Performance enhancement on the basis of identified model uncertainty sets with application to a CD mechanism

PERFORMANCE ENHANCEMENT ON THE BASIS OF IDENTIFIED...

14th World Congress ofIFAC

G-2e-07-5

C;opyright If' 1999 IF AC 14th Triennial World Congress~ Beijing, P.R. China

PERFORMANCE ENHANCEMENT ON THE BASIS OF IDENTIFIED MODEL UNCERTAINTY SETS WITH APPLICATION TO A CD MECHANISM

Hans Dotsch +,1 ~ Paul Van den HorS, Okko Bosgra t and Maarten Steinbuch §

Mechanical Engineering Systems and Control Group Dellt University of Technology Mekelweg 2, 2628 CD Deljt, The Netherlands E-mail: [email protected] § Philips Centre for Jtvfan-ufacturing Technology P. O. Box 218, 5600 MD Eindhoven, The Netherlands. :I:

Abstract. We address the design of robust feedback cont.rollers b&~ed on Ilolllinal rnodels and luodcl error bounds that arc obtained from measurement data. In order to keep controller conlplexity liluited an approa.ch is adopted ",rhere controller synthesis is performed on the basis of a low order (approxirnate) Ilorniual rnodel; prior to controller implementation the robust performance of the eontroller is evaluated based 011 a (highly accurate) system uncertainty set employing a dual Youla uncertainty structure. The procedure is based on an iterative scheme of identification and control design and is experimentally verified on the radial tracking control of a Cornpact Disc mechanism. Copyright © 1999 IFAC' Keywords. ldentification, robust systenls.

control~

1. INTRODUCTION Nowadays servo systerns in consumer electronic products must satisfy harsher requirernents rega.rding tracking performance and disturbance rejection at decreasing cost. Application of advanced feedback control offers an opportunity to comply with these requirements. .A. procedure for design of feedback controllers, directed towards an enha.nced tracking perforulance of the ~ervo system; requires the a.vailabilit.y of mathematical rnodels that describe the dynamical properties of the system. A model hO'~iever inherently constitutes an approxima.te 1 The research reported here ha.,; been support.ed by Philips R.esearch Laboratories, Eindhoven 1 The Netherlands. Hans DotRch is no ..,\, with A:\/fP-Holland B,V,) P.G. Box 288, 5201 ~~G '8Hertogpnbo8ch I The ~etherlands,

modelling

errors~

uncertainty, mechanical

system description due to the fact that knowledge of system dynamics is inaccurate or the dynamics is too complex to describe exactly. Hence, a model based control design inherently confronts us "\-vith the issue hoV\r

well a controller that is designed on the basis of an approxilnate model, v/ill do for the physical system. rrhc paradigTIl of robust control de::iign has offered viable tools to take model inaccuracies into account in the design in vicvv of producing a robust controller (Zhou et al., 1996). Instrumental in robust control is the availability of uncertainty rnodels that describe the mismateh bet'Neen a nominal model and the phy~ical plant. In this lvork a. model-based robust control design procedure is pr'oposed and experinlentally verified on the radial actuator of a CUlnpact Disc nlechanism. A full procedure \vill be presented for the following problem:

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PERFORMANCE ENHANCEMENT ON THE BASIS OF IDENTIFIED...

given a plant \vith unknoV\~n dynamics that is controlled by a given controller and that is available for taking experimental data; design a controller on the basis of identified dynalnical Inodel inforruatioll, that leads to an improved performance of the controlled plant. ..4.. driving argument is that a controller should be of linlited cornplexity, yet establishing a high robust, performance. Vle adopt an approach ~~here lo'U) cornJllexity (hence approximate) nominal models are utilized for controllpr synthesis. This is motivated by t\VO arguments . .A. s for many model-based control design approaches the cOluplexity of the resulting controller is directly deterulined by the complexity of the nominal model (and corresponding Vt~eight.ing functions), this approach directly limits the eornplexity of the controller# Secondly, it Call be observed that for many applications a reduced order nonlinal model can provide a high performance modelbased controller, as long as the nOlninal nlodel accurately reflects the control-relevant part of th~ systern d.ynamics. To ensure robust perforruance of a designed controller prior to its implementation, the performance that is achieved for the plant is evaluated utilizing an accurate SyStC111 ullcertaint:y~ set. rI'hc ingredients of the control design procedure hence are~

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Performance evaluation

Fig. 1. Schematic diagram of the several steps in the procedure

2. CONrTROL DESIGN 2.1 Introduction \\re consider a feedback systenl configuration depicted in the block diagram of figure 2. A general feedback

