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6.1991-2736

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USING CIRCUIT ANALOGIES*
L. Lcnriing
u. Ozgiillert
Department of Electrical Engineering
The Ohio State Uiiiversity
Columbus, OH 43210
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Abstract
Circuit analogies of mechanical systems are used to establish a framework in which large flexible structures
may be analyzed. Impedance and scattering parameters of flexible structures are derived directly from either partial differential equation or finite element models of the structures. A cantilevered beam example is
used for illustration and comparison of the two derivations. Component mode synthesis and controlled component synthesis of structures are discussed and related
t o circuits. Various design methods for shaping scattering parameters of flexible structures for desired scattering properties are formulated including H,-based
designs.
Introduction
Large space structures (LSS) may be viewed as a n
interconnection of several substructures or networks.
Essentially, the vibration control problem of flexible structures can be approached as a power transfer
problem-similar t o problems in transmission line theory or electrical circuits. Thus, each of the substructures may be described in terms of scattering parameters similar t o those of an electrical network.
T h e scattering parameters can be derived from either a partial differential equation (PDE) model of the
structure or from the finite element model (FEM) of
the structure. Like transmission lines, LSS are distributed parameter systems; therefore, they exhibit
wave modes similar t o those encountered in the P D E
approach. FEM’s are merely approximations of P D E
models. For lumped parameter models such as FEhI’s,
lumped parameter circuit analogies of LSS may be obtained directly from the FEM equations. Therefore,
the scattering parameters of the structures may be obtained using circuit analysis techniques.
T h e P D E models have been used in various control
approaches such as the traveling wave approach [I, 2, 3,
41. In the traveling wave approach t o flexible structure
“This paper is based on work sponsored by the Air Force
Office of Scientific Research (AFSC) under Contract F49620-S9C-0046.
t Member AIAA
Copyright 0 1 9 9 1 by the American Institute of
Aeronautics and Astronautics, Inc. All rights reserved.
control, the disturbances t o a structure are viewed as
waves traveling through a medium. These waves can
be reflected, partially reflected, or dissipated as they
travel across the structure. At interfaces within the
structure, scattering properties may be examined-how
much of an incoming wave is reflected and how much
is absorbed. With this perspective, the flow of these
disturbances through the structure may be more easily
understood.
T h e propagation of disturbances in a mechanical system is mathematically similar t o t h a t of electrical or
microwave circuits. T h e motivation for this work is to
reconsider circuit analogies of mechanical systems and
use existing results from circuit theory and network
theory in the analysis and design of LSS.
Inlthis paper, circuit and network theory will be reviewed with emphasis on circuit analogies of mechanical systems. Modeling of flexible structures will be
considered beginning with existing P D E modeling and
finite element modeling. T h e impedance and scattering
parameters of a flexible structure will then be determined both from the wave formulation from the P D E
model and from a lumped circuit analogy of the FERI.
Component mode synthesis (CMS) techniques for modeling LSS will be considered in the framework of impedance and scattering parameters. Scattering parameter
design methods will be discussed including the formulation of an H , design problem and a discussion of
controlled component synthesis (CCS) techniques.
Review of Circuit and Network Theory
For decades, circuit analysis and network synthesis has
received much attention in the research community.
T h e theory in these areas is very well developed and
several excellent texts are available [5, GI.
A general electrical network is comprised of several interconnected circuit elements. Terminal pairs or
ports allow voltage and current signals to be measured
or applied a t various points in the network. A network
with n ports is commonly called an n-port network or
simply an n-port.
For a general n-port network as shoirn i n Figure 1,
the port-voltage and port-current vectors are defined
1208
I
Scattering parameters exist for near
parameter or distributed parameter passi
works. Several quantities of interest t o powe
mission may be simply and concisely repres
terms of the scattering parameters. Thus, scattering
parameters provide a convenient way t o represent the
wave reflecting or absorbing behavior of a network.
