Interactive Model Reduction by Minimizing the Weighted Equation Error

INTERACTIVE MODEL REDUCTION BY MINIMIZING THE WEIGHTED EQUATION ERROR E. Eitelberg Institut fur Regelungs· und Steuerungssysteme, Universitiit (TH) Karlsruhe, Karlsruhe, Federal Republic of Germany

Abstract . A method for model reduction is proposed , that minimizes the equation error and yields a stationary exact model. With this model, the desired state variables or their physically meaningful linear combinations a r e reproduced . This approach to the reduction problem leads to linear equations, thus no search routines are needed . Moreover, a time-varying weighting matrix is intr oduced , so at even unstable linear time-invariant models can be reducec Especially , the characteristics of very low orde r reauced models depend highly on the weighting matrix. This influence of the weighting can be best exploited in an interactive reduction on a digital computer. The reduction for a distillation column is carried out. Keywords . System order reduction; large-scale systems; linear systems; computer aided system design; state - space methods; least squares approximations; linear differential equations.

INTRODUCTION

example Dec oster and van Cauwenberghe (1976) for a review. Most of the universaly applicable methods can be ordered into two categories. The methods of the first category conserve the dominant eigenva l ues of the original system. Litz (1979) has arrived t o very good results this way. But a question has never been answered clearly: for which purposes should a good reduced model obtain its eigenvalues from the original model? The methods of the second most common category minimize an integrated quadratic error between the original and the reduced models. Here the problem is essentially nonlinear ar.d in practical cases the minimization can be very difficult if not impossible. A good initial approximation may be crucial .

Often the mathematical modelling of physical systems or processes leads to very large models. It lays in the nature of the mathematical modelling, that simplifying interactions between different parts of a large system can not be detected and counted for in the course of the modelling. When input - output behaviour is considered , then very probably a much simpler model with on ly a few differential equations would describe the same physical process . The reaction to an arbitrary initial state can not be described by a simpler model. But the "arbitrary" initial state is , in many cases, just a substitute for disturbances in the state of the physical system and these are not so arbitrary . For many prac~ical purposes the input - output behaviour is the most important one. So the model reducti on is often possible. On the other hand the practical application of such modern concepts as state estimation, optimal state feedback and even numerical simulation is limited by the capacity of the available digital computer . In this case the model reduction is the only way out .

Obinata and Ino oka (1976) have taken a different approach . Inst ead of minimizing the output error they minimize the equation error . This kind of minimization leads to a linear problem with all of its advantages. However, in the mentioned paper only necessary conditions have been derived and instead of the original system matrices, the input and output vectors have been used. In Ei telberg (1978a) the same approach has lead to much simpler formulas , where only the matrices of

Here the reduction of linear time invariant models is considered. There already exist many methods, see for 47

E. Eitelberg

48

the original state-space model are needed. However, no weighting of the equation error was proposed and the stability of the original model was a requirement.

this system. Now, our problem is to find the model (2) so that

In the present paper i t is proved that a reduced model with a correct steadystate has the correct steady-state with any proportional or dynamic feedback, provided the closed-loop system has no zero-eigenvalues. This does not depend on the stability of the original and reduced models. Also a time-varying matrix weightingfunction is introduced. This allowes the reduction of unstable models. But even more important is, that with different weightings different reduction results emerge. So one can "play" with the computer so l ong as a satisfactory reduced model is achieved.

holds. Many physical systems remain most of the time in some steadystates or in their vicinity. Therefore the reduced model should be stationary exact. Most of the technical systems are used with some sort of proportional or dynamic feedback. So, appart from Eq. (5) the requirement, that

x

C

~

(t)

R

x (t)

(5 )

x -st

( 6)

must hold with any linear feedback, is added.

PROBLEM SOLUTION The two requirements (5) and (6) are handled separately. Stationary exactness is a universal requirement and is considered first. The satisfaction of the dynamical requirement (5) is highly dependent on the mathematical approach and is considered thereafter. Then the basis for the interactive reduction is discussed.

We have the n-dimensional model ~

-r

R

-st

PROBLEM STATEMENT

A x + B

x

x{t) ~

( 1 a) (l b)

with constant matrices ~, Band C. The reduced m-dimensional model is described by

Stationary Exactness A

x

C

x

-r-

+

B u -r-

( 2a) (2b)

-r-

The m components of the state vector should reproduce the desired components of ~ and /o r their physically meaningful linear combinations , which form the vector x

-r

Consider the structure displayed on Fig. 1.

R ~

x

(3 )

..:..

!

