on simultaneous ion of smart structures - part i_theory

8/2/2019 On Simultaneous ion of Smart Structures - Part I_Theory

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On simultaneous optimisation of smart structures Part I:

Theory

Xiaojian Liu *, David William Begg

Department of Civil Engineering, Burnaby Building, University of Portsmouth, Burnaby Road, Portsmouth PO1 3QL, England, UK

Received 23 April 1998; received in revised form 30 September 1998

Abstract

The emerging space missions not only support the idea of smart structures but also call for fast development and application of

advanced structures having highly distributed sensors (analogous to a nervous system) and actuators (muscle-like materials) to yield

structural functionality and distributed control function. A major concern of the development of the smart system is how to make the

multidisciplinary system work eciently and optimally. This problem is considered in the present paper. The optimal control, sen-

sitivity analysis and integrated optimisation of such a multidisciplinary system is presented. In particular the structural shape and

topology are used as design variables, as augmenting the traditional optimal design concept for smart structures and leading to a

relatively new eld with a high potential for innovation. Discussion of the major problem solving methodologies is also presented to

address the truly simultaneous optimisation of smart structures. 2000 Elsevier Science S.A. All rights reserved.

Keywords: Smart structure; Linear quadratic regulator; Simultaneous structural/control system optimisation; Shape/topology

optimisation; Optimal placement of actuators/sensors; Sensitivity analysis; Robustness; Controllability/observability

1. Introduction

There has been a strong continuing trend toward the concept of smart structures in aerospace engi-

neering and the literature related to the eld has increased greatly. The smart structures are sometimes

called ``intelligent'' or ``adaptive'' structures, being a class of advanced structures having highly distributed

actuators and sensors combined with structural functionality, distributed control functions and even

computing architectures [1]. The structures are able to vary their geometric congurations as well as their

physical characteristics subject to control laws. The very nature of the research is interdisciplinary, in-

volving material science, computer technology, information processing, articial intelligence, and control

disciplines. Over the last 10 years, basic investigations into smart structural systems have been carried out

all over the world. Balas [2] presented a mathematical framework for the control of space structures. Theareas emphasised in his survey have played an important role in understanding the structural optimal

control theory and in forming the concept of smart structures in the last decade. This concept was further

classied by Rao et al. [3]. Miura [4] reviewed the research and developments of intelligent structural

systems in the Institute of Space and Astronautical Science (ISAS) in Japan from 1984 to 1990. The subjects

covered vibration control for exible structures, shape control for precision antenna reectors and variable

geometry trusses (VGT) that became one of the rst real demonstrations of the ``smart'' structure

Comput. Methods Appl. Mech. Engrg. 184 (2000) 1524www.elsevier.com/locate/cma

*Corresponding author. Tel.: +44-1705-846-005; fax: +44-1705-842-521.

E-mail address: [email protected] (X. Liu).

0045-7825/00/$ - see front matter 2000 Elsevier Science S.A. All rights reserved.

PII: S 0 0 4 5 - 7 8 2 5 ( 9 9 ) 0 0 0 1 0 - 9


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worldwide. It is now realised that smart structures hold a promise of meeting the requirements of current

and future missions both in space and on the ground [57].

A major concern in the development of smart structures is the simultaneous optimal design of the

structure and its control system. The traditional design methodology treats the structure and the controller

design as two separate procedures. Because there is a very strong interaction between the structure and its

controller [8], though individually the design may be an optimum, the integration might not be optimum in

the global sense. Therefore a great deal of research is currently in progress on developing simultaneous

optimal design methods which hopefully can be used to accommodate the interaction between the structureand the control system. The integrated design problem was rst reported by Hale et al. [9]. In formulating

the combined optimisation Onoda and Haftka [10] and Salama et al. [11] established a precedent followed

by many others. Lim and Junkins [12] presented a design algorithm with numerical application. Three

objective functions, i.e. the total mass, stability robustness, and the eigenvalue sensitivity, are optimised

with respect to a unied set of design parameters which include structural and control parameters. A

numerical approach to the design of the structure, its regulator and observer has been described by Onoda

and Watanabe [13]. Simultaneous minimum weight and robust active control was presented by Khot and

Heise [14] and Khot [15]. The robust control design is achieved by specifying appropriate constraints on the

singular values of a closed-loop transfer matrix. Dhingra and Lee [16] proposed a formulation laying

emphasis on the robustness considerations for controller design, as well as a simultaneous determination of

actuator locations. A mixed discretecontinuous optimisation was solved using a hybrid optimiser of ar-ticial genetic search and gradient-based programming techniques.

