Feedback Control of Dynamic Systems. Yves Briere

Transcription

1 Feedback Control of Dynamic Sytem Yve Briere

2 I. Introduction

3 Introduction Aim of the coure Give a general overview of claical and modern control theory Give a general overview of modern control tool Prerequiite Mathematic : complex number, linear algebra 9/3/9 I. Introduction 3

4 Introduction Tool Matlab / Simulink Book «Feedback Control of Dynamic Sytem», Franklin, Powell, Amami-Naeini, Addion-Weley Pub Co Many many book, webite and free reference... 9/3/9 I. Introduction 4

5 Introduction 7 BC : the clepydra and other hydraulically regulated device for time meaurement (Kteibio) 9/3/9 I. Introduction 5

6 Introduction 36-6 : Ibn al-razzaz al- Jazari The Book of Knowledge of Ingeniou Mechanical Device crank mechanim, connecting rod, programmable automaton, humanoid robot, reciprocating piton engine, uction pipe, uction pump, double-acting pump, valve, combination lock, cam, camhaft, egmental gear, the firt mechanical clock driven by water and weight, and epecially the crankhaft, which i conidered the mot important mechanical invention in hitory after the wheel 9/3/9 I. Introduction 6

7 Introduction 6-9 : pre-indutrial revolution Thermotatic regulator (Corneliu Drebbel ) Water level regulation (fluh toilet, team machine) Windmill peed regulation. 588 : mill hoper ; 745 : fantail by Lee ; 78 : peed regulation by Mead Steam engine preure regulation (D. Papin 77) Centrifugal mechanical governor ( Jame Watt, 788) 9/3/9 I. Introduction 7

8 Introduction : mathematic, bai for control theory Differential equation firt analyi and proof of tability condition for feedback ytem (Lagrange, Hamilton, Poncelet, Airy-84, Hermite-854, Maxwell-868, Routh-877, Vyhnegradky-877, Hurwitz-895, Lyapunov-89) Frequency domain approach (Minorky-9, Black-97, Nyquit-93, Hazen-934) : claical period Frequency domain theory : (Hall-94, Nichol-94, Bode-938) Stochatic approach (Kolmogorov-94, Wiener and Bigelow-94) Information theory (Shannon-948) and cybernetic (Wiener-949) 9/3/9 I. Introduction 8

9 Introduction : modern period, aeronautic and patial indutry Non linear and time varying problem (Hamel-949, Typkin-955, Popov- 96, Yakubovich-96, Sandberg-964, Narendra-964, Deoer-965,Zame- 966) Optimal control and Etimation theory (Bellman-957, Pontryagin-958,Kalman- 96) Control by computer, dicrete ytem theory : (Shannon-95, Jury-96, Ragazzini and Zadeh-95, Ragazzini and Franklin-958,(Kuo-963, Atröm-97) : imulation, computer, etc... 9/3/9 I. Introduction 9

10 Introduction What i automatic control? Baic idea i to enhance open loop control with feedback control Thi eemingly idea i tremendouly powerfull Feedback i a key idea in control Input reference Open loop Controler Input Proce Perturbation Output Perturbation Input reference Cloed loop Controler Input Proce Output Meaurement 9/3/9 I. Introduction

11 Introduction Example : the feedback amplifier Harold Black, 97 R V R - A V Amplifier A ha a high gain (ay 4dB) V V R R A R R R R Reulting gain i determined by paive component! amplification i linear reduced delay noie reduction 9/3/9 I. Introduction

12 Introduction Ue of block diagram Capture the eence of behaviour tandard drawing abtraction information hiding point imilaritie between ytem Same tool for : generation and tranmiion of energy tranmiion of informaiton tranportation (car, aeropace, etc...) indutrial procee, manufacturing mechatronic, intrumentation Biology, medicine, finance, economy... 9/3/9 I. Introduction

13 Introduction Baic propertie of feedback () k - V V A V V V V A ( V k V) ( A k) A k A k V k Reulting gain i determined by feedback! 9/3/9 I. Introduction 3

14 Introduction Baic propertie of feedback () : tatic propertie r - e e k c k c d u u k p k p y y r : reference e : error d : diturbance y : output k c : control gain K p : proce gain d Open loop control : y k p k c e k p d Cloed loop control : kp kc kp y r d k k k k p c p c If k c i big enough y tend to r and d i rejected 9/3/9 I. Introduction 4

15 Introduction Baic propertie of feedback () : dynamic propertie Cloed loop control can : enhance ytem dynamic tabilize an untable ytem make untable a table ytem! 9/3/9 I. Introduction 5

