A ACT

The controls-structures integrated design is a com- plicated multi-disciplinary design optimization problem which involves the state equations pertaining to open- loop eigenvalues and control1 laws. A great deal of com- putational resources are required to analyze these types of problems. In order to alleviate the intensity of the computation, this study plans to use the adjoint variable method to derive sensitivity equations for the eigenvalue, Lyapunov and Riccati equations. These individual sensi- tivity equations are then combined together to form the multi-disciplinary sensitivity equations for the controls- structures integrated design problems. In this proposed approach, a set of linear sensitivity equations, propor- tional in number to the number of performance functions involved in the optimization process, are to be solved.This proposed approach may provide a great saving in com- puter sources (cpu time and memory requirements), compared to the current practice to perform the sensitivity analysis by the direct differentiation and f i t e difference methods. The validity of the newly developed sensitivity equations will be verified by numerical examples.

I. UCTIO The integrated design of controls and structures (CSI

Design) of a spacecraft is a multi-disciplinary problem [l, 21. The design variables of the problems range from the dimension and c o d p a t i o n of structural members to the locations and performance of actuators and sensors. The system equations typically involved, include the sym- metric eigenvalue problem of the structure, the closed loop eigenvalue problem for stability analysis, and the Lyapunov and Riccati equations for the computation of steady state pointing performance and steady state control efforts. In addition to such complexity, intensive compu- tational requkment is another concern of the integrated controls-structures design. For example, it takes about 600 full analysis of ADS 131, a gradient-based design op- timization algorithm, to obtain an improved design of the space structure characterized by 21 structural design vari- ables and 12 control variables [4]. This structure article, called the CSI evolutionary model, is shown in Fig. 1. The objective of this CSI design exercise is to minimize ' A d a t e Professor, AIAA and ASME member 2 ~ a d u a t e student Copyright 0 1993 by Hou and Koganti. Published by the American Institute of Aeronautics and Astronautics, Im. with permission.

the steady state pointing performance, subjected to mass and limited control effort constraints. Mathematically, the objective function is expressed as the root mean square (RMS) of the pointing error. Using the static dissipative control law to prescribe the controllers, the CSI problem is stated as,

Min trace (@,,S@:) (1)

subject to the constraints

Structural mass < 2.0

and the constraint on steady state average control power,

trace(GRGT) < 2.04 (2)

where G is a function of the gain matrices and the open- loop eigenvector components, @,, is a function of the open-loop eigenvector components, S is composed of a quadrant of the matrix R, which is a solution of the Lyapunov equation

A R + R A ~ + c = o (3)

The coefficient matrices A and C of the Lyapunov equa- tion are functions of the control gain matrices, open-loop eigenvalues, w i . and eigenvectors, x i . of the structure. Further, w i and xi are the solutions of the eigenvalue equation

K(b)xi = w;M(b)x;, i = 1 , 2 , ... m (4)

where the stiffness matrix, depend on the structural design variable, b, and m is the number of modes selected to represent the dynamic characteristics of the structure.

, and the mass matrix,

It's worthwhile to note that Lyapunov equation is just a special case of the steady state algebraic Riccati equation, which frequently appears in the CSI design applications when the LQR and LQG controllers are used [51. Similar to the Lyapunov equations, the coefficient matrices of the Riccati equations are also functions of the open-loop eigen solutions and the gain matrices.

For the example discussed in reference [4], the dimen- sions of Eqs. (3) and (4) are of the order of 32 and 3050, respectively. Considering the total number of analyses need to be done in a design optimization process, the in- tegrated design problem is undoubtedly computationally intensive. Efforts have been undertaken recently, such as

1906

approximation and sensitivity analysis [61 and use of par- allel and vector machines and algorithms [71 to reduce the computational cost. This paper also represents an effort in this regard, although the focus of the paper is solely on sensitivity analysis.

