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Actuator Selection and Placement for Control of Elastic Continua
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SENSOR/ACTUATOR SELECTION AND PLACEMENT FOR CONTROL OF ELASTIC CONTINUA
J.
de Lafontaine* and M. E. Stieber**
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Abstract. General controllability and observability conditions are developed for a flexible body whose dynamics are modelled by partial differential equations. These conditions reveal the types, number and location of actuators and sensors necessary to ensure controllability and observability of all modes of vibration . It is proved that the number and types of sensors/actuators required are directly related to the order of the partial differential equation. Controllability and observability can be guaranteed with the sensors/actuators located at an arbitrary point on the structure, although the degree of modal excitation/detection depends on their particular position.
The implementation of the above principles relies on co-location of all the sensors, and co-location of all the actuators. In general, it also requires devices which may not be easily mechanized . Fortunately, the general controllability and observability criteria are quite conservative and it is shown that, under realistic boundary conditions, they are satisfied with just two types of conventional sensors and actuators. Moreover, in most cases, complete controllability and observability of the flexiblebody modes can be achieved with only one sensor/actuator pair located at a free boundary. For illustration, the above principles are applied to an undamped flexible beam with arbitrary boundary conditions. Extensions to damped and gyroscopic systems are pointed out . The case of distributed sensors and actuators is modelled using the definitions of 'Distributed Identity' and 'Distributed Slope' operators. It is verified analytically that the distributed action of these devices leads to an attenuated detection/excitation of higher-frequency vibrations. Although the structure was idealized by using a continuum modelling approach with a theoretically infinite number of modes, the above results can be applied to the identification and control of more complex structures modelled by other techniques, such as the finite element method. Keywords. Controllability; observability; distributed parameter systems; actuators; sensors; modal control; flexible structure control; space vehicles.
INTRODUCTION
the discretization has the disadvantage that the spatial derivatives of the vibrations are not directly available in the model, which, in this paper, will be shown to be of significance.
Satellites for communication and surveillance require the control of large antenna systems to very stringent attitude and shape tolerances. The philosophy of spacecraft control design therefore has evolved from one of minimal excitation of structural vibrations to one of their active control. For this problem, the selection/placement of sensors and actuators is an important step in the overall control design process. This issue has not yet been resolved and is often dealt with in an ad-hoc fashion, while a considerable amount of research has been devoted to the improvement of modelling and control synthesis procedures.
Recently, the modelling of structures as DistributedParameter Systems (DPS) has gained the attention of the controls community. Starting from a simplified model of the structure by assuming a continuous spatial distribution of its mass and stiffness, an 'exact' analysis can proceed with a theoretically infinite number of vibration modes. The existence of the high-frequency modes is questionable (Hughes, 1984) but their inclusion into the model avoids the spillover problem. Although not always practical for real structures, the DPS approach provides interesting insights that can guide the development of the finitedimensional approach . For instance, it is known that only one pair of sensors and actuators will provide control and observation of a very large number of modes of a flexible beam (Delfour, Polis and Payre, 1985) . However, controllability and observability theorems for DPS are not straightforward since one is now dealing with infinitedimensional matrices. On the other hand, the natural modelling of the spatial derivatives of the vibrations will be shown to compensate for this difficulty. This paper combines the above two techniques and derives general controllability and observability criteria for the 'infinite' number of vibration modes of a flexible structure. These criteria will in turn provide definite answers to three fundamental questions of control-system design: what types of sensors/actuators are required, how many, and where must they be located.
Most of the work on flexible structure control reported in the literature is based on structural models of finite dimensions, which are obtained by spatial discretization e.g. using the finite element method. This modelling approach generates relatively accurate models (10,000 degrees of freedom) of complex flexible structures. However, analysis and control-system design are only practical on a truncated version of the model, giving rise to control energy 'spillover' to residual (unmodelled) modes (Balas, 1976) . Some of the control techniques proposed to overcome thIS problem require as many sensors and actuators as there are modelled vibration modes (Meirovitch, VanLandingham and Oz, 1979; Oz and Meirovitch, 1980) although a more recent formulation (Lindberg and Longman, 1984) has relaxed this requirement. The controllability and observability criteria that provide the types, number and locations of these devices are defined in terms of the rank of finite-dimensional matrices (Kailath, 1980) . However,
22S
MODEL OF ELASTIC CONTINUA Assumptions
An elastic body contained in a spatial domain [) spanned by the position column matrix r is modelled by assuming a continuous distribution of its mass and stiffness properties over [) . A static force distribution f(r.) on the body will induce deformations !!.(r:) that we as-;ume small. Stress is linearly related to strain through the linear, positive semidefinite, self-adjoint stiffness operator K:
(1)
where the modal coordinates qn(t) represent the timevarying contribution of mode n to the total response. After substitution of solution (7) into Eq. (3), the orthonormality conditions of Eqs. (6) are invoked to arrive at a set of uncoupled equations:
The modal excitations fn (t) are the components of the forcing function along the basis functions ctn (r:) :
fn(t) =
Jct~(rJL(r,t)
(9)
dr·
/)
which contains differentials with respect to r up to order = 1 while p = 2 for a slender flexible beam. Over the boundary of [), these deformations are constrained by conditions of the form
By defining the infinite-dimensional matrices 9,(t) = T,..,2 2' " }, _t F() = [ql(t) q2(t) '], !! = d'lag {2 W l , W2
(2)
[Idt) hit) the harmonic equations shown in Eq. (8) can be condensed into the familiar form :
2p. For example, a vibrating string has p
ai!!.=Q,
i=1,2," ' ,p
y,
a
where the i are also linear differential operators of order through 2p - 1. Structural damping and gyroscopic effects are neglected here but we will point out how these effects can be considered . Equations of Motion
When a dynamic force distribution fir, t) is applied on the flexible body, the dynamics of its vibrations are governed by the following equations (Meirovitch, 1967) :
M !i.(r, t)
+ K !!.(r, t) = 1..(r, t),
ai!!.(r, t) = Q,
i = 1,2" .. ,p.
(la) The physicai coordmates that describe the vibrations can be recovered from the modal coordinates through the matrix equivalent of Eq. (7):
!!.(r, t) = ~T (r)9,(t)
(11)
where the infinite-dimensional (00 x 00) matrix of mode shapes ~(r) is defined as
(3)
(4)
The mass operator M is a linear, positive-definite, selfadjoint operator of order less than 2p . Overdots denot e time differentiation.
Equations (10) and (11) form the basis upon which the controllability and observability properties of the system will be developed.
Solution (Homogeneous System)
Under homogeneous condition, the force distribution fir, t) in Eq . (3) is set to zero. The spatial and ten:;-poral coordinates can be separated by substitution of the harmonic solution !!.(r, t) = 4>(r)e iwt into the boundaryvalue problem (4) and the f;:;-llowing eigenvalue problem results:
(5) This last differential equation, of order 2p, together with the boundary conditions, define the infinite-dimensional set of mode shapes 4> (r) that characterize the vibrations at frequencies W n . U-;in~ the self-adjoint property of operators M and K, the following orthonormality conditions can be easily proved :
Jct~ Jct~Kctn
Mct n dr = om,n
/)
(6) dr = wmwnom,n,
/)
CONTROLLABILITY AND OBSERVABILITY Basic Principle
Our analysis of the controllability and observability of a flexible body is based on one definition and two theorems that are best illustrated by consideration of a one-dimensional structure. Not only does this assumption simplify the notation and presentation of the arguments but this type of structure represents a well-known and often used reference example in the study of distributedparameter system. Furthermore, many real structures with symmetric mass and stiffness distribution can be decomposed into a series of such (single-axis), interconnected, one-dimensional elements. The vibrations v(x, t) are assumed along the z direction . The domain [) of the spatial coordinate is 0 ~ x ~ L . Following the notation of the preceding section, the displacements are expanded into:
v(x, t) =
L
4>n(x)qn(t) = ~T(x)9,(t)
(13)
where om ,n is the Kronecker delta. These conditions allow the expansion of the forced response into an infinite series of uncoupled modes.
and Eq. (10) still applies. For this system, the concept of a generalized node is now introduced.