16 Introduction The On-Off or bang-bang controller : u {u max, u min } u u u e e e The proportional controller : uk c.(r y) 9/3/9 I. Introduction 6

17 Introduction The proportional derivative controller u ( t ) k e ( t ) p k d de ( t ) dt Give an idea of future : phae advance The proportional integral controller u t ( t) k e( t) k e( τ) p i dτ e(t) tend to zero! 9/3/9 I. Introduction 7

18 II. A firt controller deign 9/3/9 II. A firt controller deign 8

19 A firt control deign Ue of block diagram Compare feedback and feedforward control Inight feedback propertie : Reduce effect of diturbance Make ytem inenitive to variation Stabilize untable ytem Create well defined relationhip between output and reference Rik of untability PID controler : u( t) k e( t) p k d de dt ( t) k i t e ( τ) dτ 9/3/9 II. A firt controller deign 9

20 Cruie control A cruie control problem : Proce input : ga pedal u Proce output : velocity v F Reference : deired velocity v r Diturbance : lope θ θ mg Contruct a block diagram Undertand how the ytem work Identify the major component and the relevant ignal Key quetion are : Where i the eential dynamic? What are the appropriate abtraction? Decribe the dynamic of the block 9/3/9 II. A firt controller deign

21 Cruie control v r - Controller Throttle Engine F Body v ext. force We made the aumption : Eential dynamic relate velocity to force The force repond intantly to a change in the throttle Relation are linear We can now draw the proce equation 9/3/9 II. A firt controller deign

22 Cruie control Proce linear equation : dv m dt ( t) k v F m g θ Reaonable parameter according to experience : dv dt Where : v in m. - ( t). v u θ u : normalized throttle < u < θ lope in rad 9/3/9 II. A firt controller deign

23 Cruie control Proce linear equation : dv dt PI controller : u ( t). v u θ t ( t) k ( v v( t) ) k ( v v( τ) ) r i r dτ Combining equation lead to : d e dt ( t) ( t) de (. k) ki e( t) dt ( t) dt dθ Integral action Steady tate and θ e! 9/3/9 II. A firt controller deign 3

24 Cruie control Now we can tune k and k i in order to achieve a given dynamic d e dt ( t) How to chooe ω and σ? (. k) de dt ( t) ( t) dx( t) k d x σω ω dt dt i e ( t) x dθ dt ( t) ( t) 9/3/9 II. A firt controller deign 4

25 Cruie control Compare open loop and cloed loop Open loop Cloed loop 9/3/9 II. A firt controller deign 5

26 Cruie control Compare different damping σ (ω.) σ.5 σ σ 9/3/9 II. A firt controller deign 6

27 Cruie control Compare different natural frequencie ω (σ ) ω.5 ω. ω. 9/3/9 II. A firt controller deign 7

28 Cruie control Control tool and method help to : Derive equation from the ytem Manipulate the equation Undertand the equation (tandard model) Qualitative undertanding concept Inight Standard form Computation Find controller parameter Validate the reult by imulation END 9/3/9 II. A firt controller deign 8

29 Standard model Standard model are foundation of the control language Important to : Learn to deal with tandard model Tranform problem to tandard model The tandard model deal with Linear Time Invariant proce (LTI), modelized with Ordinary Differential Equation (ODE) : d n y dt n ( t) d y( t) n a dt n... a n y ( t) b d n dt u n ( t)... b n u( t) 9/3/9 II. A firt controller deign 9

30 Standard model Example (fundamental) : the firt order equation dy ( t) dt dy dt y ( t) y a y ( t) at ( t) y( ) e a y ( t) b u( t) at t a ( ) ( ) ( t τ t y e b e ) u( τ) dτ Input ignal Initial condition 9/3/9 II. A firt controller deign 3

31 Standard model A higher degree model i not o different : d n y dt n ( t) d y( t) n a dt n... a Characteritic polynomial i : n n ( ) a... a n A n y ( t) If polynomial ha n ditinct root α k then the time olution i : y ( t) n k C k e α k t 9/3/9 II. A firt controller deign 3

32 Standard model Real α k root give firt order repone : Complex α k σ±i.ω root give econd order repone : 9/3/9 II. A firt controller deign 3

33 9/3/9 II. A firt controller deign 33 Standard model General cae (input u) : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) τ τ α t n k t k n n n n n n n n d t g e t C t y t u b... dt t u d b t y a... dt t y d a dt t y d k Where : C k (t) are polynomial of t ( ) ( ) α n k t k ' k e t C t g A ytem i table if all pole have negative real part

