The design of a structural feedback control system is inherently an exercise ... CY or equivalently the time T required for the envelope of the modal .... F,(r) = rth discrete control force located at point P, ..... Ogata, K. Modem Control Engineering.
The compromise inherent in structural feedback control Larry M. Silverberg and Isam S. Yunis Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, USA Structural feedback control problems can be formulated as underdetermined, as uniquely determined, or as overdetermined problems. This paper shows that compromises in the structuralfeedback control are endemic to the formulation. The compromises are made between the control effort, the dynamic performance, and the sensitivity of the control system. As an illustration, the decentralized control of a circular membrane is carried out by using a minimum gain optimal control method (underdetermined), a pole placement method (uniquely determined), and a uniform damping control method (overdetermined). The illustration indicates the inherent compromises among the formulations. Keywords: feedback control, uniform damping control, dynamics, structural control
Introduction The design of a structural feedback control system is inherently an exercise in compromise. Whether the design process is carried out for aircraft structures, mechanical structures, spacecraft structures, or civil engineering structures, compromises are made between the level of effort associated with the applied forces, the sensitivity of the control system, and the time within which the structural motion is brought to rest. These unavoidable compromises explain why no one technique is always suitable for a given structure. An implication is that it is important for the designer to correctly choose the most appropriate technique for a given structure. But there is another implication that can be even more severe. As an illustration, assume that the reader wishes to design a feedback control system to dampen 90% of the structural motion in T seconds. Invariably, the modal motions of the structure will be damped oscillatory as shown in Figure I. The problem of how fast to dampen the structural motion arises. An equivalent problem is how quickly to dampen the individual motions associated with each mode of vibration. Specifically, the question arises how to choose each modal decay rate CY or equivalently the time T required for the envelope of the modal motion to decrease 90%. The decay rate a and the time Tare related by (Y = (In 10)/T. As shown Address reprint requests to Dr. Silverberg at Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695-7910, USA. Received 23 May 1989; accepted 29 September 1989
362
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Math. Modelling, 1990, Vol. 14, July
in Figure 2, the relation is divided into three regions. In the first region, for times T between zero and T,, = 1.036 s, the decay rate (Y is sensitive to changes in T, Idcz/dTl > 65”. The region between TO and T, is an intermediate region. In the third region, for times T greater than T, = 2.221 s, the decay rate LY is insensitive to changes in T, IdcxldTI < 25”. Because the reader may be directly concerned with the period of time required to dampen the motion, the parameter T is chosen first, and (Y follows. For control systems operating in region 1, the decay rate (Y can be prescribed with relatively low accuracy without affecting T significantly. In region 3, on the other hand, (Y must be prescribed with relatively high accuracy. Otherwise, a change in LY will correspond to a forbiddingly large change in the time T to dampen the structural motion. Indeed, the nature of the structural control technique depends on the region within which we wish to operate. Hence the decision to use one control technique rather than another depends not only on the structure we wish to control, but also on the time within which we wish the motion to dampen. Although many techniques exist for structural feedback control, they all represent either underdetermined formulations, uniquely determined formulations, or overdetermined formulations. The underdetermined formulations include techniques that specify fewer characteristics of the controlled structure and the control system than would be necessary to render a unique control law. Therefore an infinite number of solutions would exist. Owing to this lack of specificity, an additional performance functional is defined, and stationary values are sought, leading to the determination of a unique set of control gains. Perhaps the most pop0 1990 Butterworth-Heinemann
Compromise in structural feedback control: L. M. Silverberg and I. S. Yunis
Figure 1.
Time T for vibratory motion to dampen 90%
~65’
