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Adaptive Control Strategies for Vibration Suppression in Flexible Structures
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653
A.M. Annaswamy
Department of Aerospace and Mechanical EngineeringBoston University
Boston, MA 02215
ana
D.J. Clancy
Missile Systems Division
Raytheon Company
Tewksbury, MA 01876
study if the adaptive system which is stable in the absence of
these perturbations, will behave satisfactorily in their presence.
The results obtained using this approach have been grouped under the heading of Tobust adaptive c o n t ~ o P - ~Parallel
~.
results
have also been obtained for systems with multi-inputs and multio u t p ~ t s ' ~ -as
~~
well
, as for stochastic systems16. In all cases, the
aim has been to develop conditions for global boundedness of the
underlying adaptive system.
Requiring that the order of the system be known is quite a restrictive condition, especially in the context of flexible structures.
Even if this number is known, often, it tends to be quite large. As
a result, it is not feasible to implement the requisite controller,
since the order of the controller depends directly on the order of
the system. Considerable effort has been directed towards establishing satisfactory behavior of adaptive systems using low-order
controllers. Despite this, the existing results still prove to be
Active control of large flexible structures is an important topic
inadequate for flexible structures.
that has been studied by researchers over the past two d e ~ a d e s l - ~ .
The control of flexible structures is a highly researched area
Due to the large size and light weight of these structures, proband has received considerable attention during the past decade.
lems related to their control assume paramount importance, and
In fact, optimal control theory has served as the cornerstone for
control issues such as vibration suppression, and shaping need to
designing control systems for flexible space structures17. Modal
be adequately dealt with. Many of these structures possess indecoupling procedures have demonstrated robustness to paramherently low damping, thereby necessitating control procedures
eter uncertainties, along with eliminating spillover effects from
that will introduce damping using active means.
residual modes, but require that each controlled mode have arL
In addition to being poorly damped, the elastic modes of flexindividual actuator associated with it'8. The use of positivity
ible structures are quite often large, and possess uncertainties. It
concepts is another common approach used for flexible structures.
is well known that modal models of large structures derived using
It is well known that the use of colocated velocity output feedfinite-element methods contain about 10% errors in both modal
back results in a stable design without requiring high order truth
frequencies and mode shapes4. Active control methods based on
models, modal truncation, or accurate knowledge of the modal
finite-dimensional techniques unavoidably introduce modal trunfrequencies or modal
However, in such cases, since
cations at some point. Due to these reasons, not only are the
the underlying requirement is the positive realness of the sysmodal frequencies and the modal shapes unknown, but since diftem's transfer function, it implies that the controlled outputs
ferent modes may be excited during different operating condimust be from either velocity sensors, or velocity with scaled potions, the total number of dominant modes in any given problem
sition sensors, which makes position following impossible. LQGmay also be unknown. Structural modifications, failure of system
based controllers have demonstrated superior performance over
components, changes in the operating environment, and actuatorcompensators using colocated actuators with position and velocand sensor-dynamics are other sources of significant uncertainties.
ity sensors, along with requiring fewer control inputs to meet
Since large deformations are invariably present in these problems,
performance requirementsz1. Control systems synthesis in the
the effect of nonlinearities cannot be ignored. The use of adaptive
presence of modeling uncertainties and parameter errors has been
controllers, which can cope with the uncertainties and deliver the
accomplished using LQG/LTR loop shaping methods. LQG/LTR
required performance, is therefore qf considerable interest and
controllers are excellent a t alleviating spillover effects common to
importance for the control of flexible structures.
flexible space structures, but can be overly conservative when
The field of adaptive control has evolved over the past thirty
satisfying robustness constraints, which tends to produce infeyears and grew out of attempts to control systems that are parrior performancez2. Modifications to the LQG/LTR robustness
tially known. A significant part of this field has addressed dytests have been proposed which have resulted in less conservanamic systems that have parametric uncertainties. The adaptive, higher performance controllers. Unfortunately, LQG and
tive control of linear time-invariant plants with unknown transLQG/LTR controllers designed using reduced order space strucfer functions has been studied at length and i s currently well
ture models do not automatically guarantee closed-loop stability
understood6. The major landmark in this field is the proof of
of the actual space structure. In addition, fixed and LQG-based
global stability of a class of adaptive systems that arise while
controllers are extremely dependent on the accuracy of the plant
controlling such plants, which was vstablished in 1980. This remodel, i.e., modal frequencies and mode shapes, along with the
sult holds under certain conditions regarding the plant transfer
structure and order of the reduced order plant model used to defunction, one of which is that its order must be known. Also, the
velop the compensatorz3. All of the above indicates that control
order of the controller that *stabilizes the plant is almost twice
methodologies involving fixed compensators may not be adequate
as large as that of the plant. Since 1980, several extensions to
for the control of flexible space structures, and that new and
adaptive control theory have been attempted to include timeinnovative strategies, which utilize adaptive compensation with
variations, unmodeled dynamics, and nonlinearities in the plant.
time-varying parameters, may be warranted.
The P-rnach used was to model these as perturbations, and to
Flexible structural systems are high dimensional and lightly damped,
and invariably contain significant uncertainties in their dynamic
behavior. Adaptive controllers, which are capable of overcoming
such uncertainties and delivering high performance by providing
a time-varying compensation on-line, are therefore desirable for
such systems. In this paper, we present a new adaptive controller which can globally stabilize a class of flexible structures.
This controller is applicable whether position measurements, rate
measurements, or combinations thereof are available, as well as
for colocated and non-colocated actuator-sensor pairs. The superior performance generated using such controllers is demonstrated
using two practical structural systems.
