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# Industrial Control Systems - PID Controllers

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### Industrial Control Systems - PID Controllers

1. 1. Industrial Control Behzad Samadi Department of Electrical Engineering Amirkabir University of Technology Winter 2010 Tehran, Iran Behzad Samadi (Amirkabir University) Industrial Control 1 / 95
2. 2. Feedback Control Loop r: reference signal y: process (controlled) variable u: manipulated (control) variable e: control error d: load disturbance signal n: measurement noise signal F: feedforward ﬁlter C: controller P: plant [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 2 / 95
3. 3. On-Oﬀ Control One of the simplest control laws: u = { umax if e > 0 umin if e < 0 Behzad Samadi (Amirkabir University) Industrial Control 3 / 95
4. 4. On-Oﬀ Control One of the simplest control laws: u = { umax if e > 0 umin if e < 0 Disadvantage: persistent oscillation of the process variable P = 1 10s + 1 e−2s , umax = 2, umin = 0 Behzad Samadi (Amirkabir University) Industrial Control 3 / 95
5. 5. On-Oﬀ Control One of the simplest control laws: u = { umax if e > 0 umin if e < 0 Disadvantage: persistent oscillation of the process variable P = 1 10s + 1 e−2s , umax = 2, umin = 0 Behzad Samadi (Amirkabir University) Industrial Control 3 / 95
6. 6. On-Oﬀ Control One of the simplest control laws: u = { umax if e > 0 umin if e < 0 Disadvantage: persistent oscillation of the process variable P = 1 10s + 1 e−2s , umax = 2, umin = 0 Behzad Samadi (Amirkabir University) Industrial Control 3 / 95
7. 7. On-Oﬀ Control a) Ideal on-oﬀ controller b) Modiﬁed with a dead zone c) Modiﬁed with hysteresys [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 4 / 95
8. 8. PID Control 1 Proportional action 2 Integral action 3 Derivative action [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 5 / 95
9. 9. Proportional Action Proportional control action: u(t) = Kpe(t) = Kp(r(t) − y(t)), [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 6 / 95
10. 10. Proportional Action Proportional control action: u(t) = Kpe(t) = Kp(r(t) − y(t)), Kp: proportional gain [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 6 / 95
11. 11. Proportional Action Proportional control action: u(t) = Kpe(t) = Kp(r(t) − y(t)), Kp: proportional gain Controller transfer function: C(s) = Kp [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 6 / 95
12. 12. Proportional Action Proportional control action: u(t) = Kpe(t) = Kp(r(t) − y(t)), Kp: proportional gain Controller transfer function: C(s) = Kp Advantage: small control signal for a small error signal [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 6 / 95
13. 13. Proportional Action Proportional control action: u(t) = Kpe(t) = Kp(r(t) − y(t)), Kp: proportional gain Controller transfer function: C(s) = Kp Advantage: small control signal for a small error signal Disadvantage: steady state error [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 6 / 95
14. 14. Proportional Action Steady state error occurs even if the process presents an integrating dynamics, in case a constant load disturbance occurs. [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 7 / 95
15. 15. Proportional Action Steady state error occurs even if the process presents an integrating dynamics, in case a constant load disturbance occurs. Adding a bias (or reset) term: u(t) = Kpe + ub The value of ub can be ﬁxed or can be adjusted manually until the steady state error is zero. [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 7 / 95
16. 16. Proportional Action Proportional Band (PB): PB = 100% Kp [Astrom and Hagglund, 1995] Behzad Samadi (Amirkabir University) Industrial Control 8 / 95
17. 17. Integral Action Integral control action: u(t) = Ki ∫ t 0 e(𝜏)d𝜏, [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 9 / 95
18. 18. Integral Action Integral control action: u(t) = Ki ∫ t 0 e(𝜏)d𝜏, Ki : integral gain [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 9 / 95
19. 19. Integral Action Integral control action: u(t) = Ki ∫ t 0 e(𝜏)d𝜏, Ki : integral gain Controller transfer function: C(s) = Ki s [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 9 / 95
20. 20. Integral Action Integral control action: u(t) = Ki ∫ t 0 e(𝜏)d𝜏, Ki : integral gain Controller transfer function: C(s) = Ki s Advantage: zero steady state error [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 9 / 95
21. 21. Integral Action Integral control action: u(t) = Ki ∫ t 0 e(𝜏)d𝜏, Ki : integral gain Controller transfer function: C(s) = Ki s Advantage: zero steady state error Disadvantage: integrator windup in the presence of saturation [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 9 / 95
22. 22. PI Controller Proportional Integrator Controller: Transfer function: C(s) = Kp(1 + 1 Ti s ) [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 10 / 95
23. 23. PI Controller Proportional Integrator Controller: Transfer function: C(s) = Kp(1 + 1 Ti s ) Integral action is able to set automatically the value of ub. [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 10 / 95
24. 24. PI Controller Proportional Integrator Controller: Transfer function: C(s) = Kp(1 + 1 Ti s ) Integral action is able to set automatically the value of ub. The integral action is also called automatic reset. [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 10 / 95
25. 25. Derivative Action Derivative control action: u(t) = Kd de(t) dt , [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 11 / 95
26. 26. Derivative Action Derivative control action: u(t) = Kd de(t) dt , Kd : derivative gain [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 11 / 95
27. 27. Derivative Action Derivative control action: u(t) = Kd de(t) dt , Kd : derivative gain Controller transfer function: C(s) = Kd s [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 11 / 95
28. 28. Derivative Action Derivative control action: u(t) = Kd de(t) dt , Kd : derivative gain Controller transfer function: C(s) = Kd s Advantage: Derivative action is an instance of predictive control. [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 11 / 95
29. 29. Derivative Action Derivative control action: u(t) = Kd de(t) dt , Kd : derivative gain Controller transfer function: C(s) = Kd s Advantage: Derivative action is an instance of predictive control. Disadvantage: Sensitive to the measurement noise in the manipulated variable [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 11 / 95
30. 30. Derivative Action Derivative action is an instance of predictive control. [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 12 / 95
31. 31. Derivative Action Derivative action is an instance of predictive control. Taylor series expansion of the control error at time Td ahead: e(t + Td ) ≈ e(t) + Td de(t) dt [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 12 / 95
32. 32. Derivative Action Derivative action is an instance of predictive control. Taylor series expansion of the control error at time Td ahead: e(t + Td ) ≈ e(t) + Td de(t) dt A control law proportional to e(t + Td ) u(t) = Kp ( e(t) + Td de(t) dt ) [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 12 / 95
33. 33. Derivative Action Derivative action is an instance of predictive control. Taylor series expansion of the control error at time Td ahead: e(t + Td ) ≈ e(t) + Td de(t) dt A control law proportional to e(t + Td ) u(t) = Kp ( e(t) + Td de(t) dt ) Derivative action is also called anticipatory control, or rate action, or pre-act. [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 12 / 95
34. 34. PID Controller Transfer function: C(s) = Kp ( 1 + 1 Ti s + Td s ) Time windows: Proportional action responds to current error. Integrator action responds to accumulated past error. Derivative action anticipated future error. Peter Woolf umich.edu Behzad Samadi (Amirkabir University) Industrial Control 13 / 95
35. 35. PID Controller Transfer function: C(s) = Kp + Ki s + Kd s Frequency band: Proportional action: all-band Integrator action: low pass Derivative action: high pass [Li et al., 2006] Behzad Samadi (Amirkabir University) Industrial Control 14 / 95
36. 36. PID Controller Transfer function: C(s) = Kp + Ki s + Kd s [Li et al., 2006] Behzad Samadi (Amirkabir University) Industrial Control 15 / 95
37. 37. PID Controller Implementation methods: Pneumatic Hydraulic Electronic Digital Behzad Samadi (Amirkabir University) Industrial Control 16 / 95
