NASA Technical Memorandum 10 16 23 

INTEGRATED STRUCTURE/CONTROL LAW 

DESIGN BY MULTILEVEL OPTIMIZATION 

(NASA-TH- 10 1623) INTEGRATED N89-26623 

STBUCTURE/CONTHOL LAW DESIGN BY MULTILEVEL 

OPTIfiIZATION [NASA. Langley Research 

Center) 12 p CSCL 09B Unclas 

63/63 02227 19 

Michael G. Gilbert 

David K. Schmidt 

JUNE 1989 

National Aeronautics and 

Space Administration 

Langley Resetarch Center 

Hampton, Virginia 23665-5225 
INTEGRATED STRUCTURE/CONTROL LAW DESIGN BY MULTILEVEL OPTIMIZATION 

Michael G. Gilbert* 

NASA Langley Research Center 

Hampton, VA 23665 

David K. Schmidtt 

Arizona State University 

Tempe,AZ 85287 

Abstract 

A new approach to integrated structure/control law 

design based on multilevel optimization is presented. 

This new approach is applicable to aircraft and spacecraft 

and allows for the independent design of the structure and 

control law. Integration of the designs is achieved 

through use of an upper level coordination problem 

formulation within the multilevel optimization 

framework. The method requires the use of structure and 

contiol law design sensitivity information. A general 

multilevel structure/control law design problem 

formulation is given, and the use of LQG control law 

design and design sensitivity methods within the 

formulation is illustrated. Results of three simple 

integrated structure/control law design examples are 

presented. These results show the capability of structure 

and control law design tradeoffs to improve controlled 

system performance within the multilevel approach. 

Introduction 

Modem day air and space vehicles are complex 

systems made up of components which are often in and of 

themselves high-order dynamic systems. For example, 

the airframe, engine, and flight controls of a modem jet 

fighter aircraft individually require sophisticated 

mathematical modeling, simulation, and testing to achieve 

good performance with appropriate dynamic 

characteristics. 

8 

. The integration of good individual dynamic 

components into a single aerospace vehicle can, however, 

result in dynamic interactions which lead to unsatisfactory 

* Aerospace Engineer, Aeroservoelasticity Branch, 

Structural Dynamics Division. Senior Member AIAA. 

t Professor, Department of Mechanical and Aerospace 

Engineering. Associate Fellow AIAA. 

vehicle performance. The F-16 and F-18 exhibited adverse 

dynamic interactions between the airframe aeroelastic 

characteristics and flight control systems in flight. The 

X-29 aircraft was predicted to have adverse dynamic 

interactions as well. These actual and predicted dynamics 

problems were ultimately fixed by expensive and time 

consuming flight control system redesigns. Similar 

dynamic interaction problems are predicted for such future 

large-scale space vehicles as the proposed space station and 

satellites with large solar arrays and antennas. 

Analytical methods to predict aeroservoelastic 

stability of aircraft and servoelastic stability of spacecraft 

have been developed. The ability to predict servoelastic or 

aeroservoelastic interactions leads naturally to the desire to 

integrate, that is, include, dynamic interaction effects in 

aerodynamic, structure, and/or control system designs, 

such that adverse dynamic coupling in the final vehicle is 

eliminated. Numerous papers outlining integrated 

structure/control law design methods for large space 

structure applications have ape in recent years.1-7 

The purpose of this work is to develop the means to 

include dynamic vehicle response characteristics and well 

known control law design methods in an existing general 

framework for multidisciplinary (aerodynamics, 

propulsion, structures, etc.) design. The existing general 

framework is based on formal design problem 

decomposition, multilevel optimization techniques, and 

design sensitivity information. The formulation of a 

design problem decomposition for dynamic aerospace 

vehicles and the use of Linear Quadratic Gaussian (LQG) 

optimal control law design and sensitivity analysis 

methods leads to an alternative structure/control law 

design method which is applicable to both air and space 

vehicles. Development within the general framework 

retains the use of independent design tools for both the 

structure and the control law, and provides for the future 

addition of other design tools for aerodynamics, 

propulsion, and the like. 
The paper is organized as follows. A summary of 

existing structure/control law design methods and a 

summary of multilevel optimization theory are given 

first. These are followed by a description of the 

multilevel structure/control law design algorithm. Three 

structure/ control law design cases are presented for a twobar truss example problem. The frrst design case was to 

improve the closed-loop system stability robustness using 

control law and structural design freedoms, while the 

second and third cases were formulated to expressly 

examine structure and control law design tradeoffs. 

