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Model reduction in limited time and frequency intervalsWodek Gawronskia; Jer-Nan Juangb

a Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, U.S.A. b SpacecraftDynamics Branch, NASA Langley Research Center, Hampton, VA, U.S.A.

To cite this Article Gawronski, Wodek and Juang, Jer-Nan(1990) 'Model reduction in limited time and frequency intervals',International Journal of Systems Science, 21: 2, 349 — 376To link to this Article: DOI: 10.1080/00207729008910366URL: http://dx.doi.org/10.1080/00207729008910366

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Model reduction in limited time and frequency intervals

WODEK GAWRONSKI t and JER-NAN JUANGf

The controllability and observability gramians in limited time and frequency intervals are studied. and used for model reduction. In balanced and modal uoard~n:~lcs. n near-opttm.il reductton procedure i i uscd, )leld~ng t h rrduc~ion error Inorm of lhe J~Ercncc hcluccn the ourpul of the imaln;il s)stem and lhc reduced model) almost minimal. several example; arc given to ilustraie the concept of model reduction in limited time orland frequency intervals, for continuous- and discrete- time systems, as well as stable and unstable systems. In modal coordinates. the reduced model obtained from a stable system is always stable. In balanced coordinates it is not necessarily true, and stability conditions for the balanced reduced model are presented. Finally, model reduction is applied to advanced supersonic transport and a flexible truss structure.

Nomenclature X* complex conjugate transpose of X

trace of a square matrix X expectation operator scalar product of X , , X,, ( X , , X , ) = tr E (Xl X:) euclidean norm of X, IIX/I2 = (X, X ) representation of the full system, A is n x n, B is n x p. C is ' l x n representatton of the reduced model, A , is k x k, k < n, 6, is k x p, C, is q x k n x 1, state vector of (A, B, C) ( n - k) x I , state vector of the deleted sub-system k x I, state vector of the reduced model p x I , white noise o r impulse input of ( A , B, C) and

(An, BR, C R ) q x I, output of ( A . 6, C), and (A,, B,. C,) absolute reduction error relative (normalized) reduction error ootimalitv index ith eigenvalue of A, A = diag (ii), i = 1, ..., 11

ith balancing coefficient; 7, 2 0, r = diag (y:), i = I, ..., n

: ith reduction coefficient, a, > 0, X = diag (a i ) , i = 1 , ..., n W, = E(sx*), W,, = E(.Y,.Y:), W,,, = E(.x,x*)-controllability gramians

Received I I November 1988. t Jet Propulsion Laboratory, California Institute of Technology, Pasadena. CA 91 109,

U.S.A. f Spacecraft Dynamics Branch. NASA Langley ~esearch Center, MS. 230, Hampton. VA

13665, USA.

W20-7721iYO SJ W ,D 1990 Taylor d Francs Ltd

Downloaded By: [National Cheng Kung University] At: 05:39 25 September 2010

350 W Cuwronski and Jer-Nan Juang

1. Introduction Owing to the large number of vibrational modes in flexible space structures, there

is a need for model reduction in the analysis and control of these systems. Many approaches to the model reduction problem, such as those of Wilson (1970, 1974) and Hyland-Bernstein (1985), the component cost analysis of Skelton (1980), Skelton and YousuR(1983) and YousuRet a/. (1985), and the balancing technique of Moore (1981) (see also Gawronski and Natke 1987 a, b) have proved to be of great value for general linear systems. All of these methods use gramians for the system reduction; gramians are also used in system balancing, and also have applications to the analysis of linear systems with the stochastic input (Kwakernaak and Sivan 1972, Maybeck 1979, Stengel 1986). The gramians for stable systems are classically defined as integrals over infinite time or frequency intervals. While such definitions are of great theoretical interest, it is important to note that no realistic sensor or actuator can operate over an infinite frequency interval; nor can infinite time histories ever be obtained. Conseq- uently, the classical gramians are only approximations to the quantities that actually describe any physical system.

This paper introduces the new concepts of gramians defined over a finite time and/or frequency interval, and derives computational procedures for evaluating them. Applying the model reduction technique of Gawronski and Juang (1988) to these gramians, rather than to the classical ones, leads to near-optimal reduced models that closely reproduce the full system output, precisely in the time and/or frequency interval of interest. This approach has clear connections with filter design, and can be applied to unstable systems without encountering the divergence problems inevitable with infinite interval.

Extensive examples are given in the paper to illustrate these results for both discrete- and continuous-time systems. It should be noted that, while these examples and the original motivation for this work are drawn from structural dynamics, the results derived are in fact applicable to any linear system.

2. Gramians, balanced representations and model reduction The triple (A, B, C) represents either a continuous- or a discrete-time linear

system:

For continuous-time systems the condition i., + i.: # 0 is imposed, whereas for discrete-time systems the condition I - i.,i.: # O is imposed for every i, k = I, ..., n. The state of the system dual to the above (Kailath 1980) is denoted by 2. The controllability and observability gramians are the state covariance matrices of the system and are defined as W, = E(xx*) and W, = E(E*) , respectively. They are determined either from the equations

with zero initial conditions, or from the integrals

w = 1 exp (Ar)BB* exp (A'r) dr, W = e x ( A 8 ) C * e x ( A ) d (2.1)

For discrete-time systems, similar formulae apply:

W,(n + I) = AW,(n)A* + BB* = 0, W,(n + I) = A*W,(n)A + C'C = 0 (2.4)


Model reducrion in limited timeflrequency interuols 351

with zero initial values, or

Define a stationary solution, such that = w0 = 0 in the continuous time case, and W,(n + I) = W,(n), Wo(n + I) = Wo(n), lor the discrete-time case. For the continuous- time system the stationary solutions of (2.2) are determined from the Lyapunov equations

AW,+ W,A* + BB* = 0 , A*Wo + WoA + C L C = O (2.6)

For discrete-time systems, the stationary solutions are determined from the following equations

Note that the stationary solutions exist for stable, as well as unstable systems; for stable systems they are positive definite.