38. 38. PID Controller Structures of PID controllers: Ideal or noninteracting form: Ci (s) = Kp ( 1 + 1 Ti s + Td s ) [Visioli, 2006] and [Astrom and Hagglund, 1995] Behzad Samadi (Amirkabir University) Industrial Control 17 / 95
39. 39. PID Controller Structures of PID controllers: Ideal or noninteracting form: Ci (s) = Kp ( 1 + 1 Ti s + Td s ) Series or interacting from: Cs(s) = K ′ p ( 1 + 1 T ′ i s ) (1 + T ′ d s) [Visioli, 2006] and [Astrom and Hagglund, 1995] Behzad Samadi (Amirkabir University) Industrial Control 17 / 95
40. 40. PID Controller Structures of PID controllers: Ideal or noninteracting form: Ci (s) = Kp ( 1 + 1 Ti s + Td s ) Series or interacting from: Cs(s) = K ′ p ( 1 + 1 T ′ i s ) (1 + T ′ d s) Parallel form: Ci (s) = Kp + Ki s + Kd s [Visioli, 2006] and [Astrom and Hagglund, 1995] Behzad Samadi (Amirkabir University) Industrial Control 17 / 95
41. 41. PID Controller Structures of PID controllers: [Astrom and Hagglund, 1995] Behzad Samadi (Amirkabir University) Industrial Control 18 / 95
42. 42. PID Controller Structures of PID controllers: Series to ideal form conversion: Kp =K ′ p T ′ i + T ′ d T ′ i Ti =T ′ i + Td ′ Td = T ′ i T ′ d T ′ i + T ′ d Behzad Samadi (Amirkabir University) Industrial Control 19 / 95
43. 43. PID Controller Structures of PID controllers: Ideal to series form conversion: Only if Ti ≥ 4Td K ′ p = Kp 2 ( 1 + √ 1 − 4 Td Ti ) T ′ i = Ti 2 ( 1 + √ 1 − 4 Td Ti ) T ′ d = Ti 2 ( 1 − √ 1 − 4 Td Ti ) [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 20 / 95
44. 44. PID Controller Alternative series form: Cs(s) = Kp(1 + 1 𝛼Ti s )(𝛼 + Td s) Ideal to alternative series form conversion: Only if Ti ≥ 4Td 𝛼 = 1 ± √ 1 − 4Td Ti 2 > 0 [Li et al., 2006] Behzad Samadi (Amirkabir University) Industrial Control 21 / 95
45. 45. PID Controller A PID controller has two zeros and one pole at the origin. Ti > 4Td : two real zeros Ti = 4Td : two coincident zeros Ti < 4Td : two complex conjugate zeros [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 22 / 95
46. 46. Problems with the Derivative Action Noise: n(t) = A sin(𝜔t) [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 23 / 95
47. 47. Problems with the Derivative Action Noise: n(t) = A sin(𝜔t) Derivative action: u(t) = A𝜔 cos(𝜔t) [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 23 / 95
48. 48. Problems with the Derivative Action Noise: n(t) = A sin(𝜔t) Derivative action: u(t) = A𝜔 cos(𝜔t) u(t) is large for high frequencies. [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 23 / 95
49. 49. Problems with the Derivative Action Noise: n(t) = A sin(𝜔t) Derivative action: u(t) = A𝜔 cos(𝜔t) u(t) is large for high frequencies. In practice, a (very) noisy control signal might lead to a damage of the actuator. [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 23 / 95
50. 50. Modiﬁed Derivative Action Modiﬁed ideal form: Ci1a(s) = Kp ( 1 + 1 Ti s + Td s Td N s + 1 ) [Visioli, 2006]Behzad Samadi (Amirkabir University) Industrial Control 24 / 95
51. 51. Modiﬁed Derivative Action Modiﬁed ideal form: Ci1a(s) = Kp ( 1 + 1 Ti s + Td s Td N s + 1 ) Gerry and Shinskey, 2005: Ci1b(s) = Kp ⎛ ⎜ ⎝1 + 1 Ti s + Td s 1 + Td N s + 0.5 ( Td N s )2 ⎞ ⎟ ⎠ [Visioli, 2006]Behzad Samadi (Amirkabir University) Industrial Control 24 / 95
52. 52. Modiﬁed Derivative Action Modiﬁed ideal form: Ci1a(s) = Kp ( 1 + 1 Ti s + Td s Td N s + 1 ) Gerry and Shinskey, 2005: Ci1b(s) = Kp ⎛ ⎜ ⎝1 + 1 Ti s + Td s 1 + Td N s + 0.5 ( Td N s )2 ⎞ ⎟ ⎠ Modiﬁed series form: Cs(s) = K ′ p ( 1 + 1 T ′ i s ) ⎛ ⎝T ′ d s + 1 T ′ d N s + 1 ⎞ ⎠ [Visioli, 2006]Behzad Samadi (Amirkabir University) Industrial Control 24 / 95
53. 53. Modiﬁed Derivative Action Modiﬁed ideal form: Ci1a(s) = Kp ( 1 + 1 Ti s + Td s Td N s + 1 ) Gerry and Shinskey, 2005: Ci1b(s) = Kp ⎛ ⎜ ⎝1 + 1 Ti s + Td s 1 + Td N s + 0.5 ( Td N s )2 ⎞ ⎟ ⎠ Modiﬁed series form: Cs(s) = K ′ p ( 1 + 1 T ′ i s ) ⎛ ⎝T ′ d s + 1 T ′ d N s + 1 ⎞ ⎠ N generally assumes a value between 1 and 33, although in the majority of the practical cases its setting falls between 8 and 16 (Ang et al., 2005). [Visioli, 2006]Behzad Samadi (Amirkabir University) Industrial Control 24 / 95
54. 54. Modiﬁed Derivative Action Alternative modiﬁed ideal form: Ci2a(s) = Kp ( 1 + 1 Ti s + Td s ) 1 Tf s + 1 [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 25 / 95
55. 55. Modiﬁed Derivative Action Alternative modiﬁed ideal form: Ci2a(s) = Kp ( 1 + 1 Ti s + Td s ) 1 Tf s + 1 ˚Astr¨om and H¨agglund, 2004: Ci2b(s) = Kp ( 1 + 1 Ti s + Td s ) 1 (Tf s + 1)2 [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 25 / 95
56. 56. Modiﬁed Derivative Action Alternative modiﬁed ideal form: Ci2a(s) = Kp ( 1 + 1 Ti s + Td s ) 1 Tf s + 1 ˚Astr¨om and H¨agglund, 2004: Ci2b(s) = Kp ( 1 + 1 Ti s + Td s ) 1 (Tf s + 1)2 Derivative kick: A spike in the control signal due to an abrupt (stepwise) change of the set-point signal. [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 25 / 95
57. 57. Modiﬁed Derivative Action Alternative modiﬁed ideal form: Ci2a(s) = Kp ( 1 + 1 Ti s + Td s ) 1 Tf s + 1 ˚Astr¨om and H¨agglund, 2004: Ci2b(s) = Kp ( 1 + 1 Ti s + Td s ) 1 (Tf s + 1)2 Derivative kick: A spike in the control signal due to an abrupt (stepwise) change of the set-point signal. If the set-point is constant, the derivative action can be applied only to the process variable: u(t) = −Kd dy(t) dt [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 25 / 95
58. 58. Derivative Action 80% of the employed PID controllers have the derivative part switched-oﬀ (Ang et al., 2005). [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 26 / 95
59. 59. Derivative Action 80% of the employed PID controllers have the derivative part switched-oﬀ (Ang et al., 2005). Derivative action is the most diﬃcult to tune, why? [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 26 / 95
60. 60. Derivative Action 80% of the employed PID controllers have the derivative part switched-oﬀ (Ang et al., 2005). Derivative action is the most diﬃcult to tune, why? Consider a ﬁrst-order-plus-dead-time (FOPDT) plant: P(s) = K Ts + 1 e−Ls and a PD controller: C(s) = Kp(1 + Td s) [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 26 / 95
61. 61. Derivative Action 80% of the employed PID controllers have the derivative part switched-oﬀ (Ang et al., 2005). Derivative action is the most diﬃcult to tune, why? Consider a ﬁrst-order-plus-dead-time (FOPDT) plant: P(s) = K Ts + 1 e−Ls and a PD controller: C(s) = Kp(1 + Td s) Open loop frequency response: ∣C(j𝜔)P(j𝜔)∣ = KKp √ 1 + T2 d 𝜔2 1 + T2𝜔2 [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 26 / 95
62. 62. Derivative Action Open loop frequency response: KKp √ 1 + T2 d 𝜔2 1 + T2𝜔2 ≥ KKp min ( 1, Td T ) [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 27 / 95
63. 63. Derivative Action Open loop frequency response: KKp √ 1 + T2 d 𝜔2 1 + T2𝜔2 ≥ KKp min ( 1, Td T ) Td ≥ T ⇒ min ( 1, Td T ) = 1 [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 27 / 95
64. 64. Derivative Action Open loop frequency response: KKp √ 1 + T2 d 𝜔2 1 + T2𝜔2 ≥ KKp min ( 1, Td T ) Td ≥ T ⇒ min ( 1, Td T ) = 1 If Td ≥ T and KKp > 1, then ∣C(j𝜔)P(j𝜔)∣ ≥ 1 [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 27 / 95
65. 65. Derivative Action Open loop frequency response: KKp √ 1 + T2 d 𝜔2 1 + T2𝜔2 ≥ KKp min ( 1, Td T ) Td ≥ T ⇒ min ( 1, Td T ) = 1 If Td ≥ T and KKp > 1, then ∣C(j𝜔)P(j𝜔)∣ ≥ 1 Td ≤ T ⇒ min ( 1, Td T ) = Td T [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 27 / 95
66. 66. Derivative Action Open loop frequency response: KKp √ 1 + T2 d 𝜔2 1 + T2𝜔2 ≥ KKp min ( 1, Td T ) Td ≥ T ⇒ min ( 1, Td T ) = 1 If Td ≥ T and KKp > 1, then ∣C(j𝜔)P(j𝜔)∣ ≥ 1 Td ≤ T ⇒ min ( 1, Td T ) = Td T If Td ≤ T and KKp Td T > 1, then ∣C(j𝜔)P(j𝜔)∣ ≥ 1 [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 27 / 95
67. 67. Frequency Response The magnitude of the open-loop transfer function is not less than 0 dB. As a consequence, since the phase decreases when the frequency increases because of the time delay, the closed-loop system will be unstable. [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 28 / 95
68. 68. Derivative Action Consider the following process: P(s) = 2 s + 1 e−0.2s controlled by a PID controller in series form with Kp = 1 and Ti = 1. Behzad Samadi (Amirkabir University) Industrial Control 29 / 95
69. 69. Derivative Action Consider the following process: P(s) = 2 s + 1 e−0.2s controlled by a PID controller in series form with Kp = 1 and Ti = 1. If Td = 0.01, GM=12.3dB, PM=68.2 deg. Behzad Samadi (Amirkabir University) Industrial Control 29 / 95
