Robust discrete time control Design of robust discrete time controllers

Transcription

1 obust iscrete time control Design of robust iscrete time controllers I.D.Lanau course on robust iscrete time control, part II

2 Outline ole placement tracking an regulation Tracking an regulation with inepenent objectives Internal moel control tracking an regulation ole placement with sensitivity function shaping Design examples I.D.Lanau course on robust iscrete time control, part II

3 Computer control iscrete time controllers ossibilities an avantages Large choice of strategies for controller esign Use of more complex algorithms but with better performance than the ID Techniues well suite for the control of: - systems with elay ea time - systems characterie by high orer ynamic moels - systems with low ampe vibration moes Easy combination of control esign an system ientification I.D.Lanau course on robust iscrete time control, part II 3

4 Digital controllers Design methos ole placement tracking an regulation Tracking an regulation with inepenent objectives Internal moel control tracking an regulation ole placement with sensitivity function shaping emarks: ll the controllers witll have the --T structure two egrees of freeom controller The «memory» number of parameters epens upon the complexity of the moel use for esign ll the esign methos can be wiewe as particular cases of the pole placement The esign an tuning of the controllers reuire the knowlege of a iscrete time moel of the plant I.D.Lanau course on robust iscrete time control, part II 4

5 ole placement The pole placement allows to esign a --T controller for stable or unstable systems without restriction upon the egrees of an polynomials without restrictions upon the plant moel eros stable or unstable It is a metho which oes not simplify the plant moel eros The igital ID can be esigne using pole placement I.D.Lanau course on robust iscrete time control, part II 5

6 tructure pt rt T OCEDE yt lant moel: G H a... a n n n * b b... bn I.D.Lanau course on robust iscrete time control, part II 6

7 I.D.Lanau course on robust iscrete time control, part II 7 T T H... p p yp Close loop T.. r y reference tracking Defines the esire close loop poles Close loop T..p y isturbance rejection Output sensitivity function ole placement

8 I.D.Lanau course on robust iscrete time control, part II 8 Digital control in the presence of isturbances an noise Output sensitivity function p y yp up yb yv Input sensitivity function p u Noise-output sensitivity function b y Input isturbance-output sensitivity function v y ll four sensitivityfunctions shoul be stable! see book pg. - 3 T / / LNT rt ut yt - pt bt isturbance measurement noise vt

9 Choice of esire close loop poles polynomial pecification in continuous time t M, M D Dominant poles Choice of D - ominant poles uxiliary poles uxiliary poles iscrétisation n orer ω, ζ T e.5 ωte.5.7 ζ uxiliary poles are introuce for robustness purposes They usually are selecte to be faster than the ominant poles D I.D.Lanau course on robust iscrete time control, part II 9

10 I.D.Lanau course on robust iscrete time control, part II egulation computation of - an -?? eg n eg n an o not have common factors eout uniue minimal solution for : eg n n n eg n n eg n n *... s s n n n r n r r... *

11 Computation of - an - Euation * is written as: Mx p x M - p T x [, s,..., s, r,..., r n n n ] T p [, p,..., pi,..., pn,,...,] n... a. a a a n a.... a n b' b' b'. b'.. b' n.... b' n n n n n b' i pour i,... ; b' i b i - pour i > Use of Wineg or beout.sci.m for solving * I.D.Lanau course on robust iscrete time control, part II

12 I.D.Lanau course on robust iscrete time control, part II tructure of - an - et inclue pre-specifie fixe parts ex: intégrator ' H ' H H, H, - pre specifie polynomials ' ' '... ' ' ' n n r r r ' ' '... ' ' n n s s The pre specifie filters H an H will allow to impose certain properties of the close loop. They can influence performance an/or robustness

13 I.D.Lanau course on robust iscrete time control, part II 3 arties fixes H, H. Exemples Zero steay state error yp shoul be null at certain freuencies tep isturbance : Harmonic istrubance : e T H ω α α cos ; H ignal blocking up shoul be null at certain freuencies T e H ω β β cos ; Harmonic signal: locking at.5f :, ; n H n H yp H up

