1. Examples from personal experience
Five times exam: course Dynamic Systems including PID control ...
Robot control : Still done by PID controllers ...
Greenhouse climate control : Still done by PID controllers ...
Alternative title : War on PID terror !
First landing on the moon in 1969: What does optimal mean?
My wife (who is always right): “Do not fully turn off the heater at night”. Is that optimal?
5, 5
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

2. PID control: I do not understand !
Questions
A temperature error is needed to heat the room?
Tm(t) is always late compared to Td(t)?
Why do I need to measure the room temperature?
If I know the heater and room I can compute H(t), or not?
3, 8
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

3. PID control of greenhouse
Questions
Window opening, Heater or a combination?
How to get Td(t), Hd(t)?
Is realizing Td(t), Hd(t) the real goal?
2, 10
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

4. The “full” greenhouse control problem
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What is the control challenge/objective?
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

5. Verbal optimal control problem formulation
Find the heating, CO2 supply and ventilation patterns that produce maximum profit.
Patterns: time functions.
Profit: Benefit – Costs.
Benefit is obtained from selling crops.
Costs are associated with heating (energy supply), CO2 supply, ventilation and other investments.
Simplifications: ignore ventilation and humidity and consider perfectly isolated greenhouse.
2, 15
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

6. Quantitative mathematical formulation
What do the mathematical symbols represent?
What other information is needed to solve the mathematical problem?
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HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

7. Mathematical model of the greenhouse (1)
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HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

8. Mathematical model of the greenhouse (2)
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HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

9. Mathematical model of the greenhouse (3)
The mathematical model consists of (first-order) differential equations recognisable by presence of rates of change dx/dt.
Models containing differential equations are called dynamic models.
Dynamic models are obtained from knowledge of the system, often scientific knowledge (first-principles models / white box models).
Using scientific knowledge allows us to understand and interpret the model!
Black box models do not allow for this.
3, 29
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

10. Halfway summary (1)
Optimal control problem: Given a mathematical dynamic model of the system find the control patterns (time-functions/trajectories/histories) that maximize (minimize) a cost function (performance measure/cost criterion/penalty function).
Problem formulation is still missing one thing ...
We also have to know “where the system starts”
We have to know the initial state : values of W(0), T(0), CO2(0).
3, 32
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

11. Halfway summary (2)
Control problem has been turned into a mathematical computational problem.
No need for measurements or errors!
General control objectives (profitability, sustainability) can be quantitatively expressed in the cost function!
General non-linear, multivariable systems can be handled!
Have a break ... of no more than 15 minutes!
2, 34
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

12. Three optimal control examples
Heating of living room at night
Simplified greenhouse
Robot control
Examples included in Tomlab Propt optimal control toolbox for Matlab. Tomlab is commercially available. A free demo license is easily obtained:
See also
3, 3
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

13. Heating of living room at night (1)
Control objectives
Minimize heat energy supply.
Room temperature at least 20 [oC] at 7.00 AM.
Cost function:
Dynamic model:
Initial state:
2, 5
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

14. Heating of living room at night (2)
Constraints (equalities or inequalities that have to be satisfied exactly):
External input:
Demo!
My wife is almost always right ...
1 + 10, 16
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

15. Simplified greenhouse (1)
Control objectives: maximize profit.
Dynamic model:
Cost function = profit:
Initial state:
Constraints: demo!
4 + 10, 30
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

16. Simplified greenhouse (2)
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

17. MK2 Robot Control objectives Constraints Dynamic Model Result
Minimum time pick and place; pick and place locations specified
Constraints
Limited DC motor currents proportional to motor torque
Dynamic Model
Generated automatically from CAD drawing or a language specifying the robot topology, the associated link masses and moments of inertia
Result
Optimal DC motor currents & optimal robot motion = optimal path planning! Demo!
3 + 5, 38
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

18. Theory & Practice
Theory: optimal controls computed off-line (in advance) using the dynamic systems model:
Practice: Apply control to the real system:
2, 40
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

19. Measurements (1) Second control objective: limit magnitude
How? Is known in advance?
Perturbation state
feedback:
2, 42
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

20. Measurements (2) Is it necessary to measure all states ?
Suppose we measure
Then
Perturbation
output feedback:
How to get ?
2, 44
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

21. Perturbation state estimator: Kalman filter
How to get ?
Kalman Filter:
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HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

22. Output feedback part HAN 7/19/2019
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HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

23. Optimal Control System
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HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

24. Summary (1) With optimal control: This requires:
All types of systems can be handled.
All types of control objectives can be considered.
All types of constraints can be handled.
This requires:
A quantitative mathematical dynamic model of the system obtained by exploiting scientific knowledge.
A quantitative mathematical description of the control objectives.
A quantitative mathematical description of the constraints.
What you get: The best control & performance!
2, 52
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

25. Summary (2) Measurements:
are only required to realize feedback to counteract model and other types of uncertainty.
If the model (including the initial state and external inputs) is known perfectly optimal control:
Is errorless, as opposed to PID control.
Shows no latency, as opposed to PID control.
Takes into account all system-interactions, as opposed to PID control.
Is optimal, as opposed to PID control.
2, 54
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

26. Summary (3) The Future? or Not?
Society increasingly wants to control everything (health, environment, food, economy) and quantify and optimize everything (costs, emissions, footprints, energy, weights).
Systems become larger, more complicated and interactive.
or Not?
Robots, greenhouses & most industrial processes are mostly still controlled by PID controllers ...
Neils Armstrong & George Aldrin landed manually ... !
Optimal is only optimal with respect to the model!
If the model is poor, if the process contains highly uncertain parts, the benefits of optimal control vanish.
2, 56
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

27. Messages Design Modelling (when being a scientist)
Get rid of uncertainty!
Use optimal control as a design tool to investigate the influence of system (design) parameters on optimal performance.
Modelling (when being a scientist)
Use scientific knowledge (first principles, white box modelling) not black box approaches.
Work hard and you will be rewarded :
a white box model needs to be developed only once and not over and over again when the system is changed or extended.
You will be able to interpret and understand results and errors.
2, 58
HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,

28. Final message If you do not understand you may be wright! Questions ?
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HAN
7/19/2019
Gerard van Willigenburg, Wageningen University,