Robust Piecewise-Linear State Observers for Flexible Link Mechanisms

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Roberto Caracciolo

Dario Richiedei

Alberto Trevisani1

e-mail: [email protected]

Dipartimento di Tecnica e Gestione dei SistemiIndustriali,

Università di Padova,Stradella S. Nicola 3-36100 Vicenza, Italy

Robust Piecewise-Linear StateObservers for Flexible LinkMechanismsThis paper tackles the problem of designing state observers for flexible link mechanisms:An investigation is made on the possibility of employing observers making use of suitablepiecewise-linear truncated dynamics models. A general and novel approach is proposed,which provides an objective way of synthesizing observers preventing the instability thatmay arise from using reduced-order linearized models. The approach leads to the iden-tification of the regions of the domain of the state variables where the linear approxima-tions of the nonlinear model can be considered acceptable. To this purpose, first of all,the stability of the equilibrium points of the closed-loop system is assessed by applyingthe eigenvalue analysis to appropriate piecewise-linear models. Admittedly, the dynamicsof such a closed-loop system is affected by the perturbation of the poles caused byspillover and by the discrepancies between the linearized models of the plant and the oneof the observer. Additionally, when nodal elastic displacements and velocities are notbounded in the infinitesimal neighborhoods of the equilibrium points, the difference be-tween the nonlinear model and the locally linearized one is expressed in terms of un-structured uncertainty and stability is assessed through H� robust analysis. The method isdemonstrated by applying it to a closed-chain flexible link mechanism.�DOI: 10.1115/1.2909600�

IntroductionThe design of state observers is an essential step toward the

ynthesis of high-performance state controllers for flexible linkechanisms. These controllers are model based and need a high

umber of coordinates �i.e., the state variables� to reproduce theynamic behavior of a system. Since the direct measurement of allhe state variables is not usually possible or convenient, designingccurate state observers becomes of paramount importance.

Designing state observers for flexible link mechanisms is a con-iderable problem due to the nonlinear behavior exhibited by suchechanisms. In particular, direct use of nonlinear models appears

o be inconvenient in the synthesis of observers since the real-timeomputation of the state variables may become very costly. On thether hand, the use of linear models is not straightforward: Whenmechanism moves from an initial to a final configuration, dif-

erent linear models should generally be employed to locally re-roduce the system dynamics. This fact, in addition to the in-reased availability of control system design �1,2� and stabilitynalysis methods �3,4� both for hybrid and for switching linearystems, makes the use of piecewise-linear observers attractivehen dealing with flexible link mechanisms.Basically, a piecewise-linear observer consists in a discrete and

nite set of linear candidate observers, each tuned about a specificperating point, and a “high-level” controller called a supervisor,hich selects one of the linear time invariant �LTI� observers as

he source of the estimated state vector. Switching piecewise-inear control laws are now being extensively suggested in litera-ure for the control of a wide class of systems since they allowvercoming the limitations of conventional adaptive controlchemes �5�.

Recent theoretical results in this field are focused on the defi-

1Corresponding author.Contributed by the Dynamic Systems, Measurement, and Control Division of

SME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CON-

ROL. Manuscript received October 16, 2006; final manuscript received May 29,007; published online May 12, 2008. Assoc. Editor: Yossi Chait. Paper presented athe 8th Biennial ASME Conference on Engineering Systems Design and Analysis
ESDA2006�, Torino, Italy, July 4–7, 2006.
ournal of Dynamic Systems, Measurement, and ControlCopyright © 20

om: http://dynamicsystems.asmedigitalcollection.asme.org/ on 08/29/2013

nition of appropriate switching strategies capable of both stabiliz-ing the system and guaranteeing the desired performances of thecontrol scheme. Most of the approaches suggest performing anon-line evaluation of each candidate controller �see, e.g., Refs.�2,6,7��. However, when the complexity of the system increases,the search for the stabilizing controller can be time consuming andreal-time hardware implementation may lead to very poor perfor-mances �8�.

There follows that when dealing with real and complex plants,switching strategies are generally based on the values assumed bysome variables, which show strong correlation with plant dynam-ics. The state variable domain is then split into regions and an apriori definition of switching points is obtained. An interestingapplication of this approach to an industrial boiler system is pre-sented in Ref. �9�, where the switching points have been selectedby means of numerical simulations.

By referring to flexible link mechanisms, the objective of thepresent work is to identify significant variables to be employed inthe decomposition into regions of the domain of the state variablesof a generic mechanism, and to define practical criteria for select-ing appropriate switching points ensuring stability and accuratestate vector estimates. The scheme adopted for the observer makesuse of a steady-state Kalman filter together with a piecewise-linear model obtained by linearizing a fully coupled nonlinearmodel about a suitable set of operating points �equilibriumconfigurations�.

