DynamicSystems_VIII_IntroductionToControl_2023.pdf
- 1. Representation and Analysis of Dynamical Systems Introduction to Feedback Control 1
- 2. 2 Introduction to Feedback Control Outline 1. Introduction
- 3. Autocruise of a car ● From the control engineer point of view: – Velocity depends on throttle position – Experience says that 1° of throttle deflection increase the speed by 10 𝑘𝑚/ℎ – a grade of 1% affects the speed by 1𝑘𝑚/ℎ ● Mathematical (simplified) model 𝑦 = 10 × 𝑢 + 𝑝 velocity 10 u p y 𝑢: input (throttle angle) 𝑝: perturbation (wind, slope, etc…) 𝑦: output (velocity) 3
- 4. Performance of an “open loop controller” ● The controller is based on the system’s model: – System gain: 𝐺 = 10 – Controller gain: 𝐾 = 1 𝐺 = 1 10 – 𝑦 = 𝐾 × 𝐺 × 𝑦𝑟𝑒𝑓 + 𝑝 ● Errors analysis – 10% error on model ⇒ 10% error on output – 1° of slope ⇒ 1𝑘𝑚/ℎ error on output 4 𝐺 𝑢 𝑝 𝑦 𝑦𝑟𝑒𝑓 𝐾
- 5. 𝑦 = 1 1 + 𝐾𝐺 𝑝 + 𝐾𝐺 1 + 𝐾𝐺 𝑦𝑟𝑒𝑓 • “Proportional controller” 𝑢 = 𝐾 × (𝑦𝑟𝑒𝑓 − 𝑦) • Closed loop control with “𝐾 large” (e.g 𝐾𝐺 = 100) – 𝑦 = 0.99 × 𝑦𝑟𝑒𝑓 – 10% error on 𝐺 ⇒ almost no effect on 𝑦 – 1° of slope ⇒ almost no effect on 𝑦 – The bigger 𝐾 the best it seems to be 𝐺 𝑢 𝑝 𝑦 𝐾 𝜖 𝑦𝑟𝑒𝑓 + − Performance of a proportional controller 5
- 6. • Static analysis + Good tracking performance (𝑦𝑟𝑒𝑓 ≈ 𝑦) + Good perturbation rejection performance - 𝑢 may saturate if 𝐾 too large • Dynamic analysis + System’s dynamic faster + Unstable system may be stabilized - Stable system may become unstable 𝐺 𝑢 𝑝 𝑦 𝐾 𝜖 𝑦𝑟𝑒𝑓 + − Limits of the proportional controller ? 6
- 7. Introduction 7 ● We need a (mathematical) tool to manipulate signals and systems – Signals 𝑢(𝑡) and 𝑦(𝑡) (input and output) – System ℎ ● Everything is based on (only) two assumptions: – System ℎ is Linear 𝑢1 𝑡 → 𝑦1 𝑡 𝑢2 𝑡 → 𝑦2 𝑡 ⇒ 𝛼𝑢1 𝑡 + 𝛽𝑢2 𝑡 → 𝛼𝑦1 𝑡 + 𝛽𝑦2 𝑡 – Systems are Time Invariant 𝑢 𝑡 → 𝑦 𝑡 ⇒ 𝑢 𝑡 − 𝜏 → 𝑦(𝑡 − 𝜏) ● Laplace transform 𝑌 𝑠 = 𝐻 𝑠 × 𝑈(𝑠) ℎ 𝑦 𝑢
- 8. Introduction 8 What have we learnt? • We know how to describe an input/output dynamic system • Differential equations • Block diagrams • Transfer functions • State space equations • We can give main properties of a dynamic system • Poles, modes • Stability • Time response • Bandwidth • Dominant dynamics • We can predict closed loop behavior • Stability / Stability margin • Bandwidth We are ready to re-interpret the introductory example
