Dynamic Effects of Feedback Control!

Transcription

1 Dynamic Effecs of Feedback Conrol! Rober Sengel! Roboics and Inelligen Sysems MAE 345, Princeon Universiy, 2017 Inner, Middle, and Ouer Feedback Conrol Loops Sep Response of Linear, Time- Invarian (LTI) Sysems Posiion and Rae Conrol Transien and Seady-Sae Response o Sinusoidal Inpus Copyrigh 2017 by Rober Sengel. All righs reserved. For educaional use only. hp:// 1 Ouer-o-Inner-Loop Conrol Hierarchy Inner Loop! Small Ampliude! Fas Response! High Bandwidh Middle Loop! Moderae Ampliude! Medium Response! Moderae Bandwidh Ouer Loop! Large Ampliude! Slow Response! Low Bandwidh Feedback! Error beween command and feedback signal drives nex inner-mos loop 2

2 Naural Feedback Conrol Inner Loop Chicken Head Conrol - 1 hp:// Middle Loop Hovering Red-Tail Hawks hp:// VPVZMSEvwU Ouer Loop Osprey Diving for Fish hp:// v=qrgpl9-n6jy 3 Ouer-o-Inner-Loop Conrol Hierarchy of an Indusrial Robo Inner Loop! Focus on conrol of individual joins Middle Loop! Focus on operaion of he robo Ouer Loop! Focus on goals for robo operaion 4

3 Inner-Loop Feedback Conrol Feedback conrol design mus accoun for acuaor-sysem-sensor dynamics Single-Inpu/Single-Oupu Example, wih forward and feedback conrol logic ( compensaion ) 5 Thermosaic Temperaure Conrol Dynamics! Delays! Dead Zones! Sauraion! Coupling Srucure! Layou! Insulaion! Circulaion! Muliple Spaces Exernal Effecs! Solar Radiaion! Air Temperaure! Wind! Rain, Humidiy... all conrolled by a simple (bu nonlinear) on/off swich 6

4 Thermosa Conrol Logic Conrol Law [i.e., logic ha drives he conrol variable, u] e =! y = u c! u b % u = % < Thermosa > 1(on), e > 0 0 (off ), e " 0 : Desired oupu variable (command) y: Acual oupu u: Conrol variable (forcing funcion) e: Conrol error 7 Thermosa Conrol Logic " u = % 1 (on), e > 0 0 (off ), e! 0...bu conrol signal would chaer wih slighes change of emperaure Soluion: Inroduce lag o slow he swiching cycle, e.g., hyseresis % u = % 1 (on), e! T > 0 0 (off ), e + T " 0 8

5 Thermosa Conrol Logic wih Hyseresis Hyseresis delays he response Sysem responds wih a limi cycle Cooling conrol is similar wih sign reversal 9 Speed Conrol of Direc-Curren Moor Angular Rae Linear Feedback Conrol Law (c = Conrol Gain) u = ce where e =! y How would y be measured? 10

6 Characerisics of he Model 1/J Simplified Dynamic Model! Roary ineria, J, is he sum of moor and load inerias! Inernal damping negleced! Oupu speed, y, rad/s, is an inegral of he conrol inpu, u! Moor conrol orque is proporional o u! Desired speed,, rad/s, is consan 11 c Angular Rae Model of Dynamics and Speed Conrol Firs-order LTI ordinary differenial equaion dy d = 1 J u = c J e = c [ J! y ], y( 0) given Inegral of he equaion, wih y(0) = 0 y = 1 J! ud = c J 0 = " c J! ed = c J 0![ y ]d + c J 0! 0! 0 [ ]d [ " y ]d Posiive inegraion of Negaive feedback of y 12

7 Angular Rae Soluion of he inegral Sep Response of Speed Conroller Sep inpu : " 0, < 0 y C = 1,! 0 % ( y = * 1! exp! )* " c % J ' + -,- = y () c 1! exp. +, = y ( 1! exp! / + c )*,- where!! = c/j = eigenvalue or roo of he sysem (rad/sec)! " = J/c = ime consan of he response (sec) Wha does he shaf angle response look like? 13 Angle Conrol of Direc-Curren Moor Angular Posiion Simplified Dynamic Model! Roary ineria, J, is he sum of moor and load inerias! Oupu angle, y, is a double inegral of he conrol, u! Desired angle,, is consan Feedback Conrol Law u = ce where e =! y How would y be measured? 14

8 Model of Dynamics and Angle Conrol Angular Posiion 2 nd -order, linear, ime-invarian ordinary differenial equaion d 2 y d 2 =!!y = 1 J u( ) = c J e( ) = c [ J y! ] Oupu angle, y, as a funcion of ime y = c J " "[! y ] d d 0 0 [ ] =! c y J " " d 2 + c J 0 0 " [ ] 0 0 " d 2 15 Model of Dynamics and Angle Conrol Corresponding se of 1 s -order equaions, wih! Angle: x 1 = y! Angular rae: x 2 = dy/d!x 2 = u J!x 1 = x 2 = c [ J! y ] = c [ J! x 1 ] Angular Posiion 16

