A simple model order reduction technique is proposed for z-transfer functions. This method is ... transfer functions are matched at a number of frequency points in ...
Reduced order modelling of discrete-time systems Somnath Pan and Jayanta Pal Department
of Electrical
Engineering,
Indian Institute
of Technology,
Kharagpur,
India
A simple model order reduction technique is proposed for z-transfer functions. This method is based on approximate model matching in the frequency domain. The entire procedure is carried out in the z-domain and the resultant linear algebraic equations are solved to find the unknown parameters of the reduced-order model. An example favorably compares this method with some prevalent techniques. Keywords: discrete
modelling,
reduced-order
modelling,
1. Introduction The simplification of high-order transfer functions by low-order models is often helpful in the analysis, simulation, and design of complex control systems. The fast development and usage of microcomputers in the design and implementation of control systems increases the importance of reduced-order modelling methods for discrete time systems. Since the theory of linear discrete time systems very closely parallels the theory of linear continuous time systems, many results on the model reduction problem and the practical applications of reduced models are similar. Several methods available in the s-domain are often extended to reduce z-domain transfer functions. In some methods, the reduction procedure is carried out in the w-domain by using a bilinear transformation z = (1 + w)/(l - w), or the zdomain version of the s-domain technique is used. Hwang and Shih’ used the simplified Routh approximation method in the w-domain. Bistritz2 used the discrete version of the Routh approximation method, while Bistritz3 used the discrete version of the Michialov criterion for model order reduction. Hwang et a1.4 used the Routh approximation, dominant poles retention, and stability equations in w-domain to derive various denominators for the reduced-order models. The numerator in each case is found by using an optimization technique to yield a minimum unit step response error. Therapos5 formed the denominator of the reduced model by retaining the dominant poles while the numerator is found by preserving some approximate time moments and Markov parameters of the original system. In Therapos,6 some approximate time moments have been used to obtain the numerator of the reduced model while the denominator is found by using a discrete stability criterion.
Address reprint requests to Dr. Pal at the Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721 302, India Received 1994
20 October
1993; revised
11 April
1994; accepted
Appl. Math. Modelling 1995, Vol. 19, March 0 1995 by Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010
8 June
model matching Hwang and Chow’ simplified the high-order discrete time model in the bilinear-transformed w-domain by approximating the tangent phase function using the Pade technique. Morgan* retained the dominant poles and zeros of the original discrete time model and further used the Matsuburu time delay theorem to add another pole or zero to the low-order model. Al-Assadi’ used optimization techniques in the w-domain to find the reduced model that gives a minimum weighted least square error between the original and reduced models for the frequency range under consideration. In the method of Pal and Prasad,” two computationally attractive techniques to reduce the order of large-scale discrete time systems are given. The methods use the advantage of the Routh-Hurwitz array, bilinear transformation z = (1 + w)/(l - w), and the reciprocal transformation D(w) = wnD(l/w). Both these methods preserve the dynamic characteristics of the original systems satisfactorily and guarantee to have the zero initial condition of the original step response. In the time domain, the concept of balanced realization for the reduction of linear discrete systems has been introduced by Pernebo and Silverman.” Sreeram and Agathoklis” reduced high-order discrete models based on weighted impulse response gramians by retaining the states corresponding to the dominant eigenvalues of these gramians. In this paper a frequency domain direct method is proposed for the model order reduction problem. This is based on approximate frequency response matching between the high- and low-order models. The two transfer functions are matched at a number of frequency points in the low-frequency zone and the resultant linear algebraic equations are solved to arrive at the simplified low-order model. This method is general in the sense that the frequency points of interest may be chosen quite arbitrarily without resorting to any search procedure and it does not involve choosing some critical frequency points as does the method of Shieh et a1.13 This method avoids using the bilinear transformation and yields reduced models that compare favorably with some prevalent model order reduction techniques. 0307-904x/95/$10.00 SSDI 0307-904X(94)00010-4
Discrete
model reduction: S. Pan and J. Pal
2. Mathematical
or,
preliminaries
cg + CIZ+ ... + C,Z”
Consider a real function f(x) with derivatives f”‘(x), i= 1,2,..., n in some region around the point x”. Let the values of f(x) be given for the distinct -real numbers xi of the variable x; where xi = x0 + hi, i = Cl, n] and of calculus of divided h > 0. Using the notation differences we may define f[x,] = f(xo) and the following divided differences of arguments 2 to n + 1:
=(f[x,x,..~xk-,l
It may then be shown14
f[x,x,x,...x,]
; kEC1,~1
-f~x;X2~~~xkl)/(x0
= qQ
(x) >
i E [0, n]
-do@
e+j
the above
e)@ +jb)... n - 10 +j sin n - ie)(a +jb) 0 + j sin
G(z) =a+jb z = a, b
exp(j8)
are real functions
The relation
-do(So
G(z) =
of a high order
of
8.