The system triple is balanced if its controllability and observability gramians are equal and diagonal. In order to find the balanced representation, the semi-gramians P, Q are determined from the decomposition of controllability and observability gramians (for example, singular value, or Cholesky decomposition)

and next the matrix H = QP is obtained. The singular value decomposition of H gives

where VT V = I, UTU = I, and is a positive semi-definite diagonal matrix. Denoting

the balanced representation is obtained from

since the gramians in balanced coordinates are W,, = R-' W,R-T = r2 = RT WoR =

W,,. This indicates that the controllability and observability gramians, after transformation, are equal and diagonal.

Next consider the system reduction. Let the system state vector x be partitioned as follows xT = [x: x;], and ( A , B, C) be partitioned accordingly:

The reduced model is obtained by deleting the last n - k rows of A. B and the last n - k columns of A. C, i.e.

A , = L A L ~ , B , = L B , c , = c L ~ , L = [ I , 01 (2.13)

such that the gramians We,, WoR of the reduced model are positive definite. I f there is a need to delete rows and columns other than the last ones, the system variables are sorted in order to obtain the variables to be deleted at the last position. The reduction index J (the absolute reduction error). and the normalized index 6 (the relative


352 W Gawronski and Jer-Nan Jwng

reduction error) are defined as follows:

J = l ly -Y~I I . & = Jl l l~l l (2.14)

where y, y, are the impulse or white noise responses of the original and the reduced system, respectively. The reduction index i s determined from

J 2 = 1 1 ~ 1 1 ~ + l l ~n11~ - 2 tr(C*CnWno) (2.15)

I n order to check how far the obtained solution (A,, B,, C,) is from the optimal one, the index c from Gawronski and Juang (1988) i s used:

~ = ~ ( Y - S R ~ Y R ) ~ / ( ~ ~ Y - ~ n I l IlynII)

= Itr (C*CR WRO) - l l~n l I~ lMJ l I~n l I ) (2.16)

where llyl12 = tr (C*CW,). Ily,l12 = tr (CiC, W,,). For the optimal solution, i t i s necessary that y, and y - y, are orthgonal, so that E = lcos 41 = 0, where 6 i s the angle between y, and y - y,. Obviously 0 < e < I, and for the near-optimal reduction E << 1.

The sorting procedures that give the near-optimal reduction have been presented by Gawronski and Juang (1988), and one of them is summarized as follows. Let the system be in the balanced or modal representation, and matrix Z be one of the following Z = -2AW, W,, or Z = C'CW,, or Z = BE* Wo. Denote a: the ith diagonal entry of Z, i.e. af = z , , . Next, sort ai i n decreasing order, oi $ oi+ , , and sort accordingly the state vector x, and the state triple (A, 6, C). Either in the modal or i n the balanced coordinates, deleting the last n - k columns of A and C and the last n - k rows of A and B gives the near-optimal reduced model. Note that a# i s Skelton's component cost, (Skelton 1980, Skelton and YousuK 1983). Note, also, that the reduction in modal coordinates, unlike in the balanced coordinates, does not shift the system poles. However, i f the Schur balanced representation i s used (Gawronski and Juang 1988), the system poles are not shifted either.

3. Model reduction in a limited time interval Assume that the system is excited, and the response i s measured within the time

interval T = [ t , , t,], r, > 1, $ 0. The gramians over the time interval T = [r,, r,] are defined as

K ( T ) = P B * e x A * d Wo(Q = l:leXp (A*r)c8C exp (AT) dr I:' (3.1)

Note that W,(T) $0. Wo(T) BO, i f r, > 1;. The gramians are determined .from the formulae

S ( r ) = exp (Ar) (3.4)

where W,, W, are the solutions of (2.6). The formulae (3.2) and (3.3) are derived as follows. Denote

B,(t) = exp(Ar)BB* exp (A*r) dr 0


Model reduction in limited time,$requency inreruuls 353

then, from Kailath (1980) (see also Appendix A), we have

Since W,(7') is given by W,(T) = B,(r,) - O,(r,), therefore, from (3.5) WJT) - W,(r,) - W,(t,). The observability gramian Wo(T) is obtained similarly.

Computations of the limited-time gramians from (3.2)-(3.4) can be ineffective for large-scale systems. In this case, the modal version of (3.2)-(3.4) is used, as given in Appendix B.

Stability conditions for the reduced balanced model are studied in a limited time interval, assuming the original system stable. Denote

then the following condition is valid:

Condirion I Let A be stable, and QJT), Qo(T) positive semi-definite, then the reduced

balanced model is stable.