Solution (Forced System)
Definition 1 : Generalized Node
When an external force distribution fir, t) is exerted on the body, Eqs. (3)-(4) apply. From-their linearity, the expansion theorem tells us that the response of the system can be represented as a linear combination of admissible functions that form a complete basis in the domain of the operators (Meirovitch, 1967) . The natural modes ctn (r:) can be used in this expansion :
n=1
A mode 4>n(x)qn(t) has a th-order (generalized) node at a position xo if and only if its th spatial derivative at xo vanishes, i.e.: ph-order node at
TO
{=>
[Bi4>n(x)qn(t)] == 0, 'rIt. Bx' "' 0
00
!!.(r,t) =
L ctn(r:) qn(t) "=1
(7)
***
For instance, the familiar displacement node of a vibrating structure is a oth-order node. This concept helps simplify the following theorem: Theorem
=0)
l
[ a 4>n(x)qn(t)] ax} Xo
<==:>
j=0,1, . . . ,2p-l,Vt :,; x
*
The proof of this theorem follows directly from the Wronskian (Kreyszig, 1972) of the 2p independent solutions of the eigenvalue problem. Details can be found in (de Lafontaine and Stieber, 1986). To illustrate this result, let us consider the 4 th -order differential equation that describes the mode shapes associated with the vibrations of a uniform, slender beam of length L:
ddx~X) + f344>(X)
= 0,
(14) where .Y'.x is the curl matrix operator. For j = 0, this equation represents the modal displacement at XQ, measured along the z-axis; when j = 1, it gives the modal slope at Xo, measured along the y-axis, and so on . For a non-trivial mode, Theorem 1 tells us that the 2p column matrices (.Y'.X)lt,(ro) cannot all vanish, irrespectively of the choice of ~, unless mode 4>n(x)qn(t) is not excited. This basic prinCiple allows the development of the following controllability and observability results.
Controllability Let us assume a force distribution f(r., t) composed of 2p point actuators that can each excite one of the first 2p spatial derivatives of a vibrating structure:
2p-l
0:-:::: x :-: : L.
Dr., t) = The general solution for the mode shapes 4>n(x) can be written as:
4>n(X) = acoshf3nx
+ bcosf3nx + csinhf3nx + dsinf3n x.
If the first four derivatives of 4>n(x) vanish at, say, XQ, we get the following 4 algebraic equations:
L
cosf3XQ - sin f3xQ - cosf3xQ sin f3xQ
(j
~ I)! (.Y'.xY
[L(i)6(r. -
J
4>n(x)f(x,t) dx
D
2f:l (-1)1 1=0 (J
°
The analysis only requires the knowledge of the first 2p nodes since those of order greater than 2p can be expressed as linear combinations of the first 2p ones. Theorem 1 implies that an excited mode must have at least one nonzero spatial derivative in the first 2p ones. This, in turn, means that a mode can be excited or sensed through actuation or measurement of that nonzero derivative. This theorem can be generalized to consider modes associated with repeated eigenvalues.
(15)
Using the properties of the Dirac function 6, the modal excitations fn(t) can be easily derived after insertion of Eq. (15) into Eq. (9). In one-dimensional form, we have (de Lafontaine and Stieber, 1986):
sinh f3xQ cosh f3xQ sinh f3xQ cosh f3xQ
It can be easily shown that the above 4 x 4 Wronskian matrix is nonsingular for all XQ (determinant = 4) and consequently, only the trivial solution a = b = c = d = exists.
+ I)!
(16)
(d l 4>r:) dx}
fU}(t). x.
Extending definitIOn (12), we introduce:
j = 0,1,· . . ,2p - 1
(17) to arrive at the matrix of modal excitations: (18) where
(d2P-l~/dx2P-llx.]
2:
Two modes 4>n(x)q(t) and 4>m(x)q(t) associated with the same eigenvalue and derived from an eigenvalue problem of order 2p have their first 2p derivatives linearly related at XQ if and only if their mode shapes 4>n (x) and 4>m( x) are linearly related, i.e.
l axl [a 4>n(x)q(t)]
r.,,)].
}=o
fn(t) = cosh f3xQ sinh f3xQ [ cosh f3xQ sinh f3xQ
4>n(X)
° XQ IT are:
1:
A mode 4>n(x)qn(t), derived from an eigenvalue problem of order 2p, has Its first 2p nodes located at the same (arbitrary) position XQ if and only if this mode is the trivial one, i.e.:
Theorem
ro = [0
the derivatives at
_ k -
[a l 4>m(x)q(t)] axl
X I)
._ J-O,I, ... ,2p-l,
(19) 1
12Pi1
f(2 P-I)]T
.
(20)
By substitution of Eq. (18) into Eq. (10), the forcedsystem equation becomes: (21)
% ('1
=k4>m(x), "Ix E D,
k = nonzero constant.
* ** This theorem is proved by forming the function 4>(x) = 4>n(X)_- k4>m(x) and applying Theorem 1 to show that 4>(x) = 0, "Ix E P. In terms of the 3-dimensional notation presented earlier,
The controllability of system (19)-(21) is determined from the theorems and corollaries proved by Hughes and Skelton (1980a, 1980b). For distinct eigenvalues, this reference states that all the rows in the ll(xa) matrix above must have at least one nonzero entry. This is guaranteed by Theorem 1. For repeated eigenvalues, the rank of the associated partition of the I1 matrix must be equal to the multiplicity of the eigenvalues. The orthogonality of the mode shapes and Theorem 2 guarantee the satisfaction of this requirement. In fact, all the rows in the 1l matrix are linearly independent, thus ensuring complete controllability for any multiplicity of eigenvalues.
This theoretical result proves that 2p co-located actuators, each one exciting a specific spatial derivative of the mode shapes, are sufficient to ensure controllability of all the vibration modes of the structure. This result is quite general since it does not imply a particular system order nor does it assume a particular set of boundary conditions or a particular location for the co-located actuators. The requirement for as many actuators as there are modes is now replaced by a requirement to have as many actuators as the order of the eigenvalue problem. A consequence of this result is that the multiplicity of eigenvalues can not exceed 2p. In practice however, when the order of the PDE exceeds 2, the above results require the mechanization of 2p - 2 actuators that may not be physically realizable. For instance, a slender flexible beam (2p = 4 > 2) would require actuators of the second and third spatial derivatives of the mode shapes, in addition to conventional force and torque actuators. Fortunately, we shall soon see that, for most physical systems, this does not restrict the usefulness of the above theory since only one and possibly two types of conventional actuators (otb and pt-order derivatives) are necessary.
Ob8ervability In a similar development, we assume that 2p point sensors can measure each of the 2p spatial derivatives of the modes at a position 1:. = [0 0 x.]T. The measurement column matrix takes the form:
actuators and displacement/rotation sensors, respectively, can be co-located since their action is on different axes of the structure (see Eqs. (14)). Consequently, the first two limitations of the theory can be overcome if controllability /observability can be proved with only the otb and pt-order derivative devices. This task is considered in the next section, along with the investigation of the effects of distributed rather than point actuators/sensors, thereby taking care of limitation 3 as well. Extensions to three-dimensional structures can be formally carried out with the generalized version of Eqs. (14). Finally, Theorems 1 and 2 could be rewritten in terms of uncoupled, gyroscopic and damped flexible modes (Vigneron, 1981, 1985) and the associated controllability and observability criteria similarly derived.
DISPLACEMENT AND ROTATION DEVICES Definition8 The previous section emphasized the need to determine the controllability and observability of a flexible structure using actuation and measurement of only the otb and pt-order spatial derivatives of the vibrations. This section presents the models for these devices. Since the action of real devices is not concentrated on one point but is distributed over a small region, we begin their modelling with the following definitions. Defini tion 2 : The Distributed-Identity Operator
l{(x"t) = [v(x"t) le [4>(x)]
= Q(x.)2.(t)
Q(x.) =
[~(x.) (d~dx)%.
(£!2P-l ~ dx 2p -
1
)",
r.
(23) Examination of Eqs. (19) and (23) shows that
Q(x) = JiT(x)
The theory just described has provided definitive conclusions on the types, number and location of sensors/actuators necessary to achieve observability and controllability. When confronted with the physical implementation of its principles, the following potential problems become evident: 1. requirement to detect/excite all 2p spatial derivatives of the structural deformations;
2. requirement to co-locate all actuators at Xa and all sensors at x. (location Xa not necessarily the same as that of x.); 3. assumption of point sensors and actuators; 4. assumpt: on of a one-dimensional structure; 5. no damping or gyroscopic effects. The most familiar types of sensors and actuators are those that measure displacement and rotation and apply force and torque. It is worth mentioning that recent work (Bailey and Hubbard, 1985) has shown that some types of actuator can directly excite mode-shape spatial derivatives of order greater than one. However, even if all these higher-order devices could be mechanized, their physical co-location may not be possible. On the other hand, force/torque
4>(0 de
Defini tion 3 : The Distributed-Slope Operator
Se [4>(x)]
Theory V8 Reality
!
***
(24)
which is expected since the measurements assumed here are the dual operation of the excitations described earlier. Consequently, the same arguments will prove the observability of system (21)-(22) . Again, the required number of sensors is 2p.
%+e/2
~
%-e/2
(22) where
6
6
~ [4>(x + t:/2)
- 4>(X - t:/2)]
* •• The names of these two operators stem from the following properties: lim le [4>(x)] = 4>(x),
(25)
e~O
.
d4>(x),
hm Se [4>(x)] = -d- = X
E'-O
4> (x),
(26)
where' denotes differentiation with respect to the spatial coordinate x. The following properties of these operators will prove useful in later developments: d
dx le!4>(x)] = Se!4>(x)], dd Se[4>(x)] = Se[4>'(x)], x
l e[4>'(x)] = Se[4>(x)],
~le[4>(x)] = dx
le[4>'(x)] .
