Optimal pointwise control of flexible structures

Mathl. Comput. Modelling Vol. 17, No. 9, pp. 89-99, 1993 Printed in Great Britain. All rights reserved

Copyright@

0895-7177193 $6.00 + 0.00 1993 Pergamon Press Ltd

OPTIMAL POINTWISE CONTROL OF FLEXIBLE STRUCTURES IBRAHIM Department

of Mathematical

S. SADEK

Sciences, University of North Carolina at Wilmington

Wilmington,

NC 28403, U.S.A.

(Received and accepted June 1992)

Abstract-A class of optimal control problems for self-adjoint distributed-parameter systems is considered. An approach is proposed to damp the undesirable vibrations in the structures actively by means of pointwise controllers (actuators). Necessary conditions of optimality are derived as a set of independent integral equations which lead to explicit expressions for the pointwise controllers. A numerical example is presented to demonstrate the effectiveness of the method.

1. INTRODUCTION In most control problems, systems to be controlled optimally are distributed parameter systems which are usually formulated in terms of partial differential equations. Several methods from classical and modern control theory have been used to derive a continuous optimal control law which optimizes the system response according to a specified quadratic criterion (see, e.g., [l-3]). Although one can determine this optimal control, one faces the difficulty of implementing it through using an infinite number of actuators. To overcome this difficulty, one can use finitedimensional controls by means of point actuators which is a commonly used approach (see, e.g., [4-61). A finite dimensional control approximation to the distributed state can be obtained using discretization techniques (see, e.g., [7]) or modal approaches [8,9]. One can distinguish between two types of modal control methods, namely, coupled controls [lo] and independent modal-space control (IMSC) [ll-131. In the method IMSC, the controller is designed for each mode independently of other modes of vibration and consequently the computational burden for determining state feedback coefficients is considerably reduced [14]. Recently, a reduced order modeling procedure is used in the control of distributed parameter systems [15] which is based on the Karhunen-Loeve (KL) expansion. A comparison is also given between the performance of a control system using the KL and modal approaches. In [16], a single root-locus plot is used to determine the optimal closed-loop frequency for each controlled mode of vibration of a flexible structure when the modal decay rate is specified. Methods were formulated [17] to determine the optimal location of actuators for the control of distributed parameter systems. The recent book [18] provides a state-of-the-art review of the structural control field. The present study proposes an approach for the optimal control of self-adjoint distributedparameter systems. The approach proposed here involves the use of open-loop pointwise control, which provides an effective means of suppressing the vibrations of highly flexible structures. The effects of the proposed method are investigated by means of numerical studies on the control of a simply supported beam. In particular, a class of self-adjoint systems described by an evolutiontype linear partial differential equation is considered. The problem of damping out the vibrations of such structures by means of a finite number of actuators is studied. The basic control problem The author would like to express his deep gratitude to Sarp Adali for his valuable guidance and constant encouragement during the preparation of this paper. The author also wishes to express his appreciation to Teresa M. Geohagan for her assistance in writing programs for the numerical example.

Tyw=t 89

by 44-W

I.S. SADEK

90

is to minimize a given performance index of the structure in a given period of time. The performance index comprises the total energy of the structure together with the expenditure of control force. The control force over the structures is exercised by force actuators. For self-adjoint systems, the eigenfunctions are orthogonal where a linear combination of these eigenfunctions are used to convert the problem to that of the optimal control of lumped-parameter systems. Necessary conditions of optimality which the open-loop control must satisfy are derived in the form of integral equations using a variational approach. Since the system of integral equations is degenerate, closed-form solutions can be obtained by solving a system of linear algebraic equations. The method proposed here is carried out by modal control [12] in which the structure is controlled through its modes. This method requires that the number of modes that can be controlled to reduce to the number of control actuators. The proposed approach is illustrated by a numerical example in which a finite-dimensional control force is applied to a simply-supported beam subject to certain initial conditions. The numerical results indicate that the use of a few actuators leads to a substantial decrease in the value of the total energy function. Indeed, the designed control system results in the overall exponential decay of the vibrational energy in each mode of the structure to a certain desired value as terminal time increases with application of a limited number of actuators. Moreover, the sensitivity of the method in conjunction with the locations of the actuators is examined by numerical simulations. In this paper, a simple technique is developed to determine an optimal pointwise control such that the quadratic objective function consisting of the vibrational energy in each mode and the input energy is minimized for every initial condition. An exact closed-form solution to the control problem can be obtained. This method can be easily applied to multi-dimensional continuous structures which require less computational effort and computer storage that can be implemented on microcomputers. 2. STATEMENT

