Robust discrete time control I.D. Landau Emeritus Research Director at C.N.R. Laboratoire d utomatique de Grenoble, INPG/CNR, France pril 2004, Valencia I.D. Landau course on robust discrete time control, part
Outline Part I Introduction, examples and basic concepts Part II Design of robust discrete time controllers Part III pecial topics I.D. Landau course on robust discrete time control, part 2
Controller Design and Validation Performance specs. Robustness specs. DEIGN METOD MODEL IDENTIFICTION Ref. CONTROLLER u PLNT y Identification of the dynamic model 2 Performance and robustness specifications 3 Compatible robust controller design method 4 Controller implementation 5 Real-time controller validation and on site re-tuning 6 Controller maintenance same as 5 5 and 6 require identification in closed-loop I.D. Landau course on robust discrete time control, part 3
Part I Outline The R--T digital controller Robust control. What is it? Examples 4 applications ensitivity functions tability of closed loop discrete time systems Robustness margins Robust stability mall gain theorem/ Passivity theorem / Circle criterion Description of uncertainties and robust stability Templates for the sensitivity functions I.D. Landau course on robust discrete time control, part 4
The R--T Digital Controller CLOCK rt Computer controller ut D/ ZO PLNT /D yt Discretied Plant rt m m Controller T - R ut q d Plant Model yt y t y t I.D. Landau course on robust discrete time control, part 5 q
I.D. Landau course on robust discrete time control, part 6 rt m m T q d R ut yt Controller Plant Model - The R--T Digital Controller Plant Model: * q q q q q q q q G d d n a n q a q q... *... q q q b b q q n n R--T Controller: * t y q R d t y q T t u q Characteristic polynomial closed loop poles: q R q q q q q P d
Robust control Robustness of what? with respect to what? Robustness of an index of performance with respect to plant model uncertainties. Index of performance : closed loop stability time domain performance frequency domain performance Plant model uncertainties: low quality design model uncertainties in a frequency range variations of the dynamic characteristics of the plant I.D. Landau course on robust discrete time control, part 7
Robust control Robust controller : assures the desired index of performance of the closed loop for a set of given uncertainties Robust closed loop : the index of performance is guaranteed for a set of given uncertainties I.D. Landau course on robust discrete time control, part 8
Robustness 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 frequency domain analysis is needed The sensitivity functions play a fundamental role in robustness studies The study of closed loop stability in the frequency domain gives valuable information for characteriing robustness I.D. Landau course on robust discrete time control, part 9
Robustness. ome questions ow to characterie model uncertainty? ow to deal with model uncertainty? ow to characterie controller robustness? Model uncertainty tructured parametric uncertainty Unstructured specified in the frequency domain We will consider the uncertainties described in the frequency domain i.e. effect of parameter uncertainty should be converted in the frequency domain. Important concepts : nominal model, uncertainty model, family of plant models nominal stability, robust stability nominal performance, robust performance I.D. Landau course on robust discrete time control, part 0
Nominal model, uncertainty model, family of plant models Nominal model : the model used for design Uncertainty model : the description of the uncertainties in the frequency domain magnitude, phase Family of plant models P: characteried by the nominal plant model and the uncertainty model The different true plant models belong to the family P I.D. Landau course on robust discrete time control, part
Nominal and robust stability and performance Nominal stability : closed loop as. stable for the nominal model Robust stability: closed loop as. table for all the plant models belonging to the family set P Nominal performance: performance for the nominal model Robust performance: performance guaranteed for all plant models belonging to the family set P I.D. Landau course on robust discrete time control, part 2
Robust stability and robust performance - Robust stability: The closed loop poles remain inside the unit circle discrete time case for all the models belonging to the family P - r c Robust performance: The closed loop poles remain inside the circle c,r for all the models belonging to the family P robust performance condition can be translated in a robust stability condition this is not the only way for solving robust performance design I.D. Landau course on robust discrete time control, part 3