Proof For positive semi-definite Q,(T), Qo(T) one can find B(T), C(T) such that

Q,(T) = B(T)B*(T), Qo(T) = C8(T)C(T). Next, one can show that gramians W,(T). Wo(T) satisfy the Lyapunov equations

According to Appendix C, the reduced balanced model, obtained for the triple (A.B(T), C(T)) is stable. 0

Denote Ar = t, - I , , and Q,(At) = BB* - S(Ar)BB*S*(Ar), Qo(Ar) = C*C - S * ( C * C S A ) . In this case one obtains Q,(T) = S(r,)Q,(Ar)S*(r,). Qo(T) = S*(r,)Q,(At)S(r,), from which it follows that Q,(T), Qo(T) are positive semi-definite if QJAr) and Qo(At) are positive semi-definite. For stable systems lim,,-, S(Ar) = lim,,-, exp ( AAr) = 0; consequently, from the definitions of Q,(At), Qo(Ar) i t follows that the reduced balanced model in the finite time interval is stable if the time interval At is long enough. I f the time interval is too short to obtain stable balanced reduction, and a stable model is required, then either the interval should be enlarged, or another reduction (modal or Schur balanced) applied. The last two reductions always give a stable model while reducing a stable system.

In the discrete-time case the controllability and observability gramians within the interval N = [n, , n,], n, > n, are computed from

where

The matrices W,, Wo are obtained from (2.7), and S(n) = A". The above formulae


354 U! Gawronski and Jer-Nan h a n g

lollow from the property (A 2) (see Appendix A). If we define

then

and from ( A 2) we obtain O,(n) = W , - A"+' W,A*"+' = W,- W,(n). The introduction of O,(n) to (3.10) gives (3.9). The observability gramian within the interval N is obtained similarly.

For the stability consideration, denote

then, for the discrete-time system the following stability condition is valid.

Condition 2 Let A be stable, and Q,(N), Qo (N) positive semi-definite; then the reduced

balanced model, is stable.

Proof For positive semi-definite QJN) , Qo (N) , one can find B(N), C (N ) such that

Q,(N) = B(N)B8(N), Qo(N) = C*(N)C(N). Next, note that the gramians W,(N), Wo(N) satisfy the Lyapunov equations

AW,(N)A* - W N ) + B(N)Bb(N) = 0. A*Wo(N)A - W,(N) + C*(N)C(N) = 0 (3.12)

According to the Theorem of Appendix C, the reduced balanced model obtained for the triple ( A , B (N ) , C ( N ) ) is stable. 0

Denote An = n, - 11,. then QJAn) = BE* - S(An)BB*S'(An), Q,(An) = C*C - S*(An)C*CS(An), and one obtains Q J N ) = S(n,)Q,(An)S*(n,), Q,(N) = S*(nl)Qo(An)S(n,). Again. QJN) , Q o ( N ) are positive semi-definite if Q,(An) and Q,(An) are positive semi-definite. For stable systems

lim S ( h ) = lim AA" = 0 An-, An-0

therefore one can always obtain a stable reduced model, providing An is large enough. The computational procedure is summarized as follows.

(a) For a given (A . B, C), determine gramians W,, W, from (2.6).

(b) Compute S(ri). W,(ti), Wo(t,), i = 1, 2, from (3.4) and (3.3). (c) Determine Lv(T) , Wo(T) from (3.2).

( d ) For W,(T), W,(T), apply the reduction procedure (2.8)-(2.13) to obtain (An. C R ) .

( e ) For the reduced model (A,, B,, C,), compute J , 6, E , from (2.14)-(2.16).


Model reduction in limired rirne/freqrrency inrer~als 355

The following examples illustrate the application of the gramians in model reduction. Since the reduction i s performed in balanced, as well as in modal coordinates, the variables determined in modal coordinates are denoted with the subscript 'm', while those in balanced coordinates are indicated by the subscript 'h'. The resulting plots present the full system response as well as the reduced model responses. The full system response is represented by the solid line, the reduced balanced response by the dotted line, and the reduced modal response by the dashed line.

Example I Figure 1 shows a system with the masses m, = I I, m, = 5, m, = 10, stiffnesses

k , = k, = 10, k , = 50, k , = 55, and dampings di = 0.01 k i , i = l,Z,3,4. The single input u i s applied giving f, = u, f, = 2u, f, = - 5u; the output i s y = 2q1 - 2q, + 3q,, where yi is the displacement of the ith mass, and5 i s the force applied to that mass. The system triple is determined from its mass (M) , stiffness ( K ) , and damping ( D ) matrices:

where BT = [I 2 - 5 , and C , = [2 -2 33. The system triple (A, B, C) i s then shown as follows:

J

BT = [0 0 0 0,0909 0.4 -0.51, C = [2 -2 3 0 0 01, and i t s poles are: i .,,, = -0.0038 f jO.8738, .,,, = -0.0297 f j2.4374, i .,,, = -0.1313 f j5.1217. The reduction is performed in the balanced and modal coordinates. Two cases are considered for the model reduction from six to four state variables.

Cnse I : reduction over the interval TI = [O,8].

and

Case 2: reduction over the inlerval T, =[lo, 181

In Case 1, the first and the third pair of poles are preserved in the reduced modal

Figure I. Simple mechanical system.


356 W Gawronski and Jer-Nan Juang

model, and A,,,, = -0,0009487 f j0.8712, A,,,, = -0.1395 +j50890 are the poles of the reduced balanced model. The reduction error is 6, = 0,2954, or 6, = 0.291 1, and the optimality index is &, = 00005683, or c, = 0.01291. The optimality indices are much less than I, therefore the reduced models are near-optimal. The impulse responses lor the balanced and modal reductions are compared in Fig. 2 (a).

