Chapter III Robust Digital Controller Design Methods

Transcription

1 Chapter III obut Digital Controller Deign Metho I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3

2 Chapter 3. obut Digital Controller Deign Metho 3. Introuction 3. Digital ID Controller 3.. tructure of the Digital ID Controller 3.. Deign of the Digital ID Controller 3..3 Digital ID Controller: Example 3..4 Digital ID Controller 3..5 Effect of uxiliary ole 3..6 Digital ID Controller: Concluion 3.3 ole lacement 3.3. tructure 3.3. Choice of the Cloe Loop ole egulation Computation of - an racking Computation of ole lacement: Example 3.4 racking an egulation with Inepenent Objective 3.4. tructure 3.4. egulation Computation of - an racking Computation of racking an egulation with Inepenent Objective: Example I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3

3 Chapter 3. obut Digital Controller Deign Metho 3.5 Internal Moel Control racking an egulation 3.5. egulation 3.5. racking n Interpretation of the Internal Moel Control he enitivity Function artial Internal Moel Control racking an egulation Internal Moel Control for lant Moel with table Zero Example: Control of ytem with ime Delay 3.6 ole lacement with enitivity Function haping 3.6. ropertie of the Output enitivity Function 3.6. ropertie of the Input enitivity Function Definition of the emplate for the enitivity Function haping of the enitivity Function haping of the enitivity Function: Example haping of the enitivity Function: Example 3.7 Concluing emark 3.8 Note an eference I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 3

4 Computer control icrete-time controller oibilitie an avantage Large choice of trategie for controller eign Ue of more complex algorithm but with better performance than the ID echniue well uite for the control of: - ytem with elay ea time - ytem characterize by high orer ynamic moel - ytem with low ampe vibration moe Eay combination of control eign an ytem ientification I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 4

5 Digital controller Deign metho Digital ID controller ole placement tracking an regulation racking an regulation with inepenent objective Internal moel control tracking an regulation ole placement with enitivity function haping emark: ll the controller will have the tructure two egree of freeom controller he «memory» number of parameter epen upon the complexity of the moel ue for eign ll the eign metho can be viewe a particular cae of the pole placement he eign an tuning of the controller reuire the knowlege of a icrete time moel of the plant I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 5

6 Digital ID controller It reult from the icretization of an analog ID controller he computation can be rigorouly applie only to: - plant ecribe by a moel whoe orer i n - plant with a time elay maller than he algorithm for the parameter calculation i a particular cae of the pole placement I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 6

7 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 7 Digital ID controller nalog ID : K proportional gain, i integral action erivative action /N - filtering on the erivative action N K H i ID ; / Dicretization: Digital ID controller : N N N K H i ID

8 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 8 Digital ID controller H ID r r r ' N K r i N K r N K r i N emark: he igital ID controller ha 4 parameter a the analog ID Common factor in the enominator: integrator filtering action: factor in the enominator '

9 Digital ID controller rt ut yt rt ut yt - / LN / - / / LN.F. of the cloe loop r y tructure with H CL efine the cloe loop pole he controller introuce upplementary zero I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 9

10 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 Dicrete-time moel of the plant Z.O.H. H z - H Ge H τ e H ζω ω ω τ or < τ a a b b H he icrete-time moel i obtaine: irectly by ytem ientification general cae by icretization of the continuou-time moel

11 arameter computation of the igital ID controller erformance pecification : H CL M * M cannot be impoe a i kept an he controller introuce upplementary zero he characteritic polynomial of the cloe loop i pecifie : Continuou-time pecification t M, M p' p' icretization n orer ω, ζ.5 ω.5.7 ζ I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3

12 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 arameter computation of the igital ID controller / - Known or ientifie plant moel : - Deire performance CL pole: ; o be compute : From * lie, one olve:?? r r r b b a a p p 3 3 a a a ool : WinEG, bezout.ci.m

13 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 3 Choice of the polynomial - Fourth orer olynomial Euation. - can alo be choen a a fourth orer poly. by aing aux. pole p p p p p p α α j.5,.5 α α x x x x Dominant pole n orer ω, ζ auxiliary pole - he auxiliary pole improve the cloe loop robutne

14 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 4 Euivalent analog ID controller parameter r r r K r r r K i N 3 K r r r he continuou euivalent oe not alway exit! ' Exitence conition: /N > Digital ID controller alway can be implemente even if : no euivalent achievable performance with an analog I '