The sensitivity analysis of eigenvalue problems has long been reported in the literature [8, 9, 101, though only a few publications are available on the sensitivity analysis of Lyapunov or Riccati equations [ 11, 121. Most of these works use the direct differentiation method to derive the sensitivity equations. The Computational ex- penses in these cases are proportional to the number of design variables. In this paper, however, the adjoint vari- able method, where the computational expenses are pro- portional to the number of performance functions in a design optimization problem, is employed to derive the sensitivity equations for the eigenvalue, Lyapunov and Riccati equations. Later in the paper, these adjoint equa- tions are combined together, according to the chain rule of differentiation, to obtain the sensitivities of the entire controls-sfxuctures integrated problem.

VECTO

The problem of concern is to find the derivative of an arbitrary function of eigenvalue, wi. and eigenvector, xi, $(xi, w;). with respect to any arbitrary design variable, b. The relationship between the eigen pair and the de- sign variable is given by Eq.(4) where the eigenvector is normalized with respect to the mass matrix, i.e.,

(5) T xi Mx; = 1, i = 1 , 2 , ... m

It is a straight forward matter to find the derivative of $ with respect to b as

d$ d$dw; d$dx; + -- db dw; db ax; db - = --

To compute the above equation, however, is not simple, which includes the eigenvalue and the eigenvector deriva- tives, 9 and g, respectively.

The direct differentiation method calls to directly dif- ferentiate Eqs. (4) and (5) to obtain a composite equation of 9 and 9 as

dK K -w;M Mx;

(7) [ xTM 0

The coefficient matrix of the above equation is non sin- gular and thus a unique solution in assured. Furthermore, it should be noted that Eq. (7) needs to be solved repeat- edly n x m times for a design problem with n design variables and using m modes to represent the dynamic characteristics of the structure.

The adjoint variable method introduces an arbitrary vector, q;, and a scalar, pi, that constitute the following identity,

The direct differentiation of the above equation with E- spect to b reveals the following adjoint equation

which specifies q; and pi in such a way that the terms associated with dx;/db and dw;/db can be removed from the resultant equation obtained by differentiating Q. (8) with respect to b. Doing so, a new form of d$/db is obtained as

It is interesting to note that a. (9) is independent of the design variable b. Therefore, the number of the adjoint equations, Eq. (9). required to be solved in a design optimization process is equal to m x p where m has been dehed as the number of modes used to represent the dynamic characteristics of the structure and p is the number of performance functions considered in the design optimization formulation. In most cases, n, the number of the design variables, is much greater than p.

S OF

Lyapunov equation plays an important role in the design of controllers. Particularly, the evaluations of the steady state control power and the pointing error caused by external disturbances rely on the solution of Lyapunov equation.

The Lyapunov equation, in general, is expressed as,

where the unknown matrix and the coefficient matrices A, B and C are real matrices of dimensions m x n, m x m, n x n and m x n, respectively. It is well known that Eq. (1 1) has a unique solution if and only if the eigenvalues, ai, of A and ,Bj of B satisfy

ai + Qj # 0, i = 1 , 2 , ... m j = 1 , 2 , ..A

There are several methods available to solve the Lya- punov equations and one of the popular ones, used by the control design code, ORACLS 1131, was developed by Bartel and Stewart [141. The major step in this algorithm is to reduce matrices A and to the lower and upper Schur forms A' and B', respectively. This is usually done by the QR algorithm.

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The Lyapunov equation is a matrix equation which is in contrast to the vector equations often encountered in the sensitivity analysis literature. The direct differentiation of Eq. (11) with respect to a design variable b produces a sensitivity equation in terms of W d b as,

dB dR dR dC dA db db db db db (13)

which is another Lyapunov equation by itself [ l l , 121. From the preceding equation, d Idb always exists and is unique, since Fiq. (12) still holds.

