On model reduction by L2-optimal pole retention

On Model Reduction by L,-Optimal Retention

Pole

by W. KRAJEWSKI Systems

Research Institute,

and A. LEPSCHY, Department

Polish Academy

G. A. MIAN

of Electronics

of Sciences, 01-447 Warsaw, Poland

and u. VIARO

and Informatics,

University

of Padova, via Gradenigo 6

A, 35131 Padova, Ital?

ABSTRACT : The L,-optimal reduced model retaining a given number of poles of the original system is determined in an eficient way. In particular, an expression for evaluating the minimum of the impulse response error norm for any choice of retained poles is derived in terms of the original system parameters, which allows us to find the truly dominant poles in a simple manner. Then, the corresponding optimal model numerator is obtained using an orderrecursive procedure. An interesting interpolation property of the reduced model is also pointed out. An example shows how the method operates.

I. Introduction

The problem of constructing reduced-order models for linear systems has received continuing interest during the last years, and a variety of approaches have been adopted. Besides the methods based on the Padt technique (l), other sophisticated yet computationally simple methods, such as Moore’s balanced truncation (2) and Glover’s Hankel-norm methods (3) have been proposed recently; these techniques, in particular, have guaranteed bounds on the L2 norm of the output error for any L2 input. Nevertheless, the techniques based on the retention of poles of the original transference, usually motivated by physical considerations, are still popular. In this regard, the most spontaneous solution is probably that of retaining the so-called dominant poles, i.e. those nearest the origin or the imaginary axis. Often, the zeros belonging to the same region as the dominant poles are also preserved (as well as the Bode gain). It seems, however, more satisfactory to adjust the residues corresponding to the retained poles (or, equivalently, to move the zeros of the reduced model) so as to compensate for the elimination of the other poles in a suitable way. To evaluate the effects of the neglected poles, it is meaningful to refer to the integral of the squared deviation of the reduced model response from the original response for certain inputs. As is known, the problem of finding the overall L,-optimal reduced model of a given order is intrinsically nonlinear if the denominator is not assigned beforehand. This general problem was first considered by Meier and Luenberger (4) using the transfer-function approach and by Wilson (5) using the state-space approach.

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In the considered case in which the eigenvalues to be retained are chosen a priori, the problem is linear and can be solved using standard mean-square techniques. In this paper, an alternative order-recursive procedure to find the corresponding L,-optimal reduced model in an efficient way is presented. It must be noted, however, that for high-order systems the choice of the poles to be retained is not always straightforward. For example, with reference to the modal aggregation method of Davison (6), Chidambara (7) and Marshall (S), it has been suggested (9, 10) to eliminate those eigenvalues which minimize a quadratic criterion of the output error with either impulse or step inputs. Therefore, the procedure to find the optimal reduced model which retains k poles of the original nth order system would, in general, consist of the following steps : (i) consider

each k-tuple

of retained K(i), i = 1,2,. . , ” 0k ’

poles ;

(ii) for any i, determine the corresponding optimal reduced transference numerator ; (iii) compute the related minimum J,,,i,(i) of the considered criterion ; (iv) compare the minima for all possible choices i and find the least one Jml,(i) = mini J,,,(i). The problem greatly simplifies if step (iii) may be performed without the previous determination of the transfer function numerator, thus avoiding step (ii). In the following, it is shown how to achieve this result. Specifically, a simple expression for the evaluation of J,l,i,(i) in terms of the original system parameters is derived. In this way, the reduced transference numerator need be computed only for the optimal combination K(Q of retained poles. The suggested procedure may also be used to easily find a good starting point when an iterative method for minimizing the L2 norm of the error without constraints on the poles has to be adopted. Finally, an interesting interpolation property of the suggested technique is pointed out. II. Optimality

Condition

Let the original

nth order transfer G,(s)

where iV,_ , (s) is a polynomial

function =

be

I(4 Qh)

5:

(1)

of degree at most n - 1 and D,(s) = ir (s-PJ i= I

(2)

is a manic Hurwitz polynomial. Poles pi are assumed to be simple as the consideration of multiple poles would require a considerable increase in notation with little corresponding increase in insight. The extension of the results to the general case, however, does not entail theoretical difficulties. 62

Journal 01 the Franklin lnst~tute Pcrgamon Press plc

Model Reduction 6-y L,-Optimal Pole Retention Let us assume first that a simple pole, say pn, has to be removed. reduced model :

The optimal

where Dn-

I ($1

=

I-I i=

I

(S-Pi>

(3b)

and Nn_2(~) is a polynomial of degree at most n-2, subspace of functions r,_ ,(s) given by

is to be selected

from the

By indicating with gn(t) and gn_ ,(t) the impulse responses of the original and the optimal reduced-order model (3), respectively, the corresponding one-step reduction error is e,-l(t):=gn(t)--Sn-I(t) and its Laplace transform E,-,(s) The squared