0 To Figure 2.
=1
T
Decay rate a versus time to decay 90%
ular underdetermined formulation is known as linear optimal control.’ Linear optimal control is an extreme case in the sense that no characteristics are a priori specified and all the characteristics of the control system are inherited from the performance functional. A variance of optimal control that specifies some of the dynamic characteristics is referred to here as minimum gain optimal control. The uniquely determined formulation arises when the number of specified characteristics is sufficient to render a unique set of control gains in the associated control law. Popular uniquely determined formulations include pole placement* and independent modal space control (1MSQ.j Pole placement specifies as many closed-loop eigenvalues of the structure as the number of control gains in the associated control law. Recently, its use has been demonstrated successfully in connection with multi-input multi-output systems.4 IMSC specifies the closed-loop modes of vibration and as many closed-loop eigenvalues as the number of control forces. While pole placement can be implemented by using one control force, IMSC is implemented by using as many control forces as controlled modes of vibration. On the other hand, pole placement tends to be sensitive to parameter errors, and the effort associated
with the control force(s) is relatively high, while this is not the case for IMSC.5 The overdetermined formulation occurs when more characteristics are specified than would be necessary to render a unique set of control gains. Therefore no exact solution exists. Owing to the overspecificity, the control law cannot be implemented exactly, and an approximation of the control law is sought. Popular overdetermined formulations include Pseudo-IMSCh and uniform damping control (UDC).7.x Pseudo-IMSC specifies more closed-loop eigenvalues than the number of control forces, leading to a set of linear algebraic equations relating the specified modal control forces with the actual control forces. Then a pseudo-inverse of this set of equations yields the actual control forces. Note that in this process the actual closed-loop eigenvalues are different than their initially specified values. Going further, uniform damping control specifies the complete set of closed-loop eigenvectors and closedloop eigenvalues. The real part of the eigenvalues are specified as a single designer-chosen decay rate. The imaginary part of the eigenvalues representing the closed-loop frequencies of oscillation are specified identical to the natural frequencies of oscillation, and the closed-loop eigenvectors are specified identical to the natural modes of vibration. It turns out that the control law becomes decentralized and the control forces are distributed. The implementation of UDC using discrete forces requires approximating the UDC control law, resulting in a deviation in the closed-loop eigenvalues and closed-loop eigenvectors. This paper shows differences between the structural feedback control formulations via the decentralized control of a circular membrane. The underdetermined, uniquely determined, and overdetermined formulations are compared on the basis of the effort associated with the control forces, the sensitivity of the control system to changes in parameters, the closed-loop eigenvalues and eigenvectors, and the damping times in the three regions described earlier (Figure 2).
Structural control feedback formulations The partial differential equation describing structural motion has the form rPu(P, t)
= M(P) ~ at*
Lu(P, t) + f(P, t)
(1)
where: M(P) = mass distribution at point P u(P, t) = displacement at point P and time t L = negative, semidefinite, self-adjoint differential operator expressing the structural stiffness f(P> 0 = force distribution The stiffness operator L admits countably infinite natural modes of vibration and associated natural frequencies obeying the orthonormality conditions
Appl. Math. Modelling, 1990, Vol. 14, July 363
Compromise in structural feedback control: L. M. Silverberg and I. S. Yunis trol gain operators. For our purposes the control forces and displacement measurements are discrete, so the control law (3) has the form
F,(t) = where: C&(P) = rth natural mode of vibration w, = rth natural frequency D = domain of the structure
where:
The structural feedback control law has the form f(P, t) = Gu(P, t) +
Hy
f(P, t) =
du(P
t)
G, W,, t) + H’“--$-
>
(4)
F,(r) = rth discrete control force located at point P, G,,, H,, = scalar control gains
wps, t) = velocity of point P, at time t ~ at
(3)
where G, H = control gain operators, in which the problem at hand addresses how to best select the con-
5 s=l
NA = number of discrete control forces We can arrive at (4) from (3) by expressing the discrete forces as distributed and with the control gain operators related to the scalar control gains by
2 Fr(t)W - PJ
r=l
G = 2 $j G,,6(P - P,) j- 6(P - P,)( r=ls=l
) dD
H = s 5 H,6(P - P,) j- 6(P - P.,)( r=ls=l D
) dD
where 6(P - P,) = Dirac-delta function. In the case of decentralized control, G,,Y = 0 and H,, = 0 for r # s in (4), and the control gain operators in (4) become G = 2 G,,6(P - P,) r=l
(5)
D
H = 2 HAP - P,) r=l
where (jr(r) = rth modal control force and Qr(0 = 5 [&+-f.~(~) + ~,,4A~)l
s =1
(6)
r= 1,2,...,N,
(10)
where
In practice, only a finite number of modes participate significantly in the system response, and the remaining ones can be neglected. Expressing the displacement in terms of the participating modes (7)
u(P, t) = 2 &V%,(t) S=l
where: q,(r) = sth modal displacement NM = number of participating modes Substituting (7) into (1) and (3), premultiplying by 4,.(P), integrating the result over the domain, and considering (2), we obtain ii&) = -&q,(t) + C&(t) r =
1,2,...,N,
(8)
r =
1,2,...,N,
(9)
where Qr W =
6V’Mf’, t> dD I D
364 Appl. Math. Modelling, 1990, Vol. 14, July
where g,,, h, = modal control gains. Rather than work with the equations of motion (1) and the structural feedback control law (3), we now work with the modal equations of motion (8) and (9) and the modal control law (10). The first step can restrict our attention to a subset of the participating modes designated for control: Without loss of generality we chose the first NC modes. Accordingly, we consider only the first NC equations in (8), (9), and (lo), and we replace the summations over NM modes in (10) with summations over NC modes. This amounts to assuming that only the modes designated for control contribute significantly to the overall system response and that the remaining ones are not considered in the control system design. The next step then amounts to evaluating the performance of the control system in the presence of the complete set of NM participating modes.
