Controller design for vibration suppression of slewing flexible structures

PERGAMON

Computers and Structures 70 (1999) 119±128

Controller design for vibration suppression of slewing ¯exible structures X. Liu *, 1, J. Onoda 2 Research Division of Space Transportation, The Institute of Space and Astronautical Science, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229, Japan Received 28 January 1997; received in revised form 4 August 1998

Abstract Near optimal design of an open-loop control law for vibration suppression of slewing ¯exible structures is presented. The control law is derived in modal space. The optimal slewing time is evaluated from the control of the rigid mode of the structure, and for ¯exible modes, a quadratic control law with free parameters is suggested. Two performance indices which involves system energy and control e€ort, and control input constraints are derived based on this control law. The vibration suppression of the slewing structures is achieved through the minimization of the performance index subjected to the input constraints. Sequential nonlinear programming is employed in seeking the optimal parameters of the control law. Optimal control of a slewing ¯exible beam is given to demonstrate the e€ectiveness of the present proposal. # 1998 Elsevier Science Ltd. All rights reserved. Keywords: Controller design; Flexible structures; Open loop control law

1. Introduction As a consequence of the increasing demand for high performance positioning and rapid retargeting in space missions, research works have focused considerable attention on the control of the slewing ¯exible structures in recent years. Ben-Asher et al. [1] studied the pure time-optimal slewing problem where the cost is the transition time only. Szyszkowski and Youck [2] obtained the bang±bang feedback control law for a single link ¯exible manipulator based on rigid body dynamics and investigated the in¯uence of various ¯exibility parameters on the performance of the opti-

* Corresponding author. Tel.: +44 1705 846 005; fax: +44 1705 842 521; e-mail: [email protected]. 1 Previously Foreign Research Fellow of ISAS, present address: Department of Civil Engineering, University of Portsmouth, Burnaby Building, Burnaby Road, Portsmouth PO1 3QL, England, on leave from Dalian University of Technology, People's Republic of China 2

Professor, AIAA member

mal control. A comprehensive review of earlier and recent developments in time-optimal attitude maneuvers was given by Scrivener and Thompson [3]. As well known, the performance of the slewing ¯exible structure are severely limited by small oscillations, which persists for a period of time even after the desired rigid body con®guration is attained. The fundamental work on ¯exibility e€ects in the control of ¯exible structures has been done by Oz and Meirovitch [4], Turner and Junkins [5], and Breakwell [6] based on linear quadratic regulator (LQR) theory from which the feedback gains can be obtained in closed form for a linear system. In another paper, Meirovitch and Kwak [7] considered the control of spacecraft with multi-targeted ¯exible antennae. The maneuver of the antennae was carried out according to a minimum-time policy, while the control of the elastic vibration and of the rigid-body motions of the spacecraft were implemented by means of a proportionalplus-integral control. Lyapunov stability theory for both linear and nonlinear systems has been used to design control laws for many applications. Using this theory, Bang et al. [8] obtained a stabilizing control law for ¯exible structure maneuvering and vibration

0045-7949/99/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 8 ) 0 0 1 7 2 - 2

120

X. Liu, J. Onoda / Computers and Structures 70 (1999) 119±128

control. Furthermore, they introduced some optimality condition and optimized over the stable region of the free parameters in the control law. The present paper focuses on the vibration control during slewing of the structure from initial state to any prescribed terminal position. To simplify the problem, the nonlinear plant is linearized and then decoupled in modal space. A main idea of the study is to control the rigid slewing by time-optimal control, and then treat the elastic mode control as a minimum energy and control e€ort problem. For this purpose, the rigid mode corresponding to the slew maneuver of the structure is handled by the Pontryagin's minimum principle while for the others, a quadratic control law with a free parameter is suggested. The structural control law is then synthesized in an optimal way and applied to the control of the original nonlinear model of the structure. As an example, the optimal control of a slewing ¯exible beam is solved.