Existing Design Methods 

Numerous methods have been developed in recent 

years for structure/control law design of large space 

structure systems. These existing methods can generally 

be categorized by their approach to the problem as either 

simultaneous or sequential.1 

Simultaneous Methods - A simultaneous 

structure/control law design method is one in which the 

control law design problem and the structural design 

problem are directly combined into a single problem. 

Examples of existing structure/control law design 

approaches which can be characterized as simultaneous 

include those reported by Haftka, Martinovic, and 

Hallauer2, Salama, Hamidi, and Demsetz3, Hale4, and 

Miller et. al? 

Sequential Methods - A sequential structure/control 

law design method is one in which the control law design 

and the structural design are conducted in an iterative, 

multiple step manner. Examples of reported sequential 

structure/control law design methods are those by Messac, 

Turner, and Soosd, and Khot et.al.7 

Multilevel Optimization Theory 

A theory of multilevel optimization for large scale 

engineering design problems has been developed in recent 

year~.~*9 This theory is based on hierarchal problem 

decomposition methods8-l and the use of the sensitivity 

of optimization problem solutions to fixed problem 

parameters.12-14 

The theory is perhaps best explained by use of a 

conceptual multilevel optimization problem as shown in 

Figure 1. The conceptual problem is to minimize the 

performance index J by selection of the design variable a 

at the upper level, and to minimize the performance 

indices K and L by selection of the design variables p and 

x respectively at the lower level. Note that in this 

example, K and L are functions of a and p and a and x 

respectively, while J is a function of a, p. and x. 

The basic idea behind multilevel optimization is to 

treat the design variable a as a fixed constant during the 

minimization of K and L. With a fixed, the 

minimizations of K and L are decoupled, and are performed 

independently. Once the design variables p = p* and x = 

x* which minimize K and L are known, the performance 

index J of the upper level minimization can be evaluated 

and the gradient of J with respect to a computed. 

Clearly the p = p* and x = x* which minimize K and 

L respectively are functions of the design variable a. The 

evaluation of J at the upper level is for the fixed value of 

a and p = p* and x = x*. Thus, proper calculation of the 

gradient of J with respect to a must also include the 

effects of the changes in p* and x* with respect to changes 

in a. This gradient of J with respect to a can of course be 

calculated by a direct finite difference method at the upper 

level. In this case, J is evaluated for small perturbations 

in a, where the minimizations of K and L for p* and x* 

are repeated for each perturbation. The necessary gradient 

is then obtained by finite differencing of the perturbed 

values of J. 

Another means to calculate the gradient of J with 

respect to a is to use a chain rule approach. This results 

in a calculation for the gradient of J at the upper level 

which directly involves terms which are the gradients of 

p* and x* with respect to a. In this approach, the 

gradients of p* and x* with respect to a are calculated 

during the lower level minimizations of K and L, and are 

passed to the upper level along with the optimized design 

variables p* and x* as illustrated in the figure. This can 

simplify the overall multilevel optimization algorithm 

because the lower level minimizations do not have to be 

repeated for every perturbation of a at the upper level. 

(Multiple solutions of the lower level optimization 

problems may still be required however, if the gradients of 

p* or x* with respect to a are calculated by a finite 

difference method, but they are independent of upper level 

perturbations of a.) 

* 

In many cases, analytical expressions for the gradients 

of p* and x* with respect to a can be derived from the 

necessary conditions of ~ptimalityl~-~~ for the 

minimization of K and L, also as shown in Figure 1. 