(5)

is given by

= G,(s)-G,_,(s)

=

N,~l(s)-(s-P,)N,-*(s) Dn 6)

L2 norm of e,_ l(t), i.e. me,_,(t)e,*_l(t)dt s0

)Ie,_,(t)I12:=

where e,*_ l(t) is the complex conjugate r,- r(s) EM,_ I, the orthogonality condition

s

of e,_ l(t), (11) :

(7) is minimum

iff, for any

m

e,_ (t)y,f- I(t) dt = 0

0

I

is satisfied, with ynp l(t) = LT-‘(T,l(s)}. Now, by recalling that LT{y,*_ ,(t)) = r,*_ l(s*), from the Parseval obtain

s cc

en_ (t)y,*_ (I) dt = I

I

0

L 27cj

theorem,

we

s +I=-

E,- , (s)T,*_ 1(-as*) ds

_jm

1 - 2nj s

r,*_1(-s*)

+‘“N,~I(~)-(s--P~)N,-~(s) _jm

Dn

ds.

(9)

(4

The last integral is equal to zero for all r,- l(s) if all the poles of the integrand (in minimal form) belong to the left half-plane, i.e. if the zeros of the denominator D,*_ l(s) of r,*_ l( -s*) (which are in the right half-plane) coincide with those of Vol.327,

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W. Krujewski

et al.

N,.. r(s) -(SFpn)N,_z(.r). It follows model satisfies the equation : (s--y,)N,_*(s)

that

the numerator

N,_,(s)

of the optimal

= N,_,(s)+h,D,*~,(-s*)

(10)

with

Moreover,

from (6), (10) and (1 I), we have

(12) so that the minimum

value of the L2 norm squared

is

(13)

III. Reduction Procedure Ifpolespn, pn- ,, . . . dn-ht , are to be successively eliminated, apply (n-k) times relation (10) Lhus arriving at the G&Y) = Nk- ,(s)/D&) Since the adopted

we may recursively kth order model

where &(s) procedure

= n (s--p,). i= 1 ensures that ‘x ej(t)e;“_ r(t) dt = 0

(14)

s0

where ej(t) and e,_ ,(t) are the errors incurred in two consecutive steps, model Gk($) is L,-optimal for the given choice of eliminated poles (independently of the order of elimination). Consequently, the value of the squared norm of the (n-k)-step reduction error d,(t) is given by

Of course, two or more steps may be merged into a single stride. In the case of ;1 steps starting from G,,,(s) = N,,, ,(s)/D,,,(s) and arriving at G,, ;(s) = N,,,_; I (.~)lD,,,~j(s), we find

i

,cmf&(s-_Fi)

where polynomial 64

Qi

i

N~-J.~I(s) =N,~,(s)+Qn~,(s)D;1:,(-s*)

, (s) of degree j_- 1 is uniquely

(16)

defined by the A conditions Journal ofthe Frankhn

:

lnst~tute Pergarnon Press plc

Model Reduction by L,-Optimal Pole Retention

(17) In practice, expression (16) with 1 = 2 will be used whenever a pair of complex conjugate poles is to be eliminated. In this way, only real polynomials Di(s) need be considered, SO that O:( - s*) = Oj( -s). In conclusion, one may either eliminate successively each real pole and pair of conjugate poles (thus generating a family of models of descending degree) or eliminate simultaneously the whole set of n -k poles by means of (16), which only entails a polynomial division. It is worth noting that the &-optimal reduced-order model G,(S) having simple polesp,, i = 1,2,. . . , k, enjoys the interesting property (cJ Appendix) of taking on the same values as G,(s) at points -pi, i = 1,2, . . . , k, i.e. G&) interpolates the k points [-pi, G,( -pJ]. This suggests the alternative, even if less efficient, procedure to compute iVk_ ,(s) via Lagrange’s or Newton’s interpolation formulae. Concerning the evaluation of I/d,(t) 11 *, instead of (15), one can use an equivalent expression in terms of the poles of the original transfer function :

(18)

G,,(s) = ~ * I=

1 s-Pj.

Denoting by A&) the Laplace transform d,(t) = gn(t) -gk(t) and using (16), we have

of the

Qn-k- 16) Z-s*)

Ads)=-

n

j=lJ, Cs-Pj) By expressing

the first fraction _

and recalling

(n-k)-step

Dk(S) ’

reduction

error

(19)

in (19) as

a-k- 1(s)

j=lj,Q-Pi) =j_g,&J

G-9)

(17), we obtain rj = R,

Dk(P,) Dk*(-p:)

’

(21)

Therefore,

= $,

j_g,

R~Ri* Dk@i)Dk*@i) Pi+P; Dk*(-pj*)Dk( -p:)

(22)

where Vol 321, No I_ pp. 61-70, Printed tn Great Bntam

1990

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W. Krujewski

et al.