The proposed original approach is based on the analysis of theregion of the domain of the state variables where the linear ap-proximation of the nonlinear model is considered acceptable. Thisanalysis is carried out by analyzing the perturbation of the polesand the estimation error caused by spillover and by the discrep-ancies between the linearized models of the mechanism and of theobserver. Moreover, since elastic displacements and velocities arenot generally bounded in the infinitesimal neighborhoods of theequilibrium points when flexible link mechanisms move, it is alsointeresting to asses the stability of the closed-loop system underthese conditions. In this work, performing such an investigationby expressing the difference between the nonlinear model and the
locally linearized one in terms of unstructured uncertainty and by
MAY 2008, Vol. 130 / 031011-108 by ASME

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ssessing stability through H� robust analysis is suggested. Theethod proposed also provides an objective way of verifying the

dequacy of the dimensions of the reduced-order linear modelsmployed in the observer and chosen at the design stage.

Dynamic Model: BackgroundThe piecewise-linear model adopted in the synthesis of the ob-

erver derives from the very accurate and fully coupled nonlinearodel presented in Ref. �10� and experimentally validated with

eference to both four-link �10� and five-link �11� planar mecha-isms with all the links flexible. In Ref. �12�, a linear model intate space form has been obtained by linearizing the nonlinearodel proposed in Ref. �10� about a generic operating point. A

umerical and experimental validation of such a linearized modelas been provided in Ref. �13� with reference to a planar four-barinkage with all the links flexible. Both small and large displace-

ents of the mechanism with respect to an equilibrium configu-ation have been considered, proving that when large displace-ents are to be reproduced, a piecewise-linear model can be

uccessfully employed.So as to get a dynamic model with a finite number of degrees of

reedom, the flexible links can be subdivided into finite elements.oreover, if the total motion of each flexible link is separated into

he large rigid-body motion of an equivalent rigid-link systemERLS� and the small elastic deflection of the link with respect tohe ERLS itself, it can be proved �see Ref. �10�� that by applyinghe principle of virtual work it is possible to get this final expres-ion of the equations of motion in matrix form:

�M MS 0 0

STM STMS 0 0

0 0 I 0

0 0 0 I��

u

q

u

q�

= �M I

STM ST

0 0

0 0��g

f�

+ �− 2MG − �M − �K − MS − K 0

ST�− 2MG − �M� STMS 0 0

I 0 0 0

0 I 0 0��u

q

u

q� �1�

A more compact form for Eq. �1� is A�x�x=B�x�x+C�x�v,here x= �u q u q �T and v= �g f �T.A linearization procedure �described in detail in Ref. �12�� has

een applied to the above nonlinear model, so as to get a statepace linear model capable of reproducing the dynamic behaviorf a flexible link mechanism about an equilibrium configuration.he linearization procedure considers only the linear terms of aaylor series expansion of Eq. �1� about a generic operating pointoinciding with an equilibrium configuration where x=xe, v=ve,nd xe=0. In the neighborhood of the operating point, the stateector and the input vector can therefore be expressed in the form�t�=xe+�x�t�, v�t�=ve+�v�t�. By introducing these relationsnto Eq. �1� and by making use of the approximation A�xe

�x��xA�xe��x, after some algebraic computations, the fol-owing linear expression can be obtained, which holds in theeighborhood of any operating point:

A�xe��x = �B�xe� + ��B

�x�

x=xe

� xe� + ��C

�x�

x=xe

� ve��x

+ C�xe��v = B�xe,ve��x + C�xe��v �2�
here the symbol � indicates the inner product of
31011-2 / Vol. 130, MAY 2008


��Bi,1 /�xj¯�Bi,n /�xj�x=xeand ��Ci,1 /�xj¯�Ci,n /�xj�x=xe

with, re-spectively, xe and ve, for all the subscripts i and j.

Two aspects are to be pointed out. First, since a symbolic formof the linearized model has been obtained, once an equilibrium

point is set, the matrices A, B, and C can be immediately com-puted. Second, since an appropriate choice of the ERLS positionwith respect to the actual mechanism allows keeping the ampli-tude of the elastic displacements small, the equilibrium values ofall the matrices in Eq. �2� at any operating point only depend onthe equilibrium values of the ERLS generalized coordinates con-tained in vector qe, which prevents using additional linearizedmodels when the elastic terms of xe change. There follows that thesole knowledge of qe allows the off-line computation of the lin-earized dynamic matrices and therefore the switch between linear-ized models can be made on the basis of the value taken by apiecewise-constant function of q. In other words, the availabilityof a linearized model for any ERLS configuration and the smallamplitude of the elastic displacements with respect to the ERLSallow associating an equilibrium point to any possible system con-figuration, i.e., to associate a path of equilibrium points with thepath of the real state. In practice that implies that the nonlinearmodel may be approximated by means of a discrete and finite setof linear models.

The sensed output �y� of a flexible link mechanism does notgenerally coincide with state of the system, but it is a function ofsuch a state. The relation between the state variables x and thesensed output y is in general nonlinear and therefore it should belinearized about an equilibrium point so as to obtain a traditionalstate space model for the linearized system:

�x�t� = F�x�t� + G�v�t�

�y�t� = H�x�t� �3�

where t is the time variable, F=A−1 B and G=A−1 C.System �3� may be easily transformed into modal canonical

form by means of the modal transformation matrix Tzx relating thenodal state vector �x to the modal state vector �z=Tzx�x.