Note that the above equation needs to be solved n times if n design variables are present in the prob- lem. However, since the Schur forms of A have already been obtained when the original Lyapunov equa- tion was solved, solving the above sensitivity equation will hence be much faster than solving the original Lya- punov equation.

To proceed with the adjoint variable method for sensi- tivity analysis, it is convenient to break up the Lyapunov equation into a vector form. To do so, however, it is necessary to d e h e the following notations:

1. ag, a component at the i th row and the j th column of matrix A

2. ai., a vector made of the i th row of matrix A 3. a+ a vector made of the j th column of matrix A

With these notations, the Lyapunov equation can be ex- pressed in a vector form as

A-..,.-- + -B = - - -n - R-

Ar.; + Rb,; = c.i , i = 1 ,2 , ... n

One can then establish the following identity involving a performance function of , with n arbitrary vectors, A,;,

$(R, b) = $(r.i, r . 2 , ...r. n , b) = g(r.1 r.2, . . . r . n , b) "

i= I

The direct differentiation of the above equation gives an expression as,

= a + p (14)

where the scalars a and /3 are dehed as

d b ; dci (E 'I db db A: - r . + R - + -

The unknowns, dri /db and dR/db are all included in p. The fist term, g,, may be dropped from the equation of a, if $ is not explicitly dependent on b. Furthermore,

the terms in the summation in a can be proved to be a trace of the matrix products, i.e.,

As far as ,O is concerned, the last term can be simplified as,

As a result, the terms in ,O can be rewritten as

i= I

Hence, A,; can be selected as the solution of the following equation to make ,O zero

The transpose of the above equation gives

or

Collectively, the above equation can be expressed in a matrix form

a$ A'A + AB' + - = o d R

where is a matrix with $ being the i'h row and column component. The adjoint equation, Eq. (15).

is now again a Lyapunov equation, which warrants a unique solution since the condition of Eq. (12) holds true here as well. Furthermore, the adjoint equation is independent of b and is required to be solved p times if there are p performance functions, such as $, considered in the design process. Finally, the sensitivity in Eq. (14) becomes,

(16)

Note that the coefficient matrices in the adjoint equa- tion, Fiq. (15), are the transposes of those appearing in

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the original Lyapunov equation. Consequently, only the upper and lower real Schur forms of AT and than the lower and upper ones are available. In this case, however, one can still take advantage of the availability

' by transferring Eq. (15) into

a$ aR

(DATD)A* + A*(DBTD) + D-D = 0

is defined as the matrix with ones agonal and zeros elsewhere and A*

(17)

realized that per real Schu Therefore, Eq. (17) can be solved efficiently without go- ing through any real Schur decomposition. Once A* of Eq. (17) is found, the solution, A, of the adjoint equation can be recovered as

A = DA'D

The preceding derivation is quite general and can be applied to other types of matrix equations such as the symmetric Lyapunov equation, Eq. (2) , and the algebraic Riccati equation. The algebraic Riccati equation is very similar to Eq. (4) but with a quadratic term in sensitivity analysis of an algebraic Riccati equation is given in the Appendix.

I

The coefficient matrices of a Lyapunov equation in controls-structures integrated (CSI) design usually in- volve open-loop eigenvalues and eigenvectors of the structure. Typical examples are the matrices A and C of the Lyapunov equation defined in Eq. (3). Symbolically, this type of Lyapunov equation can be represented as

A(Ui,q)r,,i + R a i . ( w i , q ) = c.,i(wi,xi) , j 1 ,2 , .....y (18)

where y is twice the number of modes, m. The as the Lyapunov equation are thus implicitly dependent upon the structural variable b, as the eigen pair ( wi, xi) is the solution of the eigenvalue problem

K(b)xi = w;M(b)x;, i = 1 , 2 , . . .