Copyright @ 1991 American Institute of Aeronautics and
Astronautics, Inc. All rights reserved.
501
.
I
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653
Adaptive control of flexible structures has been considered
elsewhere in [24-261. The approach that has been used in these
papers is based on what is termed as the Command Generator
Tracking (CGT)
For adaptive control based on the
CGT method to result in global stability, several assumptions
have to be satisfied". Of these, the most restrictive one, is that
the outputs used for feedback must include rate measurements.
As in the case of the fixed compensators, this arises because an
underlying transfer matrix in the system is required to be strictly
positive real. Other conditions that have to be met which also
prove restrictive are that the plant has to be (i) invertible, (ii)
stabilizable using output feedback, and (iii) minimum phase, (iv)
that no transmission-zero of the plant can be equal to an eigenvalue of the reference model, and (vi) that the model outputs
must be generated from the solution of homogeneous differential
equations. Some attempts have been made to relax these conditions, but invariably they have led to other assumptions which
are either more restrictive, or unverifiable prior to the application in a practical problem. While this theory has been applied
to flexible structures, many of the assumptions mentioned above
have either not been shown to be satisfied in the structure, or
clearly invalid. For instance, when rigid body modes are present,
pure velocity feedback cannot be used due to assumptions (i),
(iv), and (v). In [ 2 5 ] ,this difficulty was avoided by introducing
an inner-loop augmentation, i.e., an inner loop control gain matrix is added to alter the modal characteristics of the plant and to
convert the rigid body modes into fbite frequency modes. It was
claimed that this matrix can be implemented without having accurate knowledge of the plant. Clearly, this approach ignores the
destabilizing interaction that has been known to occur in many
instances between rigid body modes and the flexible modes of a
structural system. If the outputs from velocity -I- scaled position
sensors are used for feedback, even though the rigid body modes
are accomodated, position following cannot be ensured. Most
importantly, in almost all results related to the CGT method,
velocity sensing is a must, and has even been claimed to be a
necessity.
In this paper, we develop new adaptive control strategies for
vibration suppression as well as shape control in flexible structures with unknown mode shapes, modal frequencies, and number of dominant modes. These adaptive controllers do not require
prior knowledge of the order of the system, are of low order, have
a simple structure, and guarantee stability as well as satisfactory
performance. The main result in this paper is that regulation and
tracking can be achieved using these new adaptive controllers for
flexible structures whether (i) position measurements, (ii) rate
measurements, or (iii) a combination of both, are available. The
result also implies that satisfactory performance can be achieved
not only with colocated actuator-sensor pairs, but also with a
class of non-colocated actuator-sensor pairs.
In section 2, we discuss the underlying dynamics of a flexible
structure. We consider a linear finite-dimensional model, whose
parameters correspond to the modal frequencies and the mode
shapes. We then derive the input-output representations for different cases that depend upon the number, the locations, and
the types of actuators and sensors. In section 3, we present the
new adaptive controller, and state the main result of this paper
in Theorem 1, where the global boundedness of all signals in the
system is established. In section 4, the application of the new
controller to flexible structures is considered. It is shown that
the main result in Theorem 1 leads to vibration suppression as
well as shape control in flexible structures. Due to space limitations, all results are stated without proofs. For the latter, we refer
the reader to [30]. Finally, in section 5 , the performance of these
controllers is evaluated in the context two flexible structures, an
experimental control facility a t JPL, which has 30 modes whose
frequencies lie between 0.1 and 5 Hz., and a flexible space station
with 2 rigid body modes and 4 flexible modes31.
The Dynamic Model
2.
For a flexible space structure with small displacements, the linear
dynamic equations describing the system are given by
f -I- diag (2(iwi) i.
+ diag (w!) r
= Bau + Bdv
[
Y =
[;I
c:a
(1)
where T is the vector of n modal coordinates, and wi and (; are
the natural frequency and damping ratio of the ith mode, respectively. u E IRna is a vector of actuator inputs, v E End
is a vector
is a vector of poof external disturbances, and y E I R n p + n c + n w
sition, velocity plus scaled position, and velocity sensor outputs.
The input matrices are given by
[
Bo =
b:],
Bd
b,
=
[ '1
dn
where bi E ELna,d ; E I R n d , i = 1,. .. ,n,are the mode shapes evaluated a t the no actuator locations, and n d disturbance locations,
respectively. The output matrices are defined as
where hp; E E.'+,
hc; E lR."c, h,; E IRnV,i = 1,...,n,are the
mode shapes a t the np position sensor locations, n, scaled position
velocity sensor locations, with Q > 0 as the scaling factor,
and nu velocity sensor locations, respectively. It is assumed that
the system parameters in (1) as well as the number of modes
n are unknown, and that an arbitrarily small, usually smaller
than 0.005 in flexible structures1, but nonzero amount of modal
damping ( i is present.
Depending on the number, the locations, and the types of actuators and sensors used, numerous input-output representations
of the system in (1)c a n be derived. Por instance, if the actuators
and sensors are colocated', it follows that
+
h,;
V i = 1, ...,n
= bi
where the subscript z = p , c , or v , depends on whether position,
scaled position velocity, or velocity measurements are used. On
the other hand, if the actuators and sensors are non-colocated but
sufficiently close, the mode shapes at these different locations are
similar. In such a case, the relation
+-
h,;
= (k+q)bi
where
k+ci>O V i = l ,
...,n ( 3 )
may be satisfied. In such a case, i.e., if the locations are such that
Eq. (3) is satisfied, we shall define the actuators and sensors to be
prozimally located. The results developed in this paper are applicable for both colocated and proximally located actuator-sensor
pairs. We now derive the underlying input-output representations
and their properties.
