IDENTIFICATION IN CLOSED LOOP A powerful design tool (theory, algorithms, applications)

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1 IDENTIFICTION IN CLOED LOOP powerful design tool theory, algorithms, applications better models, simpler controllers I.D. Landau Laboratoire d utomatiue de renoble, INP/CN, France Part 2: obust digital control brief review Prepared for Marie Curie ction TOK 3092 evilla November 2004 I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK 3092

2 Controller Design and Validation Performance specs. obustness specs. DEIN METHOD MODEL IDENTIFICTION ef. CONTOLLE u PLNT y Identification of the dynamic model 2 Performance and robustness specifications 3 Compatible controller design method 4 Controller implementation 5 eal-time controller validation and on site re-tuning 6 Controller maintenance same as 5 I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

3 Outline obust digital control -The --T digital controller -asic design -obustness issues -The sensitivity functions and their properties -obustness margins -obust stability -n example I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

4 The --T Digital Controller CLOCK rt Computer controller ut D/ ZOH PLNT /D yt Discretied Plant rt m m Controller T - ut d Plant Model yt y t y t I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

5 I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK rt m m T d ut yt Controller Plant Model - The --T Digital Controller Plant Model: * d d n a n a... *... b b n n --T Controller: * t y d t y T t u Characteristic polynomial closed loop poles: P d

6 Pole Placement with --T Controller rt m m * y td T - ut -d yt -d -d -d - - m * - m P - - * - * Controller : H ' ; H egulation: and solutions of: ' H, H : fixed parts d H ' H ' P P D P F Tracking : T P / eference trajectory: y* computer file y* m / m r I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK 3092 dominant poles auxiliary poles 6

7 Connections with other Control trategies - Digital PID : n n 2; H -Tracking and regulation with independent objectivesmc: P * P P D F Hyp.: * has stable damped eros - Minimum variance tracking and regulation MVC: P * C Hyp.: * has stable damped eros noise model - Internal Model Control IMC: P P F Hyp.: has stable damped poles I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

8 I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK Digital control in the presence of disturbances and noise Output sensitivity function p y yp up yb yv Input sensitivity function p u Noise-output sensitivity function b y Input disturbance-output sensitivity function v y ll four sensitivity functions should be stable! see book pg T / / PLNT rt ut yt - pt bt disturbance measurement noise vt

9 Complementary sensitivity function rt T - vt ut / / PLNT disturbance pt yt bt measurement noise For T one has: yr yb yp yb yp yr I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

10 obustness of a control system control system is said to be robust for a set of given uncertainties upon the nominal plant model if it guarantees stability and performance for all plant models in this set. To characterie the robustness of a closed loop system a freuency domain analysis is needed The sensitivity functions play a fundamental role in robustness studies The study of closed loop stability in the freuency domain gives valuable information for characteriing robustness I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

11 tability of closed loop discrete time systems The Nyuist is used like in continuous time can be displayed with Wineg or Nyuist_OL.sci.m H OL j j j e e e j j e e Critical point - Im H π e H H yp OL - yp H e -j O 0 H e -j O Nyuist criterion discrete time O.L. is stable The Nyuist plot of the open loop transfer fct. H OL e -j traversed in the sense of growing freuencies from 0 to 0.5f leaves the critical point[-, j0] on the left I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK 3092

12 tability of closed loop discrete time systems Nyuist criterion discrete time O.L. is unstable The Nyuist plot of the open loop transfer fct. H OL e -j traversed in the sense of growing freuencies from 0 et f leaves the critical point[-, j0] on the left and the number of encirclements of the critical pointcounter clockwise should be eual to the number of unstable poles in open loop. π - 2π π Im H unstable pole in Open Loop table Closed Loop a e H table Open Loop Unstable Closed Loopb emarks: -The controller poles may become unstable if high performances are reuired without using an appropriate design method -The Nyuist plot from 0.5f to f is the symmetric with respect to the real axis of the Nyuist plot from 0 to 0.5f 0 0 I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

