A method for simultaneous structural and control design of large flexible space structures (LFSS) to reduce vibration generated by disturbances is presented. Desired natural frequencies and damping ratios for the closed loop system are achieved by using a combination of linear quadratic regulator (LQR) synthesis and numerical optimization techniques. The state and control weighing matrices (Q and R) are expressed in terms of structural parameters such as mass and stiffness. The design parameters are selected by numerical optimization so as to minimize the weight of the structure and to achieve the desired closed-loop eigenvalues. An illustrative example of the design of a two bar truss is presented

Becus, G. A.

Lui, C. Y.

Tischler, V. A.

Venkayya, V. B.

English

NASA Technical Reports Server

N88- 13622
Simultaneous Structural and Control Optimization via
Linear Quadratic Regulator Eigenstructure Assignment
G. A. Becus
C. Y. Lui
V. B. Venkayya
V. ,4. Tischler
This paper presents a method for simultaneous optimal structural and
control design of large flexible space structures (LFSS) to reduce vibration
generated by disturbances. Desired natural frequencies and damping ratios
for the closed-loop system are achieved by using a combination of linear
quadratic regulator (LQR) synthesis and numerical optimization techniques.
The state and control weighting matrices (Q and R) are expressed in terms of
structural parameters such as mass and stiffness. The design parameters are
selected by numerical optimization so as to minimize the weight of
the structure and to achieve the desired closed-loop eigenvalues. An
illustrative example of the design of a two bar truss is presented.
INTRODUCTION
Large structural systems in general and large space structures in
particular present new challenges to the structural dynamicist and the
control engineer as well. Indeed, such large systems may exhibit well over
a thousand vibrational modes usually closely spaced and with little, if any,
damping. Some form of active control is likely to be necessary in order to
meet exacting stability and pointing requirements. In fact, structural
requirements (primarily low mass) increase the need for active control.
Some optimal trade off between structural and control criteria has to be
achieved.
Until recently, the design of control systems for large structural
systems was a two-step procedures: first the structure was designed based on
structural criteria (primarily total weight); then in a second step a
control system (satisfying some desired control oDjectives) was designed for
the structure obtained in the first step. Inasmuch as a low weight (and
thus low stiffness) structure will require high control energy, the design
objectives of the two steps are to some extent contradictory so that an
optimal control design for an optimally designed structure will not in
general result in an overall control-structure optimal design. Both designs
need to be carried out simultaneously.
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LITERATURE REVIEW
The optimal structural and control design of large flexible space
structures was recently investigated by several researchers. Venkayya and
Tischler [I-2] have suggested that the performance index (PI) in optimal
control of structural systems be a measure of the system total mechanical
energy. By appropriately choosing the state and control weighting matrices,
the PI can be expressed as the (weighted) sum of the kinetic, strain and
potential (including control) energies. Knot and Venkayya [3-4] tackled the
s_ructu_al and control optimization problem by minimizing the weight of the
structure with constraints on structural frequencies and the minimum
Frobenious norm of the gain m_trix. This process has to be carried out in
an iterative fashion.
Becus and Lui [5] have proposed a general method to choose state and
control weighcing matrices in optimal control design so as to satisfy
desired closed-loop eigenvalues. This was further extended by Becus and
Sonmez [6] to allow for eigenvector assignment. In this paper we combine
both ideas in order to obtain _ method to carry out simultaneous optimal
structural and control design.
Desired dynamic structural requirements (natural frequencies and
damping ratios for example) can be expressed both in terms of desired
close_-loop eigenstructure (eigenvalues and/or eigenvectors) and structural
parameters (mass and stiffness for example). Using a PI of the form
suggested in [lJ, the elements of the state and control weighting matrices