502
A single-input single-output model: Assuming that there
is a single actuator-sensor pair {ui,y i } , and a scalar disturbance
vi a t the actuator location, and ni modes are controllable and
observable, the input-output representation of the model in (1)
is given by
+
= Wpi(B)ui(t) W d i ( s ) v i ( t ) *
'As in Ref. 1321,by colocated aensora and actuolorr, we mean that sensors
and actuators are placed not only at the same physical positions but also
along or about the name axis.
Throughout this paper, the variable 8 will be used to denote the
differentiation operator d / d t . The following conclusions can. be
drawn regarding Wpi(s)
and W d i ( s ) .
v -K< fk,.
(5)
Kri
Thus, when a position controlled dynamic system with rigid body
modes satisfies Eq. ( 5 ) , the underlying transfer function has relative degree two, with zeros inC-. A similar property can also
be derived when scaled position velocity sensors are used for a
system containing rigid body modes. However, the use of velocity
sensors in the presence of rigid body modes leads to undesirable
pole-zero cancellations and, hence, is avoided. For a given flexible structure, the smallness of (Kfi/K,.i)depends on the sum
total of the contributions from all flexible modes relative to that
of the rigid body mode a t the i t h location. If this is small, which
is the case for most flexible structures, the above discussion indicates that the underlying transfer function is minimum phase.
For the class of flexible structures considered in this paper, we
shall assume that if rigid body modes are present, they satisfy
Eq. ( 5 ) .
(4)
+
We note that W p i ( s ) is of order ni 5 n, and has relative degree
(= no. of poles - no. of zeros) two. If a transfer function
W ( s )= x y ! l ( k i / q i ( s ) ) ,where k; > 0 and q i ( s ) = s 2 + 2 ( i w ; s + w ~ ,
it can be shown that30 the zeros of W ( s )are in C-, the open
left half of the complex plane. For both colocated as well as
proximally located actuator-sensor pairs, from Eqs. ( 2 ) and ( 3 ) ,
it follows that the numerator gain hpkbk > 0 for all k = 1,. . . ,n i .
Hence, it follows that in both cases, all zeros of W p i ( s )are inC-.
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> 0 exists
the zeros of Wpi(s) has roots inC-
Case 1. Position MeasuremenkWhen y; corresponds to a position measurement, the transfer function Wpi(s)is given by
+
tively. This implies that an arbitrarily small constant k
such that
Case 2. Scaled Position Velocity Measurements:When yi corresponds to a scaled position
velocity measurement, with a
scale factor Q > 0, the transfer function Wpi(s)is given by
+
Remark 2.1: In all the above cases, except when the output corresponds to a pure velocity measurement, W , ( g ) is a minimum
phase transfer function. This property follows since (i) eachmode
is assumed to have nonzero damping, (E) the numerator gain of
the i t h mode h,;bi is positive, and (iii) the contributions from
any existing rigid body modes are relatively large. All flexible
structures have a certain amount 01 passive damping which justifies (i). Colocated actuator-sensor pairs automatically satisfy
(ii), whereas non-colocated actuator-sensor pairs satisfy (ii) provided that the locations are sufieiently close, i.e., the pair is
prozimally located. This is quantified in Eq. ( 3 ) . Theoretically,
as the number of modelled modes for any system represented by
two non-identical actuator-sensor locations becomes sufficiently
large, W p ; ( s ) can become nonminimum phase, which is pointed
out in [ 3 3 ] . However, for a given number of modes, ni, a finite,
nonzero number of non-colocated actuator-sensor locations can
be found for which the input-output transfer functions retain the
minimum phase property.
Once again, W p i ( s )is of order ni, but has a relative degree unity.
As in case 1, if the input-output pair is colocated or proximally
located, we have that the numerator gain h&bk > 0. Since a > 0,
the zeros of Wp;(s)are in@-.
Case 3. Velocity Measurements:When yi corresponds to a velocity measurement, the transfer function Wpi(s)becomes
As in the previous case, Wpi(s)is of order n;,and has relative
degree unity. Even though h,,kbk > 0 for colocated and proximally
located pairs, this only implies that all but one of the zeros of
Wpi(s) are inC-, with one zero at s = 0.
In all the three cases, the transfer function Wdi(s) between
the disturbance and the output is given by
A multi-input multi-output model: We now consider m
actuator-sensor pairs {ui,y i } , i = 1,. ,m, which include np
position sensors, n, scaled position
velocity sensors, and nu
velocity sensors, with m = np n, t np. The MIMO model can
be expressed as
+
+
where the subscript 2 = p p , or v , and p ( s ) = l , ( s a ) , or
s depending on whether position, scaled position
velocity, or
velocity measurements are used. Therefore, the order and the
relative degree of Wdi(s) are equal to those of Wpi(s).Since the
disturbance can be located anywhere on the structure, the zeros
of W d i ( s ) need not be inC-.
For the dynamic model in Eq. (l),no rigid body modes were
included, i.e., ( i , wi > 0 for all i = 1,. ,n. Obviously, the presence of rigid body modes leads to a modification in Eq. (1) and
likewise in W p i . In this case, when a position sensor is used, the
transfer functions Wr; and Wfi represent the contributions due
to rigid body modes and flexible modes, respectively, a t the ith
output. Eq. (4)is modified to be Wpi(s) = Wri(s) W f i ( s )
where W,i(s) = ( K p ; / s 2 ) ,and
+
+
..
where y ( t ) = [ y l , ..., y,,,IT, u ( t ) = [ u l ,...,urn]*,and v ( t ) =
[VI,. ,v,lT is a disturbance vector. We assume that the locations of the actuators, sensors, and disturbances are such that
..
..