Dk@jlD,*(P;)

bx-p.:)D,(-p*) < I for each Note involves and the

(23)

pair (pi,pi). that expression (22) contains only parameters of the original system and elementary arithmetical operations on them (e.g. expression (10) in (10) corresponding expression in (9) for the modal aggregation method).

IV. Choice of the Poles to be Retained Given the k poles to be retained, the above procedure leads to the L,-optima1 reduced mode1 in a computationally efficient way. As already observed, the order in which the poles are eliminated does not affect the result. However, the result does depend on the choice of the poles to be retained. In this regard, one could eliminate at any stage of the reduction process the pole giving the least contribution to the error norm. This would require computing by means of (13) n error “energies” at the first step, n- 1 at the second, and so on. Such a procedure, however reasonable, does not lead, in general, to the overall optima1 model. For instance, starting from

Go

1

= -+s+l

1

s+2

+ -++

the best second-order mode1 turns out to be one without the best mode1 of order one is G,(s)

the pole at -2,

whereas

47115 = s+2

(25)

which cannot be obtained from the best mode1 of order two. To find the overall optima1 mode1 of order k, it is therefore necessary

to compute,

to (22), 118,(t) 1)2 for all

i possible combinations of eliminated 0 poles. Observe that the error norm for the overall optima1 model of order j is by definition less than, or equal to, the error norm for any other model of order j and, in particular, it is less than, or equal to, the norms for the models of the same order from which the best mode1 of order j- 1 may be obtained. It follows that the error norm of the best model increases as the order decreases, as is expected.

e.g. according

V. Example To show the implementation of the method, we consider the eight-order transfer function considered in (12) with reference to a different pole retention technique (based on the fitting of the first time moments) :

NT(~1 DE(S)

(26)

G*(S)= ~ with

Journal

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plc

Model Reduction by L,-Optimal Pole Retention

D*(s) = (~+1)@+2)(s+3)(~+4)(~+5)(~+6)(~+7)(~+8) and N,(S) = 18(~+0.32)(~+2.45-jO.53)(~+2.45+jO.53) (s+5-jO.65)(~+5+jO.65)(~+5.89)(~+7.43). According to the procedure described in the previous sections, the best models of order from 7 to 2 and the corresponding values of the error norms are found to be:

G,(S) = --

x (s+4.97-jO.69)(~+4.97+jO.69)(~+7.40) (~+1)(~+2)(~+3)(~+4)(~+5)(~+7)(~+8) [l&(t) 11= 0.82 x lo- 5,

18(s+0.32)(s+2.49-j0.49)(s+2.49+j0.49)(s+5.01)(s+7.24) Go

=

(s+l)(s+2)(~+3)(~+4)(~+7)(~+8) II&(t) II = 0.45 x lo-3,

G (s) = 18(~+0.32)(~+2.62-j0.29)(~+2.62+j0.29)(s+6) 5 (s+l)(s+2)(~+3)(~+5)(~+8) ,l’s&),l G,(s) =

17.98(~+0.32)(~+2.44)(~+6.82) (S+1)(S+2)(s+6)(S+8)

Go = G2($

= 0.16 x 10-2,

lls4(t)II =0.49x lop21

2

17.86(~+0.33)(~+2.43), Ils,ctj I, = 0.28x 1o-I, ts+ 1)(~+2)(~+7) =

‘;;=;;;t;;),

IliS,(t)II

=

0.18 x 10’.

If the reduced model order is not specified beforehand, one could choose it on the basis of the error increments incurred at each step. In the considered case, the error norm is less than 1% of Ilg8(t) II = 4.66 up to G3(s) (included) and it is about 4% for G,(s) which may be considered as a reasonable compromise between model simplicity on the one hand and a good fit on the other. The step response of model G2@) is compared in Fig. 1 with the step responses of the original system and of the second-order model which retains the two poles nearest the origin and fits the first two time moments. It is interesting to note that the second-order model obtained via the balanced truncation method (2), which is usually considered as one of the best, turns out to be 17.77(s+ 0.26) G2’(s) = @+0.73)(~+6.64) whose step response Vol. 327. No. I, pp. 61-70. 1990 Printed in Great Britain

is practically

indistinguishable

from that of G,(s).

67

W. Krujewski et al.

FIG. 1. Step response of the original system (solid line), of the proposed (dashed

second-order model line) and of the second-order model which retains the two poles nearest the origin and fits the first two time moments (dotted line).