2. Statement of the problem A slewing ¯exible structure is attached to a rigid hub driven by a motor. During a slewing maneuver, the structure undergoes both rigid-body rotation maneuver and elastic deformation. Assume that the structure is ideally attached to the hub, so that the relative elastic displacements of the structure on the boundary of the hub are equal to zero. The equation of motion of such a structure can be derived by using a Lagrange equation such as M…q†q ‡ Kq ‡ Q  …q; q_ † ˆ B0 u…t†

…1†

where M(q) and K are, respectively, inertia and sti€ness matrices, Q* is the vector of the nonlinear term involving q and qÇ , B0 is the control input matrix and u(t) $ Rm the control input, where m is the number of actuator forces. The control input is also required to satisfy the magnitude constraint j u…t† j F

…2†

for all t $ [0,tf ], where tf is the ®nal time. This constraint means that each component of the vector u(t) must have magnitude no greater than the magnitude of the corresponding component of the vector F. The initial state of the structure, q(0) and qÇ (0), is given. It is assumed that the ®nal state of the structure, q(tf ) and qÇ (tf ), is ®xed with tf free. The minimum-time design problem is to determine the control input u(t) that minimizes J = tf. For the purpose of vibration suppression, the system energy and control e€ort should be designed as small as possible. The minimum-time control of the structure results in a nonlinear two-point boundary value problem which

is dicult to solve. For simplicity, the nonlinear terms involved are neglected and the equation of motion of the structure may be linearized as Mq ‡ Kq ˆ B0 u…t†:

…3†

However, direct solution of the linear optimal control problem is still a dicult task, because there are many possibilities for the control law. Even in such cases, where an expression for the control law can be found (if they can be found at all), physical implementation of the optimal control law may be quite complicated for the multi-degree-of-freedom (DOF) system. One possibility is to reduce the order of Eq. (3 by using modal transformation. Truncated eigenpairs can be obtained from the eigenvalue problem KF F ÿ MF FO ˆ 0

…4†

where F ˆ ff1 f2 \ldots fp g is the normalized modal matrix, O = dia{o 21 o 22o 23 . . . . . . o 2p}, and p is the number of the truncated modes. Using the normalized modes, it is readily obtained that x ‡ O x ˆ X …t†

…5†

p

where x $ R is the vector of modal coordinates, X (t) = {x1 x2 . . . xp}T = F TB0u(t) = Bu(t). The initial and ®nal states of x are determined by x…0† ˆ F T Mq…0†; T

_ x…0† ˆ F M_ q…0†

x…tf † ˆ F T Mq…tf †; and

_ f † ˆ F T M_ x…t q…tf †:

…6†

Thus, the original nonlinear optimal control problem is transformed into p single DOF control problems: ®nd X to minimize J while Eq. (6 is satis®ed. The criteria of controllability of plant Eq. (5 is (see Laub and Arnold [9]) rank`…B† ˆ p meaning that m should be at least equal to p. 3. Control law design for the eigenmodes 3.1. Evaluation of near optimal time via the control of the rigid mode Suppose that the ®rst mode of Eq. (4 is the rigid mode. A fundamental assumption is that this rigid mode dominates the slewing maneuver while the others contribute small oscillation as the structure approaches its ®nal con®guration. The optimal time tf can be evaluated approximately from the optimal-time control of the rigid mode. With zero eigenvalue, the dominant mode yields a plant which obeys Newton's law system, and the optimal control problem for the mode can be stated as:

X. Liu, J. Onoda / Computers and Structures 70 (1999) 119±128

Min J ˆ t  s:t:

x 1 ˆ x1 …t† x1 …0†; x 1 …0†; x1 …tf † and

x_ 1 …tf †

…7†

where x1(t) is a function of u(t), i.e. x1 …t† ˆ b11 u1 ‡ b12 u2 ‡ ‡b1m um

…8†

where {b11 b12 . . . b1 m} is the ®rst row of B. Generally, plant Eq. (7 can not be solved independently. From the basic assumption, we can choose x1(t) a bang±bang function being saturated at g and ÿg, so that plant Eq. (7 can be handled by Pontryagin's minimum principle. Since the driving torque of the motor dominates the rigid motion of the beam and the actuator forces at the ¯exible DOF are mainly used for vibration suppression, a candidate value of g may be (suppose u1(t) is the driving torque) g ˆj b11 f1 j