(27)
Application to Unit Step and Delta Function8 With the step function U(x) :
U(x - xa) ~ {
0,
(28) 1,
we can formally define the Delta function with the slope operator:
o(x - xa)
6
lim Se[U(x - xa)] = U'(x - xa).
e~O
(29)
Sl'llsori.\ctllator Selectioll alld I'L1CCIlll'llt
The Delta function arises naturally in the modelling of point forces and point displacement measurements. In reality however, the action of the device is distributed over a small distance, say e, around the point of interest and it is reasonable to assume that the action of the distributed force is averaged over e and that the sensor actually measures the average displacement over e. (The distribution interval does not need to be equal for sensors and actuators, or 1 and S, but the same symbol t: is used here for notational simplicity.) The Distributed Identity cperator is used to model such a distributed action:
! ! !
00
4>(x)6'(x - xa) dx = -4>'(x a),
(34)
4>(x)l, [6(x - xa)) dx = +1, [4>(xa)] ,
(35)
4>(x)l, [6'(x - xa)) dx = -Se [4>(x a)] .
(36)
-00
+00
-00
+00
-00
1,[6(x - xa)) = S,[U(x - xa)).
(30)
The graph of 1, [6(x - xa)) is shown in Fig. la and illustrates the distributed action of a device. Naturally, as t: --+ 0, this function tends to an impulse at X a , as predicted by Eq. (25) .
r
I t
-+-._---'-I-'-T__~
L
[ x a -"2-
xa+ -~
xa
Point and Distributed Actuators With the above definitions and properties, we are now equipped with the analytical tools necessary to model the point (p.) and distributed (d.) displacement and slope actuators. The equivalent force distribution I(x, t) and the associated modal excitation In(t), derived from Eq. (9), are now presented, assuming the action is centered at Xa:
.......-- .- - [ ---~
I
These integrals will help the evaluation of the modal excitations.
2
X
P. force
1(0) :
D. force
1(0) :
P. torque
1(1) :
D. torque
1(1) :
Fig. la. The distributed Dirac Delta function.
I'
i[
l cl
I(
x· X a) J
I I I
~---£---
I
1".
x +_f:_
1.----"'-X
a
2
£
n
- .-
2
Xa
f - - £---j Fig. lb. The distributed first derivative of the Dirac Delta function.
1111111 Xa
Similarly, the first derivative of the Delta function can be formally defined as:
6'(x - xa) ~ ,_0 Iim S,[6(x - xa)).
1£[6'(x - xa)] = S,[6(x - xa)) .
L
]
]'.:""
(32)
The representation of 1, [6'(x - xa)] is shown in Fig. lb. As e --+ 0, Eq. (32) degenerates to Eq. (31) and the graph on Fig. 1b becomes an impulse pair (i.e. pure torque) at the device location Xa. With the above definitions for 6 and 6', the following function inner products are easily derived (de Lafontaine and Stieber, 1986):
!
1",,',
(31)
Point torques and point slope sensors are naturally modelled by this function. When the action of these devices is distributed over an interval e, the distributed derivative of the Delta function becomes:
1101/'
1111/2,
f(11/2 ,
00
-00
4>(x)6(x - xa) dx = H(xa),
(33)
Fig. 2. Graphical representation of the actuators defined in Eqs. (37)-(40).
.I-
:.! :I II
dt' LI\'olllaillt' ;llId \1. F. Slit'itt-r
Point and Distributed Sensors
APPLICATION TO A FLEXIBLE BEAM
The point (p .) and distributed (d .) displacement/slope sensors are similarly defined . The 2 x 1 column matrix of measurements ~ contains the displacement y(O) (x" t) and rotation y(ll(x"t) of the structure at x. :
We must now determine if the controllability and observability conditions described earlier are still preserved with the reduced number of sensors and actuators. The particular case of a slender, flexible beam is investigated. This model contains many simplifying assumptions but, as mentioned earlier, it has been extensively used in the literature as a reference example and it simplifies the illustration of basic principles related to distributedparameter systems. The availability of analytic expressions for its mode shapes will help the derivation of results that may be applicable to more complex structures.
~= [ y( O I(x"t)
y(l)(X"t)]T .
(41)
Using the matrices of mode shapes ~r(x.) and 2,T(x.) = (d2 T /dx)% ,Eq. (17), we can model these sensors as follow : •
P. Displacement: D. Displacement:
y(OI(x"t) = 2 T (x.)2(t), (42) T y(O)(x"t) = I, [2 (x.)]2(t),(43)
y(1){x"t) = 2 1T {x.)2{t), (44) D. Slope : y(l){x" t) = S, [2 T {x.)]2{t) .{45)
We will first prove controllability and observability for the free-free beam with point sensors and actuators and then generalize this result to other boundary conditions and to sensors and actuators with distributed action.
P. Slope :
These models for the sensors and actuators will now enter the input /output equations of a flexible structure.
Controllability and Observability Norms The control distribution and output matrices!l and Q of Eqs. (21) and (22) can now be rewritten for the case where only the sensors and actuators for the Otb and 1st -order spatial derivatives are considered: (p. actuator) (46) (d . actuator)
[2{x.)
2'{x.)
Q{x.) =
{ [ I, [2{x.)]
r
S, [2{x.)]
!!= [f( O)(t)
r
-!f(1)(t)]T.
+ 4>:?{xa)]t, On = [4>~{x.) + 4>:?(x.) ]t.
(49)
rf
(50)
The theorems proved in the above reference state that, for distinct eigenvalues, one has:
Cn > 0 "In,
point actuators,
C,n > 0 "In,
distributed actuators, (51) point sensors,
On> 0 "In, Observability iff
{
for at least one of the modes will make the system uncontrollable and unobservable at Xo. Consequently, Eq. (54) provides the solution for the location{s) Xo where the devices must not be located. This criterion is now applied to the free-free beam.
4>n(x) = cosh,Bnx+ cosPnx-ern{sinh,Bnx+sin.Bnx). (55)
C,n = [{ I, [4>n(X a )]} 2+ { S, [4>n{X a)]} 2] t,
{
(54)
(48)
We here extend these definitions to the 'distributed norms'
Controllability iff
Conversely, any position Xo where
Free-free beam. The mode shapes of a free-free slender flexible beam of length L (Blevins, 1979) are given by
C,n and O,n:
+ { S,[4>n{x.)]
(53)
(47) (d. sensor)
Cn = [4>~(xa)
r
For point devices, Eqs. (49) and (51)-(52) state that controllability or observability of all modes is ensured if the actuating or sensing device is positioned at a location Xo for which either one of the following two conditions is met for all modes:
(p. sensor)
The general theoretical results developed earlier are now specialized for this case. Following (Hughes and Skelton, 1980a), we define the controllability norm Cn and observability norm On of mode n by :
O,n = [{ I, [4>n(X.)]
Point Sensors and Actuators
The boundary condition at x = 0, 4>~(0) = 4>~'(0) == 0, is clearly satisfied by Eq. (55) . In order to satisfy the same (free) boundary condition at x = L, the ern must be given by sinh.BnL + sin,Bn L (56) ern = cosh,Bn L - cos,Bn L and the characteristic equation provides the solutions for the ,Bn: (57) cosh.BnL cos.BnL = +1. Equation (55) does not include the two rigid-body modes of the free-free beam . Since their associated eigenvalues are repeated, we will first concentrate our attention on the distinct eigenvalues of the flexible-body modes and consider the rigid-body ones afterwards. Let us suppose that there are locations Xo that satisfy Eq. (54). Application to Eq. (55) generates the following two additional conditions: cosh.Bnxo cos.Bnxo = -1, 2
tanh .Bnxo = tanh.Bnxo tanh.BnL.
O,n > 0 "In, distributed sensors.
(52) When repeated eigenvalues exist, the rank of the associated partitions in the !l(x a ) and Q{x.) matrices must be equal to the multiplicity of the eigenvalues. Alternatively, the controllability and observability norms defined in Eqs. (49)-(50) can be generalized to normalized determinants, as is done in (Hughes and Skelton, 1980a). Conditions (51)-(52) would still apply.
(58) (59)
The common factor tanh .Bnxo in the second equation gives Xo == 0 as a first solution. However, Xo == 0 violates Eq. (58) and we conclude that Xo f. O. Dividing both sides of Eq. (59) by tanh .Bnxo, we arrive at tanh .Bnxo = tan ,BnL
(60)
which provides a second solution: Xo = L. Again, substitution of Xo = L into Eq. (58) results in a contradiction of the characteristic equation (57) . There are no solutions for Xo and we may therefore conclude that all the flexible-body m odes are controllable and observable with displacement and slope devices at any location Xo.
~ :\
For the repeated eigenvalues of the rigid-body modes, it is easy to show that the same two devices will meet the rank condition on the control distribution and output matrices. Hence, they will ensure controllability and observability of the rigid modes as well. We may therefore state the following general result: Observation
Distributed Sensors and Actuators When the action of the devices is distributed over an interval e around an arbitrary location Xo, the controllability conditions (50)-(52) require the following condition be satisfied for all modes:
1 :
All the modes of a free-free beam are controllable (observable) if two point actuators (sensors) of the Olh (i.e. displacement) and 1'1 (i.e. slope) derivatives are co-located at an arbitrary point Xa (x,) (location Xa not necessarily the same as that of x,) .
