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6.1991-2610

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ROBUST STABILIZING CONTROLLERS FOR LINEAR
FLEXIBLE MECHANICAL SYSTEMS+
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2610
D. Da
M. Corless
School of Aeronautics & Astronautics
Purdue University
West Lafayette, Indiana 47907, USA
u. q1 and q2 are the displacements of P1 and P2 from fixed
points 0 1 and 0 2 , respectively.
If the bar B is modelled as a rigid bar of length l o ,
q1 = q2 and the motion of the system can be described by
ABSTRACT
In this paper, the problem of stability robustness of
linear memoryless feedback controllers applied to linear
mechanical systems containing flexible components is treated.
In general, if a stabilizing controller is designed based on a
‘rigid’ model, i.e., a model in which some of the flexible
components are assumed rigid, it is not true that this controller
also stabilizes the ‘real’ flexible system. We present a
condition which guarantees that a stabilizing controller whose
design is based on a ‘rigid’ model also stabilizes the flexible
system, provided that the components, whose flexibilities are
neglected in the ‘rigid‘ model, are sufficiently stiff. It is
shown that this condition involves the location of the rate
sensors and is independent of the location of the displacement
sensors. It is also shown that the condition is satisfied when
the rate sensors are collocated with the actuators. The results
are illustrated by examples.
2&=u
Suppose a controller is given by
with f l , f i > 0. Then, the feedback-controlled rigid model
subject to (1.2) is
( 1.3)
which is asymptotically stable.
Suppose now the bar is modelled as a flexible bar with
linear stiffness coefficient K > O and linear damping
coefficient c > 0. With controller (1.2), the feedbackcontrolled flexible model is described by
41
- q2) + 441 - 42)
.
(1.4a)
(1.4b)
It can be readily shown by using the root-locus technique that
if f2 > 4c, then system (1.4) is unstable for K sufficiently large
; Le., even if the bar is ‘very stiff‘, the behavior of the flexible
model (1.4) is qualitatively different from that of the rigid
model (1.3).
In designing a stabilizing controller for a flexible
mechanical system one usually considers a reduced order
model in which some of the flexible components m
considered rigid. The natural question that arises is whether a
controller, whose design is based on the reduced order model,
also stabilizes the original system. The following simplc
example illustrates that this question is not trivial.
A singular perturbation approach is one approach to
answer the above robustness question for flexible mechanical
systems in which the ‘neglected‘ elements are sufficiently
stiff. By ‘neglected’ elements we mean those flexible
elements which are considered rigid in the reduced order
model. Ficola ef a1 [6] appear to be the first to use a singular
perturbation approach to treat the control problem for elastic
robots. After that, many papers have been published on
utilizing singular perturbation theory to deal with flexible
Example. Consider a mechanical system consisting of
two particles of unit mass connected by a massless bar B; see
Figure 1. The system is constrained to move along an
inertially fixed line and particle P1 is subject to a control force
Based on research supported by the U.S.National Science Foundation
under grants MSM-87-06927 and MSS-90-57079.
Copyright c 1991. by American I n s t i t u t e of Aeron a u t i c s & A s t r o n a u t i c s , I n c . A l l r i g h t s reserved
= - ~ ( q l - 92) - c(q1 - 42) - f 2 b - flq2
42 = K(ql
1. INTRODUCTION
t
.
83
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mechanical control systems; some of them are Kokotovic [8101 , Khorasani [7] , Marino [ l l ] , Spong 113,141 ,
McClamroch [121 and Corless [2,3] .
A singularly perturbed system is one whose description
depends on a scalar parameter p > 0; this dependency is such
that setting p = 0 results in a system of lower order than that
for p > 0. For small p > 0, it is customary to determine the
qualitative behavior of a singularly perturbed system by
studying two associated lower order subsystems, namely, the
reduced order system (p = 0) and the boundary layer system.