(Q and R respectively) are also expressed in terms of structural parameters.
Thus, when choosing the Q and R m_trices (using She method of [5-6]) to
satisfy a desired closed-loop eigenstructure (i.e. dynamic structural
requirements), one in fact chooses new strucutral parameters and therefore
carries out a simultaneous optimal control structure design.
In this paper a new design algorithm is developed so that a minimum
weight structure with desired damping and natural frequency of the closed-
loop system can be obtained. We compare the results with [3] in the last
section.
SIMULTANEOUS STRUCTURAL AND CONTROL OPTIMIZATION
Consider a controlled structural dynamic system described by the
discrete (finite element) model
.o
Mr + Kr _ Du (I)
where r is a vector of n physical displacements and the number of control
inputs (forces) u is m. M, K and D are the mass, stiffness and applied load
distribution matrices of appropriate dimensions respectively. Assume that M
and K are positive definite.
The state space representation of Eq. (I) can be written as
= Ax + Bu (2)
where x _ [_T rTl T , (3)
226
- A(p) , (4)
IM 'DIand B ....... = B(p) , (5)
where p is a vector of structural parameters of dimension 1.
The optimal steady-state control is a linear state feedback
u - -Gx . (6)
The state feedback gain matrix G is obtained from LQR synthesis and the
closed-loop system is given by
- (A - BG)x . (7)
LQR synthesis determines a control u which minimizes the quadratic
performance index [I]
PI - f0® [QmrTM_ + Qk rTKr + OruTDTK-IDu] dt (8)
or in the state space coordinates
PI - f_ [xTQx + uTRu] dt (9)
where
- Q(P) (10)
and R - [erDTK-ID] - R(p) (11)
for positive scaling parameters Om, @k and @r" In Eq. (8), PI is the
absolute weighted sum of the kinetic, strain and potential energies.
The relationship between characteristic polynomial of the optimal
system and weighting matrices is obtained as follows [7]
I sI - A I BR-IBT
det(sI Z) l
I Q I sI + AT
(12)
or ¢c(S)¢c(-S) - ¢o(s)@o(-s)det[I+R-IHT(-s)QH(s)] (13)
where H(s) is the open-loop transfer function matrix
227
H(s) A (sI - A)-IB , (14)
Z is the canonical system matrix, ¢c(S) and _o(S) are the closed-loop and
open-loop characteristic polynomials respectively.
For a given desired closed-loop pole s = sd which is not an open-loop
pole, the determinant in the right-hand side of Eq. (13) must equal zero
when the weighting matrices Q and R take values which yield the desired
closed-loop eigenvalues. In order to use numerical optimization techniques
to solve Eq. (13) for Q and R, we, as in Ref. [8], set the objective
function as
obj = det[I + R-IHT(-sd)QH(Sd )] - 0 (15)
The desired characteristic equation corresponding to Eq. (15) is
j_1 " (s - Sd.)(s + Sd ) = 0
J J
(16)
where Sd. is the j-th desired closed-loop eigenvalue.
J
Q and R are determined by equating coefficients of the terms involving equal
powers of s in Eqs. (15) and (16). This yields
f1(p) = 0
fk(p) = 0
(17)
where k is the n_mber of equality constraints which involve equal powers of
s in Eqs. (15) and (16).
The objective in structural and control optimization is to make the
selection of design parameters so that the structure weight is a minimum and
the specified closed-loop eigenvalues are satisfied. The optimization
problem can be stated as
Minimize the weight W = W(p)
subject to Eq. (17) (18)
and P > P , s - I, -.-, I,
s s
where P denote minimum allowable values of the structural design
s
parameters •
228
I LLUSTRATIVE EXAMPLE
In order to illustrate the feasibility of the above algorithm, the
structural two bar truss model shown in Fig. I was considered as a simple
design example. For the geometry shown, the dynamical equations of motion(Eq. (I)) are
2 0
0 2
rl I + kl I (A,+A2) 2(AI-A2)
ra 12(AI-Aa) 4(AI+A2) r2! Isin(e)
u (19)
K = k z
2 0
2
(Al + A2)
2(Al - A2)
2(AI - A2)
4(Al + A_)
(20)
(21)
are the optimal mass and stiffness of the structure respectively. In Eq.
(19), A I and A2 are the cross-sectional areas of the bars and k_ = E/(5L) is
a stiffness coefficient, E representing the elastic modulus of the bars and
L the length of the members. A control force u is located at the vertex
with 8 being the angle between its line of action and the horizontal, r_
and r2 are the horizontal and vertical displacements of the vertex
respectively.
The dimensions of the structure were given in unspecified consistent
units. The elastic modulus of the members was assumed to be I and the
density p of the structural material was assumed to be 0.001. A
nonstructural mass of 2 units was attached at node 2 and the structural mass
of the members was ignored for simplicity (thus the mass matrix of Eq.