…9†

in which f1, the 1st element of the vector F, is the bound of u1. It is known from Pontryagin's minimum principle that the phase-plane trajectory is a family of parabolas that go through the initial and ®nal states and are given by 8 _ 21 x_ 2 …0† < x1 ˆ ÿ x2g ‡ 12g ‡ x1 …0† for x1 …t† ˆ ÿg; …10† : x_ 2 x_ 2 …0† x1 ˆ 2g1 ÿ 12g ‡ x1 …0† for x1 …t† ˆ g; and 8 _ 21 x_ 2 …t † < x1 ˆ ÿ x2g ‡ 12gf ‡ x1 …tf † for x1 …t† ˆ ÿg; :

x_ 2

x1 ˆ 2g1 ÿ

x_ 21 …tf † 2g

‡ x1 …tf †

for x1 …t† ˆ g:

…11†

By using Eqs. (10) and (11) the control routine, the number of switchings (at most 1), the switching time ts and the minimum ®nal time tf can be evaluated. 3.2. Quadratic control law for ¯exible modes

x i ‡

ˆ xi …t†

with the known xi(0), xÇ i(0), xi(tf ) and xÇ i(tf ). From the Duhamel's integral, one has   Z 1 t xi …t† ˆ xi …0† ÿ xi …t†sin oi t dt cos ot oi 0   Z x_ i …0† 1 t ‡ ‡ xi …t†cos oi t dt sin ot; oi oi 0

0

…12†

0

Z

tf 0

xi …t†cos oi t dt ˆ r2

…16†

where r1 ˆ oi xi …0† ÿ oi xi …tf †cos oi tf ‡ x_ i …tf †sin oi tf r2 ˆ ÿx_ i …0† ‡ oi xi …tf †sin oi tf ‡ x_ i …tf †cos oi tf : It can be found that, at least, two parameters are needed to ensure these two Eqs. (15) and (16) are satis®ed, and one more parameter is required to enforce the control input to satisfy the input constraints. The reasonable form of xi(t) should then be a combination of three independent functions of time. For simplicity, let xi …t† ˆ ai ‡ bi t ‡ gi t2

…17†

where ai, bi, and gi are constants to be determined. Substituting Eq. (17 into Eqs. (15) and (16), and representing ai and bi in terms of gi gives  ai ˆ ‰d3 c2 ÿ c3 d2 †gi ‡ …r1 d2 ÿ r2 c2 †Š=d; …18† bi ˆ ‰…d1 c3 ÿ c1 d3 †gi ‡ …r2 c1 ÿ r1 d1 †Š=d; where c1 c3

d3

  1 1 1 ˆ …1 ÿ cos oi tf †; c2 ˆ sin oi tf ÿ tf cos oi tf ; oi oi oi   1 2tf 2 ˆ sin oi tf ÿ t2f cos oi tf ‡ 2 …cos oi tf ÿ 1† ; oi oi oi   1 1 1 ˆ sin oi tf ; d2 ˆ …cos oi tf ÿ 1† ‡ tf sin oi tf ; oi oi oi   1 2tf 2 ˆ cos oi tf ‡ t2f sin oi tf ÿ 2 sin oi tf ; oi oi oi

and d = 1/o 3i (Z sin Z + 2 cos Z ÿ 2), where Z = oitf . Eq. (17 de®nes a quadratic control law for plant Eq. (12. Applying Eq. (18 gives X …t† ˆ e0 ‡ e1 t ‡ …s0 ‡ s1 t ‡ s2 t2 †gg