Examples to illustrate robust control design Continuous steel casting ot dip galvaniing Flexible transmission 360 flexible arm I.D. Landau course on robust discrete time control, part 4
Continuous steel casting Level to be controlled haking to prevent adherence I.D. Landau course on robust discrete time control, part 5
Continuos steel casting Plant integrator: q ; 0.5q ; d 2 ; T s Low frequency disturbances inusoidal disturbance 0.25 ut q -d yt pecifications:. No attenuation of the sinusoidal disturbance 0.25 2. ttenuation band in low frequencies: 0 à 0.03 3. Disturbance amplification at 0.07 : < 3d 4. Modulus margin > -6 d and Delay margin > T 5. No integrator in the controller I.D. Landau course on robust discrete time control, part 6
ot dip galvaniing. Control of the deposited inc teel strip Preheat oven ir knife Zinc bath Finished product Measurement of deposited mass air air inc steel strip I.D. Landau course on robust discrete time control, part 7
ot dip galvaniing. The control loops Desired deposited inc mass per cm 2 m* RT controller P0 PR ± P Pressure controller Pressuried Ga supply Coating process Measured deposited inc mass per cm 2 Measurement delay m ± m important time delay with respect to process dynamics time delay depends upon the steel strip speed sampling frequency tied to the steel strip speed constant integer delay in discrete time parameter variations of the process as a function of the type of product I.D. Landau course on robust discrete time control, part 8
ot dip galvaniing. Model and specifications Plant model for a type of product : q 7 bq a q Model : sec; b 0.3; a 0.2 0.3 T 2 pecifications performance : Modulus margin: M 0.5 Delay margin: τ 2T Integrator I.D. Landau course on robust discrete time control, part 9
ot dip galvaniing. Performance I.D. Landau course on robust discrete time control, part 20
Control of a Flexible Transmission The flexible transmission motor axis load Φ m d.c. motor axis position Controller D C ut R--T controller Position transducer yt D C Φ ref I.D. Landau course on robust discrete time control, part 2
ampling frequency : 20 Control of a Flexible Transmission q q.609555q 0.3053q d 2 0.3943q.87644q 2 2.49879q 3 0.88574q 4 Model Vibration modes:.949 rad /sec, ζ 0.042; 3.462 rad / sec, ζ 0.023 2 2 pecifications: Tracking:.94 rad / sec, ζ 0 0.9 Dominant poles:.94 rad / sec, ζ 0 0.8 Robustness margins: M 0.5 Zero steady state error integrator τ Constraints on up : 0 d for f 0.35 f 7 up max 2T I.D. Landau course on robust discrete time control, part 22
360 Flexible rm LIGT OURCE RIGID FRME MIRROR LUMINIUM DETECTOR TC. POT. ENCODER LOCL POITION ERVO. COMPUTER I.D. Landau course on robust discrete time control, part 23
360 Flexible rm Frequency characteristics Identified Model Poles-Zeros Unstable eros! I.D. Landau course on robust discrete time control, part 24
ampling frequency : 20 Model q q 2.049 q.542 q 0.0064 q 0.0382 q 5.0485 q 0.7987 q 0.046 q 5 0.007 q 2 6 6 2 0.33836 q 0.0697 q 3 3 0.044 q 0.46 q d 0 Vibration modes: 2.67rad /sec, ζ 0.08; 4.402rad / sec, ζ 0.025; 48.7rad /sec, ζ 0.038 2 2 3 3 pecifications: Tracking: 2.673 rad /sec, ζ 0.9 0 Dominant poles: 2.673 rad /sec, ζ 0.8 0 M 0.5 Robustness margins: Zero steady state error integrator Constraints on up : up up 360 Flexible rm τ 2T 5 d for f < 4 ; < 5d for 6.5 f < 8 ; up 4 4 0 d for 4.5 f < 6.5 ; up < 0 d for 8 f 0 I.D. Landau course on robust discrete time control, part 25
Robust Control. asic concepts I.D. Landau course on robust discrete time control, part 26
I.D. Landau course on robust discrete time control, part 27 Digital control in the presence of disturbances and noise Output sensitivity function p y R yp R R up R R yb R 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.02-03 T R / / PLNT rt ut yt - pt bt disturbance measurement noise vt
I.D. Landau course on robust discrete time control, part 28 ll four sensitivity functions must be stable! d Plant model: is unstable upose : R poles compensation by the controller eros ] [ yv yb up yp yv is unstable while yp, up,and yb may be stable if : 0 <
Complementary sensitivity function rt T - vt ut / / PLNT disturbance pt yt R bt measurement noise For T R one has: yr R yb R yp yb yp yr I.D. Landau course on robust discrete time control, part 29