Case 2 is obtained from Case I by shifting the interval TI by 10. In this case, the first two pairs of poles are preserved in modal reduction, and the poles i.,,,, =

-0.003865 k jO.87424, i.,,,, = -0.03047+j2.4439 are obtained in the balanced reduction. The reduction error in this case is 6 , = 0.2919, or 6 , = 0.3012, and the reduced model is near-optimal, since &, = 0.06253, &. = 0,04900. The impulse responses of the reduced and original systems are presented in Fig. 2 (b ) for Case 2. Comparison of Figs 2 (a) and 2 (b ) shows that the third mode is less visible lor f > 10.

Figure 2. Impulse responses of the mechanical system and its reduced models, for ( a ) T = [O, 81. (b) T = [lo, 181.


Model rrducfion in limiled rin~e//requenc) inleruals 357

Exomple 2 The limited-time model reduction of the discrete-time system i s discussed for the

interval N = [O. 2001. Its triple (A, B. C) i s

and i t s poles are i.,,, = 0.9901 +j0.0865, i.,,, = 0.9290 + j0.2223, i.,,, = 0.7345 t j0.3621. The system is reduced to four state variables. The poles for the reduced balanced model are A,,,, = 0.9901 + jO.08667, i.,,,, = 0.8855 f j0.23435, and are equal to i.,,, and A,,, lor the reduced modal system. The reduction error i s 6, = 0.1 123, or 6, = 0.1792, and the optimalily index c, = 0,008149, or c, = 0.08526. The indices 6 , and c, indicate that the reduced model i s near-optimal. The impulse responses o l the reduced and original systems are shown in Fig. 3.

Figure 3. Impulse responses of the mechanical system, and i t s reduced models in discrete time.

The gramians in a limited time interval also allow one to reduce unslable systems such that the response of the reduced model closely fits the response o l the original one within the prescribed time interval. The following example illustrates this statement.


358 W Cawronski and Jer-Nan Juunt

Example 3 Example 1 i s re-examined with d j = -0.006ki, i = 1, 2, 3.4, which makes the

system unstable, with the poles A , , , = 0,0023 + j0.8738, i.,,, = 0.0178 f jZ.4375, L,,, = 0.0787 f j5.1228. The system gramians are determined over the interval [O,8]. The reduced balanced model of order four has the following poles: I.,,,, =0.00594+ j0.8716, A,,,, = 0,0748 + j5.1238, whereas the first and the third pairs of poles are preserved in the reduced modal model. The reduction error i s 6,=0.2117, or 6. = 0.2122, and optimality index i s E , = 0.03201, or 6. = 0.0200. The reduced models are obviously near-optimal. The impulse responses of the original and reduced models, within the interval [O,8], are shown in Fig. 4.

Figure4. lmpulsc respones of an unstable system, and i ts reduced models

The reduced modal model, obtained for a stable system, i s stable because i t preserves the system poles. As was said before, balanced reduction within a limited time interval does not guarantee the stability of the reduced model. This can be explained heuristically by the tact that within a finite time interval, especially a short interval, the response of an unstable system can approximate the response of a stable system with acceptable error. The following example illustrates the case.

Example I i s re-examined, with damping d = 0,006k, so that the system is stable. The reduced balanced model obtained has the following poles: i , , , =0.0008736 & j0.8713, i.,,, = -0.08569 & j5.1 127, i.e. i t i s unstable. The reduced modal model is stable, with poles i,,,, = -0.002291 kj0.8738, i,,,, = -0.07875 lt jS.1228. The reduction error i s 6, = 0.2807, or 6, = 0.2810, and the optimality index c, = 0.01 192, or c, = 0.008634. The impulse responses olthe original and reduced models are shown in Fig. 5. The matrices QJT), Q,(T) for the reduced balanced model are not positive semi-definite, and their eigenvalues are i.,,, = {0.4168, -0.1999,0,0,0,0), ipor = { 16.6741, -4~7071,0,0,0, O), which indicate an unstable reduced balanced model.


Model reduction in limited timeljrequency intervals

Figure 5.

T IME

Impulse responses or a modified mechanical system and iu reduced models.

4. Model reduction in a limited frequency interval In this section, an approach is developed for determining the reduced model whose

output fits the full model output within a limited frequency interval R = [w , , w,] , o, 3 w , 3 0. A limited frequency output is obtained, for example, for a system excited by a limited frequency input. According to the Parseval theorem, the integrals (3.1) over the time interval 10, m) can be determined in the frequency domain as follows:

where

H(v) = ( j v l - A ) - ' (4.2)

is the Fourier transform of exp(At). In order to obtain gramians in the limited frequency band R, let the impulse responses pass through the frequency window [ o , , w,] , so that from (4.1) one obtains

where

Note that W,(o), W,(o) are positive definite, and that they act as low-pass filters for signals. The partial fraction decomposition of the first integral in (4.4) gives

(H(v)W, + W,H8(v)) dv/2n

which produces


360 W Gawronski and Jer-Nan Juung

where

S(w) = j:u H(v) dv/2n = ( j / 2 x ) in ( ( & I + A)( - jwl + A ) - ' ) (4.7)

and W, is the controllability gramian obtained from (2.6). In (4.6) the controllability gramian in the finite frequency interval is obtained from the stationary controllability gramian by a frequency-dependent weighting. The observability gramian in the limited frequency interval is determined similarly:

where W, is the observability gramian obtained from (2.6). For large scale systems, calculations ofa matrix logarithm (4.7) is ineffective unless

in modal coordinates. A modal version of limited frequency gramians is given in Appendix B.

