Proceedings of the 45th IEEE Conference on Decision & Control FrIP12.4Manchester Grand Hyatt HotelSan Diego, CA, USA, December 13-15, 2006

Damping Parameter Design in Structural Systems Using an AnalyticalH,, Norm Bound Approach

Yuanqiang Bai*, Karolos M. Grigoriadist and Michael A. Demetriout

Abstract-This paper examines the damping parameter de-sign problem for structural systems with collocated measure-ments and inputs so that H,, norm bound constraints aresatisfied. We utilize a particular solution of the Bounded RealLemma that provides an explicit upper bound of the H,,norm of a collocated structural system. Using this upper boundresult the damping design is formulated as a linear matrixinequality (LMI) optimization problem with respect to thedamping coefficients of the structural system. Numerical exam-ples demonstrate the benefits and computational advantages ofthe proposed damping design method.

I. INTRODUCTION

As an important component of passive structural systemsdesign, the problem of damping parameters design has beenstudied over a century. The use of dashpots and tuned massdampers (TMD) are two widely used methods of passivevibration attenuation. These simple devices have been provenvery effective for reducing severe vibrations of machinery,buildings, bridges and many other mechanical systems withrelatively low cost [2]. There are numerous optimization-based methods that have been proposed for optimal place-ment and optimal value assignment of such damping devices,yet many of them are not widely adopted due to their com-plexity and computational inefficiency [3][4][5][6][7]. Theproblem becomes an extremely challenging one for large-scale structural systems with hundreds or thousands degreesof freedom where the above methods become prohibitive.

Recently, structural parameter design techniques have beenproposed by viewing the structural parameter selection andoptimization of passive mechanical components as a controlsystem design problem. This approach allows the use of rig-orous system theoretic and system gain mathematical tools toquantify system performance and to provide a connection toclosed-loop response in controlled structural systems. Thus,optimal and robust control techniques, such as the H2/LQRand Ho, optimization approaches, have been proposed forstructural parameter design. The state space Ho, norm con-trol method based on the standard Riccati equation approachor the linear matrix inequality (LMI) formulation are nowwell developed control synthesis tools. The optimal staticstate feedback and full-order dynamic output feedback Ho,control synthesis problems can be solved using iterations on

*Sea Engineering Inc. 15915 Katy Freeway, Suite 300, Houston, TX77094, USA

tCorresponding author. Department of Mechanical Engineering, Univer-sity of Houston, Houston, TX 77204; Tel: 713-743-4387, Fax: 713-743-4503, Email: karolos@uh. edu

tDepartment of Mechanical Engineering, Worcester Polytechnic Institute,Worcester, MA 01609, USA

the corresponding Riccati solutions or via the computationalsolution of a convex LMI optimization problem [13][14][15].On the other hand, the static output feedback and thefixed-order dynamic output feedback Ho, control synthesisproblems are difficult computational problems since they re-quire the solution of (nonconvex) bilinear matrix inequalities(BMIs) or LMIs with coupling rank constraints [16][17].The optimal dynamic output feedback H2 control synthesisproblems is connected to the solution of the LQG problem,which is a combination of state estimation and state feedbackcontrol. However, the use of such control-oriented methodsfor strcurural parameter design still leads to complex numer-ically cumbersome optimization problems[26].

Having strain or displacement information as the systemoutput and feedback of the measured output rate has beenproved to be very effective in adding local damping to thestructure. The control of structural systems with collocatedsensors and actuators has been shown to provide greatadvantages from a stability, passivity, robustness and animplementation viewpoint. For example, collocated controlcan easily be achieved in a space structure when an attituderate sensor is placed at the same location as a torque actuator[1][8]. Collocation of sensors and actuators leads to symmet-ric transfer functions. Several other classes of engineeringsystems, such as circuit systems, chemical reactors andpower networks, can be modelled as systems with symmetrictransfer functions. Stabilization, robustness, model reductionand control of such systems has been examined recently[9][1O][1 1].

In this work, we use recently developed control-orientedalgebraic tools to formulate the damping coefficient designproblem as an efficient convex linear matrix inequality (LMI)optimization problem. By exploiting the particular structureof collocated structural systems, explicit upper bounds forthe Ho, norm of the system can be obtained. To this end,particular solutions of the Bounded Real Lemma are usedand explicit expression for an upper bound of the Ho, normof such external symmetric system are obtained that requireonly the computation of the maximum eigenvalue of a sym-metric matrix. Subsequently, we damping parameter designproblem is formulated as an LMI optimization problem ofminimizing the Ho, norm bound with respect to the unknowndamping coefficients.