Again, consider the n-port shown in Figure 1 and let
the following vectors and matrices be defined:
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NETWORK
- Incident voltage vector
V, - Reflected voltage vector
V’ - “Ordinary” voltage vector
I’ - “Ordinary” current vector
2’ - “Ordinary” open circuit impedance matrix
R, - Normalizing matrix
V - Normalized voltage vector
I - Normalized current vector
V, - Source voltage
- vector
2, - Input (source) impedance matrix (diagonal)
S - Scattering matrix
2 - Normalized impedance matrix
Figure 1: General Representation of an n-port Network
with Attached Sources and Loads
Furthermore, let
as
Often, one of these vectors is taken t o be the excitation
signal and the other is taken t o be the response signal.
In the frequency domain, V(s) and I(.) represent the
transforms of w(t) and i ( t ) ,respectively.
Standard definitions for linearity, time-invariance,
passivity, and causality may be found in most texts
on network theory. Several theorems and excellent
thorough discussions of these topics may be found
in [7, 6, 51.
Impedance Parameters
In general, sources with internal impedances may be
attached to the n-port network as shown in Figure 1.
The open circuit impedance matrix, Z ( s ) , is defined
such that
V ( s )= Z ( s ) I ( s )
(3)
V’
=
&(E
+K)
I’
=
&-l(v,-Vr)
*here V’ and I’ are the “ordinary” voltage and current vectors which would actually be measured at the
Ports, and RO is the normalizing matrix. Commonly,
the vectors are normalized either with respect to the
input impedances such that the normalizing matrix is
chosen to be
1
Ro = a[zs(s)+ zs(-s)],
(7)
or with respect to unity such that R, = U where U
is the identity matrix. This is equivalent to assuming
unit input resistors.
and’ V, are fictitious voltages
representing incident and reflected voltage vectors.
The normalized port-voltage and port-current vectors, V and I , are defined as
where
v
=
&-lv/
I
=
6 s .
Therefore, the incident and reflected voltage vectors
(4) satisfy
v i = -1 v + I )
For LTI n-port networks, the impedance matrices have
interesting properties relating passivity and positive realness [7, 61.
Scatterine Parameters
Scattering parameters have been used in transmission
line theory and circuit theory since the early 1920’s.
v,
=
2
1
2
v -I)
4
where
= v,.
The scattering matrix, S, is defined such that
v,
1209
= sv,
Fezt = M X + P X + ICX
Text =
J e +P i +K 6
where
0 5 ISijl L: 1
(13)
A scattering parameter with a magnitude of 1 indicates that all incident power is completely reflected.
Conversely, a scattering parameter with a magnitude
of 0 indicates that all incident power is completely absorbed by the network.
If the normalized impedance matrix, Z,is defined
such that
=ZI
(‘4)
then
v
S = (2+ U)-’(Z - U ) = ( 2 - U ) ( Z + U)-’
Modeling of Flexible Structures
The flexible beam is a typical example of a flexible
structure and has been widely used for flexible structure research. Therefore, the flexible beam will be used
throughout this discussion as an illustrative example.
(15)
X
where U is the identity matrix. This is equivalent t o
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(18)
(19)
S = (Z’+Ro)-’(Z’-Ro) = (Z’-Ro)(Z’+Ro)-’ (16)
where 2’is the “ordinary” open circuit impedance matrix.
Circuit analogies of mechanical systems have existed
for many years. From the beginning of the study
of electricity, electrical quantities such as inductances
and capacitances were described in terms of mechanical quantities such as springs and masses. When the
history of mechanical and electrical systems is considered, the complementary progress of the two areas is
evident.
Typical examples of circuit analogies for mechanical systems are shown in Table 1. Using these circuit
Electrical
voltage, V
current, I , q
charge, q
Mechanical
(Linear)
velocity, v , x
force, F
momentum, P
Mechanical
(Rotational)
angular
velocity, w , 0
torque, r
angular
The Euler-Bernoulli beam equation for a onedimensional beam is given by
where E is Young’s modulus, I is the area moment
of inertia, p is the density of the material, and A is
the cross sectional area of the beam. In the EulerBernoulli model, v ( z , t ) is defined t o be the transverse
displacement of the beam at position x along the beam
and at time t . Also, a translational force, f(x,t), is
defined as the total external force acting on the beam
as a function of position and time. This model assumes
that the beam is homogeneous and the deflections are
small. The model also ignores rotary inertia, shear
effects, and gravity.