34 Standard model Tranfer function without knowing anything about Laplace tranform it can be ueful to tore a k and b k coefficient in a convenient way, the tranfer function : ( ) F n n ( ) a... n ( ) b... bn B A a n 9/3/9 II. A firt controller deign 34

35 III. The Laplace tranform 9/3/9 III. Laplace tranform 35

36 Laplace tranform () : convolution u Sytem y We aume the ytem to be LINEAR and TIME INVARIANT y(t) The output (y) of the the ytem i related to the input (u) by the convolution : u(τ) h(t τ) dτ Example : u(t) i an impulion ( everywhere except in t ) y(t) h(t) h(t) i called the impule repone, h(t) decribe completely the ytem Cauality : h(t) if t < 9/3/9 III. Laplace tranform 36

37 Laplace tranform () : definition u(t) h(t) y(t) U() H() Y() Time pace t : real (time) Laplace pace : complex (frequency) x(t) π j c j t x (t) X() e d X() c j y(t) u(τ) h(t τ) dτ X() Y() H(). U() t x(t) e dt Mathematical formula are never ued! 9/3/9 III. Laplace tranform 37

38 Laplace tranform () : propertie Impule fonction t : x(t) infinite x(t) Step fonction : t< : x(t) t> : x(t) X() X() / Derivation : d y(t) x(t) Y ().X() x( ) dt Sinuoïdal fonction : Y() y(t) in( ω t) ω 9/3/9 III. Laplace tranform 38

39 Laplace tranform (3) : propertie Delay : y(t) t x(t t ) Y() X() e d d Initial value theorem : y( ) lim ( Y( ) ) Final value theorem (if limit exit) : y( ) lim ( Y( ) ) 9/3/9 III. Laplace tranform 39

40 Laplace tranform (4) : table From t to 9/3/9 III. Laplace tranform 4

41 Laplace tranform (4) : table From to t 9/3/9 III. Laplace tranform 4

42 Laplace tranform and differential equation ( t) a x ( t) a x( t) b u( t) b u ( t) a ox Theorem of differentiation a o b ( ) a ( X( ) x( )) a ( ( X( ) x( )) x ( )) U( ) b ( U( ) u( )) X 9/3/9 III. Laplace tranform 4

43 9/3/9 III. Laplace tranform 43 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) u U b U b x x X a x X a X a o ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) u b U b b x a x a a X a a a o ( ) ( ) ( ) ( ) ( ) ( ) o o a a a u b x a x a a U a a a b b X Initial condition Tranfer function Laplace tranform and differential equation

44 9/3/9 III. Laplace tranform 44 ( ) ( ) ( ) ( ) I U H X ( ) ( ) ( ) ( ) ( ) ( ) o o a a a u b x a x a a U a a a b b X ( ) ( ) ( ) ( ) t i t u t h t x Initial condition are generally aumed to be null! Laplace tranform and differential equation

45 Finding output repone with Laplace tranform u(t) Sytem y(t) What i the output y(t) from a given input u(t)? u(t) Table of tranform U() Y() H(). U() y(t) Table of tranform Y() 9/3/9 III. Laplace tranform 45

46 Finding final value with Laplace tranform u(t) Sytem y(t) What i the final output y(inf) from a given input u(t)? u(t) y( ) lim ( Y( ) ) Table of tranform Theorem of final value U() Y() H(). U() Y() 9/3/9 III. Laplace tranform 46

47 Pole and zero U H() Y Tranfer function i a ratio of polynomial : Y() U() H() N() D() Pole and zero : Y() b H() U() a Im a b b b a a a ( z ) ( z ) ( p ) ( p ) ( p ) p z p 3 Re 3 zero : z, z, pole : p, p, p 3 Pole and zero are either into the left plane ore into the right plane p z Complex pole and zero have a conjugate 9/3/9 III. Laplace tranform 47

48 Pole are the root of Tranfer function denominator Real value or conjugate complex pair Pole are alo the eigenvalue of matrix A Pole mode 9/3/9 III. Laplace tranform 48

49 Pole and zero : decompoition U H() Y Tranfer function can be expaned into a um of elementary term : Y() U() Y() U() H() H() a α b p b b a a p and p are conjugate : p Y() U() H() α p ω a /3/9 III. Laplace tranform 49 3 α3 p jθ jθ ω e,p ω e α, coθ ω α 3 p Second order 3... Firt order Complex ytem repone i the um of firt order and econd order ytem repone

50 Dynamic repone of firt order ytem U H() p Y p ( ) U( ) Y Example : impule repone u(t) i an impulion ( everywhere, except in : ) u(t) y(t) e p t Table of tranform U() Y ( ) H( ) U( ) Table of tranform ( ) Y p 9/3/9 III. Laplace tranform 5