13 obustness margins The minimal distance with respect to the critical point characteries the robustness of the CL with respect to uncertainties on the plant model parameters or their variations Im H - Μ Φ e H -ain margin -Phase margin φ -Delay margin Dt -Modulus margin DM Crossover freuency H OL C I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

14 obustness margins ain margin H OL j 80 pour φ o Phase margin φ 80 0 φ pour H j Delay margin τ φ cr φ M H OL cr φ min If there are several intersections with the unit circle i Modulus margin i O everal intersections points: τ min j cr i j j min yp min yp max φ i i cr I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

15 obustness margins typical values ain margin : 2 6 d [min :,6 4 d] Phase margin : 30 φ 60 Delay margin : fraction of system delay 0% or of time response 0% often.t Modulus margin : M d [min : 0,4-8 d] modulus margin M 0.5 implies 2 et φ > 29 ttention! The converse is not generally true The modulus margin defines also the tolerance with respect to nonlinearities I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

16 obustness margins. ood gain and phase margin ad modulus margin ood gain and phase margin ad delay margin I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

17 M Modulus margin and sensitivity function H OL min yp max min yp pour max e j 2πf yp e j max d M d M d d - yp yp max - M Critical region for design 0 yp - yp min - yp M max Minimum distance with respect to the critical point I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

18 Correspondance Output ensitivity Nyuist Plot d - yp Im H yp max - M - e H 0 Μ Φ yp - yp - yp min M max Crossover freuency C H OL I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

19 Properties of the output sensitivity function The open loop being stable, one has the property: 0. 5 f j 2pf/f log e df 0 yp 0 The sum of the areas between the curve of yp and the axis 0d taken with their sign is null Disturbance attenuation in a freuency region implies amplification of the disturbances in other freuency regions! I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

20 Properties of the output sensitivity function ugmenting the attenuation or widening the attenuation one Higher amplification of disturbances ouside the attenuation one eduction of the robustness reduction of the modulus margin I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

21 obust stability To assure stability in the presence of uncertainties or variations on the dynamic chatacteristics of the plant model H OL nominal F.T.; H OL Different from H OL perturbed obust stability condition sufficient cond.: - H OL H ' OL H OL H OL H OL < H Im H e H yp d 0 OL - yp yp yp P - yp yp min max ; < H H OL OL - M - yp e M max j * ie of the tolerated uncertainity on H OL at each freuency radius I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

22 I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK < < P up up d Limitation of the actuator stress 0 0.5f e up - ie of tolerated uncertainties Tolerance to plant additive uncertainty H OL H OL * From previous slide : ** ' < up

23 I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK Tolerance to plant normalied uncertainty multiplicative uncertainty < P yb From **, previous slide: The inverse of the modulus of the complementary sensitivity function gives at each freuency the tolerance with respect to normalied multiplicative uncertainty elation between additive and multiplicative uncertainty: ' ' '

24 Important message Large values of the modulus of the sensitivity functions in a certain freuency region Low tolerance to model uncertainty Critical regions for control design Need for a good model in these regions I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

25 mall gain theorem - u < y y : linear time invariant state x 2 : Then: 2 < u 2 lim x t 0 ; lim u t 0 ; lim y t t t t 0 It will be used to characterie robust stability I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

26 Description of uncertainties in the freuency domain Im H e H Uncertainty disk at a certain freuency It needs a description by a transfer function which may have any phase but a modulus < 2 The sie of the radius will vary with the freuency and is characteried by a transfer function I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

27 I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK dditive uncertainty ' W a δ δ any stable transfer function with δ W a a stable transfer function ' ' max W a H K d / ; / δ a W K - W a up δ - < W a up obust stability condition: pply small gain theorem

28 ' Multiplicative uncertainties [ δ W ] δ any stable transfer function with δ W m a stable transfer function W a H W m m δ W m - K - yb δ W m obust stability condition: yb W m < I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

29 I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK Feedback uncertainties on the input [ ] ' W r δ δ any stable transfer function with δ W r a stable transfer function W r yp δ - < W r yp obust stability condition: δ K - - W m