It is not generally useful and possible to control all the modesof the system. Hence, by reordering the modes and partitioningthem into controlled and residual �unestimated and uncontrolled�modes, the dynamics of the actual system about a generic operat-ing point is described by the following system of equations:

�zC = AC�zC + BC�v

�zR = AR�zR + BR�v

�y = CC�zC + CR�zR �4�

3 Equilibrium Model of the Control SchemeThe method proposed in this work is applied to a control

scheme based on a linear and continuous-time Luenberger ob-server, which is designed as a steady-state Kalman filter, and on alinear state variable feedback control law. It is well know that theKalman filter is the optimal state observer for linear and uncon-strained systems subject to normally distributed state and mea-surement noise. The use of Kalman filter is widely suggested inliterature for the control of flexible link mechanisms, since it leadsto a systematic approach for the design of observers once an es-timate of the noise covariance of the sensors and of the process isavailable.

When a tracking problem is considered, the dynamics of theoverall continuous-time controller is described by the followingsystem of equations:

�zC = ACob�zC + BC

ob�v + L��y − �y�

ˆ ˆ
�zR = 0, �zR = 0
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�y = CCob�zC

e = �zC − �zC

�v = − W��x − �xref�

�x = Txz��zCT�zR

T� = TxzC�zC �5�

here TxzCis the n�nC matrix whose columns are the eigenvec-

ors of the selected modes. TxzCis therefore obtained by selecting

ppropriate nc columns of the n�n transformation matrix Txz

elating the estimate of the nodal state vector �x to the estimate ofhe modal vector �z.

The observer-gain matrix L is chosen so as to minimize theean square error between the estimated and the actual values of

he state variables in the presence of process noise �P�t� andeasurement noise �M�t� affecting the plant. Both �P�t� and

M�t� are assumed to be uncorrelated zero-mean, Gaussian, white-oise vectors, with covariance matrices �correlation matrices� Rnd Q. Therefore, the following equations hold:

E��P�t�� = E��M�t�� = 0, E��M�t��P�t�T� = 0

E��P�t��P�t�T� = R, E��M�t��M�t�T� = Q �6�

y including �P�t� and �M�t� in the open-loop linearized model,t holds

erturbation matrix �B implies that limt→� ey�t��0, for any

ournal of Dynamic Systems, Measurement, and Control


�x�t� = F�x�t� + G�v�t� + G�P�t�

�y�t� = H�x�t� + �M�t� �7�

where it has been supposed that the process noise acts on the plantas an input disturbance. It is well known �14� that the solution ofsuch a problem is given by L=PCC

obTR−1, where, if an infinitetime horizon is assumed, P is the symmetric and positivesemidefinite solution of the continuous-time algebraic Riccatiequation: 0=AC

obL+ACobTL+R−LCC

obTQ−1CCobL.

While moving from an initial to a final configuration, a set oflinearized models can be employed to approximate the state tra-jectory. The matrices of the local linearized representation of thenonlinear system �AC ,BC ,CC� generally differ from those of theobserver �AC

ob,BCob,CC

ob�. The difference between the dynamics ofthe system at the equilibrium point and the dynamics of the ob-server can be made explicit by defining the perturbation matrices:

�A = ACob − AC, �B = BC

ob − BC

�C = CCob − CC, �TxzC

= TxzC− TxzC

�8�

where TxzCis obtained by selecting appropriate nc columns of

Tzx−1.The overall system dynamics is obtained by combining Eqs.

�4�, �5�, and �8�:

��zC

�zR

e� = �BC

BR

�B�W�xref + � AC − BCW 0 − BCW

− BRW AR − BRW

�A − L�C − �BW LCR AC − LCC + �A − L�C − �BW��zC

�zR

e� �9�

here W=WTxzC.

Let N be the system dynamics matrix in Eq. �9�. N can be rearranged to highlight the contribution of the unperturbed matrix N0, ofhe matrix Nsp containing the control and observation spillover terms �15� and of the matrix �N containing perturbation matrices.

N = N0 + Nsp + �N = �AC − BCWTxzC0 − BCWTxzC

0 AR 0

0 0 AC − LCC� + � 0 0 0

− BRWTxzC0 − BRWTxzC

0 LCR 0�

+ � − BCW�TxzC0 − BCW�TxzC

− BRW�TxzC0 − BRW�TxzC

�A − L�C − �BWTxzC− �B�TxzC

0 �A − L�C − �BWTxzC− �B�TxzC

� �10�

Two aspects are to be pointed out. First, the separation principleetween the controlled modes �zC and the residual modes �zR,olding in the unperturbed system with dynamics matrix N0, doesot hold, due to the coupling terms introduced by Nsp and �N.econd, the same coupling terms cause perturbation of the poles,hich can lead to slower convergence of the state and can even

ause instability. It should be noticed that the effect of �N cannote eliminated by the standard spillover reduction techniques, i.e.y prefiltering the sensor output and by locating the actuators andhe sensors at the zeros of the mode shapes of the residual modes.