I
(61) Therefore, the system will be uncontrollable and unobservable by a device at location Xo if 3; 0+e/2
I
4>n(x)dx
=
[4>n(Xo + e/2) - 4>n(XO - e/2)] = 0 (62)
%0-e/2
Furthermore, inspection of Eq . (55) will show that we always have 4>n(O) i 0, 4>n(L) i 0, 'In . The following additional conclusion results. Observation
for at least one of the modes. A necessary condition for Eq. (62) to be satisfied is a device distribution e that covers two Olh-order nodes (recall Definition 1) of that mode:
2 :
Observation
All the flexible-body modes of a free-free beam are controllable (observable) with one point actuator (sensor) of the displacement at any one of the two boundaries. The norms defined in Eq. (49) provide a measure of the controllability and observability of each distinct-eigenvalue mode. The presence of the two repeated (zero) eigenvalues of the rigid-body modes requires two actuators (sensors) to meet the rank condition of the J1 (Q) matrix partitions. Therefore, controllability (observability) of the rigid-body modes requires a second actuator (sensor) at another (arbitrary) point on the stucture . Clamped-Free Beam. For a clamped-free beam, for which 4>n(O) = 4>~(0) = 0, a similar derivation clearly shows that a unique double-node solution exists at Xo = o. This is not surprising since th~ derivatives 4>n(O) and 4>~(0), n = 1,2,··· are constramed to zero by the boundary conditions at x = o. We may therefore state the following result: Observation
3 :
All the modes of a clamped-free beam are controllable (observable) if two point actuators (sensors) of the Olh (i .e. displacement) and 1'1 (i .e. slope) derivatives are co-located at an arbitrary point Xa (x,) that does not correspond to the point where the beam is clamped (location Xa not necessarily the same as that of x.) . At the free end of the beam, one can show that the mode shapes never vanish and, in parallel with Theorem 4, we have: Observation
4:
All the vibration modes of a clamped-free beam are controllable (observable) with only one point actuator (sensor) of the displacement at the free boundary x = L. Unlike the free-free beam, the absence of rigid-body modes in the clamped-free case avoids the requirement for two devices. Other boundary conditions. Although a general proof for any type of boundary conditions may exist but has not been derived by the authors, the above analytical proofs can be duplicated for any particular case of interest. It is expected that controllability and observability will be ensured by the two co-located types of device at any unclamped position along a flexible beam and that only one type (displacement sensor/actuator) will be required at free boundaries.
5:
If the distributed controllability (observability) norm Cen (Ocn) of a mode vanishes when the Olh and pI-order derivatives actuator (sensor) are distributed over the same interval [xo - e/2, Xo + e/2], this interval must contain at least two Olh_ order nodes of that mode.
This result is deduced from Eq. (62) . First, we must have 4>n(xo - e/2) = 4>n(xo + e/2) for the distributed slope to vanish. On the other hand, if the distributed (i .e. averaged) displacement vanishes, the mode shape must have both negative and positive displacements between Xo - e/2 and Xo + e/2 . The combination of these two conditions proves the theorem. A practical consequence of this result applies to the control (observation) the first k modes of a flexible beam with a pair of distributed actuators (sensors). If the minimum distance d between the Olh-order nodes of the klh mode is computed, designing the devices with e < d will ensure controllability and observability of the first k modes. This minimum distan ce tends to decrease with increasing frequency so that, for a given extent e of the devices, the occurrance of unobservable and uncontrollable modes becomes more likely with increasing frequency. The distributed norms defined in Eqs. (50) will provide a useful measure of controllability and observability. Finally, from the general mode-shape equation of a flexible beam, one can verify that the distributed norm associated with displacement varies according to:
le [4>n(xo)] - 1/(frequency)1/2 that the action on high-frequency modes is attenuated by the distribution of the device. These analytical results confirm that high-frequency modes are not easily excited or measured by actuators and sensors which is well known in modal testing. 80
FEEDBACK CONTROL Although it has been shown in this paper that it is not necessary to co-locate sensors and actuators in order to achieve complete system controllability and observability, co-location (x, = xa) is usually desireable in order to obtain closed-loop stability for system control. With dual sensors and actuators ('sensuators' with Q = J1T, which implies x, = x a ), asymptotic stability can usually be achieved by rate output feedback, as easily proved by a Liapunov argument . However, this assumes that the controller (as well as the sensors and actuators) has an infinite bandwidth, covering all structural modes. The design of realistic, limited bandwidth feedback control laws still remains a challenge, in spite of the encouraging results on controllability/observability.
J.
dl' L,fOIlUilll' ;t lld \1. F. SI il'iln
CONCLUSIONS This paper has developed controllability and observability conditions for distributed-parameter systems and has a:Tived at the types (spatial derivatives), number (2p = order of the PDE) and location (anywhere on the structure) of sensors/actuators necessary to meet these conditions for the theoretically infinite number of modes. This theory was applied to the vibrations of a flexible beam and revealed that only two realizable types of sensors and actuators (displacement and rotation) are required to ensure controllability and observability at any uncon3trained location. In addition, only one displacement sensor/actuator is necessary at free boundaries. The distributed (rather than point) action of realistic devices leads to a degradation of controllability and observability and an attenuation of the excit ations/measurements of high-frequency modes. Although this theory was derived using a continuum modelling approach with a theoretically infinite number of modes, the results can be applied to structural models obtained by other techniques, such as finite elements. The controllability and observability criteria presented in this paper rely mainly on a knowledge of the mode shapes and are independent of the techniques used to obtain ther:l .
REFERENCES Bailey, T ., Hubbard, J .E. (1985) . Distributed PiezoelectricPolymer Active Vibration Control of a Cantilevered Beam. J. Guidance, Control and Dynamics, Vo!. 8, No. 5, pp. 605-611.
Delfour, M.C., Polis, M.P., Payre, G. (1985) . Sensor and Actuator Location for Large Flexible Space Structures. Report DOC-CR-SP-85-033, Department of Communications, Ottawa, Canada. Hughes, P.C. (1984) . Space Structure Vibration Modes: How many exist? Which ones are important? Technical 252, University of Toronto Institute for Note No. Aerospace Studies, Toronto, Canada. Hughes, P.C ., Skelton , R.E. (1980a) . Controllability and Observability of Linear Matnx-Second-Order Systems. J. Applied Mechanics, Vo!. 47, pp. 415-420. Hughes, P.C., Skelton, R.E. (1980b) . Controllability and Observability for Flexible Spacecraft. J. Guidance and Control, Vo!. 3, No. 5, pp. 452-459. Kailath, T . (1980). Linear Systems. Prentice-Hall, Inc., Englewood Cliffs, N.J. Kreyszig, E. (1972) . Advanced Engineering Mathematics, Third Edition. John Wiley &- Sons, Inc., pp . 74-79. Lindberg, R.E., Longman, R.W. (1984) . On the Number and Placement of Actuators for Independent Modal Space Contro!. J. Guidance, Control and Dynamics, Vo!. 7, No. 2, pp. 215-222 . Meirovitch, L. (1967) . Analytical Methods in Vibrations. The MacMillan Co., New York, N.V. Meirovitch, L. , VanLandingham, H.F., OZ, H. U979) . Distributed Control of Spinning Flexible Spacecraft. J. Gllidance and Control, Vo!. 2, No. 5, pp. 407-415.
Balas, M.J. (1976) . Feedback Control of Flexible Systems. IEEE 'frans. Automatic Control, Vo!. AC-23, No. 4, pp. 673-679
OZ,
Blevins, R.D. (1979) . Formulas for Natural Frequency and Mode Shape. Van Nostrand Reinhold, New York, N.Y.
Vigneron, F.R. (1981). Natural Modes and Real Modal Variables for Flexible Spacecraft. CRC Report No. 1348, Department of Communications, Ottawa, Canada.
de Lafontaine, J ., Stieber, M.E. (1986) . Controllability and Observability for Distributed-Parameter Systems. CRC Report to appear, Department of Communications, Ottawa, Canada.
H., Meirovitch, L. (1980) . Optimal Modal Space Control of Flexible Gyroscopic Systems. J. Guidance and Control, Vo!. 3, No. 3, pp. 215-221.
Vigneron, F .R. (1985) . A Natural Modes Formulation and Modal Identities for Structures with Linear Viscous Damping. CRC Report No. 1382, Department of Communications, Ottawa, Canada.
C()p ~ riglll S ~S It'IlI "',
1' , 1l'. lllH·1t' l
SENSOR/ACTUATOR SELECTION AND PLACEMENT FOR CONTROL OF ELASTIC CONTINUA
J.
de Lafontaine* and M. E. Stieber**
"' Dl ji' II(1' R I'II'II/( /i /:'I III "'I.I /i lll /'/II ()IIII'I'II. /) 1'/){III IIII'1I 1
0/
X " lilllIlI/ O I'/ I'I Il'/' . 011"«,,, .