These expressions are evaluated once for p = p* and x = 

x*, and the gradient information is passed on to the upper 

level. Multiple solutions of the lower level optimization 

problems about a fixed value of the design variable a are 

eliminated by use of these analytical gradient expressions. 

, 

2 
A geometrical interpretation of the sensitivity of 

optimized solutions is shown in Figure 2a. In this figure, 

M(u,p) represents a generic performance index to be 

minimized by selection of a design variable u, and p is a 

parameter with some fixed nominal value for a solution of 

the optimization problem. A locus of optimal solutions 

as a function of the value of p is shown in the figure 

labeled as M*@). 

The sensitivity derivative of an optimal solution with 

respect to p gives the slope and direction of the locus of 

optimized solutions at the parameter value corresponding 

to the optimal solution. This derivative has a component 

in both the p and u directions as shown in Figures 2.b and 

2.c respectively. The projection of the sensitivity 

derivative onto the M,p plane gives the change in the 

optimized performance index due to changes in the fixed 

value of the parameter p. The projection of the sensitivity 

derivative into the M,u plane gives the change in the 

optimized perfonnance index which is due to the change in 

the (optimal) u. It would appear that this projection 

violates the necessary condition of the optimization 

problem since the derivative of M with respect to arbitrary 

variations in u must be zero at the optimum. In other 

words, the derivative dM/du must be zero along a line 

parallel to the u axis. This is in fact the case here also. 

The nonzero projection in the M,u plane is caused by the 

constraint that the sensitivity derivative be calculated for u 

= u*(p). That is, the M,u projection is really aM/ap 

mapped from the M,p plane by u = u*(p). 

The sensitivity of the optimized performance index is 

an incomplete measure of the change in optimal solutions 

since there is no information on the change in the design 

variables due to parameter variations. This is particularly 

evident in the simple example of Figure 1, where the 

lower level minimization of K is considered. In that 

problem, the optimized performance index is K* = 0 for 

all values of the parameter a. but the optimizing design 

variable is p* = a. Use of the sensitivity of K* with 

respect to the parameter a would not yield any useable 

information for the upper level optimization, since that 

sensitivity is identically zero for all values of a. In the 

example then, the square difference of p from a desired 

value was used as the criteria at the upper level which is 

related to the lower level K minimization problem. The 

introduction and use of additional criteria to measure lower 

level performance at the upper levels is a requirement of 

multilevel optimization algorithms. 

. 

Several iterations of the conceptual problem of Figure 

1 for the case B =a = 1 are shown below as a final 

illus.tration of multilevel algorithms. Note in particular 

that once an a is selected, the lower level solutions for p* 

and x* are decoupled and are obtained independently of 

each other. (For the purposes of this illustration, a 

different scale factor in the computation for Aa was used 

at each step to limit the number of iterations.) 

Iteration 1: (ai = 1 assumed to start) 

1 

3 Aal = - J, Qc 

Iteration 2: 

Iteration 3: 

J3=~,x=% dJ + (p.3 - 1)(1) + (x; - 1)(2) = 0 

Analytical Solution: (Obtained by substituting the 

expressions for p*, x*, and their derivatives into the 

gradient of J with respect to a and setting the result equal 

to zero) 

=a + (a - )(1) + (2a - x)(2) = 6a - p - 2 =O 

. 3 
Multilevel Structure/Control Law Design 

Approach 

A multilevel optimization based method can be 

developed for integrated structure/ control law design. 

Specific formulations for such algorithms are of course 

highly problem dependent, however, a general formulation 

which uses Linear Quadratic Gaussian optimal control law 

design methods to determine the control law is outlined 

below. 