• Identification of a nominal model that is designed to capture the control-relevant dynamics of the plant; • QuantifiL:ation of a rnodel uncertainty set that comprises the mismatch between a norninal rrlodel and the physical plant; • C,ontroller synthesis using nOlninal 1110del informar.ion on the basis of a loop-shaped Hex procedure and quantified model error bounds in order to evaluate robustness properties. The several steps of the procedure are indicated in Figure 1. i\ s an approximate nominal model put~ restric-

tions on a maximum achievable robust performance enhan(:elnent~ we can not achieve our design objective in one synthesis step. \\re have to explore the limitations of the underlying approximate model by conducting synthesis and performance evaluation for a nunlber of prespecified, gradual per-fofInance enhancement objectives.. The identification of approximate models for design of high robust performance controllers has received extensive attention in literature, in particular in the format of iterative identification and control (Gevers~ 1993; Van den Hof and Schrama., 1995) . The approach presented

here is one example of an iterative scheme . The particular deBign of a robust controller is the topic of section 2. In section 3 the identification of nominal models and model error bounds is addressed. The effectivcness of the proposed procedure will be illustr ated in section 4 by an applicat.ion to the optical radial servo system in an industrial eornpaet disc player.

d y

u

+

Ye

F'ig. 2. Block diagraul of a feedback system, consisting of a system Go in configuration with a feedback

controller C. perforrnance objective can be expressed in terrrlS of the closed loop traJlsfer funct.ioIl that rnaps [1"2 'rl]T --t [y u]T

T(C o: C) :=

l~u

]

[1

+ CGo]-l[C

1],

v,,~hi(".h embodies a.ll relevant feedback properties of the controlled plant, among \\-~hich disturbance rejection, track-

ing properties and control effort.

As a measure of performance, we ~rill require that a designed controller complies with performance specifications 2 : (1) 2 In this paper the notation J~1.)l A E
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Copyright 1999 IFAC

ISBN: 0 08 043248 4

PERFORMANCE ENHANCEMENT ON THE BASIS OF IDENTIFIED...

14th World Congress of IFAC

tion Inethod ('fan den Hof et al., 1995)~ \\lith this approach a possibly unsta.ble plant can be identified while the model order can be controlled to be limited. Additionally, a model error expressed in terms of perturbed cop rime faetors have favourable continuity properties in vie,,,- of robustness of the corresponding closed-loop system. Considering the follov.iing system description 3 :

2.2 Controller synthesis Based on a nominal model G of the plant G 01 the controller synthesis is carried out by means of the following \veigherl optiluization: C(;lVb ==

arg c min

11

[W(~b)G]

[1 + CG]-l [6 jW(Vb)

IJ[L(2)

+ Sod

y == GoSor 'tl.

where TV1fb is a filter, designed for a nominal design banchvidth of Vb Hz. The controller synthesis is similar to a H oo loop shape de~ign according to McFarlane and Glover (1990); which is known to have favourable

=== SoT -

(3)

G"YSod

(4)

\\dth r == Tl, So ::= 1/(1 + eGo), an appropriate filter F is constructed a.ccording to F :== l/(D x + C f\lx) , ~~here (Nx , D x ) are normalized coprime factors of an auxiliary lllodel G x that is presumed to be an accurate systerD deseriptioIl. \\7ith this filt.er an auxiliary signal .'1; == Fr is obtained, leading to

robustness properties in vie\~' of model errors structured in a fraetional (coprinH:~ factor) rnodel franlevlork. The \'veighting filter ltl/-JJb is designed to Hhape the loop transfer 1~rl/b G to achieve a prespecified bandwidth~ and additionaL robustness propp.rties are achieved by solving the optimization problem (2). Ho"\vever the achieved perforrnance robustnes~ will be the crucial argulllent in deciding which loop shape filter can be safely applied) so as to achieve a performance that is similar to the designed perfornlance.

y

=::

IVo x

+ Sod

u:::=: Dox -

(5)

CSod

(6)

with (No, Do) ;= (GoSoF- J , SoF- 1 ) being a normalized coprinle factorization of Go~ The two factors No, Do are then identified from experimental data by exciting the external excitation signal r using a least squares prediction error approach (Ljung, 1987). rrhe final plant model is obtained by taking the quotient of the t.'~iO estinlates, G === .l\l/ b. Due to the normalization, the :V1cI\.1illan degree of the final model is equal to the fvfclV1il1an degree of the separate factors. Particular cont.rol-relevancy of the norninal model i~ achieved by applying the identification criterion: 1

2.3 Performance 'robustness analysis ~41though

the synthesis is known to be optimal1y robust in vi€~~ of normalized coprime factor perturbations, there is no guarantee that the synthesized controller actually is robu~tly stable or robustly performing. This has to be checked prior to controller implenlentation. Evaluation of robust performance using an identified model uncert.ainty set~ in this \vork amounts to checking whether the corresponding feedback system magnitudes IT(G o , C)I comply with our specifications (1) ...o\.s there is no exact knowledge available on the plant Go~ this test can not be perforrned directly, bllt it will be performed on the basis of a system uncertainty set P of \vhich Go is considered to be an element:

'riG E

\vhere the weightings

V~ut

and

Vin.

are chosen to satisfY

·P.