(8) can be conveniently + cr(R,
a,+a,z+...+a,z”
+ c,(R, + jr,)
+jJ,_l)ZS,+jJ,
(9)
(6)
d, + d,z + ... + zn
F(z) z G(z) 1995,
cos
itl
Ii = sin itl
cg + CIZ + ... + c,z”
Modelling,
+ jr,-,)
+ jJA
. ..-d._,(S,_,
Ri =
b, + b,z + ... + b,Zrn
Appl. Math.
as
where,
where n is the order of the reduced model and n < m; c’s and d’s are the (2n + 1) parameters of the reduced model F(z) that are to be determined. This method ensures that the reduced-order model will approximate the frequency response of the original high-order model. Hence, we may write,
134
written
+j11)
+ jJo) - d,(S,
SISO
where m is the order of the system. For generality, the orders of the numerator and denominator are taken to be the same. The pole zero excess of G(z) may also be 2 1. Let F(z) be the reduced order model of the original system in the form, F(z) =
(8)
where,
... + c,_i(R,-r
Let G(z) be the transfer function stable discrete system given by,
may be
z (cos n0 + j sin no)@ + jb)
co@, +j10)
3. The reduction method
relation
sin 0) + . . . + c,(cos n0 + j sin no)
+ jb) - d,(cos -d,_,(cos
(4)
Thus for a suitably small value of the parameter h for a given f(x), another real valued function Y(x) may always be constructed using equation (3) so that the approximate relations in equation (4) are satisfied.
After simplification, as
CO + cl(cos
(3)
Then from equation (2), ‘I’(“)(t) = f(“) (q) where 5 lies in the interval x0 I 4 I x,, + nh. If the parameter h takes a very small nonnegative value, we have” f”’ (x) Z W
(1)
xk)
period. written
ie [0, n]
=f(xi)3
-
that:
where r] lies in the interval x0 I 9 I x0 + nh. Now let Y(x) be a second real function with finite and continuous derivatives W”(x) around the point x = x0 such that ‘“(xi)
(7)
For the purpose of frequency response matching, we substitute z = exp(joT) = exp(j0) = (cos 8 + j sin 0) in equation (7). Here 0 = 07: and T is the sampling
fCxox11= (fCxo1- fCxJYbcl - XI) fCxox14 = (scw4 - fCx,x,lK% - x2) f[%xI...xkl
2 G(z)
d, + d,z + ... + z”
Vol. 19, March
Si =
u cos
$9 - b sin i0
_ri = a sin i0 + b cos i0, Separating yields c,R,+c,R,
i E [0, n]
the real and imaginary
parts in relation
(9)
+ . ..+c.R,-doSo...-d,_,S,_,~S,
c,I,+c,!,+...+c,I,-d,J,...-d,_,J,_,~J,
I (10)
Here, the left-hand side expressions of equation (10) are real functions of 8 with unknown coefficients c’s and d’s S, and J, are also two real (known) functions of 13. Hence, designating the 1.h.s. functions as 4,(e) and b,(e), respectively, relations (10) may be written in
Discrete model reduction: S. Pan and J. Pal compact
reduced solution
form as:
$R(@ g S,(0)
and
model may be determined of equations (14) as
by the least squares
x = [ATA]-iA=& MO) z J”(O)
In order to force the equivalence of two real functions, 4,(d) and 4,(e), with their approximants, S,(0) and J,(d), respectively, one may equate the appropriate number of initial terms of the corresponding Taylor’s series expansion about 8 = 0. Thus, to accomplish approximate matching of the 1.h.s. functions in equation (11) with the corresponding functions on the r.h.s., the initial N derivatives of the corresponding functions are equated at 8 = 0 to give
rdhm
f
5 csm 0=0
e=. =
and
(12a)
kE[O,N
;
- l]
e=o
(12’4 Now, using the mathematical above, 4R approximately matches
preliminaries S, if
kc[O,
4,(e) where 8, are small postive I
h(e)
N -
values around
given
l]
(13a)
0. Similarly,
= J,(e)
;
k E [0, N - l]
(13b)
t?=e,,
The relations (13a) and (13b) may be combined be expressed in a matrix form as shown below.
Here, A is a (2N) x (2n + 1) matrix
=
to
Example
Rn-1.0
‘..
Ro,o
-L1.0
-so,0
I
In-l.0
..’