In terms of system output, the estimation error can be evaluateds follows:

ey = �y − �y = �C�zC − CR�zR + �CC + �C�e �11�

y computing the transfer function of the input-output realizationf the system given by Eqs. �9� and �11�, the presence of the

�xref�0, and that the actual value depends both on spillover andperturbation matrices.

4 Robustness Analysis of the Control SchemeThe linear model proposed in the section above yields truthful

results as long as nodal elastic displacements and the velocitiesare bounded in infinitesimal neighborhoods of the equilibriumpoints. When this is not the case, the difference between the non-linear model and the locally linearized one can be expressed interms of uncertainty, and the selection of the number of models tobe employed in the synthesis of the piecewise-linear observershould account for the robustness margin that the system exhibits.

After choosing an appropriate uncertainty �y�s�, robust stabil-
ity is achieved if the following norm inequality holds:
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�y�s��v�s� � � 1 �12�

here ��v�s� is the transfer function of the closed-loop nominalystem from the disturbance ��t� to �v�t�. Therefore, the norm��v�s� � is a measure of the degree of stability of the systemhose poles have been shifted by matrices Nsp and �N. Figure 1epicts the open-loop system with uncertainty; the closed-loopncertain system is represented in Fig. 2.

In this work, an additive representation of the uncertainty haseen chosen; such an analysis could be, however, extended to anyther representations.

The truthful definition of the nominal linear system assumed inq. �9� allows including the perturbation matrices and the spill-ver terms into the nominal system, and therefore reduces modelncertainty and prevents overly conservative design. The uncer-ainty with respect to which the control should always be madeobust is the one arising from employing a linear representation ofhe system, instead of a nonlinear one. Further uncertaintiesaused by perturbations of the geometric and inertial parametersf the mechanism could also be considered. However, it should beointed out that the nominal linear system could also include annalytical linear approximation of such uncertainties by means ofhe perturbation matrices �8�.

The state space representation of ��v�s� is

��s� = � N �0

L�

�− W 0 − W� 0� �13�

Once the uncertainty has been identified, or a desired degree ofobustness has been defined, the robustness of the piecewise-inear observer designed through the procedure proposed abovehould be verified. An approach for the definition of the uncer-ainty, suitable for flexible link mechanisms, is given in Ref. �19�.

Piecewise-Linear Observer Synthesis ProcedureWhen piecewise-linear observers are to be designed, the num-

er of regions �and hence the number of linearized models em-loyed in the observer synthesis� should be minimized since thearameters used to represent these regions may take a fairly largemount of controller memory. Besides, the time interval betweenhe two consecutive switches must take a time long enough toissipate observer transient effects. Under this assumption �the soalled “slow switching condition” �5��, the switch between stableocal observers ensures stability of the whole system.

A meaningful measure of the width of the region where a linearpproximation may be considered accurate is given in Ref. �16�.

ig. 1 Representation of the open-loop system withncertainty

ig. 2 Representation of the closed-loop system with
ncertainty
31011-4 / Vol. 130, MAY 2008


In such a work, an upper bound of the L2 norm of the modelnonlinearities is given by using the Hamilton–Jacobi inequality.Unfortunately, such an inequality cannot be generally analyticallysolved. Alternatively, sufficient conditions for system asymptoticstability under generic pole perturbations are given in severalworks concerning regulation problems �e.g., Refs. �17,18��.Withthe approach proposed in the present work, the selection of theregions and of the approximated models is based on analyzing theeffects of Nsp and �N on the closed-loop dynamics performancesafter selecting independent local Kalman filters, computed withthe estimated measurement and process noise covariances. Thedecomposition into regions of the domain of the state variablesand the selection of the switching points satisfying requirementsof stability and accuracy of the estimates can be carried out fol-lowing the steps listed below:

Step 1: State and measurement noise covariances are identifiedso as to define the Kalman filtering problem; the desired locationof the poles of the regulator is then selected �e.g., by defining alinear quadratic regulation problem to be minimized�.

Step 2: The observer and the regulator are synthesized about aspecific ERLS equilibrium configuration q1, belonging to the do-main of interest.

Step 3: The neighborhood of q1 is explored, by applying thecontroller synthesized to a discrete set of operating points and bychecking each time whether the following inequalities are simul-taneously satisfied:

eig�N�q��

Ey�s� � Emax�s�

�y�s��v�s� � � 1 �14�A more conservative robustness condition, which may replace

the third one in Eq. �14� is

��v�s� � � 1/ �y�s� � �15�If all the inequalities in Eq. �14� are satisfied, the observer

synthesized in q1 may be scheduled to estimate the state when thesystem is in the neighborhood of q. Conversely, a switching pointmust be defined in the interval �q ,q1� and a new observer must besynthesized about another operating point q2, ensuring that re-quirements in Eq. �14� are met.

The third step is repeated until the whole domain of interest ofq has been analyzed.