+
The transfer function Wpi(s) of the ith subsystem between u; and
yi will therefore satisfy all the properties discussed in the SISO
case depending on the type of sensor a t the i t h location, for all i =
1 , . ,m. The same holds for W,i(s)
as well. Such a decoupled
system c a n be achieved by choosing the m locations in such a
way that the n; modes present in each Wpi(s)are controllable
and observable only through the corresponding input-output pair
..
{ u i ,V i } *
Hence
In summary, the class of flexible structures that is considered
in this paper is of the form of Eq. (6), where W p ( s )and Wd(s)
satisfy Eq. (7) and the following properties hold.
From case 1, it follows that K f ; > 0, and D j ( s ) and N f ( s ) are
polynomials with roots in C-, of degrees 2 n and 2 n - 2, respec-
( P a ) The relative degree of Wpi(s)
is equal to
(Pl) The order n; 5 n.
503
i
well as proximally located actuator-sensor pairs with either
a pure velocjty measurement, or a velocity scaled position
measurement.
+
(P3) The relative degree of W ~ ( sis)equal to two for both colocated and proximally located pairs when the output measurement corresponds to a pure position measurement.
(P4) In cases 1 and 2, all zeros are in(C-; in case 3, one zero is
at s = 0 and the remaining are in(C-.
(P5) If a rigid body mode is present, it is assumed that its contribution is large relative to those of the flexible modes, i.e.,
Eq. (5) is satisfied.
(P6) I f a disturbance is present, the relative degree of ( W d i ( s ) ) 2
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653
relative degree of ( W p ; ( s ) ) .
S t a t e m e n t of the problem: With the model of the flexible
structure given by Eqs. (6) and (7), the problem is to design a
control input u such that when a disturbance v is present, or if
there is an initial deflection on the structure, the displacements
at various points on the structure settle down to zero as quickly
as possible, Le., limt+m y ( t ) = 0.
We also consider the problem of static shape control where
it is required that the position response at different points on
the structure are displaced by a f i t e amount34. As discussed
in section 4, this problem can be posed so that the output y
follows a desired trajectory ym, where the latter is specified as the
solution of a homogeneous differential equation whose coefficients
and initial conditions are appropriately determined. If the output
error is defmed as e , where e = y, - y m , our aim is to choose the
control input u so that limt+m e ( t ) = 0.
When the assumptions (Al)-(A5) hold, we shall establish that an
adaptive controller exists which ensures global boundedness of all
signals in the system, and leads to command following as well as
disturbance rejection. Without loss of generality, we assume that
k, > 0.
We present two error models, Error Model A and Error Model
B, which illustrate the principle behind the new controller that is
being proposed. Using these error models, we derive two lemmas,
which in turn can be used to establish the main stability result
of this paper.
Error m o d e l A: In all adaptive control systems, there are two
kinds of errors present, the output error e l , and the parameter
error #. While the output error el(t) can be measured at every
instant of time, the parameter error # ( t ) is unknown but can be
adjusted a t every instant. If the adaptive system is such that the
relation between these two errors is of the form
where w ( t ) is a vector-signal present in the system that is accessible for measurement, v ( t ) is a scalar signal that arises due to
disturbances, and W ( s )is a strictly positive real transfer function,
we refer to Eq. (10) as Error Model .A. Lemma A summarizes the
various results that can be derived for this error model.
L e m m a A: In Error Model A given by Eq.
eter # b e adjusted according to the rule
(lo), let the pararn-
r = rT> 0.
i ( t ) = -rel(t)w(t)
ff the disturbance has a finite energy,
3.
A New Adaptive Controller
v E L2
In this section, we consider a single-input single-output linear
+
e l ( t ) and # ( t ) are bounded V t
t-m
lim
e1 E
f?.
e ( t ) = 0,
t+m
lirn
e l ( t ) = 0.
Error m o d e l B: In the context of flexible striictures, a special case needs to be addressed, which arises when only velocity
neasurements are available. In this case, the underlying error
:quation is once again of the form
where u is the control input, yp is the measured output, and
V I is a bounded external disturbance. The transfer functions
between the input and the output, and the disturbance and the
output, are respectively W,(s) = h," ( S I - AP)-' bp =
k,H
and W ~ ( S
= )h,' (SI- A P ) - l d p .The desired trajectory that the
output y p must follow is specified as the scalar output ym of a
homogeneous differential equation
referred to as Error Model B, where all the variables are as defined
in Error Model A with the exception that the transfer function
W ( s )is positive real. Lemma B summarizes the results obtainable from Error Model B.
(9)
where Am is any stable matrix in lR."mx"m.In the absence of
disturbances, the control objective is to ensure that the error el
between the plant output and the model output defined as el(t) =
y p ( t )- y,(t) must satisfy the condition
e l ( t ) = 0 and
if disturbances are present in the plant, then it is only required
that lel(t)l must remain bounded for t 2 t o .
Our main result requires the following assumptions regarding
the plant transfer function:
L e m m a B: Let the state-variable description corresponding to
Error Model B be given as
; =
Ae
+ b(CTw + v )
el = hTe
(13)
where W ( s ) = hT(sI - A ) - l b is positive real. If a matrix P =
PT > 0 exists such that it satisfies the equation
A T P + P A = -hhT-Mo
( A l ) The relative degree '
n of Wp(s)must be known, and '
n 5 2.
(A2) The zeros of W,(s) must lie i n C .
Pb = h
where MO,= MOT 2 0, and the parameter q5 is adjusted as
(A3) The sign of kp must be known.
i ( t ) = -rel(t)w(t)
We also make the following assumptions regarding the disturbance:
r = rT > o
(11)
then v E L2implies that e l ( t ) and + ( t ) are bounded for all t 2 t o
and el E L2.If, in addition, w ( t ) and v ( t ) are bounded, then
limt+m q ( t ) = 0.
