MM1: Basic Concept (I): System and its Variables

Transcription

1 MM1: Baic Concept (I): Sytem and it Variable A ytem i a collection of component which are coordinated together to perform a function Sytem interact with their environment. The interaction i defined in term of variable Sytem input Sytem output Environmental diturbance Dynamic ytem i a ytem ehoe performance could change according to time 9/9/2011 Claical Control 1

2 MM1: Baic Concept (II): Control Control i a proce of cauing a ytem (output) variable to conform to ome deired tatu/value Manual Control i a proce where the control i handled by human being() Automatic Control i a control proce which involve machine only 9/9/2011 Claical Control 2

3 MM1: Control Claification Open-loop Control: A control proce which doe not utilize the feedback mechanim, i.e., the output() ha no effect upon the control input() Cloed-loop Control: A control proce which utilize the feedback mechanim, i.e., the output() doe have effect upon the control input() Reference /Set-point 9/9/2011 Claical Control 3

4 MM1: Feedback Control Block Diagram Reference input + - Forward compenator actuator diturbance Plant Feedback compenator enor w r + - D() A() P() F() S() 9/9/2011 Claical Control 4

5 The Goal of thi lecture (MM2)... Eential in uing (ordinary) differential equation model Why ue ODE model Linear v. nonlinear ODE model How to olve an ODE Numerical method (Matlab) Refreh of Laplace tranform Key feature Tranformation from ODE to TF model Block diagram tranformation Compoition /decompoition Signal-flow graph 9/9/2011 Claical Control 5

6 MM2: ODE Model A general ODE model: SISO, SIMO, MISO, MIMO model Linear ytem, Time-invance, Linear Time-Invarance (LTI) Solution of ODE i an explicit decription of dynamic behavior Condition for unique olution of an ODE Solving an ODE: Time-domain method, e.g., uing exponential function Complex-domain method (Laplace tranform) Numerical olution CAD method, e.g., ode23/ode45 9/9/2011 Claical Control 6

7 MM2: Block diagram Rule 9/9/2011 Claical Control 7

8 MM2: Simulink Block diagram Sytem build-up Uing TF block Uing nonlinear block Uing math block Creat ubytem Top-down Bottom-up Uage of ode23 & ode45 9/9/2011 Claical Control 8

9 Goal for thi lecture (MM3) Time repone analyi Typical input 1t, 2nd and higher order ytem Performance pecification of time repone Tranient performance Steady-tate performance Numerical imulation of time repone 9/9/2011 Claical Control 9

10 9/9/2011 Analog and Digital Control 10 MM3: Time Repone Analyi (I) d(t)=0 Typical input u(t) Time repone y(t) TF model ) (0, 0) m( m ),, (0), (0), ( f e a m m m T T v f Laplace Tran Inv Laplace T. n i i i m i i i a b G U G Y 0 0 ) ( ) ( ) ( ) (

11 MM3: Time Repone Analyi (II) Typical input: impule, tep and ramp ignal 1t, 2nd and high-order (LTI) ytem G( ) G( ) time k, p c 1 domain, : pole : pole : p, 1, time contant time contant 1 : p : g( t) ke pt or g( t) c e 1 t Time repone = excitation repone + initial condition repone (free repone) 9/9/2011 Claical Control 11

12 MM3: Performance Specification t t r M t p p 1.8 n %, 16 %, 35 %,, d n d 0.7 n /9/2011 Claical Control 12

13 MM3: Numerical Simulation Impule repone: impule(y) Step repone: tep(y) ltiview(y) Subplot(m,n,1) EXAMPLE: y1: Sy2: num1=[1]; num2=[1 2]; den1=[1 2 1]; den2=[1 2 3]; impule(tf(num1,den1),'r',tf(num2,den2),'b') tep(tf(num1,den1),'r',tf(num2,den2),'b') 9/9/2011 Proce Control 13