15 Digital ID controller. Example lant: H τ Ge Dicretize plant: , G,, τ 3 erformance 5, ω.5 ra/, ζ.8 *** CONOL LW *** -. ut -. yt -. rt Controller : Gain Margin : 7.7 hae Margin: 67. eg Moulu Margin : Delay Margin : 45.4 nalog ID : k -.73, i -.735, -., /N. I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 5

16 erformance: ω.5 ra/, ζ.8 lant Output ime t/.5 Control ignal ime t/ Cloe Loop repone lower than Open Loop repone. he pecifie ω houl be increae I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 6

17 erformance: ω.5 ra/, ζ.8 Dicretize plant : , G,, τ 3 erformance 5, ω.5 ra/, ζ.8 *** CONOL LW *** -. ut -. yt -. rt Controller : Gain Margin : 3.68 hae Margin : 58.4 eg Moulu Margin : Delay Margin : 9.4 nalog ID : no euivalent analog ID No euivalent analog ID a >.3 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 7

18 erformance: ω.5 ra/, ζ.8 lant Output ime t/ Control ignal ime t/ - Fater repone - n overhoot appear becaue of the zero introuce by I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 8

19 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 9 he «goo» igital ID controller ID No upplementary zero i introuce H CL Deire.F. for the cloe loop: H CL ] / [ H CL an are unchange Only one coefficient intea of two coeff.

20 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 Continuou time ID correponing to igital ID k - - N - - LN K - K K i rt ut yt / N he proportional an erivative action only act on the meaure N r r r r r r r r r i r r K

21 erformance of the igital ID controller Dicretize plant : , G,, τ 3 erformance 5, ω.5 ra/, ζ.8 *** CONOL LW *** - ut - yt - rt Controller : Gain Margin : 3.68 hae Margin : 58.4 eg Moulu Margin : Delay Margin : 9.4 nalog ID : no euivalent analog ID No euivalent analog I a >.3 o be compare with ID, lie 7 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3

22 erformance of the igital ID controller ω.5 ra/, ζ.8 lant Output ime t/.5 Control ignal ime t/ euce overhoot correponing to ζ.8. ame repone for iturbance rejection o be compare with lie 8 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3

23 uxiliary pole effect he auxiliary pole reuce the input enitivity function up at high freuencie without egraing the cloe loop performance etter robutne an reuction of actuator tre I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 3

24 Digital ID controller : concluion Canonical tructure Euivalent analog ID if ' Ue with t or n orer ytem with elay < For a elay τ.5 the analog ID lea to cloe loop repone lower than open loop repone he igital ID controller give better performance for ytem with elay but there i no euivalent in continuou-time he igital ID controller lea to a tep repone with a maller overhoot than ID I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 4

25 ole placement he pole placement allow to eign a -- controller for table or untable ytem without retriction upon the egree of an polynomial without retriction upon the plant moel zero table or untable It i a metho that oe not implify the plant moel zero he igital ID can be eigne uing pole placement I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 5

26 tructure pt rt LN yt lant: H a... a n n n * b b... bn I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 6

27 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 7 ole placement Cloe loop.f. r y reference tracking H F... p p Define the eire cloe loop pole Cloe loop.f. p y iturbance rejection yp Output enitivity function

28 Choice of eire cloe loop pole polynomial D F Dominant pole Choice of D - ominant pole pecification in continuou time t M, M uxiliary pole uxiliary pole icretization n orer ω, ζ e.5 ωe.5.7 ζ uxiliary pole are introuce for robutne purpoe hey uually are electe to be fater than the ominant pole D I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 8

29 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 9 egulation computation of - an -?? eg n eg n an o not have common factor * ezout uniue minimal olution for : eg n n n eg n n eg n n *... n n n r n r r...

