An unconventional framework is described for the design of decentralized controllers for large flexible structures. In contrast to conventional control system design practice which begins with a model of the open loop plant, the controlled plant is assembled from controlled components in which the modeling phase and the control design phase are integrated at the component level. The developed framework is called controlled component synthesis (CCS) to reflect that it is motivated by the well developed Component Mode Synthesis (CMS) methods which were demonstrated to be effective for solving large complex structural analysis problems for almost three decades. The design philosophy behind CCS is also closely related to that of the subsystem decomposition approach in decentralized control

Young, K. D.

English

NASA Technical Reports Server

N90-2i071
A DISTRIBUTED FINITE-ELEMENT
MODELING AND CONTROL APPROACH FOR
LARGE FLEXIBLE STRUCTURES
K. D. Young
Lawrence Livermore National Laboratory
University of California
Livermore, California
Third Annual NASA/DOD CSI Conference
San Diego, California
January 29 - February 2, 1989
151
https://ntrs.nasa.gov/search.jsp?R=19900011755 2020-03-19T22:48:26+00:00Z
INTRODUCTION
This paper describes an unconventional framework for the design of
decentralized controllers for large flexible structures. In contrast
to conventional control system design practice which begins with a
model of the open loop plant, the controlled plant is assembled from
cont_lledcomponen_ in which the modeling phase and the control design
phase are integrated at the component level.
The developed framework is called ControlledCom_nent_n_esis (CCS)
to reflect that it is motivated by the well developed _omponent_o_
_nthes_ (CMS) methods which have been demonstrated to be effective
for solving large complex structural analysis problems for almost
three decades.
The design philosophy behind CCS is also closely related to that
of the Sub_s_mDecompos_mnAppr_ch in decentralized control.
CONTROLLED COMPONENT SYNTHESIS
CCS is a framework for an integrated, component oriented, finite-
element modeling and structural control design. Similar to CMS
methods, CCS is developed on the premise that a large complex
controlled structure is to be built from con_ol_dcomponen_: The finite
element modeilng and control design are carried out for the
individual components; the model of the large complex structure is
assembled from the controlled components only for the purpose of
performance evaluation.
The CCS method developed herein adopts the following modeling and
control design considerations at the component level: Instead of
using either th e boundary loading, or the constraint modes approach
as in CMS, we introduce a new approach called ho_dBoun_Loading for
the development of component models, For the design of controllers
for the component, an _n_kmg_ontrolconcept is developed to
minimize the motion of the nodes that are adjacent to the
boundary, thereby suppressing the transmission of mechanical
disturbance from component to component in the coupled structure.
The major ideas behind CCS are:
Component modeling using Isolated Boundary loading
Connections to Overlapping Decomposition
Intelocking Control Concept
152
COMPONENT MODELING
A two component structure, as shown below, will be used to
outline the modeling and design procedure of the CCS method. Each of
the structure component is composed of three finite elements.
Identified in the figure by Roman numerals are the finite elements,
and by solid circles are the element node points.
In the CCS method, the nodal coordinates of a component are
partitioned into three groups: The internal coordinates are
subdivided into a group of internal boundary coordinates xi_ and a
group of internal coordinates _;. The boundary coordinates Xb; remain
in a single group. The boundary coordinates are coordinates of the
boundary element, such as element III of component i, which are on
the boundary. The remaining coordinates of the boundary element are
designated the internal boundary coordinates. The remaining
coordinates of the component are the internal coordinates.
i
L,
i
X
il
A i i
ll# 'i III I
H " HI ,I '
i Ii !
14 It i
qi II I
II II I
ii -- !1 I
' "i{ '
ii t i I
.,,,<t li #l i I :
Xibl Xbl
Component 1
v
• i,# _
I
I I ,.
II
It #|
't!IV ,!, v
i | 'i
i i i_
il ,i
il l
Xb2 Xib 2 x
i2
Component 2
VI
\
d
#
A two component structure modeled with finite elements
153
ISOLATED BOUNDARY LOADING
The component mass and stiffness matrices are obtained from the
finite-element modeling of an expanded component, i.e., the original
boundary of the componentis extended one finite element into the
adjacent component. The nodes of the expanded component consist of
the original nodes of the component, and the internal boundary