C""I1r111 1\1 .4 I!I-I ~:;: C/!lIIl1lllll i ((lt i ()lI.\
N I' ... I'(l u ·/i C f' nlr(' .
J)l'j)(lrtlf/ (, lIt
of
C()lIlltll llli C(ltlO lI.\ ,
Oll" u'" . el/l/lu/" 1\211 8S2
Abstract. General controllability and observability conditions are developed for a flexible body whose dynamics are modelled by partial differential equations. These conditions reveal the types, number and location of actuators and sensors necessary to ensure controllability and observability of all modes of vibration . It is proved that the number and types of sensors/actuators required are directly related to the order of the partial differential equation. Controllability and observability can be guaranteed with the sensors/actuators located at an arbitrary point on the structure, although the degree of modal excitation/detection depends on their particular position.
The implementation of the above principles relies on co-location of all the sensors, and co-location of all the actuators. In general, it also requires devices which may not be easily mechanized . Fortunately, the general controllability and observability criteria are quite conservative and it is shown that, under realistic boundary conditions, they are satisfied with just two types of conventional sensors and actuators. Moreover, in most cases, complete controllability and observability of the flexiblebody modes can be achieved with only one sensor/actuator pair located at a free boundary. For illustration, the above principles are applied to an undamped flexible beam with arbitrary boundary conditions. Extensions to damped and gyroscopic systems are pointed out . The case of distributed sensors and actuators is modelled using the definitions of 'Distributed Identity' and 'Distributed Slope' operators. It is verified analytically that the distributed action of these devices leads to an attenuated detection/excitation of higher-frequency vibrations. Although the structure was idealized by using a continuum modelling approach with a theoretically infinite number of modes, the above results can be applied to the identification and control of more complex structures modelled by other techniques, such as the finite element method. Keywords. Controllability; observability; distributed parameter systems; actuators; sensors; modal control; flexible structure control; space vehicles.
INTRODUCTION
the discretization has the disadvantage that the spatial derivatives of the vibrations are not directly available in the model, which, in this paper, will be shown to be of significance.
Satellites for communication and surveillance require the control of large antenna systems to very stringent attitude and shape tolerances. The philosophy of spacecraft control design therefore has evolved from one of minimal excitation of structural vibrations to one of their active control. For this problem, the selection/placement of sensors and actuators is an important step in the overall control design process. This issue has not yet been resolved and is often dealt with in an ad-hoc fashion, while a considerable amount of research has been devoted to the improvement of modelling and control synthesis procedures.
Recently, the modelling of structures as DistributedParameter Systems (DPS) has gained the attention of the controls community. Starting from a simplified model of the structure by assuming a continuous spatial distribution of its mass and stiffness, an 'exact' analysis can proceed with a theoretically infinite number of vibration modes. The existence of the high-frequency modes is questionable (Hughes, 1984) but their inclusion into the model avoids the spillover problem. Although not always practical for real structures, the DPS approach provides interesting insights that can guide the development of the finitedimensional approach . For instance, it is known that only one pair of sensors and actuators will provide control and observation of a very large number of modes of a flexible beam (Delfour, Polis and Payre, 1985) . However, controllability and observability theorems for DPS are not straightforward since one is now dealing with infinitedimensional matrices. On the other hand, the natural modelling of the spatial derivatives of the vibrations will be shown to compensate for this difficulty. This paper combines the above two techniques and derives general controllability and observability criteria for the 'infinite' number of vibration modes of a flexible structure. These criteria will in turn provide definite answers to three fundamental questions of control-system design: what types of sensors/actuators are required, how many, and where must they be located.
Most of the work on flexible structure control reported in the literature is based on structural models of finite dimensions, which are obtained by spatial discretization e.g. using the finite element method. This modelling approach generates relatively accurate models (10,000 degrees of freedom) of complex flexible structures. However, analysis and control-system design are only practical on a truncated version of the model, giving rise to control energy 'spillover' to residual (unmodelled) modes (Balas, 1976) . Some of the control techniques proposed to overcome thIS problem require as many sensors and actuators as there are modelled vibration modes (Meirovitch, VanLandingham and Oz, 1979; Oz and Meirovitch, 1980) although a more recent formulation (Lindberg and Longman, 1984) has relaxed this requirement. The controllability and observability criteria that provide the types, number and locations of these devices are defined in terms of the rank of finite-dimensional matrices (Kailath, 1980) . However,
22S
MODEL OF ELASTIC CONTINUA Assumptions
An elastic body contained in a spatial domain [) spanned by the position column matrix r is modelled by assuming a continuous distribution of its mass and stiffness properties over [) . A static force distribution f(r.) on the body will induce deformations !!.(r:) that we as-;ume small. Stress is linearly related to strain through the linear, positive semidefinite, self-adjoint stiffness operator K:
(1)
where the modal coordinates qn(t) represent the timevarying contribution of mode n to the total response. After substitution of solution (7) into Eq. (3), the orthonormality conditions of Eqs. (6) are invoked to arrive at a set of uncoupled equations:
The modal excitations fn (t) are the components of the forcing function along the basis functions ctn (r:) :
fn(t) =
Jct~(rJL(r,t)
(9)
dr·
/)
which contains differentials with respect to r up to order = 1 while p = 2 for a slender flexible beam. Over the boundary of [), these deformations are constrained by conditions of the form
By defining the infinite-dimensional matrices 9,(t) = T,..,2 2' " }, _t F() = [ql(t) q2(t) '], !! = d'lag {2 W l , W2
(2)
[Idt) hit) the harmonic equations shown in Eq. (8) can be condensed into the familiar form :
2p. For example, a vibrating string has p
ai!!.=Q,
i=1,2," ' ,p
y,
a
where the i are also linear differential operators of order through 2p - 1. Structural damping and gyroscopic effects are neglected here but we will point out how these effects can be considered . Equations of Motion
When a dynamic force distribution fir, t) is applied on the flexible body, the dynamics of its vibrations are governed by the following equations (Meirovitch, 1967) :
M !i.(r, t)
+ K !!.(r, t) = 1..(r, t),
ai!!.(r, t) = Q,
i = 1,2" .. ,p.
(la) The physicai coordmates that describe the vibrations can be recovered from the modal coordinates through the matrix equivalent of Eq. (7):
!!.(r, t) = ~T (r)9,(t)
(11)
where the infinite-dimensional (00 x 00) matrix of mode shapes ~(r) is defined as
(3)
(4)
The mass operator M is a linear, positive-definite, selfadjoint operator of order less than 2p . Overdots denot e time differentiation.
Equations (10) and (11) form the basis upon which the controllability and observability properties of the system will be developed.
Solution (Homogeneous System)
Under homogeneous condition, the force distribution fir, t) in Eq . (3) is set to zero. The spatial and ten:;-poral coordinates can be separated by substitution of the harmonic solution !!.(r, t) = 4>(r)e iwt into the boundaryvalue problem (4) and the f;:;-llowing eigenvalue problem results:
(5) This last differential equation, of order 2p, together with the boundary conditions, define the infinite-dimensional set of mode shapes 4> (r) that characterize the vibrations at frequencies W n . U-;in~ the self-adjoint property of operators M and K, the following orthonormality conditions can be easily proved :
Jct~ Jct~Kctn
Mct n dr = om,n
/)
(6) dr = wmwnom,n,
/)
CONTROLLABILITY AND OBSERVABILITY Basic Principle
Our analysis of the controllability and observability of a flexible body is based on one definition and two theorems that are best illustrated by consideration of a one-dimensional structure. Not only does this assumption simplify the notation and presentation of the arguments but this type of structure represents a well-known and often used reference example in the study of distributedparameter system. Furthermore, many real structures with symmetric mass and stiffness distribution can be decomposed into a series of such (single-axis), interconnected, one-dimensional elements. The vibrations v(x, t) are assumed along the z direction . The domain [) of the spatial coordinate is 0 ~ x ~ L . Following the notation of the preceding section, the displacements are expanded into:
v(x, t) =
L
4>n(x)qn(t) = ~T(x)9,(t)
(13)
where om ,n is the Kronecker delta. These conditions allow the expansion of the forced response into an infinite series of uncoupled modes.
and Eq. (10) still applies. For this system, the concept of a generalized node is now introduced.