For the COIIlffion uncertaint.y sets based on e.g. additive, multiplicative or dual- Youla uncertainty, this test requires the evaluation of a linear fractional tra.nsformation (LFT) of the uncertainty 6, \vhich can be done \-vi thout conservatism. For Inore details see H€ct.ion 3.2.

3. IDENTIFICA'rION 3.1 J'olominal models

. A. nOlnina] IIlodel is identified from closed-loop experimental data usi.ng a so-called coprime factor identifica-

such that the

nOTln

expression in (7) equa.ls

being in accordance ,vith the control design performance criterion (2). In our approach the identificat.ion is performed in the frequency domain, employing periodic excitation signa.ls. 3

\Vithout loss of generality we consider the situation

T2 :=

rJ

O.

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Copyright 1999 IFAC

ISBN: 0 08 043248 4

PERFORMANCE ENHANCEMENT ON THE BASIS OF IDENTIFIED...

14th World Congress of IFAC

3,2 Model uncertainty

~

Additional to the nominal model, a model uncertainty ~et is e::ltirnated to bound the uncertainty that is present in the estimated nominal model. The mixed probabili~­ tic v..·orst~case bounding procedure is adopted from de \lries and \ran den Hof (1995). In this approach a probabilistic setting is taken for the effects of disturhance sig-

I

~----

---------....-------...

----------- t ---------__..

disk

~

~-=--=-C~~~ ---+- ~-M_+_i' ~1 ~~. ~. ~ ·

DC-mot.or photo diodes -------t----------~

nals (variance aspects), while a \vorst-case approach is used for unuermodelling issueH. As (unstruetllred) rnodel uncertainty structure~ use is made of the dual-·Y·oula

--I ~

pararnetrization, leading to an uncerta.inty set:

\ rat:l arm optical piek-up unIt

Fig. 3. Schematic depiction of a swing-arm type Compact Disk scrvo mechanism.

\vherc (1Vx ~ D x ) is a coprimc factorization of an auxiliary rnode.I G~r;, stabilized by the current controller C \vith cOprinl€ factorization (IVc , Dc). For each w, the uncertainty bound ,(w) is determined such that the dualY~oula. factor Rv of t.he plant Go xatisfies the uncertainty bound \vith a prespecified probability o. An attractive property of the particular uncertainty set, i~ that for all G E P

T(G, C) -

T(Gx~

C)

~

frequency range {400, 4000} Hz "vhich may prove disastrollS for tracking perfornlance when not adequately accounted for by the controller..A.n improved tracking performance, expressed in terms of the sensitivity trans~ fer function, is phrased into the performance objective of an enhanced disturba.nce rejection at higher frequencies. In the ini tial situation thp, pla.nt is controll~d by a 4th order controller: achieving l1 ba.ndwidth of around 500 Hz.

4.2 Identification

No ] dR(D x+ CNx)-l[C 1J

[ _D

(8)

\'vhich is an affine expression in ~R. Consequently, a bound OIl the error .6,.R enables a sirnple evaluation of magnitude bounds of TeG, C) for C being the actual controller during experimentation. It also implies that "'Then reducing the uncertainty in !:J..R, the uncertainty in the closcd~loop properties T(G~ C) is directly reduced.

Using the coprime factor identification procedure sketched in section 3.1, an auxiliary model is identified having order 48, and an approximate nominal model is obtained of order 8. The auxiliary model is used as a basis for the construction of a model uncertainty set, according to the procedure utilized in section 3.2. Figure 4 shows the Bode magnitude diagrarns of the error bounds of T(G, C) for the resulting uncertainty set. The magni-

For evaluation of the uncerta.inty in T(G~ C ncu)) for a nevlly designed, not yet implemented controller: the corresponding expressions become a linear fractional transformation of tJ,.R, \vhich can also be calculated with-

out conservatism. This also holds for the calculation of equivalent (open-loop) plant uncertainty, corresponding to Cnew ~ O. For IIlore details see Dotsch (1998). 3

10

)0

4

l02

frequency (Hz)

4. i\PPLICATION TO A

(~OI\f[PACT

3

10

frequency (Hz)

DISC

?vIECHANIS1JI

4.1 Problem· and initial situ.ation The model-based control design procedure presented is applied to the radial scrvo mechanism of a swing-arm type Compact Disc player. A schenlatic depictjon i~ shc)1.vn in figure 3. Characteristic for the dynamical properties of the radial actuator is that it contains a dQuble integrator aud mechanical flexibilities are present in the

-2

10 lO'----2~-~-1~O-3--

frequency (Hz)

Fig.