coordinates of the adjacent component. The mass and stiffness
matrices are obtained from the mass and stiffness matrices of the
expanded component by deleting the rows and columns corresponding to
the nodes in the expanded portion.
!
original
boundary
I
i.
s _" _ _ _ .s
!li _
t ,,,.I H ' m'i'-',_IV I_
I
Xil
1,
I
!
!
!
i I i il
: :-_. ean--_
#1 ! I I I
,, ,,; ,, ,
Xibl 9_! Xib2
I
!
[] nodes to be truncated
Expanded component 1
154
CONNECTION TO OVERLAPPING DECOMPOSITION
The component models developed using isolated boundary loading
have direct connection with the Subsystem Decomposition Approach.
These models are identical to the decoupled subsystem models if an
overlapping decomposition is applied to the finite-element model of
the coupled structure. This is a key connection which allows the use
of tools developed by Siljak I and his co-workers for evaluating the
performance of the controlled coupled structure, after the controlled
component designs have been completed.
The mass and stiffness matrix connectivity is illustrated in the
following diagram showing how the component models can be
"contracted" to form the coupled structure finite-element model.
b 1 and b2 blocks are identical
due to boundary loading
b2 and b3 blocks are identical
due to boundary loading
_CONTRACTION
Overlapping Decomposition Connection
155
INTERLOCKING CONTROL CONCEPT
The new insights gained from the developed component modeling ap-
proach in turn motivate a new component level control design concept
which we call InterLocking Control (ILC) in which collocated actu-
ator and sensors are placed at the internal boundary degrees of free-
dom, and the control law is designed, using the develoPed compQneD_
model for CCS, to minimize the internal boundary coordinate motion.
Such minimization would localize the dynamic interactions of the
coupled structure in the components. The component control action is
designed to lock up its own internal boundary to realize a boundary
condition which better approximates the one assumed in the component
modeling of its adjacent components.
A convenient control design technique for this concep t is the lin-
ear quadratic optimal regulator approach in which the internal bound-
ary coordinates are considered as regulated outputs of the component
to be weighted together with the component control inputs in the
quadratic performance index. The resulting component control law min-
imizes this index.
The ILC concept translates into a two step component control de-
sign process summarized below:
i. For the sth component, use the component model
0 M_i b M s K_i _bb,iL I. xb, I(:,_,,iL
_I s = Zib s )
;rib s
Xbs :
for control system design, where u8 and v8 denotes respectively the
control force exerted by the actuators, and the sensor outputs, at
the internal boundary coordinates.
2. Derive the component control law by minimizing the performance
index,
+ u'T R'u')dt .
0
156
77:7:T 2:::::_:
g
APPLICATION TO TRUSS STRUCTURE CONTROL
The developed CCS method is applied to the design of structural
control laws for a planar truss structure for a preliminary
assessment of its feasibility toward solving more complex structural
control design problems. This truss structure which is depicted below
has six bays, and the nodal coordinates are defined as the vertical
and horizontal displacements at the joints.
! !
2 ' 6 , 10
Disturbance _ ', '
forces , ,1 5 9
11 _ 1 _ i _ i33 7 11
I I
Xbl Xbl Xb2 Xb3 Xb3
1 1 2 2 3
u ,y u ,Y u3, y
Planar truss for CCS evaluation
External forces applied at the nodes are decomposed into
orthogonal components. The assumptions made are that the truss
members are subjected to axial forces alone, and not bending moments;
and the members are uniform rods of identical lengths L, mass per
unit length m, cross-section area per unit length A and modulus of
elasticity E.
The six bay truss can be viewed as a structure that consists of
three identical components, namely the left component, the center
component, and the right component, which are composed of the left-
most, the middle, and the right-most two bays respectively. The
six bay/three component truss structure is chosen to capture the
essential characteristics of a truss consists of an arbitrary number
of identical components, i.e., a truss structure with an arbitarily
large number of two bay components is composed of the three same
types of components identified in the six bay truss, with the center
component duplicated as necessary. Thus, conclusions from the six
bay/three component design apply equally well to the design of
structural controls for a multiple bay truss.
157
TRUSS STRUCTURE CONTROL - CCS MODELING
For CCS, the component models are developed using the expanded
component introduced in Isolated Boundary Loading. The mass and