The general multilevel structure/control law design 

formulation is shown in Figure 3. In Figure 3, the 

structural design and the control law design are 

independent lower level design problems. These lower 

level designs are coordinated through the upper level 

optimization problem. The upper level optimization 

problem reflects the desired objectives of the integrated 

structure/control law design. For example, the upper level 

objective might be to reduce peak transient responses of 

the controlled system and to reduce the weight of the 

structure. The actual peak transient responses of the 

controlled system would come from analysis of the 

control law design at the lower level, while the actual 

structural weight would come from the lower level 

structural optimization. These could then be combined as 

a weighted sum of square errors between the actual and 

desired values to form a single upper level performance 

index. The upper level design variables would then be 

selected to minimize the objective function. 

The values of the upper level design variables at any 

time are treated as fixed parameters for the lower level 

optimizations. These parameters define either the 

mathematical model of the structure or dynamic system to 

be controlled, or they define the performance index and/or 

constraints of the optimization problems, or both. The 

sensitivities of the optimized lower level solutions to 

these fixed parameters are computed and used in turn to 

compute the sensitivity of the related part of the upper 

level performance index. That is, these final sensitivities 

are the gradients necessary to complete the top level 

optimization. In the earlier example, one of the upper 

level design variables may be a local structural stiffness 

requirement, which appears as an equality constraint in the 

lower level structural optimization. The sensitivity of the 

structural weight to this parameter is computed at the 

lower level and returned for use in computing the part of 

the gradient of the upper level performance index which is 

related to structural weight. Another of the upper level 

design variables might be a mean square weight on control 

effort, which would appear directly in the performance 

index of the lower level LQG control law design problem. 

The sensitivity of the optimized LQG control law with 

respect to this parameter would then be used to compute 

the sensitivity of the peak transient response of the 

controlled system, as required to complete the upper level 

optimization. 

For the purposes of this paper, the use of existing 

nonlinear programming based structural optimization and 

design sensitivity analysis methods is assumed. These 

methods may themselves be multilevel optimization 

algorithms, such as those of references 15 and 16, 

Also for this paper, the use of Linear Quadratic 

Gaussian optimal control law design methods is assumed. 

Expressions for the sensitivity of controlled system time, 

frequency, and stochastic responses in terms of state-space 

coefficient sensitivity matrices are available from the 

literature. The sensitivity of the optimized LQG control 

law to fixed parameters must be known to compute the 

necessary state-space coefficient sensitivity matrices. 

Analytical expressions for the sensitivity of the LQG gain 

matrices to fixed problem parameters have been derived 

elsewhere from the necessary conditions of optimality. l7 

The analytical expressions for the LQG sensitivity as well 

as analytical sensitivity equations for quency responses, 

time responses, covariance responses, eigenvalues and 

singular values are summarized in reference 18. 

Two-Bar Truss Example 

, 

Description - A two-bar truss structure/control law 

design problem was taken directly from the literature7 and 

used to validate both the analytical sensitivity analysis 

developments and the multilevel structure/control law 

design method. The nominal truss structure geometry, 

definitions, and second-order differential equations of 

motion are shown in Figure 4. Note that the bar cross 

sectional areas and the Young's Modulus are scaled in this 

example, although the natural frequencies of vibration are 

consistent with an actual structure of this size and 

concentrated mass. With these scalings, the material mass 

density (= 2.59 x lb-sec2/in4) is such that the mass 

of the bars is negligible compared to the concentrated 

mass for the vibration equations of motion. The bar 

cross-sectional areas are, however, a direct measure of the 

actual truss structure weight, which is to be reduced as the 

structural design objective. - 

4 
State-space equations of motion for the truss are 

Gaussian distributed, "white" noise as well, with intensity 

W = w, w = 1.0 nominally. The noises w and v were 

assumed to be uncorrelated. 

{;} = 

or 

0100 ,o O O1 

z1 2} 

z2 

X = AX + Bu + DW 

y=cx 

where the definitions of x, y, u, w, A, B. C, and D follow 

from the state equations, I is a 2 x 2 identity matrix, and 

the stiffness matrix K is 

Note that structural damping of 0.01 has been assumed in 

this model. 