14 I.D.Lanau course on robust iscrete time control, part II 4 olving pole placement whith pre-specifie filters in the controller ' H ' H H, H, - pre specifie polynomials ' ' '... ' ' ' n n r r r ' ' '... ' ' n n s s E.* transp. becomes: H H ** eg n n n n n H H ' eg ' n n n H ' eg ' H n n n Use of Wineg or beout.sci.m for solving ** with H, H

15 Tracking reference moel H m pecification in continuous time t M, M Tracking computation of T - r t Ieal case - m m y* t esire trajectory for y t iscrertiation n orer ω, ζ I.D.Lanau course on robust iscrete time control, part II r y T e.5 ωte.5.7 ζ t * H m H m The ieal case can not be attaine elay, plant eros Objective : to approach y*t y * t m m m r t m a m b m m m b a m m

16 Tracking computation of T - uil: y * t m m r t Choice of T - : Imposing unit static gain between y* an y Compensation of regulation ynamics - T - G - G / si si.t. r y: H m m * articular case : m T G si si I.D.Lanau course on robust iscrete time control, part II 6

17 ole placement. Tracking an regulation rt m m y * t T - ut - yt m - m - - * - - * - * u t y t T y * t I.D.Lanau course on robust iscrete time control, part II 7

18 I.D.Lanau course on robust iscrete time control, part II 8 * t y t y T t u * * t y T t y G t y t u * * * t y t u t Gy t u * t r t y m m * m m * * t r t y t y m m... b b m m m... a a m m m ole placement. Control law

19 ole placement. Example lant : m Tracking ynamics --> m Te s, w.5 ra/s,.9 egulation ynamics --> Te s, w.4 ra/s,.9 re-specifications : Integrator *** CONTOL LW *** - ut - yt T - y*t y*t [m-/m-] rt Controller : T Gain margin :.73 hase margin : 65.4 eg Moulus margin : Delay margin:.. s I.D.Lanau course on robust iscrete time control, part II 9

20 ole placement. Example I.D.Lanau course on robust iscrete time control, part II

21 Tracking an regulation with inepenent objectives It is a particular case of pole placement the close loop poles contain the palant eros It is a metho which simplifies the palnt eros llows exact achievement of impose performances llows to esign a --T controller for: stable ou unstables systems without restrictions upon the egrees of the polynomials et without restriction upon the integer elay of the plant moel iscrete tim plant moels with stable eros! Does not tolerate fractional elay >.5 T unstable ero I.D.Lanau course on robust iscrete time control, part II

22 Tracking an regulation with inepenent objectives The moel eros shoul be stable an enough ampe missibility omain for the eros of the iscrete time moel I.D.Lanau course on robust iscrete time control, part II

23 Tracking an regulation with inepenent objectives. tructure y * t ut yt rt m m T m - m - D Desire close loop poles are specifie as for pole placement eference trajectory: tracking y * t m m r t I.D.Lanau course on robust iscrete time control, part II 3

24 I.D.Lanau course on robust iscrete time control, part II 4 Tracking computation of - an - Close loop T.. without T: * * * * H * * One nees to solve : shoul has the form:... * s s s n n fter simplification by *, * becomes: ' * n eg - n ; eg ' - ; eg - n - Uniue solution:... n r n r r s s '... ' ' **

25 x T egulation computation of - an - ** is written as: Mx p n a a a : : a a -... a a a a a a a..... a n [, s,..., s, r, r,..., r ] n x M - p n n T p [, p, p,..., pn, pn,..., pn Use of Wineg or preisol.sci.m for solving ** Insertion of pre specifie parts in an same as for pole placement ] I.D.Lanau course on robust iscrete time control, part II 5

26 I.D.Lanau course on robust iscrete time control, part II 6 Tracking computation of T - Close loop T..: r y T H m m m m Desire T.. It results : T - - Controller euation: * t y t y t u * t y t y t u [ ] * * t y t u t y b t u s b

