THE UMIST CONTROL SYSTEMDESIGN AND SYNTHESIS SUITES

N. Munro

Control Systems Centre, UM1ST, Manchester, England

AbstrAct. Two suites of Fortran programs, which allow the design andsynthesis of multivariable control systems are discussed. Both suitesoperate in an interactive conve~sational node, and make appropriate use ofgraphic eutput. Powertul data input-output facilities and dalo l!"dnsformationroutines are j:roviced. These are augmented by flexible analysis and automaticsystem simulation programs.

Keyworjs. Computer-aided design; computer-aided synthesis) interactive pr3grams;inverse Iyquist array; characteristic locus; pole assignment; decoupling;cptimal control.

HTRJDUCTIm

The digital computer with graphical displaytermina:s offers a very powerful facIlIty forcontrol system deeign and synthesis stUdies.The well established design techniquesdeveloped by Nyqu:st, Bode, Nichols, andEvans all rely heavily on informationpresented in graphical form, and are :deallysuited for implementation on interact~vecomputing fAcilities with graphical displays.Recent develop~en:s in frequency-domaindesign methods for ~ul:ivariable systems,whicr essentially consist of generalisationsof the "c LessLcel,methods', are also wellsuited to such implementations. Theselatter techniqLes depend entirely on :he useof a digital cemputer to carry out thenecessary calculations, which are toth complexand tedious. The prime results of thesecalculations are presented to the controlsystem desig1er i~ graohical forms, w~iehprovide informati3n about t'1eperformance ofthe system being studied, a~d giue usefulguidance on ow tnis performance can bechanged by t e uee of suita~le compensatoraAnd feedbac action.

In addition to design tools many usefulsynthesis procedures, such as pole assign­men", decoupling. and optimisation algorithms.play an essent'al rart in the achievement ofa desired performance from a syste~. Theseapproaches arose largely from state-spacetheory and usua y require the des:gner tos~ecify exactly what he wents to achieve.Tris, in turn. implies the solution which istren found algorithmically. Although thes~nthesis procedures were originally intendedfor use in a batch-mode environmen:, theirimplementation in an on-line conversationalmode offe~c considerable advantages to thecontrol system designer.

In orde~ to carry out co~pu:er-aided studieson control systems, the control engineerneeds various ether facilities. TheseconsisL of a means cf reedily entering datadefining a system model, or definirg a trialcompensator, into the computer in 6 varietyof forms. This data may then need to bema1ipulated into Jther forms for sUbsequen:analysis, design, and 5imul~tion studie~.It ie also important tJ have available theoutput ~outines necessary for the preserta­tion of frequency-response and time-responsedata 1n a meaningful manner.

A large set of modu:ar overlaid FORTRA.N IVprograms, all operating :n an interactiveconversational mode, have been imp:ementedfor these purposes on a DEC-IO computersituated in the Control Systens Centre atU~IST. These fecilitie5, to be describedbriefly in the remainder of this paper.consist of two separate suites of software.Ore of these provides the frequency-domaindesign techniques, ane the other providesthe syrthesis methods. Certoir facilitiesarc ~ommon to both suites of prcgrams andwill only be described once. However,before examini1g the software suites. thenature of the contro problem giving rise tothe need for these facilities is firstconsidered.

THE NATURE OF THE CONT.'lDL "'ROBLEM

F:g. 1 shOWS t e general nature of thecontro: proble and thus highlights thefacilities needed for the computer-aideJstudy of such systems. As a result of asystem not meeting a set of 'desiredspecifications', a 'control prOblem' exists.Before we can start 0 control study, so~cform of description 0: the system (usuallyin the form of a mathematical odell is

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344 Ii.11unro

required. This model is usually nonlinearand has to be linearlsed ebou t one, or more,operating pOints to obtain a set ~f li,earmodels for the subsequent control studies.

Simulation of the nonlinear system modsl andof the associated linear mOdels providesvaluuble time-domain characterication of thesystem. Analysis of the linear ~odelsprovides the designer with further systemcharacteristics; e.g. order, stability,controllab:llty, observability, structure;which prov~de initia: guidance as to howeffective certa:n desien or synthesis toolswill be.