Compromise in structural feedback control: L. M. Silverberg and I. S. Yunis
The eigenvalue problem associated with the structure, equations (8) and (9), together with the feedback control system, equation (IO), becomes
s=I
J(PI,Pz,.
s=I
r = 1,2,...,N,
(12)
where: &) = modal coupling coefficients A, = (Y, + ipr = rth closed-loop eigenvalues = rth modal damping rate E: = rth controlled frequency The performance of the control system is described by the dynamic performance parameters and the control performance parameters listed below. Dynamic performance parameters: a, Pr a:’
(r = 1,2, . . . , NC) (r = 1,2, . . . , N,-) (r,s= 1,2 ,..., NC)
grS
(r, s = 1,2, . . . , N,-) (r, s = 1,2, . . . , N,)
(r # s) Control performance parameters: h IS
The modal coupling coefficients are, in general, complex and can be normalized so that up’ = 1. Therefore the modal coupling coefticients represent 2N5 2Nc real independent parameters, and the number of real independent dynamic performance parameters is equal to 2N$ - 2Nc + 2Nc = 2N$. The number of real independent control parameters is designated by NP. In all, the number of real independent parameters, and hence the number of real unknowns, is given by N,=2N$+N,
(13)
where: N, = number of real independent control parameter NU = number of real unknowns We now let the control system designer specify a subset of the NU real independent parameters in the form of NS homogeneous constraint equations. Also the eigenvalue problem (12) can be cast in the form of 2Nf real homogeneous constraint equations. In all, the equations governing the control problem can be written in the functional form fr(j+,PZ,. where
. . ,PNr/) = 0
NE = 2N; + NS
r = 1,2,...,N,
The underdetermined formulation would admit an infinite number of possible solutions to (14). Hence we introduce an additional measure of performance written in the functional form (16)
,PN,,)
where J = performance functional for underdetermined formulations, and we seek stationary values of J subject to the homogeneous constraint equations (14). The uniquely determined formulation admits a unique solution to (14) and may be directly solved. The overdetermined formulation admits no exact solution to (14). Hence we introduce a measure of error written in the functional form (17) e = 4PlrP2,. .*,PNJ where E = error functional for overdetermined formulations, and we seek minimum values of E. The next sections describe in more detail some of the various methods for control system design within the context of the formulations described above.
Underdetermined formulations: Minimum gain optimal control (MGOC) MGOC, a variance of optimal control, represents an underdetermined formulation. When implemented as a decentralized control, a performance functional expressing a measure of the control effort is defined by
(18) r=
I
in which the number of independent control parameters G, and H,, (r = 1, 2, . . . , NA) is equal to Np = 2N, in (6), so the number of unknowns is equal to NU = 2N& + 2N,. According to MGOC, the modal decay rates are specified, so NS = Nc in (15), and f, = a, - %I, (19) where (Y,, = rth specified modal decay rate. The number of equations is then NE = 2N:. + NC.. Indeed, NE < NU, so MGOC represents an underdetermined formulation. (We assumed that NA 5 NC, which is generally the case.)
(14) (15)
where: f, = rth real homogeneous constraint ps = sth independent parameter NS = number of specified parameters NE = number of equations We now distinguish between formulations as follows: NE < NU = underdetermined formulation NE = Nu = uniquely determined formulation NE > NU = overdetermined formulation
Uniquely determined formulations: Pole placement (PP) Pole placement, when implemented as a decentralized control, is associated with Np = 2N, independent control parameters in (6), so the number of unknowns is equal to N,., = 2N,5 + 2N,. According to pole placement the modal decay rates and the controlled frequencies are specified, implying that NS = 2Nc in (15) and r = 1,2,...,Nc
(20)
Appl. Math. Modelling, 1990, Vol. 14, July 365
Compromise in structural feedback control: L. M. Silverberg and I. S. Yunis
where: ffor = rth specified modal decay rate pal = rth specified controlled frequency
Overdetermined formulations: Uniform damping control (UDC)
The number of equations is then equal to NE = 2N$ + 2Nc. Indeed, NE = N”, provided that NA = NC. Hence pole placement represents a uniquely determined formulation when implemented as a decentralized control.
