Integrated design of structural and control systems with a

International Journal of ControlVol. 79, No. 9, September 2006, 1062–1073

Integrated design of structural and control systemswith a homotopy like iterative method

K. HIRAMOTO*y and K. M. GRIGORIADISz

yDepartment of Mechanical Engineering, Akita University,1-1 Tegata-gakuen-machi Akita 010-8502, Japan

zDepartment of Mechanical Engineering, University of Houston,Houston, TX 77204, USA

(Received 15 January 2006; in final form 11 April 2006)

We consider an integrated design problem of structural and control systems. It is well known

that even the simplest formulation of this problem results in a kind of a BMI problem. In this

paper, homotopy-like iterative design method based on LMIs is proposed to obtain an

optimal plant and a controller simultaneously for a full-order output feedback problem.

We can deal with a multiobjective problem, e.g., H2=H1 control problem etc. in the proposed

design method. We can also optimize structural design parameters appearing non-linearly in

the coefficient matrices of the plant state space form using the first order approximation of the

non-linear dependence in the proposed algorithm. Several design examples show that the

proposed algorithm works quite effectively for various integrated design problems.

1. Introduction

The objective of control system design is to shape thebehaviour of the closed-loop system satisfactory.Generally such optimization has been carried out bysynthesizing the optimal controller (in some sense) for agiven plant and information about the plant uncertainty.Enormous methodologies have been proposed to obtainthe optimal controller for the fixed plant so far andwe are able to choose the most appropriatedesign scheme fitting the objective of the controlsystem designer from the huge catalogue of the designtechniques. Especially, the recent development of theLMI approach (Boyd et al. 1994, Scherer et al. 1997,Skelton et al. 1998) has enabled us to design a controllerthat satisfied various types of closed-loop constraintsquite efficiently in the case of a state feedback or full-order controller design problem.Meanwhile design parameters in the plant are

generally determined with some other specifications,e.g., the strength, the weight and the cost of thestructure, etc., before the controller design in mostcases. In the above two-step design methodology,

i.e., the plant is firstly fixed and the controller isobtained for the fixed plant, we do not utilize the full

design freedom to obtain the optimal control system.In other words, we could not have used the designfreedom of the plant to optimize the performance of the

controlled closed-loop system. This limitation of thedesign freedom may become a serious obstacle when ahighly tight specification is imposed on the closed-loop

system and should be resolved in some sense for thegenuine optimal control systems design.

The concept of integrated design of structural andcontrol systems (shortly, we may use the term integrateddesign in the rest of the paper), is a prescription for the

above problem in the general and current frameworkof control system design. In the integrateddesign scheme design parameters both in the plant and

the controller are dealt with equally and optimizedsimultaneously. Therefore, we can expect to obtain abetter closed-loop performance than that of the con-

ventional two step design. A lot of design methods, mostof which are basically an iterative and sequential designof the plant and the controller, have been proposed forthe integrated design problem and each of them showed

the advantage of its proposed design method over theconventional two step design scheme with their*Corresponding author. Email: [email protected]

International Journal of ControlISSN 0020–7179 print/ISSN 1366–5820 online � 2006 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/00207170600750105

numerical examples (Onoda and Haftka 1987, Rao et al.1988, Kajiwara et al. 1994, Grigoriadis et al. 1996,Hiramoto et al. 2000, Hiramoto and Doki 2004).However, unfortunately all integrated design problemsresult in a kind of BMI problem even if we assumethat the coefficient matrices of the plant state space(or descriptor) form are linear functions on structuraldesign parameters (Tanaka and Sugie 1998). It is wellknown that the BMI problem is an NP hard problem,i.e., roughly speaking, there are no efficient methods toobtain the global optimal solution, in contrast thesefor LMI problem (Blondel and Tsitsiklis 2000). In theintegrated design problem some researches obtained alocal optimal pair of the plant and the controller bysequential LMI optimization for the formulated BMIs(Tanaka and Sugie 1998, Chien 2000, Lu and Skelton2000). However, these methods required all coefficientmatrices of the state space (or the descriptor) form of theplant to be linear functions on the structural designparameter. In general, the structural design parameterenters the matrices in a non-linear manner and themethods are not capable in such a general case. Notethat Iwasaki et al. (2000) proposed a plant designframework based on a finite frequency positive realnessproperty and derived a design condition in terms ofLMIs. However it could only change zeros of thesystem, not poles. Furthermore it also required thecoefficient matrices of the state-space form to be linearfunctions on structural design parameters.In this paper we propose an integrated design method

with a homotopy approach to optimize the structuraland control design parameters simultaneously. Thehomotopy technique has been applied to some controlsystem design problems which are formulated as BMIproblems (Hassibi et al. 1999, Zhai et al. 2001,Mehendale and Grigoriadis 2003) and has shownsuccessful results numerically. In this paper the homo-topy method is applied to the matrix inequalities, whichoriginally were LMIs in the case of controller synthesisproblem, representing the norm constraints of theclosed-loop system. The overview of the proposedmethod is presented as follows. First, the optimalcontroller is obtained for a fixed plant with the LMIsrepresenting the closed-loop specification(s). This is aconvex optimization problem. Second, we assume asmall amount of perturbations both for the coefficientmatrices of the state space form of the plant, matricesrelated to the controller and the performance index inthe LMIs. The obtained conditions are no longer LMIson those perturbations. However, in the case wherethose perturbations are sufficiently small, we canapproximate the matrix inequalities to LMIs on theperturbations by ignoring their second or the higherorder products. Third, those approximated LMIs aresolved with standard LMI solvers to obtain the optimal

perturbations with a constraint such that the perturba-

tion of the performance index becomes less than zero.