Once an appropriate design or synthesismethod has been selected, the procedure :0be followed to try to meet the desiredsystem behaviour usually becomes iterativewith respect to the resulting :inear systemcontrol studies. The testing of the:-esultlng control scheme on the ful: non­linear model often involves the examinationof such problems as fine tuning, scheduling,and implementat:on. Again, simulation isrecuired before final consideration can begiven to the implementation of the proposedcortrol scheme on the phy~ical system.

The computing facilities developed at UMISTto examine some of these aspects of a controlsystem stUdy are considered in the remainderof this poper.

THE FREQUENCY-DOMAIN DESIGN SU::TE

This corrpu:er-aided design suite sonSists ofseveral major software packages (see fig.2),which are briefly described in the followingparagraphs. All communication with thesuite is via the user's (graphics) terminal,which activates the various facilit:es bymeans of predefined mnemonic command words.The supervisor program (SUPER) simply actsas an overall communications module I.hichbrings the desired major package into useand passes the appropriate information toits local supervisory program (called an'executive', bu~ not shown explicitly infig.21. The main facilities provided inthis suite consist 0: a flexible data input­output pac~age (I/O), a data manipulationJackage (DATA), a single-input Single-outputsystem cesign pockage (SISO), 0 rrultivoriab e-ystem design package (MmOI, and asimulation package (TI~E). In addition,there is a graphics pac age (GRAF) and a setof data base management routines; the latteraeing inVisible tc the user.

Data IIe

The data input-outpu~ pac age accepts a systemdescription in four different ~orms; namely,measurec f~equency-respanse data, transfer­function descriptions. Rosenbrock's systemmatrix (Rosenbrock, 1970), or state-spaceequattors. Constan: data can be enteredinto matrices or vec~ors either element-oy­element, o~ by :-ow,or by col mn, or in

diagonal form. Polynomial data ~an beentered as a series of polynomial factorsor as a single polynomial, a.g.,(s3+4s2+O.Ools+10.0)(2s2+3.6) would beentered on-line as (s3+4.0s2+1.oE-3s+l0.0)(2.052+3.6), where the round brackets areused to de:lm1t the factors.

All data describing a system is stored incompact fi es on the system disk-store ithe originally entered form. However,polynomial data in factored form isnultiplied out ~nto e single polynomial forin-core usage. Transfer-function descrip­tions including delay-terms and/or commondenom:nators are readily handled. Mostinput errors are trapped by the programs,and it is Virtually ~mpossible for ron:Jomtelatype character errors to causetermination of execution. Data input/output is also provided with respect tJ thesystem dis~-store and other peripherals.In addition, a data nodif1cation facility i~provided which allows individual elements ofdata to be changed without hav:ng to re-enterthe entire data-file for the item concerned.

Data Manipulation (DATA)

The data manipulation package providesautomatic transformation of data from anyone of the above forms to any of the others.This facility makes use of techniques such05 cu~ve fitting (Zaman and Griffin, 1970),system matrix manipulations (Rosenbrock,1970), the Faddeev algori:hm (1963), and amirimal realisation algorithm ~Munro andMcLeod, 1971). Inversion rou:ines for bothcomplex mdlrlces drill l'dtiondl polynomialmatrices (Murro and Zakian, 1970) are alsoprcvidec. These manipulations are primarilyneeded to put the data into the correct formfor any subsequent analysis, deSign, orsimulation required. The local executiveprogram determines the sequence of operationsreouired for a particular transfcrmation.from a 100 -LP table, and brings thenecessary rOLtines intc use automatically.Certain transformations are undefined whentime-delays are present in the systemdescription. (See Fig. 3).

Single-Input Single-Output System DeSign(5150)

Thjs package provldp.s thp.user with the wellestablished linear system design techniquesJf yquist, Bode, ichols ard Evans (seeO'AzzJ and Houpis, 1960). In each case,the user can specify and change the:Jescription of a cascade or feedbac'compelsator or the system, 2S required.The implemantation of the root-locus facilityused) due to Ash and Ash (196B); allowssystems with time-delays to be handled, ":husmaking this app~oach fully compatible wi":hthe other frequency-domain metlods.

ultivariaJle S~stem Design ( IMO)

The multivariable control system design

The UST Control System Design

fa:ilities provided consist of Rosenbrock'sinverse Nyquist array design techni4ue(R:lsGnbroek, 1974; Munro, 1972). andMacFarlane's c~aracteristic locus designtechrique (Bellet,utti and MacFa,lane. 1971;MacFarlane and Kouvaritakis, 1977).

