rejection of fixed direction disturbances in multivariable ... · motion systems electromechanical...
TRANSCRIPT
Mechatronics 20 (2010) 45–52
Contents lists available at ScienceDirect
Mechatronics
journal homepage: www.elsevier .com/ locate/mechatronics
Rejection of fixed direction disturbances in multivariable electromechanicalmotion systems q
Matthijs Boerlage a,*, Rick Middleton b, Maarten Steinbuch a, Bram de Jager a
a Technische Universiteit Eindhoven, Eindhoven 5600 MB, The Netherlandsb Hamilton Institute, The National University of Ireland Maynooth, Co. Kildare, Ireland
a r t i c l e i n f o
Keywords:Multivariable controlDisturbance rejectionDirectionMotion systemsElectromechanical
0957-4158/$ - see front matter Crown Copyright � 2doi:10.1016/j.mechatronics.2009.06.015
I Part of this work was presented at the IFAC World* Corresponding author. Address: Technische U
ktuigbouwkunde, DCT, Eindhoven 5600 MB, The NetheE-mail addresses: [email protected] (M. Boerlage)
(R. Middleton), [email protected] (M. Steinbuch).
a b s t r a c t
This work discusses rejection of disturbances with known directions in multivariable motion systems. Itis shown how frequency domain tradeoffs in multivariable control motivate a design where the direc-tions of disturbance are considered explicitly. A design method is presented to design multivariable cen-tralized controllers to reject disturbances only in relevant directions. A model of an industrial motionsystem is used to demonstrate this method. It is shown how the proposed design method resemblesthe solution of a competing H1 design and offers the ability to interpret H1 centralized control solu-tions understanding the inherent tradeoffs of multivariable feedback control.
Crown Copyright � 2009 Published by Elsevier Ltd. All rights reserved.
1. Introduction and output directions do not vary with frequency, hence decou-
Advanced motion systems are widely used in the high tech sys-tems industry. Examples are active vibration platforms, waferstep-pers, robotic manipulators, and atomic force microscopes, Fig. 1.The behavior of high performance motion systems is typicallydominated by mechanical dynamics. Therefore, mechanical sys-tems are constructed to be light and stiff, with the objective tomove bandwidth limiting phenomena to high frequencies. A com-mon trend is to equip systems with more actuators and sensors inorder to improve dynamical behavior [12]. In other applications, itis required to actuate motion systems in multiple degrees of free-dom [20]. These systems then naturally have multiple inputs andmultiple outputs (MIMO). The mechanics of such systems are typ-ically constructed with the objective to allow each degree of free-dom to be controlled independently. With increasing performancerequirements, multivariable control is a necessity.
Multivariable control design problems pose complexity issuesas gain, phase and directions in different closed loop transfer func-tions are strongly interrelated. Therefore, interpretation and devel-opment of multivariable control strategies is a challengingproblem. In many cases, the structure of the plant can be exploitedto reduce this complexity. In [8,9] it is discussed how MIMO con-trol problems can reduce to SIMO or MISO control problems in cer-tain frequency regions or as an effect of certain bandwidthlimitations. In [13] a class of plants is studied where the input
009 Published by Elsevier Ltd. All r
Congress in Seoul 2008.niversiteit Eindhoven, Wer-rlands. Tel.: +31 40 2474817., [email protected]
pling methods can be applied to reduce the MIMO control problemto a set of SISO control problems. However, in some cases, distur-bance have specific directional structure that can give rise to fullcomplexity MIMO control issues, even when plant dynamics aredecoupled. This was recognized by some authors [7,14, p. 85],but there has been little discussion on how this may be exploitedin multivariable control design.
In many practical applications, the directions of disturbancesare constant (fixed) for all frequencies. Typical examples are dis-turbances that originate from causes that have a fixed location,e.g., floor vibrations, pumps, fans. Also, the control system architec-ture can give rise to fixed direction disturbances, see, e.g., [17]. In[3] a method is discussed to identify the direction and cause ofsuch disturbances. A design that exploits this information for atwo input two output MIMO control system was shown in [2]. Itis of interest to investigate if the structure of fixed direction distur-bances allows the development of insightful design strategies forMIMO systems with more inputs and outputs.
In this work, it is shown that exploiting directions of distur-bances is motivated from design limitations induced by integralrelations such as the Bode sensitivity integral. Using the structureof a class of electromechanical motion systems, a method is pro-posed to design controllers that reject fixed direction disturbancesonly in relevant directions. This method allows frequency wisetradeoffs to be made in each disturbance direction independently,leading to transparent design of a centralized MIMO controller. Themethod is demonstrated on a design problem of an industrial highperformance positioning system. A competing controller is de-signed using H1 synthesis. It is shown that the proposed designmethod can be used to reverse engineer, hence interpret, theMIMO H1 controller.
ights reserved.
