modeling and control of vibration in mechanical structures

8/10/2019 Modeling and Control of Vibration in Mechanical Structures

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IT Licentiate theses2005-005

Modeling and Control of Vibrationin Mechanical Structures

PETER NAUCLER

UPPSALA UNIVERSITYDepartment of Information Technology


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BY

PETERNAUCLER

October 2005

DIVISION OFS YSTEMS AND C ONTROL

DEPARTMENT OF I NFORMATION T ECHNOLOGY

UPPSALAU NIVERSITY

UPPSALA

SWEDEN

Dissertation for the degree of Licentiate of Philosophy in Electrical Engineering with

Specialization in Automatic Controlat Uppsala University 2005


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Peter Naucler

[email protected]

Division of Systems and Control

Department of Information Technology

Uppsala University

Box 337

SE-751 05 Uppsala

Sweden

http://www.it.uu.se/

c Peter Naucler 2005

ISSN 1404-5117

Printed by the Department of Information Technology, Uppsala University, Sweden


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Abstract

All mechanical systems exhibit vibrational response when exposed to exter-nal disturbances. In many engineering applications vibrations are undesir-able and may even have harmful effects. Therefore, control of mechanicalvibration is an important topic and extensive research has been going on inthe field over the years.

In active control of vibration, the ability to actuate the system in a controlledmanner is incorporated into the structure. Sensors are used to measure thevibrations and secondary inputs to the system are used to actuate the flexible

body in order to obtain some desired structural response.

In this thesis, feedback and feedforward control of active structures are ad-dressed. The thesis is divided into four parts. The first part contains ageneral introduction to the subject of active vibration control and also pro-vides an outline of the thesis.

The second part of the thesis treats modeling and feedback control of a beamsystem with strain sensors and piezoelectric actuators. Physical modelingand system identification techniques are utilized in order to obtain a loworder parametric model that is suitable for controller design.

The third part introduces the concept of a mechanical wave diode, whichis based on feedforward control. A controller is derived on the basis ofequations that describe elastic waves in bars. The obtained controller isshown to have poor noise properties and is therefore modified and furtheranalyzed.

The final part of the thesis treats the same type of wave diode system aspart three, but with more general feedforward filters. Especially, a controllerbased on Wiener filter theory is derived and shown to drastically improvethe performance of the mechanical wave diode.


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List of Publications

The thesis is based on the following publications, which will be referred toin the following order:

[I] PETERNAUCLER, HANSNORLANDER, ANDERSJANSSON ANDTORSTEN SODERSTROM, Modeling and Control of a ViscoelasticPiezolaminated Beam, Proc. 16th IFAC World Congress, Prague,Czech Republic, July 3-8, 2005.

[II] PETERNAUCLER, BENGT LUNDBERG AND TORSTENSODERSTROM,

A Mechanical Wave Diode: Using Feedforward Control for One-WayTransmission of Elastic Extensional Waves, Submitted to IEEETransactions on Control Systems Technology, 2005.

[III] PETERNAUCLER AND TORSTENSODERSTROM, Polynomial Feed-forward Design Techniques for a Mechanical Wave Diode System,Submitted to Control Engineering Practice, 2005.

A version of Paper I was also presented as

PETERNAUCLER, HANSNORLANDER, ANDERSJANSSON ANDTORSTENSODERSTROM, Modeling and Control of a ViscoelasticPiezolaminated Beam, Preprint of Reglermote, Gothenburg, Sweden,May 26-27, 2004.


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Acknowledgments

I would like to thank my supervisor, Professor Torsten Soderstrom for of-fering me the position as a PhD student and for his guidance during thiswork. I would also like to thank my colleagues for providing a nice workingatmosphere.

Finally, I thank my wife Maria and my daughter Emma for all their loveand support.


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Summary 1

1 Introduction

The presence of vibration is an important problem in many engineering ap-plications. Traditionally, various passive techniques have been utilized tomitigate such problems. Examples include adding mass and damping inorder to reduce structural vibration [29]. Passive techniques are, however,difficult to apply in the low frequency region [15]. Furthermore, in manyapplications it is desirable to keep the weight as low as possible, which canmake passive solutions unattractive [3]. In fact, it is a growing trend inmanufacturing of engineering systems to reduce the weight of mechanicalstructures. This is particularly so in spacecraft and aircraft engineering,

where it is possible to substantially decrease costs by use of lighter materi-als and/or weaker structures [32]. However, this will in turn lead to evenmore flexible structural dynamics which may limit the performance of thestructure. Other applications that can be mentioned are vibration controlin cars [34], telescopes [38], marine applications [7, 11] and helicopters [36].

All mechanical systems exhibit vibrational response when exposed to distur-bances. In order to modify the system dynamics in a desired way, one mayuse secondary inputs to the system. This is performed in active vibrationcontrol, or smart structures, where the ability to actuate the system in acontrolled manner is incorporated into the structure.

Modeling and control of active structures have received a lot of attention inthe past decades and there are several text-books on the subject, see e.g.[32, 14, 38]. Active control of beam vibration has been the subject of intenseresearch. This is due to the fact that the beam is a fundamental element ofmany engineering structures and that its characteristics are well understood[6].

An active structure is a structure that is self-sensing and self-compensating.Sensor measurements are fed to a controller (feedback, feedforward) whichmanipulates the control variables that actuates the flexible body in order

to obtain some desired structural response. Thus, a way of transformingelectrical energy to mechanical energy and vice versa is needed [32].

Piezoelectric materials exhibit a significant deformation when an electricfield is applied and they produce an electric field when deformed. There-fore, they can be used as both acting and sensing devices in smart structures.The evolvement of materials science has resulted in the availability of inex-pensive piezoelectric materials [18] and special interest has been devoted tothe ability of applying them in smart structures, see e.g. [15, 33, 18, 17, 35].Strain gauges [46, 31] and accelerometers [35] are other sensors that are


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2 Peter Naucler

typically used to measure vibrations. Common devices used as actuators

are electromagnetic transducers [38, 13] and structures actuated by a mo-tor attached to the flexible body, as for instance in control of flexible robotmanipulators [46, 31].

2 Modeling of Mechanical Structures

In order to design and analyze an active structure it is important to havea good understanding of the vibration characteristics of the system under

consideration. An important task is therefore to construct some kind ofmodel for the actual system. This can be performed in various ways. Themost common approach is to utilize physical relations to obtain a mathemat-ical model that describes the system. Recently, however, tools from systemidentification [42, 22] have been employed in the design of active structures[12, 27].

2.1 Physical Modeling

The theory of vibrations has attracted mathematicians, physicists and me-chanical engineers for many years [30]. Often, the obtained models areordinary differential equations of one or several degrees of freedom; but notseldom they are partial differential equations which means that the systemis of infinite order. An example of a simple mechanical system that can beeasily modeled by use of physical relations is shown in Figure 1.

Example 1. Physical modeling

Consider the single degree of freedom mass-spring-dashpot system in Figure1. The deviationw from the rest position is modeled using Newtons secondlaw of motion,

md2w

dt2 = c

dw

dt kw+f(t),

where m, k and c are, respectively, the mass, the spring constant and thedamping coefficient. This is an example of viscous damping, where the forceis proportional the velocity. By use of Laplace transformation, the systemcan be written in an input-output fashion,

W(s) = 1/m

s2 + cm

s+ km

F(s)


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Summary 3

k

c

m

f(t)

w

Figure 1: Mechanical system with a mass, a spring and a dashpot.

where W(s) and F(s) are the Laplace transforms ofw(t) and f(t), respec-tively. It assumed that the system is in rest position at time t = 0.

Although very simple, the system in the example can be generalized toarbitrary degree of freedom by interconnection of several such parts.

Structural modeling of vibrating beams and plates etc. are more complexand requires partial differential equation descriptions. Then, the modalanalysis procedure is often employed to solve the equations [30, 32]. Thisapproach generates infinite order transfer functions of the form

G(s, r) =

k=1

k(r)

s2

+2

k

(1)

where r is a spatial coordinate, k(r) is a function of space and k are theeigen (or natural) frequencies of the system. It is noticeable that in general,the modal analysis procedure does not allow damping [30]. A term thatintroduces a small damping is most often added to the denominator in anad hoc manner in a final step of the modeling.

For a model to be employed for simulations it must be of finite order. Inaddition, for controller design purposes it is desirable to have a model ofquite low order. Therefore, (1) needs to be reduced in some way. The most

direct approach is to simply truncate the model and ignore all terms suchthat k > M for some positive integer M. However, the truncation mayhave an adverse effect on the low frequency dynamics since the reducedorder model will have incorrectly placed zeros. To overcome this issue, afeed-through term can be added to the truncated model [32, 19].

Finite element modeling is a technique for spatial discretization of dis-tributed parameter systems (such as beams and plates) which yields spa-tially discrete models of very high order [30, 32]. The method has been usedfor simulation and control of active structures [21].