As a result, the solution of the Lyapunov equation can be viewed as an implicit function in the form of either

(wi(b), xi(b)), depending upon whether the eigen solutions, wi and are taken as intermediate variables. The derivative of with respect to a design variable b can be then expressed as,

(dR dw; dR dxi) (19) +-- 8w; db ax; db

where the derivatives and g, are associated with Eq. (18) and the derivatives, % and %. are the eigenvalue and eigenvector derivatives resulting from F,q. (7).

___-

To start the derivation, select a function, $(R,w;, xi),

to represent a general performance function encountered in the controls-structures integrated design. Let the eigen solutions be considered as the intermediate design vari- ables, Le., $(R(x;, q), x;, w;). The derivative of 11, with respect to b is given as

m

i= I

where the relation of Eq. (19) is used here to replace

There are two types of unknowns in Eq. (20), the eigenvalue and eigenvector derivatives, dw;/db and dx;/db, respectively and the derivatives of r.,i, dr,,i/au; and dr.,i/dxi. The eigenvalue and the eigenvector deriva- tives can be directly obtained by solving Eq. (9) m x n times for n structural design variables. As for the terms, dr,,i/dwi and dr.,i/dxi, they can be obtained by solving Eq. (13) repeatedly for the number of design variables occumd. In this case, however the design variables are the eigenvalues, wi, and the components of eigenvectors at the locations of the actuators and sensors. If the num- ber of the modes selected is m and the number of the actuators, which are collocated with the sensors, is q, the number of sensitivity equations such as Eq. (13) to be solved is then m+m x q.

Next, a sensitivity equation Werent from Eq. (20) will be derived hereafter using the adjoint variable method. The terms in the brackets of .Eq. (20) can be viewed as the derivatives of $ with respect to wi and components of xi. That is,

d ~ , , i /d b .

db db i= I

where

Note that fi and gi are scalar and vector quantities, re- spectively. Most components in gi, however, are zero except for those degrees of freedom corresponding to the

1909

actuator/sensor locations. More specifically, the f; and can be viewed as the derivatives of $(R, b) in which b is taken as w ; and xi, respectively. Therefore, focussing on the terms d r / d w ; and dr.i/dxi. om can directly use Eqs. (14) and (16) to find the values of f i and gi as

a* f. -- + dW; I -

(24)

+ R- + - dw; dw;

(25) where A in both equations is the solution of the adjoint equation

(26) a* dR

A ~ A + AB^ + _. = o

With fi and gi being found, the form of Eq. (21) is now identical to that of Eq. (6). Hence, the adjoint equa- tion given in Eq. (9) can be repeatedly used herein for eigenmades 1 through m to obtain a sensitivity equation of a as, d ,$

where the m pairs of adjoint variables, vi and pi, can be obtained by solving the following adjoint equation m times,

i = 1 , 2 , ... m

Finally, one may summarize the procedure just de- rived for computing the derivative of a performance func- tion, $(R, w;, xi) in the CSI design as 1. Compute d$/dR, d$/dxi and d$/dw;. 2. Solve the Lyapunov equation, Eq. (26), once for the

adjoint variable A. 3. Compute f; and gi based upon Eqs. (24) and (25). 4. Solve the linear equation of Eq. (28) m times for the

adjoint variables, vi and pi, i=1,2. ..... m. 5. Compute d$/db based upon Eq. (27). In order to use the above procedure to evaluate g, one solution of a Lyapunov equation of Eq. (26) and m solutions of m sets of linear equations of Eq. (28) are needed. If the number of performance functions whose derivatives are sought is smaller than the number of design variables, the above procedure indeed provides a quicker and more memory-saving alternative than Eq. (20) to find the sensitivities.