An LQG optimal control law problem was formulated 

for the two-bar truss example. A cost function 

1 d 

was assumed, where E denotes expected value, and the 

weighting matrices Q and R are Q = qI and R = r, where 

the nominal values of q and r are 1.0 and 0.01, 

respectively. Noisy measured feedback signals were 

assumed as z = y + v, where the noise v is a zero mean, 

Gaussian distributed, "white" noise with intensity matrix 

V = VI, with the nominal value of v = 0.01. The 

disturbance input w was assumed to be a zero mean, 

Four parameters were selected for use as design 

variables at the top level of the multilevel 

structure/control law design algorithm for the three design 

cases presented below. These are the scale factor q (PI), 

the scale factor v (E), the cross-sectional area A1 of truss 

bar 1 b), and the force application angle 0 (p4). The 

upper level objective function for the three design cases 

was written to minimize the sum of the square difference 

between actual and desired design criteria as 

i= 1 

The design criteria for each design case are given below. 

Design Case 1 - Design case one was formulated to 

improve closed-loop stability robustness. Seven measures 

of controlled system stability robustness were used in the 

top level objective function. These included increasing 

the damping of the two structural modes by moving their 

eigenvalues to the left in the complex plane, raising the 

minimum singular value of the return difference at three 

frequencies, and reducing the magnitude of the loop 

transfer function at two frequencies. These criteria are 

further described in Table 1. 

The initial LQG optimal control law solution was 

computed for the nominal values of the design parameters. 

The initial values of the seven criteria and their sensitivity 

to each of the design variables were also computed. The 

sensitivity results for this initial iteration are summarized 

in Table 2. The sensitivity data was arranged accordingly 

in a gradient mamx and an incremental change in the four 

design variables was computed as 

where p is a vector of the four upper level design 

variables, c is a vector of the seven design criteria. the 

subscripts d and a denote desired and actual values, and t 

denotes the matrix psuedo-inverse. New values of the four 

design variables were then selected as 

Pn = Po +TAP 1 

where the subscripts n and o denote new and old and the 

factor of 1/2 was selected to reduce the effects of the 

linearization error. This process was repeated for five 

iterations. 

5 
The design results are shown in Figures 5 and 6. In 

Figure 5, the design criteria iteration history is shown 

with the results normalized by the desired value, so that 

satisfaction of a criterion occurs for a value of 1.0. All 

seven criteria are moving toward satisfaction with each 

iteration. Figure 6 shows the history of the upper level 

design variables, normalized by their initial (starting) 

value. There is a large increase in the V matrix scale 

factor v, which is effectively tuning the Kalman Filter 

design, a slight increase in the Q matrix scale factor q, 

which is tuning the regulator design, a slight increase in 

structural weight, and a decrease in the control force 

application angle. These results indicate that a 

combination of structure and control law design changes 

can be used to improve the overall stability robustness of 

a controlled system. 

Design Case 2 - The second design case was 

formulated to tradeoff structure and control design 

objectives. The upper level design objective was written 

to improve six integrated design criteria. The criteria 

included the mean-square response and control energy due 

to a random disturbance input w(t), the static controlled 

system structural deflections due to a static load, and the 

structural weight (directly related to the cross-sectional 

area of bar 1). The objective was formulated as in Case 1, 

with the desired values of the six criteria set to 95% of the 

initial values. The criteria are further described in Table 3. 

The normalized results are shown in the Figures 7 and 

8. Both the control force and the structural weight in 

Figure 7 were reduced toward their desired values, while 

the other four criteria were actually increased. Looking at 

the upper level design variable iteration history in Figure 

8, the cross-sectional area was. reduced to 95% of the 

original value, reflecting the direct influence of it's 

magnitude in the upper level objective. The control force 

application angle was increased, the Q matrix scale factor 

q was slightly decreased, and the V matrix scale factor v 

was increased 31%. The conclusion here is that the actual 

magnitude of the cross-sectional area dominates the 

design. 