The plant is updated to the new one by incorporating

the obtained perturbation and go back to the first step.

The above algorithm is iterated until a kind of

convergence is obtained. In this design algorithm we

can show a local convergence of the design parameters.

The details of the design method will be mentioned later.The advantages of the proposed design method are

enumerated as follows.

. We can employ all LMIs, which have been obtained

for the controller synthesis, for the integrated design.

This means that we can utilize the variety of the LMI

conditions for the controller design problem. In fact,

we will consider a multiobjective problem like the

H2=H1 problem given in Scherer et al. (1997) in

the design example. There are no additional factors of

the conservatism that comes from the integrated

design formulation itself in the proposed method.. The obtained closed-loop system is always optimal

in the sense of the LMIs of interest in each iteration

because the optimal controller is computed for all

perturbed plant by solving the same LMIs obtained

with the fixed plant. This means that we can obtain

the optimal perturbation of the plant design

parameter in the neighbourhood of the optimal

closed-loop system and are able to obtain a better

local optimal solution.. The non-linear dependence of the plant state space

matrices on the structural design parameters can be

dealt with by approximating the non-linear depen-

dence to a linear function. The approximation is

simply carried out by taking the first order term of

the series expansion of the non-linear function. The

simple approximation scheme is always valid because

we can restrict the amount of the perturbation

arbitrarily small in the proposed design method such

that the linear approximation of the non-linear

function has a sufficient accuracy. (The price for the

limitation is increasing the number of iterations for

the convergence. The numerical example will be

presented to describe the details.) With this advantage

we can deal with a more general class of the integrated

design problem whose structural design parameters

enter non-linearly to the plant state space matrices.

Those advantages of the proposed design method will be

demonstrated in design examples.The rest of the paper is organized as follows: in x 2, the

integrated design problem of structural and control

system is formulated; the homotopy-like design algo-

rithm is presented in x 3. Several features of the proposed

design method are also stated in this section. Various

design examples to show the effectiveness of the

Integrated design of structural and control systems 1063

proposed design scheme are given in x 4. In x 5, theconclusion of this paper is presented.

2. Integrated design problem

The control object is represented as the followinggeneral state space form:

_xðtÞ ¼ AðpÞxðtÞ þ B1ðpÞwðtÞ þ B2ðpÞuðtÞ

zðtÞ ¼ C1ðpÞxðtÞ þD11ðpÞwðtÞ þD12ðpÞuðtÞ

yðtÞ ¼ C2ðpÞxðtÞ þD21ðpÞwðtÞ,

8><>: ð1Þ

where xðtÞ 2 Rnx, wðtÞ 2 Rnw, uðtÞ 2 Rnu, zðtÞ 2 Rnz andyðtÞ 2 Rny are the state vector, the disturbance, and thecontrol effort, the controlled output and the measure-ment respectively. Matrices A(p), B1(p), B2(p), C1(p),C2(p), D11(p), D12(p) and D21(p) have conformabledimensions and are (not necessarily linear) functionsof np dimensional structural design parameter vectordenoted as p :¼ ½p1, . . . , pnp�

T. In mechanical systems tobe controlled, the damping, the stiffness and the sensor/actuator placement, etc. are supposed to be elements ofvector p. In this paper vector p is assumed to be a realvector in a set P defined as the following:

P :¼ p 2 Rnp: pl � p � pu, pl, pu 2 Rnx� �

, ð2Þ

where pl and pu are the lower and the upper bounds ofthe vector p respectively.For the control object possessing the adjustable

design parameters given in (1) a following full orderfeedback controller is synthesised:

_xKðtÞ ¼ AKxKðtÞ þ BKyðtÞ

uðtÞ ¼ CKxKðtÞ,

�ð3Þ

where xKðtÞ 2 Rnx is the state vector of the controllerand all coefficient matrices in (3) have appropriatedimensions. Let the transfer function matrix of thecontroller as K(s). Note that those coefficient matricesalso can be considered as functions on the structuraldesign parameter vector p.The closed-loop system with the plant in (1) and the

controller in (3) is given as

_xclðtÞ ¼ AclxclðtÞ þ BclwðtÞ

zðtÞ ¼ CclxclðtÞ þDclwðtÞ,

�ð4Þ

xclðtÞ :¼xðtÞ

xKðtÞ

� �, Acl :¼

A B2CK

BKC2 AK

� �,

Bcl :¼B1

BKD21

� �, Ccl :¼ ½C1 D12CK �, Dcl :¼ D11:

Define the transfer function matrix of the closed-loopsystem (4) as Gcl(s). For the closed-loop system we

define a scalar performance index J as

J :¼ kGclðsÞk�, ð5Þ

where kHðsÞk� denotes a norm of a transfer functionH(s). We can assign the performance index J for varioustypes of norm of Gcl(s), e.g., H2, H1 and L1 norm etc.or the weighted sum of such closed-loop norms (multi-objective case). Now the integrated design problem ofthe structural and control systems is formulated as thefollowing:

Integrated design problem of structural and controlsystems: Find the structural design parameter popt 2 Pand the controller Kopt(s) which minimize J or satisfyJ� Ju (Ju>0) where Ju is a scaler representing theperformance specification determined by the designer.

Remark 1: In the case of the multiobjective problem, itis often the case of minimizing a norm of a closed-looptransfer function from the components of wðsÞ :¼LðwðtÞÞ to those of zðsÞ :¼ LðzðtÞÞ subject to inequalityconstraints on other norms of another closed-looptransfer matrix from (other) components of w(s) to(other) ones of z(s). The formulated problem is alsocapable in this situation.

