Lecture 8 - SISO Loop Design

Transcription

1 Lecture 8 - SISO Loop Deign Deign approache, given pec Loophaping: in-band and out-of-band pec Fundamental deign limitation for the loop Gorinevky Control Engineering 8-1

2 Modern Control Theory Appy reult of EE25 etc Obervable and controllable ytem Can put pole anywhere Can drive tate anywhere Why cannot we jut do thi? Large control Error peaking oor robutne, margin Obervability and controllability = matrix rank Accuracy of olution i defined by condition number Analyi i valid for any LTI control, including advanced Gorinevky Control Engineering 8-2

3 Feedback controller deign Conflicting requirement Engineer look for a reaonable trade-off Educated gue, trial and error controller parameter choice Satify key pec Find a reaonable tradeoff between everal conflicting requirement k k k I D u = k Deign proce Stability erformance Robutne D τ + 1 D + lant model Indexe Contraint Spec k + ki e Analyi and imulation Gorinevky Control Engineering 8-3

4 Gorinevky Control Engineering 8-4 Tranfer function in control loop Controller C lant - e e y y d v u d diturbance feedforward reference output control error n noie Senitivity Complementary enitivity Noie enitivity Load enitivity [ ] [ ] [ ] [ ] C S C C S C C T C S y u + = + = + = + = S + T = 1

5 Loop hape performance requirement Loop gain: erformance Diturbance rejection and reference tracking S iω <<1 for the diturbance d; atified if L iω >>1 Noie rejection Tiω = [1+ Liω] -1 Liω < 1 i atified unle 1+ Liω i mall near the croover Limited control effort Ciω Siω < 1 Can be a problem if iω < 1 high frequency Gorinevky L iω = S iω = iω C iω [ 1+ L iω ] 1 d 1 C L Control Engineering 8-5 y

6 Loop hape robutne requirement y T S u Robutne C C Multiplicative uncertainty Tiω iω < 1, where iω i the uncertainty magnitude at high frequencie, relative uncertainty iω can be large, hence, Tiω mut be kept mall mut have Liω <<1 for high frequency, where iω i large Additive uncertainty Ciω Siω < 1/ iω Gain margin of 1-12db and phae margin of 45-5 deg thi correpond to relative uncertainty of the plant tranfer function in the 6-8% range around the croover y Gorinevky Control Engineering 8-6

7 Senitivity v. margin Margin are ueful for deciding upon the loop hape modification Can ue uncertainty characterization and noie or complementary enitivity intead = C S u Thi i done by modern advanced control deign method Gorinevky [ 1+ C ] 1 [ 1 + C ] 1 T = 1-1 enitivity peak margin ϕ m 1/g m Im L Re L Additive uncertainty iω radiu Control Engineering 8-7

8 Loop Shape Requirement Low frequency: Liω erformance high gain L = mall S croover lope High frequency: mall gain L mall T large Bandwidth db Bandwidth Robutne performance can be only achieved in a limited frequency band: ω ω B ω B i the bandwidth ω B ω gc Fundamental tradeoff: performance v. robutne Gorinevky Control Engineering 8-8

9 Loophaping deign Loop deign Ue,I, and D feedback to hape the loop gain Loop modification and bandwidth Low-pa filter - get rid of high-frequency tuff - robutne Notch filter - get rid of ocillatory tuff - robutne Lead-lag to improve phae around the croover - bandwidth +D in the ID together have a lead-lag effect Need to maintain tability while haping the magnitude of the loop gain Formal deign tool H 2, H, LMI, H loophaping cannot go pat the fundamental limitation Gorinevky Control Engineering 8-9

10 Example - dik drive ervo The problem from HW Aignment 2 data in dikid.m, dikdata.mat Deign model: i an uncertainty g = + 2 Analyi model: decription for Deign approach: ID control baed on the implified model ki C = k + + kd τ + 1 D Dik ervo control J ϕ& & = T VCM + T DISTURBANCE Voice Coil Motor Gorinevky Control Engineering 8-1