1i '·K .. 1.1

w

u

-- L~ -

-r

physical system is described by the state differential equation s (la) or (2 a) and the output equat i ons (lb) or (2b) only show, what one does with

= ~r~

B L:_ _r

d

"Q"

,

X

= A-:.

-; y =

+

1.

.';::i

Bu

eX

CONTROLLER

Fig.

holds. In practical problems finding of C is trivial. In fact, the

y

+

LINEAR

£)

(4 )

= 6r~

ij

that influence the output y directly (these components are not multiplied by zeros in and it may contain other components of interest. Often the measured states should also be reproduced in the reduced model. Under these conditions C R -r-

X

&..I_ _ _ _ _ _ _ _. . . . . .

~

R is the so-called reduction matrix. ~r must contain all the components

C

"i ./

I;

!i

;

li

'/,---JI .'-.j

Structure of models with feedback.

An arbitrary linear dynamic controller is described by ( 7a)

(7b)

49

Interactive Model Reduction z is the state vector of the controller and the matrices ~, ~ have appropriate dimensions.

f, !,

For the beginning it is supposed, that the matrices A and ~r are regular. Eq. (6) must hold for any feedback, also for K = 0 and L O. In this case one can write - 1

x

A

X

A B u -r -r-st

-st

and

B u -st

A R A

- 1

-r-

A

x -st

-

is substituted

the proof is completed. A R A

B -r

-r-

-1

Eq.

So

(10) ( lOa)

B

( 10)

B

From Eqs.

(7b)

B w + B K ~s t

~ ~ ~)~st -

( 14)

Eq. (15) is identical to Eq. (12a) multiplied from left by (~r~ A- 1 )

and

(la)

+ B ~ Ys t. (11)

is a necessary and sufficient condition for the stationary exactness in a closed loop system. However the regularity of the matrix ~ was required. In the case where A is singular, the reduction problem is modified in the following manner. An arbitrary proportional feedback ~ (with K 0) defines a new original system

Now Eq. (lb) is substituted into Eqs. (11) and (7a):

(A

(10)

-r-

( 15)

It will be shown immediately, that Eq. (10) is also a sufficient condition. For this purpose equations are derived, which define the steadystates ~st and ~st for the original closed loop. one gets

Now the condition into Eq. (14):

-B w

( 9)

Because of Eq. (6) we get a necessary condition for the stationary exactness: B -r

(A R-B L C)x -B K z - r - -r- - -st -r- -st

(13a)

( 8)

- 1

-st

the same substitution into Eq. we have

~ K ~st

-B '!!..,

x

A

Y

C

~

+

B w

( 16a)

x

(16b)

(12a) and a new reduced system

o

F C x + E z -st -st

( 12b)

,

-

~

A x + B w -r-r-

Now, on Fig. 1, the loop is closed over the reduced model "R". The equations for the steady-states ~st

Y

-r-

and z are derived analogically to Eqs. -H2)

Where only

(A -B L C)x -B K z -r -r- -r -st -r- -st

x -r-st

F C

+ E

z -st

-B w -r-

o

A

A -r

A -r

B L C

( 18)

and ( 1 3b)

then (~st = ~ ~st' i st = ~st) is the solution of Eqs. (13)

(12),

z

x R x and = z are -st - -st -st -st substituted into Eq. (13b) and the relation C = C R (Eq. (4)) is counted If

A

( 1 7b)

( 1 3a)

The two systems of equations, (12) and (13), have unique solutions, when the closed loop systems have no zeroeigenvalues. Now the following must be demonstrated: If (~st'~st) is the solution of Eqs.

x

C

( 17 a)

-r-

for, then it becomes clear that Eqs. (12b) and (13b) are identical. After

B L C - r - -r

( 19)

have changed. The only task of the proportional feedback ~ is to remove the zero-eigenvalues of A. With any feedback ~, in Eq. (18) regular, B

-r

that leaves A the equation ( 20)

is a necessary and sufficient condition for the stationary exactness in any closed loop system, provided neither the original nor the reduced closed loop systems have any zero-eigenvalues. That means, both

Eo Eitelberg

50

(e~t _ I)A- 1 B U

Dynamic Exactness

If

x+

A

x

-r

( t)

we have

(2a) ( 2 1)

B u

-r-

-r-

(29)

- 0

We proceed from Eq. x

the step response

S~~st ituting

systems must have unique steady states.

x(t)

R ~(t)

dO

held fo r

(~r~

+

A- 1B

-

~r ) ] ~o

( 30)

then also

all t, x -r

( 22)

A x + B u -r- r-r

Beca use of Eq . exactness)

(10)

(stationary

( 31)

d

would be true. Of course, Eq. (22) does not hold. But i t would be good already, i f the equation error

holds.