Finally, i t is easy to see that for the frequency band R = [ o , , w,], o, > o , , the gramians are alternatively obtained from (4.3), (4.7), (4.8) in the form

where S(R) = S (o , ) - S (w , ) . For stability consideration, note that H(v , ) , H(v2) commute for any v , and v,:

H(v,)H(v,) = H(v,)H(v,). From this property, and definitions (4.2), (4.7) one obtains

and from (4.6)

where Q,(w) = S(w)BBt + BB*St(w) (4.12)

For stable A, the integral (4.1 1 ) is obtained as the solution of the following Lyapunov equation

AW,(w) + K ( w ) A * + Q,(o) = 0 (4.13)

Consequently, from (4.3) the gramian W,(R) is obtained from

AW,(R) + W,(R)A* + Q,(R) = 0 (4.14)

where QJnJ = QJ%) - Q,(o,) (4.15)

In a similar way, the observability gramian W o ( 0 ) is obtained as a solution of the following Laypunov equation:


Model reduction in limired timelfrequency inreruals 361

With these establishments, the stability condition for balanced reduction in a limited frequency interval is as follows.

Condition 3 Let A be stable and Q,(R), Qo(R) positive semi-definite, then the reduced balanced

model is stable.

Proof For positive semi-definite QJR), Qo(R), one can find B(R), C(R) such that

QJR) = B(fl)B*(fl), Qo(R) = C*(R)C(R). From the Lyapunov equations (4.14), (4.16) according to the Theorem in Appendix C, the reduced balanced model for the triple (A, B(R), C(R)), is stable. 0

Note that S(w) = 0 for o = 0, and S(o) = 0.51 for o+ CQ; therefore, for the frequency interval R = [O, w] one can find large enough o such that the reduced balanced model obtained is stable.

The computational procedure is summarized as follows.

(a) For given ( A , B, C), determine gramians W,, W,, from (2.6).

(b ) Determine W,(w,), W,(o,) from (4.6) and Wo(o,), Wo(o,) from (4.8). (c) Determine W,(R). Wo(R) from (4.3).

(d) For W,(R), W,(R), apply the reduction procedure (2.8)-(2.13) to obtain (An, Bn, Cn).

(e) For the reduced model (A,, B,, C,), determine J, 6, e, from (2.14)-(2.16).

Example 5 Example I is considered again in two cases:

Case 1: in the frequency interval R, = [0.7,3.2],

and

Case 2: in the frequency interval R, = C1.5, 3.21.

In the first case, the reduced model with four state variables is determined, whereas in the second case the reduced model with two state variables is obtained. The results are as follows.

In Case 1, the reduction error is 6, = 0.03405, or 6, = 0,03531, and the optimality index e, = 0.002773, or em = 0.02340, while in Case 2 6, = 0.1086, or 6, = 0.133 1, and E, = 0.01087, or em = 0.1 145.

In both cases, the obtained reduced model is close to the optimal one (ccc I). Figures 6(a) and 6(b) show the good fit of the power spectral density functions

within the considered frequency interval, indicating that the reduction in the limited frequency interval can serve as a filter design method. For example, in Case 2 the output signal is filtered such that the resulting output of the reduced model fits in the best way the output of the original system within the band R, = 11.5, 3.21.


W Cawronski and Jer-Nan Juang

1 10

OMEGA

( b )

Figure 6. Power spectral density functions of the mechanical system and its reduced models for ( a ) n = 10.7, 3.21, ( b ) R = [ 1 5 , 3.21.

5. Model reduction in limited time and frequency intervals Combination of the two methods presented above gives the gramians and model

reduction in limited time and frequency intervals. The gramians are determined either first in the time and then in the frequency domain, or vice versa. The results are identical in both cases, since the frequency and time domain operators commute, which is shown below.

Consider the controllability gramian in the limited time interval, defined by (3.1) and determined from (3.2). According to the Parseval theorem, the gramian in the infinite time interval [0, w ) can also be determined in the infinite frequency domain ( - w, w ) from (4.1). Assuming that signals H,(w) = H(o)B in (4.1) can be measured only inside the limited frequency interval [-w, o], W, can be determined from (4.6).


Model reduction in limited rime/frequency interuols 363

Introducing (4.6) to (3.3) yields

where S(r) is given by (3.4) and W,(w) by (4.6). I n the next case, starting with the controllability gramian in the frequency domain, by applying the Parseval theorem in the time domain one then obtains

where W,(r) i s given in (3.3). and S(w) i s i n (4.7). The question arises whether W,(t, w) in (5.1) and (5.2) are identical. I n order to show this, note that S(r) and S(w) commute

for any w and 1. Indeed, since

where F( . ) denotes the Fourier transformation operator, therefore

S(r)S(w) = (exp Ar)F(exp AT) dv I: F(exp Ar)(exp At) d\ ,= S(w)S(r) (5.5)

Next, introduction of (4.6) to (%I), with the aid of (5.3), yields

q ( t , W) = S(t)W,S*(t)S*(w) + S(w)S(r)W,S*(t) (5.6)

Recalling (3.3), the equality of (5.1) and (5.2) i s thus proved. As a consequence of the commuting property, the controllability gramian over the

limited time interval T = [r,, r,], r, > r , , and the frequency interval Q = [w,, w,], w, o, , i s determined from

where

and

Similarly the observability gramian over the time interval T and frequency interval R i s obtained from

The modal gramians in limited time and frequency intervals are given in Appendix B.