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2.2 System Identification

In the field of system identification, dynamical models are constructed fromexperimental data. Traditionally, system identification experiments are car-ried out in the time domain. In the area of active structures most resultsare, however, reported from frequency domain identification techniques [37].The advantage of frequency domain identification is that a large amountof time domain data can be compressed into a small amount of frequencyresponse data [26]. The frequency data can be obtained from e.g. steppedsine excitation or Fourier transformation.

Often, lightly damped structures with several inputs and outputs are en-countered. For such systems, subspace based frequency domain algorithmshave been successfully used [26, 45, 27]. These methods identify state-spacemodels by use of geometrical relations between the input and output dataand are well suited to describe high order multivariable systems.

Example 2. System identification

Let us return to the system in Example 1. Assume that we would like toestimate the unknown parametersm,k and c. From an identification exper-iment with F(k) as input, we have retrieved the noisy frequency domainoutput dataY(k) =W(k) + (k) for M frequency points1 M. We

postulate the parametric model

G(s) = b0

s2 +a1s+a2

and try to minimize the quadratic criterion

J=Mk=1

G(ik) Y(k)F(k)2

,

with respect toa1, a2 and b0. This requires a nonlinear least squares search

procedure. For a subspace based approach, there is no need for a explicitmodel parameterization. The only parameter needed is the model order. Inaddition, there is no need for nonlinear optimization.

3 Active Control of Mechanical Structures

In the literature, a variety of control strategies for active structures has beenreported. These strategies include both feedback and feedforward control.


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Summary 5

For feedback control, particular interest has been devoted to the design and

analysis of positive real (dissipative, passive) systems. This is an attractiveproperty that applies to e.g. beams with collocated actuator and sensorpairs. If a dissipative system is controlled with a strictly positive real con-troller the closed loop system will be guaranteed to be stable. This is a wayof avoiding instability due to spillover dynamics. Early work on such robustcontrol strategies include [24, 25] and more recently the text book [23].

Other control approaches that can be mentioned is spatial control, wherevibration is minimized over an entire structure [18]; and LQG and Hcontroller synthesis [35, 2, 40].

If disturbances are measurable, one alternative is to apply feedforward con-trol. Early work reporting on feedforward control for damping vibration in-clude [43, 39, 28]. Adaptive feedforward control strategies based on Widrowsleast mean square algorithm have gained considerable popularity [9, 20, 10, 8].

Wiener filter theory has been used to construct ideal feedforward filters.Typically, these are not causal and cannot be implemented but have beenused to derive some performance limitations for the designed systems, [9].

4 Outline of the Thesis

This thesis treats modeling and control of waves and vibration in mechanicalsystems. Two main problems are considered; feedback control of flexuralwaves in a beam and feedforward control of extensional waves in a bar.

4.1 Paper I

The paper concerns modeling and control of flexural waves in a viscoelastic

beam. The beam is relatively thin and equipped with a piezoelectric actu-ator and a strain gauge. Under the assumption that the shear deformationis negligible compared to the bending deformation, elastic beams can bemodeled using the Euler-Bernoulli beam equation [44],

2

2

EI

2w(t, )

2

+A

2w(t, )

t2 = 0

wherew(t, ) is the transversal displacement of the beam at time t and spa-tial coordinate ; and E, I, and Adenote, respectively, Youngs modulusof elasticity, moment of inertia, density and cross-sectional area of the beam.


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The Youngs modulus of elasticity is real valued and constant in the elastic

case but acts as a transfer function operator for viscoelastic materials. It isdenoted by thecomplex modulusin the mechanical engineering literature [5].In addition, different parts of the beam have separate structural propertiesdue to the actuator material. Taking these issues into account makes themodeling more difficult compared to previous modeling approaches, such ase.g. [18, 17].

The modeling results in infinite dimensional frequency responses. In order toperform model based controller design, system identification techniques areemployed for model reduction. Thus, a combination of physical modelingand system identification is used to obtain a parametric model of relatively

low order. Finally, the system is controlled with a feedback controller whichis designed by use of LQG control theory.

4.2 Paper II

Paper II introduces the concept of a mechanical wave diode, which is de-signed to control elastic extensional waves in mechanical networks. The sys-tem is controlled using feedforward control with the purpose to fully rejectwaves traveling in one direction of a bar and fully transmit waves traveling

in the opposite direction. The device is equipped with strain sensors tomeasure disturbances and a piezoelectric actuator.

Elastic extensional waves in bars can be described by the partial differentialequation

E2(t, )

2

2(t, )

t2 = 0,

where is the strain as a function of time and space. This equation is thebasis for the derivation of the mechanical wave diode. An ideal feedforwarddesign is derived for the noise-free case; but is shown to have poor noise

properties. A modified feedforward design is proposed upon the basis of theideal design and analyzed for different disturbance scenarios.

4.3 Paper III

The third paper extends the results from Paper II with more general types offeedforward design techniques. Especially, a Wiener feedforward controller[41, 1] is derived and shown to improve the performance compared to the


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Summary 7

previous findings. Three different controller types are compared with respect

to output variance, control signal variance and signal model uncertainty.

5 Future Work

The results presented in this thesis rely on analytic results and computersimulations. It would be valuable to perform real-life experiments to validatethe models and control strategies presented. There are several aspects of thisissue. For the beam system in Paper I for example, it would be interestingto compare the computed transfer functions to experimental data. Then, it

would be possible to see how much is gained by using the detailed modelingcompared to less descriptive approaches. In addition, the controller shouldbe implemented and tested.

The optimality of the Wiener feedforward controller in Paper III is onlyvalid in the idealized situation that the models which describe the systemare true. Therefore, it would be of interest to test the performance of thedifferent feedforward structures on an experimental set-up.

Other ideas of coming research include:

Due to the forgiving nature of the feedback mechanism [16], it is pos-sible to use only approximative models for control design. Therefore,one approach could be to use the modal analysis [32] procedure formodeling of the viscoelastic beam, and employ the obtained model forcontroller design. This can then be compared to what can be achievedby use of the models in Paper I.

The beam system in Paper I can be extended with several actuatorsand sensors in order to improve the control effort and minimize vibra-tion over the entire structure. Especially, the use of several actuatorswould increase the complexity of the descriptive modeling approachwhere the structural properties of the piezo-elements are taken into

account.

Optimal sensor placement [4] can be used to improve the control offlexible beams and can be an interesting topic for further research.

In order to measure disturbances in the mechanical wave diode system,a design with two sensor equipments is utilized. This is the minimumnumber of sensors needed to separate waves traveling back and forthin the bar. It would be interesting to investigate how much can begained by using a larger number of sensors.


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References

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[2] S. S. Ahmad, J. S. Lew, and L. H. Keel. Robust control of flexiblestructures against structural damage. IEEE Trans. on Control SystemsTechnology, 8(1):170182, 2000.

[3] M. J. Brennan, S. J. Elliott, and R. J. Pinnington. Active control

of vibrations transmitted through struts. In International Conf. on

Motion and Vibration Control, Yokohama, September 1992.

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[5] R. M. Christensen. Theory of Viscoelasticity. An Introduction. Aca-demic Press, New York, NY, 1971.

[6] R. L. Clark, J. Pan, and C. H. Hansen. An experimental study of theactive control of multiple-wave types in an elastic beam. J. Acoust. Soc.

Am., 92(2):871876, 1992.

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[8] S. J. Elliott. Optimal controllers and adaptive controllers for multi-channel feedforward control of stochastic disturbances. IEEE Trans.on Signal Processing, 48(4):10531060, 2000.

[9] S. J. Elliott and L. Billet. Adaptive control of flexural waves prop-agating in a beam. Journal of Sound and Vibration, 163(2):295310,

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[10] S. J. Elliott and B. Rafaely. Frequency-domain adaptation of causal dig-ital filters. IEEE Trans. on Signal Processing, 48(5):13541364, 2000.

[11] M. P. Fard. Passivity analysis of nonlinear Euler-Bernoulli beams.Mod-eling Identification and Control, 23(4):239258, 2002.

[12] A. J. Fleming and S. O. R. Moheimani. Spatial system identificationof a simply supported beam and a trapetzoidal cantilever plate. IEEETrans. on Control Systems Technology, 11(5):726736, 2003.


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[13] A. J. Fleming, S. O. R. Moheimani, and S. Behrens. Synthesis and im-

plementation of sensor-less active shunt controllers for electromagneti-cally actuated systems. IEEE Trans. on Control Systems Technology,13(2):246261, 2005.

[14] C. R. Fuller, S. J. Elliott, and P. A. Nelson.Active Control of Vibration.Academic Press, London, 1996.

[15] C. R. Fuller, G. P. Gibbs, and R. J. Silcox. Simultaneous active controlof flexural and extensional waves in beams. Journal of Mater. Syst. andStruct., 1:235247, 1990.

[16] T. Glad and L. Ljung. Control Theory. Taylor and Francis, UK, 2000.

[17] S. V. Gosavi and A. G. Kelkar. Passivity based robust control of piezo-actuated flexible beam. In Proc. of the American Control Conference,Arlington, VA, June 2001.