30 Computation of - an - Euation * i written a: Mx p x M - p x [,,...,, r,..., r n n n ] p [, p,..., pi,..., pn,,...,] n... a. a an a a.... an b' b' b'. b'.. b' n.... b'n n n n n b' i pour i,... ; b' i b i - pour i > Ue of Wineg or bezout.ci.m for olving * I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 3

31 tructure of - an - an inclue pre-pecifie fixe part ex: integrator ' H H, H, - pre-pecifie polynomial ' H ' ' r' r'... r' n ' n ' ' '... ' n ' n he pre pecifie filter H an H will allow to impoe certain propertie of the cloe loop. hey can influence performance an/or robutne I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 3

32 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 3 Fixe part H, H. Example Zero teay tate error yp houl be null at certain freuencie H yp tep iturbance : inuoial iturbance : H ω α α co ; H ignal blocking up houl be null at certain freuencie H up H ω β β co ;, ; inuoial ignal: locking at.5f : n H n

33 racking reference moel H m pecification in continuou time t M, M racking computation of - r t Ieal cae - m m y* t eire trajectory for y t I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 33 r y t * icretization n orer ω, ζ m.5 ω.5.7 ζ H m m a H m he ieal cae can not be obtaine elay, plant zero Objective : to approach y*t y * t m m r t b m m m b a m m m......

34 racking computation of - uil: y * t m m r t Choice of - : Impoing unit tatic gain between y* an y Compenation of regulation ynamic - - G - G / i i F.. r y: H F m m * articular cae : m G i i I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 34

35 ole placement. racking an regulation rt m m * y t - ut - yt - m m - * - - * - - * u t y t y * t I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 35

36 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 36 ole placement. Control law * t y t y t u * * t y 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

37 ole placement. Example lant : m racking ynamic m , ω.5 ra/, ζ.9 egulation ynamic , ω.4 ra/, ζ.9 re-pecification : Integrator *** CONOL LW *** - ut - yt - y*t y*t [m-/m-] rt Controller : Gain margin :.73 hae margin : 65.4 eg Moulu margin : Delay margin:.. I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 37

38 ole placement. Example lant Output ime t/ Control ignal ime t/ I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 38

39 racking an regulation with inepenent objective It i a particular cae of pole placement the cloe loop pole contain the plant zero It i a metho which implifie the plant zero llow exact achievement of impoe performance llow to eign a controller for: table or untable ytem without retriction upon the egree of the polynomial et without retriction upon the integer elay of the plant moel icrete-time plant moel with table zero! Doe not tolerate fractional elay >.5 untable zero I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 39

40 racking an regulation with inepenent objective he moel zero houl be table an enough ampe.3 Zero miible Zone..8.6 f /f Imag xi -. ζ. ζ eal xi miibility omain for the zero of the icrete time moel I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 4

41 racking an regulation with inepenent objective rt m m * y t - ut - yt m - m * - * - * D F eference ignal: tracking y * t m m r t I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 4

42 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 4 egulation computation of - an -.F. of the cloe loop without : * * * * H CL he following euation ha to be olve : * * * houl be in the form:... * n n fter implification by *,* become: ' ** Uniue olution if: n eg - n ; eg ' - ; eg - n -... n r n r r '... ' '

43 egulation computation of - an - ** i written a: Mx p n a a a : : a a -... a a a a a a a..... a n x M - p n x [,,...,, r, r,..., r ] n n p [, p, p,..., pn, pn,..., pn Ue of Wineg or preiol.ci.m for olving ** Inertion of pre pecifie part in an ame a for pole placement ] I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 43

44 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 44 racking computation of - Cloe loop.f.: r y H m m m m F Deire.F. It reult : - - Controller euation: * t y t y t u * t y t y t u [ ] * * t y t u t y b t u b

45 racking an regulation with inepenent objective. Example lant : m racking ynamic m , ω.5 ra/, ζ.9 egulation ynamic , ω.4 ra/, ζ.9 re-pecification : Integrator *** CONOL LW *** - ut - yt - y*t y*t [m -/m -]. rt Controller : Gain margin :.9 hae margin : 65.3 eg Moulu margin : Delay margin :. I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 45

46 racking an regulation with inepenent objective. lant Output ime t/.5 Control ignal Effect of low ampe zero ime t/ he ocillation on the control input when there are low ampe zero can be reuce by introucing auxiliary pole I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 46

47 Internal moel control -racking an regulation It i a particular cae of the pole placement he ominant pole are thoe of the plant moel Doe not allow to accelerate the cloe loop repone llow to eign a controller for: well ampe table ytem without retriction upon the egree of the polynomial an without retriction upon the elay of the icrete time moel he plant moel houl be table an well ampe! Often ue for the ytem featuring a large elay emark: he name i mileaing ince it ha nothing in common with the internal moel principle I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 47

48 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 48 egulation computation of - an - F Dominant pole F n F α * typical choice houl be in the form : fter the cancellation of the common factor -,* become: F typical choice olution for: F F F