UDC is implemented as a decentralized control, so from (6) NP = 2N,. Thus the number of unknowns is equal to NU = 2N$ + 2N,. According to UDC, all of the dynamic performance parameters are specified as their open-loop values so that NS = 2N$ in (15), and
r = 1,2,...,Nc r,s= 1,2 ,..., Nc I)Nc = Im{&’ al”!) fN;+r+(.VP r = 1,2,...,N, f N;+r+(r~ I)Nc = pr - par where: (Y,, = rth specified modal decay rate par = rth specified controlled frequency a!$ = specified modal coupling coefficients
fr+(r-l,Nc = a, - ffor
Using UDC, we let a$’ = 0 for r # s, par = w,, a,,,. = a in (21) with cx representing a designer-chosen uniform damping rate. The number of equations is equal to NE = 4N’,.. Hence NE > N”, so the formulation is overdetermined, and we minimize the error functional NI
E=
zj-,z r=
I
Illustrative example
and j’(r, 0, t) = distributed control force per area. The boundary conditions are L((u, 0,t) = 0
(24) The eigenvalue problem associated with equations (23) and (24) admit closed-form natural frequencies of oscillation and natural modes of vibration, denoted by w,,, and +mn(r, 0), respectively. The natural frequencies are found from solutions to the characteristic equation J,,
Dynamics of a circular membrane
To elicit distinguishing characteristics of the different formulations, the control of a pinned circular membrane is considered. The membrane radius is denoted by a, the mass per unit area is uniform and denoted by p, and the tension is uniform and denoted by T. The motion of the membrane is governed by equation (1) in polar coordinates, written as d2u(r, 8, t) = Lu(r, 0, t) + f(r, 0, t) P at2
(23)
where
366 Appt. Math. Modelling, 1990, Vol. 14, July
( )
wn, a - = C
0
n = 1,2,...
m = 0, 1,2,. . .
(25)
When m = 0, 1
&Jr, 0) =
u(7qp.J,
Jo %r (
7
( c
)
)
n = 1,2,...
(26) where: Jo, J, = Bessel functions of the first and second kind, respectively C= (T/P) ‘j2, the wave propagation velocity F o r m = 1, 2, . . , we obtain the pairs of solutions’
cos (me) sin (me)
where J, = Bessel functions of order m. The two counting indices m and n are replaced by one index, as shown in Table 1, by reordering the natural frequencies and natural modes according to ascending values of w,,. The associated natural modes of vibration are shown in Figure 3.
(21)
(r f s)
m,n = 1,2,...
(27)
In each of the following cases we let three actuators carry out the control, so NA = 3, and they are located at (r, 6) = (0.3, 0.2), (0.2, 4.7), and (0.7, 3.3) in which the membrane radius a = 1.0 and 8 is measured in radians. Also, for each of the cases considered, the membrane is initially excited by an impulse function of magnitude 0.3 applied at (0.29, 0.79).
Compromise in structural feedback control: L. M. Silverberg and I. S. Yunis Table 1. Ordered natural freauencies Mode
m
n
w had/s)
1 2 3 4 5 6
0
1 1 1 2 1 2
2.405 3.832 5.136 5.520 6.380
1 2 0 3 1
7.016
Table 2. Eigenvalues for MGOC with 3 modes controlled and 3 modes participating Mode
cy = 0.1
ry = 1.0
cr = 3.0
1 2 3
-0.100 2.405 -0.100 3.828 -0.100 5.133
-1.000 2.323 -1.000 3.574 -1.000 4.875
-2.998 0.301 -2.997 3.331 -2.997 4.736
Table 3. Eigenvalues for PP with 3 modes controlled and 3 modes participating
Figure 3. brane
Mode
a = 0.1
(Y = 1.0
a = 3.0
1 2 3
-0.100 2.405 -0.100 3.832 -0.100 5.136
-1.000 2.405 -1.000 3.832 -1.000 5.136
-3.000 2.405 -3.000 3.832 -3.000 5.136
The first six mode shapes of a pinned circular memTable 4. Eigenvalues for UDC with 3 modes controlled and 3 modes participating
Minimum gain optimal control
In our example of MGOC, equations (19) were specified such that Ly 01 = a r = 1,2,...,Nc. (28) where (Y = uniform designer specified damping rate. Pole placement
In implementing PP, a,,,. and pal in equations (20) were specified such that cq,,- = ff
P