The natural response of the beam can be found by
considering the unforced system where f(x,t ) = 0. Using separation of variables, the solution is assumed to
be of the form
v(z,t)= R ( z ) T ( t ) .
(21)
The solution t o the spatial ODE is
R ( z ) = a sin lcz + b cos k z + csinh lcz + d cosh lcz (22)
where a , b , e, and d are constants to be determined
from the boundary conditions, and
analogies, the analogous systems of equations for electrical and mechanical systems are:
Ie,t
= CV+
(‘7)
(23)
is real.
12 10
Applying appropriate boundary conditions, the spatial solution for the clamped-free beam is found t o be
R ( x ) = d (cosh kx - cos k x - cr(sinh k x - sin kx))
(24)
where d is the constant from Equation 22. Assuming
zero shear force at the tip of the beam, the frequency
generating equation is found t o be
cosh knL cos k,L
+1=0
(25)
where n = 0 , 1 , 2 , 3 , .. .is the mode number. This equation can be solved for k , and the angular frequencies,
w,, can be determined from
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w,
=ki
:/
The impedance for the beam is defined usin velocit
and force as the mechanical equivalents of vo tag
current, respectively:
ff
(34)
Evaluating at the tip of the beam, x = L , the impedance is
Z(L) = EjIw
k3
[
sin kL cosh kL - cos kL sinh kL
1 +coskLcoshkL
Recall that k is a function of frequency as shown in
Equation 23. Thus, this expression for the impedance
is an irrational function in w .
-&=,
The impedance found in the previous section can be
used to find the scattering parameter. The scattering
parameter evaluated at the tip of the beam is given by
S(L) =
1
R n ( z )= - (coshk,z - cosk,z - gn(sinhknz - sink,x))
Jz
(27)
+
+
cosh k , L cosk, L
(28)
sinh k , L sin k , L '
These spatial solutions are orthonormal such that
u,
1'
=
R i ( z ) R j ( z ) d x= 6ij
s=-j w - c
j w 3- c
A?
The impedance of a beam can be used to model the
system. Using an impedance model, several circuit and
network synthesis methods may be applied in the design of structures and controllers.
Using the circuit analogies of mechanical systems
from Table 1, the impedance parameters for the mechanical systems [8] will be defined in terms of the
electrical analogies.
A lateral shear force f applied to the beam imposes
the boundary condition
= -EI- a3v
ax3
&
Using Equations 21 and 24 where d =
and taking
the Fourier transform with respect to time,
JI;
[sin kx - sinh kz
+ u(cosh kz + cos kz)] n ( w )
(31)
where V , F , and 0 are the Fourier transforms of v, f,
and T , respectively.
The lateral velocity of the beam in the frequency
domain is found by taking the Fourier transform:
.T
it --+ j w v
= jwR(z)n(w)
(32)
j
w
= -[cosh kz - cos k z - u(sinh k z - sin kz)]n ( w ) ( 3 3 )
dz
+
(37)
where
V i , j = 0 , 1 , 2 , 3 , .. .
ImDedance Parameters
F=-
Z(L) - R
Z(L) R
where R is the real para-hermitian part of the source
impedance. Using Equation 35, the scattering parameter can be written as
C = REIk3
(29)
f
(35)
Scat terine: Parameters
-
By using these values of k and w and letting d =
the normalized spatial solutions are found from Equation 24 t o be
where
].
+
1 cos kL cosh kL
sinkLcoshkL - coskLsinhkL
is real. Thus, the scattering parameter is all-pass with
unit magnitude. Like the expression for the impedance
parameter, this expression for the scattering parameter
is an irrational function of w . The scattering parameter
characterizes the scattering properties of the beam at
the tip.