…13†

…14†

It follows, by substituting xi(tf ) and xÇ i(tf ) to Eqs. (13) and (14), that Z tf xi …t†sin oi t dt ˆ r1 …15†

d1

The control problem for the ith mode (2E iE p) in Eq. (5 is to ®nd the control input xi(t) to satisfy o2i xi

  Zt xi …t†sin oi t dt sin ot x_ i …t† ˆ ÿxi …0†oi ‡ 0   Zt ‡ x_ i …0† ‡ xi …t†cos oi t dt cos ot:

121

…19†

where e0 and e1 are p vectors, s0, s1 and s2 are p p diagonal matrices. Their elements can be obtained from Eq. (18 as

122

8 < :

s0ii

X. Liu, J. Onoda / Computers and Structures 70 (1999) 119±128

e01 saturated between g1 and ÿ g1 ; and e11 ˆ s011 ˆ s111 ˆ s211  0 e0i ˆ …r1 d2 ÿ r2 c2 †=d; e1i ˆ …r2 c1 ÿ r1 d1 †=d; ˆ …d3 c2 ÿ c3 d2 †=d; s1ii ˆ …d1 c3 ÿ c1 d3 †=d; s2ii ˆ 1; for i ˆ 2; 3; . . . ; p:



4. Synthesis and optimal design of structural controller 4.1. Synthesis of structural control law

exp…H
t† ˆ

Ct ~t S

St Ct

u…t; g † ˆ a  ‡b b  t ‡ g  t2 or u…t; g † ˆ e  …t† ‡ s  …t†gg

2

3 0 ... 0 cos o2 t 0 7 0 cos o3 t 7 7; .. .. .. 5 . . . 0 0 . . . cos op t 2 1 3 0 ... 0 o2 sin o2 t 6 7 6 7 1 0 0 6 7 o3 sin o3 t 6 7 St ˆ 6 7 . . . 6 7 . . . . . . 4 5 1 0 0 . . . op sin op t

6 6 Ct ˆ 6 4

…21†

+

+

where a * = B (e0 + s0g ), b* = B (e1 + s1g ), g * = B + s2g , e*(t) = B + (e0 + e1t), and s*(t) = B + (s0 + s1t + s2t 2), in which B + is the generalized inverse of B. 4.2. Performance index for the vibration suppression

2

By introducing variable   xf nˆ x_ f the ¯exible part of Eq. (5 can be written as a ®rstorder form n_ ˆ Hnn ‡ f

…22†

ÿo2 sin o2 t 0 6 ~t ˆ 6 S 6 .. 4 . 0

0 ÿo3 sin o3 t

... ..

0

. ...

xf ˆ fx2 ; x3 ; . . . ; xp gT ;

Hˆ

0 ÿO Of

I 0



n p …t† ˆ a0 ‡ a1 t ‡ a2 t 2

~f2 g a2 ˆ ÿHÿ1~ ~ ~ g a1 ˆ ÿHÿ1 ‰~f1 ‡ …~f1 ‡ 2Hÿ1~f2 †gŠ ~f ‡ Hÿ1 …~ ~f ‡ 2Hÿ1~ ~f2 †Šgg a0 ˆ ÿHÿ1 f…~f0 ‡ Hÿ1~f1 † ‡ ‰~ 0 1

~f0 ‡ ~ ~f1 t ‡ ~ ~f2 t 2 †g f ˆ ~f0 ‡ ~f1 t ‡ …~

~f1 ˆ 



 0 ; e1

~ ~f0 ˆ

…25†

in which

o 23 . . . . . . o 2p g:

The control input is given by

where   ~f0 ˆ 0 ; e0

…24†

The particular solution n p under the quadratic control law has the form

and O f ˆ diafo 22

3 0 0 7 7 7: .. 5 . ÿop sin op t

Let n p be the particular integral of the inhomogeneous term of Eq. (22, then the general solution has the form n ˆ n h ‡ n p ˆ exp…H
t†c0 ‡ n p :

where 



in which

It can be deduced from Eqs. (19) and (20) that u(t) is also a quadratic control law and can be expressed as