- 9. The canonical feedback block diagram 9 A complex system, seen by the control engineer: 𝑆𝑦𝑠𝑡𝑒𝑚 𝐼𝑛𝑝𝑢𝑡 𝑂𝑢𝑡𝑝𝑢𝑡 𝑃𝑒𝑟𝑡𝑢𝑟𝑏𝑎𝑡𝑖𝑜𝑛 𝐹1 𝐹2 𝑂𝑢𝑡𝑝𝑢𝑡 𝐼𝑛𝑝𝑢𝑡 𝑃𝑒𝑟𝑡𝑢𝑟𝑏𝑎𝑡𝑖𝑜𝑛 The system is modeled as a linear system
- 10. 𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑙𝑒𝑟 10 A complex system, seen by the control engineer: 𝑆𝑦𝑠𝑡𝑒𝑚 𝐼𝑛𝑝𝑢𝑡 𝑂𝑢𝑡𝑝𝑢𝑡 𝑃𝑒𝑟𝑡𝑢𝑟𝑏𝑎𝑡𝑖𝑜𝑛 𝐹1 𝐹2 𝑂𝑢𝑡𝑝𝑢𝑡 𝐼𝑛𝑝𝑢𝑡 𝑃𝑒𝑟𝑡𝑢𝑟𝑏𝑎𝑡𝑖𝑜𝑛 𝑅𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 Controller: control the output according to the reference and feedback The canonical feedback block diagram
- 11. 𝐶2 𝐶1 11 A complex system, seen by the control engineer: 𝑆𝑦𝑠𝑡𝑒𝑚 𝐼𝑛𝑝𝑢𝑡 𝑂𝑢𝑡𝑝𝑢𝑡 𝑃𝑒𝑟𝑡𝑢𝑟𝑏𝑎𝑡𝑖𝑜𝑛 𝐹1 𝐹2 𝑂𝑢𝑡𝑝𝑢𝑡 𝐼𝑛𝑝𝑢𝑡 𝑃𝑒𝑟𝑡𝑢𝑟𝑏𝑎𝑡𝑖𝑜𝑛 𝑅𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 The controller will be linear − The canonical feedback block diagram
- 12. 𝐶2 𝐶1 12 𝐹1 𝐹2 𝑌 𝑈 𝑃 𝑅 Input / output transfer functions (input: reference and perturbation) − 𝑌 𝑠 = 𝐹2 𝑠 1 + 𝐶2 𝑠 𝐹1 𝑠 𝑃 𝑠 + 𝐶1 𝑠 𝐹1 𝑠 1 + 𝐶2 𝑠 𝐹1 𝑠 𝑅(𝑠) 𝑌 𝑠 = 𝐹2 𝑠 1 + 𝐶2 𝑠 𝐹1 𝑠 𝑃 𝑠 + 𝐶1 𝑠 𝐶2 𝑠 𝐶2 𝑠 𝐹1 𝑠 1 + 𝐶2 𝑠 𝐹1 𝑠 𝑅(𝑠) The canonical feedback block diagram
- 13. 13 A (more) canonical representation 𝐹1(𝑠) 𝐶2(𝑠) 𝐶1 𝑠 𝐶2(𝑠) 𝑌 𝑠 = 𝐹1 𝑠 1 + 𝐶2 𝑠 𝐹1 𝑠 𝑃 𝑠 + 𝐶1 𝑠 𝐶2 𝑠 𝐶2 𝑠 𝐹1 𝑠 1 + 𝐶2 𝑠 𝐹1 𝑠 𝑅(𝑠) 𝐹2(𝑠) 𝑃 𝑠 𝑅 𝑠 𝑌 𝑠 𝑈 𝑠 − 𝑅∗ 𝑠 This is why we study the “canonical” feedback: 𝐹1(𝑠) 𝐶2(𝑠) 𝑈 𝑠 𝑌 𝑠 𝑅∗ 𝑠 The canonical feedback block diagram
- 14. 14 The controller is designed in order to fulfill two kinds of requirements: • Tracking: the output 𝑦(𝑡) must follow as well as possible the reference 𝑟(𝑡) • Perturbation rejection: the output 𝑦(𝑡) must remain constant for any kind of perturbation 𝑝(𝑡) • For both cases the denominator of the transfer function (stability, performance) is CL s = 1 + 𝐶2 𝑠 𝐹1(𝑠) The performance will be illustrated through: • Bode diagrams (bandwidth, stability margin) • Step responses (damping, time response) 𝐹1(𝑠) 𝐶2(𝑠) 𝐶1 𝑠 𝐶2(𝑠) 𝐹2(𝑠) 𝑃 𝑠 𝑅 𝑠 𝑌 𝑠 𝑈 𝑠 − 𝑅∗ 𝑠 The canonical performance test
- 15. 15 Introduction to Feedback Control Outline 1. Introduction 2. The proportional controller