tability of closed loop discrete time systems The Nyquist is used like in continuous time can be displayed with WinReg ou Nyquist_OL.sci.m OL j j j e R e e j j e e Critical point - Im π Re R yp OL - yp e -j O 0 e -j O Nyquist criterion discrete time O.L. is stable The Nyquist plot of the open loop transfer fct. OL e -j traversed in the sense of growing frequencies from 0 to 0.5f leaves the critical point[-, j0] on the left I.D. Landau course on robust discrete time control, part 30
tability of closed loop discrete time systems Nyquist criterion discrete time O.L. is unstable The Nyquist plot of the open loop transfer fct. OL e -j traversed in the sense of growing frequencies from 0 et f leaves the critical point[-, j0] on the left and the number of encirclements of the critical pointcounter clockwise should be equal to the number of unstable poles in open loop. π - 2π π Im unstable pole in Open Loop table Closed Loop a Re table Open Loop Unstable Closed Loopb Remarks: -The controller poles may become unstable if high performances are required without using an appropriate design method -The Nyquist plot from 0.5f to f is the symmetric with respect to the real axis of the Nyquist plot from 0 to 0.5f 0 0 I.D. Landau course on robust discrete time control, part 3
Marges de robustesse 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 - Μ G Φ Re -Gain margin G -Phase margin φ -Delay margin Dt -Modulus margin DM Crossover frequency OL CR I.D. Landau course on robust discrete time control, part 32
Robustness margins Gain margin G O j 80 pour φ 80 80 o Phase margin φ 80 0 φ pour j Delay margin τ M φ cr φ cr φ min If there several intersections with the unit circle i Modulus margin i O j min yp O j min cr everal intersections points: τ I.D. Landau course on robust discrete time control, part 33 yp j min i max φ i i cr
Robustness margins typical values Gain margin : G 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 0.5-6 d [min : 0,4-8 d] modulus margin M 0.5 implies G 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 robust discrete time control, part 34
Robustness margins Good gain and phase margin ad modulus margin Good gain and phase margin ad delay margin I.D. Landau course on robust discrete time control, part 35
M Modulus margin and sensitivity function OL min yp R max min yp pour max e j 2πf yp e j max d M d M d d - yp yp max - M 0 yp - yp - yp M max min I.D. Landau course on robust discrete time control, part 36
Robust stability To assure stability in the presence of uncertainties or variations on the dynamic chatacteristics of the plant model OL nominal F.T.; OL Different from OL perturbed Robust stability condition sufficient cond.: OL OL < R OL yp P ; e j * Im d - yp - Re yp max - M OL ' OL OL 0 yp - yp - yp min M max ie of the tolerated uncertainity on OL at each frequency radius I.D. Landau course on robust discrete time control, part 37
I.D. Landau course on robust discrete time control, part 38 < R R R R < R P R R up up d Limitation of the actuator stress 0 0.5f e up - ie of tolerated uncertainties Tolerance to plant additive uncertainty OL OL * From previous slide : **
I.D. Landau course on robust discrete time control, part 39 Tolerance to plant normalied uncertainty multiplicative uncertainty < R P R R yb From **, previous slide: The inverse of the modulus of the complementary sensitivity function gives at each frequency the tolerance with respect to normalied multiplicative uncertainty Relation between additive and multiplicative uncertainty: ' ' '
Passivity yperstability Theorem - u strictly positive real y y 2 2 passive u 2 : trictly positive real transfer function state x 2 : linear or nonlinear, time invariant or time-varying Then : η 2 t T 2 2 0, t y t u t γ ; γ < ; t 2 2 2 t 0 2 lim x t 0 ; lim u t 0 ; lim y t t t t 0 0 I.D. Landau course on robust discrete time control, part 40
trictly positive real transfer function PR Frequency domain j s σ s j continuous Im j Re - Im j discrete Im e j Re j e Re Re < 0 Re > 0 - j j e Re < 0Re > 0 - asimptotycally stable j j - Re e > 0 for all e, 0 < < π discrete time case Input output property time domain t T 2 2 η 0, t y t u t γ κ u ; γ < ; κ > 0 ; t t 0 2 2T 0 I.D. Landau course on robust discrete time control, part 4
tability criteria for nonlinear time/varying feedback system y 2 Im e -j ku 2 - k - Re e -j u 2 e -j Re > / k ; critical circle Im e -j β u u 2 α u y 2 - α - - β Re e -j e -j Circle criterion I.D. Landau course on robust discrete time control, part 42
tability in the presence of nonlinearities tolerance of nonlinearities critical circle Im e -j - u βu αu - O M - α - β - Re e -j e -j The modulus margin defines the tolerance with respect to nonlinear and/or time varying elements. The tolerance sector is defined by : Min gain: / M Max gain: / M I.D. Landau course on robust discrete time control, part 43