70. 70. Derivative Action Consider the following process: P(s) = 2 s + 1 e−0.2s controlled by a PID controller in series form with Kp = 1 and Ti = 1. If Td = 0.01, GM=12.3dB, PM=68.2 deg. If Td = 0.05, GM=13.2dB, PM=72.7 deg. Behzad Samadi (Amirkabir University) Industrial Control 29 / 95
71. 71. Derivative Action Consider the following process: P(s) = 2 s + 1 e−0.2s controlled by a PID controller in series form with Kp = 1 and Ti = 1. If Td = 0.01, GM=12.3dB, PM=68.2 deg. If Td = 0.05, GM=13.2dB, PM=72.7 deg. If Td = 0.5, the system stability is lost! Behzad Samadi (Amirkabir University) Industrial Control 29 / 95
72. 72. Derivative Action Consider the following process: P(s) = 2 s + 1 e−0.2s controlled by a PID controller in series form with Kp = 1 and Ti = 1. If Td = 0.01, GM=12.3dB, PM=68.2 deg. If Td = 0.05, GM=13.2dB, PM=72.7 deg. If Td = 0.5, the system stability is lost! Behzad Samadi (Amirkabir University) Industrial Control 29 / 95
73. 73. Derivative Action Consider the following process: P(s) = 2 s + 1 e−0.2s controlled by a PID controller in series form with Kp = 1 and Ti = 1. If Td = 0.01, GM=12.3dB, PM=68.2 deg. If Td = 0.05, GM=13.2dB, PM=72.7 deg. If Td = 0.5, the system stability is lost! In summary: Sensitive to noise Hard to tune (4 parameters) [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 29 / 95
74. 74. Integral Windup Behzad Samadi (Amirkabir University) Industrial Control 30 / 95
75. 75. Integral Windup Solid: process output, Dashed: process input, Dotted: integral term [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 30 / 95
76. 76. Conditional Integration The integral term is limited to a predeﬁned value. [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 31 / 95
77. 77. Conditional Integration The integral term is limited to a predeﬁned value. The integration is stopped when the error is greater than a predeﬁned threshold, namely, when the process variable value is far from the setpoint value. [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 31 / 95
78. 78. Conditional Integration The integral term is limited to a predeﬁned value. The integration is stopped when the error is greater than a predeﬁned threshold, namely, when the process variable value is far from the setpoint value. The integration is stopped when the control variable saturates, i.e., when u ′ ∕= u. [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 31 / 95
79. 79. Conditional Integration The integral term is limited to a predeﬁned value. The integration is stopped when the error is greater than a predeﬁned threshold, namely, when the process variable value is far from the setpoint value. The integration is stopped when the control variable saturates, i.e., when u ′ ∕= u. The integration is stopped when the control variable saturates and the control error and the control variable have the same sign (i.e., when ue > 0). [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 31 / 95
80. 80. Anti-windup for Automatic Reset Conﬁguration Automatic reset: [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 32 / 95
81. 81. Anti-windup for Automatic Reset Conﬁguration Automatic reset: Limiting the controller output: [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 32 / 95
82. 82. Anti-windup for Automatic Reset Conﬁguration Automatic reset: Limiting the controller output: Limiting the integrator output: [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 32 / 95
83. 83. Back-calculation Integrator input: ei = Kp Ti e + 1 Tt (u ′ − u) Behzad Samadi (Amirkabir University) Industrial Control 33 / 95
84. 84. Back-calculation Integrator input: ei = Kp Ti e + 1 Tt (u ′ − u) Tuning rule for Tt (˚Astr¨om and H¨agglund 1995): Tt = √ Td Ti Behzad Samadi (Amirkabir University) Industrial Control 33 / 95
85. 85. Back-calculation Integrator input: ei = Kp Ti e + 1 Tt (u ′ − u) Tuning rule for Tt (˚Astr¨om and H¨agglund 1995): Tt = √ Td Ti Behzad Samadi (Amirkabir University) Industrial Control 33 / 95
86. 86. Back-calculation Integrator input: ei = Kp Ti e + 1 Tt (u ′ − u) Tuning rule for Tt (˚Astr¨om and H¨agglund 1995): Tt = √ Td Ti Not useful for a PI controller [Visioli, 2006]Behzad Samadi (Amirkabir University) Industrial Control 33 / 95
87. 87. Back-calculation Integrator input: ei = Kp Ti e + 1 Tt (u ′ − u) [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 34 / 95
88. 88. Back-calculation Integrator input: ei = Kp Ti e + 1 Tt (u ′ − u) Bohn and Atherton, 1995 suggest Tt = Ti . [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 34 / 95
89. 89. Back-calculation Integrator input: ei = Kp Ti e + 1 Tt (u ′ − u) Bohn and Atherton, 1995 suggest Tt = Ti . Conditioning technique (Hanus et al., 1987;Walgama et al., 1991): This is a tracking rule (u tracks u ′ ). In this framework: Tt = Kp [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 34 / 95
90. 90. PID with Tracking Input SP: Setpoint, MV: Manipulated Variable, TR: Tracking Input [Astrom and Hagglund, 1995] Behzad Samadi (Amirkabir University) Industrial Control 35 / 95
91. 91. Bumpless Transfer Bumpless Transfer between Manual (M) and Automatic (A) A is to track M in this case. [Visioli, 2006] Behzad Samadi (Amirkabir University) Industrial Control 36 / 95
92. 92. Bumpless Transfer Bumpless Transfer between Manual (M) and Automatic (A) Behzad Samadi (Amirkabir University) Industrial Control 37 / 95
93. 93. Bumpless Transfer Bumpless Transfer between Manual (M) and Automatic (A) Incremental manual input [Astrom and Hagglund, 1995] Behzad Samadi (Amirkabir University) Industrial Control 37 / 95
94. 94. Bumpless Transfer Manual Control Module: [Astrom and Hagglund, 1995] Behzad Samadi (Amirkabir University) Industrial Control 38 / 95
95. 95. Bumpless Transfer PID with Manual Switch: [Astrom and Hagglund, 1995] Behzad Samadi (Amirkabir University) Industrial Control 39 / 95
96. 96. PID Controller Design In process industries, more than 97% of the regulatory controllers are of the PID type. [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 40 / 95
97. 97. PID Controller Design In process industries, more than 97% of the regulatory controllers are of the PID type. Most loops are actually under PI control (as a result of the large number of ﬂow loops). [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 40 / 95
98. 98. PID Controller Design In process industries, more than 97% of the regulatory controllers are of the PID type. Most loops are actually under PI control (as a result of the large number of ﬂow loops). Pulp and paper industry over 2000 loops: [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 40 / 95
99. 99. PID Controller Design In process industries, more than 97% of the regulatory controllers are of the PID type. Most loops are actually under PI control (as a result of the large number of ﬂow loops). Pulp and paper industry over 2000 loops: Only 20% of loops worked well (i.e. less variability in the automatic mode over the manual mode). [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 40 / 95
100. 100. PID Controller Design In process industries, more than 97% of the regulatory controllers are of the PID type. Most loops are actually under PI control (as a result of the large number of ﬂow loops). Pulp and paper industry over 2000 loops: Only 20% of loops worked well (i.e. less variability in the automatic mode over the manual mode). 30% gave poor performance due to poor controller tuning. [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 40 / 95
101. 101. PID Controller Design In process industries, more than 97% of the regulatory controllers are of the PID type. Most loops are actually under PI control (as a result of the large number of ﬂow loops). Pulp and paper industry over 2000 loops: Only 20% of loops worked well (i.e. less variability in the automatic mode over the manual mode). 30% gave poor performance due to poor controller tuning. 30% gave poor performance due to control valve problems (e.g. control valve stick-slip, dead band, backlash). [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 40 / 95
102. 102. PID Controller Design In process industries, more than 97% of the regulatory controllers are of the PID type. Most loops are actually under PI control (as a result of the large number of ﬂow loops). Pulp and paper industry over 2000 loops: Only 20% of loops worked well (i.e. less variability in the automatic mode over the manual mode). 30% gave poor performance due to poor controller tuning. 30% gave poor performance due to control valve problems (e.g. control valve stick-slip, dead band, backlash). 20% gave poor performance due to process and/or control system design problems. [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 40 / 95