OF THE CONTROL

PROBLEM

The equation of motion of a self-adjoint distributed-parameter in the form M(Z) w(:,

t) + A W(L, t) = f(~, t),

system (DPS) can be written

(g,t) E &E x G,

(I) 0 0 where W(Z, t) is the displacement of an arbitrary point 5 = (21, x2,. . . ,x,) E fl:, R, is the interior of some domain fit, c R”, and at time t E Q2t= [te,ti] where to is the initial time, and t1 is the terminal time. A iis a linear spatial differential operator expressing the system stiffness, M(z) is the distributed mass, and f(g,t) is the distributed control force. Since we consider the pointwise control, it is assumed that

i=l

that is, m point force actuators are located at fixed points GUI E R,, and S(c - ZX~)is the multidimensional Dirac delta function. Here fi(t) are discrete forces applied at points z = zi. The displacement r&t) is subject to the boundary conditions

(3) where dR, is the boundary of the domain QZ and B is a linear spatial matrix differential operator. The initia? conditions are given by +,t0)

=

w&)

in Q,,

w&to)

=

WI(~)

in fL.

(4)

To state precisely the optimization problem, the following real function spaces are introduced: (i) The Hilbert function space H = Lz(s2) of all real-valued square-integrable functions on fl(Q = 0, or 0,) in the Lebesque sense with the usual inner product (., .)a and associated norm I].]];.

Control of flexible structures

91

(ii) The Hilbert function space G = Lz(St, x 52t) equipped with the usual inner product

(iii) The Hilbert function space Y = [Lz(R,)]” (the Cartesian product of m copies of the Ls(Rt) space) equipped with the usual inner product

where the tuplets p = (pi,pz, . . . ,p,,,) and g = (41, qz, . . . , qm) are elements in Y. Here ri denote weighting f’&tors with ri > 0, 1 I i-5 m. Let the state v(z,tr) the energy norm- -

of (1) be taken in the Hilbert space H x H with

= [~(g,ti),~~(z,ti)]~ ll?lllE =

(Mw,

w)n

Z

+

(20, Aw)n

(5)

Z

at t = ti, where the right-hand side of expression (5) represents the total (kinetic plus potential) energy of the distributed system. The optimization problem to be considered in this paper is the following. Find the control f*(t) = Vi’@), f;(t), . . *9f*m (t)lT E Y with t E Rt and so as to minimize the functional J(f)

= lldl”E + Ilfll$

(6)

as a measure of performance, with the state function zu(g:,t) satisfying (l)-(4). 3. MODAL

EQUATIONS

FOR THE DPS

Assume that {A, B} is self-adjoint and there is a complete set of orthogonal eigenfunctions {~,!~~(r,)}~>i of the operator A such that

and

n = 1,2,3,. . . .

&h = PI on a$,

(8)

The eigenvalues X, are related to the natural frequencies wn of the undamped oscillation by &=wi(n=1,2,... ). The eigenfunctions {&(:)} _ can be normalized so as to satisfy ,,>I

(Mllrm 1Clk)Q .%h)n2 = ha &k, z = h&k, ($+a,

72,k = 1,2,3,. . . .

(9)

Let ~(z,t) E G, then it is clear from the definition of the Hilbert space H that every w E G and for almost every t E Rt, the function m(g, t) is an element of H = Lz(R,). Thus it follows that the Fourier series expansion

holds, where {&(g)} nil is the complete orthogonal basis for H = Lz(s2,) and {Zn(t)}nll the Fourier coefficients of the function w given by

-&a(t)= (20,lll,)H =

s,W(g,t>$)n(:>dzc, 2

t

E

cl,.

are

(11)

I.S. SADEK

92

By Parseval’s identity, the equality

is satisfied almost everywhere in S&. It directly follows from equation (12) and the definition of G that 2, E Ls(C&), n = 1,2,3,. . . and the tuplet z = (21, 22,. . . , Z,, . . .) E 12. Introducing equation (10) into the system (1) and applying the integral transformation Jo, (. . . ) +k(g) dg and using the orthogonality relations, it can be easily shown that the modal dicplacements 2, (t) is -$&z(t)