In case the number of observers synthesized is too high, andtherefore slow switching conditions can no longer be satisfied,some of the assumptions made at the early stages of the designprocess should be reconsidered and verified. Such assumptionstypically concern the dimensions of the reduced-order linear mod-els employed in the observer �which are usually established bymeans of the classical model reduction approach �14��, the valuesof state and measurement noise covariances �which are usuallytuned in an iterative process of trial and error starting from theidentified values�, and the choice of the regulator.

6 Case Study

6.1 Description of the Mechanism. The theory developedabove has been applied to the synthesis of an observer adopted inthe motion control of the four-bar planar linkage considered inRef. �19�. Such a mechanism is assumed to be driven by a singlemotor and to have all the links flexible. Its chief geometric andinertial characteristics are summarized in Table 1. The finite ele-ment representation adopted for the mechanism is illustrated inFig. 3, where the links, the joints, the elements, and the nodes areshown. The elements employed are classical two-node and six-degree-of-freedom beam elements. Two elements of the samelength are used to model Links 2 �the coupler� and 3 �the fol-
lower�, while a single beam element is employed for Link 1 �the


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rank�, which is the shortest link. Lumped masses and inertias aresed to account for the joints and the motor driving the mecha-ism at Joint A. The finite element representation employed forhe mechanism leads to a dynamic model with 32 state variables.

The ERLS generalized coordinate �q� is the �rigid-body� rota-ion of the crank, while the elastic degree of freedom forced toero to define the position of the ERLS with respect to the de-ormed mechanism is the horizontal displacement at Node 5. Asor the sensed output, three variables are supposed to be mea-ured: the crank angle �A measured at Joint A �which is the sumf q and the elastic angular displacement at Node 1� and theurvatures C1, C3 at the midpoints of Links 1 and 3. The vector ofhe output variables therefore takes the form:

y = ��A C1 C3�T �16�These variables have been chosen because they ensure a com-

lete observability of the state of the system, and consequently thetate reconstruction. Besides being completely state observable,he system is also completely controllable.

The mechanism is supposed to move on a vertical plane, and sohe effect of the force of gravity cannot be neglected. This implieshat the open-loop system is unstable.

6.2 Implementation of the State Observer. It is well knownhat the error associated with discarding a vibrational mode alsoepends on the input matrix G and the output matrix H �see Eq.3��. As a consequence, in order to guarantee a desired level ofobustness and to prevent control and observation spillover with-ut excessively increasing the state dimension, the selection of theritical modes must not only consider the modal decay rate butlso the input-output characteristics of the model. Clearly, theodes that are less controllable and less observable should be

able 1 Mechanical parameters of the flexible link mechanism

Link 0 1 2 3

ength �m� 0.361 0.3885 0.535 0.632

lexural stiffness: 21.6 N m2 Cross-sectional area: 36E−6 m2

Joint A B C D

ass �kg� — 71E−3 70E−3 —nertia �kg m2� 4.9E−4 — — 12E−6

Fig. 3 Finite element representation of the four-bar linkage



discarded. The models adopted in this work are of the eighthorder. This choice allows selecting three couples of complex con-jugated poles in addition to the two real poles of the linearizedmodels.

Both process and measurement noise signals have been as-sumed white and uncorrelated; matrices R and Q are thereforediagonal. The process noise matrix R has been evaluated throughnumerical simulations on simple control tasks, by comparing thecontrol signals of the nonlinear simulator �which includes the non-linear model of the mechanism and of the actuating servomotor�,with the ones of a set of locally linearized models. In these tests,the rigid-body motion control has been performed using a PIDregulator. The sensor noise covariance matrix Q has been deter-mined on the basis of the chief characteristics of the sensors thatare supposed to be employed on the experimental setup �i.e., thenumber of radial lines for the encoder, the resolution of the ADCcards, the characteristics of the low-pass filters�. These valueshave been subsequently tuned with a small amount of trial anderror.

When implementing piecewise-linear observers, attentionshould be paid on the fact that when a switch between observersoccurs, accurate initial conditions must be defined for the observerselected. The equilibrium values of the state variables cannot beimposed as initial conditions because they would generally resultin state discontinuity and large velocity estimation transient error.The most general procedure dynamically consists in defining theinitial conditions as the best available estimate of the state vector�i.e., the last estimated state values, expressed with respect to thenew state vector�. Let t be a generic instant at which the switchfrom observer k to observer h occurs, and let : �0,��→ �h ,k� bethe switching signal �i.e., �t− tc�=k, �t�=h�. The initial condi-tions of observer h are computed as �zc

h0 =Thk�zck�t− tc��−zc

he,where zc

k�t− tc� is the last state vector estimated by the observer kbefore the switch occurs, and Thk is the state transformation ma-trix relating the new state vector zh to the old state vector zk �zh

=Thkzk�.A double threshold relay may be also included to avoid

chattering.The regulator has been synthesized in accordance with the lin-

ear quadratic �LQ� optimal control theory. The use of LQ regula-tors is widely suggested in literature, since it leads to a systematicapproach for the design of control schemes ensuring the achieve-ment of the required performances expressed in terms of a costfunction. The LQR problem has been defined not to produce anexcessive damping of the amplitudes of the link elastic vibrations,in order to observe large displacements from the equilibriumpoints of the elastic variables and therefore to reproduce a rathercritical configuration for testing the effectiveness of the approachproposed.

