System Identification Using Laguerre Functions: Simple Examples
Philip D. Olivier
Department of Electrical and Computer Engineering
Mercer University
Macon, GA 31211
Olivier_pd@mercer.edu
Abstract
A system identification technique based on Laguerre series expansions of filtered
input/output data is briefly summarized. Some simple examples are presented that
demonstrate the power of the technique. The examples verify that the procedure works
well for lightly damped (i.e. flexible) systems, for MIMO systems as well as SISO systems,
and that non positive real systems pose no problems.
1. Introduction
There are many situations in which it is necessary to approximate the transfer function
model of a physical system from input/output data. This problem arises in adaptive
control, deconvolution, fault detection and many other areas. This paper describes a new
way to find transfer function models based on input/output time domain data.
Procedure Description. Transfer function models of systems are obtained from
input/output data based on the Laguerre series expansions of filtered input/output data.
The steps in the technique are as follows:
1) find the coefficients for the Laguerre series representation of the filtered input and
output data, this can be done using either time domain or frequency domain data;
2) if time domain data is used, the time domain representations are Laplace
transformed yielding frequency domain representations of the filtered input and output,
U (s ) , and Y (s ) ;
3) the unknown transfer function matrix is assumed to have the form A( s) B( s) 1
where A(s) and B(s) are assumed to have Laguerre series expansions with unknown
coefficients;
4) the products B ( s )Y ( s ) and A( s )U ( s ) are expanded in terms of products of
Laguerre functions;
5) the products of Laguerre functions are replaced by differences of Laguerre
functions;
6) coefficients of like Laguerre functions are equated; and
7) solving these equations yield the coefficients of A(s) and B(s) .
Paper organization. The remainder of this paper is organized as follows: Section 2
summarizes the pertinent facts about Laguerre functions; Section 3 presents the
examples; Section 4 provides the summary and discussion of the results
1
2. Laguerre Functions
The Laguerre functions are an orthonormal set of functions that are complete in the
function space L2 . The Laplace transform of any of these functions is a rational function
of the Laplace variable s , that has all of its poles at the a point on the negative real axis.
In the time domain, the Laguerre functions are polynomials multiplied by a decaying
exponential. As such, the Laguerre functions can be used to approximate stable transfer
functions, and/or reasonably behaved functions that decay to zero in the time domain.
The basic facts about these functions are summarized below.
Frequency domain description. The k th Laguerre function is
( s  p) k 1
Lk ( s)  2 p
( s  p) k
Time domain description. The k th Laguerre function is
k !( pt )n1
2
n 1 (n  1)!) ( k  n  1)!
k 1
Lk (t )  2 pe pt / 2 
Expansion equations. A reasonably well behaved function can be expanded in terms
of the Laguerre functions as


k 1
k 1
f (t )   a k Lk (t ) F ( s )   a k Lk ( s )
where the coefficients a k can be obtained in either the time or frequency domain via
standard inner product computations.
ak  f (t ), Lk (t )  t  F ( s), Lk ( s)  s
where the time domain inner product is the standard L2 inner product and the s domain
inner product is the L2 inner product induced on the s domain by Parseval’s theorem,
i.e.