9 Open-loop dynamic equaion Sae-Space Model of he DC Moor! "!x 1!x 2 % =! 0 1 " 0 0! % " x 1 x 2 % +! 0 " 1 / J u % Feedback conrol law u = c[! y 1 ]= c! x 1 [ ] Closed-loop dynamic equaion! "!x 1!x 2 % =! 0 1 " 'c / J 0! % " x 1 x 2 % +! 0 " c / J % 17 Sep Response wih Angle Feedback! "!x 1!x 2 % =! 0 1 " 'c / J 0! % " x 1 x 2 % +! 0 " c / J % % Sep Response of Undamped Angle Conrol F1 = [0 1;-1 0]; G1 = [0;1]; F2 = [0 1;-0.5 0]; G2 = [0;0.5]; F3 = [0 1; ]; G3 = [0;0.25]; Hx = [1 0;0 1]; c/j = 1, 0.5, and 0.25 Sys1 Sys2 Sys3 = ss(f1,g1,hx,0); = ss(f2,g2,hx,0); = ss(f3,g3,hx,0); sep(sys1,sys2,sys3) 18

10 Conrol law wih rae feedback Wha Wen Wrong? No damping! Soluion: Add rae feedback in he conrol law u = c 1 [! y 1 ]! c 2 y 2 Closed-loop dynamic equaion! "!x 1!x 2 % =! 0 1 'c 1 / J 'c 2 / J "! % " x 1 x 2 % +! 0 c 1 / J " % 19 Alernaive Implemenaions of Rae Feedback dy u = c 1 [! y 1 ]! c 2 y 2 = c 1 [! y 1 ]! c 1 2 d One inpu, wo oupus One inpu, one oupu 20

11 Sep Response wih Angle and Rae Feedback c 1 /J = 1 c 2 /J = 0, 1.414, % Sep Response of Damped Angle Conrol F1 = [0 1;-1 0]; G1 = [0;1]; F1a = [0 1; ]; F1b = [0 1; ]; Hx = [1 0;0 1]; Sys1 Sys2 Sys3 = ss(f1,g1,hx,0); = ss(f1a,g1,hx,0); = ss(f1b,g1,hx,0); sep(sys1,sys2,sys3) 21 LTI Model wih Feedback Conrol Command inpu, u c, has dimensions of u u = u c! Cy x = F x + Gu + Lw y = H x x + H u u 22

12 LTI Conrol wih Forward-Loop Gain u = C[! y ] x = F x + Gu + Lw y = H x x + H u u Wih C c = C, command inpu,, has dimensions of y 23 Effec of Feedback Conrol on he LTI Model [ ] [ ]!x = Fx + Gu = Fx + G u c! Cy { } = F open loop x + G u c! C H x x = [ F! GCH x ]x + Gu c! F closed loop x + Gu c Feedback modifies he sabiliy marix of he closed-loop sysem Convergence or divergence Envelope of ransien response 24

13 LTI Model wih Feedback Conrol and Command Gain Command inpu,, is shaped by C c u = u c! Cy = C c! Cy 25 Effec of Command Gain on LTI Model { } [ ]!x = Fx + Gu = Fx + G C c! Cy { } = Fx + G C c! C H x x = [ F! GCH x ]x + GC c Seady-sae response of he sysem!x = 0 x * =![ F! GCH x ]!1 GC c * Command gain adjuss he seady-sae response Has no effec on he sabiliy of he sysem 26

14 = sin (! ) = sin( 6.28 ), deg Response o Sine Wave Inpu wih Angle Feedback: No Damping c 1 /J = 1; c 2 /J = 0 Why are here 2 oscillaions in he oupu?! Undamped ransien response o he inpu! Long-erm dynamic response o he inpu Sysem has a naural frequency of oscillaion, n Long-erm response o a sine wave is a sine wave 27 Response o Sine Wave Inpu wih Rae Damping = sin (! ) = sin( 6.28 ), deg c 1 /J = 1; c 2 /J = c 1 /J = 1; c 2 /J = Wih damping, ransien response decays In his case, damping has negligible effec on long-erm response 28

15 Sysem Dynamics Produces Differences in Ampliude and Phase Angle of Inpu and Oupu Ampliude Raio (AR) = y Oupu Peak y Inpu Peak ( ) Phase Angle = 360! Inpu Peak Oupu Peak, deg Period of Inpu Ampliude raio and phase angle characerize he sysem model 29 Effec of Inpu Frequency on Oupu Ampliude and Phase Angle = sin( / 6.28), deg Wih low inpu frequency, inpu and oupu ampliudes are abou he same Oupu angle oscillaion lags inpu by a few degrees Rae oscillaion leads angle oscillaion by ~90 deg c 1 /J = 1; c 2 /J =

16 A Higher Frequency, Oupu Ampliude Decreases, Phase Angle Lag Increases = sin( ), deg c 1 /J = 1; c 2 /J = = sin( 6.28 ), deg A Even Higher Frequency, Ampliude Raio Decreases c 1 /J = 1; c 2 /J =

17 Frequency Response of he DC Moor wih Feedback Conrol No Damping Damping More Damping!! Long-erm response o sinusoidal inpus over range of frequencies!! Deermine experimenally!! or from he ransfer funcion!! Frequency response depiced by Bode Plo!! Transfer funcion!! Laplace ransform of sysem!! TBD 33 Nex Time:! Conrol Sysems! 34