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

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Title: 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
  • 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
Disturbance (d)
Setpoint (ys)
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
Example
1
d
Gd
G
u
y
Plant (uncontrolled system)
14
(No Transcript)
15
Feedforward (FF) control
d
Gd
G
u
y
Nominal GGd ? 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
eys-y
Feedback controller
G
u
y
Negative feedback uf(e) Counteract error in y
by change in u
22
Feedback (FB) control
  • Simplest On/off-controller
  • u varies between umin (off) and umax (on)
  • Problem Continous cycling

23
Feedback (FB) control
Most common in industrial systems PI-controller
24
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
FB control change in time constant
29
FB control simultaneous change in gain and time
constant
30
FB control change in time delay
31
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 (H8)
  • Nonlinear control (1950 - )
  • Feedforward thinking, Mechanical systems
  • Adaptive control (1970-1985) (Åstrøm)

39
Control theory
Design
40
Relationship to system biochemistry/biologyWhat
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 ?eff
  • 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 ?
  • Effective delay PI-control original delay
    inverse response half of second time
    constant all smaller time constants

46
PI-control
47
Improve control?
  • Some improvement possible with more complex
    controller
  • For example, add derivative action
    (PID-controller)
  • May reduce ?eff 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 PIs)
d
ys
y2
u
y2s
G1
G2
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 gain1 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)
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)
  • 1000s of loops
  • Simple components
  • on-off P-control PI-control nonlinear
    fixes some feedforward

Same in biological systems
53
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

54
Hierarchical structure
Brain
Organs
Local control in cells
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
Brain
What should we control?
Cells
Hierarchical Centralized
57
Alternatives structures for optimizing control
What should we control? (Control theory has
little to offer)
Control theory has a lot to offer
Hierarchical Centralized
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)
J cost (overall objective to be minimized)
Loss L J - Jopt (d)
Self-optimizing control is achieved when a
constant setpoint policy results in an
acceptable loss L (without the need to reoptimize
when disturbances occur)
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
Stepwise procedure 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!

62
Stepwise procedure chemical plant
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
Stepwise procedure chemical plant
  • 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
Stepwise procedure chemical plant
  • Nm no. of dynamic (control) DOFs (valves)

65
Step 3. What should we control? (primary
controlled variables)
Stepwise procedure chemical plant
  • 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)
Stepwise procedure chemical plant
5
4
1
Given feedrate F0 and column pressure
2
3
Nm 5 3 economic DOFs
Constraints Mr lt Mrmax, xB gt xBmin 0.98
67
Stepwise procedure chemical plant
Recycle process Loss with constant setpoint, cs
Large loss with c F (Luyben rule)
Negligible loss with c L/F or c temperature
68
Recycle process Proposed control structurefor
case with J V (minimize energy)
Stepwise procedure chemical plant
Active constraint Mr Mrmax
Active constraint xB xBmin
69
Stepwise procedure chemical plant
Effect of implementation error on cost
70
II. Bottom-up assignment of loops in control layer
Stepwise procedure chemical plant
  • 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
Stepwise procedure chemical plant
  • Purpose Stabilize the plant using local SISO
    PID controllers to enable manual operation (by
    operators)
  • Main structural issues
  • What more should we control? (secondary cvs, y2)
  • Pairing with manipulated variables (mvs)

72
Selection of secondary controlled variables (y2)
Stepwise procedure chemical plant
  • 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
Stepwise procedure chemical plant
  • 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. ESCAPE12
    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
  • 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

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)
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