In considering the complex structurecontrol interaction (SCI), the optimisation can be formulated as a

multi-objective programming problem where the structure and the controller design variables are equally

treated as independent design variables. Most of the previous work concerning structural design concerns

structural sizingrather than structural layout in terms ofshape and topology. It has been recognised that the

structural topology is an important parameter and that optimal topology can greatly improve the design of

a static structure [17]. It is therefore foreseeable that layout optimisation could improve the performance of

a controlled structure far more than does structural sizing only, and therefore greatly impact on the

conceptual design of smart structures. In topology design, members are added or removed from the initial

structure and hence both the nite element model and the design space change from time to time. Fur-

thermore, the intermediate topologies could be mechanisms leading to termination of the optimisation

process due to unexpected singularities. There is no doubt that optimal design of layout of a smart structure

is far more complex than that of a static structure.

The present paper is concerned with an integrated approach to structure/controller design. This is a

typical multidisciplinary optimisation (MDO) problem. One of the major results is the novel way in which

the structural optimisation is extended from structural sizing to optimal layout. The design sensitivities

accordance with this augmentation are derived and they, apparently, facilitate the establishment of the

ecient algorithms for the problem solving. The use of these algorithms and their hybrid mixtures are

further shown to provide a unique way in which to address the truly simultaneous optimisation of smart

structures.

2. Review of structural control

The equations of motion of a adaptive structure can be written as

MU C U KU B0u FY 1

where M, Cand Kare mass, damping and stiness matrices, and F is the vector of external applied forces.

The structure is regulated by control input u P Rma via matrix B0, where ma is the number of actuators.Making use of the transformation

UU !

U 0

0 U !xY 2

16 X. Liu, D.W. Begg / Comput. Methods Appl. Mech. Engrg. 184 (2000) 1524


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where U f/1/2 g is a truncated eigenvector matrix normalised with respect to M, and x P Rn is

generalised co-ordinate vector, Eq. (1) can be arranged as

x Ax Bu fY 3

where

A 0 I

diagx2i diag2nixi !Y B 0

UTB0 ! and f 0

UTF !X

Let Bu Bu f and assume that the system is controllable, then

u u BTB1BTfX 4

From linear quadratic regulator (LQR) theory, the observed x is fed back to generate the control forces as

necessary

u GSxY 5

in which SP Rmsn is the observation matrix and G is a time-invariant gain matrix. The general solution ofthe structure under this control logic is given by

xt exp Atx0X 6

To determine G, the following quadratic function is employed as a criterion

J 1

2

I0

xTQx uTRu dtY 7

where Q and R must be positive semi-denite and positive denite matrices, respectively. Substituting

Eqs. (5) and (6) into Eq. (7) gives

J 1

2xT0

I0

expATtQ expAt dt

!x0Y 8

where A A BGS and Q Q STGTRGS. It is noted that the value of J depends on specic initialstate x0. This dependence can be eliminated using an average performance function proposed by Levine and

Athans [20] such that the minimisation of J is equivalent to that of

J 1

2tr

I0

expATtQ expAt dtX 9

The gradient oJaoG is given by

oJ

oG RGSLST BTPLSTX 10

Let oJaoG 0 (note: it is not always correct to do so ifG is constrained) then G can be expressed as

G R1BTPLSTSLST1Y 11

in which Pand L can be obtained from the following equations

PA ATP Q 0Y LAT AL I 0X 12

An iterative algorithm is needed to compute P, L and G from Eqs. (11) and (12). IfS is a unity matrix,

further simplication can be made so that G R1BTPwith Psatisfying the following well-known Riccatiequation

ATP PA PBR1BTP Q 0X 13

X. Liu, D.W. Begg / Comput. Methods Appl. Mech. Engrg. 184 (2000) 1524 17


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3. Optimal design of the smart structures

A general formulation for the multidisciplinary optimisation can be dened as

min fbsY bc

sXtX gbsY bcf6 or g0Y14

where f is a weighted objective function, gthe vector of constraints, bs P Rns the vector of structural design

variables and bc P Rnc is the vector of controller parameters including locations of actuators/sensors. As the

locations of actuators/sensors are discrete in nature, Eq. (14) is therefore a mixed integer and continuous

programming problem.

3.1. Objectives

Multiple objectives have been widely explored in previous studies [16,21]. They are, but not limited to,

the following criteria in addition to the index dened in Eq. (9).