51 Dynamic repone of firt order ytem U H() p Y p ( ) U( ) Y Example : tep repone u(t) i a tep u(t) y(t) p pt ( e ) Table of tranform ( ) U Y ( ) H( ) U( ) Table of tranform ( ) Y p 9/3/9 III. Laplace tranform 5

52 Propertie of firt order ytem U H() p Y p ( ) U( ) Y Step repone t /p i the time contant of the ytem : after t t, 63% of the final value i obtained 9/3/9 III. Laplace tranform 5

53 Dynamic repone of econd order ytem U H() σ ω ω Y Example : impule repone u(t) i an impulion ( everywhere, except in : ) u(t) y(t) ω σ e σω t à in ( ) ω σ t Table of tranform Table of tranform U() Y ( ) H( ) U( ) Y() σ ω ω 9/3/9 III. Laplace tranform 53

54 Dynamic repone of econd order ytem U H() σ ω ω Y Example : tep repone u(t) i an tep u(t) y(t) ω σ e σω t à in ( ( )) ω σ t ar co σ Table of tranform Table of tranform U()/ Y ( ) H( ) U( ) Y() σ ω ω 9/3/9 III. Laplace tranform 54

55 Propertie of econd order ytem Step repone : U H() y(t) ω σ ω σ Y e ω σω t à in ( ( )) ω σ t ar co σ σ i the damping factor ω i the natural frequency ω σ i the peudo-frequency overhoot 5% of the final value i obtained after : t 5% 3 σ ω Overhoot increae a σ decreae 9/3/9 III. Laplace tranform 55

56 Propertie of econd order ytem U H() σ ω ω Y Step repone (continued) : pole : p ±θ, ω e co(θ)σ Im ω θ Re 9/3/9 III. Laplace tranform 56

57 Stability Stable Im Untable Re untable pole, deverge like exp(t) Any pole with poitive real part i untable table pole, decay like exp(-4.t) Any input (even mall) will lead to intability See animation 9/3/9 III. Laplace tranform 57

58 «fat pole» v «low pole» fat low Im Re Fat pole can be neglected low pole, decay like exp(-t) contant time : t fat pole, decay like exp(-4.t) contant time : t 4 See animation 9/3/9 III. Laplace tranform 58

59 Effect of zero See animation Im Fat zero : neglected Re Slow zero : tranient repone affected Poitive zero : non minimal phae ytem, tep repone tart out in the wrong direction Zero modify the tranient repone 9/3/9 III. Laplace tranform 59

60 Ex. analyi of a feedback ytem Proce model : dv ( t) dt Tranfer function : V U V θ. v u θ V( ). V( ) U( ) V( ). V( ) θ( ) ( ) F( ) ( ). ( ) ( ). 9/3/9 III. Laplace tranform 6

61 Ex. analyi of a feedback ytem Tranfer function of the controller (PID) : ( t) de t u( t) k e( t) k d ki e dt U t E ( ) k k d ki ( t) ( t) dτ 9/3/9 III. Laplace tranform 6

62 Ex. analyi of a feedback ytem We can now combine tranfer function : v r - e PID u - F v V( ) Vr ( ) F F ( ) PID( ) θ ( ) PID( ) E( ) 9/3/9 III. Laplace tranform 6

63 IV. Deign of imple feedback 9/3/9 IV. Deign of imple feedback 63

64 Introduction Standard problem are often firt order or econd order Standard problem tandard olution d r C() u P() y - P b a ( ) P( ) b a b a 9/3/9 IV. Deign of imple feedback 64

65 9/3/9 IV. Deign of imple feedback 65 Control of a firt order ytem Mot phyical problem can be modeled a firt order ytem ( ) a b P Step : tranform your problem in a firt order problem : ( ) k k C i Step : chooe a PI controller Step 3 : combine equation and tune k and in k i in order to achieve the deired cloed loop behavior (ma-pring damper analogy) ( ) ( ) ( ) ( ) ( ) i i b' K k k a b k k a b C P C P CL ω ω σ

66 Dynamic repone of econd order ytem U H() σ ω ω Y Example : impule repone u(t) i an impulion ( everywhere, except in : ) u(t) y(t) ω σ e σω t à in ( ) ω σ t Table of tranform Table of tranform U() Y ( ) H( ) U( ) Y() σ ω ω 9/3/9 IV. Deign of imple feedback 66

67 Dynamic repone of econd order ytem U H() σ ω ω Y Example : tep repone u(t) i an tep u(t) y(t) ω σ e σω t à in ( ( )) ω σ t ar co σ Table of tranform Table of tranform U()/ Y ( ) H( ) U( ) Y() σ ω ω 9/3/9 IV. Deign of imple feedback 67

