Steen Toffner-Clausen, Palle Andersen and Jakob Stoustrup U96-4153· April 18,2001 4th edition Dep artment of Control Engineering, Institute of Electronic Systems Aalborg University, Fredrik Bajers Vej 7, DK-9220 Aalborg O, Denmark
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Robus t (b This note has been written for a b asic course in robust and optimal control at 9th term of the ystem Construction Line, nstitute of Electronic Sy stems, Aalborg University. Originally the note was intended for a course consisting of six modules of four hours each. Currently, the courses at Aalborg University has been reduced to five modules, and hence, the note cont ains material, that can not be included in the course. The note has been adapted to a level, whid can be expected of 9th term students at the System Construction Line. The students are expected to be acquainted with classical feedback control theory The purposes of the note is to provide an introduction to mo dern robust and optimal control especially in Hoo and i theory. n Chapter 1, a short introduction to the concept of robust control is given; m Chapter 2 nominal and robust stability for single variable(SoSO)systems is described. n Chapter 3 nominal and robust performance for sso sy stems is analyzed, and the concepts Hoo and H2 optimal control are introduced. Chapter 4 gives an introduction to the analysis of multi variable systems, and in Chapter 5, st ability and performance of multi variable systems are studied. n Chapter 6, a solution to the Hoo control problem presented. Finally, in Chapter 7, an introduction to the structured singular value I is given, and controller design with i is treated. Robust optimal control; robust st ability; robust performance; Hoo optimal control; I ana I synthes afigution of thb authors The authors are with Grundfos A/S, and with the Department of Control Engineering, an tute of Electronic Systems, Aalborg University, DK-9220 Aalborg o, Denmark. The depart lenthasthefollowinghomepagehttp://www.conTroL.auc.dk.Theemailsofthelasttwo authors are: ipa, js]@control. auc.dk
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Contents 1 Robust feedback Conti 1. 1 Feedback and Uncertainty 1. 2 Classical Com pensator De sign Metho ds 1. 3 Mo del Uncertainty. 1. 4 Feedback Systems with Model Uncertainties 1.5 Robust Control Nominal and Robust Stability 1357799 2.1 A Model of the proces 2.2 Mo del Uncertainty 2.3 Nominal Stability 2.4 Robust Stability. 3 Nominal and robust performance 13 3.1 Signal No 3.2 Norms for Systems and Transfer Functions 3.3 Specification of inputs 3.4 Requirements for Perform ance 3.5 H2 Optimal Control(LQ) 3.6H。 Optim al 3.7 Robust Perform an ce 3.8 H2 Robust Performance 3.9 Hoo Robust P n ce 3.10 Loop Shaping 4 An Introduction to Multivariable Systems 26 4.1 Poles and zeros of multi ariable Sy stems 4.1.1 Smith-McMillan Form of a Transfer matrix 4.2 Nominal Stability for Multivariable Systems 4.2.1 Internal Stability 4.2.2 The Generalized Ny quist Theorem 4.3 Frequency Responses for Multiv ariable Systems 3 4.3. 2 Induced Norms 4.3.3 Singular values
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5 Robustness Analysis for Multi ble Systems 5. 1 Nominal perfo 44 5. 1.1 Norms of Signals for Multivariable Sy stem 45 5.1 5.2 Robust Stability 5. 2.1 The Sm all Gain Theorem 5.3 Robust performan 5.3. 1 Specifications with Mixed Sensitivity Functions 5.3.2 The Significance of Zeros and Poles in the Right Half Plane Robust Design for Multivariable Systems 6.1 Loop Shaping 6.2 Mo deling Individual Channels 6.3 H Control 6.3. 1 Remarks to the Hoo solution 6.3.2 The MAtLaB Toolboxes 7 7 Design of Robust Comp ensat ors based on A Theory 7.1 7.1.1 Robust Stability 7.1 7. 1.3 Computing p esIs 7.2.1 omplex u Sy nthesis - D-K iter ation
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Chapter 1 Robust feedback Control Chapl er n conl ains a general presenl al ion of classical feedback conl rol, wilh an emphasis on robusl ness l o mo del uncerl ainl y of feedb ack sysl ems The need lo develop design mel hods, hal explicil ly can handle mo del uncerl ainl y is demons ral ed 1.1 Feedback and Uncertainty Conl rol of a dy namical process by feedb ack of a measured oul pul is a well(known principle pically wil h i he primary ob jeclive of keeping I he oul pul of I he process close lo a given e(s K(s) G(s Standard feedback configuration Fh Figure nn, a sl andard feedback configur al ion is shown, consis ing of a process G(s)and a compensa or K(s). For sy sl ems wil h feedb ack I he following imp orl anl properl ies can ofl en oys ems I hal are unsl able from nal ure, can be sl abilized fecl of exl ernal disl t The I ime cons anl s or chara eris ic frequencies of I he sysl em can be shifl ed Finally, ofl en Ihe above properlies can be obl ained even wil h an in complel e knowledge