Fig. 1. Left: Multi-degree of freedom positioning stage developed by Philips Applied Technologies [1]. Right: Active vibration isolation system developed by IntegratedDynamics Engineering.
46 M. Boerlage et al. / Mechatronics 20 (2010) 45–52
The paper is organized as follows: the next section shows howdesign limitations motivate the study of disturbance directions inmultivariable control design. Section 3 discusses structural prop-erties of electromechanical motion systems that can be exploitedin multivariable control design. The proposed design method toreject disturbances in fixed directions is presented in Section 4.The theory is illustrated by application to an industrial controldesign problem in Section 5. The last section closes withconclusions.
2. Design limitations
A feedback control architecture is considered as depicted inFig. 2. Herein, di; do;n, denote the input disturbance, output distur-bance and sensor noise respectively. Furthermore, we define theperformance variable z, plant input u and the plant output y. Theobjective is to keep z as small as possible in presence of di; do
and n. As G is linear, disturbances at the input of the plant, di,can be considered at the output of the plant as, do ¼ Gdi. It followsthat,
z ¼ GSodi þ Sodo � Ton; ð1Þ
where So ¼ ðI þ GKÞ�1 is the output sensitivity function andTo ¼ GKSo the output complementary sensitivity function. Bothare algebraically related as,
So þ To ¼ I: ð2Þ
The direction of a vector valued signal d 2 D is defined as,
�d ¼ 1kdk2
d: ð3Þ
The direction �d of disturbance d can be interpreted as the normal-ized ratio in which d acts at some point in the feedback system.Let us consider the disturbance do with direction �do at the outputof the plant. And at a single frequency p, the singular value decom-position of SoðpÞ is SoðpÞ ¼ VSoRSoUH
So. Then, zðpÞ ¼ SoðpÞdoðpÞ is smallif the last column of USo is aligned with �do, so that the disturbance isonly amplified by the smallest singular value of So. The disturbanceis rejected completely when SoðpÞdoðpÞ ¼ 0. Then, from the algebraic
z+
do
+
+di
n
- GK u y
Fig. 2. Control system architecture.
relation, (2), it follows that, at the same frequency, sensor noisewith direction �n ¼ �do is amplified with factor 1, as T�do ¼ �do. Thispoints out an inherent design tradeoff due to algebraic relationsin linear feedback control.
In cases where the direction of sensor noise and the direction ofoutput disturbances are orthogonal, (2), does not necessary imply adesign tradeoff. An approach may be to apply high gain feedback toreject disturbances. However, use of high gain feedback for all fre-quencies is restricted due to analytical relations [7]. The role ofdirections in these analytical relations is illustrated with thefollowing.
Consider a stable, non-minimum phase, n� n multivariableoutput sensitivity function, SoðsÞ, resulting from a feedback systemwith stable open loop with relative degree two or more. Then, bypre and post multiplication with vectors v ;u 2 Cn;vHu – 0 the fol-lowing scalar transfer function is obtained,
SvuðsÞ ¼ vHSoðsÞu: ð4Þ
The response to a disturbance entering the system in a directionspanned by u is given by SoðsÞu. Then, vHSoðsÞu is the componentof that disturbance that appears in the output direction spannedby v.
Theorem 1. From [6]. For Svy and u;v defined above, the followingintegral relation applies,
Z 1
0log
SvuðjxÞvHu
��������dx ¼ p
XNz
i¼1
Rezi ð5Þ
with zi; i ¼ f1; . . . ;Nz the closed right half plane zeros of Svu.