Design Case 3 - Design Case 3 was the same as 

Design Case 2 except that the upper level design criteria 

were normalized by their initial values before the upper 

level objective function was formulated. The results for 

this case show that five of the six criteria are improved 

(Figure 9), the exception now being the structural weight 

which is increasing. The design parameter history in 

Figure 10 shows the 2% increase in bar cross-sectional 

area and also an 8-9% increase in force application angle. 

The results show the dependence of the design on the 

magnitudes of the design criteria relative to one another. 

They also indicate that small increases in structural weight 

may be worthwhile in terms of controlled system 

performance. This is intuitively satisfying since decreases 

in structural deformations are often associated with 

increased structural weight. 

Conclusion 

A new approach to integrated structure/control law 

design of aeroelastic aircraft and large space structures has 

been presented. This approach uses multilevel 

optimization techniques which are are based on the 

theories of hierarchal problem decomposition, 

optimization, and sensitivity of optimum solutions. 

Unlike existing sequential or simultaneous methods, the 

structure and control law designs are obtained 

independently. Integration of the structure and control law 

designs is achieved through an optimization problem 

formulation in which the independent disciplinary designs 

are coordinated at an upper level. Multilevel optimization 

has been explained by use of a simple conceptual 

example, and a general multilevel structure/control law 

design algorithm outlined. Results for three integrated 

structure/control law designs of a two-bar truss example 

problem were presented. These results illustrate the 

multilevel design method and show the potential tradeoffs . that are possible between the structure and control law 

designs. The examples point out once more the 

importance of the relative magnitudes of the design criteria 

to the final results. 

* 

References 

Junkins, J. L., and Rew, D. W., "Unified 

Optimization of Structures and Controllers", Large 

Space Structures: Dynamics and Control, Atluri, S. 

N. and Amos, A. K. editors, Springer-Verlag 1988. 

Haftka, R., Martinovic, Z., and Hallauer, W., 

"Enhanced Vibration Controllability by Minor 

Structural Modifications", AIAA Journal, Vol. 23, 

NO. 8, August 1985, pg. 1260-1266. 

Salama, M., Hamidi, M., and Demsetz, L., 

"Optimization of Controlled Structures", JPL 

Proceedings of the Workshop on Identification and 

Control of Flexible Space Structures, 1984, pg. 

31 1-327. 

Hale, A., "Integrated Structural/Control Synthesis 

via Set-Theoretic Methods", AIAA 85-0806, 26th 

Structures, Structural Dynamics, and Materials 

Conference, April 1985. 

Miller, D. F., Venkayya, V. B., and Tischler, V. 

A., "Integration of Structures and Controls - Some 

Computational Issues", 24th Conference on 

Decision and Control, Ft. Lauderdale FL., Dec. 

1985. 

. 
[a Messac, A., Turner, J., and Soosar, K., "An 

Integrated Control and Minimum Mass Structural 

Optimization Algorithm for Large Space 

Structures", JPL Proceedings of the Workshop on 

Identification and Control of Flexible Space 

Structures, 1984, pg. 231-266. 

Khot, N. S., Venkayya, V. B., Oz, H., Grandhi, R. 

V., and Eastep. F. E., "Optimal Structural Design 

with Control Gain Norm Constraint", AIAA 87- 

0019,25* Aerospace Sciences Meeting, January 

1987, Reno, NV. 

[I 

* 

[8] Sobieszczanski-Sobieski, J., "A Linear 

Decomposition Method for Large Optimization 

Problems - A Blueprint for Development", NASA 

TM-83248. Feb. 1982. 

[9] Sobieszczanski-Sobieski. J., Barthelemy, J., and 

Giles, G., "Aerospace Engineering Design by 

Systematic Decomposition and Multilevel 

Optimization", ICAS-84-4.7.3, Sept. 1984. 

[lo] Mesarovic, M., Macko, D., and Takahara, Y., 

Theory of Hierarchical, Multilevel, Systems, 

Academic Press, 1970. 

1113 Bemussou, J., and Titli. A., Interconnected 

Dynamical Systems: Stability. Decomposition, and 

Decentralization, North-Holland Publishing Co., 

1982. 