27 Tracking an regulation with inepenent obnjectives. Examples lant : > m Tracking ynamics > m Te s, w.5 ra/s,.9 egulation ynamics ---> Te s, w.4 ra/s,.9 re-specifications : Integrator *** CONTOL LW *** - ut - yt T - y*t y*t [m -/m -]. rt Controller : T- - Gain margin :.9 hase margin : 65.3 eg Moululus margin : Delay margin :. I.D.Lanau course on robust iscrete time control, part II 7

28 Tracking an regulation with inepenent objectives. Effect of low ampe eros The oscillations on the control input when there are low ampe eros can be reuce by introucing auxiliary poles see book pg I.D.Lanau course on robust iscrete time control, part II 8

29 Tracking an regulation with inepenent objectives. 3 lant : > m Tracking ynamics > m Te s, ω.5 ra/s, ζ.9 egulation ynamics --> Te s, ω.4 ra/s, ζ.9 re-specifications : Intégrator *** CONTOL LW *** - ut - yt T - y*t y*t [m -/m - ].reft Controller : T- - Gain margin:.78 hase margin: 58 eg Moulus margin : Dealay margin :.7 s The elay margin can be improve by aing auxiliary poles I.D.Lanau course on robust iscrete time control, part II 9

30 Tracking an regulation with inepenent objectives. 3 I.D.Lanau course on robust iscrete time control, part II 3

31 Tracking an regulation with inepenent objectives. 3 euction of the ringing Three auxiliary poles at. have been introuce; I.D.Lanau course on robust iscrete time control, part II 3

32 Tracking an regulation with inepenent objectives. 3 Output sensitivity function : effect of auxiliary poles Delay margin without auxiliary poles :.7s Delay margin with auxiliary poles :.9s I.D.Lanau course on robust iscrete time control, part II 3

33 Internal moel control -Tracking an regulation It is a particular caseof the pole placement The ominant poles are those of the plant moel Does not allow to accelerate the close loop reponse llows to esign a --T controller for: well ampe stable systems without restrictions upon the egrees of the polynomial an without restrictions upon the elay of the iscrete time moel The plant moel shoul be stable an well ampe! Often use for the systems featuring a large elay emark: The name is misleaing since it has nothing in common with the internal moel principle I.D.Lanau course on robust iscrete time control, part II 33

34 I.D.Lanau course on robust iscrete time control, part II 34 Dominant poles n α typical choice * shoul has the form: fter elimination of the common factor -,* evient: typical choice olution for: or other cases see book pg egulation computation of - an -

35 Tracking computation of T - T / articular case : m tracking ynamics regulation ynamics T T suppression of the tracking reference moel I.D.Lanau course on robust iscrete time control, part II 35

36 I.D.Lanau course on robust iscrete time control, part II 36 / T H H H H / T H Internal Moel Control H

37 Interpretation of the internal moel control y*t T - Euivalent scheme ut / lant yt The plant moel preiction moel is an element of the control scheme Moel ŷt - eeback on the reicition error for H T em.: or all the strategies one can show the presence of the plant moel in the controller I.D.Lanau course on robust iscrete time control, part II 37

38 I.D.Lanau course on robust iscrete time control, part II 38 * t y t y t u t u H T [ ] * t u t y t y t u * * t u t y t y T t u t y t y t u t u stable asymptotically is Interpretation of the internal moel control

39 I.D.Lanau course on robust iscrete time control, part II 39 IMC ensitivity functions H H yp H H yb H H up The plant moel has to be stable irectly influences the sensitivity functions yp yv H

40 I.D.Lanau course on robust iscrete time control, part II 4 artial IMC Motivation: We woul like to moify the ominant poles of the moel an leave unchange the seconary poles often outsie the attenuation ban D D eout euation pole placement: ut: implifie eout euation: ominant poles