68 Propertie of econd order ytem Step repone : U H() y(t) ω σ ω σ Y e ω σω t à in ( ( )) ω σ t ar co σ σ i the damping factor ω i the natural frequency ω σ i the peudo-frequency overhoot 5% of the final value i obtained after : t 5% 3 σ ω Overhoot increae a σ decreae 9/3/9 IV. Deign of imple feedback 68

69 Propertie of econd order ytem U H() σ ω ω Y Step repone (continued) : pole : p ±θ, ω e co(θ)σ Im ω θ Re 9/3/9 IV. Deign of imple feedback 69

70 Control of a econd order ytem Step, tep : idem (PI controller) Step 3 : Tranfer function i now third order ( ) CL P ( ) C( ) P( ) C( ) K ( a ) b' σ ω ω dof (k and k i ) : the full dynamic (order 3) cannot be totally choen 9/3/9 IV. Deign of imple feedback 7

71 Simulation tool Matlab or Scilab Tranfer function i a Matlab object Adapted to tranfer function algebra (addition, multiplication ) Simulation, time domain analyi 9/3/9 IV. Deign of imple feedback 7

72 Concluion Laplace Tranform Simulation tool Deign of imple feedback 9/3/9 VI. Deign of imple feedback (Ctd) 7

73 V. Frequency repone 9/3/9 V. Frequency repone 73

74 Introduction Frequency repone : One way to view dynamic Heritage of electrical engineering (Bode) Fit well block diagram Deal with ytem having large order electronic feedback amplifier have order 5-! input output dynamic, black box model, external decription Adapted to experimental determination of dynamic 9/3/9 V. Frequency repone 74

75 The idea of black box u Sytem y The ytem i a black box : forget about the internal detail and focu only on the input-output behavior Frequency repone make a giant table of poible input-output pair Tet entrie are enough to fully decribe LTI ytem - Step repone - Impule repone - inuoid 9/3/9 V. Frequency repone 75

76 What i a LTI ytem A Linear Time Invariant Sytem i : Linear If (u,y ) and (u,y ) are input-output pair then (a.u b.u, a.y b.y ) i an input-output pair : Theorem of uperpoition Time Invariant (u (t),y (t)) i an input-output pair then (u (t-t),y (t-t)) i an inputoutput pair The giant table i dratically implified : y(t) h( t τ) u( τ) ( ) H( ) U( ) Y dτ 9/3/9 V. Frequency repone 76

77 What i the Fourier Tranform Fourier idea : an LTI ytem i completely determined by it repone to inuoidal ignal Tranmiion of inuoid i given by G(jω) The tranfer function G() i uniquely given by it value on the imaginary axe Frequency repone can be experimentally determined The complex number G(jω) tell how a inuoid propagate through the ytem in teady tate : u(t) in y(t) ( ω t) G( jω) in( ω t arg( G( jω) )) 9/3/9 V. Frequency repone 77

78 Steady tate repone Fourier tranform deal with Steady State Repone : u(t) co ( ) U Y y ( ω t) i in( ω t) i ω ( ) G( ) i ω iωt ( t) G( i ω ) e G i iω ( ω ) i ω R k e e α t k t R k α k (Sytem ha ditinct pole α k ) Decay if all α k are negative 9/3/9 V. Frequency repone 78

79 Steady tate repone 9/3/9 V. Frequency repone 79

80 Nyquit tability theorem Nyquit tability theorem tell if a ytem WILL BE table (or not) with a imple feedback - C(). P(). u(t) y(t) L()C().P() - y(t) () Standard ytem with negative unitary feedback () Nyquit tandard form () : if L(i.ω ) - then ocillation will be maintained 9/3/9 V. Frequency repone 8

81 Nyquit tability theorem Step : draw Nyquit curve Imag ω ω- Real ω L(i.ω) Step : where i (-,)? 9/3/9 V. Frequency repone 8

82 Nyquit theorem When the tranfer loop function L doe not have pole in the right half plane the cloed loop ytem i table if the complete Nyquit curve doe not encircle the critical (-,) point. When the tranfer loop function L ha N pole in the right half plane the cloed loop ytem i table if the complete Nyquit curve encircle the critical (-,) point N time. 9/3/9 V. Frequency repone 8

83 Nyquit tability theorem Nyquit tability theorem compare L(i.ω) with (-,) Imag Real (-,) ω L(i.ω) (-,) (-,) STABLE UNSTABLE 9/3/9 V. Frequency repone 83