In order to determine the stability conditions for the reduced balanced model in limited time and frequency intervals, one can prove, similarly t o the previous sections, that W,(T R) . Wo(7; 0) satisfy the ~ y a ~ u n o " equations

W,(7;R)A*+AW,(7;R)+QC(7;R)=O, Wo(7;R)A+ A*Wo(7;R)+Qo(7;Q)=0

(5.9)

where

Coiidirior~ 4 Let A be stable. and Q,(7; R ) , Qo(T R ) positive semi-definite; then the reduced

balanced model is stable.

Prolf For positive semi-definite Q,(T R ) , Qo(7; R ) one can find B(7: R ) , C(7; R ) such

that Q,(T, R ) = B(T, R)B*(T, R ) and Q0(T R ) = C*(T R ) = C*(T, R)C(T, R) . Accord- ing to the Theorem of Appendix C, the reduced balanced model obtained lor the triple ( A . B(T. 0). C(T. R ) ) is stable. 0

Again. large time and frequency intervals result in a stable reduced balanced model.

As was seen above, the computational procedure can be set up in alternative ways: either by first applying frequency and then time transformations of gramians, o r by lirsr applying time and then frequency transformations. Only the first procedure is considered further, since the second is similar.

( 1 1 ) For given ( A , 5, C), determine gramians W,. Wo from (2.6).

( h ) Compute W , ( o , ) , U:(w,), W,(w,). W,(w,) from (4.6).

( c ) Compute W,(r,, w,), W,,(1*, wg) from (5.7 c ) , (5.8 c), for a, P = 1, 2.

( d ) Compute W,(r,, R ) , Wo(t,, R ) from (5.7 h) , (5.8 h), for x = 1, 2. ( e ) Compute W,(7; R) , W0(T R ) from (5.7 u). (5.8 a).

( f ) For W,(7; R) , W0(T R) , apply the reduction procedure (2.8)-(2.13). obtaining ( A n , Bn, Cn).

(g) For the reduced model (A,, B,, C,), compute J , 6 , E. from (2.14)-(2.16).

E.~unlple 6 Example I is considered in the finite time interval T = [O, 81. and the finite

frequency interval R = [0.7,3.2]. The reduction errors are 6, = 0.1 153. o r 6, = 0.1358,


,Woodel reduction in limited time/jrcquency inlervuls 365

and the oplimality indices c, = 0.0406, or &, = 0.02595. Figures 7 (u) and 7 ( h ) show the impulse responses and the power spectral density functions of the original system, the reduced balanced model, and the reduced modal model. This example is a combination of Examples I and 5; however, the results are different. In Example I for T = [O, 81, the second pole was deleted. In the present example, the third pole is deleted, a s a result of the additional restriction on the frequency interval.

I 10 O M E G A

( h )

Figure 7. Impulse responses and power spectral density functions of the mechanical system and its reduced model for T = [0, 81, and n = [0-7. 3.21.

6. Applications In this section, model reduction of advanced supersonic transport and the flexible

truss structure are analysed. The model of advance supersonic transport given by Colgren (1988) is a linear non-stable eight-order system, with four inputs and eight outputs. In order to make our presentation concise, the model presented here is restricted to one output. The system triple ( A . B, C ) is as follows:



The system poles are A = diag (0,6687, - 1,7756. -0.0150 f jO.0886, -0.3122 f j4.4485. -0.7257 + 6,7018). so there is one unstable mode. The classical approach to model reduction with an unstable mode consists in removing the unstable pole from the model and applying the reduction procedure in the infinite time interval, (Colgren 1988). Finally the unstable part is included in the reduced model. Here, the finite time balanced reduction is applied to the aircraft model, without removal of the unstable mode. The time interval T = [0, 1.41 is chosen. Model reduction from eight to five state variables within the time interval T is performed. The obtained reduced model ( A , , B,, C,) is as follows:

C, = [- 1.1348 -0.74659 -0.94062 0.31464 0.419583

with its poles A, = diag (0.6151, - 1.4217 + j3,255l, -0.5260 f j5.2109). The reduction error is 6, = 0.008462, with the optimality index e, = 0.01947, so that the obtained reduced model is near-optimal. Its step responses (dotted line) are compared with the step responses of the full model (solid line) in Figure 8.


Model reduction in limited timeljrequency intervals

TlME INPUT- 1 .OUTPUT- 1

8 ( 4

TlME ,CIPuT-3.OuTPUT- 1

8(c)


W Cawronski and Jer-Nan Juong

INPUT-4 .0UTPUT-1

(4 Figures. Step responses lor advanced supersonic transport and its reduced model lor

T = [O, I 41.

Next, model reduction o f a flexible truss structure given in Figure 9 is investigated. Every truss node has two degrees of freedom, and is numbered as in Figure 9. The first number denotes the horizontal displacement, and the second one the vertical displacement. Let /, be the force applied to the ith degree of freedom, and qi and u; be the displacement and the velocity at the ith degree of freedom, respectively. The system has one input u, applied as follows: J, = J3 = J, , = -u, J,. = J,, = J,, = u, and two outputs J,, = q,,, g, = nu,, , where the weighting coefficient 1 = 0-001 is used to scale y , and y,. From Fig. 10 one can see that after weighting llg,ll is of order of llg, 11. The system poles are shown as follows:

i.9,10 = -4.7173 f j3070.9, I., ,,,, = -4.9259 kj3138.8,

i., ,,,. = - 5.6674 f j3366.7, I., s , l n = - 6.4074 + j3579.8,

Figure 9. Truss structure.