3.1.1. Robustness

Robustness improvement of the actively controlled system through structural/control modication was

considered by Rao et al. [22]. Let Et be the uncertainty ofA* then the system is stable if

Etk k26 1a maxfk2LgY 15

in which L is the solution of the Lyapunov equation, Eq. (12). Robust control is therefore to maximise

Mr 1a max kL so that the margin for the variation ofEt can be maximised.

3.1.2. Controllability/observability

Controllability/observability could also be a criterion for the design of smart structural systems. From

the work by Liu et al. [23], controllability can be measured using the singular-value of B. Taking

decomposition ofB gives

B UcScVTc Y 16

where UTc Uc IY VTc Vc I, and

Sc R 0

0 0

!

diagri 00 0

!X

The larger the value of the Euclidean norm Mc Rk kE, the less energy is required to produce control forces.Similarly, taking the singular value decomposition of Syields

S U0S0VT0 Y 17

where UT0U0 IY VT0V0 I and

S0

K 0

0 0 !

diagki 00 0

!X

The observability can then be measured using the Euclidean norm M0 Kk kE. The larger the value ofKk kE, the greater the weighting of the observed output in the performance index equation (7).

3.2. Structural design variables

The structural design variables can be member sizes. Member size can be either continuous or discrete. If

the members are manufactured with standard sizes, which is the real practice for some circumstances, then

the selection of optimal member size is discrete. If zero lower bound on the member sizes is allowed, then

members with zero size can be deleted from the ground structure and the joint connection may therefore

change, this is often used for structural topology design purposes.



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The co-ordinates of joints are usually used as design variables for shape optimisation. If however, a large

number of joints are involved in the design process, then shape optimisation could be an unbearable

computing burden. An alternative to this is to use parametric shape functions. In this way, the joints

concerned are regulated by only a few parameters and the number of shape design variables can be sig-

nicantly reduced.

The shape and topology determine the layout of a structure. Optimum structural layout/control design

has been of major concern in the previous studies [24]. It has been found that the potential benets from

layout optimisation are generally more signicant than those from xed-layout optimisation of adaptivetruss structures. However, the elimination of members may make both stiness and mass matrices positive

semi-denite and the optimal topology may be a mechanism, making this perhaps the most challenging of

this research area [25].

3.3. Controller design variables

Controller design variables can be the feedback gain matrix G and the locations of sensors/actuators.

The feedback gain matrix Gcan be obtained using LQR theory as described in the previous section if there

is no restriction on it, otherwise its entries must be treated as independent design variables and should be

managed using a general minimisation procedure.

The placement of actuators/sensors on a discrete host structure falls into the class of combinatorialoptimisation, for which the solution becomes exceedingly intractable as the problem size increases. Heu-

ristic-based methods have been developed for the selection of active member locations for the shape control

[26]. The risk of using heuristic-based algorithms is that the solution may not be globally optimal. To

reduce this risk, both SA and GA can be employed. Comparison of heuristic and guided random search-

based algorithms can be found in the papers by Onoda and Hanawa [27] and Anderson and Hagood [28].

3.4. Constraints

Structural weight is usually used as the objective function for static structural optimisation. For

structural control problems, especially for space structure control, the structural weight is still a critical fact

to be considered. It can be used as either an objective or a constraint for this purpose. The weight constraint

can be written as

g1 Wbs W0 6 0 or g1 Wbs W0 0Y 18

where W0 is a given structural weight.

Although members can be removed from the current structure, a mechanism is not expected for most

practical applications. This restriction is related to the fundamental eigenvalue of the structure, i.e. the

lowest eigenvalue of the structure cannot be zero. If for some reason, a prescribed eigenvalue k0 is desired,

the requirement can be incorporated in an optimal design procedure with the constraint

g2 k0 x216 0X 19

Other constraints that may be considered are displacement and control force as presented in Ref. [10].

4. Sensitivity analysis

As the behaviour of the control system relies strongly on the structural design variables as well as the

feedback control logic, a systematic sensitivity analysis is essential for the development of well-behaved

algorithms for the solution of a problem of this complexity. The rst-order sensitivity presented in [29] is

summarised and used for the second-order sensitivity derivation.