(A4) vi E L2.
2
and
tf, in addition, w ( t ) and v ( t ) are bounded, then
time-invariant plant whose order as well as parameters are unknown. The plant is represented by a differential equation
(A5) Relative degree of W d ( s )
2 to,
n*.
R e m a r k 3.1: In the adaptive control literature, the concept of
error models has been extensively used. The use of error models
504
no longer holds, since Lemma A is no longer applicable. In fact,
bounded disturbances can be found for which the controller in
Eq. (16) leads to unbounded solutions [ 5 , Chapter 81. To ensure
boundedness in such a case, modifications to the adaptive law
have to be introduced. For instance, the adaptive laws in Eq.
(16) have to be modified as
enables one to develop an intuitive understanding of the stability
questions that may arise, without a detailed analysis of the overall
system dynamics. These error models have been developed thus
far mainly for the disturbance-free case. In this paper, we have
developed two error models, both of which include a disturbance
as well as a dynamical system which is either strictly positive real
or positive real.
&(t)= -ueo(t)-el(t)y,(t),
Remark 3.2: The results of Lemma A and Lemma B hold provided the disturbance v E L 2 . Disturbances due to measure-
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653
(Al)-(A5), and the model is given by Eq. (9), then a control
input u(.) can be found such that all the signals in the loop are
globally bounded, and
(15)
Remark 3.8: If additional prior information is available regarding the poles and zeros of Wp(s), the controller in Eq. (17) can
be simplified further. In the context of flexible structures, often
the poles (pi's) and zeros ( x i ' s ) of W p ( s )satisfy the relation
The input is given by [Fig. 11
+
u(t) = e ~ ( t ) ~ p ( t )C(t)zrn(t)
e O ( t ) = -7Oel(t)Y~(~)
70 > 0
em(t) = -rmei(t)zm(t)
rm
>o
(16)
k(pi)-F
j=l( x j ) < o
i=l
when n' = 1, and by [Fig. 21
This in turn enables us to simplify the control input as
+
4 ( t ) = -xcw1(t)
u(t)
u ( t ) = Bo(t)~p(t) p(t)wl(t) e$(t)zm(t)
- el(t) ( i ~ : ( t ) ~ : ( t ) zz(t)zrn(t))
e O ( t ) = -7Oel(t)Vp(t)
70 > 0
P ( t ) = -7pel(t)vl(t)
7P > 0
ern(t) = -rmel(t)zm(t)
rm> o
+
+
+
+
+
Remark 3.3: In contrast to the standard adaptive controller
structure used in the literature, we note that the controller in
Eq. (16) is simple, consists of n, + 1 adjustable parameters, and,
most importantly, does not require the knowledge of the order n
of the plant.
Remark 3.4: The control parameters in Eq. (16) can be modified to include a proportional adjustment as
"(t)
= eo(t)Yp(t)
+e W m ( t )
+ eo@)
= eop(t)
eop(t) = -el(t)yp(t)
where p, = x,
(17)
when 71' = 2, where a , ( t ) = ( l / ( s + a ) ) w i ( t ) , Vp(t)(l/(s+a))yp(t),
zm(t)(l/(s ~ ) ) z , ( t ) , and a , ~>, 0.
To facilitate the understanding of the main result in Theorem 1, we provide below a few qualitative remarks. While all
comments are made in reference to the controller in Eq. (16) for
n* = 1, they are equally applicable to the controller in Eq. (17)
which is used when n* = 2.
-
>o
Remark 3.7: It should be pointed out that the assumption (A2),
which requires all zeros of the plant to lie inC-, limits the applicability of this controller. In the context of flexible structures, it is
well known that nonminimum phase transfer functions are quite
common, especially in the case of non-colocated actuator-sensor
pairs33. Extensions of the approach presented here to plants with
arbitrary zeros are currently under investigation.
Theorem 1: When the plant in Eq. (8) satisfies assumptions
c'.
(T
Remark 3.6: Elsewhere in the literature, boundedness has been
established for adaptive systems with relative degree unity. We
are however establishing for the first time that boundedness follows even when the underlying system has relative degree two,
and therefore represents a significant breakthrough in this area.
We now state the main stability result in Theorem 1.
E
-cre,(t)-el(t)z,(t),
:
-
which can be shown to result in boundedness of all solutions. A
similar modification to the adaptive laws in Eq. (17) will assure
boundedness when n* = 2 as well.
ment noise, or actuator errors invariably contain finite energy,
and hence, E L2. However, if disturbances are present due to
other sources of uncertainties such as nonlinear dynamics, such
an L2 assumption may not hold. If these uncertainties are such
that v E Cm, the results of Lemma A and Lemma B are no longer
valid. The adaptive law for adjusting the parameter error 4 has
to be appropriately modified to assure boundedness of the errors
e and 4.
el
e,(t)
-
&(t) = Qmp(t) t ern(t)
OmP(t) = -ei(t)zm(t)
with 00 and 0, adjusted as in Eq. (16). The proof of stability
can once again be established along very similar lines to that in
Theorem 1.
Remark 3.5: The global stability in Theorem 1 follows provided
the disturbance u E L2.If on the other hand, u E Coo,this result
+ 6, and 6 is an arbitrary positive constant3'.
Remark 3.9: All our discussions in this section pertain to SISO
plants. While the result in Theorem 1 can be extended simply to
totally decoupled MIMO plants and with relative ease to diagonally dominant MIMO plants, the problem proves to be nontrivial
for the case when the MIMO plant is symmetric and strongly coupled. It is the latter case that is of interest in a flexible structure.