Solution (Forced System)
Definition 1 : Generalized Node
When an external force distribution fir, t) is exerted on the body, Eqs. (3)-(4) apply. From-their linearity, the expansion theorem tells us that the response of the system can be represented as a linear combination of admissible functions that form a complete basis in the domain of the operators (Meirovitch, 1967) . The natural modes ctn (r:) can be used in this expansion :
n=1
A mode 4>n(x)qn(t) has a th-order (generalized) node at a position xo if and only if its th spatial derivative at xo vanishes, i.e.: ph-order node at
TO
{=>
[Bi4>n(x)qn(t)] == 0, 'rIt. Bx' "' 0
00
!!.(r,t) =
L ctn(r:) qn(t) "=1
(7)
***
For instance, the familiar displacement node of a vibrating structure is a oth-order node. This concept helps simplify the following theorem: Theorem
=0)
l
[ a 4>n(x)qn(t)] ax} Xo
<==:>
j=0,1, . . . ,2p-l,Vt :,; x
*
The proof of this theorem follows directly from the Wronskian (Kreyszig, 1972) of the 2p independent solutions of the eigenvalue problem. Details can be found in (de Lafontaine and Stieber, 1986). To illustrate this result, let us consider the 4 th -order differential equation that describes the mode shapes associated with the vibrations of a uniform, slender beam of length L:
ddx~X) + f344>(X)
= 0,
(14) where .Y'.x is the curl matrix operator. For j = 0, this equation represents the modal displacement at XQ, measured along the z-axis; when j = 1, it gives the modal slope at Xo, measured along the y-axis, and so on . For a non-trivial mode, Theorem 1 tells us that the 2p column matrices (.Y'.X)lt,(ro) cannot all vanish, irrespectively of the choice of ~, unless mode 4>n(x)qn(t) is not excited. This basic prinCiple allows the development of the following controllability and observability results.
Controllability Let us assume a force distribution f(r., t) composed of 2p point actuators that can each excite one of the first 2p spatial derivatives of a vibrating structure:
2p-l
0:-:::: x :-: : L.
Dr., t) = The general solution for the mode shapes 4>n(x) can be written as:
4>n(X) = acoshf3nx
+ bcosf3nx + csinhf3nx + dsinf3n x.
If the first four derivatives of 4>n(x) vanish at, say, XQ, we get the following 4 algebraic equations:
L
cosf3XQ - sin f3xQ - cosf3xQ sin f3xQ
(j
~ I)! (.Y'.xY
[L(i)6(r. -
J
4>n(x)f(x,t) dx
D
2f:l (-1)1 1=0 (J
°
The analysis only requires the knowledge of the first 2p nodes since those of order greater than 2p can be expressed as linear combinations of the first 2p ones. Theorem 1 implies that an excited mode must have at least one nonzero spatial derivative in the first 2p ones. This, in turn, means that a mode can be excited or sensed through actuation or measurement of that nonzero derivative. This theorem can be generalized to consider modes associated with repeated eigenvalues.
(15)
Using the properties of the Dirac function 6, the modal excitations fn(t) can be easily derived after insertion of Eq. (15) into Eq. (9). In one-dimensional form, we have (de Lafontaine and Stieber, 1986):
sinh f3xQ cosh f3xQ sinh f3xQ cosh f3xQ
It can be easily shown that the above 4 x 4 Wronskian matrix is nonsingular for all XQ (determinant = 4) and consequently, only the trivial solution a = b = c = d = exists.
+ I)!
(16)
(d l 4>r:) dx}
fU}(t). x.
Extending definitIOn (12), we introduce:
j = 0,1,· . . ,2p - 1
(17) to arrive at the matrix of modal excitations: (18) where
(d2P-l~/dx2P-llx.]
2:
Two modes 4>n(x)q(t) and 4>m(x)q(t) associated with the same eigenvalue and derived from an eigenvalue problem of order 2p have their first 2p derivatives linearly related at XQ if and only if their mode shapes 4>n (x) and 4>m( x) are linearly related, i.e.
l axl [a 4>n(x)q(t)]
r.,,)].
}=o
fn(t) = cosh f3xQ sinh f3xQ [ cosh f3xQ sinh f3xQ
4>n(X)
° XQ IT are:
1:
A mode 4>n(x)qn(t), derived from an eigenvalue problem of order 2p, has Its first 2p nodes located at the same (arbitrary) position XQ if and only if this mode is the trivial one, i.e.:
Theorem
ro = [0
the derivatives at
_ k -
[a l 4>m(x)q(t)] axl
X I)
._ J-O,I, ... ,2p-l,
(19) 1
12Pi1
f(2 P-I)]T
.
(20)
By substitution of Eq. (18) into Eq. (10), the forcedsystem equation becomes: (21)
% ('1
=k4>m(x), "Ix E D,
k = nonzero constant.
* ** This theorem is proved by forming the function 4>(x) = 4>n(X)_- k4>m(x) and applying Theorem 1 to show that 4>(x) = 0, "Ix E P. In terms of the 3-dimensional notation presented earlier,
The controllability of system (19)-(21) is determined from the theorems and corollaries proved by Hughes and Skelton (1980a, 1980b). For distinct eigenvalues, this reference states that all the rows in the ll(xa) matrix above must have at least one nonzero entry. This is guaranteed by Theorem 1. For repeated eigenvalues, the rank of the associated partition of the I1 matrix must be equal to the multiplicity of the eigenvalues. The orthogonality of the mode shapes and Theorem 2 guarantee the satisfaction of this requirement. In fact, all the rows in the 1l matrix are linearly independent, thus ensuring complete controllability for any multiplicity of eigenvalues.
This theoretical result proves that 2p co-located actuators, each one exciting a specific spatial derivative of the mode shapes, are sufficient to ensure controllability of all the vibration modes of the structure. This result is quite general since it does not imply a particular system order nor does it assume a particular set of boundary conditions or a particular location for the co-located actuators. The requirement for as many actuators as there are modes is now replaced by a requirement to have as many actuators as the order of the eigenvalue problem. A consequence of this result is that the multiplicity of eigenvalues can not exceed 2p. In practice however, when the order of the PDE exceeds 2, the above results require the mechanization of 2p - 2 actuators that may not be physically realizable. For instance, a slender flexible beam (2p = 4 > 2) would require actuators of the second and third spatial derivatives of the mode shapes, in addition to conventional force and torque actuators. Fortunately, we shall soon see that, for most physical systems, this does not restrict the usefulness of the above theory since only one and possibly two types of conventional actuators (otb and pt-order derivatives) are necessary.
Ob8ervability In a similar development, we assume that 2p point sensors can measure each of the 2p spatial derivatives of the modes at a position 1:. = [0 0 x.]T. The measurement column matrix takes the form:
actuators and displacement/rotation sensors, respectively, can be co-located since their action is on different axes of the structure (see Eqs. (14)). Consequently, the first two limitations of the theory can be overcome if controllability /observability can be proved with only the otb and pt-order derivative devices. This task is considered in the next section, along with the investigation of the effects of distributed rather than point actuators/sensors, thereby taking care of limitation 3 as well. Extensions to three-dimensional structures can be formally carried out with the generalized version of Eqs. (14). Finally, Theorems 1 and 2 could be rewritten in terms of uncoupled, gyroscopic and damped flexible modes (Vigneron, 1981, 1985) and the associated controllability and observability criteria similarly derived.
DISPLACEMENT AND ROTATION DEVICES Definition8 The previous section emphasized the need to determine the controllability and observability of a flexible structure using actuation and measurement of only the otb and pt-order spatial derivatives of the vibrations. This section presents the models for these devices. Since the action of real devices is not concentrated on one point but is distributed over a small region, we begin their modelling with the following definitions. Defini tion 2 : The Distributed-Identity Operator
l{(x"t) = [v(x"t) le [4>(x)]
= Q(x.)2.(t)
Q(x.) =
[~(x.) (d~dx)%.
(£!2P-l ~ dx 2p -
1
)",
r.
(23) Examination of Eqs. (19) and (23) shows that
Q(x) = JiT(x)
The theory just described has provided definitive conclusions on the types, number and location of sensors/actuators necessary to achieve observability and controllability. When confronted with the physical implementation of its principles, the following potential problems become evident: 1. requirement to detect/excite all 2p spatial derivatives of the structural deformations;
2. requirement to co-locate all actuators at Xa and all sensors at x. (location Xa not necessarily the same as that of x.); 3. assumption of point sensors and actuators; 4. assumpt: on of a one-dimensional structure; 5. no damping or gyroscopic effects. The most familiar types of sensors and actuators are those that measure displacement and rotation and apply force and torque. It is worth mentioning that recent work (Bailey and Hubbard, 1985) has shown that some types of actuator can directly excite mode-shape spatial derivatives of order greater than one. However, even if all these higher-order devices could be mechanized, their physical co-location may not be possible. On the other hand, force/torque
4>(0 de
Defini tion 3 : The Distributed-Slope Operator
Se [4>(x)]
Theory V8 Reality
!
***
(24)
which is expected since the measurements assumed here are the dual operation of the excitations described earlier. Consequently, the same arguments will prove the observability of system (21)-(22) . Again, the required number of sensors is 2p.
%+e/2
~
%-e/2
(22) where
6
6
~ [4>(x + t:/2)
- 4>(X - t:/2)]
* •• The names of these two operators stem from the following properties: lim le [4>(x)] = 4>(x),
(25)
e~O
.
d4>(x),
hm Se [4>(x)] = -d- = X
E'-O
4> (x),
(26)
where' denotes differentiation with respect to the spatial coordinate x. The following properties of these operators will prove useful in later developments: d
dx le!4>(x)] = Se!4>(x)], dd Se[4>(x)] = Se[4>'(x)], x
l e[4>'(x)] = Se[4>(x)],
~le[4>(x)] = dx
le[4>'(x)] .