Cantilevered Beam Example
Consider the cantilevered beam shown in Figure 2. As
an example, the parameters given in Table 2 for an aluminum beam have been used. Assuming a unit source
impedance ( R = l), frequency responses of the open
circuit impedance and the scattering parameter have
been generated numerically using a FORTRAN program. The phase variation of the impedance is shown
in Figure 4. The phase of the scattering parameter is
shown in Figure 5. Modal frequencies for the first ten
modes of the beam are shown in Table 3.
As expected, the scattering parameter has unit amplitude over the entire frequency range. This indicates
that all energy applied to the structure over all frequency ranges is completely reflected-as expected for
a cantilevered beam. Note that the scattering paranieter has zero phase at the resonant frequencies. This
indicates that the reflection is in phase with the driving
force at the tip which results in a standing wave.
12 11
inductance matrix, but is the matrix of the reciprocals
of the inductances; that is,
Table 2: Parameters for Aluminum Beam
The upper left element in the impedance matrix, 211,
corresponds t o the impedance observed at the tip of the
beam.
The impedance matrix can be used t o find the scattering matrix from
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S(s) =
Frequency (Hz)
P D E I FEM
0.0988
0.0988
0.6192
0.6193
1.7339
1.7350
3.3977
3.4054
5.6167
5.6492
+
where R is the source impedance. Like the impedance
matrix, the scattering matrix is composed of rational
functions of s. The upper left element of the scattering
matrix, S11, characterizes the scattering properties of
the beam at the tip.
Table 3: Modal Frequencies of Beam from EulerBernoulli and Finite Element Models
Mode
Number
1
2
3
4
5
Z(S) - R
Z(s) R
J
Circuit Eauivalents
4
Finite Element Model
As mentioned previously, finite element modeling is
another common method for modeling flexible structures [9, 101. The equations of motion for the beam
can be represented as
(39)
where M and I< are the mass and stiffness matrices of
the structure, respectively. Fezt is the external force
applied t o the nodes. The vectors $c and qc are the
nodal acceleration and displacement vectors, respectively.
The impedance parameters can be derived from the
finite element model as they were from the PDE model.
Using the mass and stiffness analogies of capacitance
and inverse of inductance, the impedance matrix is defined as
Z(s) = (sM
+
4).
(40)
where s is the Laplace variable of complex frequency.
Thus, the impedance matrix is composed of rational
polynomials in s. Note that in the matrix case, the
analogy for the stiffness matrix is not the inverse of the
Figure 3: Circuit Equivalent of Single Element
Usin.g the consistent mass matrix [9], the circuit equivalent of a single finite element is shown in Figure 3.
The values for each capacitor and inductor are shown
in the figure where
I<,
I<l
=
pAh
420 ’
h3
= EI ’
(43)
(44)
and the other parameters are as defined in Table 2.
From left to right, the node voltages of this circuit are
analogous to the generalized nodal velocities of the element: 6,6, ir’, and
respectively. Note that the two
ports on the left correspond the the left nodal variables
and the two ports on the right correspond t o the right
nodal variables.
In the circuit shown, some capacitances and inductances are negative. This indicates that active elements
1212
e’,
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Fmqucncy (Hz)
Figure 4: Phase of Impedance Parameter of Cantilevered Beam at Tip (PDE)
Figure 6: Phase of Impedance Parameter of Cantilevered Beam at Tip (FEM)
must be employed to implement this realization. The
impedance matrix of the element, however, is positive
real. Thus, the circuit is passive and a circuit realization exists which uses only passive elements.
Imposing the boundary conditions for coupling elements together is analogous to connecting the rightmost ports of one element to the leftmost ports of the
next element.
phase of the impedance is shown in Figure 6. The
phase of the scattering parameter is shown in Figure 7.
