Feedback control theory: An overview and connections to biochemical systems theory

1. Feedback control theory: An overview and connections to biochemical systems theory Sigurd Skogestad Department of Chemical Engineering Norwegian University of Science and Tecnology (NTNU) Trondheim, Norway VIIth International Symposium on Biochemical Systems Theory Averøy, Norway, 17-20 June 2002

2. Motivation • I have co-authored a book: ”Multivariable feedback control – analysis and design” (Wiley, 1996) • What parts could be useful for systems biochemistry? • Control as a field is closely related to systems theory • The more general systems theory concepts are assumed known • Here: Focus on the use of negative feedback • Some other areas where control may contribute (Not covered): • Identification of dynamic models from data (not in my book anyway) • Model reduction • Nonlinear control (also not in my book)

3. Outline • Introduction: Negative feedforward and feedback control • Introductory examples • Feedback is an extremely powerful tool (BUT: So simple that it is frequently overlooked) • Control theory and possible contributions • Fundamental limitation on negative feedback control • Cascade control and control of complex large-scale engineering system • Hierarchy (cascades) of single-input-single-output (SISO) control loops • Design of hierarchical control systems • Overall operational objectives • Which variable to control (primary output) ? • Self-optimizing control • Summary and concluding remarks

4. Important control concepts • Cause-effect relationship • Classification of variables: • ”Causes”: Disturbances (d) and inputs (u) • ”Effects”: Internal states (x) and outputs (y) • Typical state-space models: • Linearized models (useful for control!):

5. Typical chemical plant: Tennessee Eastman process Recycle and natural phenomena give positive feedback

6. xAs xA FA XC Control uses negative feedback

7. Control • Active adjustment of inputs (available degrees of freedom, u) to achieve the operational objectives of the system • Most cases: Acceptable operation = ”Output (y) close to desired setpoint (ys)”

8. Disturbance (d) Plant (uncontrolled system) Output (y) Input (u) • Acceptable operation = ”Output (y) close to desired setpoint (ys)” • Control: Use input (u) to counteract effect of disturbance (d) on y • Two main principles: • Feedforward control (measure d, predict and correct ahead) • (Negative) Feedback control (measure y and correct afterwards)

9. Disturbance (d) Plant (uncontrolled system) Output (y) Input (u) No control: Output (y) drifts away from setpoint (ys)

10. Setpoint (ys) Disturbance (d) Predict FF-Controller≈ Plant model-1 Plant (uncontrolled system) Offset Input (u) Output (y) • Feedforward control: • Measure d, predict and correct (ahead) • Main problem: Offset due to model error

11. Disturbance (d) Setpoint (ys) Input (u) Plant (uncontrolled system) FB Controller ≈ High gain e Output (y) • Feedback control: • Measure y, compare and correct (afterwards) • Main problem: Potential instability

12. Outline • Introduction: Feedforward and feedback control • Introductory examples • (Negative) Feedback is an extremely powerful tool (BUT: So simple that it is frequently overlooked) • Control theory and possible contributions • Fundamental limitation on control • Cascade control and control of complex large-scale engineering system • Hierarchy (cascades) of single-input-single-output (SISO) control loops • Design of hierarchical control systems • Overall operational objectives • Which variable to control (primary output) ? • Self-optimizing control • Summary and concluding remarks

13. k=10 time 25 Example 1 d Gd G u y Plant (uncontrolled system)

14. d Gd G u y

15. Feedforward (FF) control d Gd G u y Nominal G=Gd → Use u = -d

16. d Gd G u y FF control: Nominal case (perfect model)

17. d Gd G u y FF control: change in gain in G

18. d Gd G u y FF control: change in time constant

19. d Gd G u y FF control: simultaneous change in gain and time constant

20. d Gd G u y FF control: change in time delay

21. Feedback (FB) control d Gd ys e=ys-y Feedback controller G u y Negative feedback: u=f(e) ”Counteract error in y by change in u’’

22. e=ys-y u Feedback controller Feedback (FB) control • Simplest: On/off-controller • u varies between umin (off) and umax (on) • Problem: Continous cycling

23. e=ys-y u Feedback controller Feedback (FB) control • Most common in industrial systems: PI-controller

24. d Gd G u y Back to the example

25. d Gd ys e C G u y Output y Input u Feedback PI-control: Nominal case

26. d Gd ys e C G u y offset Integral (I) action removes offset

27. d Gd ys e C G u y Feedback PI control: change in gain

28. d Gd ys e C G u y FB control: change in time constant

29. d Gd ys e C G u y FB control: simultaneous change in gain and time constant

30. d Gd ys e C G u y FB control: change in time delay

31. d Gd ys e C G u y FB control: all cases

32. d Gd G u y FF control: all cases

33. Summary example • Feedforward control is NOT ROBUST (it is sensitive to plant changes, e.g. in gain and time constant) • Feedforward control: gradual performance degradation • Feedback control is ROBUST (it is insensitive to plant changes, e.g. in gain and time constant) • Feedback control: sudden performance degradation (instability) Instability occurs if we over-react (loop gain is too large compared to effective time delay). • Feedback control: Changes system dynamics (eigenvalues) • Example was for single input - single output (SISO) case • Differences may be more striking in multivariable (MIMO) case