84 Nyquit theorem Focu on the characteritic equation Difficult to ee how the characteritic equation L i influenced by the controller C Quetion i : how to change C? Strong practical application Poibility to introduce tability margin : how cloe to intability are we? 9/3/9 V. Frequency repone 84

85 Stability margin Stability margin definition : ϕ M Phae margin /g M 45-6 g M Gain margin - 6 (-,) d ϕ M d Shortet ditance to critical point /3/9 V. Frequency repone 85

86 The Bode plot Nyquit theorem i pectacular but not very efficient Impoible to ditinguih C() and P() Bode plot two curve : one for gain, one for phae : db cale for gain Linear cale for phae Logarithmic frequency axi 9/3/9 V. Frequency repone 86

87 The Bode plot Bode plot main propertie : Aymptotic curve (gain multiple of db/dec) are ok Simple interpretation of C() and P() in cacade : Gain db (C().P() Gain db (C()) Gain db (P()) Phae(C().P() Phae(C()) Phae(P()) 9/3/9 V. Frequency repone 87

88 The Bode tability criteria Gain Margin > : cloed loop ytem will be table Phae Margin > : cloed loop ytem will be table One criteria i ufficient in mot cae becaue gain and margin are cloely related 9/3/9 V. Frequency repone 88

89 Cloe loop frequency repone Typical property of H bo () are : H H H H bo bo bf bf ( jω) >> for ω << ωc ( jω) << for ω >> ωc ( jω) ( jω) Hbo H Hbo H ( jω) for ω << ωc bo( jω) ( jω) H jω for ( jω) bo ω >> ω bo( ) c 9/3/9 V. Frequency repone 89

90 Cloe loop frequency repone AdB ω c Log ω Open loop Bode plot Cloe loop Bode plot 9/3/9 V. Frequency repone 9

91 Cloe loop frequency repone AdB Phae margin effect ω c Log ω Open loop Bode plot Cloe loop Bode plot (phae margin 9 ) Cloe loop Bode plot (phae margin 3 ) 9/3/9 V. Frequency repone 9

92 Static gain H bf ( jω) Hbo H ( jω) ( jω) bo For mall value of ω (low frequency) : If H bo (ω) << then H bf (ω) H bo (ω) If H bo (ω) >> then H bf (ω) If H bo (ω) then H bf (ω) 9/3/9 V. Frequency repone 9

93 Controller pecification e() - ε C() u() G() y() Open loop tranfer function : OLTF C.G Cloe loop tranfer function : CLTF C.G / ( C.G) 9/3/9 V. Frequency repone 93

94 Controller pecification Static gain cloe to Low frequency : high gain Perturbation rejection High frequency : low gain Stability phae margin > Bandwith C(jω).G(jω) ω c Cro over frequency ω c Overhoot 5% phae margin 45 Gentle lope in tranition region 9/3/9 V. Frequency repone 94

95 Controller deign Proportionnal feedback Effect : lift gain with no change in phae Bode : hift gain by factor of K C ( ) K 9/3/9 V. Frequency repone 95

96 Controller deign Lead compenation Effect : lift phae by increaing gain at high frequency Very uefull controller : increae phae margin C Bode : add phae between zero and pole ( ) K τ Aτ 9/3/9 V. Frequency repone 96

97 Modern loop haping Ue of rltool (Matlab Control Toolboxe) 9/3/9 V. Frequency repone 97

98 VI. Deign of imple feedback (Ctd) 9/3/9 VI. Deign of imple feedback (Ctd) 98

99 Introduction More complete tandard problem : d n r F() e C() u P() x y - Controller : feedback C() and feedforward F() Load diturbance d : drive the ytem from it deired tate x Meaurement diturbance n : corrupt information about x Main requirement i that proce variable x hould follow reference r 9/3/9 VI. Deign of imple feedback (Ctd) 99

100 Introduction Controller pecification : A. Reduce effect of load diturbance B. Doe not inject too much meaurement noie into the ytem C. Make the cloed loop inenitive to variation in the proce D. Make output follow reference ignal Claical approach : deal with A,B and C with controller C() and deal with D with feedforward F() 9/3/9 VI. Deign of imple feedback (Ctd)

101 Introduction Controller pecification : A. Reduce effect of load diturbance B. Doe not inject too much meaurement noie into the ytem C. Make the cloed loop inenitive to variation in the proce D. Make output follow reference ignal Claical approach : deal with A,B and C with controller C() and deal with D with feedforward F() : Deign procedure Deign the feedback C() too achieve Small enitivity to load diturbance d Low injection of meaurement noie n High robutne to proce variation Then deign F() to achieve deired repone to reference ignal r 9/3/9 VI. Deign of imple feedback (Ctd)