14 Goal for thi lecture (MM4) Sytem pole v. time repone Pole and zero Time repone v. Pole location Feedback characteritic Characteritic A imple feedback deign Block diagram decompoition (imulink) 9/9/2011 Claical Control 14

15 MM4 : Pole v Performance Pole location Time repone 9/9/2011 Claical Control 15

16 MM4: Firt-order Sytem 1 G( ), 1 1 pole:, Impulerepone: aume 0 time contant:, 1 y(t) L( ) e 1 1 Steprepone: y(t) L( ( 1) 1 t ) 1e 1 t 63% Time contant why? 16 Time repone i determined by the time contant Sytem pole i the negative of invere time contant 9/9/2011 Claical Control

17 G( ) pole: real(different) real(identical) complex pole: complex pole: 2 MM4: Second-Order Sytem p 2 1,2 n 2 n pole: pole: 1,2 1,2 2, p p n n p p n 1,2 1,2 2 1 j, aume 0,, n n n n j n n n 2 1, 2 1, 0 if 1 if ξ 1 if 0 1 if 0 t t r M t p 1. 8 n %, 0.7 p 16 %, %, 0.3, d n 1 d n 2 9/9/2011 Claical Control 17

18 MM4: Summary of Pole v Performance 9/9/2011 Claical Control 18

19 MM4: Plot of Pole Location 1=tf(1,[1 2 1]); 2=tf(1,[ ]); 3=tf(1,[ ]); 4=tf(1,[1 0 1]); pzmap(1,2,3,4) grid 9/9/2011 Claical Control 19

20 Goal for thi lecture (MM5) Stability analyi Definition of BIBO Pole location Routh criteron Steady-tate error Final Theorem DC-Gain Stead-tate error Effect of zero and additional pole 9/9/2011 Claical Control 20

21 MM5 : BIBO Stability A ytem i aid to have bounded input-bounded output (BIBO) tability if every bounded input reult in a bounded output (regardle of what goe on inide the ytem) The continuou (LTI) ytem with impue repone h(t) i BIBO table if and only if h(t) i abolutely integrallable All ytem pole locate in the left half -plane - aymptotic internal tability Routh Criterion: For a table ytem, there i no change in ign and no zero in the firt column of the Routh array 9/9/2011 Claical Control 21

22 MM5 : Steady-State Error Objective: to know whether or not the repone of a ytem can approach to the reference ignal a time increae Aumption: The conidered ytem i table Analyi method: Tranfer function + final-value Theorem Poition-error contant Velocity contant Acceleration contant 9/9/2011 Claical Control 22 (0) 1 )) ( lim (1 1 ) ( ), ( )) ( (1 lim )) ( ) ( ) ( ( lim )) ( ) ( ( lim ) ( G G R R G R G R Y R e DC-Gain ) ( lim ) ( lim ) ( lim G K G K G K o a o v o p

23 MM5 : Effect of Additional Zero & Pole Chapter 6 An additional zero in the left half-plane will increae the overhoot If the zero i within a factor of 4 of the real part of the complex pole An additional zero in the right half-plane will depre the overhoot and may caue the tep repone to tart out in the wrong direction An additional pole in the left half-plane will increae the rie time ignificantly if the extra pole i within a factor of 4 of the real part of the complex pole 9/9/2011 Claical Control 23

24 BIBO Stability Execie (I) Are thee ytem BIBO table? Intuitive explanation Theoretical analyi 9/9/2011 Claical Control 24

25 BIBO Stability Execie (II) How about the tability of your project ytem? 9/9/2011 Claical Control 25

26 Reviit of example: Firt-order Sytem (II) 10/9 9/0.95 What the tpye of original ytem? Derive the tranfer function of the cloed-loop ytem What the time contant and DC-gain of the CL ytem? What the feedforward gain o that there i no teady-tate error? 9/9/2011 Claical Control 26

27 Goal for thi lecture (MM6) Definition characteriitc of PID control P- controller PI- controller PID controller Ziegler-Nichol tuning method Quarter decay ratio method Ultimate enitivity method 9/9/2011 Claical Control 27