(27)
Application to Unit Step and Delta Function8 With the step function U(x) :
U(x - xa) ~ {
0,
(28) 1,
we can formally define the Delta function with the slope operator:
o(x - xa)
6
lim Se[U(x - xa)] = U'(x - xa).
e~O
(29)
Sl'llsori.\ctllator Selectioll alld I'L1CCIlll'llt
The Delta function arises naturally in the modelling of point forces and point displacement measurements. In reality however, the action of the device is distributed over a small distance, say e, around the point of interest and it is reasonable to assume that the action of the distributed force is averaged over e and that the sensor actually measures the average displacement over e. (The distribution interval does not need to be equal for sensors and actuators, or 1 and S, but the same symbol t: is used here for notational simplicity.) The Distributed Identity cperator is used to model such a distributed action:
! ! !
00
4>(x)6'(x - xa) dx = -4>'(x a),
(34)
4>(x)l, [6(x - xa)) dx = +1, [4>(xa)] ,
(35)
4>(x)l, [6'(x - xa)) dx = -Se [4>(x a)] .
(36)
-00
+00
-00
+00
-00
1,[6(x - xa)) = S,[U(x - xa)).
(30)
The graph of 1, [6(x - xa)) is shown in Fig. la and illustrates the distributed action of a device. Naturally, as t: --+ 0, this function tends to an impulse at X a , as predicted by Eq. (25) .
r
I t
-+-._---'-I-'-T__~
L
[ x a -"2-
xa+ -~
xa
Point and Distributed Actuators With the above definitions and properties, we are now equipped with the analytical tools necessary to model the point (p.) and distributed (d.) displacement and slope actuators. The equivalent force distribution I(x, t) and the associated modal excitation In(t), derived from Eq. (9), are now presented, assuming the action is centered at Xa:
.......-- .- - [ ---~
I
These integrals will help the evaluation of the modal excitations.
2
X
P. force
1(0) :
D. force
1(0) :
P. torque
1(1) :
D. torque
1(1) :
Fig. la. The distributed Dirac Delta function.
I'
i[
l cl
I(
x· X a) J
I I I
~---£---
I
1".
x +_f:_
1.----"'-X
a
2
£
n
- .-
2
Xa
f - - £---j Fig. lb. The distributed first derivative of the Dirac Delta function.
1111111 Xa
Similarly, the first derivative of the Delta function can be formally defined as:
6'(x - xa) ~ ,_0 Iim S,[6(x - xa)).
1£[6'(x - xa)] = S,[6(x - xa)) .
L
]
]'.:""
(32)
The representation of 1, [6'(x - xa)] is shown in Fig. lb. As e --+ 0, Eq. (32) degenerates to Eq. (31) and the graph on Fig. 1b becomes an impulse pair (i.e. pure torque) at the device location Xa. With the above definitions for 6 and 6', the following function inner products are easily derived (de Lafontaine and Stieber, 1986):
!
1",,',
(31)
Point torques and point slope sensors are naturally modelled by this function. When the action of these devices is distributed over an interval e, the distributed derivative of the Delta function becomes:
1101/'
1111/2,
f(11/2 ,
00
-00
4>(x)6(x - xa) dx = H(xa),
(33)
Fig. 2. Graphical representation of the actuators defined in Eqs. (37)-(40).
.I-
:.! :I II
dt' LI\'olllaillt' ;llId \1. F. Slit'itt-r
Point and Distributed Sensors
APPLICATION TO A FLEXIBLE BEAM
The point (p .) and distributed (d .) displacement/slope sensors are similarly defined . The 2 x 1 column matrix of measurements ~ contains the displacement y(O) (x" t) and rotation y(ll(x"t) of the structure at x. :
We must now determine if the controllability and observability conditions described earlier are still preserved with the reduced number of sensors and actuators. The particular case of a slender, flexible beam is investigated. This model contains many simplifying assumptions but, as mentioned earlier, it has been extensively used in the literature as a reference example and it simplifies the illustration of basic principles related to distributedparameter systems. The availability of analytic expressions for its mode shapes will help the derivation of results that may be applicable to more complex structures.
~= [ y( O I(x"t)
y(l)(X"t)]T .
(41)
Using the matrices of mode shapes ~r(x.) and 2,T(x.) = (d2 T /dx)% ,Eq. (17), we can model these sensors as follow : •
P. Displacement: D. Displacement:
y(OI(x"t) = 2 T (x.)2(t), (42) T y(O)(x"t) = I, [2 (x.)]2(t),(43)
y(1){x"t) = 2 1T {x.)2{t), (44) D. Slope : y(l){x" t) = S, [2 T {x.)]2{t) .{45)
We will first prove controllability and observability for the free-free beam with point sensors and actuators and then generalize this result to other boundary conditions and to sensors and actuators with distributed action.
P. Slope :
These models for the sensors and actuators will now enter the input /output equations of a flexible structure.
Controllability and Observability Norms The control distribution and output matrices!l and Q of Eqs. (21) and (22) can now be rewritten for the case where only the sensors and actuators for the Otb and 1st -order spatial derivatives are considered: (p. actuator) (46) (d . actuator)
[2{x.)
2'{x.)
Q{x.) =
{ [ I, [2{x.)]
r
S, [2{x.)]
!!= [f( O)(t)
r
-!f(1)(t)]T.
+ 4>:?{xa)]t, On = [4>~{x.) + 4>:?(x.) ]t.
(49)
rf
(50)
The theorems proved in the above reference state that, for distinct eigenvalues, one has:
Cn > 0 "In,
point actuators,
C,n > 0 "In,
distributed actuators, (51) point sensors,
On> 0 "In, Observability iff
{
for at least one of the modes will make the system uncontrollable and unobservable at Xo. Consequently, Eq. (54) provides the solution for the location{s) Xo where the devices must not be located. This criterion is now applied to the free-free beam.
4>n(x) = cosh,Bnx+ cosPnx-ern{sinh,Bnx+sin.Bnx). (55)
C,n = [{ I, [4>n(X a )]} 2+ { S, [4>n{X a)]} 2] t,
{
(54)
(48)
We here extend these definitions to the 'distributed norms'
Controllability iff
Conversely, any position Xo where
Free-free beam. The mode shapes of a free-free slender flexible beam of length L (Blevins, 1979) are given by
C,n and O,n:
+ { S,[4>n{x.)]
(53)
(47) (d. sensor)
Cn = [4>~(xa)
r
For point devices, Eqs. (49) and (51)-(52) state that controllability or observability of all modes is ensured if the actuating or sensing device is positioned at a location Xo for which either one of the following two conditions is met for all modes:
(p. sensor)
The general theoretical results developed earlier are now specialized for this case. Following (Hughes and Skelton, 1980a), we define the controllability norm Cn and observability norm On of mode n by :
O,n = [{ I, [4>n(X.)]
Point Sensors and Actuators
The boundary condition at x = 0, 4>~(0) = 4>~'(0) == 0, is clearly satisfied by Eq. (55) . In order to satisfy the same (free) boundary condition at x = L, the ern must be given by sinh.BnL + sin,Bn L (56) ern = cosh,Bn L - cos,Bn L and the characteristic equation provides the solutions for the ,Bn: (57) cosh.BnL cos.BnL = +1. Equation (55) does not include the two rigid-body modes of the free-free beam . Since their associated eigenvalues are repeated, we will first concentrate our attention on the distinct eigenvalues of the flexible-body modes and consider the rigid-body ones afterwards. Let us suppose that there are locations Xo that satisfy Eq. (54). Application to Eq. (55) generates the following two additional conditions: cosh.Bnxo cos.Bnxo = -1, 2
tanh .Bnxo = tanh.Bnxo tanh.BnL.
O,n > 0 "In, distributed sensors.
(52) When repeated eigenvalues exist, the rank of the associated partitions in the !l(x a ) and Q{x.) matrices must be equal to the multiplicity of the eigenvalues. Alternatively, the controllability and observability norms defined in Eqs. (49)-(50) can be generalized to normalized determinants, as is done in (Hughes and Skelton, 1980a). Conditions (51)-(52) would still apply.
(58) (59)
The common factor tanh .Bnxo in the second equation gives Xo == 0 as a first solution. However, Xo == 0 violates Eq. (58) and we conclude that Xo f. O. Dividing both sides of Eq. (59) by tanh .Bnxo, we arrive at tanh .Bnxo = tan ,BnL
(60)
which provides a second solution: Xo = L. Again, substitution of Xo = L into Eq. (58) results in a contradiction of the characteristic equation (57) . There are no solutions for Xo and we may therefore conclude that all the flexible-body m odes are controllable and observable with displacement and slope devices at any location Xo.