Proof 1. As So is stable, the scalar transfer function Svu is stable,and has at most the relative degree of SoðsÞ. Therefore, the residualat high frequencies equals,
limx!1
vHSoðjxÞu ¼ vHu: ð6Þ
See also [5]. Even if So is minimum phase, Svu can have zeros in theclosed right half plane. These zeros can be removed using the Blas-chke product,
BzðsÞ ¼Ynz
i¼1
zi � s�zi þ s
; ð7Þ
where nz is the number of right half plane zeros in Svu , and,
SvuðsÞ ¼ S�vuðsÞBzðsÞ: ð8Þ
Now eSvu does not contain right half plane zeros and because BzðsÞ isall-pass, we have that jSvuðjxÞj ¼ eSvuðjxÞj 8x. Following the same
M. Boerlage et al. / Mechatronics 20 (2010) 45–52 47
reasoning as the derivation of the Bode integral relation for unstableplants in [6] leads us to (5). h
This shows that an area where jSvuðjxÞj is small, must be bal-anced by an equal area where jSvuðjxÞj is large. This is similar tothe tradeoffs induced by integral relations in scalar systems [6].In practical applications, the bandwidth of the feedback system islimited, e.g., by plant uncertainty at high frequencies. Hence, thefrequency range with small jSvuðjxÞj is constrained. Then, Theorem1, shows that rejection of disturbances at other frequencies, wherejSvuðjxÞj must be small, has to be compromised. This may implyperformance limiting tradeoffs. Specific choices of v ;u explainthose tradeoffs for some typical cases in multivariable controldesign.
Corollary 1. When v,u are chosen to be elementary vectors ei,SvuðsÞ ¼ So;iiðsÞ equals the ith diagonal element of the sensitivitytransfer function matrix. As eT
i ei ¼ 1, Theorem 1 reduces to the Bodeintegral relation for scalar systems. Hence, for each scalar loop in amultivariable system, the scalar Bode integral relation must besatisfied, even if a centralized linear time invariant controller isapplied.
+
EG
yu
Fig. 3. Multiplicative perturbation at the output of the plant to model influence offlexible dynamics.
Corollary 2. To study non-diagonal terms of the sensitivity transferfunction matrix, v,u can be chosen to approach orthogonal elementaryvectors, eH
i ej ¼ �;0 < �� 1; SvuðsÞ ¼ eTi SoðsÞej. Then,
lim�!0
logðjeHi ejjÞ ¼ �1: ð9Þ
This shows that a single non-diagonal term of a sensitivity func-tion matrix can be made arbitrarily small, even in cases where theopen loop is non-diagonal. However, as will be shown later, it maybe desired to increase non-diagonal terms of a sensitivity functionfor the sake of disturbance rejection.
Corollary 3. By choosing M ¼ Spanfv1;v2; . . . ;vng with ui ¼ v j 2Rn;vT
i v j ¼ 0 an orthogonal transformation So;mðsÞ ¼ MT SoðsÞM can bemade. As this is a non-singular input output transformation, So;m isminimum phase if So is minimum phase. Then on each new baseTheorem 1 holds with Nz ¼ 0.
This shows that a multivariable sensitivity function can betransformed to any new orthogonal base without increasing thenumber of right half plane zeros. The performance limitations inthe new base will then be the same as the performance limitationsin the original base.
If the direction of a disturbance is constant for all frequencies,and the performance is defined in the direction spanned by z, see(1), rejection of that disturbance at one frequency implies thatthe sensitivity function has to be increased at other frequenciesin that same direction. Hence, as specifications become tighter, itis crucial for design to reject disturbances only at frequenciesand in directions that are relevant. This may imply that centralizedcontrollers (with non-diagonal terms) are to be designed, even incases where the plant is decoupled.
3. Electromechanical motion systems
We first show that the multivariable plants we consider have aspecific structure. Here, we focus on the control of linear timeinvariant electromechanical motion systems that have the samenumber of actuators and sensors as rigid body modes. Typicalapplications are high performance positioning stages used in semi-conductor manufacturing, electron microscopy or componentplacement machines. The dynamics of such systems are often dom-inated by the mechanics, which are therefore constructed to belight and stiff, so that resonance modes due to flexible dynamics
appear only at high frequencies. Typically, flexible dynamics havelow internal damping, so that it is justified to assume proportionaldamping. Then, the following model can be used to describe thedynamics of the plant [10],
GmðsÞ ¼XNrb
i¼1
cibTi
s2 þXN
i¼Nrbþ1
cibTi
s2 þ 2fixisþx2i
ð10Þ
Herein, Nrb denotes the number of rigid body modes. The parame-ters fi;xi are the relative damping and resonance frequency ofthe flexible modes. The vectors ci; bi span the directions of the ithmode shape and are constant for all frequencies. The resonance fre-quencies xi are high, hence the plant can be approximately decou-pled using static input (and/or output) transformations, Tu; Ty
respectively so that,
