Introduction to Feedback Systems 1999 2007 nder YKSEL

Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL 5/20/2021 PID Controllers 1

Desired control outcomes Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Reduction in s. s. e. Reduction in overshoot Reduction in rise-time Reduction in settling time 5/20/2021 PID Controllers 2

Effect of proportional feedback Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL K - 5/20/2021 PID Controllers 3

Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Thus as K increases, İncreases, remains unchanged, 5/20/2021 PID Controllers 4

As K increases Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Error constant K s. s. e. decreases 5/20/2021 n TR TS POS s. s. e. PID Controllers increases decreases unchanged increases decreases 5

Example Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL 1 Amplitude 0. 8 K=2 0. 6 >> sys=tf(1, [1 2 2 1]) >> hold off, for k=[. 2, . 5, 1, 2], K=0. 5 step(feedback(k*sys, 1)), hold K=0. 2 on, end K=1 0. 4 0. 2 0 0 10 20 30 40 50 60 Time (sec) 5/20/2021 PID Controllers 6

Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL TR 90, 00 Ts POS sse 80, 00 70, 00 60, 00 50, 00 40, 00 30, 00 20, 00 10, 00 0, 2 5/20/2021 0, 5 PID Controllers 1 2 7

Frequency sensitive compensation Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL 5/20/2021 PID Controllers 8

P-D controller Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL GC(s)=KP + KDs Proportional + derivative control + 5/20/2021 PID Controllers 9

Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL + - 5/20/2021 PID Controllers 10

Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL increases with KD 5/20/2021 PID Controllers inceases with KP 11

By looking at the denominator alone: Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL increasing KP increasing KD n increases unchanged decreases increases TR decreases increases TS unchanged decreases POS increases decreases s. s. e. decreases unchanged 5/20/2021 PID Controllers 12

Example Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL KP+KDs 5/20/2021 PID Controllers 13

As KP increases Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Step Response 0. 9 0. 8 0. 7 Amplitude 0. 6 KP sse 0. 5 0. 4 0. 3 0. 2 0. 1 0 0 2 4 6 8 s. s. e. decreases 10 12 14 16 18 Time (sec) 5/20/2021 PID Controllers 14

As KP increases, overshoot Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Step Response 0. 9 System: untitled 3 Peak amplitude: 0. 891 Overshoot (%): 38. 6 At time (sec): 2. 35 0. 8 0. 7 System: untitled 2 Peak amplitude: 0. 652 Overshoot (%): 30. 4 At time (sec): 2. 43 Amplitude 0. 6 0. 5 0. 4 System: untitled 1 Peak amplitude: 0. 38 Overshoot (%): 128 At time (sec): 1. 93 0. 2 0. 1 0 0 2 4 6 8 10 12 14 16 18 Time (sec) 5/20/2021 PID Controllers 15

As KP increases, rise-time Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Step Response 0. 9 0. 8 0. 7 increases slightly Amplitude 0. 6 0. 5 0. 4 0. 3 0. 2 0. 1 0 0 2 4 6 8 10 12 14 16 18 Time (sec) 5/20/2021 PID Controllers 16

TS as KP increases Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Step Response 0. 9 0. 8 0. 7 Amplitude 0. 6 0. 5 0. 4 0. 3 0. 2 0. 1 0 0 2 4 6 8 10 12 14 16 18 Time (sec) 5/20/2021 PID Controllers 17

As KD increases Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Step Response 0. 8 0. 7 0. 6 Amplitude 0. 5 s. s. e. is not affected! 0. 4 0. 3 0. 2 0. 1 0 0 2 4 6 8 10 12 14 16 18 20 Time (sec) 5/20/2021 PID Controllers 18

As KD increases, rise-time Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL 0. 8 0. 7 0. 6 Amplitude 0. 5 0. 4 Decreases! 0. 3 0. 2 0. 1 0 0 2 4 6 8 10 12 14 16 18 20 Time (sec) 5/20/2021 PID Controllers 19

As KD increases, settling-time Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Step Response 0. 8 0. 7 0. 6 Amplitude 0. 5 0. 4 decreases! 0. 3 0. 2 0. 1 0 0 2 4 6 8 10 12 14 16 18 20 Time (sec) 5/20/2021 PID Controllers 20