- 16. Pure integrator+P controller 16 The simple (but very important) case: 𝐴 𝑠 𝐾 𝑈 𝑠 𝑌 𝑠 𝑅 𝑠 − 𝑌 𝑠 𝑅(𝑠) = 𝐾𝐴 𝐾𝐴 + 𝑠 = 1 1 + 1 𝐾𝐴 𝑠 = 1 1 + 𝜏′𝑠 𝐾 “large” → 𝜏′𝑠𝑚𝑎𝑙𝑙 : system fast ☺ Reference-output transfer:
- 17. First order system +P controller 17 Another simple (but very important) case: 𝐴 1 + 𝜏𝑠 𝐾 𝑈 𝑠 𝑌 𝑠 𝑅 𝑠 − 𝑌 𝑠 𝑅(𝑠) = 𝐾𝐴 1 + 𝐾𝐴 + 𝜏𝑠 = 𝐾𝐴 1 + 𝐾𝐴 × 1 1 + 𝜏 1 + 𝐾𝐴 𝑠 = 𝐺 1 + 𝜏′𝑠 𝐾 “large” → 𝐺 ≈ 1: good tracking performance ☺ → 𝜏′ < 𝜏 : system faster ☺ Reference-output transfer:
- 18. 18 Reference-control transfer: 𝐴 1 + 𝜏𝑠 𝐾 𝑈 𝑠 𝑌 𝑠 𝑅 𝑠 − Let’s consider the step response: 𝑅 𝑠 = 1/𝑠 𝑈 𝑠 = 𝐾 1 + 𝐾𝐴 × 1 + 𝜏𝑠 1 + 𝜏 1 + 𝐾𝐴 𝑠 × 1 𝑠 Initial value theorem: 𝑢 0 = lim 𝑠→∞ 𝑠𝑈 𝑠 = 𝐾 → K “large” means strong transient control effort 𝑈 𝑠 𝑅(𝑠) = 𝐾(1 + 𝜏𝑠) 1 + 𝐾𝐴 + 𝜏𝑠 = 𝐾 1 + 𝐾𝐴 × 1 + 𝜏𝑠 1 + 𝜏 1 + 𝐾𝐴 𝑠 First order system +P controller
- 19. Closed loop: 𝐹 𝑠 1+𝐹(𝑠) 19 The same example from a “frequency domain” point of view: 𝐹 𝑠 = 𝐾 1 + 𝑠 Open loop: 𝐹 𝑠 Bandwidth Low frequency: 𝐹 𝑠 « large » → Closed loop ≈ 1 High frequency: 𝐹(𝑠) « small » → Closed loop ≈ Open loop Limit between « low » and « high » frequency given by intersection with 0𝑑𝐵 line 20𝐿𝑜𝑔𝐾 First order system +P controller
- 20. The system is an “unstable” first order system: 𝐹 𝑠 = 𝐴 𝜏𝑠 − 1 𝑌 𝑠 𝑅(𝑠) = 𝐾𝐴 −1 + 𝐾𝐴 + 𝜏𝑠 = 𝐾𝐴 −1 + 𝐾𝐴 × 1 1 + 𝜏 −1 + 𝐾𝐴 𝑠 = 𝐺 1 + 𝜏′𝑠 The closed loop system becomes stable when 𝐾𝐴 > 1 Unstable First order system +P controller
- 21. 21 Proportional control of a second order system 𝐴 1 + 2𝜎 𝜔0 𝑠 + 𝑠2 𝜔0 2 𝐾 𝑌 𝑠 𝑅 𝑠 − 𝑌 𝑠 𝑅(𝑠) = 𝐾𝐴 1+𝐾𝐴+ 2𝜎 𝜔0 𝑠+ 𝑠2 𝜔0 2 = 𝐾𝐴 1+𝐾𝐴 × 1 1+ 2𝜎′ 𝜔0′ 𝑠+ 𝑠2 𝜔0 ′2 = 𝐺 1+ 2𝜎′ 𝜔0′ 𝑠+ 𝑠2 𝜔0 ′2 𝐾 “large” → 𝐺 ≈ 1: good tracking performance ☺ → 𝜔0 ′ > 𝜔0: system faster ☺ → 𝜎′ < 𝜎: system less damped Input-output transfer: Second order system +P controller
- 22. 22 Root locus >>rlocus(F) 𝐴 1 + 2𝜎 𝜔0 𝑠 + 𝑠2 𝜔0 2 𝐾 𝑌 𝑠 𝑅 𝑠 − Poles of K𝐹(𝑠) (open loop) Locus of poles of 𝐾𝐹 𝑠 1+𝐾𝐹(𝑠) (closed loop) when 𝐾 increases Second order system +P controller
- 23. 23 Frequency domain analysis Case 1: closed loop 𝐾1 → Phase margin ≈ 45° Case 2: closed loop 𝐾2 > 𝐾1 → Bandwidth ↗ ☺ → System faster → Phase margin ↘ → More oscillations Second order system +P controller