Work is under progress to determine the least restrictive set of assumptions on the plant transfer matrix under which a stabilizing
controller exists.
4.
Adaptive Control Strategies for a Flexible Structure
In section 2, the dynamic models that describe the behavior of a
flexible structure were discussed at length. It was seen that these
structures have high order, low damping, unknown moaal mequencies and/or modal shapes, and even an indeterminate number of dominant modes. The control of these structures has to
therefore be carried out in the presence of such uncertainties.
The new adaptive controller developed in the previous section is
ideal for realizing the control objectives commonly encountered
in these structures such as vibration suppression and shape control. In this section, we show that indeed the results of Theorem
1 are applicable to the control of a flexible structure, and consider these two control objectives in sections 4.1 and 4.2. In each
case, we show that satisfactory control is achievable whether (i)
only position measurements are available, (ii) a combination of
position and velocity measurements are available, or (Gi)I only
505
uniform settling time, it follows that the real parts of the (2n-2)
zeros are identical to that of 2n- 2 poles which once again leads to
the relation in Eq. (20). In such cases, as pointed out in Remark
3.8, the parameter p ( t ) no longer needs to be adaptively adjusted
and can be fixed as p ( t ) E p , = zc 6,6 > 0. This simplifies the
control input as
velocity measurements are available. Also, we show that these
objectives are attainable using colocated as well as proximally
located actuator-sensor pairs.
4.1
+
Vibration Suppression
In section 2, it was shown that the input-output representation
can be expressed as
y = Wp(s)u
+
Wd(S)Y
(6)
where y , u , v : lR+ -+ IR", and that the locations of the m
actuator-sensor pairs can be chosen in such a way that the transfer matrices Wp(s)and Wd(a) are diagonal. It was also seen
satisfies properties
in section 2 that each diagonal entry W+(s)
(Pl)-(P5) for i = 1,.. ,m. The control objective is to choose an
input u in Eq. (6) so that the displacements in various locations
on the structure settle down to zero as quickly as possible, even
and Wd(s) are unknown. In Theorem 2, we describe
when W,(s)
the adaptive controller that generates the control input which ensures global stability as well as vibration suppression in all the
modes that can be controlled through the m actuators in Eq. (6).
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653
.
Theorem 2: Let the dynamic model in Eq. ( 6 ) denote the inputoutput relation of a flexible structure. Let the m sensors consist
of np position sensors, n, velocity scaled position sensors, and
nu velocity sensors. The transfer matrix Wp(s) can be arranged in
such a way that the sensor outputs yi, i = 1,.. . ,np, correspond to
position measurements, y;, i = (n, l), ..,(n, t n,) correspond
scaled position sensors, and y;, i = (np n,
to velocity
11,. . ,m, correspond to velocity sensors, where m = n,+n, tn,.
Let u = [q,.
..,umIT and y = [yl, . ,ymIT. If
+
+ .
+
.
+ +
..
-zcWi(t) t u;(t)
2, > 0
Boi(t)yi(t) +pi(t)wi(t) - yi(t) ($(t) t @ ( t ) )
-7i~i(t)jii(t)
ri > 0
-rpi~i(t)gi(t)
rpi > 0
(18)
% ( t ) = (Wo(s))~;(t) gi(t) = ( W o ( s ) ) w ; ( t )
1
W.(S)
= a>O
a+a
hi(t) =
~ ( t =
)
$i(t) =
?ii(t) =
..,np,and
for i = 1,.
ui(t) = eoi(t)yi(t)
= -7i~"t)
&i(t)
ri > 0
(19)
+ .
for i = np 1,. .,m, global boundedness of all signals follows
and l i ~ - , - y ( t ) = 0.
Remark 4.1: The adaptive controller suggested in this paper is
applicable to the control of all flexible structures which can be
described by decoupled, linear, finite-dimensional, MIMO models, including rigid body modes as well as lightly damped flexible
modes. Most importantly, the results of Theorem 2 imply that
stable control of flexible structures can be achieved using only position measurements, with either colocated actuator-sensor pairs,
or sufficiently close non-colocated pairs.
Without loss of generality, this input can also be expressed as
In simulation studies, it is seen that such a control input leads to
good performance with reasonable values of BO;.
It is obvious that extensions to strongly coupled MIMO plants
as well as to plants with arbitrary zeros need to be established before the approach presented in this paper can be considered as a
viable practical methodology. We strongly believe that the controller suggested here will result in boundedness even for these
extensions. In fact, in section 5 , we provide simulation studies
of two practical flexible structures that have been studied extensively in the literature, which illustrate that superior performance can be obtained using this controller not only for decoupled
MIMO plants but also for strongly coupled ones.
4.2
Shape Control
We now consider the problem of shape control where it is required
that the displacements at various points on the structure have to
achieve certain steady-state values so as to obtain a static shape.
The desired values for the various points can then be represented
as ym, the output of a homogeneous differential equation, as in
Eq. (8). The problem is then posed as the choice of u in Eq. (6)
so that y, follows ym asymptotically. The results of Theorem 1
once again enable us to determine such an input which is only
a slight modification of that in Theorem 2. An additional term
OZz, is added to the control input, where z, is the state of the
model in Eq. (8), and 0, is a time-varying parameter that is
adaptively adjusted. This is stated in Theorem 3. Once again,
the disturbance Y is assumed to have a finite energy. For ease of
exposition, we assume that Eq. (%a) is satisfied by the flexible
structure.