GyuðsÞ ¼ TyGmðsÞTu ¼ GðsÞ þ GflexðsÞ; GðsÞ ¼ 1ms2 I ð11Þ
where m 2 R1 and GflexðsÞ contains the flexible dynamics of the plantand is often non-diagonal. Systems with rigid body dynamics intranslation and rotation directions can be scaled to a similar struc-tured model as (11). In that case, performance definitions should bescaled accordingly. In many applications, the frequencies anddamping of the resonance modes changes in the lifetime of theplant and are sensitive to changes in operating point. Therefore,inversion of these dynamics often leads to robustness problems.The objective is to control the rigid body behavior of the plant withhigh fidelity. The influence of the flexible dynamics can then mod-eled as a multiplicative perturbation at the output of the plant GðsÞ,
GyuðsÞ ¼ ðI þ EðsÞÞGðsÞ ð12Þ
with EðsÞ ¼ GflexðsÞG�1ðsÞ, see also Fig. 3. With a multivariable con-troller KðsÞ, the plant GðsÞ ¼ 1
ms2 I, and stable sensitivity functionSo ¼ ðI þ GKÞ�1, the closed loop system is stable whenqðEðjxÞToðjxÞÞ < 1 8x [11]. Herein, qð�Þ is the spectral radius,qð�Þ ¼ maxijkið�Þj and To ¼ I � So. The objective is to show the influ-ence of, neglected, flexible dynamics on the achievable performanceof a design based on the rigid body dynamics only. Therefore, wesplit up qðEToÞ using qðEToÞ 6 lðEToÞ and the propertylðEToÞ 6 lToðEÞ�rðToÞ. Herein, l; Toð�Þmeans that the structured sin-gular value, l, is calculated with respect to the structure of To. Thisresults in the following sufficient condition for closed loop stability.
Theorem 2. The MIMO system ðI þ GyuKÞ�1 is closed loop stable ifSo ¼ ðI þ GKÞ�1 is stable, Gflex is stable and if,
�rðToðjxÞÞ < l�1To
EðjxÞÞ; ð13Þ
is satisfied for all frequencies x.
Proof 2. See [15, p. 371]. h
When To is a diagonal transfer function matrix, we have that,�rðToðjxÞÞ ¼maxijTo;iðjxÞj for i ¼ f1; . . . ;ng. Herein, To;i is the com-plementary sensitivity function of the ith individual loop. Then,(13) provides a single bound on all n individual loops. Usually,the structured singular value can be approximated by calculatingan upperbound, see e.g. [14]. When E is unstructured, one mayuse l�1
ToðEÞ ¼ �rðEÞ�1.
For electromechanical systems this implies that a simple struc-tured MIMO model can be used for control design in the low and
48 M. Boerlage et al. / Mechatronics 20 (2010) 45–52
mid frequency region. The influence of flexible modes can be cap-tured in an additional (sufficient) condition for closed loop stabil-ity. The influence of flexible modes, E, will be large at highfrequencies and therefore limit the maximum achievable band-width of the feedback system. This means that the frequency re-gion where To can be large is limited, so that one is forced tomake frequency wise tradeoffs when disturbances are to be re-jected at low frequencies.
4. Design for fixed direction disturbances
A method is presented to design a controller that rejects distur-bances with a fixed direction for the class of electromechanicalmotion systems discussed earlier. We focus on the rejection of out-put disturbances that can be represented by the following model,
doðsÞ ¼ UVdðsÞwðsÞ: ð14Þ
Herein, VdðsÞ is a square diagonal transfer function matrix, wðsÞ an� 1 vector valued unity gain white noise signal and U is a non-sin-gular matrix with constant elements for all frequencies. The matrixU can be obtained using either classical identification techniques,e.g., [19], or more straightforwardly, the blind identification methoddescribed and illustrated in [3,4]. In the following, we restrict our-selves to the case where U is an orthogonal matrix. The theory is notlimited to this, a few remarks on more general choices of U will begiven at the end of this section.
Originally, the plant is defined in the control coordinates. A coor-dinate transformation yd ¼ UT y so that, UT doðsÞ ¼ VdðsÞ, can beused to express the variables in disturbance coordinates. This isequivalent to choosing a controller KðsÞ ¼ UKdðsÞUT , withU 2 Rn�n;UT U ¼ I and KdðsÞ a diagonal matrix with scalar transferfunctions. For the class of plants considered here holds that,GðsÞ ¼ gðsÞI. Hence, the transfer functions at the right hand sideof (1), become,
So ¼ USdoðsÞU
T
To ¼ UgðsÞKdðsÞSdoðsÞU
T ð15ÞSoG ¼ USd
oðsÞgðsÞUT
where SdoðsÞ ¼ ðI þ gðsÞKdðsÞÞ�1, is the (diagonal) sensitivity function
in disturbance coordinates. Therefore, all relevant closed loop trans-fer functions are decoupled in the disturbance coordinates. Thismeans that a disturbance with direction aligned to a column ofUT , is only affected by a single controller on the diagonal of KdðsÞ.Also, by Corollary 3, the frequency domain tradeoff, implied byrejecting such a disturbance, only manifests itself in the diagonalelement of Sd
oðsÞ corresponding to the diagonal element of Kd(s).At the same time, a single element of KdðsÞ can change more ele-ments of SoðsÞ, which is in control coordinates.