The equations derived in the pervious sections will be validated in this section via a cantilever beam example. The sensitivities of the performance functions, the control energy and the root mean square W S ) pointing error, with respect to structural and control design variables will be determined. The cantilever beam considered here is made up of two 2-dimensional Euler beam elements and three nodes, node 1 being rigidly fixed as shown in Fig. (2). The collocated actuators and sensors are placed in X and Y directions of node 2. The white noise external disturbance is introduced at the same locations as the ac- tuators/sensors and the error is measured in the X and Y directions of node 3. The beam members are assumed to be circular in cross-section and the diameters of these elements are the structural design variables. The compo- nents of the displacement and rate feedback gain matrices are the control variables. This section is further divided into four subsections. In the first subsection, the deriva- tions of the closed loop control matrices, the calculations of the control energy and the RMS pointing error will be shown. The second subsection will deal with the sensitiv- ities of the open-loop eigenvalue and eigenvectors with respect to the structural design variables. In the third subsection, the sensitivities of the performance functions with respect to the control design variables will be shown. In the final subsection, the multi-disciplinary sensitivity equations for the integrated design problem with respect to structural design variables will be shown. The numer- ical results presented here are obtained by the software tool, MATLAB 1161.

V.l Controller Formulations

In this section, the closed loop control matrices and the equations for the control energy, Eq. (l), and the RMS pointing error, Eq. (3, are derived based upon the static dissipative control law [5].

The equations of motion for a controlled structure are,

My + K y = Bu

are the mass and stifhess matrices of the structure, B is the actuator location matrix, y is the structural displacement vector and 11 is the control force vector. Further, using the static dissipative control law, the control force vector is defined as

where G, and 6, are the displacement and rate feedback gain matrices, respectively and y, and yr are the displace- ment and velocity vectors, respectively. Note that yp and yr are the subsets of y.

Transforming the equations of motion into modal co- ordinates, they can be written as,

~ + D G + o ~ = @ ~ B u

19 10

where @ is the matrix of open-loop eigenvectors, D is the modal damping matrix, which is assumed to tional to the open-loop eigenvalues, and R is a diagonal matrix with open-loop eigenvalues as it’s components. Further, @ and s1 are the solutions of the following open- loop eigenvalue equation,

K @ - C2M@ = 0

The analysis of the closed loop performance is facilitated by a state space description. Consequently, the second order differential equations should be rewritten into a first order form.

X = A,lx + BC1w

where the state vector x is defined as,

x = { ;} with the closed loop matrices,

and w is the white noise disturbance. Since the system is stable, the covariance equation

can be used to compute the control energy and the RMS pointing error. The expected value of (x x? may be calculated from the Lyapunov equation of Eq. ( 3). where matrix A is nothing but A, and the right-hand-side matrix is defined as

o @ T ~ ~ ~ T @ c = [ 0

where W is the covariance of the white noise input w. Finally. the RMS pointing error is calculated as

rc = trace(@TB,R1lBTiP)

the error measurement location second quadrant submatrix of the

is the solution of the Lyapunov equation. that the terms S and QP in Eq. (1) are

TB,, respectively. On the other hand, the control energy is calculated as

C, = trace(GRGT)

where the G matrix defined above and in Eq. (2) is a part of the closed loop cl matrix , defined as,

G = [ aTBGP @TBG,]

In this example, the beam members are chosen to be circular in cross-section with their nominal diameters be- ing 0.65 l l inches, the moduli of elasticity of the material is lo7 Psi, the density of the material is 3.633 x lo4

lb/inch3 and the length of each element is 5 inches. The k s t two modes have been chosen to represent the dy- namic characteristics of the structure. With the aid of above described properties of the material and the con- figuration of the structure, matrices @, Q and D are cal- culated to be

@ =

-0.000 0.000 17.07 36.87 5.934 1.111 0.000 0.000 52.15 -24.1

-7.559 -18.9

0 = l o7 1.922202 O I equals 0.02 times R.

The nominal values for the gain matrices are

and

0.5 0.0 G p = [ 0.0 0.51

0.7 0.0 G r = [o.o 0.71

and the white noise is of unit intensity, thus the W in the equation of C is an identity matrix. The actuator and error measurement location matrices B and B, are defined as

I 1 0 0 0 0 0 0 1 0 0 0 0 B = [

It may be noted that the number of rows in the location matrices equals the number of actuators/error measure- ment locations and the number of columns equals the total number of degrees of freedom in the system. Most of the elements of these matrices are zeros, except in the columns where actuators/ error measurements are located.