Theorem 3: Let the dynamic model in Eq. (6) denote the inputoutput relation of a flexible structure, and Eq. (8) specify the
desired response. Let the output y consist of n, position sensors
and nc velocity t scaled position sensors, so that y;, i = 1 , . . . ,n,,
correspond t o position measurements, yi, i = (n, I ) , . . .,(np
ne) correspond to velocity + scaled position measurements. We
define 0, = [Bo;, BZilT, and W; = [yi(t), z f ( t ) l T . If the various
components ui of the control input u are chosen as
+
A Special Case:
Often in many flexible structures, the damping characteristics are
satisfy the relation
such that the poles and zeros of W+(s)
. . ,n,,
for i = 1,.
where pi and zj are the real parts of the ith pole and the j t h
zero. For instance, when there is uniform modal damping, using
simple extensions of the result in [32], it can be shown that
Pi
< zj < P;+I
v i , j = 1,...,n - 1
and
u;(t) =
(20)
B?(t)w;(t)
ei(t) = -r;e;(t)yi(t)
Bmi(t) = -I'miei(t)zm(t)
+
+
.
ri > o
rmi> o
(2%)
for i = np 1,. . ,m, global boundedness of all signals follows,
, ~ = 0.
and l i ~ - ei(t)
which in turn implies Eq. (20). Similarly, when all modes have a
506
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653
In this section, simulation studies of two popular flexible structures are presented. The f i s t is an experimental facility called
the Large Spacecraft Control Laboratory developed jointly by
JPL and AFAL, which replicates the main properties of a flexible
space structure that are most relevant when implementing active
control methods. The second is a flexible space station with a
two-panel configuration. In both cases, we discuss the dynamic
model, the number of dominant modes, the modal frequencies
and the mode shapes. We then choose the actuator-sensor locations, their number, and the type of sensors. We incorporate
the new controller proposed in this paper, and present the resulting system performance. In each case, we discuss whether the
assumptions under which the proposed controller is stable are
indeed satisfied by the structures.
1. The LSCL: The structure is a large, 20 foot diameter, 1 2
rib circular antenna-like flexible structure with a gimballed central hub and a long flexible feed-boom assembly (Fig. 3). The
ribs are very flexible in the vertical, out-of-plane direction, and
are coupled to one another by tensioned wires which dynamically
simulate the coupling effect of a mesh on a real antenna. A 10
degree-of-freedom finite-element model of this structure yields 30
flexible modes, all of which are below 5 Hz and no rigid body
modes. All modes are assummed to have a uniform modal damping of 0.001. A total of 6 actuators can be placed at the locations
H1, H10 (hub torquers), Rl,R4,R7, and R10 (rib root torquers).
30 sensors can be placed throughout the structure, including the
6 actuator locations and 24 locations on the 12 ribs. The disturbance input is a single, 0.20 Hz pulse with an amplitude of 1.5,
which is introduced a t the hub torquers. The control objectives
are to minimize the the displacements at various locations on the
structure, while returning the structure to equilibrium as quickly
as possible. Table 1 summarizes the 2 cases examined for the
LSCL.
Case Transfer
Sensor
Adaptive
Functions
Type
Gains
from to
pposition,
u-velocity
1
H1-H1
P
71 = l E l l
R1-R1
u+p
72 = l E l l
__
2
U
R1-R1
R4-R4
R7-R7
R10-R10
P
~
+
U
71 = 1E6
p 72=1E6
73 = 1E6
7 4 = 1E6
a decoupled 2x2 system and hence is a direct application of Theorem 2. All but 8 of the 30 modes were controllable. In this case
with position measurements and with velocity+ scaled position
measurements, shown in Figures 4 and 5 , as the adaptive gains
71 and 72 increased, the settling times of the position responses
decreased, but the magnitude of the actuator inputs increased as
well.
Case 2 leads to a strongly coupled 4 x 4 plant transfer matrix,
and hence the theory developed in this paper is not directly applicable. However, when we used the same controller structure as
in Theorem 2, we obtained very satisfactory performance. Similar observations and conclusions to those in case 1 can be drawn.
The responses using position sensors and velocity -t scaled position sensors are presented in Figs. 6 and 7. The amplitudes of
the actuator inputs and the steady-state feedback gains 01 and 82
were in fact slightly lower than those for case 1, since the burden
of control was spread over 4 actuators, decreasing the requirements on any one actuator. These results give credence to OUT
belief that the approach reported here can be extended to general
multivariable systems that occur in flexible structures.
Next, case 1was modified to include the same f i s t colocated
actuator-position sensor pair and a proximally located actuatorposition sensor pair at R1-LI1, with the resulting 2x2 system
being totally decoupled. With 71 = 1E6 and 72 = 1E12, the responses were excellent only at the locations controlled by actuator
H1, and at position sensor location, LI1, which was controlled by
actuator R1. However, the responses at all other sensor locations
affected by actuator R1 were unsatisfactory. Similar observations
were made when 4 proximally located actuator-position sensor
pairs, R1-LI1, R4-L14, R7-LI7,and R10-LI10, were used. This indicates that while the theory developed in this paper is applicable
to non-colocated actuator-sensor pairs, more work remains to be
done before realizing results that are practically reasonable.
2. The Flexible Space Station: Various configurations have
been developed by NASA for the proposed space station. One
of them has a two-panel planar configuration, whose dynamic
model consists of 2 rigid modes, and 4 flexible modes between
0.04 Hz and 0.3947 Hz. The problem is to determine a control
strategy to contain the effect due to initial condition deflections.