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4.1. First-order sensitivity

4.1.1. Derivative oJaobsThe rst-order derivatives of the performance functional J with respect to bj P bs is given by

oJ

obj

oJ

oA

oA

obj

oJ

oB

oB

obj

oJ

oS

oS

objj 1Y 2Y F F F Y nsY 20

where ``'' means summation of all the products of two corresponding entries of the matrices and

oJ

oA PLY 21

oJ

oB PLSTGTY 22

oJ

oS GTBTPL GTRGSLX 23

4.1.2. Sensitivity of control input

Making use of Eqs. (4) and (5) one has

ou

obj

oG

objSx G

oS

objx GS

ox

obj BTB

1 oBT

objB

BT

oB

obj

u u

oBT

objf BT

of

obj

!X 24

The sensitivity of the state variable can readily be found by dierentiating Eq. (6)

ox

obj

oA

objxt eA

tox0

objY 25

in which ox0aob can be represented as a function ofU0 and U0, the initial conditions of original controlproblem equation (1).

4.1.3. Sensitivity of measure of robustness

Because L dened in Eq. (12) is a symmetric positive denite matrix, the sensitivity of its eigenvalue is

oklmax

obj uTl

oL

objulY 26

where klmax is the largest eigenvalue ofL, ul is the eigenvector and oLaobj can be found by dierentiatingEq. (12). Then the sensitivity of the robustness measurement is

oMr

obj

1

2k2l max

oklmax

objX 27

4.1.4. Sensitivity of measures of controllability/observability

If it is assumed that the closed-loop system described by Eq. (3) is controllable and observable, it can be

readily derived that

oMc

obj

okRkEobj

1

2

mai1

or2iobj

2 3 mai1

r2i

2 31a2DY 28

where fr2vcg is an eigenpair ofBTB and

or2

obj vTc

oBTB

objvcX 29



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Similarly, the sensitivity of Kk kE can be evaluated as

oM0

obj

okKkEobj

1

2

msi1

ok2i

obj

2 3 msi1

k2i

2 31a2DY 30

in which ki is the ith eigenvalue ofSST.

The present expressions for oJaobs is in a fairly general form and associated with P, L and G only,

making it suciently dierent from that of other researchers (see [3032]).

4.2. Second-order derivatives

Estimating nonlinear eects including interactions between variables motivates the second-order sensi-

tivity analysis. This second-order derivative information can also be used to develop more ecient opti-

misation algorithms, for instance the quadratic programming method. The second-order sensitivities may

be obtained directly from the previous results.

4.2.1. Second-order design sensitivity o2Jaob2sDierentiating Eq. (17) with respect to bk P bs gives

o2J

obj obk

o

obk

oJ

oA

oA

obj

oJ

oA

o2A

objobk

o

obk

oJ

oB

oB

obj

oJ

oB

o2B

objobk

o

obk

oJ

oS

oS

obj

oJ

oS

o2S

obj obkY 31

where

o

obk

oJ

oA

oP

obkL P

oL

obkY 32

o

obk

oJ

oB

oP

obkLS

T

G

T

P

oL

obkS

T

G

T

PL

oS

obk

T

G

T

PLS

T oG

obk

T

Y 33

o

obk

oJ

oS

oGT

obkBTPL GT

oBT

obkPL GTBT

oP

obkL GTBTP

oL

obk

oGT

obkRGSL GTR

oG

obkSL GTRG

oS

obkL GTRGS

oL

obk34

and oPaobk is the solution of the following Lyapunov equation

AToP

obk

oP

obkA

oQ

obk oAT

obkP P

oA

obk X 35The second-order sensitivities of the parametric matrices are as follows:

o2A

obj obk

0 0

diago

2x2iobj obk

! 2diag ni

o2xi

obj obk

!PTTR

QUUSY 36

o2B

obj obk

0

o2U

T

objbkB0

PTTR

QUUSY 37



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o2S

obj obk Sd

o2U

objbkSv

o2U

objbk

!Y 38

where the second-order sensitivity of structural eigenvalues and eigenvectors can be obtained using Nel-

son's formulas [33].

4.2.2. Sensitivity of measure of robustnesso2

Mraob

2

s

o2Mr

obj obk

1

k3lmax

oklmax

obj

oklmax

obk

1

2k2lmax

o2klmax

obj obkY 39

where

o2klmax

obj obk uTl

o2L

obj obkul 2u

Tl

oL

obj

oulobk

40

and o2Laobj obk can be obtained by dierentiating Eq. (12) twice.