[18] D. Halim and S. O. R. Moheimani. Spatial resonant control of flexiblestructures - application to a piezoelectric laminate beam. IEEE Trans.on Control Systems Technology, 9(1):3753, 2001.

[19] D. Halim and S. O. R. Moheimani. Compensating for the truncationerror in models of resonant systems that include damping. In Proc. ofthe 41st Conf. on Decision and Control, Las Vegas, USA, December2002.

[20] C. R. Halkyard and B. R. Mace. Feedforward adaptive control of flex-ural vibration in a beam using wave amplitudes. Journal of Sound andVibration, 254(1):117141, 02.

[21] Y-H Lim, S. V. Gopinathan, V. V. Varadan, and V. K. Varadan. Finiteelement simulation of smart structures using and optimal output feed-back controller for vibration and noise control. Smart Materials andStructures, 8(3):324337, 1999.

[22] L. Ljung. System Identification. PrenticeHall, Upper Saddle River,NJ, USA, 2nd edition, 1999.

[23] R. Lozano, B. Brogliato, O. Egeland, and B. Maschke. DissipativeSystems Analysis and Control. Theory and Applications. Springer, Hei-delberg, 2000.

[24] R. Lozano-Leal. On the design of dissipative LQG-type controllers. InProc. of the 27st Conf. on Decision and Control, Austin, USA, Decem-ber 1988.

[25] R. Lozano-Leal and S. M. Joshi. Strictly positive real transfer functionsrevisited. IEEE Trans. on Automatic Control, 35(11):12431245, 1990.


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[26] T. McKelvey, H. Akcay, and L. Ljung. Subspace-based identification

of infinite-dimensional multivariable systems from frequency-responsedata. Automatica, 32(6):885902, 1996.

[27] T. McKelvey and S. O. R. Moheimani. Estimation of phase constrainedmimo transfer functions with application to flexible structures withmixed collocated and non-collocated actuators and sensors. In 16thIFAC World Congress, Prague, July 2005.

[28] R. J. McKinnell. Active vibration isolation by cancelling bending waves.Proceedings of the Royal Society, A421:357393, 1989.

[29] D. J. Mead. Passive Vibration Control, volume 141-2. Wiley, UK, 1979.

[30] L. Meirovitch. Principles and Techniques of Vibrations. Prentice-Hall,Upper Saddle River, NJ, 1997.

[31] Z. Mohamed, J. M. Martins, M. O. Tokhi, J. S. da Costa, and M. A.Botto. Vibration control of a very flexible manipulator system. ControlEngineering Practice, 13:267277, 2005.

[32] S. O. R. Moheimani, D. Halim, and A. J. Fleming. Spatial Control ofVibration: Theory and Experiments. World Scientific Publishing Co.Pte. Ltd., Singapore, 2003.

[33] S. O. R. Moheimani, H. R. Pota, and I. R. Petersen. Spatial control for

active vibration control of piezoelectric laminates. In Proc. of the 37stConf. on Decision and Control, Tampa, USA, December 1998.

[34] C. Olsson. Active Vibration Control of Multibody Systems. PhD thesis,Uppsala University, Department of Information Technology, Uppsala,Sweden, 2005.

[35] I. R. Petersen and H. R. Pota. Minimax LQG optimal control of aflexible beam. Control Engineering Practice, 11:12731287, 2003.

[36] B. Petitjean, I. Legrain, F. Simon, and S. Pauzin. Active control ex-periments for acoustic radiation reduction of a sandwich panel: Feed-

back and feedforward investigations. Journal of Sound and Vibration,252(1):1936, 2002.

[37] R. Pintelon and J. Schoukens. System Identification: A Frequency Do-main Approach. IEEE Press, New York, NY, 2001.

[38] A. Preumont. Vibration Control of Active Structures: An Introduction.Kluwer Academic Publisher, Dordrecht, 2002.

[39] J. Scheuren. Active control of bending waves in beams. InProceedingsof Inter Noise 85, pages 591595, 1985.


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[40] R. S. Smith, C. C. Chu, and J. L. Fanson. The design ofHcontrollers

for an experimental non-collocated flexible structure problem. IEEETrans. on Control Systems Technology, 2(2):101109, 1994.

[41] T. Soderstrom.Discrete-time Stochastic Systems. Springer-Verlag, Lon-don, UK, 2nd edition, 2002.

[42] T. Soderstrom and P. Stoica. System Identification. Prentice Hall In-ternational, Hemel Hempstead, United Kingdom, 1989.

[43] M. A. Swinbanks. The active control of sound propagation in longducts. Journal of Sound and Vibration, 27(3):411436, 1973.

[44] S. Timoshenko. Vibration Problems in Engineering. D. Van NostrandCompany, Princeton, NJ, 3rd edition, 1955.

[45] P. Van Overshee and B. De Moor. Continuous-time frequency domainsubspace system identification. Signal Processing, 52:179194, 1996.

[46] X. Zhang, W. Xu, S. S. Nair, and V. Chellaboina. PDE modeling andcontrol of a flexible two-link manipulator. IEEE Trans. on ControlSystems Technology, 13(2):301312, 2005.


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PAPER I


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Modeling and Control of a Viscoelastic

Piezolaminated Beam

Peter Naucler Hans Norlander Anders Jansson

Torsten Soderstrom

Abstract

Modeling and control of a viscoelastic beam are considered. Piezo-

electrical elements are bonded to the beam and used as actuators.

The beam is also equipped with a strain gauge that serves as a sensing

device. The beam system is described by the Euler-Bernoulli beam

equation, which is Fourier transformed and numerically solved in the

frequency domain. Then, two different approaches are evaluated to ap-

proximate the infinite-dimensional system with a low order parametric

approximation. Finally, LQG control theory is applied to control bothstrain and transversal vibrations. Especially, the control of the strain

shows promising results.

1 Introduction

The presence of vibrations is a common problem in mechanical structures,particularly in flexible parts, for instance aircraft wings or robot arms. Thiscan be reduced by making such parts strong or heavy enough. For many

applications, e.g. in aircrafts and spacecrafts, it is desirable to keep theweight as low as possible, which makes such solutions less suitable. Instead,one would like to have a device which can perform active damping of thevibrations without any substantial increase of the mass. In the case whensuch a device is embedded in the structure it is often referred to as a smartmaterial or a smart structure [10].

Department of Information Technology, Uppsala University. P.O. Box 337, SE-751 05

Uppsala, Sweden. Email: {Peter.Naucler, Hans Norlander, Torsten.Soderstrom}@it.uu.seDepartment of Engineering Sciences, Uppsala University, Sweden. Email: An-

[email protected]


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One way to design smart structures is to use piezoelectric elements that are

attached to the material. Piezoelectric elements exhibit a significant defor-mation when an electric field is applied, and they produce an electric fieldwhen deformed. Therefore they can be used as both actuators and sensorsin a smart structure. Using piezoelectric patches is a simple and cheap wayof integrating actuating and sensing devices in mechanical structures. Mod-eling and control of simple flexible structures have received a lot of attentionin recent years; see for example [9], [8] and the references therein.

A popular setup is to use a steel beam that is simply supported at both ends.Often the beam is equipped with collocated piezoelectric actuator-sensorpairs which conveys that the tractable passivity property can be used for

controller design. This is utilized in [4], which employs the popular modalanalysis technique, or assumed modes approach, for describing the dynamicsof the system. In this method the orthogonality between vibration modes isused to obtain transfer functions which have the form of infinite sums; eachterm describing one vibrational mode. The sum is then truncated to obtaina low order approximation of the infinite-dimensional system.

There are, however, a number of disadvantages related to the modal analy-sis technique. First of all the method assumes pure elasticity which meansthat no damping is present. This is most often compensated for by adding asmall damping term to each vibrational mode in an ad-hoc manner. In addi-

tion, the piezoelectric sensor/actuator is assumed not to affect the structuralproperties of the beam. To take the piezoelectrical patches into account thesystem gets a more complex structure and the modal analysis technique canno longer be applied.

In this paper we present a procedure for how to model a viscoelastic beamstructure where the piezoelements are taken into account. Furthermore,damping is included in the modeling phase by using a complex valuedYoungs modulus which was experimentally determined in [5]. Piezoelec-trical patches are bonded to each side of the beam and used as actuators.As a sensing device a strain gauge is attached to the beam. It consists of a

thin wire whose effect on the structural properties of the beam is negligible.


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Paper I 3

u(t)

u(t)

y(t)

y(t)

e(t)

e(t)

H(s)

G(s)

f(t)

f(t)

Fy(s)

Fy(s)

1 2 3

Figure 1: An experimental setup of a viscoelastic beam with piezoelectricpatches (actuator) and a strain sensor.

2 Problem Formulation

Consider an experimental setup as in Figure 1. A viscoelastic beam is fixedat one end and free at the other. The beam is divided into three sectionsand piezoelectric patches are attached to the beam at its middle section. Adisturbance forcef(t) enters at the tip of the beam, possibly as an impulse.The beam is also equipped with one sensor that measures the strain, y(t) =(t, ), at a spatial position , that arises due to the impact of f(t). The

measured signal is corrupted with additive noise, e(t).