Since U

- 0

characteristic, dO

~(t)

were sma ll . The equation error d O has a very clear influence on the reduction err or

x

-

-r

the e xpressi on

(23)

B u -r-

A X - r -r

X

-r

is not a model

(24)

x

(R

At - 1

A R)e- A -r-

A

will be minimized in the sense

I

Jtrace{ ~(t)~(t)~T(t)

min. (33)

}dt

A - r

o

substituting

into Eq.

~

x

x

A

gives G(t) is an appropriate weighting matrix. Eitelberg (1979) has shown, that the minimization in Eq . (33) is a generalization of the Gaussian least squares algorithm . Mo re over , i t was proved , that a solution ~r always

B u -r-

-r-

- r

(21)

x)+(x-r -A-r-r x -B u) -r-

(A X - A -r-r -r+ dO

e

A

- r-

(25)

Eq. (25) underlines the usefulness of the equa tion error ~ 0, but further the definition (23) wil l be used . It has already been mentioned that in practical systems the input-output behaviour is of the primary interest. Very often the states of the system move only between some stationary points. This situation is well represented by choosing for to = 0 ~o

=

Q,

therefore also

~r o

O.

And

the input ~(t)

U

-0

a

(26)

(t)

makes the following simple . For t

~

R

dO

B

from Eq.

(R A-A R)x + -

-r- -

with

S

At

Jr e-

(~

- 1

~)~ (t)

(A

- 1

B)

T ATt e- dt. (35)

Cl

It can be shown, that a very strong sufficient condition for the regularity of the matrix (R S RT ) in Eq. (34) is the controllability of the original model (1) and the maximal rank of the reduction matrix R (Eitelberg, 1 979) . In practically all cases one has a unique solution (36 )

A

U

-r-o ~

(34 )

- r

x

substituting

exists and satisfies the equation

formulas very

0 we have now

dO

( 32)

B

(la)

(R B-B )U -

-r - 0

( 27) we have (28 )

The direct integration in Eq. (35) a difficult task. When a special we ighting matrix 2a ~(t)

2. (e

1

t

2C1. +

Be

2

is

t )

( 37)

is introduced , then by integrating Eq . (35) partially one sees , that

51

Interactive Model Reduction That means:

the matrix 5 -

=

can be found by solving two Lyapunov equations: (~+a 1.!.) ~1 +~1 (~+a 1.!.) -( ~

- 1

~)2(~

- 1

T

~)

is

found

and with this matrix B - r Eq. ( 10) At the end A - r Eq. ( 19)

(38 )

5 -1

~~

A -r

=

A -r

+

from Eq.

(36)

is

found

from

is

found

from

(4 1)

B L C -r-

In case of a regular matrix A the and B are found reduced matrices A

T

-r

(39)

directly

and

from Eqs.

- r

(36)

and

(10) .

EXAMPLE

(A

( 40) Three most effective algorithms for numerical solution of Lyapunov equations hav e b een c o mp a red by Ko upan and MUll e r (1976) Interactive Reduction The constant weighting matrix 2 in Eq. (37) allows the use of different weightings for different inputs . The constants a and a must be chosen 2 1 so that the matrices (A + a .!.) and

Trilling and Klein ( 1979) have made a rather unsuccessful attempt to reduce high order models of distillation c o lumns . Modal methods were used. One of the examples was a three - component column with 27 plates. The linearized model of the 45-th order describes the dynamics of the temperatures , c o ncentrations and bottom product in the c o lumn. This model has 10 inputs. Four of them - heating vapour , withdrawn bott o m product , withdrawn vapour at the side and reflux - are considered as c o ntrol inputs. The other six are consi d ered as disturbance inputs .

1

(A + a

.!.) in Eqs. 2

(39)

and

(40)

have

all eigenvalues with negative real parts . This way even unstable original systems ( 1 ) can be reduced . When Ai is an e ig envalue of ~ and Ali an eigenvalue of

(A + a .!.) 1

then A1i=Ai+a

1

holds for all i = l, . . . , n. Appart from guaranteeing the convergence of the inteqral in Eq. (35) the scalar 2a t 2a t 1 2 function (e + Se ) may be used to match the parameters of the reduced system to some special parts in the step response. Namely, the maximum of this function can be placed on any point of the t - axis. Knowing the influence of the weighting parameters, they are modified so long as a satisfactory reduced mo del is achieved . In an interactive reduction one has to decide , whether a model is good or ~ot . This can be done by comparing the step resFonses of the original and reduced models. For the simulation of large systems an L- or at least Astable method should be used because of the very probable stiffness . The author has had no problems with a second order L - stable meth o d from Eitelberg (1978b). In case of a singular matrix A a closed loop system with ~ ' = A - B L C according to Eq. (18) is reduced.