28 Control objective Reference/ Set-point Control i a proce of cauing a ytem (output) variable to conform to ome deired tatu/value (MM1) Control Objective Stable (MM5) Quick reponding (MM3, 4) Adequate diturbance rejection Inenitive to model & meaurement error Avoid exceive control action Suitable for a wide range of operating condition 9/9/2011 (extra Claical reading: Control Goodwin lecture) 28

29 MM6:Characteritic of PID Controller Proportional gain, K p larger value typically mean fater repone. An exceively large proportional gain will lead to proce intability and ocillation. Integral gain, K i larger value imply teady tate error are eliminated more quickly. The trade-off i larger overhoot Derivative gain, K d larger value decreae overhoot, but low down tranient repone and may lead to intability due to ignal noie amplification in the differentiation of the error. R() + - E() K(1+1/T i + T D ) Plant G() Y() 9/9/2011 Claical Control 29

30 MM6: PID Tuning Method- Trial-Error See Hou Ming lexture note 9/9/2011 Claical Control 30

31 MM6: PID Tuning Zieglor Niechol (I) Pre-condition: ytem ha no overhoot of tep repone See Hou Ming lexture note 9/9/2011 Claical Control 31

32 MM6: PID Tuning Zieglor Niechol (II) Pre-condition: ytem order > 2 See Hou Ming lexture note 9/9/2011 Claical Control 32

33 Goal for thi lecture (MM7) Some practical iue when developing a PID controler: Integral windup & Anti-windup method Derivertive kick When to ue which controller? Operational Amplifier Implementation Other tuning method 9/9/2011 Claical Control 33

34 Anti-windup Technique 9/9/2011 Claical Control 34

35 Derivative Kick 1 t u( t) K( e( t) e( ) d TD y(t) ) T t0 1 U() K(1 ) E( ) T T Y( ) Derivative kick: if we have a etpoint change, a pike will be caued by D controller, which i called derivative kick. Derivative kick can be removed by replacing the derivative term with jut output (y), intead of (ret-y) Derivative kick can be reduced by introducing a lowpa filter before the et-point enter the ytem The bandwidth of the filter hould be much larger than the cloed-loop ytem bandwidth I I D 9/9/2011 Claical Control 35

36 Cohen-Coon Tuning Method Ke G ( ) (1t order) 1 Pre-condition: firt-order ytem with ome time delay Objective: ¼ decay ratio & minimum offet 9/9/2011 Claical Control 36

37 Ke G ( ) (1t order) 1 9/9/2011 Claical Control 37

38 Goal for thi lecture (MM8) Eential for frequency domain deign method Bode plot Bode plot analyi How to get a Bode plot What we can gain from Bode plot How to ue bode plot for deign purpoe Stability margin (Gain margin and phae margin) Tranient performance Steady-tate performance Matlab function: bode(), margin() 9/9/2011 Claical Control 38

39 Frequency Repone X() G() Y() The frequency repone G(j) (=G() =j) i a repreentation of the ytem' repone to inuoidal input at varying frequencie G(j) = G(j) e G(j), Input x(n) and output y(n) relationhip Y(j) = H(j) X(j) Y(j) = H(j) + X(j) The frequency repone of a ytem can be viewed via the Bode plot (H.W. Bode ) via the Nyquit diagram 9/9/2011 Analog and Digital Control 39

40 Open-Loop Tranfer Function Motivation Predict the cloed-loop ytem propertie uing the openloop ytem frequency repone Open-loop TF (Loop gain) : L()=KD()G() KD() G() KD() G() Cloed-loop: G cl ()=L()/(1+L()), or G cl ()=G()/(1+L()) 9/9/2011 Analog and Digital Control 40