Overshoot as KD increases Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Step Response 0. 8 0. 7 0. 6 Amplitude 0. 5 0. 4 0. 3 0. 2 0. 1 0 0 2 4 6 8 10 12 14 16 18 20 Time (sec) 5/20/2021 PID Controllers 21

Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL increasing KP increasing KD TR decreases? decreases TS ? decreases POS ? ? s. s. e. decreases unchanged 5/20/2021 PID Controllers 22

P-I controller Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL GC(s)=KP + KI/s Proportional + integral control + 5/20/2021 PID Controllers 23

Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL + - 5/20/2021 PID Controllers 24

Loop gain is multiplied by: Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Adds a finite zero Modifies d. c. gain 5/20/2021 Adds a pole at the origin PID Controllers 25

An additional pole at the origin Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL • Increases the “Type” of the system – Any “finite” s. s. e. is reduced to zero – Can follow more complex inputs – Stability is weakened! 5/20/2021 PID Controllers 26

An additional finite zero Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL • Advantages of a PD controller can be enjoyed – Rise-time – Settling time 5/20/2021 PID Controllers 27

Modifiable d. c. gain Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL • “finite” steady state error can be improved • Attention!! Stability can easily be lost! 5/20/2021 PID Controllers 28

1 st order example Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL + - 5/20/2021 PID Controllers 29

Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL n increases with KP n increases with KI 5/20/2021 PID Controllers 30

An increase in proportional gain KP Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL • Increases n – Decreases time constant • Decreases settling time TS – Increases damping factor • Decreases overshoot – Rise-time TR increases somewhat – Steady state error is not affected 5/20/2021 PID Controllers 31

An increase in integral gain KI Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL • Increases undamped natural frequency n – Decreases damping factor • Increases overshoot • Decreases rise-time TR – Settling time TS is not affected – Steady-state error to ramp input reduces 5/20/2021 PID Controllers 32

Remarks Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL • In the previous discussion, the effect of the finite zero is overlooked. – It is a function of both KI and KP – May change all results • Finite s. s. e. is reduced to zero by integral control – Improvement provided by KI is on error for higher type input 5/20/2021 PID Controllers 33

3 rd order (previous) example Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL _ 5/20/2021 KP+KI/s PID Controllers 34

For increasing KP Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Step Response 1. 5 s. s. e. is zero Amplitude 1 (for step input) 0. 5 0 0 5 10 15 20 25 30 Time (sec) 5/20/2021 PID Controllers 35

For increasing KP, the settling time TS Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Step Response 1. 5 decreases Amplitude 1 and then ? . . . . 0. 5 0 0 5 10 15 20 25 30 Time (sec) 5/20/2021 PID Controllers 36

For increasing KP, the rise-time TR Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Step Response 1. 5 1 Amplitude decreases 0. 5 0 0 5 10 15 20 25 30 Time (sec) 5/20/2021 PID Controllers 37

For increasing KP, the overshoot Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Step Response 1. 5 decreases Amplitude 1 and …………. ? 0. 5 0 0 5 10 15 20 25 30 Time (sec) 5/20/2021 PID Controllers 38

For increasing KI Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Step Response 1. 5 Settling time increases Amplitude 1 Rise-time decreases Overshoot increases 0. 5 s. s. e. remains zero 0 0 5 10 15 20 25 30 35 Time (sec) 5/20/2021 PID Controllers 39

P-I-D controller Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL GC(s)=KP + KI/s + KDs Proportional + integral + derivative control + - 5/20/2021 + PID Controllers 40

1 st order example Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL KP+KI/s+KDs 5/20/2021 PID Controllers 41

Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL 5/20/2021 PID Controllers 42

KD=Ki=1, KP: 0→ 2 Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL 1. 3 1. 2 1. 1 POS decreases! 1 0. 9 0. 8 0. 7 0. 6 0. 5 0. 4 0 5 10 15 20 25 Time (sec) 5/20/2021 PID Controllers 43

KD=Ki=1, KP: 0→ 2 Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL 1. 3 1. 2 1. 1 POS decreases! 1 Rise-time increases 0. 9 0. 8 0. 7 TS=1. 6, 2. 3, 3. 0, 3. 8, 4. 4 0. 6 0. 5 0. 4 0 5 10 15 20 25 Time (sec) 5/20/2021 PID Controllers 44