Remark 2: In some cases we need to optimize orconstrain the open-loop plant for a structural objectivefunction by adjusting the p 2 P. We can also deal withsuch a structural optimization problem without anydifficulties.

3. Homotopy like design algorithm

3.1 LMI conditions for controller design

The problem formulation in the previous section doesnot assume the specific norm of the closed-loop system.Actually we can deal with all kinds of norm if the optimalcontroller synthesis conditions are given as LMIs oncontroller design parameters (Scherer et al. 1997).In this subsection we introduce LMI conditions for theoptimalH2 andH1 controllers, as examples, to mentionthe following homotopy like method concretely.Furthermore, we show the principle for a design of amultiobjective controller, say, H2=H1 controller, in thesense of a sufficient condition by taking commonLyapunov matrices that appear in LMI conditions onsingle objective problem, i.e., each of H2 or H1.

H2 norm (Scherer et al. 1997): A controller K2(s)yielding the kGclðsÞk

22 5 �2 ð�2 4 0Þ exists if and only

if the following LMIs have solution matrices X2 ¼ XT2 ,

Y2 ¼ YT2 2 Rnx�nx, Q ¼ QT 2 Rnz�nz, A2 2 Rnx�nx,

1064 K. Hiramoto and K. M. Grigoriadis

B2 2 Rnx�ny and C2 2 Rnu�nx:

�11 �T21 �T

31

�21 �22 �T32

�31 �32 �33

264

3755 0, ð6Þ

�11 �T21 �T

31

�21 �22 �T32

�31 �32 �33

264

3754 0, ð7Þ

traceðQÞ5 �2, Dcl ¼ 0 ð8Þ

where

�11 ¼ AX2 þX2AT þB2C2 þ ðB2C2Þ

T,

�21 ¼ A2 þAT,

�22 ¼ ATY2 þY2Aþ B2C2 þ ðB2C2ÞT,

�31 ¼ BT1 , �32 ¼ ðY2B1 þ B2D21Þ

T, �33 ¼ �I,

�11 ¼ X2, �21 ¼ I, �22 ¼ Y2, �31 ¼ C1X2 þD12C2,

�32 ¼ CT1 , �33 ¼ Q:

H1 norm (Scherer et al. 1997): A controller K1(s)yielding the kGcl(s)k1< �1 (�1>0) exists if and onlyif the following LMIs have solution matrices X1 ¼ XT

1,Y1 ¼ YT

1 2 Rnx�nx, A1 2 Rnx�nx, B1 2 Rnx�ny andC1 2 Rnu�nx:

�11 �T21 �T

31 �T31

�21 �22 �T32 �T

42

�31 �32 �33 �T43

�41 �42 �43 �44

26664

377755 0, ð9Þ

X1 I

I Y1

� �4 0, ð10Þ

where

�11 ¼ AX1 þ X1AT þ B2C1 þ B2C1

� �T,

�21 ¼ A1 þ AT,

�22 ¼ ATY1 þ Y1Aþ B1C2 þ B1C2

� �T,

�31 ¼ BT1 , �32 ¼ Y1B1 þ B1D21

� �T, �33 ¼ ��1I,

�41 ¼ C1X1 þD12C1, �42 ¼ C1,

�43 ¼ D11, �44 ¼ ��1I:

In both of the above cases, the coefficient matrices of thecontroller

KiðsÞ :¼ðAKÞi ðBKÞi

ðCKÞi 0

" #ði ¼ 2 or1Þ

is given as

CKð Þi ¼ CiM�Ti , BKð Þi¼ N�1i Bi,

AKð Þi ¼ N�1i

�Ai �Ni BKð ÞiCXi � YiB CKð ÞiM

Ti

�YiAXi

�M�Ti ,

where matrices Mi 2 Rnx�nx and Ni 2 Rnx�nx are non-singular square solutions to the following decompositionproblem:

I� XiYi ¼MiNTi , i ¼ 2 or1:

Note that the above decomposition problem always hassolution matrices Mi and Ni (i¼ 2 or 1) if the LMI foreach controller design problem is feasible. Both of theabove LMIs can be solved efficiently with a standardLMI solver, e.g. Gahinet et al. (1995).

In the case of the multiobjective problem of the abovetwo criteria, i.e., the H2=H1 problem, the necessary andsufficient condition for the existence of controllerK2/1(s) yielding kGclðsÞk

22 5 �2 and kGclðsÞk15 �1 is

given by taking the common matrix variables asA2 ¼ A1 :¼ A2=1, B2 ¼ B1 :¼ B2=1 and C2 ¼ C1 :¼

C2=1 in (6)–(8) and (9). (In this case the condition in (10)is automatically accomplished if the condition in (7) isfeasible.) However those matrix inequality conditionsare no longer LMIs in this case and the optimal H2=H1controller design problem is still an open problem. For asufficient condition to exist the controller satisfyingkGclðsÞk

22 5 �2 and kGclðsÞk15 �1 is obtained with the

above variables changing and taking commonLyapunov matrices, i.e., X2¼X1¼X2/1 and Y2¼

Y1¼Y2/1 (Scherer et al. 1997, Lu and Skelton 2000).Then matrix inequality conditions in (6)–(8) and (9) areLMIs on each matrix variable and can be solved by LMIsolvers. The resulting controller may be conservativedepending on the given problem. However, this methodcurrently can be said to be a reasonable way to obtainthe multiobjective controller for the following reasons:

. We can obtain the controller taking multiple closed-loop criteria (although it is not necessarily the optimalcontroller) efficiently because the conditions are givenas LMIs on variable matrices.

. The order of the controller is always same as the plantin contrast to the method based on coprime factortechnique (Boyd and Barratt 1991). The property ofthe controller order is favourable in real applications.