-l

10 4

10 10 2

lO:~ frequency (Hz)

4. Upper and lo\ver magnitude bounds of {T(G, C), G E P} (solid) and averaged ETFE (dot); frequency range [100, 10.000] Hz.

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Copyright 1999 IFAC

ISBN: 0 08 043248 4

PERFORMANCE ENHANCEMENT ON THE BASIS OF IDENTIFIED...

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tude error bounds of T(G, C) are obt.ained [roIn expression (8) ..A.s the plant model G == (l'lx + DcR)j(D x IVc R) is a linear fractional transformation of R, a dual )'aula error bound can be transforlned into an additive error bound on G. For details \-ve refer to Dotsch (1998). In figure 5 the Bode magnitude diagram is shown of (tra.nsformed) additive error bounds of the radial actuator.

bJl t:::I

E

Id'

:....-:-::~,~._= . -:=:~-:7"~~?!"'

..;r:'

-,

~

IO~

4

10·

10

frequency (Hz)

Q

~-JOO

-a ltI'

l

-200

-"100

-. 10freQuency (Hz)

- t0 2

~

w,l l

~

_ _~ ~ _ ~ [(>.'

Fig. 5. lSpper and lo"ver magnitude error bounds of the plant Go (solid) and a Iueasured frequency response (dot); frequency range [100, 10.000] Hz (left) and [300, 1.500J Hz (right). Note that the additive plant Inodel error bounds are particularly tight. in the frequency region {300, 700} Hz \vhile the corresponding feedback sy-stem error bounds are not specifically tighter in the specified region (Vl here the sensitivity has large rnagnitudc). This observation indicates that error bound identificat.ion elnploying a dual Y~oula model parametrization correspondH to addiLive plant model error bounds that are tight specifically in the frequency region relevant for closed loop dynamics.

Fig. 6. Bode dia.grarn of controllers designed \\rith 3Td order \veighting Hl(Vb) for l/~ == 550 Hz (solid), Vb ~ 600 Hz (dot), Vb == 650 Hz (dash) and Vb :::::: 700 Hz (dash-dot).

700 (figure '7.c and cl) the error bounds indicate a peaking of the sensitivity at 2 kHz and 4 kHz (due to resonance modes of the a.ctuator) that is not ace.ounted for by the approximate model (dashed line). Consequently, these resonance modes should be captured by the nominal model in case we strive for higher design bandwidths. Also it can be observed that the error bounds becolne larger the more the designed (future) bandwidth differs fraIn the achieved (current) bandwidth.

Cb)

(a) I

I I

.;' ~

~

j

L"-'

4~3

Control design

;;=:

J( T08

+1

T08 VfJ

70

== 5'

Wo

1k15

8

2

.

== 5Vb,

f3 == O.4~

Bode diagrams of the designed controllers a.re shown in figure 6. In order to &~sess robust perfornlance of the synthesized controllers; in figure 7 error bounds are evaluated of the sensitivity function magnitudes on the basis of the identified model uncerta.inty set and the newly designed controllers. For the designs corresponding to Vb ~ 650 and

__

Ir~q ....'n~" rib)

Cc)

+ 2/3wQS + w5

"::-L----.~.

w·

III IrL·~,,~n~v 1Hz)

llased on a crude model of the radial act.uator of order 8: four controllers of order 14 are designed employing a third order \veighting 1{l3(vb) that establishes a gradually enhanced disturbanee rejection for a double integrator. The ·'tveighting function is parametrized as:

W 3(Vb)

4

10

J

-.~.1

1(~'

(d)

--~._----

III

,I)

:1<·
Fig. 7. Bode magnitude diagram of the sensitivity function: rohust eontrollers designed with VV· (Vb) for Vb == 550 (a),600 (b), 650 (c)~ 700 (d) Hz in eOIljunction with the nominal model (dash) and the system uncertainty set bounds (solid). To evaluat.e the quality of the error bounds, the pre~ dieted (a priori) perforJuance of the controller C 6,7.50 designed with Vb ;;:::;:750, is compared with frequency re-

