1 feedback control theory: an overview and connections to biochemical systems theory sigurd...

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

1. Introduction: Negative feedforward and feedback control

2. Introductory examples– Feedback is an extremely powerful tool

(BUT: So simple that it is frequently overlooked)

3. Control theory and possible contributions

4. Fundamental limitation on negative feedback control

5. Cascade control and control of complex large-scale engineering system • Hierarchy (cascades) of single-input-single-output (SISO) control loops

6. Design of hierarchical control systems• Overall operational objectives

• Which variable to control (primary output) ?

• Self-optimizing control

7. 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

Control uses negative feedback

XC

xAs

xAFA

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

Plant(uncontrolled system)

Disturbance (d)

Input (u) Output (y)

Acceptable operation = ”Output (y) close to desired setpoint (ys)”Control: Use input (u) to counteract effect of disturbance (d) on yTwo main principles:

• Feedforward control (measure d, predict and correct ahead)• (Negative) Feedback control (measure y and correct afterwards)

9

Plant (uncontrolled system)

Disturbance (d)


No control: Output (y) drifts away from setpoint (ys)

10


Disturbance (d)


Feedforward control:• Measure d, predict and correct (ahead)• Main problem: Offset due to model error

FF-Controller≈Plant model-1

Setpoint (ys)

Predict

Offset

11


Disturbance (d)

Input (u)

Output (y)

FB Controller≈ High gain

Setpoint(ys)

Feedback control:• Measure y, compare and correct (afterwards)• Main problem: Potential instability

e

12

Outline

1. Introduction: Feedforward and feedback control

2. Introductory examples– (Negative) Feedback is an extremely powerful tool



4. Fundamental limitation on control






13

Example

G

Gd

u

d

y

Plant (uncontrolled system)

1

k=10

time25

14

GGd

u

d

y

15

Feedforward (FF) control

G

Gd

u

d

y

Nominal G=Gd → Use u = -d

16

GGd

u

d

y

FF control: Nominal case (perfect model)

17

GGd

u

d

y

FF control: change in gain in G

18

GGd

u

d

y

FF control: change in time constant

19

GGd

u

d

y

FF control: simultaneous change in gain and time constant

20

GGd

u

d

y

FF control: change in time delay

21

Feedback (FB) control

G

Gd

u

d

y

Feedbackcontroller

ys e=ys-y

Negative feedback: u=f(e)”Counteract error in y by change in u’’

22


Feedback controller

e=ys-y u

Simplest: On/off-controller• u varies between umin (off) and umax (on)• Problem: Continous cycling

23


Feedback controller

e=ys-y u

Most common in industrial systems: PI-controller

24

GGd

u

d

y

Back to the example

25

GGd

u

d

yC

ys e

Feedback PI-control: Nominal case

Input u Output y

26

GGd

u

d

yC

ys e

Integral (I) action removes offset

offset

27

GGd

u

d

yC

ys e

Feedback PI control: change in gain

28 FB control: change in time constant

GGd

u

d

yC

ys e

29

FB control: simultaneous change in gain and time constant

GGd

u

d

yC

ys e

30 FB control: change in time delay

GGd

u

d

yC

ys e

31 FB control: all cases

GGd

u

d

yC

ys e

32

GGd

u

d

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

Input u Output y

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











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











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






5. Cascade control and control of complex large-scale engineering systems • Hierarchy (cascades) of single-input-single-output (SISO) control loops





45

Problem feedback: Effective delay θ

• Effective delay PI-control = ”original delay” + ”inverse response” + ”half of second time constant” + ”all smaller time constants”

46

PI-control

G1

u

d

yC

ys eG2

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

Cascade control w/ extra meas. (2 PI’s)

G1u

d

yC1

ys G2C2

y2

Without cascade

With cascade

y2s

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

Control configuration with two layers of cascade controly1 - 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)

More complex cascades

51

Hierarchical structure in chemical industry

52

Engineering systems

• Most (all?) large-scale engineering systems are controlled using hierarchies of quite simple single-loop controllers – Commercial aircraft

– Large-scale chemical plant (refinery)

• 1000’s of loops

• Simple components: on-off + P-control + PI-control + nonlinear fixes + some feedforward

Same in biological systems

53

Outline











54

Hierarchical structure

Brain

Local controlin cells

Organs

55

Alan Foss (“Critique of chemical process control theory”, AIChE Journal,1973):

The central issue to be resolved ... is the determination of control system structure. Which variables should be measured, which inputs should be manipulated and which links should be made between the two sets?

56

Alternatives structures for optimizing control

What should we control?

Hierarchical Centralized

Brain

Cells

57

Alternatives structures for optimizing control

Hierarchical Centralized

What should we control?(Control theory has little to offer)

Control theory has a lot to offer

58

WHAT SHOULD WE CONTROL?Example: 10 km Run• Overall objective: Minimum time• No major disturbances• What should we control?

– constant speed? easy to measure with clock.– constant heart beat?– constant level of sugar?– Constant level of lactic acid?

Example: 10 km cross-country skiing• Overall objective: minimum time• Disturbance = hill. • What should we control?

– Constant speed no longer optimal.– Could have a mix depending on disturbance (constant feed when flat, lactic acid in

hill?, max speed downhill turn)Example: Cell• Overall objective = optimize cell growth? • What should we control?

– constant oxygen?