102 9/3/9 VI. Deign of imple feedback (Ctd) Relation between ignal Three intereting ignal (x, y, u) Three poible input (r, d, n) Nine poible tranfer function! r C P F C n C P C d C P C P u r C P F C P n C P d C P P y r C P F C P n C P C P d C P P x Six ditinct tranfer function

103 Relation between ignal Nine frequency repone Bode Diagram From: d From: n From: r To: x - -4 Magnitude (db) To: y - -4 To: u Frequency (rad/ec) 9/3/9 VI. Deign of imple feedback (Ctd) 3

104 Relation between ignal Nine tep repone From: d Step Repone From: n From: r To: x - Amplitude To: y - To: u Time (ec) 9/3/9 VI. Deign of imple feedback (Ctd) 4

105 Relation between ignal A correct deign mean that each tranfer ha to be evaluated Need to be a little bit organized! Need le criteria Concept of enibility function 9/3/9 VI. Deign of imple feedback (Ctd) 5

106 9/3/9 VI. Deign of imple feedback (Ctd) 6 Senibility function r C P F C n C P C d C P C P u r C P F C P n C P d C P P y r C P F C P n C P C P d C P P x C P C P L L T C P L S C P L Senibility function Complementary enibility function Loop enitivity function L tell everything about tability : common denominator of each tranfer function

107 Senibility function LPC tell everything about tability : common denominator of each tranfer function Magnitude (db) Bode Diagram Phae (deg) Frequency (rad/ec) 9/3/9 VI. Deign of imple feedback (Ctd) 7

108 Senibility function S/(L) tell about noie reduction d n r e C u P x y - Without feedback : y ol n P d With feedback control : P y n d P C P C S cl y ol Diturbance with S(iω) < are reduced by feedback Diturbance with S(iω) > are amplified by feedback 9/3/9 VI. Deign of imple feedback (Ctd) 8

109 Senibility function It would be nice to have S(iω) < for all frequencie! Cauchy Integral Theorem : for table open loop ytem : log S( i ω) ( ) For untable or time delayed ytem : Concluion : water bed effect log S(iω) log S i ω > ω 9/3/9 VI. Deign of imple feedback (Ctd) 9

110 Senibility function Nyquit tability criteria : ω, C P < L (-,) P P < T d L /T tell how much P i allowed to vary until ytem become untable C P 9/3/9 VI. Deign of imple feedback (Ctd)

111 Senibility function Nyquit tability criteria : Minimum value of d tell how cloe of intability i the ytem d min i a meaure of robutne : the bigger i M/d the more robut i the ytem (-,) log S(iω) d L C P M ω 9/3/9 VI. Deign of imple feedback (Ctd)

112 VII. Feedforward deign 9/3/9 VII. Feedforward deign

113 Introduction Feedforward i a ueful complement to feedback. Baic propertie are: Reduce effect of diturbance that can be meaured Improve repone to reference ignal No rik for intability - Deign of feedforward i imple but require good model and/or meaurement Beneficial when combined with feedback 9/3/9 VII. Feedforward deign 3

114 Attenuation of meaured diturbance d F u y P P Y P P F ( ) D Diturbance i eliminated if F i choen uch a: F P - Need to meaure d P need to be inverible 9/3/9 VII. Feedforward deign 4

115 Combined Feedback and Feedforward F d r C u P P y Diturbance d i attenuated both by F and C : ( ) Y P P F D P C 9/3/9 VII. Feedforward deign 5

116 Sytem invere The ideal feedforward need to compute the invere of P. That might be tricky Example: ( ) P ( ) P ( ) P ( ) ( ) e F P ( ) ( ) ( ) F P e F( ) P ( ) Differentiation Prediction Untable 9/3/9 VII. Feedforward deign 6

117 Approximate ytem invere The ideal feedforward need to compute the invere of P. That might be tricky Example: ( ) P ( ) P ( ) P ( ) ( ) e F P ( ) ( ) ( ) F P e F( ) P ( ) Differentiation Prediction Untable 9/3/9 VII. Feedforward deign 7

118 Approximate ytem invere Since it i difficult to obtain an exact invere we have to approximate. One poibility i to find the tranfer function which minimize : ( ( ) ( )) J u t v t dt V Where: P X U And where U i a particular input (ex: a tep ignal). Thi give for intance: ( ) P P ( ) ( ) e ( ) P ( ) P T P ( ) ( ) F P 9/3/9 VII. Feedforward deign 8