49 racking computation of - F / articular cae : m F tracking ynamic regulation ynamic F cancellation of the tracking reference moel I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 49

50 Interpretation of the internal moel control y*t - Euivalent cheme ut / lant yt he plant moel preiction moel i an element of the control cheme Moel ŷt - Feeback on the reiction error F F for H F em.: For all the trategie one can how the preence of the plant moel in the controller I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 5

51 Internal moel control of a ytem with large elay lant: 7;. - ; - he «elay margin» can be atifie by introucing auxiliary pole F α -< α < a yb Magnitue Freuency epone Magnitue emplate for Delay margin α -. α α Freuency f/f b yp Magnitue Freuency epone α.;.3;.333 Goo value Magnitue α -. α emplate for Moulu margin emplate for Delay margin Freuency f/f I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 5

52 Internal moel control of a ytem with large elay Magnitue a yb Magnitue Freuency epone emplate for Delay margin H, F H -, F Freuency f/f b yp Magnitue Freuency epone Magnitue - H, F H -, F emplate for Moulu margin emplate for Delay margin Freuency f/f H correpon to the opening of the loop at.5f ee alo: I.D. Lanau 995 : obut igital control of ytem with time elay the mith preictor reviite Int. J. of Control, v.6,no. pp I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 5

53 ole placement with enitivity function haping erformance pecification for pole placement : Deire ominant pole for the cloe loop he reference trajectory tracking reference moel Quetion: How to take into account the pecification in certain freuency region? How to guarantee the robutne of the controller? How to take avantage from the egree of freeom for the maximum number of pole which can be aigne? nwer: haping the enitivity function by: - introucing auxiliary pole - introucing filter in the controller I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 53

54 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 54 enitivity function - review Output enitivity function: yp Input enitivity function: up Controller tructure : ' H ' H F D re pecifie part filter Dominant an auxiliary filter: tuy of the propertie of the enitivity function in the freuency omain: ze jω

55 ropertie of the output enitivity function.- he moulu of the output enitivity function at a certain freuency give the amplification or attenuation factor of the iturbance on the output yp ω < attenuation yp ω > yp ω operation in open loop amplification. M Moulu margin jω yp max I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 55

56 ropertie of the output enitivity function.3 he open loop KG being table one ha the property:. 5 f j πf/f log e f yp he um of the area between the curve of yp an the axi taken with their ign i null Diturbance attenuation in a freuency region implie amplification of the iturbance in other freuency region! I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 56

57 ropertie of the output enitivity function ugmenting the attenuation or wiening the attenuation zone Higher amplification of iturbance outie the attenuation zone yp Magnitue Freuency epone euction of the robutne reuction of the moulu margin 5 Magnitue ω.4 ra/ec - ω.6 ra/ec ω ra/ec emplate for Moulu margin emplate for Delay margin Freuency f/f I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 57

58 ropertie of the output enitivity function.4 Cancellation of the iturbance effect at a certain freuency: e jω { e Zero of yp jω e jω H e jω e jω ; ω π f / llow introuction of zero at eire freuencie yp Magnitue Freuency epone f 5-5 Magnitue H - - H Freuency f/f I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 58

59 ropertie of the output enitivity function.5 - yp jω at freuencie where: * e jω e jω * e jω H e jω e jω ; ω π f / f llow introuction of zero at eire freuencie yp Magnitue Freuency epone 5 Magnitue H H Freuency f/f I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 59

60 ropertie of the output enitivity function.6 ymptotically table auxiliary pole F lea in general to the reuction of yp jω in the attenuation ban of / F n F F p.5 p. 5 n n F nd yp Magnitue Freuency epone 5 Magnitue F F emplate for Moulu margin emplate for Delay margin Freuency f/f In many application, introuction of high freuency auxiliary pole i enough for auring the reuire robutne margin I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 6

61 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 6 ropertie of the output enitivity function.7 imultaneou introuction of a fixe part H i an of a pair of auxiliary pole Fi having the form: H i i F α α β β reulting from the icretization of : ω ω ζ ω ω ζ F en num z z e with: introuce an attenuation at the normalize icretize freuency: arctan e ic ω ω with the attenuation: en num M t ζ ζ log en num ζ ζ < an with negligible effect at f << f ic an at f >> f ic

62 ropertie of the output enitivity function yp Magnitue Freuency epone 5 Magnitue H, F H ω.5, ζ., F ω.5, ζ Freuency f/f Effective computation with the function: filter.ci.m I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 6