103. 103. PID Controller Design Process industries: [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 41 / 95
104. 104. PID Controller Design Process industries: 30% of loops operated on manual mode. [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 41 / 95
105. 105. PID Controller Design Process industries: 30% of loops operated on manual mode. 20% of controllers used factory tuning. [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 41 / 95
106. 106. PID Controller Design Process industries: 30% of loops operated on manual mode. 20% of controllers used factory tuning. 30% gave poor performance due to sensor and control valve problems. [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 41 / 95
107. 107. PID Controller Design Process industries: 30% of loops operated on manual mode. 20% of controllers used factory tuning. 30% gave poor performance due to sensor and control valve problems. Chemical process industry: [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 41 / 95
108. 108. PID Controller Design Process industries: 30% of loops operated on manual mode. 20% of controllers used factory tuning. 30% gave poor performance due to sensor and control valve problems. Chemical process industry: Half of the control valves needed to be ﬁxed (results of the Fisher diagnostic valve package). [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 41 / 95
109. 109. PID Controller Design Process industries: 30% of loops operated on manual mode. 20% of controllers used factory tuning. 30% gave poor performance due to sensor and control valve problems. Chemical process industry: Half of the control valves needed to be ﬁxed (results of the Fisher diagnostic valve package). Most poor tuning was due to control valve problems. [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 41 / 95
110. 110. PID Controller Design Process industries: 30% of loops operated on manual mode. 20% of controllers used factory tuning. 30% gave poor performance due to sensor and control valve problems. Chemical process industry: Half of the control valves needed to be ﬁxed (results of the Fisher diagnostic valve package). Most poor tuning was due to control valve problems. Reﬁning, chemicals, and pulp and paper industries over 26,000 controllers: [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 41 / 95
111. 111. PID Controller Design Process industries: 30% of loops operated on manual mode. 20% of controllers used factory tuning. 30% gave poor performance due to sensor and control valve problems. Chemical process industry: Half of the control valves needed to be ﬁxed (results of the Fisher diagnostic valve package). Most poor tuning was due to control valve problems. Reﬁning, chemicals, and pulp and paper industries over 26,000 controllers: Only 32% of loops were classiﬁed as excellent or acceptable. [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 41 / 95
112. 112. PID Controller Design Process industries: 30% of loops operated on manual mode. 20% of controllers used factory tuning. 30% gave poor performance due to sensor and control valve problems. Chemical process industry: Half of the control valves needed to be ﬁxed (results of the Fisher diagnostic valve package). Most poor tuning was due to control valve problems. Reﬁning, chemicals, and pulp and paper industries over 26,000 controllers: Only 32% of loops were classiﬁed as excellent or acceptable. 32% of controllers were classiﬁed as fair or poor, which indicates unacceptably sluggish or oscillatory responses. [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 41 / 95
113. 113. PID Controller Design Process industries: 30% of loops operated on manual mode. 20% of controllers used factory tuning. 30% gave poor performance due to sensor and control valve problems. Chemical process industry: Half of the control valves needed to be ﬁxed (results of the Fisher diagnostic valve package). Most poor tuning was due to control valve problems. Reﬁning, chemicals, and pulp and paper industries over 26,000 controllers: Only 32% of loops were classiﬁed as excellent or acceptable. 32% of controllers were classiﬁed as fair or poor, which indicates unacceptably sluggish or oscillatory responses. 36% of controllers were on open- loop, which implies that the controllers were either in manual or virtually saturated. [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 41 / 95
114. 114. PID Controller Design Process industries: 30% of loops operated on manual mode. 20% of controllers used factory tuning. 30% gave poor performance due to sensor and control valve problems. Chemical process industry: Half of the control valves needed to be ﬁxed (results of the Fisher diagnostic valve package). Most poor tuning was due to control valve problems. Reﬁning, chemicals, and pulp and paper industries over 26,000 controllers: Only 32% of loops were classiﬁed as excellent or acceptable. 32% of controllers were classiﬁed as fair or poor, which indicates unacceptably sluggish or oscillatory responses. 36% of controllers were on open- loop, which implies that the controllers were either in manual or virtually saturated. PID algorithms are used in vast majority of applications (97%). For the rare cases of complex dynamics or signiﬁcant dead time, other algorithms are used. [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 41 / 95
115. 115. PID Controller Design Ziegler-Nichols PID Tuning Method: Ziegler-Nichols closed-loop tuning method Ziegler-Nichols open-loop tuning method Ziegler, Nichols, “Optimum settings for automatic controllers”, Trans. ASME, 64, pp. 759-768, 1942 [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 42 / 95
116. 116. PID Controller Design Ziegler-Nichols closed-loop tuning method: 1 Remove integral and derivative action. Behzad Samadi (Amirkabir University) Industrial Control 43 / 95
117. 117. PID Controller Design Ziegler-Nichols closed-loop tuning method: 1 Remove integral and derivative action. 2 Create a small disturbance in the loop by changing the set point. Adjust the proportional, increasing and/or decreasing, the gain until the oscillations have constant amplitude. Behzad Samadi (Amirkabir University) Industrial Control 43 / 95
118. 118. PID Controller Design Ziegler-Nichols closed-loop tuning method: 1 Remove integral and derivative action. 2 Create a small disturbance in the loop by changing the set point. Adjust the proportional, increasing and/or decreasing, the gain until the oscillations have constant amplitude. 3 Record the gain value (Ku) and period of oscillation (Pu). Behzad Samadi (Amirkabir University) Industrial Control 43 / 95
119. 119. PID Controller Design Ziegler-Nichols closed-loop tuning method: 1 Remove integral and derivative action. 2 Create a small disturbance in the loop by changing the set point. Adjust the proportional, increasing and/or decreasing, the gain until the oscillations have constant amplitude. 3 Record the gain value (Ku) and period of oscillation (Pu). Behzad Samadi (Amirkabir University) Industrial Control 43 / 95
120. 120. PID Controller Design Ziegler-Nichols closed-loop tuning method: 1 Remove integral and derivative action. 2 Create a small disturbance in the loop by changing the set point. Adjust the proportional, increasing and/or decreasing, the gain until the oscillations have constant amplitude. 3 Record the gain value (Ku) and period of oscillation (Pu). [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 43 / 95
121. 121. PID Controller Design Ziegler-Nichols closed-loop tuning method: [Love, 2007] Behzad Samadi (Amirkabir University) Industrial Control 44 / 95
122. 122. PID Controller Design Ziegler-Nichols closed-loop tuning method (ultimate cycle method): C(s) = Kp(1 + 1 Ti s + Td s) Kp Ti Td P Controller Ku/2 - - PI Controller Ku/2.2 Pu/1.2 - PID Controller Ku/1.7 Pu/2 Pu/8 [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 45 / 95
123. 123. PID Controller Design Ziegler-Nichols closed-loop tuning method: Advantages: [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 46 / 95
124. 124. PID Controller Design Ziegler-Nichols closed-loop tuning method: Advantages: Easy experiment; only need to change the P controller [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 46 / 95