41 Definition of Phae Margin (PM) Bode plot of the openloop TF The phae margin i the difference in phae between the phae curve and -180 deg at the point correponding to the frequency that give u a gain of 0dB (the gain cro over frequency, Wgc). 9/9/2011 Analog and Digital Control 41

42 Remark of Uing Bode Plot Precondition: The ytem mut be table in open loop if we are going to deign via Bode plot Stability: If the gain croover frequency i le than the phae croover frequency (i.e. Wgc < Wpc), then the cloedloop ytem will be table Damping Ratio: For econd-order ytem, the cloed-loop damping ratio i approximately equal to the phae margin divided by 100 if the phae margin i between 0 and 60 deg A very rough etimate that you can ue i that the bandwidth i approximately equal to the natural frequency 9/9/2011 Analog and Digital Control 42

43 Goal for thi lecture (MM9) A deign example baed on Bode plot Open-loop ytem feature analyi Bode plot baed deign Nyquit Diagram What Nyquit diagram? What we can gain from Nyquit diagram Matlab function: nyquit() 9/9/2011 Claical Control 43

44 Nyquit Diagram: Definition The Nyquit diagram i a plot of G(j), where G() i the open-loop tranfer function and i a vector of frequencie which encloe the entire right-half plane G(j) = G(j) e G(j), The Nyquit diagram plot the poition it the complex plane, while the Bode plot plot it magnitude and phae eparately. 9/9/2011 Claical Control 44

45 Nyquit Criterion for Stability (MM9) The Nyquit criterion tate that: P = the number of open-loop (untable) pole of G()H() N = the number of time the Nyquit diagram encircle 1 clockwie encirclement of -1 count a poitive encirclement counter-clockwie (or anti-clockwie) encirclement of -1 count a negative encirclement Z = the number of right half-plane (poitive, real) pole of the cloed-loop ytem The important equation: Z = P + N 9/9/2011 Claical Control 45

46 Goal for thi lecture (MM10) An illutrative example Frequency repone analyi Frequency repone deign Lead and lag compenator What a lead/lag compenator? Their frequency feature A ytematical procedure for lead compenator deign A practical deign example Beam and Ball Control 9/9/2011 Claical Control 46

47 What have we talked in lecture (MM10)? Lead and lag compenator D()=(+z)/(+p) with z < p or z > p D()=K(T+1)/(T+1), with <1 or >1 A ytematical procedure for lead compenator deign max T 1 1 in 1 in max max Controller KD() Plant G() 9/9/2011 Claical Control 47

48 Exercie Could you repeat the antenna deign uing 1. Continuou lead compenation; 2. Emulation method for digital control; Such that the deign pecification: Overhoot to a tep input le than 5%; Settling time to 1% to be le than 14 ec.; Tracking error to a ramp input of lope 0.01rad/ec to be le than 0.01rad; Sampling time to give at at leat 10 ample in a rie time. (Write your analyi and program on a paper!) 9/9/2011 Claical Control 48

49 1. Introduction - Root Locu K G() Open-loop tran. Func.: KG(); Cloed-loop tran. Func.: KG()/(1+KG()) Senitivity function: 1/(1+KG()) The root locu of an (open-loop) tranfer function KG() i a plot of the location (locu) of all poible cloed loop pole with proportional gain K and unity feedback From the root locu we can elect a gain uch that our cloedloop ytem will perform the way we want 9/9/2011 Claical Control 49

50 Control Deign Uing Root Locu (I) Objective: elect a particular value of K that will meet the pecification for tatic and dynamic 1+KG()=0 Magnitude condition: K=1/ G() 9/9/2011 Claical Control 50

51 Exercie Quetion 5.2 on FC page.321; Conider a DC motor control uing a PI controler D() G() Where the motor i modeled a G()=K/(+1) and PI controller i D()=K p (T i +1)/T i, with parameter K=30, =0.35, T i = Through the root locu method determine the larget vaule of K p uch that =0.45 Try to ue the root locu method to deign a lead compenator for the examplifed attenna ytem. 9/9/2011 Claical Control 51