63 ropertie of the input enitivity function. Cancellation of the iturbance effect on the input at a certain freuency up : e jω H e jω jω e llow introuction of zero at eire freuencie up Magnitue Freuency epone ; ω π f H β < β active at.5f / f Magnitue H H.5 - H Freuency f/f em: he ytem operate in open loop at thi freuency I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 63

64 ropertie of the input enitivity function. t freuencie where: jω e H One ha: yp jω e jω jω e up e jω ; ω π f e e jω jω / f Invere of the ytem gain Coneuence : trong attenuation of the iturbance houl be one only in the freuency region where the ytem gain i enough large in orer to preerve robutne an avoi too much tre on the actuator emember: up jω give the tolerance with repect to aitive uncertaintie on the moel high jω weak robutne up I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 64

65 I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 65 ropertie of the input enitivity function.3 imultaneou introuction of a fixe part H i an of a pair of auxiliary pole Fi having the form: reulting from the icretization of : H i i F α α β β ω ω ζ ω ω ζ F en num z z with: introuce an attenuation at the normalize icretize freuency: arctan e ic ω ω with the attenuation: en num M t ζ ζ log en num ζ ζ < an with negligible effect at f << f ic an at f >> f ic

66 emplate for the output enitivity function yp yp yp - M max obutne erformance,5f e yp yp - M max opening the loop,5f e ttenuation zone I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 66

67 emplate for the input enitivity function up yp yp - M max loop opening.5f attenuation region I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 67

68 haping the enitivity function. Choice of the ominant et auxiliary pole of the cloe loop. Choice of the fixe part of the controller H an H 3. imultaneou choice of the fixe part an the auxiliary pole roceure: aic haping : ue an Fine haping: ue 3 ool for enitivity haping: Wineg aptech an ppmater.m here exit alo tool for automatic enitivity function haping bae on convex optimization Optreg from aptech I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 68

69 haping the enitivity function - Example I lant:.7 ;.3 ; ; e pecification: Integrator Dominant pole: icretization of a cont. time n orer ytem : ω ra/, ζ.9 Controller : ttenuation ban: up to.58 Hz but M < -6 an τ < Objective: ame attenuation ban but with M > -6 an τ > - inertion of auxiliary pole:.4 Controller : goo margin but reuction of the attenuation ban -inertion of pole-aero filter H / F centere at ω.4 ra/.64 Hz Controller C : goo attenuation ban but yp > 6 - larger lower auxiliary pole.4.44 Controller D : Correct F I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 69

70 haping the enitivity function - Example I yp Magnitue Freuency epone 5 Magnitue C D emplate for Moulu margin emplate for Delay margin Freuency f/f I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 7

71 haping the enitivity function - Example II lant integrator: ;.5 ; ; Low freuencie iturbance inuoial iturbance.5 Hz ut - yt pecification:. No attenuation of the inuoial iturbance at.5 Hz. ttenuation ban at low freuencie : à.3 Hz 3. Diturbance amplification at.7 Hz: < 3 4. Moulu margin > -6 an Delay margin > 5. No integrator in the controller I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 7

72 haping the enitivity function - Example II - Fixe part eign : H ; H Opening the loop at.5 Hz -Dominant pole: icretization of a cont. time n orer ytem: ω.68 ra/, ζ.9 Controller : the pec. at.7 Hz are not fulfille - inertion of a pole-zero filter H / F centere at ω.44 ra/ Controller : ttenuation ban maller than that pecifie - ominant pole acceleration: ω.9 ra/ Controller C : Correct I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 7

73 haping the enitivity function - Example II yp Magnitue Freuency epone 5 Magnitue -5 - C emplate for Moulu margin emplate for Delay margin Freuency f/f I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 73

74 ome concluing remark ll the igital controller ha a three branche tructure--. hey have two egree of freeom tracking an regulation Controller eign i one in two tep: et regulation tracking Controller complexity epen upon the plant moel complexity ole placement i the baic control trategy racking an regulation with inepenent objective i applicable to icrete time moel with table zero Internal moel control i applicable only to table an well ampe plant Deign of igital ID i a particular cae of pole placement. Can be ue for the control of imple plant orer max. ll the igital controller preente implement a preictive control an they contain implicitly a preicition moel of the plant I.D. Lanau, G. Zito - "Digital Control ytem" - Chapter 3 74