Our goal is to dampen the vibrations in the beam by applying a controllerthat uses the strain measurements in a feedback loop. The control signalu(t) is fed to the piezoelectric patches which are used as actuators. Thesystem can be schematically described as in the lower part of Figure 1. Byusing the properties of linear systems, the output signal can be viewed asa superposition of the strains caused by u(t) and f(t). The control signalaffects the output through the transfer function G(s), and the force con-tributes through the transfer function H(s).


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Table 1: Properties of the beam and the piezoelement.Description Value

Beam length [m] 0.59Beam width [m] 0.01Beam thickness [m] 0.002Beam density [kg/m3] 1183Piezo. length [m] 0.0318Piezo. thickness [m] 0.00066Piezo. density [kg/m3] 7878Piezo. Youngs modulus [N/m2] 5.781010Length of first section [m] 0.202

The vibrations in the beam are mathematically described by a partial dif-ferential equation (PDE), which means that the system is of infinite order.The piezoelectric patches are considerably stiffer than the rest of the beam.This is a fact that is accounted for in the modeling. The strain sensor is,however, very small and of negligible weight. It should thus not contributeto the dynamics of the system. We aim at describing the infinite dimensionalsystem with a parametric model of finite order. This model should then bea basis for model-based control. The properties of the beam are listed inTable 1.

3 Modeling

Although the strain is measured, we first consider modeling of the transversaldeflection,w, at each section,k, of an elasticbeam. At a spatial coordinateat time instance t the deviation can be described by the PDE

2

2EkIk

2wk(t, )

2+ kAk

2wk(t, )

t2 = 0 (1)

which is called the Euler-Bernoulli beam equation [11]. The quantitiesIk,Ak and k represent the moment of inertia of a cross-section area, cross-section area and density, respectively. These parameters are assumed tobe time invariant and spatially constant for each section, k, of the beam.The Youngs modulus of elasticity is denoted by Ek. In the pure elasticcase, this quantity is constant and real-valued for each section of the beam.By Fourier transformation of (1) w.r.t. the temporal variable, the PDE istransformed to a frequency dependent ordinary differential equation in the


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spatial domain,

EkIkd4WFk (, )

d4 kAk2WFk (, ) = 0 (2)

where WFk (, ) is the Fourier transform of wk(t, ) and is the angularfrequency [rad/s]. In the sequel we will deal with both the Fourier transformand the Laplace transform. To separate the two transforms the superscriptsF (Fourier) andL(Laplace) will be employed.

A viscoelasticbeam exhibit the property that if the deformation is speci-fied, the current stress depends on the entire deformation history. This is

described by a Youngs modulus that is frequency dependent and complexvalued [1]. Based on this, viscoelasticity can be introduced in a frequencydomain context by replacing Ek in (2) with Ek(). The frequency depen-dent Youngs modulus was experimentally determined in [5]. To solve (2) anumber of boundary values and compatibility conditions are needed. The(Fourier transformed) boundary values of the cantilever beam are

WF1 (, 0) = DWF1 (, 0) = D

2WF3 (, L) = 0

E3()I3D3WF3 (, L) =F

F()

where D = dd

is the differentiation operator. Furthermore the sections of

the beam are tied together by four compatibility conditions at each pointwhere two sections meet. For example, if sections k and k+ 1 meet at thespatial coordinate k the following must hold

WFk (, k) =WFk+1(, k)

DWFk (, k) = DWFk+1(, k)

MFk (, k) =MFk+1(, k)

TFk (, k) =TFk+1(, k)

whereMandTare the bending moment and transversal force, respectively.

The disturbance force, FF() = F[f(t)](), and the actuating voltage,UF() =F[u(t)](), are treated as input signals to the system. FF()enters through one of the boundary conditions and UF() enters throughtwo of the compatibility conditions.

Using the above observations we replace Ek by Ek() and solve (2). Itscharacteristic equation now reads

r4k 2 kAkEk()Ik

= 0


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6 Peter Naucler

with the solution

rk,l = il

2

kAkEk()Ik

14

, l= 1, . . . , 4

where i =1. The solution to (2) is readily written

WFk (, ) = [erk,1 . . . erk,4 ][ck,1() . . . ck,4()]

T

RTk (, )Ck() (3)

where ck,l() are unknown parameters that are to be determined by usingthe boundary values and compatibility conditions. Now, denote

C() [CT

1() CT

2() CT

3()]T

R121

. (4)

Then, the following system of equations can be formed

A()C() = B1FF() + B2UF() (5)whereA() C1212,B1 andB2 R121 are completely determined by theboundary values and compatibility conditions (i.e. fully known). The inputsignals enters through the right hand side of the equation. Using (4) and(5) we have

Ck() = PkA1()B1FF() + B2UF() (6)

where Pk R412 is constructed such that PkC() = Ck(). Finally, com-bining (6) and (3) the frequency responses from the inputs to the transversaldeflection at the coordinate is extracted

WFk (, ) = RTk (, )PkA1()B1FF()

+RTk (, )PkA1()B2UF() HFw (, )F

F() + GFw(, )UF() (7)

The subscript w in the last line of (7) denotes that it is the frequencyresponse to the transversal deflectionand not the strainas in Figure 1. Toobtain an expression for the strain,k, as a function of frequency and space,

compatibility conditions are utilized [2]

YF() =Fk(, ) = hk2D

2WFk (, ) (8)

where hk is the height of the beam at section k. To find the frequencyresponses HF(, ) and GF(, ), (8) is simply applied to (7), noting thatRTk is the only term depending on .

Due to the complexity ofA() it is not possible to obtain a simple closedform expression directly from the above equations. Instead HF(, ) and


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Paper I 7

GF(, ) can be numerically computed for any frequency or spatial coor-

dinate. Typically, the spatial coordinate is held fixed and the frequency isvaried to obtain the frequency responses. Utilizing this, we drop the spa-tial dependence and introduce the shortened notations HF() and GF().Solving (7) directly is, however, not numerically sound. Instead, (5) is firstsolved by means of an LU factorization and the result is put into (3) in asecond step. Finally, (8) is applied to get an expression for the strain.

It is most often desirable to have a parametric model of finite order thatdescribes the dynamics of a system; not the least if the model should be abasis for model based control. In the following, two different strategies forfitting parametric models to the frequency responses are evaluated.

3.1 An Ad-hoc Approach

The first approach is to realize the transfer functions as proper rationalfunctions in the Laplace domain, i.e.

HL(s, H) =BH(s)

A(s) , GL(s, G) =

BG(s)

A(s)

where A(s), BH(s) and BG(s) are polynomials of order na, nbH and nbG,

respectively. The parameter vectorH contains the coefficients of the un-known polynomials A(s) and BH(s), i.e. H = [a1 . . . ana b0 . . . bnbH]

T;whereasG only contains the coefficients ofBG(s). The reason for this con-vention should soon be clear. To find Hwe attempt to minimize a quadraticcriterion

V(H) =Nk=1

HF(k) HL(ik, H)2 W(k) (9)

where N is the number of frequency points at which HF() is computedand W() is a user chosen weighting function. To minimize V(H) theMATLAB function invfreqs is used. It applies a damped Gauss-Newton

iterative search algorithm [6]. Once acceptable estimates ofA(s) andBH(s)are obtained, A(s) is used when determining GL(s, G) =BG(s)/A(s) in asecond step. This is performed by using the same type of loss function as in(9). SinceA(s) is now known this will be a linear problem which is solved byusing a standard weighted least squares procedure. It is then straightforwardto carry on and compute zero-polynomials for any sensor location along thebeam, using the pole polynomial from the first identification step.


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8 Peter Naucler

100

101

102

103

106

105

104

103

102

Magnitude

Frequency [rad/s]

HF()

HL(i,)

HF() H

L(i,)

Figure 2: Magnitude plot of the numerically computed transfer functionHF() (solid), the low order parametric approximation HL(i, ) (dashed)and the error (dash-dotted).

Example 1. Now, this method is applied to the beam-system in Section

2. A collocated actuator-sensor configuration is used, i.e. the sensor isattached on top of one of the actuators. Figure 2 shows the magnitude of thenumerically computed HF() and its low order approximation HL(s, H)evaluated on the imaginary axis; i.e. s = i. HF() is used as frequencydomain data for the first identification step. In addition, the magnitude ofthe error between the two frequency responses is depicted. The thick dash-dotted line denotes an ideal low-pass filter which is used to delimit the partof the frequency response that is utilized for the identification. The rationaltransfer function HL(s, ) has four zeros and six poles, that are fitted to thedata set.

The figure shows that a very good fit between the frequency response dataand the parametric model is obtained.