The latter subsystem is obtained by letting p = 0 in the 'fast
time' scale. One method of controller design for such systems
is to base the design on the reduced order model. To
guarantee stability robustness with respect to the dynamics
which were not taken into account in the controller design, it
is customarily required that the boundary layer system be
asymptotically stable.
In treating a flexible mechanical system as a singularly
perturbed system, one lets p = K-% where K > 0 is a stiffness
parameter associated with the 'neglected' elements; as K
increases the 'neglected' elements become stiffer. The
reduced order system corresponds to the 'rigid' model of the
system, i.e., the system model obtained by considering the
'neglected' elements rigid. To obtain a boundary layer system
(hereafter called 'neglected' elements) of the system are
modelled as rigid elements, this can be represented by linear
constraints on the generalized coordinates, Le., there exists a
matrix s E IR~* with
such that
sq=o.
(2.2)
Also we suppose that there are no other possible kinematical
constraints on the system. Then the systems under
consideration are described by
Mq+ Cq+ Kq= Wu + STh
Y1
(2.3a)
(2.3b)
=D1q
(2.3~)
Y2 = D24
where u E Rm is a vector of control inputs;
y1 E IRp' , y2 E RP2 represent measurements which are
based on displacement and rate sensors, respectively;
M E RN* is the system mass matrix, hence it is symmetric
and positive definite, i.e.,
M~=M>o;
(2.4)
the matrices C, K E RNfi are such that - Cq - Kq represents
all the generalized forces acting on the system except the
generalized forces due to the control inputs and the
'neglected' elements mentioned above; W E RNm is the
influence matrix; the vector S T h represents the generalized
which is asymptotically stable one usually assumes the
'neglected' elements have damping coefficients which are
proportional to IC". In this paper, we consider linear flexible
mechanical systems in which the damping coefficients of the
'neglected' elements remain constant as K increases. In this
case, the boundary layer system is just marginally stable; Le.,
stable, but not asymptotically stable.
We present a condition which guarantees that a
stabilizing controller whose design is based on the 'rigid'
model, a model in which the 'neglected' elements are assumed
rigid, also stabilizes the flexible system provided that the
'neglected' components are sufficiently stiff, It is shown that
this condition involves the location of the rate sensors and is
independent of the location of the displacement sensors. It is
also shown that the condition is satisfied when the rate sensors
are collocated with the actnators. Two examples are given to
illustrate the results.
forces exerted by the 'neglected' elements; and D1 E IRpl*,
D2 E RpzXN
are determined by the type and location of the
displacement and rate sensors, respectively.
2.1. 'Rigid' Model
Consider first the situation in which the 'neglected'
elements are assumed rigid. Then (2.2) holds. Condition
(2.1) guarantees that there exists a matrix U E lRNx(N-L) such
that
su=o
(2.5)
and
rank (U) = N - L
2. SYSTEM DESCRIPTION
Define
For the class of flexible mechanical systems under
consideration, we let q(t) E lRN denote a vector of generalized
coordinates which describe the configuration of the system at
time t, We assume that, when some of the flexible elements
Then V E lRNxL and
84
.
(2.6)
F]M[llvI=
:]
Fz
, MkUTMU;
(2.8)
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Assumption 1. The matrix Fz, given in (3.I), is such
that the following inequality is satisfied:
(2.lob)
(2.1oc)
Y2
=
(2.1od)
(3.2)
In the following we present an assumption which
guarantees that any stabilizing controller (3.1) for the ‘rigid’
model (2.10) also stabilizes the flexible model (2.13),
provided the stiffness parameter K is sufficiently large.
(2.10a)
=El+
.