~f2 ˆ and ~

…20†



 0 ; s0

~ ~f1 ˆ



 0 ; s1

where ÿ1

H

 0 : s2



 0 ÿO ÿ1 f ˆ : I 0

From the theory of ordinary di€erential equation, the homogeneous solution of Eq. (22 can be expressed as

It is readily obtained that c0 = v(0)ÿ a0, which is a function of g and switches at ts. Substituting Eqs. (25) and (24 gives

n h …t† ˆ exp…H
t†c0

~~ ~ ‡ Y…t†g n …t; g† ˆ exp…H
t†c0 ‡ Y…t†

…23†

in which c0 is constant vector related to the initial conditions, and the state transition matrix is

where

…26†

X. Liu, J. Onoda / Computers and Structures 70 (1999) 119±128

~ …t† ˆ ÿHÿ1 ‰…~f0 ‡ Hÿ1~f1 † ‡ ~f1 tŠ; Y ~~ …t† ˆ ÿHÿ1 f‰…~~f0 ‡ Hÿ1 …~ ~f2 †Š ~f1 ‡ 2Hÿ1~ Y ~f †t ‡ ~ ~f ‡ 2Hÿ1~ ~f t 2 g: ‡ …~ 1

2

e 1i ‡ ti ˆ

p X

2

jˆ1

p X jˆ1

123

s1ij gj …i ˆ 1; 2; . . . ; m†

…29†

s2ij gj

In a standard way, a reasonable performance index is de®ned as the summation of the system energy and control e€ort. However, without loss of generality, the kinetic energy of the rigid motion of the structure, which is a constant, can be excluded. Thus, the performance index is Z 1 tf T J …gg † ˆ …nn Qv ‡ uT Ru† dt …27† 2 0

for a given g, in which e *1i, s *1ij, and s *2ij are, respectively the elements of e *1, s *1 and s *2. Hence, the constraint (28 is equivalent to ÿF ÿ e  …0†  s  …0†gg  F ÿ e  …0†; ÿ ÿF ÿ e  …t ÿ g  F ÿ e  …t ÿ s †  s  …t s †g s †; ‡ ‡ ÿF ÿ e  …t s †  s  …t s †gg  F ÿ e  …t ‡ s †; ÿF ÿ e  …tf †  s  …tf †gg  F ÿ e  …tf †;

…30a† …30b† …30c† …30d†

where  Of Qˆ 0

ÿfi  ui …ti ; g †  fi ;

…30e†

0 I

 and R ˆ I:

Eq. (27 emphasizes the dependence of the performance index on g . Substituting Eqs. (21) and (26), the explicit expression of the expected J (gg) can be obtained in terms of g . The involved integrals of the matrix exponential exp(H
t) in the computation of J (gg) can be evaluated as follows Z tf exp…H
t† dt ˆ ‰exp…H
t† ÿ IŠHÿ1 ; 0 Z tf exp…H
t†t dt ˆ exp…H
t†Hÿ1 …tf I ÿ Hÿ1 † ‡ Hÿ2 0 Z tf exp…H
t†t2 dt ˆ exp…H
t†Hÿ1 0

Z 0

tf

‰t2f I ÿ 2…tf I ÿ Hÿ1 †Hÿ1 Š ÿ 2Hÿ3 ; exp…H
t†T Qexp…H
t† dt ˆ Qtf :

4.3. Evaluation of optimum g using sequential nonlinear programming Any value of g in Eq. (21 can satisfy the ®nal state condition of Eq. (12 accurately and that of Eq. (3 approximately. In addition, it should also satisfy the following constraint ÿF ÿ e  …t†  s  …t†gg  F ÿ e  …t† for t 2 ‰0; tf Š

…28†

in which e*(t) and s*(t) are de®ned in Eq. (21. Note that u(t) is a quadratic control law, its extreme value may take place at any one of the instants t = 0, t = ts, t = tf or t = t, where t is the instant when uÇ (t) takes zero within (0, tf ). Values of t for di€erent elements of u(t) may be di€erent, and can be evaluated individually from