11 Dik drive ervo controller Start from deigning a D controller pole, characteritic equation 1+ C = 2 + g k D + g k Critically damped ytem k = k + k 2 D = w / g; k = w / 2 g D g = where frequency w i the cloed-loop bandwidth In the derivative term make dynamic fater k than w. Select τ D =.25/ w D τ + 1 D Gorinevky Control Engineering 8-11

12 Dik drive ervo Step up from D to ID control 1 g 1+ k + kd + ki = g k D + g k + g k I = Keep the ytem cloe to the critically damped, add integrator term to correct the teady tate error, keep the caling k = / τ w 2 3 w g; kd = aw / g; ki = bw / g D = c / where a, b, and c are the tuning parameter Tune a, b, c and w by watching performance and robutne Gorinevky Control Engineering 8-12

13 Dik drive - controller tuning Tune a, b, w, and τ D by trial and error Find a trade off taking into the account Cloed loop tep repone Loop gain - performance Robutne - enitivity Gain and phae margin Try to match the characteritic of C2 controller demo Gorinevky Control Engineering 8-13

14 Dik ervo - controller comparion ID i compared againt a reference deign th-order compenator C2 blue, dahed, ID red Reference deign: 4-th order controller C2 = lead-lag + notch filter Matlab dikdemo Data in dikid.m, dikdata.mat Amplitude Time ec Gorinevky Control Engineering 8-14

15 Loop hape, margin hae deg Magnitude db LOO GAIN - C2 blue, dahed, ID red Croover area of interet Gorinevky Frequency rad/ec Control Engineering 8-15

16 Loop hape and margin, zoomed 4 3 LOO GAIN - C2 blue, dahed, ID red C2 gain margin Magnitude db ID gain margin: 4 12dB -135 hae deg ID phae margin: 43 deg C2 phae margin -27 Gorinevky 1 3 Frequency rad/ec Control Engineering 8-16

17 Senitivitie Robutne to multiplicative uncertainty Tiω iω < 1 Magnitude db 5 COMLEMENTARY SENSITIVITY INVERSE ROBUSTNESS Tiω C2 blue, dahed, ID red, olid Diturbance rejection performance Magnitude db Siω SENSITIVITY ERFORMANCE - C2 blue, dahed, ID red yiω = Siω diω Frequency rad/ec Gorinevky Control Engineering 8-17

18 Fundamental deign limitation If we do not have a reference deign - how do we know if we are doing well. I there i a much better controller? Cannot get around the fundamental deign limitation frequency domain limitation on the loop hape ytem tructure limitation engineering deign limitation Gorinevky Control Engineering 8-18

19 Frequency domain limitation erformance v. robutne tradeoff Diturbance rejection performance: Siω <<1 Bode integral contraint - waterbed effect log S iω dω = Siω + Tiω = 1 log Siω Robutne: Tiω <<1 for minimum-phae table ytem; wore for the ret Gorinevky Control Engineering 8-19

20 Waterbed effect Gunter Stein Bode Lecture, 1989 IEEE CSM, Augut 23 Gorinevky Control Engineering 8-2

21 Waterbed effect Gorinevky Control Engineering 8-21

22 Structural deign limitation Delay and non-minimum phae r.h.. zero cannot make the repone fater than delay, et bandwidth maller Untable dynamic make Bode integral contraint wore re-deign ytem to make it table or ue advanced control deign Flexible dynamic cannot go fater than the ocillation frequency practical approach: filter out and ue low-bandwidth control wait till it ettle ue input haping feedforward Gorinevky Control Engineering 8-22

23 Advanced application Untable dynamic need advanced feedback control deign Gorinevky Control Engineering 8-23

24 Flexible dynamic Very advanced application really need control of 1-3 flexible mode Gorinevky Control Engineering 8-24

25 Engineering deign limitation Senor noie - have to reduce Tiω - reduced performance quantization - ame effect a noie bandwidth etimator - cannot make the loop fater Actuator range/aturation - limit the load enitivity Ciω Siω actuator bandwidth - cannot make the loop fater actuation increment - ticktion, quantization - effect of a load variation other control handle Modeling error have to increae robutne, decreae performance Computing, ampling time Nyquit ampling frequency limit the bandwidth Gorinevky Control Engineering 8-25