With the aid of above defined equations and matrices, the closed loop matrices are calculated to be,

r o 0 iX10-7 0 0 0 1x10-7 l o -0.072692 -0.000031 -0.001472 -0.000044

A,1= l o 7

L -0.000031 -1,922269 -0.000044 -0.038539

r o 0 0 0 1

L8.5364 18.4324 11.9509 25.80541 and the last term of the Lyapunov equation,

y o o 0 0 0 0

0 0 0.291479 0.629385 10 0 0.629385 1.359019

19 11

The A matrix of the Lyapunov equation is nothing but the closed loop &I matrix. Thus, knowing all the matrices, the solution of the Lyapunov equation, the symmetric is computed as,

I 0.000014 0.000000 0.000000 -0.000006 0.000000 0.000061 0.00000000

9.864433 1.65786802 [ 1.76138619

R =

Finally, the control energy and the RMS pointing error are calculated to be 3.54128898 and 0.00585537, respec- tively.

Open-loop Eigensystem Sensitivities

In this section, the sensitivities of open-loop eigen- values and eigenvector components with respect to the structural design variables are calculated and the results are compared with those obtained by the finite differences. The detailed procedure for the sensitivity of a function of open-loop eigenvalue and eigenvector,$(w;, xi), is shown below. For simplicity, in the first illustrated example, the entire beam is assumed to be made of members of uniform thickness, with their diameter being b. In the second example, the sensitivities of the eigenvalues and eigenvectors where the diameter of each of the elements is considered as a separate design variable are found.

Now consider the derivative of the function $J, dehed as,

whose numerical value is 10362.08868 for the problem stated in section V.1. The derivatives of the function with respect to the opn-loop eigenvalue and eigenvector are defined as

a+ a* ~ = 2x1 and - = 0.01 ax 1 dw I

and-side of Eq. (7) is {0 - 34.1455 -

* = w1/100 + xyx,

3163 - 15.1180 O.O1OO]T. Solving Eq. (91, the adjoint variables 771 and p1 are found to be

ql = { 0 0.1738 0.0594 0 0.5196 0.0738}T PI = -3104.307

Substituting these values into Eq. (lo), we have d 4 - = 12726.0154 db

and the Gnite difference result is 12722.1 127 with a per- turbation of lom4.

Similarly, the derivatives of the $J with respect to individual thicknesses of the members are computed using the same set of adjoint variables and were found to be

d* db I d 4 db 2

_. = 38292.0934

__ = -25566.078

compared to 38285.799 and -25570.099. respectively as obtained by finite Werence method with the same per- turbation.

.3 Sensitivity Analysis with the

In this section, the sensitivities of the performance functions with respect to the control design variables are derived. As stated, the components of the displacement and rate feedback matrices are the control design vari-

These variables are a part of the closed-loop A,, ol matrices. Consider the derivative of the control

energy and the Rh4S pointing error with respect to the control variable which is the G,(2, 2) component. The Last term of the adjoint equation, Eq. (15) is computed to be, for each performance function as,

1 72.8698 167.3463 102.0178 220.2849 339.7647 220.2849 476.6666

142.8249 308.3988 666.9192

r2720.48 -1255.34 o 01

ac e = GTG = aR

- dTB,B,d are -- BR

The solution of the adjoint equation , Eq. (15). for the control energy yields the symmetric adjoint variable ma- trix,

I [ 0.0009

1 [ 0.0000

3412.12 7356.19 0.0000 0.0182

0.0005 0.0008 15909.3 -0.4823 0.0000 xc, =

and the solution of the adjoint equation for the RMS pointing error yields,

27.6593 -12.2656 0.0019 0.0000 5.4468 -0.0008 0.0000

0.0000 0.0000 Are =

Substituting these adjoint variable matrices into Eq. (16), the sensitivities are found to be

and the corresponding finite difference sensitivity values, with a perturbation of lo4, are 5.0893 x 10- and - 1.3929 x IO6, respectively. The sensitivities with respect to other variables are tabulated in Table 2. Note that these results are obtained without solving additional adjoint equations.