…i ˆ 1; 2; . . . ; m†:

The constraint (30a±d is a linear constraint whilst Eq. (30e represents a nonlinear constraint of g due to Eq. (29. It should be noted that, Eq. (30 is the uniformed formula for control input u(t). From Eq. (8, the detail constraint for the driving torque is " # m X jgÿ b1j uj …t; g † =b11 j f1 for 0  t  tf : jˆ2

The optimal controller design can then be stated as follows: seek g that minimizes J (gg) subject to constraint (30a±e). This is a nonlinear mathematical programming problem in which the cost J (gg) is a quadratic function of g whilst the constraints are nonlinear ones. It can be solved sequentially by using a gradient projection method ®rst developed by Rosen in 1960 (see Refs. [10, 11]). When using this method, constraint (30e must be linearized. The ®rst derivatives of ui(ti,gg) with respect to g are given by @ui ˆ s 0il ‡ s 1il ti ‡ s 2il t2i @gl ‡

e 1i

‡

p X jˆ1

s 1ij gj

‡2

p X jˆ1

! s 2ij gj ti

@ti @gl

…i ˆ 1; 2; . . . ; m; l ˆ 1; 2; . . . ; p† where

@ti ˆ @gl

s 1il

p X jˆ1

s 2ij gj

ÿ p X jˆ1

e 1i

‡

0 X jˆ1

!2

! s 1ij gj

s 2il :

s2ij gj

Thus the value of ui(ti, g) can be approximated by

…31†

124

X. Liu, J. Onoda / Computers and Structures 70 (1999) 119±128

Fig. 1. Slewing ¯exible beam with size actuator forces.

ui …ti ; g †1ui …ti …gg…k† †; g …k† † ‡

@ui j …k† …gg ÿ g …k† † @gg g

width a = 0.02 m, thickness b = 0.002 m and length L = 1.0 m. Five identical elements were used in the FEM descritization of the beam. Polar moment of inertia of the hub is Ih = 1.510 ÿ 4 Kg/m2. For simplicity, the axial deformations and rotational angles of the beam are neglected, so that the e€ective generalised coordinate vector is q = {y, n2, n3, n4, n5, n6}T.To simulate actuators attached, a 0.2 kg added-mass is assumed at each of the 6 nodes of the beam. All the six normal modes are calculated and used to describe the dynamics of the linearized model of the beam, with the ®rst one being the rigid mode. The control input is u(t) = {u1 u2 . . . u6}T, in which u1(t) is the driving torque of the motor, and ui(t) (i = 2, 3, . . . , 6), the actua-

…32†

where g(k) is the previous solution of the nonlinear programming. 5. Numerical example A slewing ¯exible beam, as shown in Fig. 1, is driven from y = 0 to its ®nal angle y = p/2 by actuator forces. The beam is made of wrought aluminum alloy with Young's modulus E = 7.31010 N/m2, nominal admissible bending stress s0 = 1.8108 N/m2, and density r = 2800 Kg/m3. The dimensions of the beam are

Table 1 Coecients of the quadratic control law u(t) u(t) = a* + bt + g *t 2

u1 u2 u3 u4 u5 u6

a*

b*

g*

maxvuv

ÿ 1.7500/ÿ1.7515 1.9255/1.4791 0.4610/ÿ0.4319 0.8150/ÿ0.5243 1.0810/ÿ0.7048 2.2267/0.1068

5.2382 4.6389 ÿ9.5951 ÿ10.0435 9.2648 ÿ3.7137

ÿ3.2824 ÿ11.3921 14.2408 14.6405 ÿ14.1847 2.4279

1.7500 7.5647 6.6309 6.5366 8.0666 2.2267

X. Liu, J. Onoda / Computers and Structures 70 (1999) 119±128

125

Fig. 2. Variations of slewing angle y(t) with time from 0 to 1.1176 s.