stiffness matrices of the expanded component are derived using a
finite-element method with the Ritz-Rayleigh approximation. The truss
member mass and stiffness matrices used in the assembly process are:
[,,,,]
' M"_"_er = -6- L
The component model is further scaled to remove the effects of the
material properties: a new time variable
 --EIr = Lt
is introduced, and the nodal forces are scaled by L(EA) -I
The three expanded components from which the component models are
derived are shown below. The internal boundary degrees of freedom at
which collocated force actuators and displacement sensors are placed
are marked by A.
S?L
Right-most Component
Left-most Component
_ Center Component
L 7 : :z 7
L ":" L:
['_ : Nodes to be truncated/Degrees of Freedom to be constrained
,The three expanded component
158
=
TRUSS STRUCTURE CONTROL - INTERLOCKING CONTROL DESIGN
The component level control design using the Interlocking Control
concept is carried out with a 4 × 4 identical control weighting
matrix R=.001 I for all three components. The control designs for the
left-most and right-most components are identical due to symmetry.
Therefore, we only need to carry out a center component design and an
end component control design.
The controlled components' poles, as well as the poles of the
controlled truss structure, are plotted in the figure below. Since the
left-most and right-most components are identical, we plot only the
poles of one of them which are denoted by End Component Poles in
these figures. All the poles of the controlled structure have
negative real parts, indicating that the closed-loop system is
asymptotically stable.
That the pole locations of the controlled components are close to
that of the controlled coupled structure indicates that the component
models developed for CCS are effective for this structural control
design.
+ Failed Coupled Structure Poles
A End Component Poles
x Center Component Poles
Coupled Structure Open Loop Poles
4- d,
+ X _ "4" X
+++ L& &+ L& X+ +L&
$++ z& _$ n _ +L&
.i .[
d,
+
+ X A + X
+ +÷ i
I
' I ' I ' I ' I ' I ' I
1.5
1.0
0.5
m.,
i_
0.0 "-
-0.5
-1.0
-1.5
-0.030 -0.025 -0.020 -0.015 -0.010 -0.005 0.000
Real
Closed up view of pole location near the imaginary axis of the controlled components and
the controlled structure, open loop poles of the coupled structure are inserted for reference
159
TRUSS STRUCTURE CONTROL - SIMULATION RESULTS
For transient response studies, the response of the controlled
structure to three disturbance force pulses of 0.5 seconds is exam-
ined, simultaneously applied to the left-most nodes of the truss, as
shown in the three expanded component figure. The coupled structure
is assumed to be in static equilibrium initially in the simulation,
of which samples of the sensor output time responses are shown in
the top two figures below. Two of the twelve sensor channels, one
horizontal (Channel 1 in the three expanded component figure) and
one vertical nodal displacements (Channel i0), are selected. The
magnitudes of the displacement response drop by an order of magni-
tude per component for nodes that are farther away from the distur-
bances. The delay effect of the force pulses on the displacements
shown in the Channel i0 displacement figure below is typical for the
right-most component.
The developed CCS method inherits the capability to withstand
system failures from decentralized control developed using the
Subsystem Decomposition Approach. The controlled structure, in
which the center component controller failed, is simulated for the
same disturbances and initial conditions as before. The two bottom
figures below show an order-of-magnitude performance degradations
for one of the displacements (Channel 5) at the center component.
However, despite the center component controller failure, th_
neighboring components stabilize the vibrations in the center
component with interlocking controls.
Channel 5 displacement,
with center component controller failure
160
REFERENCES
i. Siljak, D. D. (1978). Large-Scale Dynamic Systems, North-Holland.
BIBLIOGRAPHY
i. Young, K. D. (1983). An Application of Decomposition Techniques to
Control of Large Structures. Proceedings of the Fourth VPI &SU /AIAA
Symposium on Dynamics and Control of Large Structures, Blacksburg,
VA, June 6-8, 1983, pp.201-214.
2. Ikeda M. and D. D. Siljak(1980). Overlapping Decompositions, Ex-
pansions, and Contractions of Dynamic Systems. Large Scale Systems,
Vol. i, pp. 29-38.
3. Ikeda M., D. D. Siljak, and D. E. White(1981) . Decentralized Con-
trol with Overlapping Information Set. J. Opt. Theory Appl., Vol.
34, pp. 279-310.
161
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A distributed finite-element modeling and control approach for large flexible structures

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