The initial conditions placed on the individual nodes of the space
station were the same as those used in [25]. In addition to being strongly coupled, the underlying system also has rigid body
modes. With the same adaptive controller structure as in Theorem 2, we obtained uniformly superior performance (see Figures
8 and 9) with a simple structure utilizing 4 adjustable parameters
when position measurements are utilized and 1adjustable parameter when velocity
scaled position measurements are used. A
comparison of Figures 8 and 9 clearly demonstates the advantages of using pure position measurements over velocity scaled
position responses. Figure 9 further clarifies the fact that a satisfactory velocity + scaled position response doesn’t always result
in a satisfactory response a t the corresponding position sensor.
The benefits of using pure position measurements with the new
adaptive controller over the approach used in [25] are obvious.
+
In Table 1, The second column indicates the input-output pairs
and their locations, the third column indicates the types of sensors used, and the fourth gives the values of the adaptive gains
used. All initial conditions are set to zero. The closed-loop responses obtained a t the various sensor locations along the structure, a t the actuator inputs, and at selected signals in the feedback loop, for the 2 cases presented in Table 1, are shown in Figs.
4-7. The adaptive controller for all cases is turned on after 5
seconds, providing a direct comparison between the amplitudes
of the open-loop and closed-loop responses. In addition, since the
open-loop settling times would be extremely slow due to assuming
a uniform modal damping of 0.001, a direct comparison between
the settling times is demonstrated as well. The units for the hub
sensor responses (Hl,H10), for the rib sensor responses (Rl, R4,
R7, R10, LI1 - LI12, and LO1 - LOlZ), and the actuator inputs
are in radians, meters, and Newton-meters, respectively. Since
the velocity with scaled position responses were exactly the same
as the pure velocity responses, only measurements from the velocity with scaled position sensors are presented. Case l leads to
+
6.
Conclusions
In this paper, a new adaptive controller is proposed to achieve
the objectives of vibration suppression as well as shape control in
flexible structures. The controller is simple, requires no knowledge of the order of the underlying dynamic model, and achieves
global boundedness as well as the desired objectives. By making
model, it is shown that satisfactory
use of the underlving- dynamic
performance can be realized using only position measurements.
Such a result, to the authors’ knowledge, is totally new and is being reported for the fist time in the literature. The same results
I
507
I
can also be derived using either only velocity measurements, or
velocity scaled position measurements. Both colocated as well
as proximally located actuator-sensor pairs can be used.
This paper represents the f i s t step in realizing a practically
viable adaptive controller for flexible structures. The main result
reported in this paper is based on the assumptions that underlying dynamic model is decoupled, and minimum-phase. These
assumptions have to be relaxed before significant practical results
can be derived using the suggested approach. Work is currently
in progress in these directions.
+
Acknowledgments
Meirovitch, L. and Baruh, H.,”On the Robustness of the Independent Modal Space Control Method,” Journal of Guidance, Control and
Dynamics, Vol. 6, No. 1, Jan.-Feb. 1983, pp. 20-25.
Arbel, A. and Gupta, N. K.,”Robust Colocated Control for Large
Flexible Space Structures,” Journal of Guidance, Control and Dynamics, Vol. 4, No. 5, Sept.-Oct. 1981, pp. 480-486.
Auburn, J. N., ”Theory ofthe Control ofstructures by Low-Authority
Controllers,” Journal of Guidance, Control and Dynamics, Vol. 3, No.
5, Sept.-Oct. 1980, pp. 444-451.
Sundararajan, N., Joshi, S. M., and Armstrong, E. S.,”Robust Controller Synthesis for a Large Flexible Space Antenna,” Journal of Guidance, Control and Dynamics, Vol. 10, No. 2, March-April 1987, pp.
201-208.
”
The work reported here was carried out under NSF Grant No.
ECS 8915276. We would also like thank Dr. Aziz Ahmed a t JPL
for providing us with the data for the LSCL structure.
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653
l7 Martin, G. D. and Bryson, A. E.,”Attitude Control of a Flexible
Spacecraft,” Journal of Guidance and Control, Vol. 3, No. 1, Jan.-F&-.
1980, pp. 37-41.
a1 Blelloch, P. A. and Mingori, D. L.,”Robust Linear Quadratic Gaussian Control for Flexible Structures,” Journal of Guidance, Control and
Dynamics, Vol. 13, No. 1,Jan.-Feb. 1990, pp. 66-72.
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34
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la
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’’Singh, R. P., and Narendra, K. S.,”Prior Information in the Design of
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508
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653
Figure 1: The new adaptive controller (n*= 1).
Figure 3: A plan view of the LSCL structure with actuators and
Figure 2: The new adaptive controller (n*= 2).
A
PY
75
1~
f
:
5~
r
0
H
5 -
.
I
: ' 1
1
'
5
2
t
L5
~
"
,.
"
,
"
,
"
"
"
'-
Figure 4: Adaptive control for a decoupled MIMO flexible structure with two colocated actuator-sensor pairs using position feedback.
509
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653
A
1.1
:
2.5
,
?
Figure 5: Adaptive control for a decoupled MIMO flexible structure with two colocated actuator-sensor pairs using velocity scaled position feedback.
+
A
P
II
R
0
A
0
L
I
1
P
A
I
P
n
A
0
I
G
0
L
0
L
I
0
Figure 6: Ad,aptive control for a strongly coupled MIMO flexible structure with four colocated
actuator-sensor pairs using position feedback.
5 10
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653
Figure 7: Adaptive control for a strongly coupled MIMO flexible structure with four colocated
actuator-sensor pairs using velocity scaled position feedback.
+
"
P
P
I
I
2
I
Y
P
P
;
I
1
I
Figure 8: Adaptive control for a space station, a strongly coupled MIMO flexible structure with
four colocated actuator-sensor pairs using position feedback.
Figure 9: Adaptive control for a space station, a strongly coupled MIMO flexible structure wit
four colocated actuator-sensor pairs using velocity scaled position feedback.
+
511
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