Mq+(C+STCoS+WF2D2)q+(K+KSTKOS+WF1 D 1)q = 0. (3.3)
Under the new coordinates system (2.2),(2.3) can be
described by
Y1
m1&)=
$
0
If one uses controller (3.1) on the flexible model (2.13), then
the feedback controlled flexible model is given by
(2.9)
O=O
RmP2.Then the feedback controlled ‘rigid’ model is
a&+ (e+ m 2 6 2 ) &+ (E+
hence the square matrix [u VI is invertible. We introduce a
coordinate transformation defined by
a&+ e&+ E+= WU
E
C,+ PTCP + PTWF2D2P> 0
(3.4)
where P E RNXL
is given by
where
-A
-A
C
= UT CU ; -KA= UT KU ; W
= UT W ;
-
P 4M-’ST(SM-’ST)-’
.
(3.5)
A
AD1U ; D2 =D2U.
Our main result is given by the following theorem.
We call this the ‘rigid’ model.
Theorem. Suppose the controller (3.1) satisfies
Assumption 1 and the feedback controlled ‘rigid‘ model (3.2)
is asymptotically stable. Then there exists K* > 0 such that for
all K > K* the feedback controlled jlexible model (3.3) is
asymptotically stable.
2.2. Flexible Model
We suppose that the effect of the flexibilities of the
‘neglected’ elements can be represented by letting
h = - KKO(Sq) - CO(S4)
(2.11)
Proof. We outline the proof in the next section. A
complete proof will be given in a forthcoming paper.
where KO E RLxLis symmetric and positive definite, Le.,
K;f=&>O,
(2.12)
Remark 1. The theorem shows thaL the satisfaction of
Assumption 1 is the only condition for a stabilizing controller
to be robust with respect to unmodelled flexibilities. The term
C, + PTCP in (3.4) represents the damping of the ‘neglected’
flexible components which is not related to the controller.
The term PTWF2D2P represents damping exerted by the
controller. Inequality (3.4) guarantees that the total damping
related to the ‘neglected’ components is positive.
and C, E IRLxL; K E (0, -) is a parameter characterizing the
stiffness of these elements; see [I] . Clearly, the larger K is,
the stiffer these elements are. Then the flexible model is given
bY
M i + (C + sTC,S)q + (K + KST&s)q = WU (2.13a)
Y1
=D1q
(2.13b)
Y2
=D24.
(2.13c)
3.2. Collocation of Rate Sensors and Actuators
3. MAIN RESULTS
If the rate sensors of the system are collocated with the
actuators, we have
3.1. The Main Theorem
D2 = WT
Suppose one designs a stabilizing controller
U=--lYl
-F2Y2
and hence F2 E R”
(3.1)
is a square matrix. Suppose
C, + P T C P > 0 .
for the ‘rigid’ model (2.10), where F1 E Rmpl and
85
(3.6)
Then we have the following corollary for this special case.
(4.3e)
Corollary. Suppose (3.6) and (3.7) hold, and F2 2 0.
Then any stabilizing controller (3.1) for the ‘rigid‘ model
(2.10) QISO stabilizes the jfexible model (2.13) provided K is
sujjiciently large.
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2610
Remark 2. By considering (3.4) and (3.6) we can see
that the collocation of the rate sensors and actuators
guarantees that the feedback control does not decrease the
(4.30
The reduced order system associated with (4.3) represents
the feedback controlled ‘rigid’ model (3.2) and the associated
boundary layer system which is described by
(4.4)
damping related to the ‘neglected’ components.
4. OUTLINE OF THE PROOF OF THE MAIN
THEOREM
The flexible mechanical system (3.3) can be represented
by a singularly perturbed system by letting
& K-v2
5. TWO EXAMPLE§
5.1. A Simple Mechanical System
and introducing the coordinate transformation (2.9), which
yields
Z $ = G ~ ~ ~ , + G ~ ~ ~(4.la)
+H
8 = (3214, + G226 + Hzl I$+ H22 8 - p-’kO
is just marginally stable.
The remainder of the proof proceeds by applying the
results in [4,5] to (4.3).