125. 125. PID Controller Design Ziegler-Nichols closed-loop tuning method: Advantages: Easy experiment; only need to change the P controller Includes dynamics of whole process, which gives a more accurate picture of how the system is behaving [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 46 / 95
126. 126. PID Controller Design Ziegler-Nichols closed-loop tuning method: Advantages: Easy experiment; only need to change the P controller Includes dynamics of whole process, which gives a more accurate picture of how the system is behaving Disadvantages: [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 46 / 95
127. 127. PID Controller Design Ziegler-Nichols closed-loop tuning method: Advantages: Easy experiment; only need to change the P controller Includes dynamics of whole process, which gives a more accurate picture of how the system is behaving Disadvantages: Experiment can be time consuming [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 46 / 95
128. 128. PID Controller Design Ziegler-Nichols closed-loop tuning method: Advantages: Easy experiment; only need to change the P controller Includes dynamics of whole process, which gives a more accurate picture of how the system is behaving Disadvantages: Experiment can be time consuming Can venture into unstable regions while testing the P controller, which could cause the system to become out of control [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 46 / 95
129. 129. PID Controller Design Ziegler-Nichols closed-loop tuning method: Advantages: Easy experiment; only need to change the P controller Includes dynamics of whole process, which gives a more accurate picture of how the system is behaving Disadvantages: Experiment can be time consuming Can venture into unstable regions while testing the P controller, which could cause the system to become out of control It does not hold for I, D and PD controllers. [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 46 / 95
130. 130. PID Controller Design Process Reaction Curve: P: the size of the step disturbance in the setpoint L: the time taken from the moment the disturbance was introduced to the ﬁrst sign of change in the output signal ΔCp: the change in output signal in response to the initial step disturbance T: the time taken for this change to occur [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 47 / 95
131. 131. PID Controller Design Process Reaction Curve: N = ΔCp T : reaction rate [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 48 / 95
132. 132. PID Controller Design Ziegler-Nichols open-loop tuning method (process reaction method): C(s) = Kp(1 + 1 Ti s + Td s) Kp Ti Td P Controller K - - PI Controller 0.9K L/0.3 - PID Controller 1.2K 2L 0.5L K = P NL [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 49 / 95
133. 133. PID Controller Design Ziegler-Nichols open-loop tuning method: Advantages: [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 50 / 95
134. 134. PID Controller Design Ziegler-Nichols open-loop tuning method: Advantages: Quick and easier to use than other methods [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 50 / 95
135. 135. PID Controller Design Ziegler-Nichols open-loop tuning method: Advantages: Quick and easier to use than other methods It is a robust and popular method [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 50 / 95
136. 136. PID Controller Design Ziegler-Nichols open-loop tuning method: Advantages: Quick and easier to use than other methods It is a robust and popular method Of these two techniques, the Process Reaction Method is the easiest and least disruptive to implement [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 50 / 95
137. 137. PID Controller Design Ziegler-Nichols open-loop tuning method: Advantages: Quick and easier to use than other methods It is a robust and popular method Of these two techniques, the Process Reaction Method is the easiest and least disruptive to implement Disadvantages: [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 50 / 95
138. 138. PID Controller Design Ziegler-Nichols open-loop tuning method: Advantages: Quick and easier to use than other methods It is a robust and popular method Of these two techniques, the Process Reaction Method is the easiest and least disruptive to implement Disadvantages: It depends upon purely proportional measurement to estimate I and D controllers. [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 50 / 95
139. 139. PID Controller Design Ziegler-Nichols open-loop tuning method: Advantages: Quick and easier to use than other methods It is a robust and popular method Of these two techniques, the Process Reaction Method is the easiest and least disruptive to implement Disadvantages: It depends upon purely proportional measurement to estimate I and D controllers. Approximations for the Kc , Ti , and Td values might not be entirely accurate for diﬀerent systems. [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 50 / 95
140. 140. PID Controller Design Ziegler-Nichols open-loop tuning method: Advantages: Quick and easier to use than other methods It is a robust and popular method Of these two techniques, the Process Reaction Method is the easiest and least disruptive to implement Disadvantages: It depends upon purely proportional measurement to estimate I and D controllers. Approximations for the Kc , Ti , and Td values might not be entirely accurate for diﬀerent systems. It does not hold for I, D and PD controllers. [Woolf, 2007] Behzad Samadi (Amirkabir University) Industrial Control 50 / 95
141. 141. PID Controller Design Summary: Ziegler-Nichols closed-loop tuning method: Gain margin of 2 Ziegler-Nichols open-loop tuning method: Decay ratio of 0.25 Behzad Samadi (Amirkabir University) Industrial Control 51 / 95
142. 142. PID Controller Design Ziegler-Nichols closed-loop tuning methods: G(s) = Ke−Ds 𝜏s + 1 [Yu, 2007] Behzad Samadi (Amirkabir University) Industrial Control 52 / 95
143. 143. PID Controller Design Ziegler-Nichols closed-loop tuning methods: G(s) = Ke−td s 𝜏s + 1 Quarter decay ration for 0 < td 𝜏 < 1 [O’Dwyer, 2002] [Chau, 2002] Behzad Samadi (Amirkabir University) Industrial Control 53 / 95
144. 144. PID Controller Design Quantities to characterize the error: Maximum error: max e(t) [Astrom and Hagglund, 1995] Behzad Samadi (Amirkabir University) Industrial Control 54 / 95
145. 145. PID Controller Design Quantities to characterize the error: Maximum error: max e(t) Integrated Absolute Error: IAE = ∫ ∞ 0 ∣e(t)∣dt [Astrom and Hagglund, 1995] Behzad Samadi (Amirkabir University) Industrial Control 54 / 95
146. 146. PID Controller Design Quantities to characterize the error: Maximum error: max e(t) Integrated Absolute Error: IAE = ∫ ∞ 0 ∣e(t)∣dt Integrated Error for non-oscillatory processes: IE = ∫ ∞ 0 e(t)dt [Astrom and Hagglund, 1995] Behzad Samadi (Amirkabir University) Industrial Control 54 / 95
147. 147. PID Controller Design Quantities to characterize the error: Maximum error: max e(t) Integrated Absolute Error: IAE = ∫ ∞ 0 ∣e(t)∣dt Integrated Error for non-oscillatory processes: IE = ∫ ∞ 0 e(t)dt Integrated Squared Error: ISE = ∫ ∞ 0 e2(t)dt [Astrom and Hagglund, 1995] Behzad Samadi (Amirkabir University) Industrial Control 54 / 95
148. 148. PID Controller Design Quantities to characterize the error: Maximum error: max e(t) Integrated Absolute Error: IAE = ∫ ∞ 0 ∣e(t)∣dt Integrated Error for non-oscillatory processes: IE = ∫ ∞ 0 e(t)dt Integrated Squared Error: ISE = ∫ ∞ 0 e2(t)dt Integrated Time Absolute Error: ITAE = ∫ ∞ 0 t∣e(t)∣dt [Astrom and Hagglund, 1995] Behzad Samadi (Amirkabir University) Industrial Control 54 / 95
149. 149. PID Controller Design Quantities to characterize the error: Maximum error: max e(t) Integrated Absolute Error: IAE = ∫ ∞ 0 ∣e(t)∣dt Integrated Error for non-oscillatory processes: IE = ∫ ∞ 0 e(t)dt Integrated Squared Error: ISE = ∫ ∞ 0 e2(t)dt Integrated Time Absolute Error: ITAE = ∫ ∞ 0 t∣e(t)∣dt Integrated Time Error: ITE = ∫ ∞ 0 te(t)dt [Astrom and Hagglund, 1995] Behzad Samadi (Amirkabir University) Industrial Control 54 / 95
150. 150. PID Controller Design Quantities to characterize the error: Maximum error: max e(t) Integrated Absolute Error: IAE = ∫ ∞ 0 ∣e(t)∣dt Integrated Error for non-oscillatory processes: IE = ∫ ∞ 0 e(t)dt Integrated Squared Error: ISE = ∫ ∞ 0 e2(t)dt Integrated Time Absolute Error: ITAE = ∫ ∞ 0 t∣e(t)∣dt Integrated Time Error: ITE = ∫ ∞ 0 te(t)dt Integrated Time Squared Error: ITSE = ∫ ∞ 0 te2(t)dt [Astrom and Hagglund, 1995] Behzad Samadi (Amirkabir University) Industrial Control 54 / 95