(20)). The actuator and sensor were located in element I connecting node I
and 2. The minimum cross-sectional area was set equal to 10 units for both
member s.
Once the choice of the material is fixed, the design variables are the
cross-sectional areas of the members A_ and A2, the scaling parameters C)m,
O k and Or, and the angle B of the applied load with respect to the
horizontal. The optimal closed-loop eigenvalues are specified as Sd_ = -
w0.0228 + 1.17j and Sd2 -0.361 + 4.81j. Arbitrary lower and upper values
of 0 were set at 30 o and 60 o respectively.
Analytical and numerical computations were carried out using MACSYMA TM
[9] for symbolic algebraic manipulations, MATLAB [10] for matrix
computations and LQR synthesis, and GRG2 [11] for numerical optimization.
The numerical results for several representative optimal designs are listed
in Table I. A discussion of these results appears in the next section.
229
DESIGN
I
2
3
4
5
6
Table I. Optimal Two Bar Truss Designs
10'1.98 889.88
7_8.86 78.81
531.74 ! 5'3.02
442.86 I 44.16
311.18 31.03
11'5.36 11.50
A2 WEIGHT
22.18
19.40
7.65
2.84
e
m Ok Or
I i."i'2 '''2.78
I 1.216
I I 1.87
I i 2.25 73.68
I i 3.208 64
0
deg
ii
6O
5553.80
I73.68 60 '[
6O
73.68 60
73.68 6O
ACHIEVED CLOSED-
LOOP EIGENVALUES
-0.5074 ± 1.25j
-0.382 ± 4.82j
-0.0702 ± 1.17j
-0.361 ± 4.81j
i -0.0393 ± 1.17j
-0.361 ± 4.81j
2
F£g. 1 Two Bar Truss
230
DISCUSSIONOFRESULTS
For this simple example the six design variables were not independent.
The scaling parameters 0m and 0k appeared in the constraint equations only
as the combination k_Om/Ok. This combination was then used as one of five
independent design variables. To obtain the values of Table I, Om was
arbitrarily set equal to I then Ok was evaluated by multiplying the value
obtained by numerical optimization by k_ = 0.0089.
Since there are five independent design variables and only four
equality constraints, there are manysolutions to the optimization problem.
In order to obtain a unique solution one could arbitrarily fix the value of
one of the five independent design variables or equivalently introduce an
additional constraint.
Of all designs presented in Table I, Design 6 is the best since it
leads to the lowest value for the weight. This "optimal" design leads to a
weight of 2.84 which is less than half of the best design of Ref. [3]
(6.417).
A closer examination of Table I leads to someinteresting observations.
Designs 3 through 6 have weights which are inversely proportional to Ok. In
fact the product OkxWeignt is nearly constant for these four designs and
equal to 24.45. In addition it can be seen that for these four designs the
ratio A_/A2 is nearly constant and equal to 10. It is conjectured that many
other designs could be obtained by choosing areas satisfying this
relationship and calculating the corresponding Ok while keeping the other
design variables constant.
Design I is representative of several designs for which the ratio AI/A _
is nearly constant and equal to 0.1 while Design 2 leads to an angle less
than the upper bound value of 60o. For all designs obtained the product
OkxWeightwas nearly constant and equal to 24.45.
Finally it must be noted that as more weight is given to the control
effort the achieved closed-loop eigenvalues are closer to the desired
eigenvalues. As more weight is being given to the strain energy cost the
total weight decreases.
CONCLUSION
An algorithm for simultaneous structural and control optimization
design of a minimum weight structure with desired closed-loop eigenvalues
was proposed. It has been shownthat structural and control designs can be
obtained by LQR assignment. The design parameters were appropriately
selected by numerical optimization so as to minimize the weight of the
structure and to achieve desired natural frequencies and damping ratios.
The feasibility of the algorithm was demonstrated by applying it to a simple
example. Further work is needed to investigate the application of the
algorithm to large-order systems.
231
I •
2,
•
•
a
o
1
•
,
10.
11.
REFER ENC ES
V.B. VenKayya and V.A. Tischler, "Frequency Control and its Effect on
the Dynamic Response of Flexible Structures," AIAA Journal, Vol. 23,
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AIAA-87-2314, AIAA GNC Conference, Monterey, CA, Aug. 17-19 1987.
G.A. Becus and G. Sonmez, "Eigenstructure Assignment in LQR Design of
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presented)•
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232


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