Modal frequencies for the first ten modes from the finite element model are shown in Table 3. These frequencies are very close to the frequencies obtained from
the Euler-Bernoulli beam model as shown in the same
c table. The lower frequencies are nearly identical.
Since the FEM is a finite dimensional approximation
of
the PDE model, the models are expected to match
Cantilevered Beam Example
very closely. Indeed, the frequency responses shown in
Consider again the cantilevered beam example. Unit Figures 6 and 7 closely match the responses shown in
source impedances were assumed such that R = 7.7, Figures 4 and 5 .
where U is the identity matrix. Frequency responses
The amplitude of the scattering parameter from the
Of zll(s)
and sll(s),
corresponding to the
FEM is also unity over all frequency ranges as expected
parameter and the scattering parameter at the tip of and discussed previous~y.
the beam, have been generated using MATLAB. The
Figure 5: Phase of Scattering Parameter of Cantilevered Beam at Tip (PDE)
1213
Figure 7: Phase of Scattering Parameter of Cantile'vered Beam at Tip (FEM)
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Component Mode
Circuit analogies of large flexible structures also have
interesting applications to the well known component
mode synthesis (CMS) methods [11, 12, 13, 14, 15,
16, 171. In CMS, a large flexible structure is broken
into components or substructures. Each component
or substructure is typically modeled using finite element modeling techniques. The order of each of these
component models is reduced using the assumed-modes
method or the Rayleigh-Ritz method with constraint
equations. The reduced order components are then
mathematically coupled together with interface compatibility equations t o form a reduced order model of
the entire structure.
The basic idea in CMS is to find a set of component
modes for each component that make the greatest contribution t o the modes of the composite structure. Several methods exist t o select these modes. Perhaps the
most widely known methods are the Craig-Bampton
method, the MacNeal-Rubin method, and the BenfieldHruda method.
The Craig-Bampton method [12], a refinement of the
original CMS method proposed by Hurty [ll], uses
fixed-interface normal modes and constraiiit modes.
Fixed-interface normal modes are the natural modes
of vibration which result when all of the component
interface nodes are fixed. These are the modes which
result when a single component is allowed to vibrate
freely but all of the attached components are held
rigid. Note that there are an infinite number of normal modes. Thus, in practice, the set of normal modes
is truncated and some low frequency subset selected.
The constraint modes are the displacement or shape
functions of the component resulting from a statically
imposed displacement at a single component interface
node while keeping all other interface nodes fixed (rigid
with zero displacement). Thus, the fixed-interface normal modes and the constraint modes are linearly independent and the mode set comprised of these modes
can be used t o describe the motion of the component.
The MacNeal-Rubin method [13, 141 uses freeinterface normal modes and residual modes. Freeinterface normal modes are the natural modes of vibration which result when all of the component interface nodes are allowed to move freely. Again, note that
there are an infinite number of normal modes. Thus, in
practice, the set of normal modes is truncated and some
low frequency subset selected. Attachment modes are
the static displacement or shape functions which result
when a unit force is exerted on a single node and all
other nodes are free. The residual modes are basically
attachment modes which are linear combinations of the
truncated normal modes representing the contributions
of the truncated modes to the component motion.
The Benfield-Hruda method [15] uses loadedinterface normal modes.
Loaded-interface normal
modes are the natural modes of vibration which result
when the free component interface nodes are loaded
with representative mass and stiffness contributions
from the attached components. This method generates
component modes which more closely resemble the system modes by including approximate dynamic effects
of the remaining components.
Isolated Boundary Loadinn
Isolated boundary loading developed by Young [18] is
a component modeling method which is based on the
boundary loading ideas of the Benfield-Hruda method.
Again, dynamic contributions of other components are
included in the model of a given component.
In isolated boundary loading, the interface loading
matrices are merely the stiffness and mass matrices
corresponding t o the boundary nodes of the adjoining
component. The modified boundary submatrices are
the same for each of the components involved and may
be found either by loading the first component onto the
second or by loading the second onto the first.