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Copyright 1999 IFAC

ISBN: 0 08 043248 4

PERFORMANCE ENHANCEMENT ON THE BASIS OF IDENTIFIED...

sponse est.iruates after ill1plclllcntation of this controller (a posteriori). This is done for both sellsitivity and planttinles-sensitivity, and is shov.rn in figure 8. For frequen10

1 _.'~~-_.~-j

I j -.g .z -2

-1

colO ~

E

-2

-I

10

10 _~~...........c-~-olL-...~

1

4

3

10

10

10

frcqucncv (Hz)

10

2

3

10 freuuency (Hz)

4

10

Fig. 8. G /(1 + CC) (left) and 1/ (1 + CG) (right) vlith controller C G, 750 in the loop: system uncertainty set (solid), identified with controller C 6 ,500 ~ and frequency response estimate (dot).

cics higher than 800 Hz the achieved response is Vt.~it,hin the predicted hounds. lTp to 800 Hz the bounds do not capture the frequency response. From a range of possible causes of this phenomenon, such as e.g. nonlinear plant behaviour ~ it has appeared that -in particular- a mismatch between designed (and presuITlably kno'~ln) controller and implemented controller is likely to be the cause of the mismatch that is found. Ne\v uncertainty sets a.re identified \vith the new control1er C 6 ,750 implemented in the loop. The results for the sensiti~"ity function are displayed in figure 9, showing that the error bounds no\V better capture the frequency response, although not 100% correctly.

F'ig, 9, Upper and lo\\rer nlagnitude bounds of sen~jtivity function, identified v~tith controller Ca 750 (solid)~ and estimated frequency response (dot)'.

5. CONCLUSIONS

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models and a. quantified bound of the rnodel error. Driving aspect is the need for a low comp lexity controller. .Controller synthesis is based on a low order nominal model while robust perforIIlan~e is evaluated based on an accurate system uncertainty set, employing a dual Youla lllodel structure. Conducting synthesis and analysis ,vith a gra.dually increasing performance obj~ctive indicates the utility of the nominal model underlying the design. Critical issue in the procedure follows from experimental irnpleInentation on a COlnpact Disc mechanism: the procedure hinges on the availabilit.y of kIl(r~vl­ edge of the controller in the loop during experiment, ,vhich makes the method sensitive to inaccurate controller knoVvTledge..4 related approach relying on robust control design and application of J.l-analysis procedures for perforlnancc robustness evaluation~ is presented in De Callafon (1998).

RE,FER.ENCES De Callafon~ R,A. (1998). Feedback Orip.nted Identification for E~nhanced and Robust ControL Dr. Dissertation, rvlechan. Engin. Systerns and Control Group, Delft linlv. Technology, October 1998. De Vries D.K. and P.I\1.J. \'an den Hof (1995). Quantification of uncertainty in transfer function estimation: a mixed probabilistic v.:nrst-case approach, ~4utomat­ iCQ, 31, 543-557. D6tsch, J .G.1VL (1998). Identification for Control Design with Application to a C01npact Disk Mechanism. Dr. Dissertation, rvl eeh all. Engin. SystelIls and Control Group, Delft IJniv. Technology} :Vlarc.h 1998. Gevcrs, M. (1993). Towards a joint design of identifica.tion and control? In: H.L. 'Trentelman and J .C. \ViIlems (Eds.) ~ Essays on Control: Perspectives in the Theory and its Applications. Dirkhauser, Boston, pp. 111-151. Ljung, L. (1987). System. Identification.. Theory for the [Is er. Prcnticc-Hall~ Englewood CLiffs, N J. IVlcFarlane) D. and K. Glover (1990). Robust Controller Design 1l,sing Norrnalized C()pTi:rn~ Factor Plant DF;scriptions. Springer \/erlag, Berlin. Van den Hof, P.l\1.J, and R.J,P. Schrama (1995). Identifica.tion and control - closed-loop issues~ A utomatica, 31, 17.51-1770. V'a.n den Hof~ P.M,J., R.J,P. Schrama, R.A, de Callafon and O.H. Bosgra. (1995). Identification of normalized coprime plant factors from closed-loop experimental data. European J. Control, 1, 62-74. Zhou) K.~ J.C. Doyle and K. Glover (1996). Robust and Optimal Control. Prentice-Hall, Engle\vood Cliffs, NJ. i

~t\. procedure has been proposed for an enhanced performan(:e desjgn for an electro mechanical servo systenl by means of feedback control design, employing identified

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