(4.1b)
where
Consider the simple mechanical system introduced in
Section 1; also see Figure 1. Suppose the measured output y
form+(2.3b),(2.3~),
has ~
the ~
~
H ~ ~ where
O
D1 = [dii di21 ; Dz = [dzi, dzzl ; q =
~1
;
(5.1)
and dij E IR ;i,j = 1,2.
If the bar B is modelled as a rigid bar of length l o , we
obtain a linear constraint on q1 and q2, Le.,
S q A= [ l - l ] q = O ,
(5.2)
and the system can be described by
24, = u
Let
y1 = (dl1 + d l d 4
y2 = (d21 + d2214,
A
(4.2)
Then, system (3.3) can be represented by
X
+ A12(p)z
(4.3a)
= A21 x + A22 (P)z
(4.3b)
= Allx
A
where 4, = q1 = 92. We call this the rigid model.
Suppose now the bar is modelled as a flexible bar with
linear stiffness coefficient K > 0 and linear damping
coefficient c > 0. Then the system can be described by (2.13)
with
M = I ; C=K=O ; W=
where
(4.3c)
(5.3a)
(5.3b)
(5.3c)
; &=l
We call this the flexible model.
Suppose there is a stabilizing controller
(4.3d)
for the rigid model (5.3). Since
; CO=C.
[]
I2 -11
p A M-~ST(SM-~ST)-~
=
’
(5.8)
Assumption 1 is satisfied iff
1
Co + PTWf2D2P= c + -f2(d21 - d22) > 0 .
4
and the system can be described by
(5.5)
JT$
Hence by the theorem in Section 3, the feedback controlled
flexible system
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M i = - WflDlq - Wf2D2q - KSTSq - cSTSq
or
JT = J1 + 352 + J3 + J4
M=diag (J1, 3J2, 53, J4)
C=K=O
W=[l 0 0
op
&)=I3
5.2. Flexible Dual Rudder Steering Mechanism
Co = c13 .
Consider the system illustrated in Figure 2. Each part of
the system is constrained to rotate about a line parallel to the
Consider
inertially fixed direction g. PI is a rotor with moment of
inertia J1 about L1 and its angular displacement is 91. P2 is a
gear assembly composed of three identical gears, each of
which has moment of inertia J2 about its aiis of rotation. We
let qz be the angular displacements of the gears as shown in
Figure 2. PI and P2 are connected by a massless shaft B1.
Two rudders P3 and P4 with moments of inertia J3 and J4
about
and L3 are connected to P2 by two massless shafts
J1
= J3 = J4 = 1,
now
J2
system
parameters
= -,
1 and suppose there is a controller
3
u = - fIY1 -f2y2
(5.10)
which stabilizes rigid model (5.9). Since
PAM-~sT(sM-
B2 and B3, respectively; q3 and 94 are the angular
displacements of P3 and P4, respectively. u is a control torque
exerted on PI. We suppose the measured output has the form
s
1 T -1
--
’1
- 1
(5.11)
4 -1
1-1
-1
Assumption is satisfied iff
i 1
- 1611
D2 = [d21 d22 d23 d241.
and di, E R ;i=1,2 ;j=1, ...,4.
The rigid model of the system is obtained by
considering the shafts Bi, i = 1, 2, 3, rigid. This results in
9
(5.9b)
Suppose now the shafts are considered flexible. We
model each Bi, i = 1, 2, 3, as a flexible shaft of linear torsional
stiffness coefficient K > 0 and linear damping coefficient c > 0.
Thus the flexible model of the system can be described by
(2.13) with
then, for any f2 2 0, (5.5) holds; hence, provided the feedback
controlled rigid model is asymptotically stable, the feedback
controlled flexible model (5.6) is asymptotically stable for
sufficiently large K.
A
.
(5.6)
d21 = 1 , d22=O,
@ = q 1=q2 =93 =q4
(5.9a)
where
is asymptotically stable for sufficiently large K if (5.5) holds.