Considering the influence of the flexible dynamics, modeled as amultiplicative perturbation EðsÞ, Fig. 3, the transformation impliesthat the multiplicative perturbation in disturbance coordinatesequals, EdðsÞ ¼ UT EðsÞU. As U is orthogonal, lðEdÞ ¼ lðEÞ and thesame upper bound as shown in (13) has to be satisfied for all ele-ments of the diagonal complementary sensitivity function in dis-turbance coordinates, Td
oðsÞ.The same coordinate transformation can be performed with a
general in-vertible matrix U, so that KðsÞ ¼ UKdðsÞU�1. The objec-tive will still be to choose the columns of U to align with the dis-turbance directions. However, the consequence of choosing anon-unitary U is that the bound on Td
oðsÞ induced by E, can be con-servative if the condition number of U increases. This is explainedas, lðUEU�1Þ 6 �rðEÞjðUÞ, where jðUÞ ¼ �rðUÞ=rðUÞ is the conditionnumber. Likewise, specifications as the peak value of the sensitivity
functions are difficult to carry over to disturbance coordinates ifjðUÞ is large.
As the rejection of fixed direction disturbances is studied incoordinates where the disturbances are decoupled, frequency do-main tradeoffs are more transparent. Because the controller andall closed loop transfer functions in disturbance coordinates arediagonal, scalar design techniques, such as manual loopshapingsee e.g. [18], can be facilitated. The resulting controller in controlcoordinates, KðsÞ, will in general be non-diagonal as the inputdirections of the sensitivity function in control coordinates SoðsÞwill be aligned to the disturbance directions (that can be non-canonical). The following example will show how the proposedmethod can be used in a practical design problem.
5. Application example
As a demonstration of the theory, a model of a high precisionpositioning stage is studied. A detailed description of the plantcan be found in [16]. The objective is to regulate the position of thisstage in presence of environmental disturbances, e.g. floor vibra-tions, machine oscillations, pumps, etc. As the location of the originof these disturbance is fixed, the directions of these disturbancesare fixed. Identification of the direction and the origin of such dis-turbances is discussed in [3]. The stage has six degrees of freedom,here only the cartesian axis x; y; z (and interaction in between) arestudied. It is a design requirement that the plant is expressed inthese cartesian coordinates. The Bode magnitude diagram of theGyuðsÞ is shown in Fig. 4. Note that the index of the frequency axisis removed for commercially sensitive reasons, frequencies that arediscussed in this work are indicated by numbers. We will later dis-cuss disturbance rejection at frequencies 1, 2, 3, and the implica-tions of a bandwidth restriction due to a plant resonance atfrequency 4. Following the approach in Section 3, the plant to becontrolled equals,
GðsÞ ¼ 1ms2 I ð16Þ
where m 2 R1, Fig. 5. Flexible dynamics GflexðsÞ, Fig. 5, are consideredunstructured in this case. They limit the achievable bandwidth asthe inverse of the maximum singular value of EðsÞ upper boundsthe allowable complementary sensitivity, see Fig. 9. For robustnesspurposes it is required that �rðSoÞ < 6dB. The rejection of three har-monics in orthogonal directions is studied. Each harmonic is de-scribed by a scalar model VdiðsÞ; i ¼ f1;2;3g, representing theharmonic at frequency 1, 2, 3[�] respectively. The disturbance atthe output of the plant then equals,
doðsÞ ¼ UVdðsÞwðsÞ ð17Þ
where VdðsÞ ¼ diagfVd1ðsÞ;Vd2ðsÞ;Vd3ðsÞg;wðsÞ a n� 1 vector valuedunity gain white noise signal and U is a constant orthogonal matrix,
U ¼ 13
2 �2 11 2 22 1 �2
264
375 ð18Þ
which columns span the directions of the disturbances. In the nextsubsections, three design approaches are considered that are de-signed to have the same level of rejection of these disturbances.