The advantage of this simple method include that frequency weighting iseasily performed and that continuous time parametric models are directlyobtained. In addition, the method allows arbitrary spacing of the frequencypoints. It is thus possible to use e.g. logarithmically spaced frequencyresponse data to get a natural weighting of the data. Nevertheless, thereare some disadvantages. The most obvious one is that only one data set isused to identify the pole polynomial. This could be a real problem if pole-


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Paper I 9

zero cancellation is hidden in the data set chosen for the first identification

step. It is also not obvious how to choose model orders and how to realizethe system in a state space representation; not the least if a large numberof outputs are used. Further, the numerical search procedure in the firstidentification step is quite expensive computationally.

3.2 Subspace-based Identification

The subspace based algorithm is the algorithm denoted Algorithm 1 in the

paper [7]. It utilizes frequency response data from infinite-dimensional sys-tems to identify MIMO state space models. The frequency response dataare restricted to be generated from equidistant frequency samples.

The method is based on estimating the impulse response coefficients byusing the inverse discrete Fourier transform on the frequency response data.Then, the coefficients are used to construct a block Hankel matrix on whicha singular value decomposition (SVD) is performed. The sub-matrices fromthe SVD that correspond to the most significant singular values are used toconstruct the state space representation. In contrast to the ad-hoc approachoutlined in the previous section, this method estimates discrete-time models.

To convert the models to continuous-time the zero-order hold approach isutilized.

Example 2. Now, the subspace-based approach is applied to find a loworder approximation of HF() as in Example 1. Once again, the dashed-dotted line denotes an ideal low pass filter which defines the part of thefrequency response that is used for the identification. In Figure 3 a statespace representation of order six is chosen.

The solid and dashed lines depict that a quite good fit between the para-metric model and the frequency response data is obtained. However, the

dash-dotted line indicates that the high performance of the ad-hoc approachis not attained.

The advantage of this method is that frequency response data from all trans-fer functions are used to estimate the low order parametric approximationin a single step. The retrieved model is a MIMO state space representa-tion, which is useful for controller design purposes. In addition, the singularvalues of the Hankel matrix provide a tool for determining the order ofthe representation. The disadvantage is primarily the lack of a frequencyweighting possibility.


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100

101

102

103

105

104

103

102

Frequency [rad/s]

Magnitude

HF()

HL(i,)

HF() H

L(i,)

Figure 3: Magnitude plot of the numerically computed transfer functionHF() (solid), the low order parametric approximation HL(i, ) (dashed)and the error (dash-dotted). The parametric model is generated from thesubspace-based approach.

4 Controller Design

In order to apply controller design the system is realized in state space form

x(t) = Ax(t) +Bu(t) +Nf(t)

z(t) = Mx(t) + D1u(t)

y(t) = Cx(t) + D2u(t)

y(t) =y(t) + e(t) (10)

where y(t), z (t) and e(t) are the strain, transversal deflection and measure-

ment noise, respectively. The unit of the strain is [1] and the unit of thetransversal deflection is [m]. Although the strain is measured, z(t) =w(t, )is also modeled. The reason for this is that it may be desirable to controlthe transversal deflection.

This section is divided into two parts. First, LQG-control theory is ap-plied to control the strain. Thereafter, an attempt is made to control thetransversal deflection. For simplicity, the subspace-based approach is uti-lized to model the beam system. This method directly yields a model of theform (10).


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Paper I 11

4.1 Control of the Strain

For controller design purposes the machinery of LQG control theory is uti-lized, see e.g. [3]. A controller of the form

u(t) = Lx(t)x(t) = Ax(t) +Bu(t) + K(y(t) Cx(t) D2u(t)) (11)

is then retrieved.

To find the controller, the following quadratic criterion is employed

J=

0

[qyy2(t) + qzz

2(t) + u2(t)]dt (12)

with qy = 104 and qz = 0. When determining the Kalman gain, K, thevariance of the disturbance, R1, and the variance of the measurement noise,R2, are treated as design variables. The values R1= 5 107 and R2= 1 arechosen.

The system is simulated in MATLAB, with a small noise term added tothe measured signal. Figure 4 shows the time response, y(t), to an impulsedisturbance, f(t). The disturbance is of low-pass characteristics and has its

main power within the modeled part of the spectrum.

The figure shows that the disturbance is well damped if control action isapplied. To analyze stability due to unmodeled dynamics the Bode diagramof the loop gain is drawn. Here, the numerically computed frequency re-sponse GF() is employed. It is performed by evaluating the controller (11)for different frequencies and the result is multiplied with GF(). The Bodeplot of the loop gain is depicted in Figure 5. In addition, the sensitivityfunction, S, and the complementary sensitivity function, T, are visualizedin the upper part of the figure.

In view of the figure it should be clear that the closed loop system is robustagainst the ignored dynamics, since the high frequency content ofGF() iswell damped by the controller. The sensitivity function shows how sensitivethe closed loop system is to modeling errors. Thus, in frequency regionswhere S is small the closed loop system is insensitive to modeling errors.The figure shows that S is quite small in regions where HF() is large.Often, one would like to obtain S(0) 0 (integral action in the controlloop), in order to get rid of stationary errors. However, due to the nature ofthe disturbance force (impulse), there is no need for integral action in thecontrol loop. The complementary sensitivity also has a reasonable behavior.


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12 Peter Naucler

0 1 2 3 4 5 6 7 83

2

1

0

1

2

3

4

x 105

Time [s]

Strain[1]

Closed loopOpen loop

Figure 4: The time response of the strain. Closed loop (solid) and open loop(dashed).

101

100

101

102

103

104

103

100

103

Magnitude

100

101

102

103

104

100

50

0

50

Frequency [rad/s]

Ph

ase[deg]

loop gain

ST

loop gain

Figure 5: The Bode plots of the loop gain (solid), S(dashed) and T (dash-dotted).


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Paper I 13

0 1 2 3 4 5 6 7 8

0.025

0.02

0.015

0.01

0.005

0

0.005

0.01

0.015

0.02

0.025

Time [s]

Trans.

deflection[m]

Closed loopOpen loop

Figure 6: The time response of the trans. deflection. Closed loop (solid)and open loop (dashed).

4.2 Control of the Transversal Deflection

Now, the transversal deflection, z(t), is controlled. Still, the model (10) isemployed, i.e. the strain is measured. The parametersqy = 0, qz = 1 andR1 = 10

6 are adjusted. The time response of the transversal deflection isdepicted in Figure 6.

Once again it is seen that the vibrations are nicely damped when controlaction is applied. In the same fashion as previously the loop gain, S andT are plotted for the new controller, see Figure 7. The Bode plot of theloop gain shows that a very small phase margin is obtained. The sensitivity

function and the complementary sensitivity function also have a very un-pleasant appearance. The result indicates that it is quite hard to controlthe transversal deflection from strain measurements.


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14 Peter Naucler

101

100

101

102

103

104

102

100

102

104

Magnitude

loop gainST

101

100

101

102

103

104

200

150

100

50

0

Phase[deg]

Frequency [rad/s]

loop gain

Figure 7: The Bode plots of the loop gain (solid), S(dashed) and T (dash-dotted). The transversal deflection is controlled.

5 Conclusions

In this paper, modeling and control of the vibrations in a viscoelastic beamhave been considered. For control purposes the beam is equipped with apiezoelectrical actuator and a strain sensor. The aim has been to accuratelymodel the beam system by taking the structural properties of the piezo-electrical actuator into account. In addition, viscoelasticity is introduced ina frequency domain context by employing a frequency dependent Youngsmodulus of elasticity. The system is described by equations of infinite orderand two approaches for model reduction have been treated. Both methodsshowed nice results.

Finally, LQG control theory was applied to the system. First the strainwas controlled with promising result. Then, the transversal deflection wascontrolled by using the strain measurements in a feedback loop. Even thoughthe vibrations were well damped the Bode plot of the loop gain indicatedsevere robustness problems.


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References

[1] R. M. Christensen. Theory of Viscoelasticity. An Introduction. Aca-demic Press, New York, NY, 1971.

[2] J. M. Gere and S. P. Timoshenko. Mechanics of Materials. Chapman& Hall, UK, 1991.

[3] T. Glad and L. Ljung. Control Theory. Taylor and Francis, UK, 2000.

[4] D. Halim and S. O. R. Moheimani. Spatial resonant control of flexiblestructures - application to a piezoelectric laminate beam. IEEE Trans.on Control Systems Technology, 9(1):3753, 2001.

[5] L. Hillstrom, U. Valdek, and B. Lundberg. Estimation of the statevector and identification of of the complex modulus of a beam. Journalof Sound and Vibration, 261:653673, 2003.

[6] L. Ljung. System Identification Toolbox for use with MATLAB: UsersGuide. The Mathworks, Cochituate Place, 24 Prime Park Way, Natick,MA, USA, April 1988.

[7] T. McKelvey, H. Akcay, and L. Ljung. Subspace-based identificationof infinite-dimensional multivariable systems from frequency-responsedata. Automatica, 32(6):885902, 1996.