+ w; &&) Z&o)

= %8(r),

n=l,2,3

,...,

= (ho,

&z(L?))HY

t E at - {to},

(13)

(14)

$ z, (to) =

bJl(~c),

$Jn(z))If,

where the relation between the modal forces qn(t), n = 1,2,3,. . . and the actuator forces fi(t), i= 1,2,..., m is obtained in the form

qn(t)=

2 h&i)

fdt),

n=

1,2,...,

t E a,

(15)

i=l

using the property of the Dirac Function, i.e.,

The infinite system of equations (13) can be written in the vector form as

$ z(t) + A z(t) = y(t)>

(17)

where, as before, z(t) = (Zl(t), . . . , Zn(t), . . .)T and q(t) = (41(t), . . . , qn(t), . . . )T are elements of 12, t E Rt. A = diag [wzlooxm is the infinite diagonal matrix and q(t) is the infinite-dimensional modal vector defined by (13) x(t) = Q f(t), where * is the time-invariant 00 x m matrix defined by

and f(t) = [h(t), h(t), . . a,

f&)lT-

Tlik solution of the vector equation (17) is given by

z(t) =

zh(t)+ ?w,

t E a,

where zh(t) denotes the homogenous solution given by g(t)

= C(t)g + S(t)b,

(19)

Control of flexible structures

93

with S(t) = diag [sinw,t],,,,

C(t) = diag [cosw,~]~~~,

inwhicha=(al,az ,..., a, ,... )Tandb=(bl,bz ,..., b, ,... )T are constant vectors to be determined by (14). The particular solution zp(t) is given by

g(t)

=

II-l

s

S(t - 7)c~(ddT,

f-h

is the inverse of the infinite matrix A. Inserting equations (10) into equation (6), and applying the integral transformation with the orthogonality property, one obtains where A-’

J(f_)= E(f) + K(f),

(20)

where

$,(3;))~ and R = diag[r,],,, is a positive definite in which C = diag [c,]~~~,c, = (I+!J~(zx), diagonal matrix. In order to be able to design a control force for each mode independently, we must have as many actuators as the number of controlled modes, m = nc. The optimization problem in the modal space then becomes n, minJ(f) = $; (21) c Uf)> fEG

subject

to (17).

Jn(f)

n=l

is the performance index associated with the nth mode given by n=

Jn(f) = GZ(_f) + K(f),

1,2 ,...,

n,,

(22)

where

En(f) = c,

p:(h) +4 ml,]

I

KL(f) = 7-n /

or

f:(t) 6

n=

1,2 ,...,

n,.

(23)

In equation (22), En(f) rep resent the total energy at t = tl associated with the nth mode, and K,(f) represent the?nput energy. T,( > 0) is a weighting factor which can be chosen in such a way toobtain the desired modal decay rate. 4. OPTIMALITY

CONDITIONS

We now proceed directly by taking the first variation with respect to f(t) and making use of (19), the necessary condition for the control force f(t) to be optimal is th%t it must satisfy the following integral equation

f*T(t) + [$+I)

C(tl - 7) + $+I)

where g(t) = gh(t) + Up(t) = [21(t), . . . , &,(t)] optimal control force. Let g =

(-q(h),. *., -q(tl))T,

AI(T) = R-’ C(tl - T),

AS(tl - A)] R-l = 0,

T is the solution of (17) and f*(7)

(24) denotes the

y = Kvl), ***,.$c(h))T, AZ(T) = R-' AS(tl - T),

then (24) becomes

[*CT)+ AI(T): + Az(4y = -Al(T) 3%) - h(T) #(td, MU417:9-G

for all 7 E 0,.

(25)

I.S.SADEK

94

Thus the minimization of J leads to a degenerate system of integral equations (24) or (25) (25) can be that can be solved for f*( T ) in a closed form. The system of integral equations transformed into a line& algebraic system of equations, of twice size by multiplying (25) from left first by

As(~) = C(t, - T)

(26)

A4(~) = A-l S(tl - T)

(27)

and then by

and then integrating algebraic system

over Rt.

Making

use of equation

(18), this leads to the following

MS=!,

(28)

1 s=k&7

where M12

M22+I

linear

b= k1&21T 7

’

in which

Mij =

Ai+2(t) Aj (t) dt, I f-b

4 = - s,. I =

Ai+2(7)

the identity

[A&)

$@d

+ Am

Z%)]

i, j = 1,2,

4

matrix.