The circle criterion : a particular case KM - Km- should lie inside the unit circle The modulus of the should be smaller than at all frequencies i.e.: < This is the small gain theorem I.D. Landau course on robust discrete time control, part 44
mall gain theorem - u < y y 2 2 2 : 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 robust discrete time control, part 45
I.D. Landau course on robust discrete time control, part 46 Relationship between passivity theorem and small gain theorem If is passive, If is passive Remark: Correct writing in the general operator case If is strictly passive, < < If is strictly passive 2 2 2 2 2 2 4Re 2Re 2Re j j j j e e e e int for a proof:
Relationship between passivity theorem and small gain theorem u - 2I < - I strictly positive real I - y y 2 0.5I - I 2 passive - u 2 2 I 2 I.D. Landau course on robust discrete time control, part 47
Description of uncertainties in the frequency domain Im Re Uncertainty disk at a certain frequency 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 frequency and is characteried by a transfer function I.D. Landau course on robust discrete time control, part 48
I.D. Landau course on robust discrete time control, part 49 dditive uncertainty ' W a δ δ any stable transfer function with δ W a a stable transfer function ' ' max W a R K d / ; / δ a W K - W a up δ - < W a up Robust stability condition: pply small gain theorem
' Multiplicative uncertainties [ δ W ] δ any stable transfer function with δ W m a stable transfer function W a W m m δ W m - K - yb δ W m Robust stability condition: yb W m < I.D. Landau course on robust discrete time control, part 50
I.D. Landau course on robust discrete time control, part 5 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 Robust stability condition: δ K - - W m
I.D. Landau course on robust discrete time control, part 52 Robust stability conditions, ', δ W Ρ Family set of plant models Robust 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
Robust stability and templates for the sensitivity functions Robust 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 frequency 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 robust discrete time control, part 53
I.D. Landau course on robust discrete time control, part 54 Modulus margin and robust stability π < 0 j r j yp e W e Modulus margin: M e j yp < Robust stability cond.: Possible uncertainties characteried by: 2,...,,, ; 2 2 f f M W r λ λ δ Examples of families of plant models for which robust stability is guaranteed: M M λ λ
I.D. Landau course on robust discrete time control, part 55 Delay margin and robust stability.t s τ d d π < 0 j m j yb e W e [ ] [ ] ; W W m m δ δ π < 0, ; j yb e Can be interpreted as a multiplicative uncertainty Robust stability condition: j yb e d < ; 20log Define a template on yb yb
I.D. Landau course on robust discrete time control, part 56 Delay margin template on yp R yp R R yb yb yp yb yp yb yp π < 0, ; j yb e pproximate condition
Defintion of templates for the sensitivity functions Nominal performance requirements and robust stability conditions lead to the definition of desired templates on the sensitivity functions The union of various templates Upper template: Wi e j will define an upper and a lower template j j j W e min W e,... W e sup i n i Lower template: j j j W e max W e,... W e inf i I In i I.D. Landau course on robust discrete time control, part 57
Frequency templates on the sur sensitivity functions The robust stability conditions allow to define frequency templates on the sensitivity functions which guarantee the delay margin and the modulus margin; The templates are essential for the design a good controller Frequency template on the noise-output sensitivity function yb for τ T Frequency template on the output sensitivity function yp for τ T and Μ 0.5 I.D. Landau course on robust discrete time control, part 58
Templates for the ensitivity Functions nominal perform. 0 yp d yp max - M delay margin 0.5f s Output ensitivity Function up d 0 actuator effort 0.5f s Input ensitivity Function up - sie of the tolerated additive uncertainty a W G G δw a I.D. Landau course on robust discrete time control, part 59
Templates for the ensitivity Functions yp d yp - M max 0 opening the loop 0,5f e Output ensitivity Function ttenuation one up d 0 Limitation of the actuator stress Zone with uncertainty upon the modelquality 0.5f e Input ensitivity Function Opening the loop at the frequency f < - 00 d I.D. Landau course on robust discrete time control, part 60