34. Feedback is an amazingly powerful tool

35. Stabilization requires feedback Output y Input u

36. Why feedback?(and not feedforward control) • Counteract unmeasured disturbances • Reduce effect of changes / uncertainty (robustness) • Change system dynamics (including stabilization) • No explicit model required • MAIN PROBLEM • Potential instability (may occur suddenly)

37. Outline • Introduction: Feedforward and feedback control • Introductory examples • Feedback is an extremely powerful tool (BUT: So simple that it is frequently overlooked) • Control theory and possible contributions • Fundamental limitation on control • Cascade control and control of complex large-scale engineering system • Hierarchy (cascades) of single-input-single-output (SISO) control loops • Design of hierarchical control systems • Overall operational objectives • Which variable to control (primary output) ? • Self-optimizing control • Summary and concluding remarks

38. Overview of Control theory • Classical feedback control (1930-1960) (Bode): • Single-loop (SISO) feedback control • Transfer functions, Frequency analysis (Bode-plot) • Fundamental feedback limitations (waterbed). Focus on robustness • Optimal control (1960-1980) (Kalman): • Optimal design of Multivariable (MIMO) controllers • Model-based ”feedforward” thinking; no robustness guarantees (LQG) • State-space; Advanced mathematical tools (LQG) • Robust control (1980-2000) (Zames, Doyle) • Combine classical and optimal control • Optimal design of controllers with guaranteed robustness (H∞) • Nonlinear control (1950 - ) • ”Feedforward thinking”, Mechanical systems • Adaptive control (1970-1985) (Åstrøm)

39. Control theory Design

40. Relationship to system biochemistry/biology:What can the control field contribute? • Advanced methods for model-based centralized controller design • Probably of minor interest in biological systems • Unlikely that nature has developed many multivariable control solutions • Understanding of feedback systems • Same inherent limitations apply in biological systems • Understanding and design of hierarchical control systems • Important both in engineering and biological systems • BUT: Underdeveloped area in control • ”Large scale systems community”: Out of touch with reality

41. Outline • Introduction: Feedforward and feedback control • Introductory examples • Feedback is an extremely powerful tool (BUT: So simple that it is frequently overlooked) • Control theory and possible contributions • Fundamental limitation on control • Cascade control and control of complex large-scale engineering system • Hierarchy (cascades) of single-input-single-output (SISO) control loops • Design of hierarchical control systems • Overall operational objectives • Which variable to control (primary output) ? • Self-optimizing control • Summary and concluding remarks

42. Inherent limitations • Simple measure: Effective delay θeff • Fundamental waterbed limitation (”no free lunch”) for second- or higher-order system: • Does NOT apply to first-order system

43. Inherent limitations in plant (underlying uncontrolled system) • Effective delay: Includes inverse response, high-order dynamics • Multivariable systems: RHP-zeros (unstable inverse) – generalization of inverse response • Unstable plant. Not a problem in itself, but • Requires the active use of plant inputs • Requires that we can react sufficiently fast • ”Large” disturbances are a problem when combined with • Large effective delay: Cannot react sufficiently fast • Instability: Inputs may saturate and system goes unstable • All these may be quantified: For exampe, see my book

44. Outline • Introduction: Feedforward and feedback control • Introductory examples • Feedback is an extremely powerful tool (BUT: So simple that it is frequently overlooked) • Control theory and possible contributions • Fundamental limitation on control • Cascade control and control of complex large-scale engineering systems • Hierarchy (cascades) of single-input-single-output (SISO) control loops • Design of hierarchical control systems • Overall operational objectives • Which variable to control (primary output) ? • Self-optimizing control • Summary and concluding remarks

45. Problem feedback: Effective delay θ • Effective delay PI-control = ”original delay” + ”inverse response” + ”half of second time constant” + ”all smaller time constants”

46. d u ys e G2 G1 C y PI-control

47. Improve control? • Some improvement possible with more complex controller • For example, add derivative action (PID-controller) • May reduce θeff from 5 s to 2 s • Problem: Sensitive to measurement noise • Does not remove the fundamental limitation (recall waterbed) • Add extra measurement and introduce local control • May remove the fundamental waterbed limitation • Waterbed limitation does not apply to first-order system • Cascade

48. Without cascade With cascade Cascade control w/ extra meas. (2 PI’s) d ys y2 y2s u G2 G1 C1 C2 y

49. Cascade control • Inner fast (secondary) loop: • P or PI-control • Local disturbance rejection • Much smaller effective delay (0.2 s) • Outer slower primary loop: • Reduced effective delay (2 s) • No loss in degrees of freedom • Setpoint in inner loop new degree of freedom • Time scale separation • Inner loop can be modelled as gain=1 + effective delay • Very effective for control of large-scale systems

50. More complex cascades Control configuration with two layers of cascade control y1 - primary output (with given setpoint = reference value r1) y2 - secondary output (extra measurement) u3 - main input (slow) u2 - Extra input for fast control (temporary – reset to nominal value r3)