°2s
The approximate truncated balanced realization is found from an
approximate impulse response which is obtained from the following
discrete model of this system (it uses z-inverses for the delays and
Tustin for the s term),
H(z) = 50 (z6 + z5) , T=0.001 .
2002z 21 - 1998z 2° + 50z + 50
The approximate truncated balanced realization is
H(s) = 1000 (-.0025s 2 -.3940s + 523.7) ,
s3 + 105.7s 2 + 8644s + 481446
while a Pade(1,1) approximation for the delays [9] yielded
H(s) = 50 (-s 2 + 33.33s + 13333)
s3 + 185.3s 2 + 12133s + 693333
The step responses of each of these systems is plotted in Figure 5
It can be seen that the approximate truncated balanced realization
tracks better initially but that there is a little steady state error.
This may be due to either the balanced truncation or the
approximation to the Hankel operator. Alternatively, for
comparison, the Pade approximation has a much worse transient
response, while there is no steady state error. It was also found
that higher order truncations of the balanced realization provided
more accurate approximations.
Conclusion
This paper contains an approximate method for obtaining reduced
order models of infinite dimensional systems. It is based on an
approximation to the Hilbert-Schmidt expansion of the Hankel
operator. Much additional work on this approximate approach is
necessary. In particular, determining the error bounds associated
with this approximation is of primary importance. Furthermore,
better approximations to the integral are being considered.
Preliminary results indicate that the trapezoidal rule can give a
better approximation than the rectangular rule used here. Other
methods will also be considered. Another approach is to allow
variable values of T for infinite duration impulse responses. Finally,
an approach to using frequency domain information or input/output
data should be pursued as that is all is available in some cases.
B.e.Le.ce_ 7,  
1. Lai, J.S. & Hung, J.C., "Practical Model Reduction Methods for
Control and Instrumentation," Proc. of the IEEE Industrial Electronics
Society Conference, 1985 (IECON '85).
2. Glover, K., "All Optimal Hankel-norm Approximations of Linear
Multivariable Systems and Their L_-error Bounds," Intl. Jour. of
Control, vol 39, no 6, pp 1115-1193, June 1984.
3. Carslaw, H.S. & Jeager, J.C., Qoeratiqn_l Methods in Applied
Mathematics. Oxford 1941.
4. Van Valkenburg, M.E., Intrqdvqtion to Modern Network Synthesis.
Wiley, 1960.
5. Glover, K., Curtain, R.F., & Partington, J.R., "Realisation and
Approximation of Linear Infinite Dimensional Systems with Error
Bounds," SIAM Jour. of Control and Optimization, vol 26, no 4, pp
863-898, July 1988.
6. Lovitt, W. V., Linear Integral Eouations. McGraw Hill, 1924 &
Dover, 1950.
7. Glover, K., Lam, J., & Partington, J.R., "Rational Approximation of a
Class of Infinite Dimensional Systems," Cambridge University Report
CUED/F-INFENG/TR.20, 1988.
8. Willoh, R., "A Mathematical Analysis of Supersonic Inlet
Dynamics," NASA TN D-4969, December, 1968.
9. Truxal, J.G., Control Systems Synthesis. McGraw-Hill, 1955.
Figtare 1. Truncated balanced approximation for triangular pulse and its approximation
;__......................................... _ ........................
0.8-
0.6-
0.4_-
0.2
,
0.5 1 1.5 2 2.5 3
1.2
0.8 "_'
0.6
0.4_
I
0.24-
[
i
O_
-0.2 i
Figure 2. Exact and crude (T=0.2) approximation of triangular pulse.
i
0 0.5 1 1.5 2 2.5 3
1 2- Figure3. 2ndand10th order approximations to unit block•
\
V
0.8 ", ",,
/
0.6 '_
'l
i
'"'i
0.4_-
II
t
0.2-
Ii I
/;
_0.2 i _ -._ __j--
0 1 2 3
i
4 5 6
]
Figure 4. Exact and approximate sin(t)/t.
1.2
I
1 '--: ,
0.8 !
0.6
0.4!
0.2_-
I
iOr-
-0.2
t_
i
,, -. __y f _4
I
!
I
-0.4
0 5 10 15 20
1.5_- Figure5. Discrete,Pade,andapproximateinletresponses.


Approximate truncated balanced realizations for infinite dimensional systems

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