Model reduction in limited timelfrequency intervals

12,, , ,=-14~814fj5443~1, 1 .,,, ,,=-17.137fj5854.4,

i.,,,,, = -20.846 f j6456.9, A,,,,, = -21.170 f j6506.9,

i.,,,,, = -22.994 1j6781.4, 1 .,,,,, = -25.208 kj7100.4,

i .,,,,, = - 27.386 kj7400.7, j .,,,,, = -29.647 f j7700.1,

= -38.775 f j88O6.l

Figure 10. Impulse responses of the truss structure, and its reduced models for T = [O.0.04].

The reduction from 42 to 22 stare variables is considered in five cases.

Case I Gramians and balanced representation are obtained in the rime interval

TI = [O, 0-041.



Case 2 Gramians and balanced representations are obtained within the time interval

T2 = [O. 0.21.

Case 3 The grarnians are determined and balanced in the frequency interval

n, = 10, ssool.

Case 4 Gramians and balanced representations are obtained within the frequency interval

R2 = [ I 700,70001.

1000

OMEGA

( b ) Figure I I . Power spectral density functions of the truss structure and its reduced models for

n = [o, ssool.


Model reduction in limited limel/requency intervals 371

Case 5 Balancing and reduction is accomplished in the finite time interval T, and the

finite frequency interval R,.

In Case 1, the reduction index is 6, = 0-09272, or 6, = 0.09748, and the optimality index is ~,=0.01761, or &.=0.01595. The outputs of the original as well as the reduced models are shown in Fig. 10.

In Case 2, the following indices: 6, = 0.08539, 6. = 0.089 15, E, = 0.001797, and c. = 0.003063 are obtained.

In Case 3, the results are 6, = 0.02193, 6, = 0.02242, 8, = 0.001 372, and E, = 0.008987, whereas in Case 4 6, = 0,04327, 6, = 0.08702, E, = 0.005830, and E. = 0.02107.

The Dower soectrum of both the outputs for Case 3 are shown in Fig. I I , and for Case 4 in Fig. I>.

OMEGA

(4 Figure 12. Power spectral density functions of the truss structure and its reduced models for

n= [ 1 7 ~ . 7 m ] .


W Gun~ronski and Jer-Nnn Juang


Model reduction in limited rimelfrequency interuals

Figure 13. Impulse responses and power spectral density functions of the truss structure and its reduced models lor T = [O. 0.23 and $2 = [O, 55001.

In Case 5, which is a combination of Cases 2 and 3, the results are 6, = 0.04061, or 6, = 0.03710, and E , = 0.006334, or c, = 0.01444, and the impulse responses and power spectra are shown in Fig. 13.

The reduced models, for all cases presented, are near-optimal owing to the small value of '. The reduction error is also reasonably small. The model reduction from 42 to 22 state variables caused less than 10% error in each case.

7. Concluding remarks I t has been shown in this paper that system balancing as well as model reduction

can be performed in finite time and/or frequency intervals. This fact makes the method more useful in practical cases, since data is always available in limited time and frequency intervals. In order to obtain gramians in limited time and frequency intervals, the stationary gramians are computed first lrom the standard Lyapunov equations, and then transformed into the finite time and/or frequency interval. The method presented in this paper allows one to check if the results obtained are close to the optimal ones. The reduction in the balanced or modal coordinates gives, in most cases, a reduced model close to the optimal one, and the illustrative examples support this statement.

I t is well known that if a system is stable, the reduced model in the infinite time interval obtained from the balanced coordinates is also stable. In a finite time interval, the stability of the balanced reduced model may not be assured. In the case where a reduction in balanced coordinates gives an unstable model, the use of modal coordinates, or Schur balanced coordinates, is recommended. On the other hand, an unstable system can be reduced using limited-time balancing, since the gramians exist for unstable systems within a finite time interval.

ACKNOWLEDGMENTS This work was done while the first author held a National Research

Council-NASA Langley Research Center Senior Research Associateship.


374 W Cawronski and Jer-Nan Juang

Appendix A Proprrry of gramians

From Kailath (1980) the following property is obtained. If W, satisfies (2.6), then

W, = exp ( A ! ) W, exp (A* r ) + exp ( A r ) BB* exp (A ' r ) dr I ' - 0 ( A 1)

For a discrete-time system, a similar result can be obtained. If W, is a solution of (2.7) then

Proof

The result ( A 2) can also be written in the form

Subtraction of ( A 2) from ( A 3) gives

0 = A"+' W,(A8)"+' - A " W,(A*)" + AnBB*(A*)"

Since (2.7) is fulfilled, so the above equation is satisfied. 0

Appendix B

Gromians in modal coordinares

Here, the gramians in modal coordinates are determined. Modal gramians are important from the point of view of computational efficiency: for large-scale systems computational effort in modal coordinates is much smaller than in other coordinates.