Table 2 Sensitivities with respect to control variables

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Table 2 (Continued) Sensitivities with respect to control variables

In this subsection, the sensitivities of the performance functions with respect to the structural design variables are computed using the adjoint variable methods. In this integrated sensitivity analysis, two types adjoint equations are to be solved. The first adjoint equation, the Lyapunov equation of Eq. (15), is solved to obtain the sensitivities of the performance functions with respect to the open- loop eigenvalues and eigenvectors and the second adjoint equation, E!q. (9), uses the results of the first adjoint equation to obtain the sensitivities of the performance functions with respect to the structural design variables. It may be noted that in the process, the sensitivities of open- loop eigenvalues and eigenvectors is not computed or stored anywhere, This reduces the memory requirements to a considerable extent as compared to the merentiation method.

Consider the sensitivities of control energy and the RMS pointing error with respect to the diameter of the structural members, where again the entire structure is assumed to be make of uniform material. The solutions of the first adjoint equations for control energy and RMS pointing error are exactly the same as dehed in the previ- ous sections and the sensitivities of the performance func- tions with respect to open-loop eigenvalues and eigenvec- tors are found to be,

Ar 2 = -1.GG41 x ace Ad 1 ALJ 1

aw2 aLJ2

- = -1.9488 x

- ace = -1.0846 1 0 - ~ 2 = -1.2266 x lo-'' a

- - Ace - ( 0 -0.4391 0 0 0 O } T ax I

-0.1787 0 0 0

( 0 -0.4117 0 0 -0.1336 O}T

( 0 -0.0772 0 0 -0.1112 O}T

Substituting these Eq. (91, the second

into the second adjoint equation, set of adjoint variables are obtained

as,

prel = -7.076G x lo-" p r e 2 = -2.7606 x

/reel = -3.7481 [ L C , ~ = -3.2938

qra1 = lo-'{ 0 - 0.1902 -0.0686 0 -0.4931 -O.OGIG}T qr,z = IO-" { 0 0.9432 0.159G 0 0.9181 -0.08G3}T

q ~ ~ l = { 0 -0.6663 -0.1252 0 -0.8068 0.0175}T

qc-2 = lo-" { 0 -0.0118 0.0086 0 0.1121 0.0328}T

where subscripts 1 and 2 denote the adjoint variables as- sociated with the first and second eigenvalues, respec- tively. Substituting these results into Eq. (10). the final sensitivities are calculated to be

dC -2 = -32.38937 db

db - drc = -0.035676

compared to - 32.4066 and - 0.03569, respectively as obtained by the finite difference method. The results obtained when the diameter of each of the members is considered a separate design variable are tabulated in Table 3. Again, no additional adjoint variable equations need to be solved.

VI. CONCLUSIONS

Based upon the adjoint variable method, a unified ap- proach has been developed to derive sensitivity equations suitable for the controls-structures integrated design prob- lems, which include eigenvalue, Lyapunov and Riccati equations. This approach limits the computational effort for sensitivity analysis to be proportional to the number of performance functions involved in the design optimiza- tion formulation, instead of number of design variables. Since in the context of CSI Design, the number of perfor- mance functions to be considered is usually less than that of design variables, the proposed sensitivity analysis can greatly improve the computational efficiency and reduce the computer memory requirements.

Table 3 Sensitivity in Integrated Design Formulation

This research is supported by NASA Langley task order no. NAS1-18454-139 and NSF Grant no. DDM- 8657917.