 f (t ), g (t )  t   f (t ) g (t )dt
0
 F ( s), G ( s)  s 
1
2
2
f
 F (s)G(s)ds
 f
Orthonormality conditions. The Laguerre functions are orthonormal, i.e.
 Lk (t ), Ln (t )  t  Lk (s), Ln (s)  s   ks n
where  k s n is the standard dirac delta.
Product property. In addition to these standard properties of Laguerre functions, this
paper relies on the following newer property [see Olivier, 1995] that states that a product
of two s domain Laguerre functions is the weighted difference of two other s domain
Laguerre functions.
Lk ( s ) Lm ( s ) 
Lk  m 1 ( s )  Lk  m ( s )
2p
3. Examples
Three simple examples are considered. These examples are chosen to demonstrate
that the procedure will work on examples with known solutions. The first example
mimics some properties of flexible structures, the second example is a MIMO system,
and the third example is of a non positive real system (a stumbling block for some other
system ID procedures).
Flexible structure example. NASA’s Marshall Space Flight Center has a flexible
structure testbed referred to as CASES. This structure was designed to exhibit many of
the properties of flexible structures that cause problems to control engineers. The
example considered here is motivated by the CASES structure in that the transfer
function to be identified has damping (very low), and natural frequency (low) similar to
the average damping and natural frequency of the CASES structure.
Consider a system described by the transfer function H (s)  (s  a) /( s  a) 2  b 2 )
with a  .7084 and b  7.6469 . The pole of the Laguerre functions was chosen to
bp  a . The input was chosen to be a step input and the input and output data were
prefiltered by a filter with transfer function F ( s)  s /( s  p) .
The procedure outlined in the introduction is implemented to yield the following:
Step 1. Expand filtered input:
u (t )  u1 L1 (t )  u 2 L2 (t )  ...  (1 /( 2 p) 5 ) L1 (t ) Expand the filtered output:
y (t )  y1 L1 (t )  y 2 L2 (t )  ...  y10 L1 0 (t )
Step 2. Laplace transform these time domain representations:
U ( s)  (1 /( 2 P) 5 ) L1 ( s)
Y ( s)  y1 L1 ( s)  y 2 L2 (s)  ...  y10 L10 (s)
3
Step 3. Assume Laguerre expansions for the numerator and denominator A(s) and
B(s)
A(s)  a1 L1 (s)  a2 L2 (s)  a3 L3 ( s)
B(s)  b1 L1 (s)  b2 L2 (s)  b3 L3 (s)
These expansions were chosen to be 3rd order to get 2nd order approximations.
Step 4. Expand the products A( s )U ( s ) and B ( s )Y ( s ) :
B(s)Y ( s)  b1 L1 (s)  b2 L2 ( s)  b3 L3 (s)  y1 L1 ( s)  y 2 L2 ( s)  ...  y10 L10 (s)
 b1 y1 L1 L1  b1 y 2 L1 L2  b1 y 3 L1 L3  ...  b3 y1 L3 L1  b3 y 2 L3 L2  b3b3 L3 L3
A( s )U ( s )  (1 /( 2 p) 5 )a1 L1 ( s )  a 2 L 2 ( s )  a3 L3 ( s )L1 ( s )
 (1 / 2 p) 5 )a1 L1 ( s) L1 ( s )  a 2 L2 ( s) L1 ( s )  a3 L3 ( s ) L1 ( s)
Step 5. Replace products of Laguerre functions with differences of Laguerre functions
using the product property.
B( s)Y (s)  [b1 y1 ( L1  L2 )  b1 y 2 ( L2  L3 )  b1 y3 ( L3  L4 )  ...
b3 y1 ( L3  L4 )  b3 y2 ( L4  L5 )  b3 y2 (L4  L5 )  b3 y3 (L5  L6 )]/ 2 p
A(s)U (s)  [a1 ( L1  L2 )  a2 ( L2  L3 )  a3 ( L3  L4 )]/ 2 p
 [a1L1  (a2  a1 ) L2  (a3  a2 ) L3  a3 L4 ]/ 2 p
Step 6. Equate coefficients of like Laguerre functions. This produces a set of matrix
equations of the form Y  U x  0 where the first column of Y is
Y (i,1)  y1
Y (i,1)  y2  y1 , if j  i
The other columns are shifted down so that the non square matrix Y is “lower triangular.”
The matrix U has the same structure. The unknown vector x is composed of x  [b, a, ], .
Step 7. Solve the equations. The set of equations have nontrivial solution only if they
are underdetermined. To obtain a non singular set of equations, bring all terms involving
b3 (or some other b) to the right, and arbitrarily set b3  1 . This yields
Y (:,1: 2)
 U  z  Y (:,3)
where z is x with b3 removed. This is the set of equations that is solved.
The technique identified the transfer function to 4 significant figures yielding
H a ( s)  (1.0004s  .7128) /( s 2  1.4141s  58.9963)  7.0764e5
MIMO example. Multi input and/or Multi output systems can be handled just as
easily. Consider a system described by the MIMO transfer function
4
 2 s  3 s  2
 4 s  3 s  3

H ( s)  
( s  1)( s  2)
The expansions apply without modification, except that the coefficients are matrices,
and that care must be taken to keep the order of matrix products correct. Step inputs are
applied one at a time to identify each column separately.
The technique again identified the original transfer function quite accurately, yielding
1.0000( s  2) 
2.000s  3.0002
 4.0001s  3.000 1.0000 s  3.0000

H a ( s)  
2
1.0000 s  3.0000 s  2.0000
with direct transmission terms
 7e  6  3e  6 
E a ( s)  

  2e  5  2e  6
NPR example. Non positive real transfer functions cause problems for many system
identification procedures. This example presents the results of identification of a non
positive real system. Consider the transfer function H ( s)  1/(( s  a)2  b2 ) with the
same a, and b as in the first example. The technique produced
H a ( s )  (.0003s  1.0044) /( s 2  1.4293s  59.0671)  4.9e  5
4. Summary/Discussion
This paper has outlined a system identification procedure based on Laguerre series
expansions of system input and output. It has demonstrated that the technique works well
for simple SISO, MIMO and NPR examples.
Some particular points to be made include:
a) This approach can be viewed as a numerical Laplace transform
b) All unknown parameters enter linearly, i.e. no nonlinear solution techniques are
needed.
Future work needs to include input and output noise.
Future applications appear to include: adaptive control of flexible structures; fault
detection and isolation in analog circuits; crack detection in aircraft.
5. Acknoledgements
This research was supported in part by a grant from the National Aeronautics and
Space Administration through grant no. NAG8-1301. The author greatly appreciates this
support.
5
6. References
1. Olivier, P.D., “On-line system identification using Lagueer Series,” IEEE
Proceedings D, Control Theory and Applications, 1994, Vol. 141, (4), pp. 249-251.
2. Olivier, P.D., “Approximate System Inverses,” Electronics Letters, 1995, Vol. 31,
No. 23, pp. 2050-2051.
(Presented at 29th Southeastern Symposium on System Theory in March of 1997.)
6