It follows from (24) that [*'CT,

where p and y are solutions

where

Z,(t;

f:,)

is defined

5. EXAMPLE:

= - [MT)

(i%)

+/J) +A2(7)

of (28). The corresponding

(Z"(tl,+y)],

(2%

optimal

state variable

is given by

OF A SIMPLY

SUPPORTED

BEAM

by (19).

MODAL

CONTROL

In this section, we present the results of some numerical studies on the optimal control of the bending vibration of a uniform beam hinged at both ends. Choosing for convenience unit bending stiffness, and mass per unit length, and a unit length beam. The stiffness and mass operators of equation (1) and the boundary operators of equation (3) become

A=$, Bwlan where [0] is the zero matrix. The eigenvalue problem (7)-(8)

and the initial

displacement

uJ(O,Q w(l,t)

=

admits

x, = (nII)2,

tin(z)

and velocity

Z,(O) = 0,

M= W&t) %(11t)

a closed-form

1,

1 ’ = [O]

solution:

= sin(nIIz), (14) are taken

i,(O) =

1,

(32)

n=1,3,...,

(33)

in the form n=1,2

,...)

n,.

(34)

Control of flexible structures

95

E,(i)

0.25:

‘1.

/ /

/

t,= 1.0 i

/

/ /

\ ‘1 \

\

r

\

0.001 I

0.0

0.1

’

I

/

0.2

0.5

,

.

0.L

/

0.5

0.6

I

1

0.7

2

!

0.a

.

:

0.3

’

i. 0

X

Figure 1. The energy, El(f),

plotted against I for tl = 1.0, 3.0, and 5.0.

The measure of the force spent in the control process is taken as

C(t) =

0t,; lfib-)12~~. Jc

(35)

2-l

Figure 1 shows El(f) plotted against x with tl = 1.0, 3.0 and 5.0. It is observed that the absolute minimum valise of the controlled beam energy El(f) occurs at the mid-span of the beam (xr = 0.5) for various terminal times of the control pro&s. Figure 2 shows E,(f) plotted against x for the actuators 1, 3 and 5 with ti = 1.0. The simulations indicate t&t the optimal actuators locations of the controlled beam energies are shown to be at i xi = i=1,2 ,...,% n,+l’ where nc = 1,3 or 5 for this case. Consequently, Figures 1 and 2 suggest that the absolute minimum of the controlled energy can be obtained by distributing the actuators at the equally spaced points x; given by (36) with any terminal time tl. Figure 3 compares the curves of En(f) plotted against the terminal time ti for different actuators 1, 3, 5, and 10 with ri = 1.0 5th xi(i = 1,3,5) defined by (36). It is clear that the energy of the beam decays exponentially as the number of actuators increases with the amount of decay depending on the terminal time tl. As tl increases, the effect of the actuator becomes more pronounced. Figure 1 also indicates that for the number of actuators n, 2 5, the nth mode energy E,(f) tends to the fifth mode energy Eh(f) as tl approaches 10. This suggests that the beam can b% controlled by only a few actuators (6 modes) as the terminal time is folded a little bit longer. In Figure 4, the curves of the displacement zo(1/2, t) is plotted as a function of time t for ri = 1.0, tl = 1.0 and different actuators 1, 3, 5. This figure shows the effectiveness of different actuators in reducing the dynamic displacement of a controlled beam as compared with an uncontrolled one. It is observed that the maximum deflection of the controlled beam dies out gradually

I.S. SADEK

96

En(f)

I

n_=I

r

0.05 I

0.0

0.:

I

I

I

I

0.2

0.3

0.4

0.5

’

I

I

I

I

0.6

0.7

0.8

0.9

1.0

X

Figure 2. The energy, E,,(f),

plotted against z for n, = 1, 3, and 5.

as m(n,) increase. Figure 5 shows C(t) plotted against t for modes 1 and 2. As expected, the total control force for each mode increases as t increases.

