The approximate inversion of an internally unknown linear system, given by its impulse response sequence, by an inverse system having a finite impulse response, is considered. The recursive least squares procedure is shown to have an exact initialization, based on the triangular Toeplitz structure of the matrix involved. The proposed approach also suggests solutions to the problems of system identification and compensation

Bach, Ralph E., Jr.

Baram, Yoram

English

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NASA Technical Memorandum 100070 
Recursive Inversion of Externallv 
Defined Linear Systems 
Ralph E. Bach, Jr., and Yoram Baram 
#88-195 15 (HASA-TIS-lO0070) BBCGESI9E l i V E O S I O I  O P  
bXTBBlOULY DBPlliEP L I B L I E  SPSILUIS (USA) 
13 P CSCL 12B 
Unclar 
G3/66 01287; 1 
March 1988 
National Aeronautics and 
Space Administration 
https://ntrs.nasa.gov/search.jsp?R=19880009831 2020-03-20T08:01:43+00:00Z
NASA Technical Memorandum 100070 
Recursive Inversion of Externally 
Defined Linear Systems 
Ralph E. Bach, Jr. 
Yoram Baram, Ames Research Center, Moffett Field, California 
March 1988 
National Aeronautics and 
Space Administration 
Ames Research Center 
Moffett Field, California 94035 
RECURSIVE INVERSION OF EXTERNALLY DEFINED LINEAR SYSTEMS 
Ralph E. Bach, Jr.,* and Yoram Baram** 
SUMMARY 
The approximate inversion of an internally unknown linear system, given by its 
impulse response sequence, by an inverse system having a finite impulse response, is 
considered. The recursive least squares procedure is shown to have an exact ini- 
tialization, based on the triangular Toepliz structure of the matrix involved. The 
proposed approach also suggests solutions to the problems of system identification 
and compensation. 
1. INTRODUCTION 
Linear system inversion--that is, interchanging the roles of the input and 
output signals--has been of interest in systems theory and in signal analysis in the 
past (see, e.g., [ l ] ,  [2]). Recently, interest in the system inversion problem has 
been motivated by problems of geophysical exploration (see, e.g., [3]). In many 
cases, the system at hand is internally unknown and may not even possess a finite- 
order representation. An external representation in the form of the impulse 
response function may be generated, in principle, by measuring the response to a 
high-energy pulse, as is done in geophysical exploration, or by calculating the 
cross-correlation function between the output and a pseudo-random input, as is done 
in control applications. While, as we show, the impulse response of the inverse 
system is well defined, its calculation becomes practically intractable for long 
data records. Instead, a finite-order approximation of the inverse system may be 
sought. Such an approximation will yield, simply by inversion, a solution to the 
problem of system identification. We show that when the approximate inverse system 
is restricted to have a finite impulse response, the design problem can be solved by 
a recursive least-squares procedure having an exact initialization. The approach is 
readily extendable to the problem of cascade compensation. 
*R. E. Bach, Jr., is with the NASA Ames Research Center, Moffett Field, CA 
**Y. Baram is a Senior Research Associate of the National Research Council at 
94035. 
the NASA Ames Research Center, Moffett Field, CA 94035, on sabbatical leave from the 
Department of Electrical Engineering, Technion, Israel Institute of Technology, 
Haifa 32000, Israel. 
1 
2. EXACT INVERSION 
In order to demonstrate the need for approximation in the inversion of inter- 
nally unknown linear systems, we first consider their exact inversion. Since dis- 
crete time systems often have delayed response, it is also our purpose to define a 
unified representation for the inverses of systems with and without delays that, 
without burdening the notation, will make the results applicable to both cases. 
be the impulse response of a causal time-invariant linear Let {hk), k I 0 
system. 
output, and real, i.e., that (hk) is a scalar sequence. 
obtained from its input {uk) by the convolution operation 
We shall assume for convenience that the system is single-input, single- 
The system's output {y,} is 
k 
i =O 
k I 0  
It is readily verifiable that the inverse of (2.1) is given by 
k 2 O  yk uk = hi * 
I where {h-}, k k I 0 satisfies the progressive equation 
k 
with 
(2.1) 
(2.2) 
k r  1 1 hi = - - hi-ihi 
i= 1 hO 
(2.3) 
1 - -  hi - ho 
Next, suppose that hk = 0 for k = 0, . . . , a  - 1, i.e., that the system has 
a. time-unit delays. Then, clearly, the above inversion is ill-defined. The 