41 I.D.Lanau course on robust iscrete time control, part II 4 lant moel: 7;. - ; - ; T Using IMC with: ; H... yp yb π ω ω ; j yb e π ω ω ; max j yp e Moulus margin is OK bove the template for elay margin over.7f Objective: s M.T.5; τ Conition for the elay margin: Upper template on yb Internal moel control of a system with large elay

42 Internal moel control of a system with large elay Use of auxiliary poles ; H α < α < Delay margin conition: jω < ; e yb Or: ω π α α jω < ; e ω π Worst situation: α α <.5 α.333 I.D.Lanau course on robust iscrete time control, part II 4

43 Internal moel control of a system with large elay lant moel: 7;. - ; - α -< α < a α.;.;.333 The goo value can be obtaine analytically b I.D.Lanau course on robust iscrete time control, part II 43

44 One can also use: Internal moel control of a system with large elay α α n α n n < α.5 <. 5 a b I.D.Lanau course on robust iscrete time control, part II 44

45 I.D.Lanau course on robust iscrete time control, part II 45 Use of: ; H H H H H β β β β β yb π ω β β ω < ; j e Delay margin conition:.5 β β β H Internal moel control of a system with large elay

46 Internal moel control of a system with large elay a b H correspons to the opening of the loop at.5f ee also: I.D. Lanau 995 : obust igital control of systems with time elay the mith preictor revisite Int. J. of Control, v.6,no. pp I.D.Lanau course on robust iscrete time control, part II 46

47 I.D.Lanau course on robust iscrete time control, part II 47 Digital control in the presence of isturbances an noise Output sensitivity function p y yp up yb yv Input sensitivity function p u Noise-output sensitivity function b y Input isturbance-output sensitivity function v y ll four sensitivity functions shoul be stable! see book pg. - 3 T / / LNT rt ut yt - pt bt isturbance measurement noise vt

48 reuency templates on the sur sensitivity functions The robust stability conitions allow to efine freuency templates on the sensitivity functions which guarantee the elay margin an the moulus margin; The templates are essential for esigning a goo controller reuency template on the noise-output sensitivity function yb for τ T reuency template on the output sensitivity function yp for τ T an Μ.5 I.D.Lanau course on robust iscrete time control, part II 48

49 ole placement with sensitivity functions shaping erformance specification for pole placement : Desire ominant poles for the close loop The reference trajectory tracking reference moel Questions: How to take into account the specifications in certain freuency regions? How to guarantee the robustness of the controllers? How to take avantage from the egree of freeom for the maximum number of poles which can be assigne? nswer: haping the sensitivity functions by: - introucing auxiliary poles - introucing filters in the controllers I.D.Lanau course on robust iscrete time control, part II 49

50 I.D.Lanau course on robust iscrete time control, part II 5 ensitivity functions - review yp up ' H ' H D Output sensitivity function: Input sensitivity function: Controller structure : re specifie parts filters Dominant an auxiliary filters: tuy of the properties of the sensitivity functions in the freuency omain: e jω

51 roperties of the sensitivity functions It is funamental to unerstan an to interpret the behaviour of the sensitivity functions in the freuency omain The following slies will explore the properties of the output an input sensitivity functions or the examples one uses: lant moel:.7 ;.3 ; olynomial : Define by the iscretiation of a continuous time n orer system with: ω.4;.6; ; ζ.9 I.D.Lanau course on robust iscrete time control, part II 5

52 roperties of the output sensitivity function.- The moulus of the output sensitivity function at a certain freuency gives the amplification or attenuation factor of the isturbance on the output yp ω < attenuation yp ω > amplification yp ω operation in open loop. M Moulus margin yp jω max I.D.Lanau course on robust iscrete time control, part II 5

53 roperties of the output sensitivity function.3 The open loop KG being stable one has the property:. 5 f j pf/f log e f yp The sum of the areas between the curve of yp an the axis taken with their sign is null Disturbance attenuation in a freuency region implies amplification of the isturbances in other freuency regions! I.D.Lanau course on robust iscrete time control, part II 53