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Robu r tcl r Hedke, fee Fackcldi Ga f sy stems have attiactive pi deities, that cad be maFe i bust with sthe caie The afad ages (f fee Fback cad be imstiateF by cldsi feiicg the udkeit aid ermeds id a pildess. Theie aie tw Uieaslds why, ad ttput fith a pildess is d completer. kdud a piii. Fiist, the Ey damicanpi deities tithe sy stem ale dltediier kdld i.e., a m tencad be cldsi FeieF Icy as ad appi timate Fesciiptild tf a physicanpildess. TecldE, udkdG Fstuibades cadicHuedke the piIdess.. d ttput as dt td. a fudcti ld U the iput cIdiTefby the cthpedat Ti but arLa ist uib ade Cwhich Uted caddO be measuiee The basic pieities lf fee lack(cllseFlllp cldil is exp lse Fby a clmpalisld with a fee FlwaiFcICfiguiati ld tpedillp cldill By tpedllp(OL)cldiGithe iesul FepedF Gmpretely Id the acculacy by which the pildess has beedm ufere side the clmpedsat ti Feteimides thq iput base F ldy. Id the m UfenadFthe iefelede. Deviati lds cause Fby the Estuibade CldFm UEenudkeitaidies have a fulrimpact as a Rsciepadoy betweedthe actual adFthe expecte F lutput Ui Figule 1y Feecforui rd configur vion with the teteieckis s adle taik: c s the at减ia: te dthe utpt ii Gii kdl rege availabn ld the sy stem id teims l a mathematicanm UFen the fee Fb ack clhpedsat dexpllits the kdwreFge fthe actuarbehavi li (fthe pildess adFthe simuHadelus Fistuibades, which aie imp lti E. givedby the measuiemed aHedke, it isp [ Rt UieFuce the effect T the Ist uib ade Cs welnas the effect ITthe impeifect m felid Desigd ti fee F ack cthpedsat lis emb aiks fith iequiiemeds cldceididg the st atic adF Ey damicanbehaviGi lf the cldilef'system. These iequiiemeds cllFidknfe the flwidg stability: the lpeiati ldp lid (fthe cldila f systems must be stab y. symp lic cldiurU a givedcBss I iefeiede idputs sChdF Istuib ades c'thhe eitI lust tedFt Uzei Uas the time GedES t ucficty 3. Dy damicalmequiiemeds the cldila f system must fulfil set t specificati[ds,sud as b ludes [dthe step lesp ldse, adFiequiiemeds t ls the Fegiee lf Fec lup lig betweed the vaii lis siguan, et 4. Requiiemeds Iditbust dess: The pileities that aie specifieFrtithe cldiGefsysten ab le, must be pieseive Fudfei a gived class (fi vaiiatilds idthe pildess ey damic. Idpildess c ldilnthe iequiiemeds 3 aie tftedftimulate F as specificatilds fli Output acdFfUthe step wiiamp shape Fvaiiati lds id iefeiedce t RAS GHe aNear
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disturbance. The specifications, e.g. can be bounds on overshoot Mp, rise time tRand settling ee Fi The requirements can also be formulated in the frequency domain as specifications for the open loop or closed loop transfer function, for example as conditions on a closed loop resonance ak MRor bandwidth fM see Figure V4 y +1% 1.0 0.9 Figure V3: Time domain des ign specifications: conditions on Mp, trnd tofor outp ut y by a al 0 f Figure V4: Fr cy 1.2 CI SSicL I COB Prns toA DrSign Mrthods A majority of the methods, that are applied to the design of compensators, presume an exact model of the process with an emphasis on items Vf3 1. Fecb a. cEk n
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Stability analysis in the frequency plane was developed by Ny quist and Bode. Design of the compensators were carried out by methods based on'trial and error Design rules such as Ziegler and Nichols are based on simple models, but they have succesfully been applied to pro cesses with a higher complexity. Root locus analysis was developed by W.R. Evans, and aims at obtaining prescribed dynam ical properties by determining a compensator, which provides a satisfactory closed loop pole placement Moreover,methods have been developed that provide explicit formulae for a compensator Methods, that aim at satisfying performance specifications formulated as conditions on the der. gtal of the square error, have been introduced by Newton, Gould Kaiser and further al Even though one of the ob jectives of feedb ack control is to to reduce the effects of model deviations, all these methods presume a perfect model, and only indirectly accounts for the fact, that a model can never be perfect Since several of the met hods have been applied succesfully for many years due to the inherent robustness of feedb ack compensators, it can be questioned whether it is reasonable to include model variations as an explicit design condition. There are, however, many examples where the introduction of feedback not automatically imply the robustness required to actual model riations. The following example originates from Lun891 Example 1.1(Robust Stability rocess is described by the model G(s)=1+718 which experimentally has been shown to prouide app rorimately the same step response as the real process. The process can, e.g., be controlled by a proportional compensator with the gain should be possible wi thout problems to select the gain K arbitrarily large It turns out in reality, however, that a more accurate model of the process is given by (1+718)(1+728) Now, it is apparent that the control system will become uns table, if k is chosen too large. To ensure stability, K must be chosen below the bound If for example To T2=0.1- T1, K must satisfy K< ll to ensure stability lead to a model that approximately describe the process, such a model can not be used for compensator design regie of validity of the model