inverse system can, however, be redefined. First it should be realized that the 
inverse system no longer operates in "real time" on {yk) but, instead, it operates 
on a record of (yk}, advanced time-units with respect to the original 
I sequence. Defining the sequences 
2 
and 
. 
The system can, again, be written in "causal form" as 
which has the inverse 
(E)- where {hk ), k 2 0 satisfies 
k 
with 
(2 .5)  
(2 .6 )  
Since the a-delay system now has exactly the same form as the 0-delay system, the 
index (E) can be removed with the understanding that when h system at hand has '7 as an input, as in k delays, its inverse must be interpreted as operating on y 
equation (2.5). 
It can be seen that the inverse of a causal, time-invariant, linear system 
given by an infinite-length impulse response is well defined. However, the calcula- 
tion of the inverse impulse response becomes intractable, as both the memory and the 
amount of computations required grow indefinitely. Under certain conditions, dis- 
cussed in the following section, the error resulting from taking a finite portion of 
the sequence {hi) can be shown t o  diminish as the length of this portion is 
increased. However, this will not be, in general, an optimal choice of a finite 
impulse response approximation of the inverse system. Such approximation is the 
subject of the next section. 
3 
3. RECURSIVE LEAST-SQUARES INVERSION 
Let {fk), k 1 0 ,  be the impulse response of an approximate model for Chi).  
Noting that 
hi * hk = 6k 
where 6k is the unit impulse sequence ( b k  = 1 for k = 0, 6k = 0 for k > 0) an 
approximation of hi 
between fk * hk and 6k with respect to fk. 
can be obtained by minimizing some measure of the difference 
Denoting the sum-square norm of a sequence {ak), k Z 0 ,  by 
m 1 /2 
an approximation criterion for the problem at hand is defined by 
( 3 . 1 )  
Suppose that the inverse approximating model is restricted to have a finite 
impulse response of length n, that is, fk = 0 for k Z n. The approximation norm 
then becomes 
where 
- f = (fo fl - - -  fn- 1 IT 
bk =r 0 k > O  = o  
and the n-dimensional vector sequence {hk) is given by 
(3.2) 
4 
k l n - 1  T [hk hk,l ho 0 01 
Lhk hk-l . . *  hk-n+ 1 
h = {  -k I T  k l n  
- 
ho 
h l  
m-n . h  
The least-squares problem 
yields the equation 
where 
m 
2 min (hTf - bk) -k- f - k=O 
T T T T f = T b  
m . m -  (Iw 
m -1 and b- = 
Let fm be defined by the equation 
where, for  m L n, 
hO 
h l  
hn- 1 
hn 
0 
hO 
n-2' h 
hn- 1 .. 
.. hm-2 
(3 .4)  
(3 .5)  
(3 .3)  
5 
and 
T b = [ l  0 ... 01 
111 
is m-dimensional. The matrix Tn can be seen to be lower triangular Toeplitz. It 
is readily verifiable (also see, e.g., [ 4 ] )  that such structure yields the inverse 
where, as in (2 .3 )  
k 
k r l  1 
h, hi = - - hi,ihi 
U i=l 
and 
1 - -  hi - ho 
S‘n e ho 
hi” = ha, as explained in section 2), the matrix Ti 
can be assumed to be non-zero (otherwise, jt is replaced by 
exists, yielding 
f = T’lb 
11 n-n (3.7) 
For m 2 n, we obtain from the least-squares recursion (see, e.g., [51, p. 176) 
m 
f = f  - K h’f m+l 111 m-m-m 
I with 
R,lm 
Km = 1 + hR,lm T 
( 3 . 8 )  
( 3 . 9 )  
6 
where Rm = (TTT )-l satisfies the recursion m m  
h hT 
1 + $Rm$ 
-m-m 
T Rm = Rm - Rm Rm+ 1 
* 
initialized at 
-1 -1 T Rn = Tn (Tn ) 
(3.10) 
(3.11) 
where T;' 
recursive least-squares procedure be initialized at arbitrary values of 
in order to avoid matrix inversion (see, e.g., [ 5 ] ,  p .  177), the above procedure is 
initialized at the exact values of 
is given by (3.6). We note that while it is often suggested that a 
R o  and f o  
fn and Rn, specified by (3.7) and (3.11). 
The convergence of the above recursive procedure is examined next. From (3.4) 
we directly obtain the equation 
Rf = $ 
where h0 is defined by (3.3) and 
T R = T T  = 
m m  
. . . a  
n- 1 
a1 n-2 
aO n-3 
- a  
- a  
. .  O2 
. .  
* " a o  n-1 a n-2 On-3 
where 
ai = hkhk+i 
k=O 
Suppose that the system at hand is externally (bounded input/bounded output) 
stable. Then 
m 
k=O 
7 
we have 
W m m c 
k=O 
hkhk+i  
y i e l d i n g  converg-nce of 
W = c  
k=O k=O k=O - i - 5  
t follows t h a t  t he  r e c u r s i v e  l e a s t - s q u a r e s  i n v e r s i o n  a i .  
procedure  (3.8) converges  i f  t h e  sys tem is e x t e r n a l l y  stable.  
The i n v e r s i o n  error a t  each s t e p  of t h e  r e c u r s i v e  i n v e r s i o n  procedure  can be 
c a l c u l a t e d  as 
which, by s u b s t i t u t i n g  (3.5), y i e l d s  
E z = 1 - fm,oho 
m 
where fm,o is t h e  f irst  component of fm.  
We n e x t  examine t h e  b e h a v i o r  of t h e  i n v e r s i o n  error as the  l e n g t h  (or t h e  