[121 Sobieszczanski-Sobieski, J., Barthelemy, J., and 

Riley, K., "Sensitivity of Optimum Solutions of 

Problem Parameters", AIAA Journal, Vol. 20. No. 

9, Sept. 1982, pg. 1291-1299. 

[13] Barthelemy, J., and Sobieszczanski-Sobieski, J., 

"Extrapolation of Optimum Design Based on 

Sensitivity Derivatives", AIAA Journal, Vol. 21, 

No. 5, May 1983, pg. 797-799. 

Barthelemy, J., and Sobieszczanski-Sobieski, J., 

"Optimum Sensitivity Derivatives of Objective 

Functions in Nonlinear Programming", AIAA 

Journal. Vol. 21, No. 6, June 1983, pg. 913-915. 

Sobieszczanski-Sobieski, J., James, B., and Dovi, 

A., "Structural Optimization by Multilevel 

Decomposition", AIAA 83-0832, 24th Structures, 

Structural Dynamics, and Materials Conference, 

May 1983. 

[16] Sobieszczanski-Sobieski, J., James, B., and Riley, 

M., "Structural Optimization by Generalized, 

Multilevel Optimization", AIAA 85-0697, 26th 

Structures, Structural Dynamics, and Materials 

Conference, April 1985. 

Gilbert, M. G., "Design Parameter Sensitivity of 

Optimum LQG Control Laws for Application to an 

Integrated Interdisciplinary Design Methodology", 

PhD Dissertation, hdue University, W. Lafayette 

IN, to be published, December 1989. 

[14] 

[15] 

[17] 

[18] Gilbert, M. G., "An Analytical Sensitivity 

Method For Use in Integrated Aeroservoelastic 

Aircraft Design". European Forum on 

Aeroelasticity and Structural Dynamics, Aachen, 

FRG, April 17-19, 1989. Also available as NASA 

q

TM 101583, May 1989. 

Criteria Original Desired 

Number Value Value Description 

1 -0.506 -0.531 Real Parts of Closed Loop Structural Mode Pair 1 

2 - 1.452 -1.524 Real Parts of Closed Loop Structural Mode Pair 2 

3 0.834 0.900 Rem Difference Singular Value at 1 .OO rad/=. 

4 0.832 0.900 Return Difference Singular Value at 3.35 rad/sec. 

5 0.859 0.900 Rem Difference Singular Value at 5.99 rad/sec. 

6 1.263 0.900 Loop Transfer Magnitude at 5.21 rad/=. 

7 
Table 2. Case 1 Original Design and Sensitivity Data for First Iteration 

& &i 

aP3 aP4 

Criteria Original &i kl 

Number ValUe aP 1 aP2 

1 -0.506 -0.210 0.000 -2.923 x -0.762 

2 - 1.452 -0.542 0.000 1.031 X 0.851 

3 0.834 5.148 x 10-3 4.274 2.874 x 10-4 -3.891 x 

4 0.832 -1.178 x 4.44 1 3.652 X -3.922 x 

5 0.859 -4.637 x 1.962 1.416 x -8.860 x 

6 1.263 0.451 -2.263 x lo1 4.245 x 10-3 1.874 

Table 3. Case 2 and Case 3 Design Criteria 

Number Value Value Description 

Criteria Original Desired 

1 1.194 x 1.135 x Mean-Square Deflection in x Direction 

2 3.125 x 10-2 2.968 x Mean-Square Deflection in y Direction 

4 3.629 x 3.447 x 10-2 Steady-State x Deflection to Step Load 

5 4.490 x 10-2 4.266 x Steady-State y Deflection to Step Lolad 

6 5.000 x 102 4.750 x lo2 Bar 1 Cross-Sectional Area (Truss Weight) 

3 0.646 0.614 Mean-Squm C~ntrol For~e 

Figure 1 - Conceptual Multilevel Optimization Problem 

8 
Upper Level Structure/ 

Control Law Design 

Figure 2a - Geometrical Interpretation of Sensitivity of 

Optimum Concept 

M 

*I 1 

* [ \., PmFfM*(p) 