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age 5 of 92 The example further illustrate the main problem in the design of compensators for sy st with model uncert aunties: the desire of good performance lead to a need for high gains is in contrast to robustness, since model deviations easily can lead to inst ability in sy stems with high gains The fact that practical compensators have been designed in the past, is due to experien ced engineers who design compensators with a ce conservatism, by not taking the gains to the theoretical limit. For comp ensators that ived by izing a cost function, some robustness can be achieved by incorporating signal increments in the cost function The development of methods that directly in corporate model uncertainties in compensator design, has accelerated within the past 20 years 1. 3 MRoae ui caxsmti s. Normally, the first step in a compensator design would be the derivation of a model of the pro cess to be controlled. Design of comp ensators b ased on the model derived must take into consideration under which conditions the model is valid. If certainty is required that the compensator is going to work under all conditions, it is necessary to augment the model of the pro cess with a model, which expresses the possible deviations from the nominal model There are to reasons why, the output of the pro cess can not be predicted exactly by the derived model of the preocess. The pro cess can be influenced by disturb ances, and the dynamics of the model can deviate from that of the model, see Figure 1.5 xternal signals, th dep en dent of the disturbances can be aggregated at the output of the process as an exogenous input d, added Deviations between the dynamics of the process and its model lead to discrep andies that in contrast to disturb ances are not caused by unknown disturbances, but highly depend on the input. Hence, mo del un certainties can be represented by an error model with input z and output w, see Figure 1.5b Rejection of dist urb ances is an integral part of compensator design related to items 1-3. Thus, in this context model uncert aunties are emphasized. Th e three main sources for model ledge on the process; this type of u y might be due to the fact, that the model has been derived from the laws of physics, although the exact parameters of the process can not be determined from the available knowledge on the process If e model is determined experiment ally, the accuracy of the model depends on whether the process has been excited by input s, that are suited for determining a model, and to what extent the pro cess has been influenced by dist urb ances during the experiment 2. Model simplification: even though the original sy stem might be known in great det ail he mo del might have been reduced in order to simplify the design task 3. Incomplete model structure: in general it is desirable to design compensators b ased on del. Hence, nonlinearities in actuators or sensors must be omitted. Other types of nonlinearities is caused by nonlinear dynamics of the process it self. This type of nonlinearities often result in par ameters that depend on the operating point Rtanarf tert c
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sn 68-t2 U4k4ow 4 K4ow 4 pad I(al x(akh e(alk mak (ak (ak U4k4ow 4 K4ow 4 (ak eaR Figure 1.5: Disturb- ces Us. model ul cert-i ty i. the clos ed loop system. -)Closed loop system exposed to disturb- ces. b) Closed loop system with model u. cert-i. ty 4. 5ime varyi4g parame gers: also resul9 i4 variagio4 of ghe parame gers of she mode f4 a4y circums9a4 ce, ghe model will o4ly be a4 approximate represe4gagio4 of ghe physical aysgems wigh sig4ifica49 model u4 cer Cai4 gies are simply refered 9o as u. systems irref specgive of whether ghe u4 cerga14 gies are caused by lack of i4 formagio4, model simplificagio4 S 404li4eariGies, or gime varyi4g parameters f4 ghe ligerasure, various co4 cep 9s are associated wigh special gypes of model u4 cer9ai4gies P-r-metric u cert-i ty is ghe gpe of u4 cergai49y Shag ca4 be compe4saed for by adjusgi4g ghe parame gers of ghe model. Structur- a cert-A ty is ghe gype of u4 cer gai49y Cha9 relase So a4 14 complete or 14 correc9 mo del sgrudgure, e. g. by applyi4g a li4ear mo del for process gha9 exhibi 9s 404li4ear beh by omi9G14g dy 4 amical eleme49 i4 ghe process. Moreove model u4 cergai49y is classified sub jec go whe gher o4ly ghe overall level of u4 cergai49y is k4ow 4, or whe gher also ghe character of ghe u4 cer 9 is k4ow 4. 5o ghis e4d structured a cert-i. ty is dissi4 guished from u- structured w- cert-i. t by defi414g a se 9 of possible of models, each member ca4 represe49 ghe origi4al process, bug ig is u4k4ow4 which specific member gha9 actually does. Depe4de49 o4 ghe character of ghe model u4cer9ai49, ghe se9 of possible models appear 14 differe49 ways. By mappi4g all possible models i4 ghe g yquis9 pla4e, a bou4ded regio 14 ghe plale is associated wigh each freque4 cy rasher gha4 a4 isolated po
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