119 Improved repone to reference ignal The reference ignal can be injected after the controller: M u r M y y m C u P y y m i the deired trajectory. Chooe M u M y / P Deign concern: M u approximated M y adapted uch that M y /P feaible 9/3/9 VII. Feedforward deign 9

120 Combining feedback and feedforward Feedback Cloed loop Act only when there are deviation Market driven Robut to model error Rik for intability Feedforward Open loop Act before deviation how up Planning Not robut to model error No rik for intability Feedforward mut be ued a a complement to feedback. Require good modeling. 9/3/9 VII. Feedforward deign

121 VIII. State feedback 9/3/9 VIII. State feedback

122 Introduction - Simple deign become difficult for high order ytem - What i the State concept? - State are the variable that fully ummarize the actual tate of the ytem - Future can be fully predicted from the current tate - State i the ideal bai for control 9/3/9 VIII. State feedback

123 State feedback Let u uppoe the ytem i decribed by the following equation (x i a vector, A, B and C are matrixe) : dx A x B u dt y C x The general linear controller i : u K x L u The cloed loop ytem then become : dx A x B( K x L u) ( A B K) x B L u dt y C x The cloed loop ytem ha the characteritic equation: ( ) ( ) ( ) P det I A B K 9/3/9 VIII. State feedback 3

124 State feedback Let u uppoe the ytem i decribed by the following equation (x i a vector, A, B and C are matrixe) : dx A x B u dt y C x The general linear controller i : u K x L u The cloed loop ytem then become : dx A x B( K x L u) ( A B K) x B L u dt y C x The cloed loop ytem ha the characteritic equation: ( ) ( ) ( ) P det I A B K Main mathematical tool i linear algebra and matrixe! 9/3/9 VIII. State feedback 4

125 Pole placement Original (open loop) ytem behavior depend on it pole, olution of the characteritic equation: POL ( ) det ( I A) Cloed loop ytem behavior depend on it pole, olution of the characteritic equation: CL ( ) ( ) ( ) P det I A B K Need to tune N parameter (N : dimenion of x and K) Appropriate choice of K allow to place the pole anywhere! (Need imple mathematical kill (not detailed here ) Two problem : obervability, controllability 9/3/9 VIII. State feedback 5

126 Pole placement Pole of the OL ytem Pole of the CL ytem 9/3/9 VIII. State feedback 6

127 Firt problem : obervability In the control feedback equation x i uppoed to be known. If one can acce (meaure) x, there i no problem. Sometime, x cannot be meaured but can be oberved. Sytem decribed by: dx A x B u dt y C x Only u and y acceible, A and B known. Solution i to etimate internal tate x with a tate oberver of gain K o : dxob A x B u K y y dt yob C xob ( ) ob ob ob Appropriate choice of K ob minimize y ob y : x ob tend to x ( ob ) Pole of the oberver are the pole of: P( ) det I ( A K C) 9/3/9 VIII. State feedback 7

128 Firt problem : obervability In the control feedback equation x i uppoed to be known. If one can acce (meaure) x, there i no problem. Sometime, x cannot be meaured but can be oberved. Sytem decribed by: dx A x B u dt y C x Only u and y acceible, A and B known. Solution i to etimate internal tate x with a tate oberver of gain K o : dxob A x B u K y y dt yob C xob ( ) ob ob ob Appropriate choice of K ob minimize y ob y : x ob tend to x ( ob ) Pole of the oberver are the pole of: P( ) det I ( A K C) 9/3/9 VIII. State feedback 8

129 Firt problem : obervability Pole of the oberver are the pole thoe of: ( ob ) ( ) ( ) P det I A K C Pole of the ytem Pole of the oberver 9/3/9 VIII. State feedback 9

130 Firt problem : obervability Problem : i the ytem obervable? In mot cae : ye Sometime, the tate i not obervable : The oberver doe not converge to the true tate, whatever K ob i. Can be derived from a mathematical analye of (A,C): rank(a,ac,aac,aaac ) N 9/3/9 VIII. State feedback 3

131 Combining an oberver and a tate feedback True (with x) tate feedback can be replaced by an oberved (x ob ) tate feedback: u K xob L u Pole of the OL ytem Pole of the oberver Pole of the CL ytem 9/3/9 VIII. State feedback 3

132 Second problem : controllability Sometime a tate i not controllable : mean that whatever the command u i, ome part of the tate are not controllable Can be derived from a mathematical analye of (A,B): rank(a,ab,aab,aaab ) N Problem if : - A tate i not controllable and untable - A tate i not controllable and low No problem if : - A tate i not controllable and fat (decay rapidly) 9/3/9 VIII. State feedback 3