The standard notation > (<) is used to denote the positive(negative) definite ordering of symmetric matrices. The itheigenvalue of a real symmetric matrix N will be denoted byAi(N) where the ordering of the eigenvalues is defined asAmax(N) = A1(N) >A2(N) > ... > An(N). The maximum

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singular value of a (not necessarily square) matrix N will bedenoted by omax(N), which is also its spectral norm N .

II. ANALYTICAL HOc-NORM BOUND FOR COLLOCATEDSYSTEMS

Consider the following vector second-order representationof a structural system with collocated velocity measurementsand inputs

Mq + Dq + Kq Fu (1)y = FTq

where q(t) E R' is the generalized coordinate vector, u(t) ERm is the input vector and y(t) E R' is the measured outputvector. The matrices M, D and K are symmetric positivedefinite matrices that represent the structural system mass,damping and stiffness distribution, respectively.

The system has a state-space realization as follows

;i = Ax+ Buy= Cx

with

A =

C = I

00M-1KO FT]

(2)

IM1D]j B= 01F3)

Theorem 1 Consider the vector second-order system real-ization (2)-(3). The system has an Ho, norm -y that satisfies

7<~7)=Amax(FTD- 1F) (6)

To prove this result recall the Bounded Real Lemma(BRL) characterization of the Ho, norm of a system.

Lemma 1 [22] A stable system (2) has an Ho, norm lessthan or equal to -y if and only if there exists a matrix P > 0satisfying

[ ATP+ PA PB CTBTP --yI 0 < 0C 0 -yII

(7)

Recall also the following Schur complement formula [23].

Lemma 2 The block matrix

s -- Sll S121[1S2 122]

where S11 and S22 are symmetric, is positive definite if andonly if

where x = [qT qT]T. Notice that the transfer function G(s)of the system (2)-(3)

G(s) = sFT (Ms2 + Ds + K)-1F

is symmetric, i.e., G(s) = GT(s). The system (2)-(3) isan externally symmetric state-space realization, that is, thereexists a nonsingular matrix T such that

ATT= TA, CT = TB (4)

This class of systems is more general than the class ofinternally or state space symmetric systems that they satisfythe symmetry conditions (4) with a positive definite trans-formation matrix T [12]. Obviously, state-space symmetryimplies external symmetry, but the converse is not true, thatis, there exist symmetric transfer matrices for which thereis no internally symmetric realization. An analytical solutionof the Ho, control problem for internally symmetric systemshas been presented in [18].

Recall that the Ho, norm of the system (2) is given by

CQ loo = SUp Jmax{G(j) }w R

(5)

where G(s) = C(sI-A)- 1B is the transfer function of thesystem and omax denotes the maximum singular value of amatrix. It is well known that for a stable LTI system, its Ho,norm can be approximated iteratively, for example using abisection method [19]. The following result shows that for avector second-order realization (2)-(3), an upped bound of itsHo, norm can be computed using a simple explicit formula[20].

Sii > 0 and S22 512511 S12 > 0

or

S22 > 0 and Sll -S2S2S1T2 > 0

Theorem 1 follows from the BRL condition and thefollowing algebraic result [21].

Lemma 3 Consider matrices F and Q such that F has fullcolumn rank and Q is symmetric positive definite. Then Q >ppT if and only if

Amax(FTQ lF) < 1

Proof of Theorem 1 (Sketch) The result follows from theBounded Real Lemma 1 by utilizing the following Lyapunovmatrix

(8)P= KO

Application of the Schur complement formula in Lemma 2results in the following condition

D FFT < 0.

Then, application of the Lemma 3 provides the bound (6).ENumerical examples in [20] demonstrate the validity andcomputational efficiency of the above analytical bound.

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III. DAMPING DESIGN USING THE ANALYTICAL BOUNDAPPROACH

The analytical Ho, norm upper bound (6) of the collocatedstructural system (1) is solely dependent on the dampingdistribution matrix D and the input/output distribution matrixF. For lumped parameter systems, the damping metric D canbe expressed in terms of the elemental damping coefficientsas follows m

D CiTi (10)i=l

where ci denotes the viscous damping constant of the ithdamper and Ti represents the distribution matrix of thecorresponding damper in the structural system. The Ti's aregiven symmetric matrices with elements 0, 1 and -1 thatdefine the structural connectivity of the damping elementsin the structure. Then, using the Schur complement formula,the Ho, norm upper bound condition (6) can be re-writtenas

m

,: CiTli=lFT

Fl > 0

YII(1 1)

Practical structural system design specifications impose up-per bound constraints on the values of the damping coeffi-cients, that is

0 <Ci < Cmax (12)

Also, often an upper bound on the total available dampingresources is imposed, that is

m

E? Ci < Ctotali=l

(13)

Based on the above discussion the damping design prob-lem can be formulated as follows.