- 24. Double integrator+P controller 24 The simple (but very important) case: 𝐴 𝑠2 𝐾 𝑈 𝑠 𝑌 𝑠 𝑅 𝑠 − 𝑌 𝑠 𝑅(𝑠) = 𝐾𝐴 𝐾𝐴 + 𝑠2 = 1 1 + 1 𝐾𝐴 𝑠2 = 1 1 + 𝑠2 𝜔0 2 Closed loop system will always be a purely oscillating system (second order system with damping =0) Reference-output transfer:
- 25. Neglected dynamics (open loop) 25 A system with a “slow” dynamic and a “fast” one: 𝐹1 𝑠 = 1 1 + 0.1𝑠 1 + 10𝑠 And the simplified one: 𝐹2 𝑠 = 1 1 + 10𝑠
- 26. Neglected dynamics (closed loop) 26 Comparison of closed loop dynamic with a proportional controller K=10 𝐶𝐿 𝑠 = 𝐾𝐹 𝑠 1 + 𝐾𝐹 𝑠 Phase margin ≈ 90° 20𝐿𝑜𝑔10 = 20𝑑𝐵
- 27. Neglected dynamics (closed loop) 27 Comparison of closed loop dynamic with a proportional controller K=1000 𝐶𝐿 𝑠 = 𝐾𝐹 𝑠 1 + 𝐾𝐹 𝑠 Phase margin 1 ≈ 10° Phase margin 2 ≈ 90° 20𝐿𝑜𝑔1000 = 60𝑑𝐵
- 28. Neglected dynamics (conclusion) 28 If the controller is « weak » the « fast » dynamic can be neglected If the controller is « strong » neglecting the fast dynamic may result in a poorly damped or unstable closed loop behavior
- 29. Time delay 29 Comparison of a first order system and a first order system with time delay → Phase of the system with delay goes below the − 180° line
- 30. 30 Introduction to Feedback Control Outline 1. Introduction 2. The Proportional controller (P) 3. Proportional controllers in cascade
- 31. Limits of the Proportional controller 31 The Proportional controller is ok for simple systems: • First order • Pure integrator The Proportional controller is ok for low performance requirements But when the system is of higher order or if the required performance is high then the P controller will result in poorly damped or unstable closed loop system Solution: cascade controllers
- 32. PD control of a double integrator 32 A very classical example, covers many applications • 𝑢 𝑡 is the control input (voltage) • 𝐴 represents the current (torque) controlled motor connected to a constant inertia • 𝑦𝑑𝑑(𝑡) is the acceleration • 𝑦𝑑(𝑡) is the speed • 𝑦(𝑡) is the position 𝐴 1 𝑠 1 𝑠 𝑈(𝑠) 𝑌𝑑𝑑(𝑠) 𝑌𝑑(𝑠) 𝑌(𝑠) Basic idea: control first the velocity (gives a first order system) and secondly the acceleration