7 Numerical ResultsThe test employed to validate the approach proposed consists in

a fast 80 deg rotation of the mechanism crank from an initialhorizontal static equilibrium configuration. A complete dynamicsimulator has been employed. The simulator comprises the non-linear model of the four-bar flexible linkage described in Sec. 6.1and a model of a torque controlled brushless servomotor. A 20,000line encoder has been assumed to be employed to measure thecrank angle at the drive shaft; the signals of all the measuredoutput variables are supposed to be acquired by means of a 12 bitADC board and sampled at 1000 Hz. The same resolution andsample rate are adopted for the control signal.

The requirements to be satisfied for the test, as defined in Eq.�14�, are the following:

eig�N�q�� 0

T
�Ey�s�� − 40 dB − 30 dB 30 dB�
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lims→0

E�A�0� � − 50 dB

��v�s� � � 36 dB �17�As a consequence of Eq. �16�, it holds

Ey = �E�A�s� EC1

�s� EC3�s��T �18�

The values chosen in Eq. �17� ensure satisfactory performancesor the system, in terms of nominal stability �first condition�, es-imation error �second and third conditions�, and robustness mar-in �fourth condition�. In the third inequality, E�A

�0� is the steady-tate error in the first component of vector Ey�s�. The valuehosen for the robustness margin has been evaluated on the basisf the uncertainty identified for the same mechanisms in Ref. �19�.dmittedly, these values may change according to the system

onsidered.As a first example, Fig. 4 shows the time history of the crank

ngle estimation error, when only a single observer synthesizedbout q=0 deg is employed to perform the estimation of the state.he test carried out consists in the aforementioned 80 deg rotationf the mechanism crank. The step change of the reference takeslace at 1.3 s.

The single linear observer satisfies the first, the second, and theourth requirement in Eq. �17�. However, the transfer function ofhe output estimation error of the crank angle does not allow

eeting the third requirement set in Eq. �17�. This fact implieshat the linear observer should theoretically fail to correctly repli-ate the rigid-body motion when the mechanism is far from to thenitial configuration, which is confirmed by the simulation resultshowing a considerable estimation error for the crank angle. Thisrror obviously affects the controller performances and may causesignificant steady-state tracking error.In order to synthesize a piecewise-linear observer satisfying all

he requirements in Eq. �17�, the procedure presented in Sec. 5 hashen been applied. The domain of q that is of interest for the testase has been discretized in order to allow a straightforward as-essment of the inequalities. An appropriate discretization step haseen chosen, corresponding to 1 deg. This value has been selectedince negligible nonlinearities arise when the displacement of q isithin such a discretization step.A state observer employing two linearized models has been

esigned. The equilibrium configuration about which the first ob-erver has been synthesized �q1, as defined at Step 2 of the pro-edure� has been chosen to be the final configuration to be takeny the ERLS during the test �q1=80 deg�. Since such a controllernsures that all the inequalities are concurrently satisfied until theRLS generalized coordinate is up to 33 deg, a switch has beencheduled when q=38 deg. An arbitrary “safe margin” of 5 degas been adopted.

The second observer has been synthesized about the initialquilibrium configurations of the test �q2=0 deg�. It has then beenroved that such a controller ensures that all the requirements areimultaneously satisfied as long as the ERLS generalized coordi-ate is below 44 deg. Since the two regions overlap ��33 deg,0 deg�, �0 deg, 44 deg��, only two linear observers can be effec-ively employed for the synthesis of the piecewise-linear observer.

1.2 1.4 1.6 1.8 2 2.2 2.4-1

0

1

2

Time [s]

e αA[degrees]

ig. 4 Time history of the output estimation error for the crankngle, obtained using the linear observer

n order to clarify this point, Fig. 5 shows a comparison between

31011-6 / Vol. 130, MAY 2008


the magnitude of the transfer function of the output estimationerrors, by performing the estimate of the state with the observersynthesized in q1 when q=38 deg �dashed lines� and q=30 deg�solid lines�, and by performing the estimates of the state with theobserver synthesized in q2 when q=38 deg �dotted lines�. It canbe noticed that if the observer synthesized in q1 were employed atq=30 deg, the third requirements in Eq. �17� would not be met.