5.1. Multiloop SISO design in control coordinates
This case illustrates how a conventional decentralized controlapproach handles rejection of disturbances in a multivariable sys-tem. As the plant is decoupled, a typical approach is to design amultiloop SISO controller (lead lag, second order lowpass) and ex-tend each SISO controller with a collection of inverted notches
1 2 3 4−250
−200
−150
−100
−50
Freq.[−]
dB
1 2 3 4−250
−200
−150
−100
−50
Freq.[−]
dB
1 2 3 4−250
−200
−150
−100
−50
Freq.[−]
dB
1 2 3 4−250
−200
−150
−100
−50
Freq.[−]
dB
1 2 3 4−250
−200
−150
−100
−50
Freq.[−]dB
1 2 3 4−250
−200
−150
−100
−50
Freq.[−]
dB
1 2 3 4−250
−200
−150
−100
−50
Freq.[−]
dB
1 2 3 4−250
−200
−150
−100
−50
Freq.[−]
dB
1 2 3 4−250
−200
−150
−100
−50
Freq.[−]
dBFig. 4. Bode magnitude diagram of the high precision positioning stage.
12 3 4−250
−200
−150
−100
−50
Freq.[−]
dB
12 3 4−250
−200
−150
−100
−50
Freq.[−]
dB
12 3 4−250
−200
−150
−100
−50
Freq.[−]
dB
12 3 4−250
−200
−150
−100
−50
Freq.[−]
dB
12 3 4−250
−200
−150
−100
−50
Freq.[−]
dB
12 3 4−250
−200
−150
−100
−50
Freq.[−]
dB
12 3 4−250
−200
−150
−100
−50
Freq.[−]
dB
12 3 4−250
−200
−150
−100
−50
Freq.[−]
dB
12 3 4−250
−200
−150
−100
−50
Freq.[−]
dB
GGflex
Fig. 5. Bode magnitude diagram of G and Gflex .
M. Boerlage et al. / Mechatronics 20 (2010) 45–52 49
(bandpass filters) tuned at the components of the disturbances atthe output of the plant, Fig. 6. As in this approach, the directions
of the disturbances are not taken into account explicitly, the distur-bances at each jth output of the plant are then modeled as,
1 2 3 4
130
140
150
Freq.[−]
dB
1 2 3 4
130
140
150
Freq.[−]
dB
1 2 3 4
130
140
150
Freq.[−]
dB
Ksiso,11
Ksiso,22
Ksiso,33
Fig. 6. Bode magnitude diagram of the diagonal terms of the multiloop SISOcontroller (in control coordinates).
50 M. Boerlage et al. / Mechatronics 20 (2010) 45–52
do;jðsÞ ¼X3
i¼1
Uj;iVdiðsÞwðsÞ ð19Þ
where Uj;i denotes the j; ith element of U and wðsÞ is a n� 1 unitygain white noise signal. It turns out to be very difficult to rejectthese disturbances while at the same time satisfying the other de-sign requirements, �rðSoÞ < 6½dB�; �rðToÞ < �rðEÞ�1. Increased attenua-tion of the disturbances is only possible if either the sensitivity peakincreases, or the complementary sensitivity function crosses thebound induced by the flexible dynamics, Fig. 9. This typical limita-tion can be explained as in Fig. 10 it is visible that the sensitivityfunction is reduced in directions that are not relevant for this design
1 2 3 4
80
100
120
140
Freq.[−]
dB
1 2 3
80
100
120
140
Fre
dB
1 2 3 4
80
100
120
140
Freq.[−]
dB
1 2 3
80
100
120
140
Fre
dB
1 2 3 4
80
100
120
140
Freq.[−]
dB
1 2 3
80
100
120
140
Fre
dB
Fig. 7. Bode magnitude diagram of the contr
case. As a consequence of the Bode integral relation, the sensitivityfunction is therefore increased in other frequencies. This design ap-proach does not result in a satisfactory solution for this problem.
5.2. Design in disturbance coordinates
A 3� 3 MIMO controller is designed in disturbance coordinates,following the approach in Section 4. Each element of the (diagonal)controller in disturbance coordinates affects a single (diagonal)term of Sd
oðsÞ, that has to be small at frequencies where VdiðsÞ islarge. A lead lag controller with second order low pass filter is de-signed for each loop. The harmonic disturbance is rejected using asingle inverted notch (band pass) filter, tuned to a single frequency,1, 2, 3 [�], for each loop respectively. The bandwidth is limited by�rðEÞ, Fig. 9, which was shown to be invariant under orthogonalcoordinate transformation. The bode magnitude diagram of thecontroller, KdðsÞ, is depicted in Fig. 8. The controller in control coor-dinates equals UKdðsÞUT and is shown in Fig. 7. In Fig. 10 it is visiblethat each principal gain is small at only one frequency of the dis-turbance. Hence, the sensitivity function is small only in the direc-tions that are relevant for this disturbance model.