~ :\
For the repeated eigenvalues of the rigid-body modes, it is easy to show that the same two devices will meet the rank condition on the control distribution and output matrices. Hence, they will ensure controllability and observability of the rigid modes as well. We may therefore state the following general result: Observation
Distributed Sensors and Actuators When the action of the devices is distributed over an interval e around an arbitrary location Xo, the controllability conditions (50)-(52) require the following condition be satisfied for all modes:
1 :
All the modes of a free-free beam are controllable (observable) if two point actuators (sensors) of the Olh (i.e. displacement) and 1'1 (i.e. slope) derivatives are co-located at an arbitrary point Xa (x,) (location Xa not necessarily the same as that of x,) .
I
(61) Therefore, the system will be uncontrollable and unobservable by a device at location Xo if 3; 0+e/2
I
4>n(x)dx
=
[4>n(Xo + e/2) - 4>n(XO - e/2)] = 0 (62)
%0-e/2
Furthermore, inspection of Eq . (55) will show that we always have 4>n(O) i 0, 4>n(L) i 0, 'In . The following additional conclusion results. Observation
for at least one of the modes. A necessary condition for Eq. (62) to be satisfied is a device distribution e that covers two Olh-order nodes (recall Definition 1) of that mode:
2 :
Observation
All the flexible-body modes of a free-free beam are controllable (observable) with one point actuator (sensor) of the displacement at any one of the two boundaries. The norms defined in Eq. (49) provide a measure of the controllability and observability of each distinct-eigenvalue mode. The presence of the two repeated (zero) eigenvalues of the rigid-body modes requires two actuators (sensors) to meet the rank condition of the J1 (Q) matrix partitions. Therefore, controllability (observability) of the rigid-body modes requires a second actuator (sensor) at another (arbitrary) point on the stucture . Clamped-Free Beam. For a clamped-free beam, for which 4>n(O) = 4>~(0) = 0, a similar derivation clearly shows that a unique double-node solution exists at Xo = o. This is not surprising since th~ derivatives 4>n(O) and 4>~(0), n = 1,2,··· are constramed to zero by the boundary conditions at x = o. We may therefore state the following result: Observation
3 :
All the modes of a clamped-free beam are controllable (observable) if two point actuators (sensors) of the Olh (i .e. displacement) and 1'1 (i .e. slope) derivatives are co-located at an arbitrary point Xa (x,) that does not correspond to the point where the beam is clamped (location Xa not necessarily the same as that of x.) . At the free end of the beam, one can show that the mode shapes never vanish and, in parallel with Theorem 4, we have: Observation
4:
All the vibration modes of a clamped-free beam are controllable (observable) with only one point actuator (sensor) of the displacement at the free boundary x = L. Unlike the free-free beam, the absence of rigid-body modes in the clamped-free case avoids the requirement for two devices. Other boundary conditions. Although a general proof for any type of boundary conditions may exist but has not been derived by the authors, the above analytical proofs can be duplicated for any particular case of interest. It is expected that controllability and observability will be ensured by the two co-located types of device at any unclamped position along a flexible beam and that only one type (displacement sensor/actuator) will be required at free boundaries.
5:
If the distributed controllability (observability) norm Cen (Ocn) of a mode vanishes when the Olh and pI-order derivatives actuator (sensor) are distributed over the same interval [xo - e/2, Xo + e/2], this interval must contain at least two Olh_ order nodes of that mode.
This result is deduced from Eq. (62) . First, we must have 4>n(xo - e/2) = 4>n(xo + e/2) for the distributed slope to vanish. On the other hand, if the distributed (i .e. averaged) displacement vanishes, the mode shape must have both negative and positive displacements between Xo - e/2 and Xo + e/2 . The combination of these two conditions proves the theorem. A practical consequence of this result applies to the control (observation) the first k modes of a flexible beam with a pair of distributed actuators (sensors). If the minimum distance d between the Olh-order nodes of the klh mode is computed, designing the devices with e < d will ensure controllability and observability of the first k modes. This minimum distan ce tends to decrease with increasing frequency so that, for a given extent e of the devices, the occurrance of unobservable and uncontrollable modes becomes more likely with increasing frequency. The distributed norms defined in Eqs. (50) will provide a useful measure of controllability and observability. Finally, from the general mode-shape equation of a flexible beam, one can verify that the distributed norm associated with displacement varies according to:
le [4>n(xo)] - 1/(frequency)1/2 that the action on high-frequency modes is attenuated by the distribution of the device. These analytical results confirm that high-frequency modes are not easily excited or measured by actuators and sensors which is well known in modal testing. 80
FEEDBACK CONTROL Although it has been shown in this paper that it is not necessary to co-locate sensors and actuators in order to achieve complete system controllability and observability, co-location (x, = xa) is usually desireable in order to obtain closed-loop stability for system control. With dual sensors and actuators ('sensuators' with Q = J1T, which implies x, = x a ), asymptotic stability can usually be achieved by rate output feedback, as easily proved by a Liapunov argument . However, this assumes that the controller (as well as the sensors and actuators) has an infinite bandwidth, covering all structural modes. The design of realistic, limited bandwidth feedback control laws still remains a challenge, in spite of the encouraging results on controllability/observability.
J.
dl' L,fOIlUilll' ;t lld \1. F. SI il'iln
CONCLUSIONS This paper has developed controllability and observability conditions for distributed-parameter systems and has a:Tived at the types (spatial derivatives), number (2p = order of the PDE) and location (anywhere on the structure) of sensors/actuators necessary to meet these conditions for the theoretically infinite number of modes. This theory was applied to the vibrations of a flexible beam and revealed that only two realizable types of sensors and actuators (displacement and rotation) are required to ensure controllability and observability at any uncon3trained location. In addition, only one displacement sensor/actuator is necessary at free boundaries. The distributed (rather than point) action of realistic devices leads to a degradation of controllability and observability and an attenuation of the excit ations/measurements of high-frequency modes. Although this theory was derived using a continuum modelling approach with a theoretically infinite number of modes, the results can be applied to structural models obtained by other techniques, such as finite elements. The controllability and observability criteria presented in this paper rely mainly on a knowledge of the mode shapes and are independent of the techniques used to obtain ther:l .
REFERENCES Bailey, T ., Hubbard, J .E. (1985) . Distributed PiezoelectricPolymer Active Vibration Control of a Cantilevered Beam. J. Guidance, Control and Dynamics, Vo!. 8, No. 5, pp. 605-611.
Delfour, M.C., Polis, M.P., Payre, G. (1985) . Sensor and Actuator Location for Large Flexible Space Structures. Report DOC-CR-SP-85-033, Department of Communications, Ottawa, Canada. Hughes, P.C. (1984) . Space Structure Vibration Modes: How many exist? Which ones are important? Technical 252, University of Toronto Institute for Note No. Aerospace Studies, Toronto, Canada. Hughes, P.C ., Skelton , R.E. (1980a) . Controllability and Observability of Linear Matnx-Second-Order Systems. J. Applied Mechanics, Vo!. 47, pp. 415-420. Hughes, P.C., Skelton, R.E. (1980b) . Controllability and Observability for Flexible Spacecraft. J. Guidance and Control, Vo!. 3, No. 5, pp. 452-459. Kailath, T . (1980). Linear Systems. Prentice-Hall, Inc., Englewood Cliffs, N.J. Kreyszig, E. (1972) . Advanced Engineering Mathematics, Third Edition. John Wiley &- Sons, Inc., pp . 74-79. Lindberg, R.E., Longman, R.W. (1984) . On the Number and Placement of Actuators for Independent Modal Space Contro!. J. Guidance, Control and Dynamics, Vo!. 7, No. 2, pp. 215-222 . Meirovitch, L. (1967) . Analytical Methods in Vibrations. The MacMillan Co., New York, N.V. Meirovitch, L. , VanLandingham, H.F., OZ, H. U979) . Distributed Control of Spinning Flexible Spacecraft. J. Gllidance and Control, Vo!. 2, No. 5, pp. 407-415.
Balas, M.J. (1976) . Feedback Control of Flexible Systems. IEEE 'frans. Automatic Control, Vo!. AC-23, No. 4, pp. 673-679
OZ,
Blevins, R.D. (1979) . Formulas for Natural Frequency and Mode Shape. Van Nostrand Reinhold, New York, N.Y.
Vigneron, F.R. (1981). Natural Modes and Real Modal Variables for Flexible Spacecraft. CRC Report No. 1348, Department of Communications, Ottawa, Canada.
de Lafontaine, J ., Stieber, M.E. (1986) . Controllability and Observability for Distributed-Parameter Systems. CRC Report to appear, Department of Communications, Ottawa, Canada.
H., Meirovitch, L. (1980) . Optimal Modal Space Control of Flexible Gyroscopic Systems. J. Guidance and Control, Vo!. 3, No. 3, pp. 215-221.
Vigneron, F .R. (1985) . A Natural Modes Formulation and Modal Identities for Structures with Linear Viscous Damping. CRC Report No. 1382, Department of Communications, Ottawa, Canada.