59

Self-optimizing control(Skogestad, 2000)

Self-optimizing control is achieved when a constant setpoint policy results in an acceptableloss L (without the need to reoptimize whendisturbances occur)

Loss L = J - Jopt (d)

J = cost (overall objective to be minimized)

60

Good candidate controlled variables c (for self-optimizing control)

Requirements:

• The optimal value of c should be insensitive to disturbances

• c should be easy to measure and control

• The value of c should be sensitive to changes in the degrees of freedom

(Equivalently, J as a function of c should be flat)

• For cases with more than one unconstrained degrees of freedom, the selected controlled variables should be independent.

Singular value rule (Skogestad and Postlethwaite, 1996):Look for variables that maximize the minimum singular value of the appropriately scaled steady-state gain matrix G from u to c

61

Stepwise procedure for design of control system in chemical plant

I. TOP-DOWN

Step 1. DEFINE OVERALL CONTROL OBJECTIVE

Step 2. DEGREE OF FREEDOM ANALYSIS

Step 3. WHAT TO CONTROL? (primary outputs)• control active constraints• unconstrained: “self-optimizing variables”

Mainly economic considerations: Little control knowledge required!

Stepwise procedure chemical plant

62

II. BOTTOM-UP (structure control system):

Step 4. REGULATORY CONTROL LAYER

5.1 Stabilization

5.2 Local disturbance rejection (inner cascades) ISSUE: What more to control? (secondary variables)

Step 5. SUPERVISORY CONTROL LAYER

Decentralized or multivariable control (MPC)?Pairing?

Step 6. OPTIMIZATION LAYER (RTO)


63

Step 1. Overall control objective

• What are the operational objectives?

• Quantify: Minimize scalar cost J

• Usually J = economic cost [$/h]

• + Constraints on flows, equipment constraints, product specifications, etc.


64

Step 2. Degree of freedom (DOF) analysis

• Nm : no. of dynamic (control) DOFs (valves)


65

Step 3. What should we control? (primary controlled variables)

• Intuition: “Dominant variables”

• Systematic: Define cost J and minimize w.r.t. DOFs– Control active constraints (constant setpoint is optimal)

– Remaining DOFs: Control variables c for which constant setpoints give small (economic) loss

Loss = J - Jopt(d)

when disturbances d occurs


66

Application: Recycle processJ = V (minimize energy)

Nm = 5 3 economic DOFs

1

2

3

4

5

Given feedrate F0 and column pressure:

Constraints: Mr < Mrmax, xB > xBmin = 0.98


67

Recycle process: Loss with constant setpoint, cs

Large loss with c = F (Luyben rule)

Negligible loss with c =L/For c = temperature


68

Recycle process: Proposed control structurefor case with J = V (minimize energy)

Active constraintMr = Mrmax

Active constraintxB = xBmin


69

Effect of implementation error on cost


70

II. Bottom-up assignment of loops in control layer

• Identify secondary (extra) controlled variable

• Determine structure (configuration) of control system (pairing)

• A good control configuration is insensitive to parameter changes!

Industry: most common approach is to copy old designs


71

Step 4. Regulatory control layer

• Purpose: “Stabilize” the plant using local SISO PID controllers to enable manual operation (by operators)

• Main structural issues:• What more should we control? (secondary cv’s, y2)

• Pairing with manipulated variables (mv’s)

y1 = c

y2 = ?


72

Selection of secondary controlled variables (y2)

• The variable is easy to measure and control

• For stabilization: Unstable mode is “quickly” detected in the measurement (Tool: pole vector analysis)

• For local disturbance rejection: The variable is located “close” to an important disturbance (Tool: partial control analysis).


73

Summary

Procedure plantwide control:

I. Top-down analysis to identify degrees of freedom and primary controlled variables (look for self-optimizing variables)

II. Bottom-up analysis to determine secondary controlled variables and structure of control system (pairing).


•Skogestad, S. (2000), “Plantwide control -towards a systematic procedure”, Proc. ESCAPE’12 Symposium, Haag, Netherlands, May 2002.•Larsson, T. and S. Skogestad, 2000, “Plantwide control: A review and a new design procedure”, Modeling, Identification and Control, 21, 209-240. •Skogestad, S. (2000). “Plantwide control: The search for the self-optimizing control structure”. J. Proc. Control 10, 487-507.

See also the home page of Sigurd Skogestad:http://www.chembio.ntnu.no/users/skoge/

74

Biological systems

• ”Self-optimizing” controlled variables have presumably been found by natural selection

• Need to do ”reverse engineering” :– Find the controlled variables used in nature

– From this identify what overall objective the biological system has been attempting to optimize

75

Conclusion

• Negative Feedback is an extremely powerful tool

• Complex systems can be controlled by hierarchies (cascades) of single-input-single-output (SISO) control loops

• Control extra local variables (secondary outputs) to avoid fundamental feedback control limitations

• Control the right variables (primary outputs) to achieve ”self-optimizing control”

76

Outline











77

Paper by Doyle (special issue of Science on Systems Biology, March 2002)

SUMMARY

• Robustness

• Speculation: Most of the supposedly important genes are related to control– Compare with commercial airplane or chemical plant

• HOT: mechanism for power laws that challenges the self-optimized-criticality and edge-of-chaos concepts (Santa Fe Institute)

1 feedback control theory: an overview and connections to biochemical systems theory sigurd...

Documents