1913

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181-187, 1992 Nelson, R. R., “Simplified Calculation of Eigenvector Derivatives,” AIAA Journal, Vol. 14, pp. 1201-1205, 1976. Rogers, L. C., “Derivatives of Eigenvalue and Eigen- vectors.’’ AHA Journal, Vol. 8, pp. 943-944, 1970. Huag, E. J., Komkov, V. and Choi. K. K.,

Press, 1986. Haftka, R. T., “Integrated Structure-Control Opti-

Gilbert, M. G.. “Results of an Integrated Strut- ture/Control Law Design Sensitivity Analysis,” &

timization, edited by J,-F. M. Barthelemy, NASA CP-

Bartels, R. H. and Stewart, G. W., “Algorithm 432, Solution of the Matrix equation AX+XB=C,” munications of the ACM, Vol. 15, 110.9, 1972, pp.

3031, 1988, Pt.2, pp. 727 - 746.

820 - 826.

The algebraic Riccati equation is of the type A ~ P + PA - PHP = Q

where the steady-state solution of the time-invariant Ric- cati equation, P, and the coefficient matrices A, H, and Q are all n x n square matrices. The direct differentiation of the preceding equation with respect to a design variable, b, gives a Lyapunov equation in the form of Eq. (11)

dP dP (A’ - PH)- + -(A - HP)

db db dQ dAT dA dH -- -P - P- + P-P db db db db

- -

Solving Eq. (A.2) is generally faster than solving Eq. (A.1). This can be realized when Eq. (A.l) is solved by the Newton-Raphson method [151. In this case, a Lya- punov equation similar to Eq. (A.2) is solved at each of the Newton-Raphson iterations. Furthermore, a unique solution of Eq. (A.2) can be found only when the eigen- values of (AT- PH) and (A - P) satisfy Eq. (12).

Next, to start the adjoint variable method for the sensitivity analysis, the algebraic Riccati equation should first be in a vector form. Thus, Eq. (A. 1) can be expressed as,

(A’- PI3)p.i + Pa.; = q,i, i = 1 ,2 , ... n Any function of P can then be argumented, without changing its value, by the products of the preceding equa- tion and n arbitrary vectors A.i,

$ P , b ) = $(PI,PZ,.. .Pn,b) n

X’s [(AT - PH)p , + Pa, - q ,] I= I

Following the same procedure as described in Section lII, the derivative of $ with respect to b can be abbreviated as

” d$ dp a+P - d* = - a$ + E-2

db db I= I dP, db where the scalars, CY and P, are defined as,

db db

db dA dH

P + P - - P - P - - db db db

= -+trace a$ (n. ($

1 9 14

and

Now, the adjoint variable, A,;, is specified in such a way tliat the tcrrns associated with dpj/db, be dropped from d$/db. In other words, X j is selected to make p zero. Conscqucntly, the adjoint equation of X.i is establishcd as

i = 1 , 2 , .... n z

Collectively, it represents a Lyapunov equation,

a* aP (A - HTPT)A + A(AT - PTHT) + - = 0 (A. 4)

whose solution is unique as long as Eq. ( A 4 yields a unique solution of dI'/db. Finally, the sensitivity coeffi- cicnt, d$/db, of Eq. (A.3) is simplified as

- d* = EL + trace db a b

db +P- - P-P - - d A dH

db db

The above results can also be extendcd to find the derivatives of the pcrforrnance functions in a controls- structures integrated design problem which involves the eigenvalue equation and the algebraic Riccati equation. Nevertlicless, the derivation is omitted here.

I

Figure 1 CSI Evolutionary Phase 0 Modcl

C i r c u l a r Crossec t ion 3 m 2 121

X

A c t u a t o r / S e n s o r L o c a t i o n E E r r o r m e a s u r e m e n t l o c a t i o n

Figure 2 Cantilever Bcain - Example Problem

1915