The inverse Nycuist array design techniqueessertially presents an approach to themultivariable contrcl prcblem whereby aninitial compensator is determined whichmaKes tilesys"tem 'diagonal dominant· anere~uces tre design of the originalinteractirg system wiLh m-inpLts And m­outputs to tle design of m sirgle-loop con­trollers. The design is carried cut ir thefrequen:y domain and tenes to prodLce simplecontrol schemes. Plso, engireerirgconstraints are more easily satisfied tranwith some of ha syrt esis techniques. Inthe initial design stages, the polar plotsof all of the elements of tle inverse matrixbeing cunsLdared ere displayed. P.sthedesign ~roce6S continues it becomesnecessary to display only tle polar plotsassociated with tle diagonal elements of theinverse matrix. and finally the polar plotsof sele:ted dlagolal elements. Thecharactaristic-lo~us design tochnique dependson an extension of the concept of eigenvaluesand eigenvectors Jf cOlstant matrices tomatrices of ratiolal polynomial forms. Here,the fre~uency behaviour of the claracteristicvalues snd c,aracteristic vectors of theappropriAte transfp.r-funr.ti~nmatrix Aredisplayed and modified by tle design Jf asuitable multivariable controller. 30th~ode-ty~e plots ald multiple direct Nyquistplots are us3d with this apJrOaCl. In thislatter appr03ch, the concept of dominance isnot reqJired.

Rosenbrock's HIA nathod for continuous systerrshas also been extended to deal with single­rate multivariacle samDled-Data sys~ems(I"Unroand Ibr-ehdn,19741. Here. thscontroller design is carried out in thefictitious-frequency d::>main using thebilinear tr-ens+or-net i on ~I= (z-ll/(z.l); 1na manner analogou~ to tha~ of continuDussystems.

In all cases. the user can Design pre orpost-collpensa~ors ano feedbac, compensators(see Fig. 4). and can readily determiletheir effect on ~he re5~1t.1re syste Jygenerating and exarining ~ e appropr:ategraphic representations.

System Simulation (TIME)

The simu ation paCKage (Munro and Bowland.1972; provides a ,eans of evaluating thetime-responses of 00 h open-loop and :losed­loop configurations of the type consi eredin Fig. 4. Both con~inuous and single-ratesamo:ed-data syste responses "tostepchanges on the system inpu"tscan be oJtainedfor syste~ components descrioed :y "transfer­function aescrip~ion~ s~a"te-space eq a"t10ns.er a mdx t ur-cof bD~r. In t e fomcr ccse,

345

the equations are automatically transformedto a minimal state-space realization (non­minimal, if transport delays arc present).The resultin~ equations are thenautomat:cally assembled for integrationusing a variable step-length Runge-Kutta­Merson algor:thm. The time responses soobtained can be displayed individually orsuperimposed, as rRQuired. Identical andindividual scaling of responses :s providedinteractively. ~he system conf:gurationand definition of the desired sinulationare also set up interactively.

Graphics Packas;e

The graphics facilities implemented a:10""the user te display and graphically editdirect and inverse Nyquist plots, Sodediagrams, Nichols charts, root-locus plots.inverse Nyquist arrays, characteristic lociand rrisalignment plots, and also systemtime-responses. Facilit1es such as originshifting. element magnification andau:orratic scaling are prOVided. Thegraphic facilities are classed as part ofthe I/O package.

THE MULTIVARIABLE SYSTEM SYNTHESIS SUITE

In addi:ion :0 the rrul:ivariable systemfrequency-domain design suite. a set ofsynthesis techniques is provided in the formof Cl~t;parCltefClciliLy. The structure ofthis suits of programs is shown in Fig. 5,where i: can be seen that certain fac:litiessuch as the data input-output routines, datamanipulation facilities, and simulationprograms are common to those developed forthe fr~quency-domain dcsign methods. Again.this suite of pro~rams can be seen toconsist of several rrajorpackages; eachwith its cwn local supervisory progran (or'execu"tlve').all linked oy an overal:supervisor program (SUPER). All directivesentered by a user in the form of mnemoniccommand words are inte,preted by thesupervisory mOdules. which sirrplyactivatethe required facilities. The main newfeatures to be considered in this set ofprograms are tre facilities providRd 'n theA ALYSIS and SY THESIS paCkages. Datainput-out~ut is performed by a flexible(liD) package (as previously described). ando po",erfuldata manipulotion focility (DATA)enables this data to be transformed into anyother desired for. (see Fig. 3) as requiredby tre syrthesis procedure. or analYSisprocedu,e. to te applied. These newfacilities are now briefly described. in thefo Ie ing sub-sec:ions.