The time histories of the actual and estimated output variables,when the piecewise-linear estimator is employed, are instead re-ported in Fig. 6. These plots clearly demonstrate the stability ofthe system and the accuracy of the estimates of the state, which isalso proved by the plot of the estimation error of the crank angle,reported in Fig. 7. They also prove that the closed-loop system hasgood tracking properties: admittedly the better the estimates, themore effective the control action. Finally, only a negligible tran-sient response can be discriminated in the estimated variables

10-2

100

102

104

-140-120-100-80-60-40

Frequency [rad/s]

EC3[dB]

-150

-100

-50

EC1[dB]

-150

-100

-50

EαA[dB]

Fig. 5 Magnitude of the frequency responses of the outputestimation errors

1.2 1.4 1.6 1.8 2 2.2 2.4020406080100

CrankAngle

[degrees]

1.2 1.4 1.6 1.8 2 2.2 2.4-0.4

-0.2

0

Crank

Curvature[1/m]

1.2 1.4 1.6 1.8 2 2.2 2.4-0.06-0.04-0.02

00.02

Time [s]

Follower

Curvature[1/m]

Fig. 6 Time histories of the output variables obtained usingthe piecewise-linear observer. The actual values are plotted as
solid lines; the estimated values as dashed lines.


w8t

8

smsei

mtt

pbsacist

dl

N

Ftl

Fc

J

Downloaded Fr

hen the switching between the linear observers occurs �see Fig.�. This is a consequence of the accurate dynamical definition ofhe observer initial conditions.

ConclusionsIn this paper, a design procedure has been proposed for the

ynthesis of robust piecewise-linear observers for flexible linkechanisms. The piecewise-linear structure chosen for the ob-

erver allows overcoming the difficulties related to system nonlin-arities even when large displacements with respect to an operat-ng configuration are considered.

First, starting from a very accurate model for flexible linkechanisms, it has been shown that the generalized coordinates of

he ERLS can be employed in the decomposition into regions ofhe domain of the state variables of a generic mechanism.

Moreover, practical criteria for selecting appropriate switchingoints, ensuring stability and accurate state vector estimates, haveeen defined. In particular, it has been proved that an effectiveynthesis of piecewise-linear state observers can be carried out bynalyzing the perturbation of the poles and the estimation erroraused by spillover and by the discrepancies between the linear-zed models of the mechanism and of its observer. The procedureuggested can also ensure closed-loop robustness to the uncer-ainty arising from system nonlinearities.

The effectiveness and the ease of use of the suggested proce-ure have been illustrated by numerical simulations on a four-barinkage with all the links flexible.

omenclatureAC ,BC ,CC matrices of the state space realization

of the controlled modes of the actualplant

ACob,BC

ob,CCob matrices of the modal state space

realization of the model employed inthe observer

AR ,BR ,CR matrices of the state space realizationof the neglected modes of the actualplant

A�x� ,B�x� ,C�x� matrices of the nonlinear system

B�xe ,ve� matrix of the linearized systemC1 ,C3 ,�A curvatures and crank angle of the

four-bar linkageE�A

�s� ,EC1�s� ,EC3

�s� estimation error of �A ,C1 ,C3E �s�

ig. 8 Actual „solid line… and estimated „dotted lines… values ofhe crank angle about the observer switching instant „verticaline…

ig. 7 Time histories of the output estimation error for therank angle, obtained using the piecewise-linear observer

y transfer function from �xref to ey



Emax�s� maximum allowed value of Ey�s�E�� expected value

eig�N� eigenvalues of matrix Ne ,ey state and output estimation error

F ,G ,H matrices of the nodal linearized statespace realizations of the actual plant

f vector of the concentrated externalforces and torques

g gravity acceleration vectorI identity matrix

K K�q�, stiffness matrixL observer-gain matrix

M M�q�, mass matrixMG MG�q , q�, Coriolis acceleration

matrixNsp perturbation of N0 due to spilloverN0 unperturbed system dynamics matrix

n dimension of the full order modelnC ,nR number of the controlled and of the

neglected modesP solution of the continuous-time filter

algebraic Riccati equationQ ,R measurement and process noise cova-

riance matricesq ,qe vector of the ERLS generalized coor-

dinates and its equilibrium valueq ,q1 ,q2 operating points of generic local ob-

servers under analysis in the synthe-sis procedure

S=S�q� ERLS sensitivity coefficient matrixof the nodes

t generic switching timetc sampling time

Thk state transformation matrix relatingthe state vector zh to the state vectorzk

Txz transformation matrix from �z to �xTzx transformation matrix from �x to �z

TxzC transformation matrix from �zC to�x

TxzC transformation matrix from �zC to�x

u vector of the nodal elastic displace-ments with respect to the ERLS ofall the nodes of the elements

v ,ve system input and its equilibriumvalue

W nodal gain vector of the linearregulator

W equivalent modal gain vector of thelinear regulator

x ,xe nodal state vector and its equilibriumvalue

y sensed outputzh ,zk generic modal state vector

zche equilibrium configuration of the

model adopted in observer hzc

k modal state vector estimated by theobserver k

� ,� Rayleigh coefficients�A ,�B ,�C ,�TxzC perturbation matrices

�N perturbation of N0 due to perturba-tion matrices

�x, �v, �x displacement of x, v, and x fromtheir equilibrium values

�x estimate of �x

MAY 2008, Vol. 130 / 031011-7


R

0

Downloaded Fr

�xref difference between state referencevalues and the equilibrium point statevalues