30 I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK obust stability conditions, ', δ W H H Ρ Family set of plant models obust stability : The feedback system is asymptotically stable for all the plant models belonging to the family, δ Ρ W dditive uncertainties < W a up π < 0 j a j up e W e Multiplicative uncertainties < W m yb π < 0 j m j yb e W e Feedback uncertainties on the input or output < W r yp π < 0 j r j yp e W e

31 δw uncertainty disk 0 Im π a related sensitivity function defines the sie of the tolerated uncertainty obust tability e -j e Family of plant models: ' F, δ, W nominal model; δ W xy obust stability condition: W xy xy xy W < xy < xy - sie of uncertainty a type of uncertainty defines an upper template for the modulus of the sensitivity function There also lower templates because of the relationship between various sensitivity fct. I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

32 obust stability and templates for the sensitivity functions obust stability condition: xy e j < W e j 0 π The functions W the inverse of the sie of the uncertainties define an upper template for the sensitivity functions Conversely the freuency profile of terms of tolerated uncertainties xy e j can be interpreted in d - yp up d yp max - M 0 yp - yp min - yp M max 0 up - 0.5f e Tolerated feedback uncertainty on the input Tolerated additive uncertainty I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

33 Templates for the ensitivity Functions nominal perform. 0 yp d yp max - M delay margin 0.5f s Output ensitivity Function up d Dangerous ones. Need for good models in these regions actuator effort 0 0.5f s Input ensitivity Function up - sie of the tolerated additive uncertainty a W δwa I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

34 obust Controller Design Pole placement with sensitivity functions shaping Nominal performance: P P P D F ' ' H H P D and part of H and llow to shape the sensitivity functions everal approaches to design : -Iterative Choosing P F and using band stop filters matlab toolbox «ppmaster» i Fi H H / P, H / j P Fj -Convex optimiation see Langer, Landau, utomatica, June99, Optreg daptech I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

35 Properties of the output sensitivity function The asymptotically stable auxiliary poles P F lead in general to the reduction of yp j in the freuency regions corresponding to the attenuation regions for /P F F n PF P p 0.5 p np n F P npd In many applications the introduction of damped high freuency auxiliary poles is enough for assuring the reuired robustness margins I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

36 I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK P H i i F α α β β Obtained by the discretiation of : ζ ζ s s s s s F den num 2 T s e with: produce and attenuation hole at the normalied discretied freuency: 2 2arctan 0 e disc T with attenuation: den num M t ζ ζ log 20 den num ζ ζ < and has negligible effects at f << f disc and at f >> f disc Properties of the output sensitivity function imultaneous introduction of a fixed part H i and of a pair of auxiliary poles P Fi of the form:

37 Properties of the output sensitivity function For details see Landau: Commande des ystèmes, Hermes Efective computation using: filter22.sci.m I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

38 360 Flexible rm LIHT OUCE IID FME MIO LUMINIUM DETECTO TCH. POT. ENCODE LOCL POITION EVO. COMPUTE I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

39 360 Flexible rm Freuency characteristics Poles-Zeros Identified Model I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

40 haping the ensitivity Functions Output ensitivity Function - yp Input ensitivity Function - up 25 yp - ensibilité perturbation-sortie 60 up - ensibilité perturbation-entrée gabarit template Module [d] D C Module [d] template gabarit D C Freuence [H] without auxiliary poles - with auxiliary poles C- with stop band filter D- with stop band filter Freuence [H] H P H P / F 2 / F 2 I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK

41 obust Discrete Time Controller Design More details can be found in : I.D. Landau: Commande des systèmes conception, identification, mise en œuvre Hermes, 2002, Paris, chapters 2 and 3 english translation available and ome references directly related to the course «lides» version of the chapters can be downloaded Free routines matlab, scilab can be downloaded as well as a matlab based software «ppmaster» for pole placement design with sensitivity functions shaping I.D.Landau,. Loano, M. M aad «daptive Control», pringer, 997, chap.8 I.D. Landau : course on «obust Discrete Time Control», Valencia, pril 2004 I.D. Landau : course on system identification in closed loop 2/Marie Curie ction TOK