If the rate sensor is collocated with the actuator, i.e.,
D2= WT
=u
1
-1612
where
11 = W 2 1 - d22 - d23 - d d f 2
12 = (3d23 - d2i - d22 - d d f 2
(5.7)
13 = (3d3 - d21 - d22 - du)f2
which puts a linear constraint on q, Le.,
.
Two special cases arc considered in the following.
07
31
marginally stable boundary layer systems,
Proceedings of the 28st Annual Allerton
Conference on Communication, Control, and
Computing, 1990.
( 1 ) A single rate sensor which is placed on Pd. Thus
D2=[O 0 0 11.
It can readily be obtained that Assumption 1 is satisfied if
;::
[;O
0
1.
5.
D. Da and M. Corless, Lyapunov functions for
linear singularly perturbed systems with
marginally stable boundary layer systems,
Proceedings of the American Control Conference,
1991.
6.
A. Ficola, R. Marino, and S. Nicosia, A singular
perturbation approach to the dynamic control of
elastic robots, Proceedings of the 21st Annual
Allerton Conference on Communication, Control,
and Computing, 1983.
7.
K. Khorasani, A slow manifold approach to linear
equivalents of nonlinear singulariy perturbed
systems, Automatica Vol. 25 pp. 301-306. 1989.
8.
P. V. Kokotovic, Applications of Singular
Perturbation Techniques to Control Problems,
SIAM Review Vol. 26 pp. 396-416, 1984.
9.
P.V. Kokotovic and H.K. Khalil, Singular
Perturbations in Systems and Control, IEEE
Press, New York , 1986.
10.
P.V. Kokotovic, H.K. Khalil, and J. O’Reilly,
Singular Perturbation Methods in Control:
Analysis and Design, Academic Press , 1986.
11.
R. Marino and S. Nicosia, On the feedback
control of industrial robots with elastic joints: a
singular perturbation approach, University of
Rome: R-84.01 , 1984.
12.
N. H. McClamroch, A singular perturbation
approach to modeling and control of manipulators
constrained by a stiff environment, ZEEE
Conference on Decision and Control, 1989.
13.
M. W. Spong, Modeling and control of elastic
joint robots, Journal of Dynamic Systems,
Measurement and Control Vol. 109 pp. 310-319,
1987.
14.
M. W. Spong, K. Khorasani, and P. V. Kokotovic,
An integral manifold approach to the feedback
control of flexible joint robots, IEEE Journal of
Robotics and Automation Vol. RA-3 pp. 291-300,
1987.
G
9-;
1- 30
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where 0 A= f2/16c; and as a result, a condition which assures
the satisfaction of Assumption 1 is given by
O<G<-
1
8
or
O<f2<2c
By the theorem in Section 3, this guarantees that controller
(5.10) also stabilizes the flexible model of the system for
sufficiently large K.
(2) A single rate sensor which is placed on P I . In this
case, the rate sensor and actuator are collocated, hence
D 2 = W T = [ 1 0 0 01
and
9
-3 - 3
Co + PTWf2D2P = CI-thence Assumption 1 is satisfied for any f2 2 0.
REFERENCES
1.
2.
3.
4.
M. Corless, Modelling ’flexible constraints’ in
mechanical systems, Twentieth Midwestern
Mechanics Conference, 1987.
M. Corless, Stability robustness of linear
feedback controlled mechanical systems in the
presence of unmodelled flexibilities, Proceedings
of the Conference on Decision and Control ,
1988.
M. Corless, Controllers which guarantee
robustness with respect to unmodelled flexibilities
for a class of uncertain mechanical systems,
International Journal of Adaptive Control and
Signal Processing Vol. 4 pp. 565-579, 1990.
M. Corless and D. Da, On the stability of
singularly perturbed linear systems which have
88
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Figure 1: Two Masses Connected by a Massless Flexible Bar
Figure 2: Dual Rudder Steering Mechanism
89
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