5.3. H1 design in control coordinates
The objective is to design a 3 � 3 MIMO controller using H1-synthesis, that takes into account the directions of the distur-bances. The design is formulated in control coordinates. Synthesisof the H1 controller is illustrated using the following mixed sen-sitivity formulation,
minstab:Khinf
WsSoV
WksKSoV
��������1: ð20Þ
Herein, WsðsÞ and WksðsÞ are chosen diagonal, see Fig. 11, using asimilar procedure as in [20], see also [15]. The weight VðsÞ modelsthe disturbances at the output of the plant and equals,
4q.[−]
1 2 3 4
80
100
120
140
Freq.[−]
dB
4q.[−]
1 2 3 4
80
100
120
140
Freq.[−]
dB
4q.[−]
1 2 3 4
80
100
120
140
Freq.[−]
dB
KhinfK
oller in control coordinates, Khinf and K.
1 2 3 4
0
50
100
150
Freq.[−]
dB
1 2 3 4
0
50
100
150
Freq.[−]
dB
1 2 3 4
0
50
100
150
Freq.[−]
dB
1 2 3 4
0
50
100
150
Freq.[−]
dB
1 2 3 4
0
50
100
150
Freq.[−]dB
1 2 3 4
0
50
100
150
Freq.[−]
dB
1 2 3 4
0
50
100
150
Freq.[−]
dB
1 2 3 4
0
50
100
150
Freq.[−]
dB
1 2 3 4
0
50
100
150
Freq.[−]
dB
KhinfK
Fig. 8. Bode magnitude diagram of the controller in disturbance coordinates, Khinf and K.
1 2 3 4−100
−80
−60
−40
−20
0
20
40
Freq. [−]
σ [d
B]
1/σ1(E)ToTo,sisoTo,hinf
Fig. 9. Maximum singular value of complementary sensitivity for H1 design, SISOdesign and design in disturbance coordinates compared to the bound induced byEðjxÞ.
1 2 3 4
−50
−40
−30
−20
−10
0
10
Freq. [−]
σ dB
1 2 3 4
−50
−40
−30
−20
−10
0
10
Freq. [−]
σ dB
Sosiso So
Fig. 10. Principal gains of the output sensitivity function. Multiloop SISO design(left) and design in disturbance coordinates (right).
M. Boerlage et al. / Mechatronics 20 (2010) 45–52 51
VðsÞ ¼ UVdðsÞ; ð21Þ
where VdðsÞ is the diagonal transfer function matrix from (17). Notethat as VðsÞ is non-diagonal, the generalized plant becomes coupled,and there is no reason why the resulting H1 controller should bediagonal. The resulting H1 controller indeed turns out to have largenon-diagonal terms, see Fig. 7. However, if the same H1 controlleris transformed to disturbance coordinates using, UT Khinf ðsÞU, it isvisible, Fig. 8, that H1 synthesis comes up with the same solutionas our manual design in disturbance coordinates, at least in the fre-
quency region of the disturbances. The H1 controller has a differ-ent roll off at higher frequencies. This leads to better roll off ofthe singular values of To, but seems to result in slightly largercross-terms of the complementary sensitivity function at thesefrequencies.
5.4. Discussion
Only the H1 control and the manual control design in distur-bance coordinates are shown to provide feasible solutions to thisdisturbance rejection problem. This is explained as theseapproaches allow the designer to exploit specific directionalproperties of disturbances. Hence, the sensitivity function isreduced only in those directions that are relevant. A conventionalmultiloop design approach was shown to be overly conservative
1 2 3 4−40
−20
0
20
40
60
Freq. [−]
Mag
. [dB
]WsWks
Fig. 11. Weighting filters used in H1 design.
52 M. Boerlage et al. / Mechatronics 20 (2010) 45–52
and hence dictates frequency domain tradeoffs that made it impos-sible to satisfy the design requirements. For comparison, each SISOcontroller in the multiloop approach was designed using H1 -syn-thesis (results not shown here). As these designs still have to facethe same frequency domain tradeoffs, no feasible control designcould be obtained.