Analysis Fackage (AMLYSIS)

This package contains sLch routines as areneee~sary fo~ the calculation of eigcnvalucsand eigenvectors, syste 'zeros'. andsingLla~ values. In addition. systensdescribed by state-space equations can be~ransformed to con"trollacle or observablestancord forms. Tre eigenvalue and

346 N. Munro

eigenvector calculations provided arecarried out us:ng the QR-algorithm andeigenvector routines available in EISPAC(l~78). System zeros are calculatedusing the generalised eigenva:ue approach,or QZ algorithm, as suggested by Laub andMoore (1976). An alternative method fortre determinotion of system zeros due toDavison and Wang (1974), based on eigen­value calculations using the OR-algorithm,is also provided. Although the QZ methodis numerically robust, there are oftendifficulties in interpreting the results,particularly in the case of zeros atirfinity, and :he availability of twoseparate procedures is o:ten useful inresolving such difficulties. The routineused fcr :he determination of the singularvalues of a system is also token fro~ElSPAC (197el and is par"icularly usefulin resolving certain rank questions (seeStrang, 1976) which arise in :he determin­ation cf :he controllability and observab­ility cf 0 5ys~em, and in the determino~ionof the consistency of sets of equations(Strang, 1976) which arise in connectionwith certair pole-assignment algorithms(Munro and ~ovin-Hirbcd, 1978). TheLransfcrmation technicues required tomanipulate a state-space s~stem into itscorrespondirg controllable or obse~vablestandard form are describec in Mun~o (1974;1977) and Munro and Vardulakis (1974).

Although mOEt of the algorithmc provided inthe ANALYSIS package are also used implicitlyby many of the synthesis technicues ~rovided,it is often usefLl to have them availableexplicitly.

Synt1esis Package (SYNTH)

The synthesis procedures implemented to dateconsist of pro:edures for the decoupling cfsystems f~r both the state and output-feed­back cases, procedure~ for pole assignmentusing botl state and output-feedback, andoptimal control for the quadratic cost­function :ase. Several different poleassignment algorithms are implemented, seeFig. 6, a,d provide both the spectralapproach (Munro, 1939) a,d the mappingapproaches (Young ald Levsen, 1970). Bothdyadic anj full-rank minimal jegreecomp2nsator synthesis procedures areprovided ( ovi1-Hir~od, 1978). T1ese allowa variety of pole assignment techniques tobe r9pidly applied to a given problem andafford the designer a choice of the resultingcontrol schemes. The decoupling algorithmsimplenented c01s1st of both t1e stote-feed­back approach. as c3scrioed by Falb andWolovich (1967), anj the freqJency-domainapproach si g output-fe~dbac~ as describedby Wolovich (1975). For systems laving'wed" Lnheren t coupling' (see Gilbert, 1969),the approac sug,ested by Wang (1970) canreadily be applied using the croceduresinplemented.

The optimal control facilities provided at

present cater only for the steady-s:ate, orinfinite settling time, case. The requiredoptimal controller is dcter~ined using theeigenvalue/eigenvector approach suggested bynacFa~lane (1969), once the appropriatestate and input-weighting matrices have beenspecified interactively by the user. Thevarious weighting factors can be re~dilyaltered And the r.orrRsponding controllerobtainec. The resulting feedback systemscar eacr be readily simulated to observe theeffects of trese ~ariations.

In addition to thEse synthe~is procedures,algorithms have also been implemented forthe design of botr Luerberger observers(Munro and Vardulakis, 1974) and Kalmanfilters, which provide good estimates of thestate-vector for those cases W1ere theoutput-feedback approach is not suitable.In the case of the Kalman filter only thesteady-state case is currently available ard1s implemented Jsing the 9pproach sJggestec~y MacFar19ne (lGSGl. Finally, a ~imple1ut flRxible automatic system simu19tionpackage (as previously described) providesfacilities for the open or resJlting closec­loop syste~ responses to Je eX3mined.