4.2.3. Second-order sensitivity of measures of controllability and observability

From Eqs. (31) and (33) one has

o2Mc

obj obk

1

2kRkE

4 2

okRkEobj

okRkEobk

mi1

o2r2i

obj obk

541

and

o2M0

obj obk

1

2kKkE

4 2

okKkEobj

okKkEobk

mi1

o2k

2i

obj obk

5X 42

IfG is independent variable then one has oGaobj o2Gaobj obk 0 j 1Y 2Y F F F Y nsY k 1Y 2Y F F F Y nsin above expressions. In this case, it may be not correct to simply set oJaoG 0. The gradient equation (10)must be used instead.

5. Solution methodologies

A nested strategy taking advantage of optimal linear quadratic control theory can be used in the design

process because of its simplicity. The optimal gain matrix Gcan be found from LQR theory for the given bs.

In this case, G is a dependent variable varying with bs and the actuator/sensor locations. One of the layout

design techniques for static open-loop structures is the so-called ground structure approach, which has been

developed over the past decades for optimising large truss-type structure [25]. Algorithms frequently usedfor this are based on gradient information. These algorithms are ecient in computation but may have the

risk of misleading to local optima. As an alternative, guided random search methods have been successfully

used for topological design of the discrete structures.

The above ideas form the framework for smart structure optimisation. Dierent algorithms are thus

developed. The key points of the algorithms are summarised below and the details will be given in [34].

The simulated annealing (SA) [18] had been used to simulate the thermal motion of atoms in thermal

contact with a heat bath at temperature Tto nd an equilibrium state which minimises energy. By replacing

the energy with the objective f and the state of atoms with design variable vectors bs and bc, it is

straightforward to generate a number of designs using SA, one of which is the optimal design [24]. The GA

is a stochastic optimisation technique working with a design family represented as a population of chro-

mosome-like string [19]. There are three basic manipulations on these strings, i.e. reproduction, crossover



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and mutation, used to ensure offspring inherit genetic information from their parents such that stronger

individuals are more likely to enter the reproduction process and so produce even stronger offspring.

As both the SA and the GA work with a coding string of the design variables rather than the design

variables directly, a procedure converting design variables to a string, or vice versa, is needed. Several

coding methods are available but herein binary code with 0s and 1s is used to represent member size.

Discretisation of the component bsi i 1Y 2Y F F F Y ns is made using linear mapping from a smallest possiblelower bound bLsi to the largest possible upper bound b

Usi . This mapping uses an Nb-bit binary unsigned

integer. In this coding a string code 00 00 maps to bLsi and 111 11 maps to bUsi . There are 2Nb alternativeNb-bit binary unsigned integers representing the intermediate values of bsi which is given by

bsi bLsi

bUsi bLsi

2Nb 1sY 43

where s is an intermediate decimal value of the binary number. If the strings of the ns structural design

variables are chained together, then an ns Nb-bit binary string is created, representing one of the 2nsNb

alternative structures.

The locations of actuators/sensors on the discrete structure can be represented as an integer array of size

(ma ms). Entries of the array are numbers of structural members, representing an actuator/sensor con-guration. For the integrated design purpose, a complete design string b, which is a computer representable

(nc Nb ma ms) integer array, is formed by putting the structural design string and the actuatorplacement array end-to-end.

The SA and GA dier from the normal design procedure of iterative improvement in that they allow

transition out of a local optimum to a possible global solution [18].

The SLP and SQP can also be used to nd bs from the following approximations of problem equation

(14), respectively

min fk rTfDbs

sXtX gk rTgkDbs6 0Y

bLs 6 bs6 bUs

44

and

min fk rTfDbs 1

2DbTs HDbs

sXtX gk rTgkDbs6 0Y

bLs 6 bs6 bUs Y

45

where H is a Hessian matrix. The rst- and second-order derivatives are needed to build the approxima-

tions.

It is not necessary to restrict a solution procedure to the use of the above individual algorithms. Some

hybrid methods can be constructed without diculty.

6. Conclusions

The integrated design of smart structures with the augmentation of shape and topology optimisation has

been discussed. The objectives considered are quadratic performance index (sum of the system energy and

control eort), robustness and controllability. Both the rst- and second-order design derivatives of these

measures have been presented, the formulations providing a systematic information to build well-behaved

algorithms for problem solving of such complexity.

The feedback gain matrix Gcan be found either using a nested method or from a general optimisation

procedure. The former strategy is commonly used and no doubt is an ecient approach. Algorithms

supporting this strategy are developed. These algorithms, featuring sensitivity orientation and guided

random search mechanism, are described in detail in the companion article [34].



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on simultaneous ion of smart structures - part i_theory

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