tor forces corresponding to ni, are used for vibration suppression. Let 2

l2 l3 0:1 l1 6 0 1:0 0 0 6 6 0 0 1:0 0 B0 ˆ 6 6 0 0 0 1:0 6 4 0 0 0 0 0 0 0 0

l4 0 0 0 1:0 0

3 l5 0 7 7 0 7 7; 0 7 7 0 5 1:0

then the resultant torque on the beam is Tr = u1(t) + l1u2(t) + l2u3(t) + l3u4(t) + l4u5(t) + l5u6(t), where li is the distance from the hub to node i. Assume that the bound on the driving torque u1(t) is 2s0Ib/b = 2.4 Nm, where Ib denotes the bending moment of inertia [2], and from the assumption in Section 3.1, the resultant torque should not exceed this value. So that from the solution of Eq. (7 the resultant torque is saturated at 2.4 and ÿ2.4, i.e. Tr ˆ 2:4 for 0  t  t5 ; and Tr ˆ ÿ2:4 for ts  t  tf : The driving torque is then u1 …t† ˆ 2:4 ÿ l1 u2 …t† ÿ l2 u3 …t† ÿ l3 u4 …t† ÿ l4 u5 …t† ÿ l5 u6 …t† for 0  t  ts ; and u1 …t† ˆ ÿ2:4 ÿ l1 u2 …t† ÿ l2 u3 …t† ÿ l3 u4 …t† ÿ l4 u5 …t† ÿ l5 u6 …t† for ts  t  tf : The bounds on u(t) is chosen as F = {2.4 8.5 8.5 8.5 8.5 8.5}T.

5.1. Rest-to-rest maneuver At ®rst we consider the beam slewing in rest-to-rest manner, that is q(0) = 0, qÇ (0) = 0, q(tf ) = {p/2 0.0 0.0 0.0 0.0 0.0}T, and qÇ (tf ) = 0. The optimum time is then 1.1190 s, switching at 0.5595 s. The optimal parameter is g (opt) = 0, and it follows that b * = 0 and g * = 0. The nominal energy of the beam decreases from 0.000012 to 0.0, and the control e€ort from 4.7128 to 4.6873. The optimal control input is a bang±bang form u(t) = {0.0008/ ÿ 0.0008, 0.2232/ÿ 0.2232; 0.4464/ ÿ 0.4464, 0.6697/ÿ 0.6697, 0.8929/ ÿ0.8929, 1.0599/ ÿ 1.0599}T, in which `*/]]' means switching from `*' to `]]' at ts (hereafter). Eq. (1 was solved using a Runge± Kutta±NystroÈm scheme (see Ref. [12]) under this bang±bang input. There are no elastic deformations and the phase plane trajectory of the slewing maneuver is ideally a parabola. 5.2. Maneuver with any given initial and ®nal states The e€ectiveness of the present method may be further examined from the following computation, in which q(0) = {0.0 0.002 0.003 0.004 0.005 0.006}T, qÇ (0) = 0, q(tf ) = {p/2 0.0 0.005 0.005 0.0 0.0}T and qÇ (tf ) = 0 are considered. It is known from the solution of Eq. (7 that the optimal time is tf = 1.1175 s with ts = 0.5588 s. An initial feasible design is g (0) = {0.0, 1.0, 1.0, 1.0, 1.0, 1.0}T. Minimization of the performance index J (g g) subject to constraints (30 gives g (opt) = {0.0, ÿ23.4589, ÿ16.0722, 7.4491, 50.0178, 268.0484}T, an interior stationary point. The nominal energy of the bean decreases from 0.0156 to 0.0058, and the control e€ort from 43.6415 to 20.5879. The

126

X. Liu, J. Onoda / Computers and Structures 70 (1999) 119±128

Fig. 3. Comparison of displacement of tip point of the beam for di€erent control laws.