OTHER FEATURES JF THE CAD SOFTWARE

All data, ~hether input by the user orgenerated by the design packages, is storedin data files on the system tiis"-st.on".Such a scheme may be likened to using thedisk as a Virtually unlimited dynamic storewhose contents are available to all programs.With such an arrangement it is necessary todefine suitable mnemonics by which the usermay refer to a given set of data, and ul~oto define the filenamefsl under which thatdata is to be found in the user's own disk­area. All :ndividual programs adhere t~th:s mnemonics list.

In addition to the mnemonic names used toidentify system data, a set of mnemonicconmand words is used to obtain the desiredaction from the CAO su:tes :n a conversat­ional mode. These mnemonics can be .mLt:lretiarbitrarily in reply to a program generatedrequest for either a direct:ve or data, atalmost any point in a conversation. Theyare used w en the user wishes to eitherinterrup~ the nornal conversation to branchto sane ot er part 0 the current convercation(or to so~e other part of another conversationor in reply to the q estion "1~HAT 'OW?", whichsignifies t e end of t e current conversation.

T e reply"?", which is 0 reque~t for help,en:ered by the user. causes explanatory tRxtsto be printed out. T is feature isexcremely valua 1e since the possible answersto the various questions generated by theC.A.D. suites form a rather comprehensivelist, which is di:ficult ~o remember. Insome cases. :.10 levels of explanation areprovided: one to prompt a :arriier user, andone to give a rroredctailed account ofavailable facilities.

The UMIST Control System Design

CONCLUDING REMARKS

Many of the algorlthms used throughout thefacilitie~ deccribed above are numericallyweak. This is particularly so of thetechniques used to manipUlate data inpolynomia: form. It would be ofsign1ficant value to the control communitytherefore, if some professionol effort couldbe brought to bear on these problems.

In the developnent of large facilities ofthe type described in this paper, the effortsof mdny people are involved. Thesecontributions are hereby gratefully acknow­ledged, and in part:cular the recent effortsof Dr. S. Novin-Hirbod (1978) in thedeve:opment of the synthesis facil:ties.Acknowledgements are also due to the ScienceRosearch Counc~l for providing the facilitieson which this work has been carried out.

REFERENCES

Ash, R.H. and G.R. Ash (1968). Numericalcomputation of root-loci using theNewton-Raphson technique. IEEE Trans.on Auto. Control, Vol.AC13, pp.576-582.

Belletrutti, J.J .•AnrlA.G.J. MA~FArlanR(1971). Characteristic loci techniques1n multivariable control system design.Froc. lEt, VOl.118. No.9, pp.1291-1297.

Davison, E.J.,and S.H. Wang (1974).Properties and calculation of:ransmission zeros 0: linear multivariab esystems. Automatica, Vol.IO, pp.643-658.

O'Azzo, J.J., and C.H. Houpis (1960).Feedback Control Sys:errAnalysis andSynthesis. McG:-al,-Hill,I e\o,York.

ElSPAC-Release 2 (lS7e). Eigensys:emSubroutine P~ckage, Argonne NotionalLaboratory. Argonne. Illinois. USA.

Faddee~, O.K., ard V.~. Faedeevc (1963).Computational Methods ef Linear Plgebra.Freeman, San Francisco, California.

Falb. F.L., and ~.A. Wolovich (1967).Decoupling ir the design anG synthesisof multivaricble contrel systems.IEEE Trans. cn Auto. Ccntrol, Vol.12(S), pp.651-E59.

Gilbert, E.G. (lS69). The oecocp lIng ofmultivariable systems ty state-feedback.Slam J. Control, Vol.7(1), ~p.SO-63.

Laub. A.J.,cnd B.C. Meore (19761.Calculation ef transmi~sion zeros usirgZ techrique~. Systems Control Report

No.7618, Uni~ersity of Tororto, (ontreScience and Engineering Oepartmert,Toronto. Canada.

MacFarlane, A.G.J. (1969). DLal-systemmethods in dynamical analysis.Proc. lEE, Vol. lIS, ~o.a, pp.1458-l462.

MacFarlane, A.G.J., and B. KOLvaritak1s(1977). A design techniqLe for linearmultivariable feedback systems.Int. J. Control, 25, pp.837-374.

Munro, N. (1969). Computer Aided Design of~lultivariable Control Systems, Ph.O.ThesiS, JMIST, Man:hester, E~gland.

Munro, .• and V. Zakian (1370). Inversionof rational polyno lal matri:es.Electron. Lett., Vol.6, 0.19, pp.629-630.