The coupling of substructures using isolated boundary loading is slightly different than the coupling using
CMS techniques. Rather than using boundary compatibility conditions, isolated boundary loading employs
superposition to couple the components at the boundary. Bssentially, the equation of motion for the boundary of the coupled structure is found from superposition of the forces and motions at the boundaries of the
individual components A and B such that
AB
Qi
=
fi””
=
QA + Q ?
fd” +fi”.
(45)
(46)
Using isolated boundary loading for component
modeling and this superposition-based approach for
substructure coupling, the coupled model is the exact
model for the composite system.
Circuit Aiialorries
The CMS methods which have been presented and the
issues which are addressed may be analyzed using circuit analogies. The components or substructures of
the system can be represented as subcircuits or subnetworks within a larger circuit or network. Coupling
substructures is analogous to connecting circuit networks as shown in Figure 8.
The ports of an n-port circuit are equivalent to the
interface nodes of a component. Internal nodes of a
circuit are equivalent to the interior or internal coordinates. Ports which are electrically shorted to ground
represent fixed-interface nodes. Similarly, ports which
are electrically open represent free-interface nodes.
Thus, fixed-interface normal modes of the component
are equivalent to the circuit modes which exist when
1214
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Figure 8: Structure/Circuit Coupling Analogy
all ports are shorted t o ground. Free-interface modes
are equivalent t o circuit modes when all ports are open.
A constraint mode is basically the DC voltage profile
of the circuit when a unit voltage is applied to a single port while all other ports remain grounded. An
attachment mode is basically the DC voltage profile of
the circuit when a unit current is applied t o a single
port while all other ports remain open. Rigid body
modes are equivalent t o the DC steady-state response
of the circuit when dealing with rigidly connected substructures. Models for interconnected multibody structures such as a multiple-link manipulator are nonlinear.
These can be handled with some nonlinear circuit elements.
CMS from the circuits perspective reduces to finding
a set of circuit modes for each subnetwork that make
the greatest contribution t o the modes of the composite
network. The number of internal nodes used in the
circuit is reduced and the capacitances and inductances
of the subnetwork are modified t o retain some of the
properties of the truncated modes.
The previously discussed CMS techniques can obviously be applied t o the circuit equivalents derived
from the finite element model by merely replacing the
mass and stiffness matrices with the capacitance and
inverse of inductance matrices, etc. A more interesting
issue is that of boundary loading as addressed with the
Benfield-Hruda method and isolated boundary loading
in controlled component synthesis.
From the Benfield-Hruda method, the concept of
loaded-interface normal modes is developed where a
component is loaded with approximate mass and stiffness properties of the adjoining components. This loading is equivalent to loading the corresponding subnetwork with capacitance and inductance properties of the
adjoining subnetworks. The Benfield-Hruda method
does not provide intuitive physical insight into what is
happening other than the fact that these properties are
being projected onto the component.
From the circuits perspective, isolated boundary
loading attaches all adjoining subnetworks to the principal subnetwork. Essentially, the ports or nodes of
the adjoining networks are connected to the ports of
the principal subnetwork. All parallel and series circuit element combinations are reduced to their equivalent elements. Thus, only the values of the circuit
elements attached t o the ports ar
ements attached only t o interi
After these element values are changed, t
responding t o the adjoining components
so that only the “original” no
principal subnetwork are retained.
An alternative approach t o these loading methods
might be to reduce the adjoining subnetworks to their
Thevenin or Norton equivalents or some circuit approximation of these equivalents. To generate these approximations, any of the several circuit approximation
techniques could be used. The equivalents could then
be connected to the principal subnetwork to determine
a mode set which closely resembles the modes of the
composite system. This method of loading provides
insight regarding how the loading is accomplished.
Cantilevered Beam Example
The cantilevered beam example which has been considered previously is now assumed t o be composed of
two identical components with the same material properties as before. Thus, the parameters for the components are as specified in Table 2 with the exception
that the length and mass are halved.