-*e M2 ---. 2 

Figure 2b - Mg Plane Projection 

M 

Figure 2c - M,u Plane Projection 

I I I I 

Figure 3 - General Multilevel Structure/Control Law 

Design Algorithm 

L-. 20 -_I 

Mass=m=2 

AreaBar1=A1=500 

AreaBat2=A2=1OOO 

Length Bar 1 = Length Bar 2 = 1 = 22.36 

Force Application Angle = 8 = 45O 

Young's Modulus E = 1 

Figure 4 - Two-Bar Truss Geometry and Data 

9 
- Criterion 1 

Criterion 2 

Criterion 3 

--C- Criterion 4 

- Criterion 5 Criterion6 

Criterion 7 

1 2 3 4 5 

Iteration 

Figure 5 - Case 1 Design Criteria Iteration History 

> 

- __ DesignVar.l DesignVar.2 

4 DesignVar.3 

DesignVar.4 

I 

E 

> 

- __ DesignVar.l DesignVar.2 

4 DesignVar.3 

DesignVar.4 

I 

E 

6 

1 2 5 6 Iteration 

Figure 6 - Case 1 Upper Level Design Variable History 

1 

Criterion2 

s + Criterion 3 

- criterion4 

Criterion5 

Criterion 6 

1 2 Iteration 4 5 6 

Figure 7 - Case 2 Design Criteria Iteration History 

1 .: 

1 

ql,lml,0.1 

1 2 5 6 Iteration 

Figure 8-2 Case Upper Level Design Variable History 

1 .o€ 

i i 

-C- Criterion 4 

Criterion 

+ Criterion 6 

1.oc I ~ I 1 I. I. 

1 2 4 5 6 Iteration 

Figure 9 - Case 3 Design Criteria Iteration History 

1.oz 

l'lC r 

. 

- - DesignVar. 1 

DesignVar. 2 

- DesignVar.3 DesignVar.4 

1 1.1. 

1 2 4 5 6 Iteration 

Figure 10-3 Case Upper Level Design Variable History 

10 
Report Documentation Page 

1. Report No. 

NASA TM-101623 

2. Government Accession No. 

7. Author(s1 

7. Key Words (Suggested by Authods)) 

Michael G. Gilbert 

David K. Schmidt 

NASA Langley Research Center 

Hampton, VA 23665-5225 

9. Performing Organization Name and Address 

18. Distribution Statement 

12. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

9. Security Classif. (of this report) 

Unclassified 

3. Recipient's Catalog No. 

20. Security Classif. (of this pagel 21. No. of pages 22. Price 

Unclassified 11 A03 

5. Report Date 

June 1989 

6. Performing Organization Code 

8. Performing Organization Report No. 

10. Work Unit No. 

505-63-21-04 

11. Contract or Grant No. 

13. Type of Report and Period Covered 

Technical Memorandum 

14. Sponsoring bgency Code 

5. Supplementary Notes 

To be presented at the AIAA Guidance, Navigation, and Control Conference, August 14-16,1989, in 

Boston, Massachusetts. 

6. Abstract 

A new approach to integrated structure/control law design based on multilevel optimization is presented. 

This new approach is applicable to aircraft and spacecraft and allows for the independent design of the 

structure and control law. Integration of the designs is achieved through use of an upper level 

coordination problem formulation within the multilevel optimization framework. The method requires 

the use of structure and control law design sensitivity information. A general multilevel structure/control law design problem formulation is given, and the use of LQG control law design and 

design sensitivity methods within the formulation is illustrated. Results of three simple integrated 

structure/control law design examples are presented. These results show the capability of structure and 

control law design tradmffs to improve controlled system performance within the multilevel approach. 

Aeroservoelasticity 

Sensitivity 

Integrated Design 

Optimization 

Unclassified - Unlimited 

Subject Category - 63