- 33. PD control of a double integrator 33 The “cascade” loop 𝐴 1 𝑠 1 𝑠 𝑈(𝑠) 𝑌𝑑(𝑠) 𝑌(𝑠) Tuning of the controller in two steps: • Internal loop (speed) 𝑌𝑑 𝑠 𝑌𝑑𝑟𝑒𝑓(𝑠) = 1 1 + 𝜏𝑠 𝜏 = ⋯ • External loop 𝑌 𝑠 𝑌𝑟𝑒𝑓(𝑠) = 1 1 + 2𝜎 𝜔0 𝑠 + 𝑠2 𝜔0 2 𝜎 = … 𝑎𝑛𝑑 𝜔0 = ⋯ 𝐷 𝑃 𝑌𝑑𝑟𝑒𝑓(𝑠) 𝑌𝑟𝑒𝑓 (𝑠) − −
- 34. PD control of a double integrator 34 Another point of view: 𝐴 1 𝑠2 𝑈(𝑠) 𝑌(𝑠) 𝑃 + 𝐷𝑠 𝑌𝑟𝑒𝑓 (𝑠) − Closed loop transfer function: 𝑌 𝑠 𝑌𝑟𝑒𝑓(𝑠) = 𝐴 𝑃 + 𝐷𝑠 1 + 2𝜎 𝜔0 𝑠 + 𝑠2 𝜔0 2 𝜎 = … 𝑎𝑛𝑑 𝜔0 = ⋯ the same We have modified the “open loop” transfer function with the controller
- 35. PD control of a double integrator 35 Frequency domain analysis: (example: 𝐴 = 1; 𝑃 = 1; 𝐷 = 1) • The system (solid): 𝐹 𝑠 = 𝐴 𝑠2 Phase margin: 0𝑑𝐵 • The controlled system (dashed): 𝐶𝐿 𝑠 = 𝑃 + 𝐷𝑠 𝐴 𝑠2 Phase margin: 40𝑑𝐵
- 36. PD control of a double integrator 36 Bode diagram of the PD controller: • Increases the gain • Increases the phase ☺ Idea of “loop shaping” is to modify the open loop transfer function in order to increase the phase near cut-off frequency → increase phase margin
- 37. PD control of a double integrator 37 Drawbacks of the PD controller • Amplifies high frequency feedback • Strong transient response Analysis of the transient step response of the PD controlled double integrator: 𝑈 𝑠 = 𝑃 + 𝑠𝐷 1 + 𝑃 + 𝑠𝐷 𝐴 𝑠2 × 1 𝑠 = 𝑃 + 𝑠𝐷 𝑠2 𝑠2 + 𝑃 + 𝑠𝐷 𝐴 × 1 s 𝑢 0 = lim 𝑠→∞ 𝑠𝑈(𝑠) = ∞ Solution 1: replace the PD controller by a filtered-PD controller 𝐶 𝑠 = 𝑃 + 𝑠𝐷 1 + 𝜏𝑠
- 38. PD control of a double integrator 38 Solution 2: only filter the feedback (modification of the block-diagram) 𝐴 1 𝑠2 𝑈(𝑠) 𝑌(𝑠) 𝑃 𝑌𝑟𝑒𝑓 (𝑠) − 𝑠𝐷 − Analysis of the transient step response of the PD controlled double integrator: 𝑈 𝑠 = 𝑃 1 + 𝐴𝐷𝑠 𝑠2 + 𝑃𝐴 𝑠2 × 1 𝑠 = 𝑃𝑠2 𝑠2 + 𝐴𝐷𝑠 + 𝑃𝐴 × 1 𝑠 𝑢 0 = lim 𝑠→∞ 𝑠𝑈(𝑠) = 𝑃
- 39. 39 Introduction to Feedback Control Outline 1. Introduction 2. The Proportional controller (P) 3. Proportional controllers in cascade 4. Loop shaping
- 40. Loop shaping 40 The controller modifies the “shape” of the open loop transfer function 𝐹(𝑠) 𝐾 𝑠 𝑈 𝑠 𝑌 𝑠 𝑅 𝑠 − 𝐾𝐹 1 + 𝐾𝐹 𝜔 Low frequency Cross over frequency high frequency
- 41. Loop shaping 41 High gain at low frequency → Good tracking performance Infinite gain at low frequency → Perfect tracking performance (no error) Let 𝐻 𝑠 = 𝐾 𝑠 × 𝐹(𝑠) open loop transfer function Static error (computed for a unitary step input): lim 𝑡→∞ 𝜖(𝑡) = lim 𝑠→0 𝑠 × 1 𝑠 × 1 1 + 𝐻(𝑠) Loop shaping 1: low frequency