Damping Design Optimization ProblemConsider the collocated structural system (1) with the

damping distribution (10). For a given positive scalar -y,the Ho, norm of the system is less than -y if the followingconditions with respect to the damping coefficients ci arefeasible.

m

E: CiTii=lFT

F > o

-YIJ0 K Ci K Cmax (15)m

Ci < Ctotal - (16)

The above conditions constitute an LMI feasibility problemwith respect to the damping coefficients ci. Then the opti-mization of damping coefficients can be achieved by solvingthe LMI optimization problem

min y, (17)

subject to constraints (14)-(16).Remark 1: Some dynamical system can exhibit negative

values of damping. This has been observed in the optimal

damping design of cables[25]. In such case, the designparameter cis are allowed to be negative.

The upper bound approach in the above LMI formulationprovides a computationally efficient method to compute thedamping coefficients of collocated structural systems. Sincethe assigned upper bound -y is always greater than or equalsto the exact Ho, norm of the system, the design resultis conservative. However, our computational examples andexperience with the proposed bound indicate that it indeedprovides a good approximation of the exact Ho, norm.

IV. NUMERICAL EXAMPLES

Single Degree of Freedom (1-DOF) CaseTo demonstrate and motivate the above results consider

the 1-DOF case (n = 1) where q(t), u(t) and y(t) are scalarquantities in (1). For this scalar case the magnitude of thefrequency response function (FRF) of the system (1) is

G(jw) ~~~F2wbj|G(,/=(K Mwn2)2 +D2w2where M, D and K represent the scalar mass, damping andstiffness coefficients of the system. It can be easily observedthat at the natural frequency of this dynamic system, i.e.,when

KM

the magnitude of the FRF reaches its maximum. Thus theabove FRF magnitude satisfies the following bound at allfrequencies

G(w F2~w F2IG(j) < Da-D~w D

that is, CIGo, < F2/D. This bound is precisely the oneprovided in Theorem 1 for the system. In fact, in this scalarcase the above bound provides the exact Hz, norm of thesystem, that is CIG&,, = F2/D. Therefore, the Hz, normbound CIGI,,, < -y is achieved if and only if the dampingcoefficient D is selected to satisfy the bound

D > F2/_aNotice that this result coincides with the bound obtained from(14).As an example consider the case where F = 10, M = 30

kg and K = 500 N/m. Then, for a desired Hz, norm bounda = 0.5 the designed value of the damping coefficient D =

102/0.5 = 200 N s/m. This result is confirmed from theBode diagram of the system shown in Fig. 1 that obtains amaximum of 20 * log10 (0. 5) =-6.02 dB at w = FK7/M4.08 rad/ sec.

Two-story Shear Building ModelNow let us consider a two-story shear building model with

added viscous dampers as shown in Fig. 2. This model isconsidered in reference [4]. The primary system of massr1 = rn2 = 100 has stiffness k1 = k2 = 200 relative tothe base. The system is subject to the disturbance inputs d1and d2 (such as wind gust disturbance on the structure). Our

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Bode Diagram

-10 /

-15m

-20/\0)

-25 - /\

-30-/

-350.1 1 10

Frequency (rad/sec)

Fig. 1. Bode diagram of the I-DOF example

objective is to design the damping coefficients cl and C2that attenuate the velocities of masses rn, and rn2 causedby the disturbance forces d, and d2. Specifically, we wantthe designed spring-mass-damper system to satisfy the givenH,,,, norm performance -y < 0.5 from the disturbance forcesto the velocities vl, V2 of the masses rn, and rm2.

n addition, the damping coefficients are subject to thebonstraints

ci<10, i =1,2.