Figure 9: Input Impedance at Tip of the Coupled Cantilevered Beam
Figures 9 and 10 show the impedance and scattering
parameters for the coupled structure using the CraigBampton mode set. These impedance and scattering
parameters are very close to the impedance and scattering parameters from the PDE model and the full FEM
as shown in Figures 4, 5, 6, and 7. Modal frequencies
for the first eight modes of the coupled structure using the Craig-Bampton method are shown in Table 4.
These frequencies are very close t o the frequencies obtained from the Euler-Bernoulli beam model and the
FEM a s shown in Table 3. The lower frequencies are
identical to those from the FEM.
1215
Figure 11: Cascaded Circuits/Structures
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defined such that
Figure 10: Scattering Parameter at Tip of the Coupled
Cantilevered Beam
Table 4: Modal Frequencies of Beam Using CraigBampton Method
4
5
6
0.0988
0.6193
1.7353
3.4195
5.6818
8.6777
where the superscripts i and r denote incident and reflected voltages, respectively. SA and SB are the scattering matrices for subnetworks A and B , respectively.
V2 and Vb represent the port voltage vectors for the
m ports of A and B , respectively, that are connected
together. VI represents the port voltage vector for the
n ports of A that are not connected to B but may be
connected to some other network or structure. The
scattering matrix for A can be partitioned as
=
[
s
1
1
s12
SZl
s
2
2
1
(49)
wherg the submatrices have the appropriate dimensions.
If the two subnetworks are cascaded together, the
scattering parameter, S, for the resulting n-port network can be found such that
vy = sv;.
By cascading the two networks and imposing the appropriate boundary conditions, the scattering matrix
can be written as
Scattering Parameter Control Design
s = S11 + sIZsB(I - S22sB)-1S21
As mentioned previously, scattering parameters are
closely related to power transfer properties of a network
or structure. Often, it may be important to control the
scattering of energy through the structure. Therefore,
scattering parameter design is of great interest.
This type of design objective differs fundamentally
from the approach of adding damping to a structure
or minimizing motion of a structure. The actual scattering of energy throughout the structure may be considered and controlled. Disturbances may be reflected,
trapped, redirected, or absorbed to some extent using
scattering parameter design.
Consider the cascaded networks shown in Figure 11.
Assume that subnetwork A is a n m-port network
and subnetwork B is a m-port network. In general,
subnetworks A and B may be cascaded as shown where
the m ports of B are connected to m of the ports of
A . The scattering parameters for the subnetworks are
+
(51)
assuming that SB # ST; such that ( I - S ~ ~ S Bex)-~
ists.
If subnetwork B is taken to be a controller for a
structure represented by subnetwork A, then SB may
be designed for desired properties. For example, suppose no reflection is desired at the port such that S = 0.
B would be designed such that
SB = [STiSiiSF:Szz
+ I]-'
[ST~SIIST:]
(52)
assuming that ST; and ST: exist. For a system with
# nz, S 1 2 and S 2 1 are not square and the inverses do
not exist. Even if n = m and these inverses do exist,
causality and realizability problems still may arise.
However, if some reflection is allowed such that the
scattering parameter is S, B could be designed to have
the scattering parameter
n
12 16
SB = [SGI(S - S11)S,;~S22
+ I]
-I
[S,l(S - Sll)S,-,l].
(53)
Perhaps it is important t o minimize th.e reflections
over some frequency range. It should be noted that
the model of the structure is passive, linear, and timeinvariant. Thus, the scattering matrix is bounded-real
and the scattering parameter for the cascaded network
given in Equation 51 can be minimized over the desired
frequency range using H2 or H , design techniques.
The key idea is t o make IS(jw)l small over some desired frequency ranges. A weighting function,
may be chosen such that IW(jw)l is large over the desired frequency ranges where the scattering parameter
is to be minimized. The maximum of I W ( j w ) S ( j w ) l
over all w can be minimized over SB. The maximum
over all w can be expressed in terms of the H,-norm
defined such that
Controlled ComDonent Svnt hesis
Controlled component synthesis (CCS) is a control design method in which controllers are designed for individual components based on component models [18,
191. This decentralized control strategy extends the
component mode synthesis modeling concepts into control concepts by allowing component controllers based
on component models to be developed independently.