- 42. Loop shaping 42 Closed loop transfer function: 𝐻𝑐𝑙 𝑠 = 𝐻 𝑠 1 + 𝐻(𝑠) Gain: 𝐻𝑐𝑙 𝑗𝜔 = 𝐻 𝑗𝜔 1 + 𝐻(𝑗𝜔) Gain at low frequency: 𝐻𝑐𝑙 𝜔 𝑠𝑚𝑎𝑙𝑙 = 𝐻 𝜔 𝑠𝑚𝑎𝑙𝑙 1 + 𝐻(𝜔 𝑠𝑚𝑎𝑙𝑙) ≈ 1 Gain at high frequency: 𝐻𝑐𝑙 𝜔 𝑙𝑎𝑟𝑔𝑒 = 𝐻 𝜔 𝑙𝑎𝑟𝑔𝑒 1 + 𝐻(𝜔 𝑙𝑎𝑟𝑔𝑒) ≈ 𝐻 𝜔 𝑙𝑎𝑟𝑔𝑒 Limit 𝜔 𝑠𝑚𝑎𝑙𝑙 / 𝜔 𝑙𝑎𝑟𝑔𝑒 = cross over frequency = bandwidth Loop shaping 2: cutoff (or cross over) frequency
- 43. Loop shaping 43 The steepness of the gain at crossover frequency (𝑛𝑔𝑐 ) is related to phase margin PM: 𝑃𝑀 = 𝜋 + 𝑛𝑔𝑐 2 𝜋 Example: - 𝐻 𝑗𝜔 ≈ 1 𝜔 → 𝑛𝑔𝑐 = −1 → 𝑃𝑀 = 𝜋/2 (good) - 𝐻 𝑗𝜔 ≈ 1 𝜔2 → 𝑛𝑔𝑐 = −2 → 𝑃𝑀 = 0 (bad) Loop shaping 2bis: robustness
- 44. Loop shaping 44 𝑌 𝑠 = 𝐻 𝑠 1 + 𝐻(𝑠) 𝑁(𝑠) Loop shaping 3: high frequency noise rejection 𝐹(𝑠) 𝐾 𝑠 𝑈 𝑠 𝑌 𝑠 𝑅 𝑠 − Sensor noise 𝑁(𝑠) Sensor noise is generally high frequency. 𝐻 𝑗𝜔 must be small at high frequency in order to attenuate sensor noise
- 45. Loop shaping 45 Loop shaping conclusion 𝐾𝐹 1 + 𝐾𝐹 𝜔 Low frequency gain as high as possible: reduces static error Cross over frequency = closed loop bandwidth high frequency gain as small as possible: noise rejection Gain slope at crossover frequency not too steep: phase margin (robustness)
- 46. Loop shaping 46 Lag compensator 𝐾𝑙𝑎𝑔 𝑠 = 𝑎 + 𝑠 𝑏 + 𝑠 𝑎 > 𝑏 Increase gain at low freq Don’t modify phase near cutoff frequency Increase gain at low freq → Reduce static gain
- 47. Loop shaping 47 Lead compensator (aka derivative controller) 𝐾𝑙𝑒𝑎𝑑 𝑠 = 𝑎 + 𝑠 𝑏 + 𝑠 𝑎 < 𝑏 Increase phase near cutoff frequency Don’t increase too much the gain Increase phase at cutoff freq → Improve phase margin
- 48. 48 Introduction to Feedback Control Outline 1. Introduction 2. The Proportional controller (P) 3. Proportional controllers in cascade 4. Loop shaping 5. Integral Control
- 49. First order system +PI controller 49 𝐴 1 + 𝜏𝑠 𝑃 𝑈 𝑠 𝑌 𝑠 𝑅 𝑠 − In steady state 𝜖(𝑡) is equal to 0 (otherwise error is integrated ≠ steady state) Demonstration (input is a step): 𝜖 𝑠 = 𝑠 1 + 𝜏𝑠 𝐴𝐼 + 1 + 𝐴𝑃 𝑠 + 𝜏𝑠2 × 1 𝑠 lim 𝑡→∞ 𝜖(𝑡) = lim 𝑠→0 𝑠𝜖 𝑠 = 0 𝜖 𝑠 𝐼 𝑠 Integrator 𝑅 𝑠