In this paper we take the above common Lyapunovmatrix strategy in the case of a multiobjective problemeven with other closed-loop norm constraints (e.g., L1or other quadratic constraints) are considered.


Remark 1: Recently a dilated-LMI approach for amultiobjective controller synthesis was proposed(Ebihara and Hagiwara 2004). They showed that thedilated-LMI technique is less conservative than theabove common Lyapunov matrix approach andprovided a controller synthesis condition in terms ofLMIs for a specific multiobjective control problem(H2=D-stability problem). If the control specification isgiven as the closed-loop H2=D-stability criterion we canemploy this dilated-LMI based condition in the homo-topy-like integrated design procedure which will bepresented in the next subsection. However, because theH1 synthesis problem for continuous time systems isstill open in the dilated-LMI framework (Ebihara andHagiwara 2004) and we deal with the H2=H1 problemas a representative of various multiobjective controlproblems in this paper, we still adopt the commonLyapunov matrix strategy (Scherer et al. 1997) as thereasonable option for the multiobjective controllerdesign. (In x 4 an H2=H1 integrated design problemfor a simple mechanical system will be considered.)

3.2 Homotopy-like design algorithm

In this subsection we propose a homotopy-likeintegrated design method based on LMI conditions inthe previous subsection. We show the algorithm only inthe case of the H1 problem. The method is exactly thesame when we consider the H2 problem or the multi-objective problem.Define �p:¼ ½�p1, . . . ,�pnp�

T2 Rnp, �A1 2 Rnx�nx,

�B1 2 Rnx�ny, �C1 2 Rnu�nx, �X1 ¼ �XT1 2

Rnx�nx, �Y1 ¼ �YT1 2 Rnx�nx and a scalar ��1 as

perturbations of p, A1, B1, C1, X1, Y1 and �1 in (9)and (10) respectively. With those perturbations definethe perturbed version of (9) and (10) as follows:

�11 �T21 �T

31 �T31

�21 �22 �T32 �T

42

�31 �32 �33 �T43

�41 �42 �43 �44

26664

377755 0, ð11Þ

X1 þ�X1 I

I Y1 þ�Y1

� �4 0, ð12Þ

where

�11 ¼Aðpþ�pÞðX1þ�X1Þþ ðX1þ�X1ÞAðpþ�pÞT

þB2ðpþ�pÞðC1þ�C1Þþ ðC1þ�C1ÞT

�B2ðpþ�pÞT,

�21 ¼ A1 þ�A1 þ Aðpþ�pÞT,

�22 ¼Aðpþ�pÞTðY1þ�Y1Þþ ðY1þ�Y1ÞAðpþ�pÞ

þ ðB1þ�B1ÞC2ðpþ�pÞþC2ðpþ�pÞT

�ðB1þ�B1ÞT,

�31 ¼ B1ðpþ�pÞT,

�32 ¼

nðY1 þ�Y1ÞB1ðpþ�pÞ þ ðB1 þ�B1Þ

�D21ðpþ�pÞoT

,

�33 ¼ �ð�1 þ��1ÞI,

�41 ¼ C1ðpþ�pÞðX1 þ�X1Þ þD12ðpþ�pÞ

� ðC1 þ�C1Þ,

�42 ¼ C1ðpþ�pÞ, �43 ¼ D11ðpþ�pÞ,

�44 ¼ �ð�1 þ��1ÞI:

If all coefficient matrices in (1) are linear functions on p,the perturbed coefficient matrices, say, A(pþ�p),B1(pþ�p) and B2(pþ�p) etc. are given explicitly as

?ðpþ�pÞ ¼ ?þXnpj¼1

�pj@?

@pj, ð13Þ

where the symbol ? denotes the component of eachcoefficient matrix in (1). If a component of thosecoefficient matrices is a non-linear function on p,equation (13) is satisfied approximately if �p issufficiently small such that the linear approximation,obtained by taking the first order term of the infiniteseries expansion of the non-linear function, has a certainaccuracy. We denote those (approximated) perturba-tions of coefficient matrices in (1) as �A, �B1, �B2,�C1, �C2, �D11 �D12 and �D21 respectively. Then, allcomponents of (11) can be expanded in the form of thesecond order equations in terms of perturbations.Obviously (11) gives BMIs on those perturbations andis difficult to solve. However, if we assume allperturbations are small, it is valid to approximatethose BMIs as LMIs by ignoring the 2nd order productof perturbation matrices. We take those approximatedLMIs for the integrated design. The linear approxima-tion of BMI in (11) is given as the following:

�11 �T21 �T

31 �T31

�21 �22 �T32 �T

42

�31 �32 �33 �T43

�41 �42 �43 �44

266664

3777755 0, ð14Þ


where

�11 ¼ AX1 þ X1AT þ B2C1 þ CT

1BT2 þ A�X1

þ�X1AT þ�AX1 þ X1�AT þ B2�C1

þ�CT1B

T2 þ�B2C1 þ CT

1�BT2 ,

�21 ¼ A1 þ�A1 þ ðAþ�AÞT,

�22 ¼ ATY1 þ Y1Aþ B1C2 þ CT2 B

T1 þ AT�Y1

þ�Y1Aþ�ATY1 þ Y1�Aþ B1�C2

þ�CT2 B

T1 þ�B1C2 þ CT

2�BT1,

�31 ¼ ðB1 þ�B1ÞT,

�32 ¼ BT1Y1 þDT

21BT1 þ BT

1�Y1 þ�BT1Y1

þDT21�BT

1 þ�DT21B

T1,

�33 ¼ �33,

�41 ¼ C1X1 þD12C1 þ C1�X1

þ�C1X1 þD12�C1 þ�D12C1

�42 ¼ C1 þ�C1, �43 ¼ D11 þ�D11, �44 ¼ �44:

Equations (14) and (12) are LMIs on all perturbationmatrices and can be solved efficiently.As mentioned before the linear approximation of

BMIs is reasonable only if the perturbations are small.Hence, we need to confine the amount of thoseperturbations in some sense. In this paper we take aconstraint on the perturbation matrices given by

�ð��Þ5 ��ð�Þ ð�4 0Þ, ð15Þ

where �ð�Þ denotes the maximum singular value of avariable matrix, e.g., p, X1 and Y1 etc. The constraintin (15) is also given as the following LMI:

��ð�ÞI ��

ð��ÞT ��ð�ÞI

� �4 0: ð16Þ

Therefore, we can obtain the small amount of theoptimal perturbations by combining LMI conditions(14), (12) and (16). With the above linearized LMIs wepropose the following integrated design algorithm.

Algorithm 1 (Integrated design algorithm):

Step 0: Define the iteration number i¼ 0. Set the initialvalue of the structural design parameter vectorp0 2 P. In the following we define superscript ‘i’which denotes the ith iteration of this algorithm.

Step 1: Set �>0 in (15). For the plant derived for thefixed pi 2 P obtain the optimal H1 controllerKi(s) and the performance index Ji :¼ � i

1 for thefixed plant with LMI conditions (9) and (10).

Step 2: Obtain the optimal perturbation matrices, e.g.,�pi, �Ai

1 and �Bi1 etc. minimizing the

perturbation of the performance index �� i1

subject to constraints �� i15 0 andpi þ�pi 2 P, and satisfying (approximated )LMIs in (14), (12) and (16). If such perturba-tions can not be obtained, then set the optimalstructural vector popt :¼ pi and Kopt(s) :¼Ki(s)and stop. Otherwise, update piþ1 pi þ�pi

and go to the next step.

Step 3: Obtain the optimal H1 controller Kiþ1(s) andthe performance index Jiþ1 :¼ � iþ1

1 again for theplant yielding the new structural design para-meter vector piþ1. If Jiþ1 � Ji � 0 then � ��where 0< �<1 and go to Step 2. Otherwiseupdate pi piþ1 and go to Step 1.

In the proposed algorithm at least a convergence to alocal optimal solution is assured by the iterationbetween Steps 2 and 3 because the structural designparameter p is not updated unless the conditionJiþ1 � Ji 5 0 is satisfied. In this algorithm the value of� in (15) is working like a step width of general descent-based optimization methods, e.g., steepest descentmethod etc. In the integrated design problem somedesign methods guaranteed the local convergence of theobjective function have also been proposed (Tanaka andSugie 1998, Chien 2000, Lu and Skelton 2000).However, in all of their design algorithms all coefficientmatrices of state space (or descriptor) form must belinear functions on the structural design parameters.Meanwhile in our method the structural design param-eters and the controller can be adjusted simultaneouslywith a guarantee of the local convergence even if thestructural design parameters enter non-linearly incoefficient matrices of the plant state space form (1).Therefore by taking the proposed integrateddesign method we can largely extend the class of theplant to which the integrated design technique can beapplied.

Let us make some remarks on the proposedalgorithm.

Remark 2: According to the problem we can relax thecondition in (15) on the component of the vector pas long as we can obtain a decreasing sequence of theperformance index J i. From the authors’ experience(as shown in x 4) this relaxation is possible if allcoefficient matrices of the plant state space form arelinear functions on the vector p and make theconvergence much faster. If those coefficient matricesare not linear functions on the vector p, we must alwaysset �>0 as a quite small value, say, 10�3 such that thelinear approximation of the non-linear dependencementioned before is valid. Currently we do not have


a systematic way to select an appropriate value � to

obtain a good result, therefore, a kind of trial

and error scheme may be required. These characteris-

tics from the numerical aspect will be mentioned

again in x 4.

Remark 3: The above algorithm is specifically

described in the case for minimizing the H1 norm of

the closed-loop system. However, as mentioned before,

we can deal with another closed-loop norm or a

multiobjective problem by changing the above LMI

conditions appropriately. The algorithm works as

long as the optimal controller synthesis condition is

given in terms of LMIs on controller related matrices.

Remark 4: In this paper we assume the full-order

output feedback controller which does not have any

specific structures. We note that we can also deal with

the design problem in which the controller order is less

than that of the plant (reduced-order controller design)

(e.g., Iwasaki (1999)) and/or the controller must satisfy

some structural properties like a decentralized control

problem (e.g., Zhai et al. (2001), Mehendale and

Grigoriadis (2003)) if there exists an efficient (analytical

or numerically tractable) method to obtain such

controllers. The integrated design algorithm proposed

in this paper can embed the method to obtain such

controllers. However, the above problems are generally

still open and only some iterative algorithms for a

locally optimal controller have been proposed so far.

Therefore the amount of computation for the integrated

design subject to the order and/or the structural

constraints of the controller may become much larger

than that of the current full-order case.

Remark 5: Rakowska et al. (1993) applied a homo-

topy method to an integrated design problem. They

considered a specific performance index related to the

integrated design of a space structure and discussed a

homotopy-based solution procedure. In this study the

performance index about the closed-loop performance

is quite elementary and the control law is limited to a

direct velocity feedback law which is only effective

under the collocation of sensors and actuators. On

the other hand we aim to propose a general solution

procedure to obtain a solution to various integrated

design problems which are formulated as BMIs.