IO.0
-Jn-LO
-Jo,,
-Sn-l.k
-sO,k
R -1,k I ”
pn*’
“’
Ro,k
...
I
-Jn-
&G-I
...
ly-,
-LI,N-1
-So.+I
In-l,N-I
...
IO,N-I
-Jn-m-l
-Jo,N-
n a,:,-, I,,,-
1
l,k
1.k
C~n,oJn,o~n,~Jn,~~~~~n.~-~Jn,~-~l=
In the above expressions, Ri,k =
cos
function
1 ’
-Jo,,
X = [c;~~ci~~~cod,_, ..di..dolT; and 6 =
system with 8th order z-transfer
G(z) = N(W(z)
given by
R n.0 “9 o
1
The following is considered7
(14)
Ax = b
A
The frequency points chosen are confined to lie in a small zone around the point w = 0 or fl = 0 (fl = UT). In effect, the matching is done approximately for the effective range of the frequency response in discrete domain, i.e., o = [0, wJ2] or 8 = [0, ~1. For various systems, the sampling period T may be different and so is the sampling frequency w,. But always o,T/2 is a constant and equals n. Therefore, for matching purposes the frequency points are chosen in terms of the normalized frequency variable 0 and assurance about the smallness of the values of 8, is readily checked by comparing it with rc. For simplicity it is assumed that ok = kc?, k = 1, 2,. . . , N, where the choice 6 = 0.01 seems to be a suitable value for most of the systems. Too small a value of 0, may create numerical difficulties in the solution of equations (14). Since a least squares solution (equation 15) is used for obtaining the reduced-order models, its steady-state value may not be the same as the original high-order system. For achieving zero steady state error, F(z) is multiplied by a scalar constant k, where k is given by k = G(z)IW)l,=, 4. Examples
I
e=er-
(15)
(11)
I
where,
8,;
N(z) = 1.682~~ + 1.116~~ - 0.21~~ +0.152z4 -0.516~~ - 0.262~’ +0.0442 - 0.006
Ii,k = sin 8,; and
Si.k = a cos id, - b sin itl,; Ji,k = a sin itI, + b cos itlk; iE [0, n];
k E [0, N - l]
It is clear from equation (14) that N values of 8 give 2N linear equations in the unknown parameters of the reduced model. For (2n + 1) number of unknowns, N is at least equal to (n + 1). The parameters of the
D(z) = 82’ - 5.046~~ - 3.348~~ + 0.63~~ - 0.456~~ + 1.548~~ +0.786z2 - 0.132~ +0.018 The dominant
poles of G(z) are at
0.8790 f j0.2446 For obtaining second-order reduced following two cases are considered.
Appl. Math.
Modelling,
1995,
models
of G(z) the
Vol. 19, March
135
Discrete
model
reduction:
S. Pan and J. Pal
Case 1
The dominant poles of G(z) are retained such that the denominator of the reduced model is (z - 0.8790)’ + 0.24462 and the numerator
is chosen as
where c1 and co are the two unknown parameters to be evaluated. The proposed reduction technique is employed with the value of 6 chosen as 0.01. For different values of N, different sets of values of cr and co are found and listed in Table 1. In this table, J is an index based on the cumulative sum of the squares of the step response error. This error between the oriczinal and the reduced models is considered for 60 samplmg periods. TIME(NO.OF
Case 2
Figure
In this case the reduced-order the form
1.
Step response
SAMPLING
comparison
PERIODS)
(Example
I).
SYSTEM
(ORDER=31
model is taken to be of
C1Z+ co z2 + d,z + do N
where c’s and d’s are the four unknowns to be determined. Here, the value of 6 is chosen as 0.01. For various values of N, the resultant reduced-order models are listed in Table 2. The reduced-order model obtained in Case 2 for N = 25 is finally chosen for further comparison. The unit step and frequency responses are compared with the original system in Figures 1 and 2, respectively. In Figure 2, frequency variable w has been normalized to 8 and so
ORIGINAL ______._ ___.__.______ REDUCED
Table 1. The parameters (Case 1, Example 1) N
of
Cl
reduced
order
CO (x
1 5 10 25
0.240434 0.240423 0.240389 0.240193
-0.166205 -0.166163 -0.166045 -0.165294
Table
2.
N
Appl.
order models
Reduced
I
'0.00
I
0.60
I
I
I
THETA Figure
(Case
model
2, Example
2.