54 roperties of the output sensitivity function ugmenting the attenuation or wiening the attenuation one Higher amplification of isturbances ousie the attenuation one euction of the robustness reuction of the moulus margin I.D.Lanau course on robust iscrete time control, part II 54

55 I.D.Lanau course on robust iscrete time control, part II 55.4 Cancellation of the isturbance effect at a certain freuency: e j j j j j f f e e H e e e / ; π ω ω ω ω ω ω { Zeros of yp llows introuction of eros at esire freuencies roperties of the output sensitivity function

56 I.D.Lanau course on robust iscrete time control, part II at the freuencies where: j yp ω e j j j j j f f e e H e e e / ; * * π ω ω ω ω ω ω roperties of the output sensitivity function llows introuction of eros at esire freuencies

57 roperties of the output sensitivity function.6 symptotically stable auxiliary poles lea in general to the reuction of yp jω in the attenuation ban of / n p.5 p. 5 n n nd In many applications, introuction of high freuency auxiliary poles is enough for assuring the reuire robustness margins I.D.Lanau course on robust iscrete time control, part II 57

58 I.D.Lanau course on robust iscrete time control, part II 58.7 imultaneous introuction of a fixe part H i an of a pair of auxiliary poles i having the form: H i i α α β β resulting from the icretiation of : ω ω ζ ω ω ζ s s s s s en num T s e with: introuces an attenuation at the normalie iscretie freuency: arctan e isc T ω ω with the attenuation: en num M t ζ ζ log en num ζ ζ < an with negligible effect at f << f isc an at f >> f isc roperties of the output sensitivity function

59 roperties of the output sensitivity function or computation etails see book pg Effective computation with the function: filter.sci.m I.D.Lanau course on robust iscrete time control, part II 59

60 Computation of H / for f <.7f or the freuencies below.7f s the computations can be one irectly in iscrete time with a very goo precision. In this case: M ω ω ω ω t / ζ ζ en num isc num en M t : esire attenuation at the freuency ω isc H isc ω, ζ ; isc ω, ζ ; ζ num en en min.3 ω ra / s f.59 f s ω ra / s f.59 f s I.D.Lanau course on robust iscrete time control, part II 6

61 Computation of H / general formula Central freuency for attenuation f isc ω π Desire attenuation at f isc : M t Minimum amping for :. 3 min ζ en isc f isc tep I : computation of the analog filter ω T e ω tan isc t ω π ζ num ζ en isc M / s s s ζ ζ num en ω ω s ω s ω tep II : computation of. the igital filter using the bilinear transformation s Te or etails of formulas see Commane es systèmes Lanau or effective computation use filter.sci.m or ppmaster I.D.Lanau course on robust iscrete time control, part II 6

62 roperties of the input sensitivity function. Cancellation of the isturbance effect on the input at a certain fréuency up : jω jω jω e H e e ; ω π llows introuction of eros at esire freuencies H β < β active at.5f f / f e em: The system operate in open loop at this freuency I.D.Lanau course on robust iscrete time control, part II 6

63 roperties of the input sensitivity function. t the freuencies where: jω jω jω e H e e ; ω π One has: jω jω e yp jω up e jω e f / f e Inverse of the sytem gain Conseuence : strong attenuation of the isturbances shoul be one only in the freuency regions where the system gain is enough large inorer to preserve robusteness an avoi too much stress on the actuator emember: up jω gives the tolerance with respect to aitive uncertainties on the moel high jω weak robustness up I.D.Lanau course on robust iscrete time control, part II 63

64 I.D.Lanau course on robust iscrete time control, part II 64.3 imultaneous introuction of a fixe part H i an of a pair of auxiliary poles i having the form: resulting from the icretiation of : ω ω ζ ω ω ζ s s s s s en num T s e with: introuces an attenuation at the normalie iscretie freuency: arctan e isc T ω ω wih the attenuation: en num M t ζ ζ log en num ζ ζ < an with negligible effect at f << f isc an at f >> f isc H i i α α β β roperties of the input sensitivity function