B. I . Limited-rime gramians The ikth entry of the limited-time controllability and the observability gramian in

modal coordinates is as follows:

- j k r 2 *>,dT) = m 3 0 e ( r l ) - w o e ( r 2 ) ( B 1 ) where

w , d O = *:iksi(t)s:(r), wo.(t) = wo.s~(r)sk(r) ( B 2)

and w,* = - b,b:/(L, + i.:), nSoe = -ctc,/(i.f + i.,), st(() = exp ( i , r ) , and b, is the ith row of 6: ci is the ith column of C .

In the discrete-time case, the ikth entries of modal gramians W,(N) and W o ( N ) are computed by

where w,. = b,b:/(I - i.,i.:), woik =c:c,/(I - i t i t ) , s,(n) = j.;

B.2. Limitedrfrequency gramians In the modal representation, the ikth entry of the gramians in the finite frequency


Model reduction in limited time/frequency interuals 375

interval are determined as follows:

w,.(n) = WJW~) - ). wOikw) = WO~I(WZ) - w o i k ( ~ ~ (B 4 a)

where

wCit(w) = w<ii(si(w) + s:(w)), woik(w) = WO;~(S?(W) + %(w)) ( B 4 b)

si(u) = (j/2n) In (( j w + A)/(-jw + A ; ) ) ( B 4 0 and w,,, wo, are the ikth entries of the modal gramians (B 3). Alternatively,

B.3. Limited time andfrequency gramians

Again, for computational purposes in large scale systems, the limited-time and limited-frequency gramians are determined in modal coordinates. The ikth entry of W,(T, n), Wo(T, R) in modal coordinates is

~ c i t ( T Q ) = w c f i ( T ~ 2 ) - ~ c i ) i ( T w 1 ) , ~ o i l r ( T n ) = w o a ( 7 ; ~ 2 l - w o , ( T w 1 )

( B 6 4 where

w,dT 4 = wcik(t1, 0 ) -W,~I(~~, 4, wOik(T W) = w d r , , 0 ) - wOik(t2. W)

( B 6 b )

where

wce(t, Q) = weik(t, u 2 ) - W,;~(I. w ~ ) . ntoik(T 4 = wo.(t. u 2 ) - wOik(t,

( B 7 b )

The terms w,,, w,, are given by ( B 3), si(0 by ( 6 4) and si(w) by (B 4 c).

Appendix C Stability theorem

The theorem presented is a modification or the stability theorem due to Moore (1981), Parnebo and Silverman (1982), and Glover (1984). In this theorem, the positive definite requirement of gramians is dropped.

Theorem

Let (A, B, C) be a triple satisfying

AT'+ r 2 ~ * + BB*=O, A * T ~ + T ~ A + C*C=O (C 1)

where T = diag ( y ; ) , i= I, .... n, with the entries y, in decreasing order, yi 3 yi+, 3 0 .


376 Model reduction i n limited cime/j*equency intervals

Let r be partitioned as follows: I- = diag (r,, I-,), T, i s positive definite, and the triple (A. 6, C) is partitioned conformally wi th r as in (2.12); then (A,, B,, C,) is stable.

Proof

Equations (C I) and (2.12) imply

A,~;+~;A:+B,B:=o, AAI-;+I-;A,+c;c,=o Since I-, is positive definite. (A,, B,, C,) is completely controllable and observable, and consequently, from the Theorem 3.3 o f Glover (1984), A, is stable. A similar result is obtained for discrete-time systems. 0

REFERENCES COLGREN, R. D., 1988, A.I.A.A. Guidance, Novigorion and Conrrol Conference. Minneapolis,

Minnesota. DaDer 88-4144. D. 777. GAWRONSKI, W., a n i ~UANG, 1.-N.,' i988, Near-optimal model reduction in balanced or modal

coordinates. 26th Annual Allerron Confercnce on Communicarion. Conrrol and Com~ut- ing, Monticello, Illinois.

GAWRONSKI, W., and NATKE, H. G., 1987, Inr. J . Sysrems Sci., 17,237; 1988, Balanced state space representation in the identification of dynamical systems. Applicarion of Sysrem Identification on Engineering, edited by H. G. Natke (Wien: Springer-Verlag).

GLOVER, K., 1984, Inr. J . Conrrol, 39, 11 15. HYLAND, D. C.. and BERNSITIN, D. S.. 1985, I.E.E.E. Trans. aurom. Conrrol, 30, 1201. KAILATH, T., 1980, Linear Systems (Englewood Clifis, NJ: Prentice Hall). KWAKERNAAK. H.. and SIVAN. R., 1972, Lineor Oprimol Control Sysrems (New York: Wiley). MAYRECK, P. S.. 1979, Stochastic Models. Esrimation and Conrrol, Vol. 1 (New York: Academic

Press). Moone, 6. C., 1981, I.E.E.E. Trans. aurom. Conrrol, 26. 17. PARNEBO, L., and SILVERMAN, L. M., 1982, I.E.E.E. Trans. ourom. Conrrol, 27, 382. SKBLTON, R. E., 1980, lnr. J . Control, 32, 1031. SKELmN, R. E., and YOUSUPF, A,, 1983, lnt. J . Control, 37, 285. STENGEL, R. F., 1986, Srochasric Oprimol Conrrol (New York: Wiley). WILSON, D. A,, 1970, Proc. lnsln elecr, Engrs, 119, 1161: 1974, lnr. J . Control, 20, 57. Y o u s u ~ ~ , A,, WAGIE, D. A,, and SKELTON, R. E., 1985. J . marh. Analysis Applic., 106, 91.


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