The spectral radius equals qðEðjxÞToðxÞÞ ¼ 0:70 for the designin disturbances coordinates, achieved at frequency 4[�], see alsoFig. 9. And q ¼ 0:19;q ¼ 3:13 for the H1 controller and multilooprespectively. This again shows that the multiloop design is not ableto satisfy the design requirements. The H1 controller has 42states. The SISO design must duplicate notches in each loop andtherefore results in a controller with 27 states. The controller de-signed in disturbance coordinates, does not have to repeat notchesin each loop and therefore has as little as 15 states. Aside from thisdifference, the design in disturbance coordinates is more transpar-ent as a controller can be designed for each harmonic disturbanceindependently.
Also, this approach resulted in approximately the same control-ler as the H1 design. A possible, more refined, design strategy is tochoose different bandwidths for the controllers in each disturbancedirection so that robustness margins or roll off can be increased inthose directions.
Although the manual design in disturbance coordinates is onlyapplicable for a limited class of plants, it does illustrate how rejec-tion of fixed direction disturbances can be achieved in a transpar-ent way. It is of great practical value to be able to interpret MIMOcontrollers that result from H1 synthesis in this class of problems.
6. Conclusions
It is discussed and illustrated that rejection of disturbances inonly the relevant directions is important in design problems with
tight specifications. This is motivated by a scalar integral relation,describing the disturbance response per direction. Exploiting thestructure of a class of electromechanical systems, a coordinatetransformation can be applied that allows to transparently, evenmanually, design centralized multivariable controllers that rejectdisturbances only in the relevant directions. If the same problemis approached with H1 design, and the resulting MIMO H1 con-troller is transformed to the same coordinates, the H1 controlleris shown to do exactly the same.
References
[1] Boerlage M, Steinbuch M, Angelis G. Frequency response based multivariablecontrol design for motion systems. In: Conference on control applications,Toronto; 2005. p. 980–5.
[2] Boerlage M, de Jager B, Steinbuch M. Multivariable control design for fixeddirection disturbances. In: American control conference, New York; 2007. p.4649–54.
[3] Boerlage M, de Jager B, Steinbuch M. Blind identification and allocation ofmultivariate disturbances. In: International conference on control applications,Singapore; 2007, p. 509–14.
[4] Boerlage M, de Jager B, Steinbuch M. Control relevant blind identification ofdisturbances with application to a multivariable active vibration isolationplatform. IEEE Trans Contr Syst Technol; inpress.
[5] Chen J. Logarithmic integrals interpolation bounds and performancelimitations in MIMO feedback systems. IEEE Trans Automat Contr2000;45(6):1098–115.
[6] Freudenberg J, Looze D. Right half plane poles and zeros and design tradeoffs infeedback systems. IEEE Trans Automat Contr 1985;30(6):555–65.
[7] Freudenberg J, Looze D. Frequency domain properties of scalar andmultivariable feedback systems. Lecture notes in control and informationsciences. Springer-Verlag; 1988.
[8] Freudenberg J, Middleton R. Design rules for multivariable feedback systems.In: Conference on decision and control, Kobe; 1996. p 1980–85.
[9] Freudenberg J, Middleton R. Properties of single input two output feedbacksystems. Int J Contr 1999;72(16):1446–65.
[10] Gawronski W. Advanced structural dynamics and active control ofstructures. Springer-Verlag; 2004.
[11] Grosdidier P, Morari M. Interaction measures for systems under decentralizedcontrol. Automatica 1986;22(3):309–19.
[12] Huang X, Horowitz R, Li Y. A comparative study of MEMS microactuators foruse in a dual-stage servo with an instrumented suspension. IEEE Trans Mech2006;11(5):524–32.
[13] Hovd M, Braatz R, Skogestad S. SVD controllers for H2-, H1-, l-optimalcontrol. Automatica 1997;33(3):433–9.
[14] Maciejowski J. Multivariable feedback design. Addison-Wesley; 1989.[15] Morari M, Zafiriou E. Robust process control. Prentice-Hall; 1989.[16] Oomen T, van de Wal M, Bosgra O. Design framework for high-performance
optimal sampled-data control with application to a wafer stage. Int J Contr2007;80(6):919–34.
[17] Skogestad S, Morari M. Effect of disturbance directions on closed-loopperformance. Ind Eng Chem Res 1987;26(1):2029–35.
[18] Steinbuch M, Norg M. Advanced motion control: an industrial perspective. EurJ Contr 1998:278–93.
[19] Soderstrom T, Stoica P. System identification. Prentice-Hall; 1989.[20] van de Wal M, van Baars G, Sperling F, Bosgra O. Multivariable H1=l feedback
control design for high-precision wafer stage motion. Contr Eng Pract2002;10(7):739–55.