65 Templates for the output sensitivity functions yp yp yp - M max obustness erformances,5f e yp yp - M max opening the loop,5f e ttenuation one I.D.Lanau course on robust iscrete time control, part II 65

66 Templates for the input sensitivity function up up Limitation of the actuator stress.5f e Zone with uncertainty upon the moeluality Opening the loop at the freuency f < - I.D.Lanau course on robust iscrete time control, part II 66

67 haping the sensitivity functions. Choice of the ominants et auxiliary poles of the close loop. Choice of the fixe part of the controller H an H 3. imultaneous choice of the fixe parts an the auxiliary poles roceure: asic shaping : use an ine shaping: use 3 Tools for sensitivity shaping: Wineg aptech an ppmaster.m There exist also tools for automatic sensitivity function shaping base on convex optimiation Optreg from aptech I.D.Lanau course on robust iscrete time control, part II 67

68 Continuous Continuous steel steel casting casting Level to be controlle haking to prevent aherence I.D.Lanau course on robust iscrete time control, part II 68

69 Continuos steel casting lant integrator: ;.5 ; ; T s Low freuency isturbances inusoial isturbance.5h ut - yt pecifications:. No attenuation of the sinusoial isturbance.5 H. ttenuation ban in low freuencies: à.3 H 3. Disturbance amplification at.7 H: < 3 4. Moulus margin > -6 an Delay margin > T 5. No integrator in the controller I.D.Lanau course on robust iscrete time control, part II 69

70 haping the sensitivity functions H - ynthesis of the fixe parts: H ; Opening the loop at.5 H - Dominant poles: iscretiation of n orer: ω.68 ra/s, ζ.9 Controller : specs.à.7 H are not satisfie see 7 - insertion of a ipole H / centere at ω.44 ra/s Controller : ttenuation ban smaller than specs. - acceleration the ominant poles: ω.9 ra/s Controller C : Correct see 7 I.D.Lanau course on robust iscrete time control, part II 7

71 Output sensitivity functions Continuous casting I.D.Lanau course on robust iscrete time control, part II 7

72 Hot ip galvaniing. Control of the eposite inc teel strip reheat oven ir knife Zinc bath inishe prouct Measurement of eposite mass air air inc steel strip I.D.Lanau course on robust iscrete time control, part II 7

73 Hot ip galvaniing. The control loops Desire eposite inc mass per cm m* T controller ± ressure controller ressurie Ga supply Coating process Measure eposite inc mass per cm Measurement elay m ± m important time elay with respect to process ynamics time elay epens upon the steel strip spee sampling freuency tie to the steel strip spee constant integer elay in iscrete time parameter variations of the process as a function of the type of prouct I.D.Lanau course on robust iscrete time control, part II 73

74 Hot ip galvaniing. Moel an specifications lant moel for a type of prouct : pecifications peformance : Moulus margin: M.5 Delay margin: τ T Integrator 7 b a Moel: sec; b.3; a..3 T ole placement IMC n n n eg n n n 9 H H Case I : ominant pole:. aux. pole:.3 Case II : ominant pole:. aux. poles:.3 7x. I.D.Lanau course on robust iscrete time control, part II 74

75 Hot ip galvaniing output sensitivity Delay margin :.9 < T Delay margin : 5.7 > T T s I.D.Lanau course on robust iscrete time control, part II 75

76 Hot ip galvaniing Nyuist plot ominant pole:. aux. pole:.3 ominant pole:. aux. poles:.3 7x. I.D.Lanau course on robust iscrete time control, part II 76

77 Control of a lexible Transmission The flexible transmission motor axis loa Φ m.c. motor axis position Controller D C ut --T controller osition transucer yt D C Φ ref I.D.Lanau course on robust iscrete time control, part II 77