[8] S. O. R. Moheimani, D. Halim, and A. J. Fleming. Spatial Control ofVibration: Theory and Experiments. World Scientific Publishing Co.Pte. Ltd., Singapore, 2003.

[9] H. R. Pota and T. E. Alberts. Multivariable transfer functions for aslewing piezoelectric laminate beam. ASME J. of Dyn. Syst., Measure-ment, Contr., 117(3):352359, 1995.

[10] A. Preumont. Vibration Control of Active Structures: An Introduction.Kluwer Academic Publisher, Dordrecht, 2002.

[11] S. Timoshenko. Vibration Problems in Engineering. D. Van NostrandCompany, Princeton, NJ, 3rd edition, 1955.


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PAPER II


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A Mechanical Wave Diode: Using Feedforward

Control for One-Way Transmission of Elastic

Extensional Waves

Peter Naucler Bengt Lundberg Torsten Soderstrom

Abstract

This papers concerns the design and analysis of an active deviceable of controlling mechanical waves. Its functionality is similar tothe diode in electrical networks and therefore it is referred to as amechanical wave diode. The aim is to block waves in one directionin a mechanical structure, while not affecting waves travelling in theopposite direction. The device is based on feedforward control and

the design is proposed on the basis of equations that describe elasticextensional waves in bars.

We present an ideal feedforward design which performs perfectly inthe noise-free case. However, this design has poor noise propertiesand is modified to be useful in a realistic scenario where measurementnoise is present. Theoretical analysis and computer simulations showthat the modified design is drastically less sensitive to such noise. Wealso present a mechanical diode with a noise barrier and discuss theimplication of bandwidth limitation in the control loop.

1 Introduction

In different types of structures, waves and vibrations generated by machineryor accidentally constitute an important engineering problem. Two aspects ofthis problem are radiation of noise and failure due to fatigue. If a structure

Department of Information Technology, Uppsala University. P.O. Box 337, SE-751 05Uppsala, Sweden. Email: {Peter.Naucler, Torsten.Soderstrom}@it.uu.se

Department of Engineering Sciences, Uppsala University, Sweden. Email:[email protected]


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2 Peter Naucler

is partially built up of slender bar elements, disturbances may propagate

through such elements from the source to different parts of the structureas extensional, torsional and flexural waves, [20, 15]. Even if the sourceprimarily generates only one type of wave, there will generally be secondarygeneration of the other two types due to coupling of waves at joints, bends,etc. in the structure. With regard to reduction of harmful effects, therefore,no type of wave should generally be neglected. Clearly, however, measuresagainst one type of wave only, applied between a source of such waves andthe main structure, may significantly reduce the presence of all types ofwaves and vibrations.

Traditionally, passive techniques have been used to reduce waves and vibra-

tions, and their effects. During the last decades, however, there has beenconsiderable interest in active techniques involving the use of sensor andactuator arrays, [4, 18], wave analysers, wave synthesizers and control. Thecontrol of extensional and torsional waves in elastic members is similar tothat of plane sound waves in ducts [29], as these waves are all governed by thesecond-order wave equation. The control of flexural waves is more complexas such waves are governed by a fourth-order partial differential equation,implying the presence of evanescent waves and dispersion. The principlesof active control of waves and vibrations in structures are outlined in [11],where numerous references are also given to work in the field.

There are many ways to design a controller [14]. When some disturbancesare measurable it is often an advantage to apply feedforward control. Inideal cases, the effects of the disturbances can then be totally eliminated.However, for this to happen it is necessary that the designer has a perfectmodel of the system. Further, the ideal feedforward controller may turn outto be noncausal or unstable, and therefore has to be approximated.

For feedback controllers, there are many design principles available, for ex-ample [14, 13]. When there is limited knowledge of the system dynamics,one alternative is to apply adaptive control, [2, 26]. Also here there are sev-eral options. In signal processing and communications applications variants

of Widrows LMS (least mean squares) algorithm, [30, 31], used for FIR (fi-nite impulse response) filter structures, have gained considerable popularity.As FIR filters by construction are always asymptotically stable, the risks toend up with an unstable controlled system are eliminated, even if the FIRfilter structure does not allow optimal rejection performance.

Early work reporting on feedforward control for damping vibrations includes[29, 27, 24], while [23] focuses on feedback control. Feedforward control ofwaves in a structural member signifies that the excitation by the primarysource of waves, isolated from the secondary excitation by the actuators,


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Paper II 3

can be observed. This demands upstream sensors relative to the wave to

be controlled and also a detailed knowledge of the dynamic behaviour ofthe structural member. Several papers report on successful designs of feed-forward control to reduce transmission of vibrations in beams and plates.Wiener filter theory has been used to first construct ideal feedforward filters.Typically, these are not causal and cannot be implemented but have beenused to derive some performance limitations for the designed systems, [8].

Many ways to improve the slow convergence rate of the originally proposedLMS algorithm have appeared in the literature. For adaptive feedforwardcontrol of waves and vibrations, the filtered-x LMS and filtered-error LMSalgorithms have gained considerable popularity. Examples include [8, 7, 9,

16].

Fuller et al, [12], seem to be among first to implement adaptive feedbackcontrol to both extensional and flexural sinusoidal waves in bars. Similarwork, but with control of bending in two planes, was carried out in [6, 3]and [10].

Further studies deal with the effect of sensor locations, [1, 25], extensionsto multivariate cases, [7, 18], extensions to control of 2D or 3D mechanicalstructures, [18], combination of adaptive feedforward and feedback control,[5, 25], and explicit consideration of how random noise effects the system,

[19].

This paper is concerned with the design of feedforward control of elasticextensional waves in a bar. The aim is to realize a device which fully re-flects waves travelling in one direction and fully transmits waves travellingin the opposite direction. Thus, with regard to wave transmission, this de-vice should have properties analogous to those of an electrical diode. Wetherfore can describe this device as a mechanical wave diode. An overalldescription of such a mechanical wave diode and the control principle usedis given in Section II. The modeling of the electromechanical system andan ideal feedforward design are presented in Section III. In Section IV, this

ideal feedforward design is evaluated through numerical simulation, in par-ticular with respect to its shortcomings in the presence of noise. In SectionV, a modified feedforward design which can handle noise is presented andevaluated, again through numerical simulation. Finally, main conclusionsare summarized in Section VI.


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4 Peter Naucler

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

tm tm0

N1 N2

Nn

NpStrain gauge

Ab, Eb, bBar

Actuator

Controller

Ni Nt

tc ta tc tc+ ta

A2, E2, 2

A1, Eb, b

Nc

Nr

1st interface 2nd interface

Figure 1: Bar with two strain gauge pairs, one actuator pair and feedforward

control for one-way transmission from left to right.

2 Mechanical Diode Based on Feedforward Con-

trol

Consider propagation of elastic waves in a straight bar as shown in Figure1. At two sections, the bar is equipped with strain gauge pairs. Also,along a segment of the bar, a pair of piezo-electric actuators, electricallyand mechanically in parallel, are attached. The idea with this configuration

of bar, sensors and actuator is to apply feedforward control so that wavestraveling from the sensors towards the first bar-actuator interface will befully reflected while waves traveling in the opposite direction, towards thesecond bar/specimen interface, will be transmitted undisturbed. Therefore,waves traveling from the sensors towards the first bar/actuator interface willbe considered measurable disturbances which are to be suppressed throughfeedforward control.

Two types of disturbing waves will be considered. The first consists ofisolated transient pulse-shaped disturbances that enter the system only oc-casionally, and the second are random disturbances with given statistical

properties. Typical origins of such disturbances may be accidental impactsby tools and pressure from wind, water and traffic, respectively.

Feedforward control is used to eliminate the influence ofmeasurable distur-bances. The basic idea is to introduce some suitable compensating controlaction to reject the disturbance before it reaches the output. This is themajor advantage of feedforward control compared with control based onfeedback from the output. With a successful feedforward control law therewould be no need to compensate for the disturbance by use of feedback.However, there are some difficulties related to feedforward control. First of


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Paper II 5

all, feedforward is an open-loop compensation and therefore requires an ac-

curate system model. Furthermore, design of a feedfoward controller involvescalculation of the inverse of a dynamical system. Sometimes, this inversedoes not exist due to unstable zeros of the system to be inverted. Also, adynamical system commonly has low-pass characteristics, which means thatits inverse will have excessive high frequency gain. This implies that thefeedforward filter then will be sensitive to noise.

3 Modeling and Ideal Feedforward Design

In order to design a feedforward controller for the system in Figure 1, afirst step is to model the actuator and its effect on the wave propagation inthe bar. We present such modeling in the next subsections. As the systemconsidered is linear, superposition will be used. In particular, the wavesoriginating from the disturbing source and those generated by the actuatorwill be considered one by one and superimposed.