Munro, ~., and R.S. McLeod (1971). Minimalrealisation of transfer-functionmatrices using the system matrix.Proc.IEE, Vol.118, No.9, pp.1298-1301.

Munro, N. (1972). Design of controll~rs foran open-loop unstable multivariablesystem using the inverse Nyquist array.Proc. lEE, Vol.l19, No.9, pp.1377-1382.

Munro, ., and J .B. BOv/land (1972).Transform techniques and automati:systen simulation. U.K. SimulationCouncil Synposium, UMIST, Manchester,England.

Munro. N. (19741. Furthp.rrRsults on pole­shifting using output feedback.:nt. J. Control, Va],20, No.5, pp.775-786.

Munro, N., and T.A.S. Ibrahim (1974).Computer-aided design of nultivariablesampled-data systems. lEE Int. Conf. anComputer A:ded Design, Southampton,England.

Munro. N., and A.I. Varculakis (1974).Correspondence on conputer-aided designprocedure for reduced-order observers.Froc. lEE, Vol.121, No.4, pp.310-312.

unro. N. (1977). Further results on pole­shifting using output feedback, ControlSystem Design by Pole-Zero Assignment:ed. F. Fallside), Academ:c Press,London, pp.123-l35.

Munro, N., and S. Novin-Hirbod (1979). Poledssignmelll using full- ank output-feed­back compensators. To appear in Int.J.System Science.

Novin-Hirbod, S. (1978). Computer AidedControl System Synthesis, Ph.O. Thesis,UMIST, Manchecter, England.

Rosenbrock, H.H. (1970).11ultivariable Theory.

State-Space andNelson, Landon.

Rosenbroc, H.H. (1974). Computer-AidedControl System Design. Acade ic Press,Landon.

Fig. 1 : Nature of the control problemFig. S Structure of synthesic cuite

G(j(.:j G(t) G(s) pes) ABCD~ .j.

SPECTRAL ..•.. S.S.E .ANALYSIS TRANSFORMATIO S Pole Assi nment

CURVE FAOOEEV StateFITTING ALGORITHM Feedback

S=jw MIrJIf1O.L IREALISATION I

Full Rank. DyadicINVERSION INVERSION

~O"-1- 1-

G(jw) G-l(jw) G(s) G-l(sl ABCO

Fig. 3:Data manipulationsCanonical Canonical

348

Strang, G. (1976). Linear Algebra and ItsApplications, Academic Press, New Yor+, ,

Wang. S.H. (1970). Oesign of precompensatorfor decoupling problem. Electron. Lett.,Vol.S (23), pp.739-74l.

Wolovich, W.A. (1975). Outout-feedbackdecoupling. IEEE Trans. on Auto. Control.Vol.AC20(1), pp.14B-149.

Young, P.C.,and L.O. Levsen (197J). Linearsystems design, Naval Weapons CentreReport NWC TP 4927. China Lake.California.

Zaman, M .• and A.W.J. Griffin (1970).Transfer function from sampled impulseresponse. Measurement and Control,Vol.3. aP.Tlel-TlOa.

- - - +Desired Specificationt

Control Problem~

System Description(Mathematical ~odels) ~ - I

~ II

Simulation~

AnalYSis + - ----+ I

IDeSign/Synthesis - - - -

+-- ..•.IIIIIII'- - -Tuning/Scheduling

~

Control Scheme+

Simulation+

I- - - - - -Implellentation

-I' MO--+I.1K ( s 1 11------->1.1 G(s) 1f---_-+l.1 L s)'__1..---++ 1 I I 1 1 1

'---------ll F (s) 11--------'I I

Fig. 4 Feedback configuration

N. Nunro

DATABASE

DATA DATA SISO MIMO SIMUL-1/0 TRANS DESIGN DESIGN P.TIDN

r I I r I1

SUPER-VISOR

IGRAPHICSTERMINAL

Fig. 2 Structure of frequency-domaindesign suite

DATAB.ASE

DATA DATA ANJl.LYSIS SYNTHESIS SH1JL -I/O TRANS ROUTINES ROUTINES ATION

I T I T I1

SUPER-VISOR

IGRAPHICST::R~lINAL

OutputFeedback

~n-Observer Observer

~Full Dyadic

~Spsctral 11apping

Constant Dyna~ic

Fig. 6 Pole assignment algor:thms