VI. concIusions

A computationally efficient reduction method which retains the dominant modes of the original system while minimizing the L2 norm of the reduction error, has been illustrated. Unlike other methods that simply neglect far-off poles or are based on state aggregation schemes, the numerator of the simplified transfer function takes into account the neglected poles in a proper way. The families of reduced models of decreasing order are obtained recursively and the increments of the error norm are evaluated at each step. In any case, the goodness of the best approximation increases with the model order. An interesting interpolation property of the proposed method is also pointed out. An example shows that the method leads to models whose transient response satisfactorily fits the original one. References (1) A. Bultheel and M. Van Bare], “Pade techniques for model reduction in linear system theory: a survey”, J. Cornput. appl. Math., Vol. 14, pp. 401438, 1986. (2) B. C. Moore, “Principal component analysis in linear systems : controllability, observability and model reduction”, IEEE Trans. Aut. Control, Vol. AC-26, pp. 17-32, 1981. (3) K. Clover, “All optimal Hankel-norm approximations of linear multivariable systems and their L, error bounds”, ht. J. Control, Vol. 39, pp. l115P1193, 1984. of linear constant systems”, IEEE (4) L. Meier and D. G. Luenberger, “Approximation Trans. Aut. Control, Vol. AC-12, pp. 585-588, 1967. (5) D. A. Wilson, “Optimum solution of model-reduction problem”, Proc. ZEE, Vol. 117, pp. 1161-1165, 1970. (6) E. J. Davison. “A method for simplifying linear dynamic systems”, IEEE Trans. Aut. Control, Vol. AC-l 1, pp. 93-101, 1966. (7) M. R. Chidambara, “On ‘A method for simplifying linear dynamic systems’ “, ZEEE Trans. Aut. Control, Vol. AC-12, pp. 119-120, 1967; and related discussion, pp. 120-121,213-214,214,799,799,7999800,800. Journal

68

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Franklm Pergamon

lnstmtc Press plc

Model Reduction by L,-Optimal Pole Retention (8) S. A. Marshall, “An approximate method for reducing the order of a linear system”, Control, Vol. 10, pp. 642-643, 1966. (9) C. Commault, “Optimal choice of modes for aggregation”, Automatica, Vol. 17, pp. 397-399, 198 1. (10) M. Gopal and S. I. Metha, “On the selection of the eigenvalues to be retained in the reduced order model”, IEEE Trans. Aut. Control, Vol. AC-27, pp. 688-690, 1982. (11) D. J. Luenberger, “Optimization by Vector Space Methods”, Wiley, New York, 1969. (12) Y. Shamash, “Linear system reduction using Pad& approximation to allow retention of dominant modes”, ht. J. Control, Vol. 21, pp. 257-272, 197.5.

Appendix In this Appendix we shall prove the interpolation property of the &optimal reducedorder model mentioned in Section IV. For the sake of simplicity, we refer only to the case of real distinct poles (the extension to the general case is, however, straightforward). Specificaliy, we show that the &-optimal G&) = Nk_ ,(s)/D&) with specified denominator D,(s) = n (s-p,) i= I

satisfies the interpolation G,(-p;)

conditions

= G,(-p,),

i=

: 1,2 ,...,

k

(Al)

i.e. it takes on the same values as the function G,(s) to be approximated at the opposites of its own poles. Let us denote the generic function belonging to Mk = span (I/@-p,), l/(3-p,), . ,

l/kid1

by

and define

f/X Jn := II&(t) //2 = ?ij Jk turns out to be a quadratic the residues I?; are such that

.s

I’

function

[C,(s)-r,(s)l[G,(-s)-r,(-s)lds. in the k variables

l?JI -=O, aI?,

i=1,2

,...,

R,. Therefore,

643) it is minimum

iff

k.

Now, from (A3), we have JJ, ^= (7R,

--

2

+,ic

2nj s ~_,T,

[G.,(S)-I&)]~ds.

(A51

Since

ar,(-s)

---7.-= 84

conditions

(A4) may be restated

Vol.321,No. I.pp.61-70.1990 Prmted in Chat Britain

--

1 J +pi

646)

as

69

W. Krujewski

et

al. (A7)

Consequently,

the k residues i?,, of the optimal

It is immediately

satisfy the set of k linear equations

solution

seen that

I

ds

G+PJ(s-PA and

+,Gn(&& =

-I -,= 1

2nj

Therefore,

(A9)

h+P,

”

-G,(--pi).

(A101

(A8) may be written in the form -mi,Ktk

Since the left-hand are proved.

= GA-P,)>

side of (Al 1) coincides

i=

with expression

1,2 ,...,

k.

(Al 1)

(A2) of I-,(-p,),

Journal

70

:

conditions

(Al)

of the Franklin Pergamon

Institute Press plc