KD=Ki=1, KP: 0→ 2 Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL 1. 3 1. 2 1. 1 POS decreases! 1 Rise-time increases 0. 9 0. 8 Settling-time ? ? ? 0. 7 0. 6 0. 5 0. 4 0 5 10 15 20 25 Time (sec) 5/20/2021 PID Controllers 45

KD=KP=1, Ki: 0→ 2 Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL 1. 2 1. 1 POS inceases 1 0. 9 0. 8 0. 7 0. 6 0. 5 0 5 10 15 20 25 30 Time (sec) 5/20/2021 PID Controllers 46

KD=KP=1, Ki: 0→ 2 Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL 1. 2 1. 1 POS inceases 1 0. 9 Rise time decreases considerably! 0. 8 0. 7 0. 6 0. 5 0 5 10 15 20 25 30 Time (sec) 5/20/2021 PID Controllers 47

KD=KP=1, Ki: 0→ 2 Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL 1. 2 1. 1 POS inceases 1 0. 9 Rise time decreases considerably! 0. 8 0. 7 Settling-time ? ? ? 0. 6 0. 5 0 5 10 15 20 25 30 Time (sec) 5/20/2021 PID Controllers 48

KP=Ki=1, KD: 0→ 2 Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL 1. 4 1. 2 POS increases slightly 1 0. 8 0. 6 0. 4 0. 2 0 0 2 4 6 8 10 12 14 16 18 Time (sec) 5/20/2021 PID Controllers 49

KP=Ki=1, KD: 0→ 2 Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Step Response 1. 4 1. 2 POS increases slightly Amplitude 1 0. 8 Rise time ? ? 0. 6 0. 4 TS=2. 2, 2. 8, 3. 0, 2. 9, 2. 8 0. 2 0 0 2 4 6 8 10 12 14 16 18 Time (sec) 5/20/2021 PID Controllers 50

KP=Ki=1, KD: 0→ 2 Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL Step Response 1. 4 1. 2 POS increases slightly Amplitude 1 0. 8 Rise time ? ? 0. 6 0. 4 Settling-time increases 0. 2 0 0 2 4 6 8 10 12 14 16 18 Time (sec) 5/20/2021 PID Controllers 51

Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL 1 0. 9 0. 8 KP 0. 7 KD Ki Open-loop 0. 6 TS TR sse 3. 9 2. 2 0 0. 5 0. 4 0. 3 0. 2 0. 1 0 0 5/20/2021 1 2 3 4 5 6 7 8 9 PID Controllers 10 0 0 ND 10% 10 0 1 >10 . 34 0 100 0 1 . 04. 02 0 10 52

Third order example Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL _ 5/20/2021 P. I. D. PID Controllers 53

Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL 1. 4 1. 2 Open loop 1 PID with KP=2. 2 KD=4. 2 Ki=1 0. 8 0. 6 0. 4 0. 2 0 0 5 10 15 Same overshoot Time (sec) 5/20/2021 PID Controllers 54

Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL 1. 4 1. 2 Open loop 1 PID with KP=2. 2 KD=4. 2 Ki=1 0. 8 0. 6 0. 4 0. 2 0 0 5 10 Time (sec) 5/20/2021 PID Controllers 15 Same settling-time 55

Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL 1. 4 1. 2 Open loop 1 PID with KP=2. 2 KD=4. 2 Ki=1 0. 8 0. 6 0. 4 0. 2 0 0 Rise time reduced by a factor of 3! 5 10 15 Time (sec) 5/20/2021 PID Controllers 56

Introduction to Feedback Systems / © 1999, 2007 Önder YÜKSEL sys=tf([3 1. 5 2], [1 2 2 1 0]); step(feedback(sys, 1)) >> sys=tf([6 3 3], [1 2 2 1 0]); step(feedback(sys, 1)) >> sys=tf([1 1 1], [1 2 2 1 0]); step(feedback(sys, 1)) >> sys=tf([0 1 0], [1 2 2 1 0]); step(feedback(sys, 1)) >> sys=tf([0 1 1], [1 2 2 1 0]); step(feedback(sys, 1)) 5/20/2021 PID Controllers 57