Using the above parameters we formulate the dampinglesign LMIs and solve the corresponding LMI feasibilityroblem (14)-(15). The damping coefficients are obtained as

c1 = 5.04, c2 6.49

nd the exact Ho, norm of the damped system is 0.481Which is a bit less than the desired bound 0.5. The frequency.esponses from the disturbance d1 to output v1, d2 to V2, d1o V2 and d2 to v1 are shows in Fig. 3 and Fig. 4 respectively.t can be seen that the designed system reduced the effect ofisturbances significantly and our design goal is achieved.

OAftnr rnciinr. _ A~~~~~~~~~~~~~~~~~Ml tfil Utji:Slyl

m-10 Before design

-20

'E -30

7 -40

10 100 101Frequency (rad/s)

0F- After design

Before designm -10V

) -20-

'E -30-

73 -40 r

101Frequency (rad/s)

Fig. 3. Frequency responses of the undamped and damped system: fromdl to vl (above) and from d2 to v2 (below).

100Frequency (rad/s)

101

Fig. 2. Two-story shear building model with added viscous dampers

The design parameters are cl and c2 and the damping dis-tribution matrices T1 and T2 are obtained from the expansion

D Cl + C2 -C2- C2 C2

2= ciT

111

101Frequency (rad/s)

Fig. 4. Frequency responses of the undamped and damped system: fromdl to v2 (above) and from d2 to vl (below).

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45

Five Spring-mass-damper System ExampleNext consider a serial mass, spring and damper system

shown in Fig. 5. This example is borrowed and revised fromthe papers by Sipola et al. [26] and Zuo et al. [3].

0

~00

-o

40yfrom LMI resultExact Hinf norm

35

30

25

20

158

I 10

Fig. 5. Diagram of the 5 mass-spring-damper system

Our design objective is to optimize the values of thedamping coefficients ci 1, 5) so that the Ho, normof the collocated system from the disturbance forces d1 andd2 to the velocities of masses Tf2 and rn4 is minimized. Thephysical constants of this system are mi = 1, ki 1 andthe damping coefficients ci 1, 5) are chosen as thedesign parameters.The damping distribution matrix D of this system is given

by

Cl + C2 -C2

-C2 C2 + C3D = - c3

-C3

C3 + C4

-C4 C4

-C4 C+ C5 -C5-C5 C5

5

i TiFor comparison and design trade-off purposes we consider

a family of optimal damper designs using the result ofTheorem 2. The design correspond to different values of thetotal damping capacity Ctotal ranging from 0.5 to 20. Theresults of the optimal design using the Ho, upper boundoptimization approach are shown are shown in Fig. 6 andFig. 7. Fig.6 shows the values of the Ho, norm boundobtained using our upper bound optimization approach, aswell as, the exact Ho, norm that corresponds to each designas the total damping capacity Ctotal changes.

Indeed, we observe that in each design the guaranteed Ho,norm bound of the designed system and its actual Ho, normare very close. This result demonstrates that our upper boundoptimization approach provides a very good upper boundestimate of the actual Ho, norm of the system. The valuesof the optimal damping parameter that correspond to eachdesign are shown in Fig. 7. We oberve that for each designC1 = C3 and C2 = C4 although C5 is closed to 0.The significance and benefits of the proposed Ho, upper

bound damping parameter design approach is evident in thedesign and optimization of very large scale structural systemswhere standard methods based on nonlinear optimizationapproaches are computationally prohibitive. The proposedLMI-based convex optimization formulation can address theaddress the design of structural systems with handreds or

010 5 10

Ctnta

15 20

Fig. 6. Comparison of the upper bound optimization result and the exactH,, norm of the designed systems

6

-5I)z-4a)

%4-3-.a)0

C-).5-

EcJ1

CZ 1

C1C2c3

c4

c5

0 5 10

total15 20

Fig. 7. Optimal damping coefficients using the H,, upper bound approach

thousands of states and design variables. To solve a similardesign problem as stated in Ref. [3], it takes 1070 LMIiterations to find a sub-optimal Ho, solution using alternativeminimization. By using Theorem 2, it takes 14 LMI iterationswithin one second CPU time to find the optimal result. It canbe seen that this method is of time efficient compared withother existing methods (although the method proposed byZuo et al[3] can design the damping and stiffness parameterssimultaneously).

V. CONCLUSION

We have obtained a simple explicit expression for an Ho,norm bound of structural systems with collocated sensorsand actuators. By utilizing these bounds, LMI conditions areformulated to design the damping parameters that guaranteea desired Ho, norm of such structural systems. The designmethods are particularly useful for very large scale systemswhere the existing damping coefficient design methods arecompuationally prohibitive.

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