Those BMI conditions originally derive from LMI

conditions for the controller design. Currently a large

number of LMI conditions to obtain the controller

with various properties are available. This means that

we can deal with many closed-loop specifications by

using the proposed homotopy-like integrated design

method.

4. Design examples

4.1 2DOF system

Let us consider an active control of a 2DOF system,depicted in figure 1, where q1(t), q2(t), w(t) and u(t) arethe displacement of m1 and m2, the disturbance forceand the control force respectively. The spring anddamper denoted by ki and di (i¼ 1, 2) connect betweenthe fixed wall and m1 and m1 and m2 respectively. Thedisturbance and the control force are applied to mass m1

and the displacement of the mass m2 is measured tosuppress the vibration of the same mass. By taking thestate vector x(t) as xðtÞ :¼ ½q1ðtÞ q2ðtÞ _q1ðtÞ _q2ðtÞ�

T,the state space form in (1) is

_xðtÞ ¼ AxðtÞ þ B11 B2

1

� wðtÞ

vðtÞ

� �þ B2uðtÞ

zðtÞ ¼z1ðtÞ

z2ðtÞ

� �¼

C11

01�4

� �xðtÞ þ

0

1

� �uðtÞ

yðtÞ ¼ C2xðtÞ þ ½ 0 10�3�wðtÞ

vðtÞ

� �,

8>>>>>>>><>>>>>>>>:

ð17Þ

A :¼

0 0 1 0

0 0 0 1

�k1 þ k2m1

k2m1

�d1 þ d2m1

d2m1

k2m2

�k2m2

d2m2

�d2m2

26666664

37777775,

B11 :¼

0

01

m0

266664

377775, ð18Þ

B21 :¼ 04�1, B2 :¼ B1

1, C11 ¼ ½0 103 0 0�,

C2 ¼ ½0 1 0 0�,

where v(t) is the measurement noise.We consider a multiobjective problem where the

performance index Jm is defined by

Jm :¼ J1 þ �J2, �4 0, ð19Þ

Figure 1. Active control of 2DOF system.


where J1 and J2 are upper bounds of the closed-loopH1 and the square of H2 norm from [w(t) v(t)]T to z1(t)and z2(t) respectively. The scalar �>0 is a weightingparameter between the above two performance indices.For the fixed plant, the controller optimizing Jm canbe obtained by taking X2¼X1¼X, Y2¼Y1¼Y,A2 ¼ A1 ¼ A, B2 ¼ B1 ¼ B and C2 ¼ C1 ¼ C inLMIs given as (6)–(8) and (9) and minimizing the�1þ ��2. In this example two cases of structural designparameter vectors are considered. Each result is shownin the following:

Case A: In this case we take the structural designparameter vector p as:

p :¼ ½ d1 k1 �T: ð20Þ

We set the lower and the upper bounds of the vector pas pl :¼ 10�2 � ½dn1 kn1�

T and pu :¼ 10� ½dn1 kn1�T, where

dn1 ¼ 0:01 ðNs=mÞ and kn1 ¼ 1 ðN=mÞ, respectively. From(18) only the matrix A is a linear function of eachcomponent of the vector p in this case. The otherphysical design parameters are defined as m1¼m2¼ 1(kg), d2 ¼ dn1 and k2 ¼ kn1 respectively.By taking �¼ 0.2 (in (15)) and �¼ 1, the proposed

integrated design algorithm is applied to the problem.However, we do not impose the norm constraint oneach perturbation of the vector p in the this case (seeRemark 2). The initial value of the vector p isp0 :¼ ½dn1 kn1�

T. Optimization histories for the perfor-mance index Jm, the structural design parameters d1 andk1 are shown in figures 2, 3 and 4 respectively.The damping coefficient d1 and the spring constant k1

converge to their upper bounds respectively after onlyfour times iteration of the proposed integrated designalgorithm. To check the quality of the obtained solution(as mentioned in the previous section we can guaranteeonly the local convergence of the design parameters), wecarry out the search for the whole design spaceP :¼ fp : pl � p � pug by computing the optimal control-ler at 2500 plants obtained by gridding each rangeðd1: ½10

�4, 0:1�, k1: ½10�2, 10�Þ of structural design para-

meters to 50 points. The 3D plot of the result of theexhaustive search is shown in figure 5. Clearly, in this

case we could successfully find out the global optimalsolution with the proposed homotopy based method.This result also means that we could find out the globaloptimal solution of the current BMI problem. Weconduct the optimization several times from other initialvalues which are randomly selected to check theconvergence property of the design algorithm.Fortunately, we can obtain same global optimalsolutions for almost all cases. Furthermore we carriedout the optimization by imposing the norm constraintin (15) even on �p:¼ ½�d1 �k1�

T. The optimal solutionobtained was almost the same as in the case we showed.However, it took many more iterations to obtain.

02

46

810

00.02

0.040.06 0.08

0.1

600

700

800

900

1000

1100

1200

1300

1400

1500

k1 [N/m]d1 [Ns/m]

Jm

⊗

min(Jm) = 659.17 (k1 = 10, d1 = 0.1)

Figure 5. The actual performance index Jm (Case A).

Iteration number

J m

0 1 2 3 4600

800

1000

1200

1400

Figure 2. The optimization history for the performanceindex Jm.

Iteration number

d 1 [

Ns/

m] d1

u

0 1 2 3 4

0.04

0.08

0.12

Figure 3. The optimization history for the dampingcoefficient d1.

Iteration number

k 1 [

N/m

] k1u

0 1 2 3 4

4

8

12

Figure 4. The optimization history for the spring constant k1.