order) of t h e  i n v e r s e  sys tem is i n c r e a s e d  t o  i n f i n i t y .  We assume t h a t  the  sequence 
{ h i }  is bounded ( i n  other words, t h e  i n v e r s e  system is s table) .  
sequence 
Let u s  d e f i n e  the  
k = 0, ..., n - 1 
k l n  
Then, s i n c e  { f k )  minimizes  t h e  i n v e r s i o n  norm, we have 
y i e l d i n g  
a 
hence, 
In words, the inversion error diminishes to zero as the length of the inverse filter 
is increased to infinity. 
An approximate model for the system at hand can be obtained by inverting the 
resulting in erse system. 
l/(fo + f l z  -'r + ... + fn-,z -n+ 1 1, where z is the z-transform variable. We note 
that is the given system has, say, % 
i = 0, ..., %-1, the resulting approximate model should be multiplied by 
This approach yields a solution to the problem of system identification. 
impulse response sequence may be obtained directly by applying a high-energy pulse 
at the input, or ,  by simple calculation, from the cross correlation between the 
output and a white noise input. 
The model will be of the all-pole type, specifically 
time unit delays, that is, hi = 0 for 
z - ' ,  as the 
proposed technique, in effect, approximates the given system multiplied by z a. . 
The 
Next, suppose that it is desired to compensate a system given by its impulse 
response sequence (hk} by cascading it with a relatively simple system so as to 
approximate a desired system whose impulse response (gk) is given or can be calcu- 
lated. 
parameters may be obtained by minimizing the norm I Ifk * hk - gk( I. The rec rsive 
least-squares procedure ( 3 . 8 )  is then replaced by fm+, = fm + Km(gm - hmfm) T ?  , 
with K, and &, defined by (3.9-3.11) and ( 3 . 3 ) ,  respectively. Finally, we note 
that a recent paper [ 6 ]  addresses the least-squares compensation, inversion, and 
approximation of internally known linear systems by polynomial systems. 
When the compensator is restricted to have a finite impulse response, its 
4. CONCLUSION 
The inversion of an internally unknown linear system has been considered. A 
recursive least-squares procedure having an exact initialization, for calculating 
the parameters of a finite impulse response inverse system, has been derived. The 
proposed approach also suggests solutions to the problems of system identification 
and compensation. 
9 
REFERENCES 
1. L. A. Zadeh and C. A. Desoer, Linear System Theory, McGraw-Hill, New York, 1963. 
2. A. V. Openheim and R. W. Schafer, Digital Signal Processing, Prentice-Hall, 
Englewood Cliffs, NJ, 1975. 
3 .  E. A. Robinson and S. Treitel, "Geophysical Signal Analysis,'' Prentice-Hall, 
Englewood Cliffs, NY, 1980. 
~ 
4. D. Commenges and M. Monsion, "Fast Inversion of Triangular Toeplitz Matrices," 
IEEE Trans. on Automatic Control, Vol. AC-29, No. 3 ,  March 1984. 
5. G. C. Coodwin and R. L. Payne, Dynamic System Identification, Academic Press, 
New York, 1977. 
6. Y. Baram, "Polynomial Compensation, Inversion and Approximation of Discrete Time 
Linear Systems,'' NASA TM 100027, 1987. 
10 
Report Documentation Page 
1. Report No. 
NASA TM-100070 
2. Government Accession No. 
7. AuthorM 
Ralph E. Bach, Jr. and Yoram Baram 
17. Key Words (Suggested by Author(s)) 
9. Performing Organization Name and Address 
Ames Research Center 
Moffett Field, CA 94035 
18. Distribution Statement 
12. Sponsoring Agency Name and Address 
National Aeronautics and Space Administration 
Washington, DC 20546 
19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of pages 
Unclassified Unclassified 11 
3. Recipient's Catalog No. 
22. Price 
A02 
5. Report Date 
March 1988 
6. Performing Organization Code 
8. Performing Organization Report No. 
A-8809 3 
10. Work Unit No. 
505-66-1 1 
11. Contract or Grant No. 
13. Type of Report and Period Covered 
Technical Memorandum 
14. Sponsoring Agency Code 
15. Supplementary Notes 
Point of Contact: Ralph E. Bach, Jr., MS 210-9., Ames Research Center, 
Moffett Field, CA 94035, (415) 694-5429 or FTS 464-5429 
16. Abstract 
The approximate inversion of an internally unknown linear system, given by 
its impulse response sequence, by an inverse system having a finite impulse 
response, is considered. The recursive least squares procedure is shown to have 
an exact initialization, based on the triangular Toepliz structure of the matrix 
involved. 
system identification and compensation. 
The proposed approach also suggests solutions to the problems of 
Filtering estimation 
Stochastic realizations 
Spectral factorization 
Unclassified - Unlimited 
Subject Category - 66 


Recursive inversion of externally defined linear systems

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