Frequency
response
J(x
10-S)
f jo.2690362
IO
0.2366322 - 0.163917 zz - 1.7632122 + 0.835955
0.881606
f jO.2423341
3.64134
25
s-
0.2350632 - 0.161379 I.7641362 + 0.837715
0.882068
+ j0.2442765
2.33419
50
0.2296632 - 0.153627 zz - I.7605272 + 0.836563
0.880264
f p.2483931
3.64156
Math. Modelling,
1995,
Vol. 19, March
245.937
I
I
2.40
(RAD
comparison
1)
Poles of the model
I
I .60
0.860227
s-
(ORDER=21
I
1.20
0.2793532 - 0.187413 I.7204542 + 0.812371
4
136
IL)
3.67967 3.61161 3.53028 6.79955
The reduced
MODEL
models
(Example
3
1 1).
Discrete Table
3.
Comparison
Serial
no.
of reduced
order models
(Example
1) Reduced
Method 0.2350632
1
Present
2
Therapos5
3
R.-Y.
4
C. Morgan*
5
C. Hwang
6
L. Pernebo
and L. M. Silverman”
7
V. Sreeram
and P. Agathoklis’*
8
R.-Y.
9
Y. Bistritz*
method
R.-Y.
11
Y. Bistritz3
12
C. Hwang
Shih4
C. Hwang,
and Y.-P.
C. Hwang,
and Y.-P.
0.223349z
Shih4
z - 0.2623951
- 1.730342
and Y.-P.
Shih’
the following third order system taken
- 2.62840~~ + 2.300378~ - 0.67032
226.169
+ 0.784276 - 0.215721
1.7303442 0.3644292
1-
77.6081
+ 0.832881
319.578
+ 0.784275 - 0.28918
1.626873~ 0.373124~
8z2 - 15.28289~ + 7.31213 G(z) = z3
- 0.140755
0.316331
1-
45.4093
+ 0.835445
0.2696522
Hwang,
12.5338
- 0.148224
zz - 1.75332
$-
4.25318
+ 0.8338
+ 0.8258
zz - 1.7582~
f
- 0.1969
- 0.1742
1.75142
0.2276962
Shih4
4.14130
z + 0.833638 +0.2966z
0.24862
2
We consider from,16
- 0.161872
1 - 1.75831
$-
the primary frequency range [O, wJ23 is normalized to [O, rc]. Using the error index J, this model is compared with the reduced models obtained by other present-day available techniques in Table 3. It is observed in Tables I and 2 that by increasing the number of frequency points N, at which the frequency matching is performed, a better reduced model is obtained. At some high value of N (in Case 1 it is 25 and in Case 2 it is 50; the number of unknowns being 2 and 4 in these cases, respectively), the reduced models become degraded, which may be due to numerical illconditioning. It may be noted that even for N = 1 (Case l), the resultant reduced model has a lower cumulative error J than that for many of the reduced models listed in Table 3. As seen for this example, when the poles of the reduced model are unspecified (Case 2) these tend to approach the dominant poles of the system as N is increased. As the number of unknown parameters is increased in this case, better approximation is achieved as expected. Example
0.23722
3.6326
z + 0.833638
zz - 1.75942 and C. H. Chow’
2.33419
- 0.166331
- 1.759311
-0.02531
Hwang.
10
and Y.-P.
J( x lo-s)
+ 0.837715
0.2406572
C. Hwang,
Model
- 0.161379
i? - 1.764136~
t
Hwang,
model reduction: S. Pan and J. Pal
510.599
+ 0.701497 - 0.298503
1.626873~
518.236
+ 0.701497
0.201789t
+ 0.044842~
- 0.156947
1.201789i!
- 1.955158~
+ 0.843053
739.125
the dc gain of the system is 17.7212. Pade approximation of G(z) obtained via the bilinear transformation (z = 1 + w/l - w) gives the following unstable secondorder model: -4.7554058~~ - 0.1883834~ + 4.5670224 FPade(Z)=
0.7116578~~ - 2.0106304~ + 1.2777118 (16)
Let the reduced-order
model be chosen as
k(z + 21)
F(z) =
(z + PlXZ +
P2)
For 6 = 0.01, a family of stable reduced-order models as detailed in Table 4 has been obtained for various values of N. Here, for 100 samples the error index J is computed for the normalized original and reduced models (i.e., a dc gain of 1). For this example it is seen that the Pade approximation technique gives an unstable model, whereas the present method yields a family of stable reduced-order models. In the various numerical experimentations carried out, when the poles of reduced-order models are unspecified, this method gave acceptable stable loworder models from stable high-order systems. Because this method relies on low-frequency matching, as with
Appt.
Math.
Modelling,
1995,
Vol. 19, March
137
Discrete model reduction: Table 4.
The reduced-order
S. Pan and J. Pal models
(Example
N
k
21
Pl.