In the 14 - th ord e r red u ced model 7 temperatures T , 6 concentrations i C and the height of the bottom k product hb were approximated. The reduction , which was carried out on the BURROUGH5 7700 of the University of Karlsruhe , was so unproblematic , that only 2 .!. and a a = S = 0 1 2 were tried. On Fig. 2 . a) the temperature on the 24-th plate T24 as a response to a unit st e p in the first feed concentration u is 1 displayed . No difference can be noticed . In a more exact simulation this input - output combination was the only one, where an approximation error could be discovered . This is displayed on Fig . 2.b). The results of the reduction to the 9 - th, 6 - th and 3 - rd order became gradually worse . Actually the model o f the 3 - rd order made no sense. Then the author was informed , that the inputs u and u were the most 1 3 important ones. 50 the reduction to a model that described the influence of u (first feed concentration) and u 3 1 (heating vapour) upon T , T and hb 24 28 made sense. The weighting parameters q11 = q33 = 1 and a 1 = a 2 S = 0 were taken. In all state variables there were errors in a late stage of the simulati on . C ho os in g a +0,3 1

52

E . Eite lberg

s u ppressed the l ate e rr o r s so t h a t all input - output comb i nat i ons were exact (see F ig . 3 . a)) , except T 24 as a r esponse to u

1

a(t)

=

o

~l

(see F ig . 3 . b))

hb

\

g)-~ ~-'j

. \ \

o

~

.\

'\ <'? :.

_-I

... ~I

'~ ~-------

~j

t ime

•0.0 a

I

I

0.8

I

(fl I

1.6

2.4

3.2

(t)

h

~ I' ----r---~----,-----rl----~---,I~~~~~~I~~ time in h ~-i~

o

I;

a

8

12

16

(t)

long time .

...

o o

"

~

0 I

I

!

-I I 0 CD

0

I 4

I

t ime in h I

0.04 a (t),

Fig.

2.

I

0.08

I

0012

I

0.16

U

Fig .

short time.

Step re::ponses ; --- - -- x .

b)

x;

CONCLUSIONS The efficiency of the proposed model reduction method has been demonstrated by the application to a practical problem . I ts advantages simplicity and uniqueness of the solution - allow the interactive reduction on a digita l computer , where the weighting parameters are modified as long as a satisfactory reduced model is found. Especially by very low order reduced models the weiqhting has a great influence on the result. REFERENCES Decoster , M. , and A . R . van Cauwenberghe (1976). A comparative study of different reduction methods . J.A 17, 68 - 74 and 1 25 - 134 . E itelberg, Ed . (1978a) Model l reduk tion durch Minimieren des Glei chungsfehlers. Regelungstechnik , ~, 320 - 322 .

1

=

3.

I

i

8

12

time w h I

16

a(t) . Step responses ; -- ---- x.

x;

Ei telberg, Ed . (1978b) Numer i ca l simu l ation of stiff system s wi t h a d i agonal splitting method . I n I. Troch (Ed.) , S i mula t i o n of Contro l Systems , I MACS; North Holland . pp. 95 - 96 . Eitelberg, Ed . (1979) . Model lr eduk tion linearer zeit i nva r iante r Sy steme du r ch Minimieren des Glei chungsfehlers . Dissertat i on. Universitat (TH) Karlsruhe . Koupan , A ., and P . C . Mu ll er (1976) Zur numerischen Lasung der L japunovschen Matrizeng l eichun g. Regelungstechnik , ~ , 167 -1 69 . Litz , L . (1979) . Reduktion der Ordnung linearer Zustandsraummodelle mi t t e ls modaler Verfahren . Disserta tion. Universitat (TH) Karlsruhe . Obinata , G. and H. Inooka ( 1 976). A method for modeling linear time inva ri ant systems by linea r systems of low order . IEEE T ra n s . Au t om. Contro l , AC - 21 , 602 - 603 and AC - 22 , 286 . Trill i ng , U., and H. - J. Kle in ( 1 979) Erfahrungen bei der Red u k ti on d er Ordnung linearer dynamische r P r o zeBmode l le hoher Ordn u ng mit Hi lfe von modalen Verfah r en . Rege l ung s technik, ~ , 37 - 45 .