6. CONCLUSIONS We have suggested an approach to actively control a class of flexible systems using optimal independent modal-space control which requires the number of actuators be equal to the number of controlled modes. Using the calculus of variations, necessary conditions of optimality for the pointwise control are obtained in the form of integral equations. This approach provides an optimal closed-form solution to the linear optimal control problem for multi-dimensional continuous structures. Numerical results are presented in graphical form and the following observations have been made. (1) The beam can be controlled to a desired level by a few actuators (numerical simulations show n, 5 5). (2) The actuators are distributed at equally spaced points along the length of the beam. (3) The optimal locations of the actuators are insensitive to the terminal time of the control process. (4) The proposed method exhibits no instability characteristics, and minimizes the energy of the beam. (5) This method requires less computational effort and computer storage, and can be implemented on microcomputers.

Control of flexible structures

E”(f)

0

1

2

4

3

5

6

7

9

9

10

? Figure 3. The energy, E,,(f),

0.0

0.1

0.2

0.3

plotted against tl for n, = 1, 3, 5, and 10.

0.4

0.5

0.6

0.7

= 0.9

t Figure 4. Displacement,

UJ(~, t), plotted against t for n, = 0, 1, 3, and 5.

0.9

i. 0

IS. SADEK

98 C(t.1

t Figure 5. Total control force, C(t), plotted against t for n, = 1 and 3.

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Governed

by Partial Differential

Equations,

Springer-Verlag,

New

2. S.G. Tzafestas, Distributed Parameter Control Systems, Theory and Application, Pergamon Press, New York, (1982). systems, IMA 3. J.M. Sloss, J.C. Bruch and I.S. Sadek, A maximum principle for non-conservative self&joint Journal of Mathematical Control and Information 6, 199-216 (1989). controllers for hyperbolic systems, In Proceedings of the Third VPI & SU 4. R. Gran, Finite-dimensional Symposium on Dynamics and Control of Large Flexible Spacecraft, pp. 301-318, Blacksburg, VA, (1979). 5. M.J. Balas, Active control of flexible systems, Journal of Optimization Theory and Applications 25 (3), 415-436 (1978). 6. M.J. Balas, Suboptimality and stability of linear distributed parameter systems with finite-dimensional controllers, In Proceedings of the Third VPI d SU Symposium on Dynamics and Control of Large Flexible Spacecraft, pp. 683-700, Blacksburg, VA, (1981). 7. R.E. Moore, Computational Functional Analysis, John Wiley & Sons, New York, (1985). 8. M. KGhne, Optimal feedback control of flexible mechanical systems, In Proceedings of the IFAC Symposium on the Control of Distributed Parameter Systems, Vol. 1, Paper 12-7, Banff, Canada, (1971). 9. H.H.E. Leipholz, Modal approach to the control of distributed parameter systems, Vibration Control and Active Vibration Suppression, Presented at the Conference on Mechanical Vibration and Noise, Boston, MA, September 27-30, 1987, (Edited by D.J. Inman and J.C. Simonis), pp. 71-77, The American Society of Mechanical Engineers, New York, (1988). 10. M.J. Balas, Active control of flexible systems, Journal of Optimization Theory and Applications 25 (3), 415636 (1978). 11. L. Meirovitch and L.M. Silverberg, Globally optimal control of self-adjoint distributed systems, Optimal Control Applicatiotis and Methods 4, 365-386 (1983). systems, Journal of Guidance, 12. L. Meirovitch and H. Baruh, Control of self-adjoint distributed-parameter Control and Dynamics 5 (I), 61366 (1982). 13. L. Meirovitch, Some problems associated with the control of distributed structures, Optimal Control, Lecture Notes in Control and Information Sciences, Vol. 95, (Edited by M. Thomas and A. Wyner), pp. 289-303, Springer-Verlag, Berlin, (1986).

Control of flexible structures

99

14. L. Meirovitch, H. Baruh and H.L. &, A comparison of control techniques for large flexible systems, Journal of Guidance, Control and Dynamics 6 (4), 302-310 (1983). 15. J.B. Burl, Application of the Karhunen-Loeve expansion to the reduced order control of distributed parameter systems, Control and Computers 16 (l), 12-15 (1988). 16. A. Sinha, Optimal vibration control of flexible structures for specified modal decay rates, Journal of Sound and Vibration 12.3 (l), 185-188 (1988). 17. M. Amouroux and J.P. Babary, On the optimal pointwise control and parametric optimization of distributed parameter systems, International Journal of Control 28 (5), 789-807 (1978). 18. L. Meirovitch, Dynamics and Control of Structures, John Wiley & Sons, New York, (1990).