78 ampling freuency : H Control of a lexible Transmission The moel Vibration moes: ω.949 ra /sec, ζ.4; ω 3.46 ra / sec, ζ.3 pecifications: Tracking: ω.94 ra / sec, ζ.9 Dominant poles: ω.94 ra / sec, ζ.8 obustness margins: M.5 Null static error integrator τ Constraints on up : for f.35 f 7 H up max T I.D.Lanau course on robust iscrete time control, part II 78

79 Controllers for the lexible Transmission Control strategy: ole placement n n n n n,,: n 8 H C : n H 9 H- H- Close loop poles Moulus margin Delay margin s Max up Dominant uxiliary -- ω.94 ζ ω.94 ζ.8 ω 3.46 ζ C -- - ω.94 ζ.8 ω 3.46 ζ I.D.Lanau course on robust iscrete time control, part II 79

80 Control of a lexible Transmission Output sensitivity function Input sensitivity function yp - without auxiliary poles - with auxiliary poles C- with stop ban filter H / up I.D.Lanau course on robust iscrete time control, part II 8

81 Control of a lexible Transmission eal time results - Tracking Transmission souple Transmission souple sortie [V].5 sortie [V] temps [s] T e.5s temps [s] T e.5s commane [V] temps [s] T e.5s Controller with auxiliary poles commane [V] temps [s] T e.5s Controller C with auxiliary poles an opening of the loop at.5f I.D.Lanau course on robust iscrete time control, part II 8

82 Control of a lexible Transmission eal time results - egulation sortie [V] commane [V] Transmission souple temps [s] T.5s e temps [s] T e.5s Controller with auxiliary poles sortie [V] commane [V] Transmission souple temps [s] T e.5s temps [s] T e.5s Controller C with auxiliary poles an opening of the loop at.5f I.D.Lanau course on robust iscrete time control, part II 8

83 36 lexible rm LIGHT OUCE IGID ME MIO LUMINIUM DETECTO TCH. OT. ENCODE LOCL OITION EVO. COMUTE I.D.Lanau course on robust iscrete time control, part II 83

84 36 lexible rm reuency characteristics Ientifie Moel oles-zeros Unstable eros! I.D.Lanau course on robust iscrete time control, part II 84

85 ampling freuency : H Moel Vibration moes: ω.67ra /sec, ζ.8; ω 4.4ra / sec, ζ.5; ω 48.7ra /sec, ζ pecifications: Tracking: ω.673 ra /sec, ζ.9 Dominant poles: ω.673 ra /sec, ζ.8 M.5 obustness margins: Zero steay state error integrator Constraints on up : up up 36 lexible rm τ T 5 for f < 4 H; < 5 for 6.5 f < 8 H; up 4 4 for 4.5 f < 6.5 H; up < for 8 f H I.D.Lanau course on robust iscrete time control, part II 85

86 36 lexible rm - haping the ensitivity unctions Output ensitivity unction - yp Input ensitivity unction - up 5 yp - ensibilité perturbation-sortie 6 up - ensibilité perturbation-entrée 5 5 gabarit Moule [] D C Moule [] - gabarit D C reuence [H] reuence [H] - with auxiliary poles n, an 3r vibration moes 6 - with aititionnal auxiliary poles C- with stop ban filter /.5 H D- with stop ban filter H / I.D.Lanau course on robust iscrete time control, part II 86

87 Controllers for the 36 lexible rm H - H - Close loop poles Dominant uxiliary ω.673 ζ.8 n an 3 r vibration moes ω.673 ζ.8 n an 3 r vibration moes.5-6 C D - - ω 6.8 ζ ω 6.8 ζ.44 - ω 9.57 ζ.9 ω.673 ζ.8 ω.673 ζ.8 I.D.Lanau course on robust iscrete time control, part II n an 3 r vibration moes.5-6 ω 6.8 ζ.8 n an 3 r vibration moes.5-6 ω 6.8 ζ.8 ω 4. ζ.74 87

88 36 lexible rm Input isturbance-ouput sensitivity function - yv amping Can be reuce by augmenting the amping of secon pair of close loop poles freuency of the n vibration moe I.D.Lanau course on robust iscrete time control, part II 88