The interlocking control concept is one of the key
ideas behind CCS. With interlocking control, collocated sensors are placed at the internal boundary coordinates of each component. The component controllers
are designed to minimize motion at the internal boundary nodes. One or more controllers are inserted near
the substructure interfaces. By minimizing the internal boundary motion, the transmission of disturbances
between components is suppressed.
Although any technique may be used for controller
design, the linear quadratic optimal regulator (LQR)
is perhaps the most convenient [18]. The weighted internal boundary states are t o be included in the cost
function with the weighted control inputs. Assuming
velocity measurements, the output for component A
is defined to be yA = ifbecause the sensors are located at the internal boundary coordinates. The control inputs are uA =
because the actuators are also
lecated at the internal boundary coordinates.
Using the LQR approach, the control law for component A is found t o minimize the quadratic cost function
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2736
w(~),
(54)
where 8 denotes the largest singular value.
Therefore, the objective is to find SB which, if possible, achieves
where SB stabilizes the scattering properties of the
structure in some sense and is the scattering matrix
of a realizable network or structure. If achieving infimum is impossible, perhaps it is possible to come arbitrarily close t o infimum. This problem is similar to
the four-block problem from H, theory. It should be
noted that in the FEM case, the scattering parameters
of the structure are rational matrices in s. In the PDE
case, however, the scattering parameters are complex
irrational functions of w .
sensitive
A
disturbance
sources
Figure 12: Configuration for Disturbance Isolation
Several other similar problems may be formulated.
The network shown in Figure 12 illustrates how a controller might be used to isolate disturbances from sensitive areas of the structure. In this case, the scattering
parameters representing the scattering properties from
the disturbance ports t o the sensitive area ports should
be minimized in some sense. Conversely, the scattering
parameters representing the scattering properties from
the sensitive area ports to other insensitive ports may
be maximized t o allow disturbances to escape to the
insensitive areas. Shaping the scattering parameters
as desired can be a powerful tool for vibration control
of large flexible structures.
ft
In general, a state feedback control law could be found
to minimize this performance criterion. However, optimal output feedback can also be employed to achieve
good results and is easier to implement than state feedback. CCS controller designs are inherently decentralized since only the component outputs or states are
required for the component control law.
The basis for the CCS method is closely related to
the scattering parameters of the structure. The fundamental idea in CCS, as mentioned before, is to minimize the transmission of disturbances between components. In other words, certain component scattering properties are desired. With this concept in mind,
scattering parameter design techniques such as those
discussed previously may be used at the component
level. Thus, the controllers for each component would
be designed to achieve desired scattering properties directly.
Conclusions
In this paper, circuit analogies of mechanical systems
have been used to establish a framework in which large
12 17
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2736
flexible structures may be analyzed. It has been established that impedance and scattering models of flexible
structures can be derived directly from either partial
differential equation models or finite element models of
the structures. A cantilevered beam example has been
used to parallel these discussions and illustrate the concepts. Component mode synthesis has also been discussed in terms of circuit analogies. It has been shown
that issues related to component mode synthesis can
be addressed in terms of circuit analogies.
Using circuit analogies of flexible structures, the
scattering properties of structures are characterized in
terms of the scattering parameters. The circuit analogy method of modeling large flexible structures can
provide additional insight into system characteristics
such as scattering properties that conventional modeling techniques cannot usually provide.
This knowledge can be exploited in controller design.
The scattering parameters may be “shaped” as desired
to achieve desired scattering properties. Controllers
may be designed using H2 or H, design techniques
to minimize scattering over certain frequency ranges
of interest. This type of problem has been formulated
and is very similar to the familiar four-block problem
from H, theory.
Controlled component synthesis was presented. Ties
have been drawn between the fundamental concept behind interlocking control and the scattering properties.
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