1 feedback control theory: an overview and connections to biochemical systems theory sigurd...
TRANSCRIPT
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)
Input (u) Output (y)
No control: Output (y) drifts away from setpoint (ys)
10
Plant(uncontrolled system)
Disturbance (d)
Input (u) Output (y)
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
Plant(uncontrolled system)
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
(BUT: So simple that it is frequently overlooked)
3. Control theory and possible contributions
4. Fundamental limitation on 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
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 (FB) control
Feedback controller
e=ys-y u
Simplest: On/off-controller• u varies between umin (off) and umax (on)• Problem: Continous cycling
23
Feedback (FB) control
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
1. Introduction: 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 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
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
1. Introduction: 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 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
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
1. Introduction: 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 control
5. Cascade control and control of complex large-scale engineering systems • 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
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
1. Introduction: 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 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
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)
Stepwise procedure chemical plant
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.
Stepwise procedure chemical plant
64
Step 2. Degree of freedom (DOF) analysis
• Nm : no. of dynamic (control) DOFs (valves)
Stepwise procedure chemical plant
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
Stepwise procedure chemical plant
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
Stepwise procedure chemical plant
67
Recycle process: Loss with constant setpoint, cs
Large loss with c = F (Luyben rule)
Negligible loss with c =L/For c = temperature
Stepwise procedure chemical plant
68
Recycle process: Proposed control structurefor case with J = V (minimize energy)
Active constraintMr = Mrmax
Active constraintxB = xBmin
Stepwise procedure chemical plant
69
Effect of implementation error on cost
Stepwise procedure chemical plant
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
Stepwise procedure chemical plant
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 = ?
Stepwise procedure chemical plant
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).
Stepwise procedure chemical plant
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).
Stepwise procedure chemical plant
•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
1. Introduction: 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 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
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)