151. 151. PID Controller Design Quantities to characterize the error: Maximum error: max e(t) Integrated Absolute Error: IAE = ∫ ∞ 0 ∣e(t)∣dt Integrated Error for non-oscillatory processes: IE = ∫ ∞ 0 e(t)dt Integrated Squared Error: ISE = ∫ ∞ 0 e2(t)dt Integrated Time Absolute Error: ITAE = ∫ ∞ 0 t∣e(t)∣dt Integrated Time Error: ITE = ∫ ∞ 0 te(t)dt Integrated Time Squared Error: ITSE = ∫ ∞ 0 te2(t)dt Integrated Squared Time Error: ISTE = ∫ ∞ 0 t2e2(t)dt [Astrom and Hagglund, 1995] Behzad Samadi (Amirkabir University) Industrial Control 54 / 95
152. 152. PID Controller Design ITAE optimization: G(s) = Ke−td s 𝜏s + 1 [Chau, 2002] Behzad Samadi (Amirkabir University) Industrial Control 55 / 95
153. 153. PID Controller Design Ciancone and Marlin Tuning: G(s) = Ke−td s 𝜏s + 1 Fractional dead time: Tf = td td +𝜏 Using Tf , compute 𝜇CM and 𝜏CM: Tf 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 𝜇CM 1.1 1.1 1.8 1.1 1.0 0.8 0.59 0.42 0.32 𝜏CM 0.23 0.23 0.23 0.72 0.72 0.70 0.67 0.60 0.53 Compute the controller gains: Kp = 𝜇CM K , Ti = 𝜏CM(td + 𝜏) Minimizing IAE or ISE [Chau, 2002] Behzad Samadi (Amirkabir University) Industrial Control 56 / 95
154. 154. PID Controller Design Ciancone and Marlin PI Tuning: [Chau, 2002] Behzad Samadi (Amirkabir University) Industrial Control 57 / 95
155. 155. PID Controller Design Ciancone and Marlin PID Tuning: [Chau, 2002] Behzad Samadi (Amirkabir University) Industrial Control 58 / 95
156. 156. PID Controller Design Direct Synthesis: We have: C R = GcGp 1 + GcGp Therefore: Gc = 1 Gp C/R 1 − C/R [Chau, 2002] Behzad Samadi (Amirkabir University) Industrial Control 59 / 95
157. 157. PID Controller Design Direct Synthesis: C R = 1 𝜏cs + 1 ⇒ Gc = 1 Gp ( 1 𝜏cs ) Behzad Samadi (Amirkabir University) Industrial Control 60 / 95
158. 158. PID Controller Design Direct Synthesis: C R = 1 𝜏cs + 1 ⇒ Gc = 1 Gp ( 1 𝜏cs ) Example: Gp = Kp 𝜏ps + 1 ⇒ Gc = 𝜏p Kp𝜏c ( 1 + 1 𝜏ps ) [Chau, 2002] Behzad Samadi (Amirkabir University) Industrial Control 60 / 95
159. 159. PID Controller Design Direct Synthesis: For delayed systems: C R = e−𝜃s 𝜏cs + 1 ⇒ Gc = 1 Gp ( e−𝜃s (𝜏cs + 1) − e−𝜃s ) Behzad Samadi (Amirkabir University) Industrial Control 61 / 95
160. 160. PID Controller Design Direct Synthesis: For delayed systems: C R = e−𝜃s 𝜏cs + 1 ⇒ Gc = 1 Gp ( e−𝜃s (𝜏cs + 1) − e−𝜃s ) Considering e−𝜃s ≈ 1 − 𝜃s: Gc ≈ 1 Gp ( e−𝜃s (𝜏c + 𝜃)s ) Behzad Samadi (Amirkabir University) Industrial Control 61 / 95
161. 161. PID Controller Design Direct Synthesis: For delayed systems: C R = e−𝜃s 𝜏cs + 1 ⇒ Gc = 1 Gp ( e−𝜃s (𝜏cs + 1) − e−𝜃s ) Considering e−𝜃s ≈ 1 − 𝜃s: Gc ≈ 1 Gp ( e−𝜃s (𝜏c + 𝜃)s ) Behzad Samadi (Amirkabir University) Industrial Control 61 / 95
162. 162. PID Controller Design Direct Synthesis: For delayed systems: C R = e−𝜃s 𝜏cs + 1 ⇒ Gc = 1 Gp ( e−𝜃s (𝜏cs + 1) − e−𝜃s ) Considering e−𝜃s ≈ 1 − 𝜃s: Gc ≈ 1 Gp ( e−𝜃s (𝜏c + 𝜃)s ) Example: Gp = Kpe−td s 𝜏ps + 1 ⇒ Gc = 𝜏p Kp(𝜏c + 𝜃) ( 1 + 1 𝜏ps ) for 𝜃 = td [Chau, 2002] Behzad Samadi (Amirkabir University) Industrial Control 61 / 95
163. 163. PID Controller Design Direct Synthesis: Second order underdamped desired response: C R = 1 𝜏2s2 + 2𝜁𝜏s + 1 ⇒ Gc = 1 Gp ( 1 𝜏2s2 + 2𝜁𝜏s ) Behzad Samadi (Amirkabir University) Industrial Control 62 / 95
164. 164. PID Controller Design Direct Synthesis: Second order underdamped desired response: C R = 1 𝜏2s2 + 2𝜁𝜏s + 1 ⇒ Gc = 1 Gp ( 1 𝜏2s2 + 2𝜁𝜏s ) Behzad Samadi (Amirkabir University) Industrial Control 62 / 95
165. 165. PID Controller Design Direct Synthesis: Second order underdamped desired response: C R = 1 𝜏2s2 + 2𝜁𝜏s + 1 ⇒ Gc = 1 Gp ( 1 𝜏2s2 + 2𝜁𝜏s ) Example: Gp = Kp (𝜏1s + 1)(𝜏2s + 1) ⇒ Gc = (𝜏1s + 1)(𝜏2s + 1) Kp𝜏s(𝜏s + 2𝜁) Behzad Samadi (Amirkabir University) Industrial Control 62 / 95
166. 166. PID Controller Design Direct Synthesis: Second order underdamped desired response: C R = 1 𝜏2s2 + 2𝜁𝜏s + 1 ⇒ Gc = 1 Gp ( 1 𝜏2s2 + 2𝜁𝜏s ) Example: Gp = Kp (𝜏1s + 1)(𝜏2s + 1) ⇒ Gc = (𝜏1s + 1)(𝜏2s + 1) Kp𝜏s(𝜏s + 2𝜁) Assuming 𝜏2 > 𝜏1 and 𝜏 = 2𝜁𝜏2 Gc = 𝜏1 4Kp𝜁2𝜏2 ( 1 + 1 𝜏1s ) [Chau, 2002] Behzad Samadi (Amirkabir University) Industrial Control 62 / 95
167. 167. PID Controller Design Internal Model Control: Assume that we have an approximate model ˜Gp of the process Gp Open loop: Gc = ˜G−1 p Behzad Samadi (Amirkabir University) Industrial Control 63 / 95
168. 168. PID Controller Design Internal Model Control: Assume that we have an approximate model ˜Gp of the process Gp Open loop: Gc = ˜G−1 p Closed loop: Behzad Samadi (Amirkabir University) Industrial Control 63 / 95
169. 169. PID Controller Design Internal Model Control: P = G★ c (R − C + ˜C) = G★ c (R − C + ˜GpP) Behzad Samadi (Amirkabir University) Industrial Control 64 / 95
170. 170. PID Controller Design Internal Model Control: P = G★ c (R − C + ˜C) = G★ c (R − C + ˜GpP) Therefore: P = G★ c 1 − G★ c ˜Gp (R−C) Behzad Samadi (Amirkabir University) Industrial Control 64 / 95
171. 171. PID Controller Design Internal Model Control: P = G★ c (R − C + ˜C) = G★ c (R − C + ˜GpP) Therefore: P = G★ c 1 − G★ c ˜Gp (R−C) P = Gc(R − C) Therefore: Gc = G★ c 1−G★ c ˜Gp [Chau, 2002]Behzad Samadi (Amirkabir University) Industrial Control 64 / 95
172. 172. PID Controller Design Internal Model Control: Closed loop transfer function: C = [ (1 − G★ c ˜Gp)GL 1 + G★ c (Gp − ˜Gp) ] L + [ GpG★ c 1 + G★ c (Gp − ˜Gp) ] R [Chau, 2002] Behzad Samadi (Amirkabir University) Industrial Control 65 / 95
173. 173. PID Controller Design Internal Model Control: Closed loop transfer function: C = [ (1 − G★ c ˜Gp)GL 1 + G★ c (Gp − ˜Gp) ] L + [ GpG★ c 1 + G★ c (Gp − ˜Gp) ] R How to choose G★ c : ˜Gp = ˜Gp+ ˜Gp− ˜Gp+ contains all positive zeros if any. [Chau, 2002] Behzad Samadi (Amirkabir University) Industrial Control 65 / 95
174. 174. PID Controller Design Internal Model Control: Closed loop transfer function: C = [ (1 − G★ c ˜Gp)GL 1 + G★ c (Gp − ˜Gp) ] L + [ GpG★ c 1 + G★ c (Gp − ˜Gp) ] R How to choose G★ c : ˜Gp = ˜Gp+ ˜Gp− ˜Gp+ contains all positive zeros if any. The design is based on ˜Gp− only: G★ c = 1 ˜Gp− [ 1 𝜏cs + 1 ]r r = 1, 2 [Chau, 2002] Behzad Samadi (Amirkabir University) Industrial Control 65 / 95
175. 175. PID Controller Design Pad´e Approximation of Delays: e−td s ≈ Nd (s) Dd (s) http://mathworld.wolfram.com/PadeApproximant.html Behzad Samadi (Amirkabir University) Industrial Control 66 / 95
176. 176. PID Controller Design Pad´e Approximation of Delays: e−td s ≈ Nd (s) Dd (s) Gd1/0(s) = 1 − td s http://mathworld.wolfram.com/PadeApproximant.html Behzad Samadi (Amirkabir University) Industrial Control 66 / 95
177. 177. PID Controller Design Pad´e Approximation of Delays: e−td s ≈ Nd (s) Dd (s) Gd1/0(s) = 1 − td s Gd0/1(s) = 1 1+td s http://mathworld.wolfram.com/PadeApproximant.html Behzad Samadi (Amirkabir University) Industrial Control 66 / 95
178. 178. PID Controller Design Pad´e Approximation of Delays: e−td s ≈ Nd (s) Dd (s) Gd1/0(s) = 1 − td s Gd0/1(s) = 1 1+td s Gd1/1(s) = − td 2 s+1 td 2 s+1 http://mathworld.wolfram.com/PadeApproximant.html Behzad Samadi (Amirkabir University) Industrial Control 66 / 95
179. 179. PID Controller Design Pad´e Approximation of Delays: e−td s ≈ Nd (s) Dd (s) Gd1/0(s) = 1 − td s Gd0/1(s) = 1 1+td s Gd1/1(s) = − td 2 s+1 td 2 s+1 Gd2/2(s) = t2 d 12 s2− td 2 s+1 t2 d 12 s2+ td 2 s+1 http://mathworld.wolfram.com/PadeApproximant.html Behzad Samadi (Amirkabir University) Industrial Control 66 / 95
180. 180. PID Controller Design Internal Model Control: Example: ˜Gp = Kpe−td s 𝜏ps+1 ˜Gp ≈ Kp (𝜏ps + 1)(td 2 s + 1) ( − td 2 s + 1 ) Behzad Samadi (Amirkabir University) Industrial Control 67 / 95
181. 181. PID Controller Design Internal Model Control: Example: ˜Gp = Kpe−td s 𝜏ps+1 ˜Gp ≈ Kp (𝜏ps + 1)(td 2 s + 1) ( − td 2 s + 1 ) ˜Gp− = Kp (𝜏ps + 1)(td 2 s + 1) ˜Gp+ = − td 2 s + 1 [Chau, 2002] Behzad Samadi (Amirkabir University) Industrial Control 67 / 95
182. 182. PID Controller Design Internal Model Control: Example: G★ c = ˜G−1 p− 1 𝜏cs + 1 = (𝜏ps + 1)(td 2 s + 1) Kp 1 𝜏cs + 1 Gc = G★ c 1 − G★ c ˜Gp = Kp(1 + 1 Ti s + Td s) Behzad Samadi (Amirkabir University) Industrial Control 68 / 95
183. 183. PID Controller Design Internal Model Control: Example: G★ c = ˜G−1 p− 1 𝜏cs + 1 = (𝜏ps + 1)(td 2 s + 1) Kp 1 𝜏cs + 1 Gc = G★ c 1 − G★ c ˜Gp = Kp(1 + 1 Ti s + Td s) where: Kc = 1 Kp 2 𝜏p td + 1 2𝜏c td + 1 ; Ti = 𝜏p + td 2 ; Td = 𝜏p 2 𝜏p td + 1 [Chau, 2002] Behzad Samadi (Amirkabir University) Industrial Control 68 / 95