optimal control law is given in Table 1. The control law was applied in turn to the nonlinear model Eq. (1. Using a Runge±Kutta±NystroÈm integration with step size 1.117510 ÿ 4 s, the ®nal con®guration {1.5708, 4.010 ÿ 6, 0.005007, 0.005006, 2.010 ÿ 6, ÿ10 ÿ 6}T was obtained. The error of the slewing angle is less than 0.001%, giving a precision positioning result. To compare with the quadratic control law, dynamic response of the beam was also analyzed using a rigidbody model based bang±bang control law. The ®nal

position of the beam which is then {1.5428, 0.009786, 0.016912, 0.020808, 0.023339, 0.025677}T which gives large errors especially for the ¯exible deformations. The history of the approaching beam in its desired ®nal position is depicted in Fig. 2Fig. 3. Variations of y(t) for the linear and nonlinear plants coincide (see Fig. 2) and approach smoothly the designated value, indicating that the linearized Eq. (3 is a reasonable model and the control law based on this model is e€ective, while the rigid-body model based bang±bang law

_ Fig. 4. Phase plane trajectory y±y.

X. Liu, J. Onoda / Computers and Structures 70 (1999) 119±128

127

Fig. 5. Phase plane trajectory n6±nÇ 6 for nonlinear model under quadratic control law with initial and optimum g .

leads to 2% error on the ®nal angle and results in considerable oscillations during the slewing procedure. Fig. 3 is the comparison of the deformations at the tip point of the beam under quadratic control law and the bang±bang input. Smooth and steady deformations were achieved under the quadratic control law with both initial and optimum g while for the bang±bang rule a very strong vibration took place at the end point. Fig. 4 shows the y±y_ trajectories of the beam. Unlike the rest-to-rest maneuver, the trajectory is no longer a parabola. Coupling with ¯exible deformation

_ makes the angular velocity y(t) oscillate with a tiny amplitude. The n6±nÇ 6 trajectories are depicted in Fig. 5. It can be seen that the amplitude of the velocity under the control law with g (opt) is greater than that under the law with g (0). This is because g (opt) increases the kinetic energy of the beam while the potential energy is reduced. The same trajectory under the rigid-body model based bang±bang law is shown in Fig. 6. It can be observed that the amplitude of nÇ 6 is almost 1000 times the value of that by the quadratic law shown in Fig. 5.

Fig. 6. Phase plane trajectory n6±nÇ 6 for nonlinear model under rigid model-based bang±bang control law.

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X. Liu, J. Onoda / Computers and Structures 70 (1999) 119±128

It can be concluded that the beam can reach its ®nal state with good accuracy for both initial and optimal designs. For rest-to-rest maneuvers the quadratic control law will degenerate into pure bang±bang control law and an idealized slewing maneuver could be expected for there is no elastic deformation at all.

Acknowledgements The support provided by the Ministry of Education, Science, and Culture of Japan through the ISAS Fellowship is gratefully acknowledged. References

6. Conclusions The derivation and optimization of the quadratic control law for the optimal-time design of the ¯exible slewing structure has been outlined. The method concerns the model linearization and modal space transformation. Control of the slewing mode leads to an obeying of the Newton's law system from which the optimal time was evaluated. For ¯exible modes, the quadratic control law was suggested. The control law has more ¯exibility and by varying the parameter g, laws that minimize some desired performance criteria, such a system energy and control e€ort can be obtained. Solution for a slewing beam have shown the e€ectiveness of the control law. With the control law, smooth and steady slewing maneuver and highly precise positioning have been achieved for the beam. For the rest-to-rest maneuver, the control law degenerated into a bang±bang law and has even resulted in idealized results. It should be pointed out that the proposed quadratic control law is not a unique law. There are many other laws that can satisfy Eqs. (15) and (16). The present law is an open-loop law being easily implemented and would be useful for rapid retargeting purposes where the ®nal state is always changing. The stress constraint at the end point where the beam is attached to the hub has been considered, but not for the whole beam. We leave this as a topic for future investigation.

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