3.1 Bar, sensor and actuator configuration

The operative wave lengths are assumed to be much greater than the trans-verse dimensions of the bar, which implies that one-dimensional theory isaccurate [20, 15]. Thus, the wave propagation outside the actuator region

is characterized by the wave speed cb = (Eb/b)1/2 and the characteristic

impedance Zb = Ab(Ebb)1/2, whereAb is the cross-sectional area,Eb is the

Youngs modulus and bis the density. Within the actuator region, the cross-sectional area of the bar is reduced toA1, while the total cross-sectional areaof the two actuators is A2, their Youngs modulus is E2 and their densityis 2. With the assumption that one-dimensional theory applies also within

this region, the effective wave speed and the characteristic impedance be-comeca= (Ea/a)

1/2 andZa= Aa(Eaa)1/2, respectively, where Aa= A1+

A2 is the total cross-sectional area,Ea= (A1Eb+ A2E2) /Aais the effectiveYoungs modulus, and a = (A1b+ A22) /Aa is the effective density. Forgiven properties of the bar and actuator materials, the cross-sectional areasA1and A2are chosen so that impedance matching is achieved, i.e. Za= Zb.Therefore, for zero feedforward input voltage to the actuators from an ampli-fier with negligible output impedance, waves traveling in both directions willbe transmitted undisturbed through the actuator region without generationof reflected waves.


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6 Peter Naucler

3.2 Wave Propagation

Let x be an axial coordinate and let t be time. For zero feedforward inputto the actuators (Nc= 0 in Figure 1), the wave motion in the bar, includingthe actuator region, can be expressed in terms of the normal force as

N(, ) = Np()ei + Nn()e

i (1)

where

=

x0

ds

c(s), (2)

with c(s) =ca within and c(s) =cb outside the actuator region, is a trans-formed axial co-ordinate with dimension of time, and where Np() andNn() are the Fourier transforms of the normal forces Np(t) and Nn(t)at the origin = 0 associated with waves Np(t ) and Nn(t + ) travelingin the directions of increasing and decreasing , respectively. As these waveshave unit speed, the angular frequency plays the role of wave number in(1). In terms of, the sensor positions become =tm, and the actuatorregion becomes tc ta < < tc+ ta as shown in Figure 1.

3.3 Wave Separation

Generally, the waves represented by Np() and Nn() at = 0 overlap inthe time domain. However, these waves can be separated ifN1(t) =N(1, t)and N2(t) = N(2, t) are known from measured strains at two differentbar sections 1 and 2 [22]. This can be achieved by substituting = 1,N = N1 and = 2, N = N2 into (1) and solving the resulting system oftwo equations for Np() and Nn(). If the two bar sections are taken to be1 = tm and 2 = tm as shown in Figure 1, one obtains for tm = n/2(n= 0, 1, 2,...)

Np() = 1

1 ei4tm eitm N1() e

i3tm N2() (3)and a similar result for Nn().

According to (1), the wave Np() gives rise to the delayed incident wave

Ni() =B()Np() (4)

with

B() = ei(tcta) (5)

at the first bar/actuator interface = tc ta.


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3.4 Transmitted and Reflected Waves Generated by Incident

Wave

First, it is assumed that the incident wave Ni() is the only wave actingin the actuator region, i.e. there is no control action. Then, according to(1), the incident wave Ni() at the first bar/actuator interface = tc tagenerates a delayed transmitted wave

N(i)t () =T()

Ni() (6)

with

T() = ei2ta

(7)

at the second bar/actuator interface = tc+ ta. Because of the impedancematching, there is no reflected wave generated at the first bar/actuatorinterface due to the incident wave, i.e.,

N(i)r () = 0. (8)

3.5 Out-going Waves Generated by Actuator

Next, it is assumed that the feedforward input Nc to the actuator pair isacting alone,i.e., there is no incident wave at anyone of the two bar/actuatorinterfaces. If it is imagined that such input would be supplied to the actuatorpair with rigid constraints at its ends, a normal force Nc(), assumed to beproportional to the input voltage, would be generated in each section of

the actuator region. It generates out-going waves N(c)t () and

N(c)r () at

the second and first bar/actuator interfaces, respectively. As shown in [17],these waves are

N(c)t () =

N(c)r () =C()Nc() (9)

with

C() =1

2

1 ei2ta

. (10)

3.6 Feedforward Input for One-way Transmission

Finally, the incident wave Ni() at the first bar/actuator interface and thefeedforward input Nc() are considered to act simultaneously and their re-sponses are superimposed. Thus, according to equations (6), (8) and (9)


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8 Peter Naucler

out-going waves

Nt() = N(c)t () +

N(i)t () =C()

Nc() + T()Ni() (11)

and

Nr() = N(c)r () +

N(i)r () =C()Nc() (12)

are generated at the second and first bar/actuator interfaces, respectively.Zero transmission can be achieved by putting the right-hand member of (11)equal to zero, which gives the feedforward input, c.f. (3) (5),

Nc() = D()C1

()T()B()eitm N1() ei3tm N2() (13)where

D() = 1

1 ei4tm. (14)

With this feedforward input to the actuator pair, the outgoing waves fromthe second and first bar/actuator interfaces will be

Nt() = 0 (15)

and

Nr() = T()Ni(), (16)

respectively, where the last equation has been obtained by using (11) and(15) into (12). Thus, due to the action of the actuator pair, the wave Ni()incident on the first bar/actuator interface is not transmitted through theoutput bar/actuator interface. From (7) and (16) it follows that this waveis subjected to an apparent free-end reflection, with reversal of sign, at themid-section = tc of the actuator region. Waves incident on the secondbar/actuator interface do not activate the actuators and are therefore trans-mitted undisturbed through the actuator region without reflection. Theseproperties are illustrated in Figure 2.


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Paper II 9

0000111100001111

N1(t) N2(t) Nc(t)

t1

t1

t2 > t1

t2> t1

(a)

(b)

(c)

Figure 2: (a) Bar with strain gauge pairs and actuator pair.(b) Apparent free-end reflection of wave traveling from left to right.(c) Undisturbed transmission of wave traveling from right to left.

4 Evaluation of Ideal Feedforward Design

For the purpose of a further analysis it is useful to rewrite the relationsderived in the previous section as difference equations in discrete time. Thisis straightforward since the equations derived so far basically involve timedelays,e.g. eita N() N(t ta). For simplicity we assume that thetime delays ta, tm and tc are integer multiples of the sampling interval T.We label these multiples as a, m and c, respectively, so that ta = aT,etc. This means that we can use the backward shift operator q1 so that,

e.g., qaN(t) =N(t aT) =N(t ta).

We aim at describing the system in a way that is tractable to analysis froma control perspective. Then, it is natural to consider the total controlledsystem (comprising both the mechanical part and the controller) as a twoinput, single output system with the waves Np(t) and Nn(t) as inputs andNt(t) as output. Note that Nn(t) does not affect Nt(t), see (22) below.The measured signals N1(t) and N2(t) are viewed as internal measurablevariables rather than inputs to the total system. The control signal is Nc(t),i.e., the input to the actuator.


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10 Peter Naucler

To make the analysis more realistic, we assume the measured signals N1(t)

and N2(t) to be corrupted with additive measurement noise. By use of (1),these signals can be expressed as

N1 (t) =qmNp(t) + q

mNn(t) + w1(t) (17)

N2 (t) =qmNp(t) + q

mNn(t) + w2(t) (18)

wherew1(t) and w2(t) are measurement noise sequences which are assumedmutually independent. The extraction of Np is performed by the time-domain equivalent of (3), which is

Np (t) = 1

1 q4

m

No (t) (19)

No (t) =qmN1 (t) q

3mN2 (t) (20)

In case there is no measurement noise present in (18), the reconstructedforce Np (t) would be equal to Np(t).

The relation between control force and generated wave by the piezo-elementsis given by (9). For implementation purposes a power amplifier is neededto amplify the control signal. The amplifier is modeled as linear and itstransfer function operator is denoted by G(q1). Taking this into accountwe obtain the output equation (c.f. (11))

Nt(t) y(t)

=C(q1)G(q1) Nc(t) u(t)

+T(q1)B(q1)Np (t)

= 1

2(1 q2a)G(q1)Nc(t) + q

(c+a)Np (t). (21)

The notationsu(t) andy(t) are introduced for the convenience of the controloriented reader. Hence, the output y(t) = Nt(t) is a linear combinationof the wave resulting from the control action u(t) = Nc(t) and the waveNp (t) propagating through the bar in the forward direction. By use of therelations derived so far the system is schematically described in Figure 3. In

the figure, the wave Nn propagating in the backward direction is modeledas the sum of two waves, N

(ext)n and N

(int)n . The waveN

(ext)n is due to any

external disturbance coming from the right, whereas N(int)n originates from

the control action.

The signal No is a filtered sum of the measured signals N1 and N

2 . The

filtering can be viewed as a decoupling operation as it will be seen to removethe Nn component. Therefore, it is convenient to regard N

o as a decoupled

single input to a feedforward filter, which has the control signal u(t) =Nc(t)as output.


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Paper II 11

1

Wave separation

Feedforward Amplifier

Actuator

Controller

Figure 3: A direct realization of the system derived in Section 3. The waveseparation can be divided into a decoupling part and a reconstruction part.We denote the feedforward filter as the filter with No as input.