184. 184. PID Controller Design [Chau, 2002] Behzad Samadi (Amirkabir University) Industrial Control 69 / 95
185. 185. PID Controller Design Skogestad PID Tuning Method: S. Skogestad, “Probably the best simple PID tuning rules in the world” Presented at AIChE Annual meeting, Reno, NV, USA, 04-09 Nov. 2001, Paper no. 276h [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 70 / 95
186. 186. PID Controller Design Skogestad PID Tuning Method: g(s) = k e−𝜃s (𝜏1s + 1)(𝜏2s + 1) [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 71 / 95
187. 187. PID Controller Design Skogestad Internal Model Control (SIMC): ( y ys ) desired = 1 𝜏cs + 1 e−𝜃t Behzad Samadi (Amirkabir University) Industrial Control 72 / 95
188. 188. PID Controller Design Skogestad Internal Model Control (SIMC): ( y ys ) desired = 1 𝜏cs + 1 e−𝜃t c(s) = (𝜏1s + 1)(𝜏2s + 1) k 1 (𝜏c + 𝜃)s Behzad Samadi (Amirkabir University) Industrial Control 72 / 95
189. 189. PID Controller Design Skogestad Internal Model Control (SIMC): ( y ys ) desired = 1 𝜏cs + 1 e−𝜃t c(s) = (𝜏1s + 1)(𝜏2s + 1) k 1 (𝜏c + 𝜃)s c(s) = Kc(1 + 1 𝜏I s )(𝜏Ds + 1) Behzad Samadi (Amirkabir University) Industrial Control 72 / 95
190. 190. PID Controller Design Skogestad Internal Model Control (SIMC): ( y ys ) desired = 1 𝜏cs + 1 e−𝜃t c(s) = (𝜏1s + 1)(𝜏2s + 1) k 1 (𝜏c + 𝜃)s c(s) = Kc(1 + 1 𝜏I s )(𝜏Ds + 1) Tuning Rule Kc = 1 k 𝜏1 𝜏c + 𝜃 , 𝜏I = 𝜏1. 𝜏D = 𝜏2 [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 72 / 95
191. 191. PID Controller Design Skogestad Internal Model Control (SIMC): Tuning Rule Kc = 1 k 𝜏1 𝜏c + 𝜃 , 𝜏I = 𝜏1, 𝜏D = 𝜏2 Eﬀective cancellation of the ﬁrst order dynamics by 𝜏I = 𝜏1 [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 73 / 95
192. 192. PID Controller Design Skogestad Internal Model Control (SIMC): Tuning Rule Kc = 1 k 𝜏1 𝜏c + 𝜃 , 𝜏I = 𝜏1, 𝜏D = 𝜏2 Eﬀective cancellation of the ﬁrst order dynamics by 𝜏I = 𝜏1 For processes with large 𝜏1, this choice results in a long settling time for disturbance response. [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 73 / 95
193. 193. PID Controller Design Skogestad Internal Model Control (SIMC): Tuning Rule Kc = 1 k 𝜏1 𝜏c + 𝜃 , 𝜏I = 𝜏1, 𝜏D = 𝜏2 Eﬀective cancellation of the ﬁrst order dynamics by 𝜏I = 𝜏1 For processes with large 𝜏1, this choice results in a long settling time for disturbance response. Consider: g(s) = k e−𝜃s 𝜏1s+1 ≈ k 𝜏1s for large 𝜏1. With a PI controller c(s) = Kc(1 + 1 𝜏I s ), the poles of the closed loop system can be obtained from the following equation: 𝜏I k′ Kc s2 + 𝜏I s + 1 = 0 with k ′ = k 𝜏1 [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 73 / 95
194. 194. PID Controller Design Skogestad Internal Model Control (SIMC): Comparing to s2 + 2𝜁𝜔0s + 𝜔2: 𝜔0 = √ k′ Kc 𝜏I , 𝜁 = 1 2 √ k′ Kc𝜏I [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 74 / 95
195. 195. PID Controller Design Skogestad Internal Model Control (SIMC): Comparing to s2 + 2𝜁𝜔0s + 𝜔2: 𝜔0 = √ k′ Kc 𝜏I , 𝜁 = 1 2 √ k′ Kc𝜏I To avoid slow oscillations, consider: 𝜁 ≥ 1 [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 74 / 95
196. 196. PID Controller Design Skogestad Internal Model Control (SIMC): Comparing to s2 + 2𝜁𝜔0s + 𝜔2: 𝜔0 = √ k′ Kc 𝜏I , 𝜁 = 1 2 √ k′ Kc𝜏I To avoid slow oscillations, consider: 𝜁 ≥ 1 𝜏I ≥ 4 k′ Kc [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 74 / 95
197. 197. PID Controller Design Skogestad Internal Model Control (SIMC): Disturbance rejection: Assuming input disturbance (gd (s) = g(s)) ∣y(j𝜔)∣ = ∣g(j𝜔)∣ ∣1 + g(j𝜔)c(j𝜔)∣ d ≤ ymax [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 75 / 95
198. 198. PID Controller Design Skogestad Internal Model Control (SIMC): Disturbance rejection: Assuming input disturbance (gd (s) = g(s)) ∣y(j𝜔)∣ = ∣g(j𝜔)∣ ∣1 + g(j𝜔)c(j𝜔)∣ d ≤ ymax Assuming ∣gc∣ >> 1 at low frequencies: ∣c(j𝜔)∣ > d ymax [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 75 / 95
199. 199. PID Controller Design Skogestad Internal Model Control (SIMC): Disturbance rejection: Assuming input disturbance (gd (s) = g(s)) ∣y(j𝜔)∣ = ∣g(j𝜔)∣ ∣1 + g(j𝜔)c(j𝜔)∣ d ≤ ymax Assuming ∣gc∣ >> 1 at low frequencies: ∣c(j𝜔)∣ > d ymax For P, PI and PID: c(j𝜔) ≥ Kc [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 75 / 95
200. 200. PID Controller Design Skogestad Internal Model Control (SIMC): Disturbance rejection: Assuming input disturbance (gd (s) = g(s)) ∣y(j𝜔)∣ = ∣g(j𝜔)∣ ∣1 + g(j𝜔)c(j𝜔)∣ d ≤ ymax Assuming ∣gc∣ >> 1 at low frequencies: ∣c(j𝜔)∣ > d ymax For P, PI and PID: c(j𝜔) ≥ Kc Kc ≥ d ymax [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 75 / 95
201. 201. PID Controller Design Skogestad Internal Model Control (SIMC): Skogestad Internal Model Control (SIMC) Kc = 1 k 𝜏1 𝜏c + 𝜃 = 1 k′ 1 𝜏c + 𝜃 𝜏I = min{𝜏1, 4 k′ Kc } = min{𝜏1, 4(𝜏c + 𝜃)} 𝜏D = 𝜏2 Tuning for fast response with good robustness 𝜏c = 𝜃 Tuning for slow response with acceptable disturbance rejection Kc ≥ d ymax [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 76 / 95
202. 202. PID Controller Design Rules of thumb: [McMillan, 2001] Behzad Samadi (Amirkabir University) Industrial Control 77 / 95
203. 203. Digital Control Ts: sampling time [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 78 / 95
204. 204. Digital Control Ts: sampling time Analog to Digital (A-D) conversion: y(tk) = y(kTs) [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 78 / 95
205. 205. Digital Control Ts: sampling time Analog to Digital (A-D) conversion: y(tk) = y(kTs) Digital to Analog (D-A) conversion: u(t) = u(tk) for kTs ≤ t < (k + 1)Ts [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 78 / 95
206. 206. Digital Control Diﬀerence equation: x((k + 1)Ts) = ax(kTs) + bu(kTs) [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 79 / 95
207. 207. Digital Control Diﬀerence equation: x((k + 1)Ts) = ax(kTs) + bu(kTs) Discrete time: x[k + 1] = ax[k] + bu[k] [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 79 / 95
208. 208. Digital Control Diﬀerence equation: x((k + 1)Ts) = ax(kTs) + bu(kTs) Discrete time: x[k + 1] = ax[k] + bu[k] Z-transform: zX(z) = aX(z) + bU(z) ⇒ X(z) U(z) = b z − a [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 79 / 95
209. 209. Digital Control Approximating Continuous-Time Controllers: Euler’s method (forward diﬀerence): z = eTs s ≈ 1 + Tss ⇒ s ≈ z − 1 Ts ˙x(t) ≈ x(t + Ts) − x(t) Ts [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 80 / 95
210. 210. Digital Control Approximating Continuous-Time Controllers: Euler’s method (forward diﬀerence): z = eTs s ≈ 1 + Tss ⇒ s ≈ z − 1 Ts ˙x(t) ≈ x(t + Ts) − x(t) Ts Backward diﬀerence: z = eTs s ≈ 1 1 − Tss ⇒ s ≈ 1 − z−1 Ts ˙x(t) ≈ x(t) − x(t − Ts) Ts [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 80 / 95
211. 211. Digital Control Approximating Continuous-Time Controllers: Tustin’s approximation: z = eTs s ≈ 1 + sTs/2 1 − sTs/2 ⇒ s ≈ 2 Ts z − 1 z + 1 It is also called trapezoidal method or bilinear transformation. [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 81 / 95
212. 212. Digital Control Approximating Continuous-Time Controllers: Stability: Images of the left side of the s-plane [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 82 / 95
213. 213. Digital Control Approximating Continuous-Time Controllers: Re(s) < 0 Forward −−−−−−→ Diﬀerence Re(z) < 1 [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 83 / 95
214. 214. Digital Control Approximating Continuous-Time Controllers: Re(s) < 0 Forward −−−−−−→ Diﬀerence Re(z) < 1 Stable system in s domain Forward −−−−−−→ Diﬀerence Stable or unstable system in z domain [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 83 / 95
215. 215. Digital Control Approximating Continuous-Time Controllers: Re(s) < 0 Forward −−−−−−→ Diﬀerence Re(z) < 1 Stable system in s domain Forward −−−−−−→ Diﬀerence Stable or unstable system in z domain Re(s) < 0 Backward −−−−−−→ Diﬀerence (Re(z) − 1 2)2 + Im(z)2 < 1 4 [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 83 / 95
216. 216. Digital Control Approximating Continuous-Time Controllers: Re(s) < 0 Forward −−−−−−→ Diﬀerence Re(z) < 1 Stable system in s domain Forward −−−−−−→ Diﬀerence Stable or unstable system in z domain Re(s) < 0 Backward −−−−−−→ Diﬀerence (Re(z) − 1 2)2 + Im(z)2 < 1 4 Stable or unstable system in s domain Backward −−−−−−→ Diﬀerence Stable system in z domain [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 83 / 95