- 50. First order system +PI controller 50 𝐴 1 + 𝜏𝑠 𝑃 𝑈 𝑠 𝑌 𝑠 𝑅 𝑠 − Closed loop transfer function: 𝑌 𝑠 𝑅 𝑠 = 𝐴𝐼 + 𝐴𝑃 𝑠 𝐴𝐼 + 1 + 𝐴𝑃 𝑠 + 𝜏𝑠2 = 1 + 𝜏′𝑠 1 + 2𝜎 𝜔0 𝑠 + 𝑠2 𝜔0 2 𝜖 𝑠 𝐼 𝑠
- 51. First order system +PI controller 51 Frequency domain analysis: 𝑌 𝑠 𝑅 𝑠 = 𝐶𝐿(𝑠) = 𝑂𝐿 𝑠 1 + 𝑂𝐿(𝑠) High gain (infinite) at low frequency: no steady state error Same phase margin
- 52. PI controller 52 Bode diagram of the PI controller: • Increases the gain at low frequency ☺ • Reduces the phase at intermediate frequency Idea of “loop shaping” is to modify the open loop transfer function in order to increase the low frequency gain and not too much increase the phase near cut-off frequency
- 53. 53 Introduction to Feedback Control Outline 1. Introduction 2. The Proportional controller (P) 3. Proportional controllers in cascade 4. Loop shaping 5. Integral Control 6. PID controller
- 54. PID controller 54 PID controller is a combination of P, D and I effect nicely tuned 𝐾 𝑠 = 𝑃 + 𝐼 𝑠 + 𝐷𝑠 Proportional Decrease rise time, increase overshoot, decrease static error, degrade stability Integral Increase overshoot, eliminate static error, degrade stability Derivative Decrease overshoot, improve stability
- 55. PID controller 55 Demo tuning of PID
- 56. 56 Introduction to Feedback Control Outline 1. Introduction 2. The Proportional controller (P) 3. Proportional controllers in cascade 4. Loop shaping 5. Integral Control 6. PID controller 7. Fundamental limitations
- 57. Fundamental limitations 57 Zero in the right half-plane Let 𝐹 𝑠 = 𝑧−𝑠 𝐷(𝑠) where 𝑧 > 0 Whatever the controller there will be a theoretical limitation: 𝜔𝑐 < 𝑧 𝜔𝑐 𝑧
- 58. Fundamental limitations 58 Time delay A time delay is equivalent to a RHP zero: 𝑒−𝑇𝑠 ≈ 1 − 𝑇 2 𝑠 1 + 𝑇 2 𝑠 Whatever the controller there will be a theoretical limitation: 𝜔𝑐 < 2 𝑇
- 59. Fundamental limitations 59 Pole in the RHP The faster is the RHP pole the more demanding will be the control
- 60. Fundamental limitations 60 Sensitivity function 𝑆 𝑠 = 1 1 + 𝐾 𝑠 𝐹(𝑠) (−1,0) 1 + 𝐾 𝑠 𝐹 𝑠 𝑅𝑒 𝐼𝑚 Nyquist plot of 𝐾 𝑠 𝐹(𝑠) We want min( 1 + 𝐾 𝑠 𝐹 𝑠 ) to be as large as possible We want max( 1 + 𝐾 𝑠 𝐹 𝑠 −1 ) to be as small as possible
- 61. Fundamental limitations 61 Bode’s integral formula න 0 ∞ log 1 1 + 𝐾 𝑠 𝐹 𝑠 = 0