The feedforward filter is built by three blocks. The first, D, is the recon-struction (19) ofNp, the second contains the time delays B , (4), andT, (6),

to compensate for the propagation time through the bar, the third invertsthe actuator dynamics (10).

As depicted in Figure 3, the ideal feedforward design derived in Section 3can be divided into a wave separation part and a feedforward part. Thedecoupling equations yield

No (t) =qmN1 (t) q

3mN2 (t)

=D1(q1)Np(t) + w(t), (22)

w(t) =qmw1(t) q3mw2(t). (23)

Here w(t) is the assembled effect of the two noise sources. Equation (22)shows that the waveNndoes not influenceN

o . IfN

o is viewed as a measured

signal, and by use of the above observation, the total controlled system canbe described as having Np as a single input, see Figure 4. In the figure theideal feedforward filter is introduced as

F(q1) = D(q1)B(q1)T(q1)C1(q1)

= 2q(c+a)

(1 q4m) (1 q2a). (24)


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12 Peter Naucler

B T

w(t)

Ni(t)

Np (t)

N

(i)

t (t)

N(c)t (t)

G

BTC1Np(t)

Feedforward, F(q1)

C

D1 D

No (t)

Nc(t) =u(t)

N

t(t) =y(t)

Figure 4: A simplified realization of the system. The feedforward filter isdenoted by F(q1).

Note the difference between this description of the feedforward filter andthe relation (13) derived in Section 3, where the inputs to the filter werethe true measurements N1 and N2. Here, we instead use the decoupledsignal No , which includes additive measurement noise, as a single inputto the feedforward filter F(q1). The noise part crucially influences the

performance of the mechanical diode. This will be highlighted in Section4.2.

4.1 Modeling of disturbance and measurement noise

The disturbance entering the system is the wave Np. Two kinds of dis-turbances are treated. The first type is a pulse-shaped disturbance that

enters the system relatively seldom and is well separated from other incom-ing pulses. The second type is a stationary disturbance of low frequencycontent.

One way to model a pulse disturbance stochastically is to describe it as theimpulse response from a linear system [21]. If the disturbance is a pulse asdepicted in Figure 5, it can be modeled as

Np(t) =A1 qk

1 q1p(t). (25)


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Paper II 13

Here A is the amplitude, T is the sampling interval, k is the length of the

pulse (in terms of samples) and {p(t)} is a sequence of independent randomvariables with the property

p(t) =

1 with probability p/2

1 with probability p/20 with probability 1 p,

wherep is a positive and small number. Although Np(t) has a deterministicappearance, the arrival time of pulses is stochastic and modeled by p(t).Obviously, p(t) has zero mean, and for the auto-covariance we have

Ep(t)p(s) = 0 t =sp t= s,i.e. the stochastic signal p(t) has exactly the same first and second ordercharacteristics as a white noise sequence. This means thatNp(t) will havethe same statistical properties as filtered white noise in terms of first andsecond order moments.

The stationary disturbance is modeled as a first-order AR process with ap-proximate bandwidth 0 that is small compared to the sampling frequency,

Np(t) = 1 e0T

1 e0Tq1v(t). (26)

In (26)v(t) is a white noise sequence with zero mean and variance 2v. Thisdisturbance type is the most general of the two treated in this paper. Arealization of the stationary disturbance (26) is shown in Figure 6.

The measurement noise is also modeled as a first-order AR process drivenby zero mean white noise e(t) with variance 2e, i.e.,

w(t) = 1

1 0.5q1e(t). (27)

This implies a bandwidth of about 0.7/T rad/s.


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14 Peter Naucler

Np(t)

A

to to+ kTt

Figure 5: A pulse-shaped disturbance with length kTthat enters attime to .

0 0.5 1 1.5 2 2.5 3 3.5 4

x 103

3

2

1

0

1

2

3

4

p

t [s]

Np

(t)

Figure 6: Realization of the continuous disturbance with0T = 10

2 rad.


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Paper II 15

Table 1: Simulation parameters.Description Parameter Value

Sampling interval T 5 s

System parameters {a, m, c} {2, 8, 200}

Continuous disturbance 0 2 1000 rad/s2v 100

Pulse disturbance A 10k 4p 104

Noise variance 2e 0 or 0.01

Power amplifier G(q1) 1

4.2 Simulation of Ideal Feedforward Design

So far, a number of parameters have been introduced. The values of these

parameters are summarized in Table 1. In the sequel these parameter val-ues are chosen for numerical examples and for simulation (unless otherwisestated). For the case of non-ideal power amplification, we refer to Section5.5.

For purpose of illustration, the ideal feedforward design is simulated with astationary disturbance. The power amplifier is assumed to be ideal ( G(q1) 1), and the case of no measurement noise (2e = 0) is first treated. The re-sult of the simulation is depicted in Figure 7. The figure shows that thedisturbance is completely cancelled out when control action is applied.

Next, a small amount of measurement noise w(t) is added. The measurementnoise obeys (27), with 2e = 0.01. The result of the simulation is shown inFigure 8. The figure shows that the ideal feedforward control completelybreaks down when measurement noise is present. This is due to the polesof the feedforward filter F(q1) (24), being located on the unit circle.

The output from the wave separation filter, No (t), can be divided into oneinformation signal part and one noise part. The information signal part willpass through the feedforward filter and cancel the positive wave propagatingthrough the bar. The noise part will, however, be a random walk process


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16 Peter Naucler

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 103

3

2

1

0

1

2

3

Nt*=

y

t [s]

No FF

Ideal FF

Figure 7: Simulation of the ideal feedforward design with no measurementnoise added. Transmitted wave y (t) =Nt(t)

versus time t.

after passing the filter. The variance of the output, Ey 2(t), will hence in-

crease linearly with time and be asymptotically unbounded. This impliesthat the ideal feedforward design needs to be modified to be useful in arealistic scenario where measurement noise is present.


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Paper II 17

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 103

4

3

2

1

0

1

2

3

4

5

t [s]

w,

Np

w(t)N

p(t)

(a)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.16

4

2

0

2

4

6

8

t [s]

Nt*=

y

Ideal FFNo FF

(b)

Figure 8: Simulation of the ideal feedforward design with noise added: (a)Measurement noisew(t) (solid) and disturbanceNp(t) (dashed); (b) Resultsof feedforward control (grey) and open loop control (black).


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18 Peter Naucler

5 Modified Feedforward Design

The unpleasant performance of the ideal feedforward control depicted inFigure 8 is due to the poles of the feedforward filter (24) lying on the unitcircle. Therefore, an attempt is made to modify the filter by moving the polestowards the origin to make it asymptotically stable. One such modificationis

Fmod(q1) =

2q(c+a)

(1 r1q4m) (1 r2q2a) (28)

where r1 and r2 are real numbers in the interval [0, 1). With this type ofmodification we still keep Nn from affecting the output N

t . If we assume

ideal power amplification, G(q1) 1, and use the modified feedforwardfilter, the output y = Nt satisfies (c.f. Figure 4)

q(c+a) Nt(t) y(t)

=

1

1 q2a

1 q4m

(1 r2q2a) (1 r1q4m)

Np(t)

+

1 q2a

(1 r2q2a) (1 r1q4m)

w(t). (29)

By careful choices of the parameters r1and r2, the effect of the measurementnoise can be significantly reduced and at the same time the wave Np(t) can

be considerably damped. However, perfect cancellation can not be obtainedwhen measurement noise is present. More specifically, we would like tominimize the variance of the output, Ey2(t), by choosing appropriate valuesof the parameters r1 and r2. It is obvious that r1 = 1 is a bad choice.The system then will have poles on the unit circle. It might be temptingto choose r2 = 1 and cancel the common poles and zeros in front ofw(t).However, this is an invalid operation because the output of a filter with poleson the unit circle will depend on initial conditions also as t . Thus,r2= 1 is a poor choice.

Ultimately, one would not only want the variance of the output to be min-

imized, but also include some kind of robustness criterion. Unfortunately,there is no obvious way how to include a robustness measure when mini-mizing the variance of the output. Ifr1 and r2 are treated as independentdesign variables it turns out that Ey 2(t) will decrease as r2 approaches 1.As explained above, the variance is not defined for r2= 1 and the criterionwould therefore not be continuous at r2 = 1. To circumvent this problem,the choice is made to minimize Ey 2(t) with the constraint that all polesshould have the same distance to the origin, i.e. lie on the same circle. The

relation betweenr1 and r2then will ber4m1 =r

2a2 , orr2 = r

a/2m1 . This

will in some sense also facilitate robustness since the approximative inverse


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Paper II 19

of the actuator dynamics 1 r2q2a

1will be forced to be asymptotically

stable.

5.1 Variance Analysis

IfNp(t) and w(t) are mutually independent, the variance of the output canbe written as

Ey 2(t) =Ey2sig(t) + Ey2w(t) (30)

whereysig(t) andyw(t)

modeling and control of vibration in mechanical structures

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