217. 217. Digital Control Approximating Continuous-Time Controllers: Re(s) < 0 Forward −−−−−−→ Diﬀerence Re(z) < 1 Stable system in s domain Forward −−−−−−→ Diﬀerence Stable or unstable system in z domain Re(s) < 0 Backward −−−−−−→ Diﬀerence (Re(z) − 1 2)2 + Im(z)2 < 1 4 Stable or unstable system in s domain Backward −−−−−−→ Diﬀerence Stable system in z domain Re(s) < 0 Tustin −−−−−−−−→ Approximation ∣z∣ < 1 [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 83 / 95
218. 218. Digital Control Approximating Continuous-Time Controllers: Re(s) < 0 Forward −−−−−−→ Diﬀerence Re(z) < 1 Stable system in s domain Forward −−−−−−→ Diﬀerence Stable or unstable system in z domain Re(s) < 0 Backward −−−−−−→ Diﬀerence (Re(z) − 1 2)2 + Im(z)2 < 1 4 Stable or unstable system in s domain Backward −−−−−−→ Diﬀerence Stable system in z domain Re(s) < 0 Tustin −−−−−−−−→ Approximation ∣z∣ < 1 Stable system in s domain Tustin −−−−−−−−→ Approximation Stable system in z domain [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 83 / 95
219. 219. Digital Control Digital PID: Proportional: P(t) = Kpe(t) P[k] = Kpe[k] [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 84 / 95
220. 220. Digital Control Digital PID: Proportional: P(t) = Kpe(t) P[k] = Kpe[k] Integral: I(t) = Kp Ti ∫ t 0 e(𝜏)d𝜏 I[k + 1] = I[k] + KTs Ti e[k] [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 84 / 95
221. 221. Digital Control Digital PID: Proportional: P(t) = Kpe(t) P[k] = Kpe[k] Integral: I(t) = Kp Ti ∫ t 0 e(𝜏)d𝜏 I[k + 1] = I[k] + KTs Ti e[k] Derivative: Td N dD dt + D = −KpTd dy dt D[k] = Td Td + NTs D[k − 1] − KpTd N Td + NTs (y[k] − y[k − 1]) [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 84 / 95
222. 222. Digital Control Typical sampling time: Type of variable Sampling time (sec) Flow 1-3 Level 5-10 Pressure 1-5 Temperature 10-20 [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 85 / 95
223. 223. Digital Control Typical sampling time: Type of variable Sampling time (sec) Flow 1-3 Level 5-10 Pressure 1-5 Temperature 10-20 Rule of thumb for PI controllers: Ts Ti ≈ 0.1 to 0.3 [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 85 / 95
224. 224. Digital Control Typical sampling time: Rule of thumb for PID controllers: NTs Td ≈ 0.2 to 0.6 [Astrom and Wittenmark, 1996] Behzad Samadi (Amirkabir University) Industrial Control 86 / 95
225. 225. Digital Control Inverse Response: When the initial response of a dynamic system is in a direction opposite to the ﬁnal outcome, it is called an inverse response. [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 87 / 95
226. 226. Digital Control Inverse Response: When the initial response of a dynamic system is in a direction opposite to the ﬁnal outcome, it is called an inverse response. Obtaining the eﬀective delay (Half rule): Eﬀective delay =True delay + inverse response time constant(s) + half of the largest neglected time constant + all smaller higher order time constants [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 87 / 95
227. 227. Digital Control Inverse Response: When the initial response of a dynamic system is in a direction opposite to the ﬁnal outcome, it is called an inverse response. Obtaining the eﬀective delay (Half rule): Eﬀective delay =True delay + inverse response time constant(s) + half of the largest neglected time constant + all smaller higher order time constants Time constant: 𝜏 = the largest time constant + 1 2 the second largest time constant { 𝜏1 = the largest time constant, 𝜏2 = the second largest time constant + 1 2the third largest time constant [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 87 / 95
228. 228. Digital Control Plant transfer function: G(s) = ∏ j (−Tj0s + 1) ∏ j (𝜏i0s + 1) e−td s First order approximation: ˆG1(s) = e−𝜃s 𝜏s + 1 𝜃 =td + ∑ j Tj0 + 𝜏20 2 + ∑ j≥3 𝜏i0 + Ts 2 𝜏 =𝜏10 + 𝜏20 2 [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 88 / 95
229. 229. Digital Control Second order approximation: ˆG2(s) = e−𝜃s (𝜏1s + 1)(𝜏2s + 1) 𝜃 =td + ∑ j Tj0 + 𝜏30 2 + ∑ j≥4 𝜏i0 + Ts 2 𝜏1 =𝜏10 𝜏2 =𝜏20 + 𝜏30 2 [Skogestad, 2001] Behzad Samadi (Amirkabir University) Industrial Control 89 / 95
230. 230. Digital Control Series form of the PID controller: U(s) = Kp(1 + 1 𝜏i s )(1 + 𝜏d s)E(s) Behzad Samadi (Amirkabir University) Industrial Control 90 / 95
231. 231. Digital Control Series form of the PID controller: U(s) = Kp(1 + 1 𝜏i s )(1 + 𝜏d s)E(s) U(s) = Kp(1 + 1 𝜏i s )(E(s) + 𝜏d sE(s)) Behzad Samadi (Amirkabir University) Industrial Control 90 / 95
232. 232. Digital Control Series form of the PID controller: U(s) = Kp(1 + 1 𝜏i s )(1 + 𝜏d s)E(s) U(s) = Kp(1 + 1 𝜏i s )(E(s) + 𝜏d sE(s)) U(s) = Kp(1 + 1 𝜏i s )(E(s) − 𝜏d sY (s)) Behzad Samadi (Amirkabir University) Industrial Control 90 / 95
233. 233. Digital Control Series form of the PID controller: U(s) = Kp(1 + 1 𝜏i s )(1 + 𝜏d s)E(s) U(s) = Kp(1 + 1 𝜏i s )(E(s) + 𝜏d sE(s)) U(s) = Kp(1 + 1 𝜏i s )(E(s) − 𝜏d sY (s)) U(s) = Kp(1 + 1 𝜏i s )(R(s) − (1 + 𝜏d s)Y (s)) Behzad Samadi (Amirkabir University) Industrial Control 90 / 95
234. 234. Digital Control Series form of the PID controller: U(s) = Kp(1 + 1 𝜏i s )(1 + 𝜏d s)E(s) U(s) = Kp(1 + 1 𝜏i s )(E(s) + 𝜏d sE(s)) U(s) = Kp(1 + 1 𝜏i s )(E(s) − 𝜏d sY (s)) U(s) = Kp(1 + 1 𝜏i s )(R(s) − (1 + 𝜏d s)Y (s)) U(s) = Kp(1 + 1 𝜏i s )Ed (s) Behzad Samadi (Amirkabir University) Industrial Control 90 / 95
235. 235. Digital Control Series form of the PID controller: Behzad Samadi (Amirkabir University) Industrial Control 91 / 95
236. 236. Digital Control Antiwindup digital PID: public class SimplePID { private double u,e,v,y; private double K,Ti,Td,Beta,Tr,N,h; private double ad,bd; private double D,I,yOld; public SimplePID(double nK, double nTi, double NTd, double nBeta, double nTr, double nN, double nh) { updateParameters(nK,nTi,nTd,nBeta,nTr,nN,nh); } ARTIST Graduate Course on Embedded Control Systems Behzad Samadi (Amirkabir University) Industrial Control 92 / 95
237. 237. Digital Control Antiwindup digital PID: public void updateParameters(double nK, double nTi, double NTd, double nBeta, double nTr, double nN, double nh) { K = nK; Ti = nTi; Td = nTd; Beta = nBeta; Tr = nTr N = nN; h = nh; ad = Td / (Td + N*h); bd = K*ad*N; } ARTIST Graduate Course on Embedded Control Systems Behzad Samadi (Amirkabir University) Industrial Control 93 / 95
238. 238. Digital Control Antiwindup digital PID: public double calculateOutput(double yref, double newY) { y = newY; e = yref - y; D = ad*D - bd*(y - yOld); v = K*(Beta*yref - y) + I + D; return v; } public void updateState(double u) { I = I + (K*h/Ti)*e + (h/Tr)*(u - v); yOld = y; } } ARTIST Graduate Course on Embedded Control Systems Behzad Samadi (Amirkabir University) Industrial Control 94 / 95
239. 239. Digital Control Antiwindup digital PID: public class Regul extends Thread { private SimplePID pid; public Regul() { pid = new SimplePID(1,10,0,1,10,5,0.1); } public void run() { // Other stuff while (true) { y = getY(); yref = getYref(): u = pid.calculateOutput(yref,y); u = limit(u); setU(u); pid.updateState(u); // Timing Code }}} ARTIST Graduate Course on Embedded Control Systems Behzad Samadi (Amirkabir University) Industrial Control 95 / 95
240. 240. Astrom, K. J. and Hagglund, T. (1995). PID Controllers: Theory, Design, and Tuning. Instrument Society of America. Astrom, K. J. and Wittenmark, B. (1996). Computer-Controlled Systems: Theory and Design. Prentice Hall, 3 edition. Chau, P. C. (2002). Process Control: A First Course with MATLAB (Cambridge Series in Chemical Engineering). Cambridge University Press, 1 edition. Li, Y., Ang, K. H., and Chong, G. C. Y. (2006). Pid control system analysis and design. IEEE Control Syst. Mag., 26(1):32–41. Love, J. (2007). Process Automation Handbook: A Guide to Theory and Practice. Springer, 1 edition. Behzad Samadi (Amirkabir University) Industrial Control 95 / 95
241. 241. McMillan, G. K. (2001). Good Tuning: A Pocket Guide. The Instrumentation, Systems, and Automation Society (ISA). O’Dwyer (2002). Handbook of PI & Pid Controller Tuning Rules. World Scientiﬁc Publishing. Skogestad, S. (2001). Probably the best simple pid tuning rules in the world. In AIChE Annual meeting, Reno, NV, USA. Visioli, A. (2006). Practical PID Control (Advances in Industrial Control). Springer, 1 edition. Woolf, P., editor (2007). The Michigan Chemical Process Dynamics and Controls Open Text Book. Yu, C.-C. (2007). Behzad Samadi (Amirkabir University) Industrial Control 95 / 95
242. 242. Autotuning of PID Controllers: A Relay Feedback Approach. Springer. Behzad Samadi (Amirkabir University) Industrial Control 95 / 95
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