11 12 13 14 15 16 17 18 19 20 25
2.80427 4.166 5.21395 6.0106 6.6174 7.08323 7.44693 7.73525 7.96214 8.148 8.70674
- 0.775906 - 0.839436 -0.867507 -0.883413 - 0.893696 -0.900918 -0.906285 -0.910437 -0.913754 -0.916466 -0.924981
0.95162 0.92188 0.89955 0.88302 0.87081 0.86172 0.85451 0.84973 0.84574 0.84264 0.83444
2) J
P2
k jO.18021 k jO.17786 + jO.16996 z jO.16080 + XI.1 5169 I io.14315 _+jO.13538 + jO.12845 k jO.12230 + jO.11688 k jO.097228
5.749 1.873 0.992 0.698 0.591 0.556 0.553 0.566 0.586 0.610 0.737
optimization procedure and generally yields a reduced model having excellent matching qualities. (4) As this method relies on low-frequency matching, when the poles of the reduced model are unspecified, stability of the reduced model cannot be guaranteed in general. In case of instability, one may use any stability preservation methodIU3 for obtaining stable poles and use this method for finding the zeros while assuring good frequency domain matching. (5) For the first example presented, this method yields the least value of the error J.
References 1.
other frequency domain reduction methods (equation 16), stability of the reduced model cannot be guaranteed in general.
2. 3. 4.
5. Conclusions A new model order reduction method is proposed that uses approximate frequency response matching between the original system and the reduced-order model. The matching is done at low frequencies and in the sequel, an overall good matching is achieved for the effective range of frequency response. One novel feature of this method is that there is no need to perform an elaborate frequency response analysis of the original system for the selection of frequency points at which the matching has to be performed. The proper choice of the frequency points for matching purposes is made by selecting an appropriate 6. By varying 6 and N, one may obtain a family of reduced-order models of particular order having various emphasis on the low- to high-frequency behaviors. The proposed order reduction method has the following advantages. (1) It is a direct z-domain reduction method and avoids using any bilinear transformation. (2) There are no strict requirements on the number and choice of the frequency points where the matching has to be done. (3) The method is algebraic and calls for the solution of a set of linear algebraic equations. It avoids using any
138
Appl.
Math.
Modelling,
1995,
Vol. 19, March
8. 9. 10. 11.
12.
13.
14. 15. 16.
Hwang, C. and Shih, Y.-P. Routh approximations for reducing order of discrete systems, Trans. ASME J. Dynamic Systems, Meas. Control, 1982, 104, 107-109 Bistritz, Y. A direct Routh stability method for discrete system modelling, Syst. Control Lett. 1982, 2(2), 83-87 Bistritz, Y. A discrete stability equation theorem and method of stable model reduction, Syst. Control Lett 1982, l(6), 373-381 Hwang, R.-Y., Hwang, C. and Shih, Y.-P. A stable residue method for model reduction of discrete systems, Cornput. Elec. Eng. 1983, 10(4), 259-267 Therapos, C. P. Direct method for discrete low order modelhng, Electron. Lett. 1984, 20(6), 266-268 Therapos, C. P. Low-order modelling via discrete stability equations, IEE Proc. 1984, 131, Pt-D(6) 248-252 Hwang, C. and Chow, H. C. Simplification of z-transfer function via Pade approximation of tangent phase function, IEEE Trans. 1985, AC-30(1 l), 1101-I 104 Morgan, C. A low order modelling method for z-domain transfer functions, Trans. Instru. Meas. Control 1987, 9(3) 165-168 Al-Assadi, S. A. K. An iterative method for simplification of discrete systems, J. Franklin Institute 1989. 326(4) 587-607 Pal, J. and Prasad, R. Biased reduced order models for discrete time systems, Systems Science (Poland), 1992, 18(3), 41-50 Pernebo, L. and Silverman, L. M. Model reduction via balanced state space realization, IEEE Trans. Auto Control 1982, 27, 382-387 Sreeram, V. and Agathoklis, P. Model reduction of linear discrete systems via weighted impulse response gramians, int. J. Control 1991, 53(l), 129-144 Shieh, L. S., Chang, Y. F. and Yates, R. E. A dominant data matching method for digital control systems modelling and design, IEEE Trans. Indus. Electron. Control Instrum. 1981, 28(4), 39G396 Mime-Thomson, L. M. The calculus of finite differences. Macmillan, London, 1960 Pal. J. An algorithmic method for the simplification of dynamic scalar systems, ht. J. Control 1986, 43(l), 251-269 Pal, J. Reduced order models for control studies, Ph.D. Thesis, Roorkee University, India, 1980