�z modal state vector of the linearizedmodel of the actual plant:�zT= ��zC

T �zRT�

�z estimate of �z�zC, �zR controlled and neclected modes�zC, �zR estimate of �zC and �zR

�zch0 initial conditions of observer h switching signal

�M�t�, �P�t� measurement and process noises� region of the complex plane where

eig�N� are to be located�y�s� uncertainty bound in the frequency

domain��t� disturbance due to uncertainty

��v�s� transfer function of the closed-loopnominal system from ��t� to �v�t�

� �T transpose of generic matrix� �−1 inverse of generic matrix

eferences�1� Koutsoukos, X. D., and Antsaklis, P. J., 2002, “Design of Stabilizing Switch-

ing Control Laws for Discrete and Continuous-Time Linear Systems UsingPiecewise-Linear Lyapunov Functions,” Int. J. Control, 75, pp. 932–945.

�2� Freidovich, L. B., and Khalil, H. K., 2005, “Logic-Based Switching for RobustControl of Minimum-Phase Nonlinear Systems,” Syst. Control Lett., 54�8�,pp. 713–727.

�3� Branicky, M. S., 1994, “Stability of Switched and Hybrid Systems,” Proceed-ings of 33rd IEEE Conference on Decision and Control, Lake Buena Vista, pp.3498–3503.

�4� Liberzon, D., and Morse, A. S., 1999, “Basic Problems in Stability and Designof Switched Systems,” IEEE Control Syst. Mag., 19�5�, pp. 59–70.

�5� Hespanha, J. P., Liberzon, D., and Morse, A. S., 2003, “Overcoming the Limi-

31011-8 / Vol. 130, MAY 2008


tation of Adaptive Control by Means of Logic-based Switching,” Syst. ControlLett., 49�1�, pp. 49–65.

�6� Hespanha, J. P., Liberzon, D., and Morse, A. S., 1999, “Logic-Based Switch-ing Control of a Nonholonomic System With Parametric Modeling Uncer-tainty,” Syst. Control Lett., 38�8�, pp. 167–177.

�7� Freidovich, L. B., and Khalil, H. K., 2004, “Comparison of Logic-basedSwitching Control Design for a Nonlinear System,” Proceedings of the Ameri-can Control Conference, pp. 1221–1222.

�8� Corradi, M. L., and Orlando, G., 2002, “A Switching Controller for the OutputFeedback Stabilization of Uncertain Interval Plants via Sliding Modes,” IEEETrans. Autom. Control, 47�12�, pp. 2101–2107.

�9� Marquez, H. J., and Riaz, M., 2005, “Robust State Observer Design WithApplication to an Industrial Boiler System,” Control Eng. Pract., 13�6�, pp.713–728.

�10� Giovagnoni, M., 1994, “A Numerical and Experimental Analysis of a Chain ofFlexible Bodies,” ASME J. Dyn. Syst., Meas., Control 116�1�, pp. 73–80.

�11� Caracciolo, R., Gasparetto, A., and Trevisani, A., 2001, “Experimental Valida-tion of a Dynamic Model for Flexible Link Mechanisms,” Proceedings ASMEDETC 2001, Pittsburgh, PA, pp. 461–468.

�12� Gasparetto, A., 2001, “Accurate Modelization of a Flexible-Link PlanarMechanism by Means of a Linearized Model in the State-Space Form forDesign of a Vibration Controller,” J. Sound Vib., 240, pp. 241–262.

�13� Caracciolo, R., Gasparetto, A., Rossi, A., and Trevisani, A., 2003, “Lineariz-zazione di modelli dinamici per meccanismi a membri deformabili,” Proceed-ings 16th AIMETA Conference of Theoretical and Applied Mechanics, Ferrara,Italy, in Italian.

�14� Green, M., Limebeer, D., 1996, Linear Robust Control, Prentice-Hall, Engle-wood Cliffs, NJ, Chap. 9.

�15� Balas, M. J., 1978, “Feedback Control of Flexible Systems,” IEEE Trans.Autom. Control, 23�4�, pp. 673–679.

�16� Kihas, D., and Marquez, H. J., 2004, “Computing the Distance Between aNonlinear Model and Its Linear Approximation: An L2 Approach,” Comput.Chem. Eng., 28�12�, pp. 2659–2666.

�17� Chou, J. H., Chen, S. H., Chang, M. Y., and Pan, A. J., 1997, “Active RobustVibration Control of Flexible Composite Beams With Parameter Perturbation,”Int. J. Mech. Sci., 39�7�, pp. 751–760.

�18� Liao, W. H., Chou, J. H., and Horng, I. R., 2001, “Robust Observer-BasedFrequency-Shaping Optimal Vibration Control of Uncertain Flexible LinkageMechanisms,” Appl. Math. Model., 25�11�, pp. 923–936.

�19� Caracciolo, R., Richiedei, D., Trevisani, A., and Zanotto, V., 2005, “RobustMixed-Norm Position and Vibration Control of Flexible Link Mechanisms,”Mechatronics, 15�7�, pp. 767–791.



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Robust Piecewise-Linear State Observers for Flexible Link Mechanisms