Case B: Vector p is taken in this case to examine thenon-linear dependence of the plant coefficient matriceson components of p with

p :¼ ½m1 m2�T: ð21Þ

The lower and the upper bounds of vector p arespecified pl :¼ 0:1� ½mn

1 mn2� and pu :¼ 10� ½mn

1 mn2�T,

where mn1 ¼ mn

2 ¼ 1 ðkgÞ respectively. Obviously from(18), matrices A, B1 and B2 are non-linear functions onm1 and m2 in this case. Note that the methods proposedin Tanaka and Sugie (1998), Chien (2000) and Lu andSkelton (2000) can no longer handle this case because ofthe non-linear dependence of those coefficient matriceson vector p. For the same � as that of Case A, theoptimal design is carried out. In this case we set �¼ 10�2

and the norm constraint in (15) is imposed on allperturbations. The result of the optimization is shownin figures 6–8. In contrast to Case A it takes around900 time iterations to obtain the convergence.In this case masses m1 and m2 are converged to their

lower and upper bounds respectively. An exhaustivesearch is also conducted in the whole design parameterspace P by taking the 50� 50 gridding on the range m1:[0.1, 10] and m2: [0.1, 10]. The result is shown in figure 9.We could successfully find out the global optimalsolution again even in the case that the coefficientmatrices of plant state space form are non-linearfunctions on vector p. Although the proposed algorithmis not guaranteed to obtain the global optimal solution,

this result suggests the capability of the proposed designmethod for more general integrated design problems.

4.2 Sensor/actuator placement

A sensor/actuator placement problem is solved in this

subsection. The optimal selection/placement of sensorsand actuators is quite important in control systemdesign because they strongly affect the achievableclosed-loop performance. In some cases a convexoptimization method exists for the optimal selectionand/or placement of sensors and actuators and thecorresponding optimal controller simultaneously(Geromel 1989, Oliveira and Geromel 2000). However,in general case optimal sensor/actuator selection andplacement, the problem is still open and any analytical

or numerically tractable methods to obtain the globaloptimal sensor/actuator selection and placement andthe feedback controller have not been proposed so far.In this example we deal with an optimal sensor/actuatorplacement problem minimizing the H1 norm of theclosed-loop system with the proposed homotopy-likedesign algorithm.

02

46

810 0 2 4 6 8 10

200

400

600

800

1000

1200

1400

m1 [kg]

m2 [kg]

Jm

min(Jm) = 352.31 (m1= 0.1, m2 = 10)

⊗

Figure 9. The actual performance index Jm (Case B).

Iteration number

J m

0 200 400 600 800 1000300

400

500

600

700

800

Figure 6. The optimization history for the performanceindex Jm (mass optimization).

Iteration number

m1

[kg]

m1l

0 200 400 600 800 1000

0.2

0.4

0.6

0.8

1.0

1.2

Figure 7. The optimization history for the mass m1.

Iteration number

m2

[kg]

m2u

0 200 400 600 800 1000

2

4

6

8

10

Figure 8. The optimization history for the mass m2.


Let us consider an active vibration control system ofa simply supported beam with a circular cross sectionin figure 10. The length, the diameter, the density andYoung’s modulus of the beam are denoted by L (m),d (m), � (kg/m3) and E (N/m2) respectively. The momentof inertia of the area is obtained by I :¼�d4/64. Twoactuators producing control forces u1(t) and u2(t) areinstalled at � ¼ �1a, � ¼ �2a, respectively. Two sensors areequipped for measuring displacements qð�1s , tÞ andqð�2s , tÞ. A disturbance force w(t) is injected at �¼ �w.We obtain the optimal sensor and actuator placement inthis problem with the proposed method. The structuraldesign parameter vector and its lower and upper boundsare defined by

p :¼ �1a �2a �1s �2s� T

, pl ¼ 04�1,

pu ¼ L� ½ 1 1 1 1 �T: ð22Þ

We assume that the displacement q(�, t) can beapproximated by

qð�, tÞ ’X3j¼1

qjðtÞjð�Þ, ð23Þ

where qj(t) is the jth modal displacement of the beam.The function i(�) is a normalized jth modal shape of asimply supported beam given as

jð�Þ ¼

ffiffiffiffi2

L

rsin

j��

L

� �: ð24Þ

Then the (approximated) modal equation of motion ofthe beam system is obtained as

€qf ðtÞ þ 2Z�_qf ðtÞ þ�2qf ðtÞ ¼ LwwðtÞ þ LauðtÞ, ð25Þ

where qfðtÞ :¼ ½q1ðtÞ q2ðtÞ q3ðtÞ�T is the (approxi-

mated) modal displacement vector and uðtÞ :¼½u1ðtÞ u2ðtÞ�

T respectively. Matrices Z :¼ diagð1, 2, 3Þand � :¼ diagð!1,!2,!3Þ, ð!j :¼ ð j�Þ

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEI=�SL4

p,

j¼ 1, 2, 3, S ¼ �d2/4) are the modal damping matrix

and normal frequency matrix respectively. Matrices Lw

and La are given as

Lw :¼

1ð�wÞ

2ð�wÞ

3ð�wÞ

264

375, La :¼

1ð�1aÞ 1ð�

2aÞ

2ð�1aÞ 2ð�

2aÞ

3ð�1aÞ 3ð�

2aÞ

264

375: ð26Þ

We take a controlled output z(t) as

zðtÞ :¼ qð0:3L, tÞ qð0:6L, tÞ r1u1ðtÞ r2u2ðtÞ� T

, ð27Þ

where r1 and r2 are the positive scalar weightings.By assuming the proportional damping, i.e., Z¼ ��(0<�� 1) (Meirovitch 1990) and taking the statevector xðtÞ :¼ ½qfðtÞ _qfðtÞ�

T, we derive the followingstate space form of the beam system:

_xðtÞ ¼ AxðtÞ þ B1ðtÞ þ B2uðtÞ

zðtÞ ¼ C1xðtÞ þD11wðtÞ þD12uðtÞ,

yðtÞ ¼ C2xðtÞ þD21wðtÞ

8><>: ð28Þ

A :¼03�3 I3

��2 �2��2

� �, B1 :¼

03�1

Lw

� �, B2 :¼

03�2

La

� �,

C1 :¼Lz 02�3

02�6

� �, D11 :¼ 04�1, D12 :¼

02�2

diagðr1, r2Þ

� �,

C2 : ¼ ½Ls 02�3�, D21 :¼ 02�1,

where the matrices Lz and Ls are given as

Lz :¼1ð0:3LÞ 2ð0:3LÞ 3ð0:3LÞ

1ð0:6LÞ 2ð0:6LÞ 3ð0:6LÞ

� �,

Ls :¼1ð�

1s Þ 2ð�

1s Þ 2ð�

1s Þ

1ð�2s Þ 2ð�

2s Þ 3ð�

2s Þ

" #:

In this case, matrices B2 and C2 are clearly non-linearfunctions on the vector p in (22). We take theperformance index (denoted by Jb) as the closed-loopH1 norm from w(t) to z(t). The optimal controller canbe obtained for a fixed p by minimizing �1 in LMIsin (9) and (10). The values of physical parameters aredepicted in table 1. Note that the material of the beamis assumed to be a kind of plastic which is used inDoki et al. (2002).

Figure 10. Simply supported beam system.

Table 1. Parameter values.

Value

Length L (m) 1.00

Diameter d (m) 5.00� 10�3

Density � (kg/m3) 6.21� 103

Young’s modulus (N/m2) 6.06� 106

Proportional constant � 10�3

Weighting r1 and r2 1


By taking �¼ 5� 10�3, the proposed optimal designmethod is applied to the sensor/actuator placementproblem. To obtain the better locally optimal solution,we conduct the optimization several times from differentinitial placements. In this example three local optimalsolutions are obtained from six initial placements whichare randomly selected. However, in most cases (in fourcases from six initial placements) both actuator place-ments converge to the place where the disturbance isapplied (�¼ 0.4L). Those results are shown in table 2with their initial placements. A typical result (case Iin table 2) is presented in figures 11 and 12. In thisproblem, it is found that the performance index Jb isquite insensitive to the sensor placements �1s and �2s .As shown in table 2, we can see that the values of theoptimal performance indices (Jb)opt are almost the same,irrespective of their sensor placements. This result alsosuggests we do not need to equip two actuators but needonly one actuator at the place where the disturbanceis injected.We do not check whether the obtained solutions are

globally optimal or not, in this example, because thecomputational load is too large to obtain the globaloptimal solution with the extensive search of the wholedesign parameter space. However we can claim that theobtained result is quite reasonable from the physicalconsideration because it is the efficient way to suppress

the effect of the disturbance force by collocatingactuators.

The results of this section show that the proposedintegrated design method works quite effectively evenfor the problems which earlier proposed BMI basedintegrated design approaches (Tanaka et al. 1998,Chien 2000, Lu and Skelton 2000) cannot deal with.The result of design examples indicates the capability ofthe proposed design scheme for more general andcomplex design problems in real applications.

The computational time to obtain the optimalsolution in each example is discussed. In Case A of the2DOF system example the computational time is lessthan one minute with a standard Windows OS PC(CPU: Pentium III 600MHz, 256MB RAM) withMATLAB software. In Case B of the 2DOF systemand sensor/actuator placement examples, the time toobtain the optimal solution is up to several hours withthe same PC. If a more sophisticated computer, which iscurrently available, is employed the computational timemust be dramatically reduced and this fact indicates thatthe application of the proposed homotopy-like inte-grated design method to real industrial applications isnow becoming more realistic.

5. Conclusion

We have investigated the homotopy based integrateddesign method in this paper. An iterative designalgorithm based on the homotopy method has beenproposed. In this method we can utilize all LMIconditions for controller synthesis to the integrateddesign problem. This fact means that we can deal withthe multiobjective problem in the integrated designproblem. Currently the LMI based approach is recogn-ized as one of the most effective solution methods.The proposed method can also be applied to the casewhere the coefficient matrices of the plant state spaceform are non-linear functions on their structural designparameters by taking the linear approximation of thenon-linear dependence and limiting the amount of the

Table 2. Optimal sensor/actuator placements.

Case Jb �1a �2a �1s �2s

I Initial 1.673 0.950 0.231 0.607 0.486

Optimal 0.711 0.399 0.400 0.574 0.453II Initial 0.761 0.511 0.324 0.232 0.493

Optimal 0.709 0.400 0.400 0.232 0.490

III Initial 0.975 0.144 0.518 0.243 0.482Optimal 0.708 0.400 0.400 0.247 0.451

IV Initial 2.574 0.550 0.493 0.030 0.219

Optimal 0.709 0.399 0.401 0.030 0.229

Iteration number

0 100 200 300 400 500

0.2

0.4

0.6

0.8

1.0

x a, x

s

xa1

xs1

xw

xa2

xs2

Figure 12. The optimization history for the sensor/actuator

�ia and �is (i¼ 1, 2) (Case I).

Iteration number

J b

0 100 200 300 400 5000.6

0.8

1.0

1.2

1.4

1.6

1.8

Figure 11. The optimization history for the performance

index Jb (case I).


updates of design parameters. The proposed iterativealgorithm is guaranteed to converge to a local optimalsolution. With several design examples we have shownthe capability of the proposed design methodology.

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Integrated design of structural and control systems with a