Diapositive 1

1
International Project Course in Automatic Control
A joint project between Lund University of
Technology and Ecole des Mines de Nantes
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Project organization

• Project organization
• Work locally at each university and a week long
  workshop in Sweden
• Problem specified by ABB
• Swedish open championships in robot control
• Four groups working with different approaches
• PID
• State feedback
• Output feedback
• Robust

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General problem description
Typical industrial problem Dynamic nonlinear
systems Different disturbances Regulator problem
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System description
u
z
Process
w
y
v
Inputs u control signal w motor disturbance v
tool disturbance
Outputs z tool position y motor shaft angle
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Disturbances
Disturbances acting on both tool position and
motor torque Steps Pulses Chirps Stiffness
related nonlinearities in the gears
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Performance evaluation
The controller will be evaluated for three
different model sets Mnom, M1, M2 Mnom Nominal
model M1 small variations in the process
parameters M2 large variations in the process
parameters
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Cost criterion
8
Measured performance values
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Measured performance values
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PID
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Summary

• Introduction
• Theory
• Result
• Conclusion

PID
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Introduction
-Task Creation of a discrete PID
Controller -Basis ABBs PID Controller -Goal
1st optimize the cost function of the nominal
system 2nd extend to the non-linear
model -Working method Try different methods of
PID building Develop the best one
PID
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Basic control structure
PID
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Theory Ziegler-Nichols Methods
Step response method
Frequency method

• No controller connected
• Put a unit step to the process
• Calculate K, Ti and Td with K 1.2/a Ti 2b
  , Td 0.5b

• P-part connected
• Increase Kp gradually until system oscillates
• Find K0 and measure T0
• Calculate K, Ti and Td with K 0.6K0 Ti
  0.5T0 and Td T0/8

PID
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PID-structure

• PID with modified D-part using filter
• Reason Presence of high frequency disturbances
• - Solution Use a second-order filter
  Gf1/(sFF)2

PID
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Theory Optimization Method

• Multi dimensional optimization method
• Iterate four parameters to make best combination
  to obtain lowest ABB cost
• Need to choose an evaluation range for each
  parameter
• Return a matrix with the parameters and
  associated cost

Step
Kp0
Step
Kd0
Step
Ki0
Step
FF0
PID
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Results Setting the starting parameters

• - Ziegler-Nichols Method (Step response)
• Order of the system too high
• As a consequence it did not work
• Ziegler-Nichols Method (Frequency)
• Oscillation never occurred
• As a consequence it did not work
• Consequently use of the ABB parameters

PID
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Results Linear nominal model

• The optimization method was the best tool
• Efficient but high complexity (n4)
• Robustness good
• Not working for non linear models

Results
Our parameters (ABBs) - Kp 12 (12) - Ki
2,1 (1,2) - Kd 61 (30) - FF 0,7 (0,5)
Cost 65 (82)
PID
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Results (All models)

• Still using the gradient method stability
  aspect
• Cost calculated with CheckPerformance
• Worse robustness
• uncertainty of 5

Results
Our parameters (ABBs) - Kp 14,83 (12) -
Ki 2,4 (1,2) - Kd 53 (30) - FF 0,725
(0,5)
Cost 155,1 (184,9)
PID
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Conclusion
Bad Points - Solution found but maybe not the
best one - A lot of time is needed to
optimize - Results still worse than the winner
of the competition Good Points - Better
results than the one of ABBs PID - Not so far
from the bests of the competition - Methods use
are very easy to apprehend
PID
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Output Feedback
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Output Feedback

• Group members
• Output feedback in general
• The problem
• Different Approaches
• Swedish Scaling and pole placement of all the
  poles
• French Robust pole placement

Output Feedback
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Numerical Problem
Problem with representation of poles in MATLAB
Output Feedback
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Time scaling
Output Feedback
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Controller design
Output Feedback
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Problem

• 16 poles to place
• - Not an easy task
• French approach
• Two variables to tune

• Swedish approach
• - Abandoned due to complexity

Output Feedback
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The robust pole placement

• Three advantages.
• Few parameters to manipulate (Tf and Tc).
• Just one equation The Diophantine equation
• A(s)S(s) B(s)R(s) Abf(s) With
  Abf(s)C(s)F(s)
• And T(s) C(0)/B(0)F(s)
• No restriction of the regulator degree

Output Feedback
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Construction of C(s)
Im
-1/Tc
0
Re
Output Feedback
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Construction of F(s)
Im
0
-1/Tf
Re
Output Feedback
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Improvement of the method

• The integral action
• A(s)sS'(s) B(s)R(s) C(s)F(s) with A(s)s
  A
• The delete of the poles with large imaginary
  part
• Rotation of the poles
• Use of the fonction balreal

Output Feedback
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Results
Output Feedback
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Improvement of the method

• The integral action
• A(s)sS'(s) B(s)R(s) C(s)F(s) with A(s)s
  A
• The delete of the poles with large imaginary
  part
• Rotation of the poles
• Use of the fonction balreal

Output Feedback
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Conclusion

• Pole placement is hard and require lots of
  experience
• A great experience working in a international
  project

Output Feedback
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State Feedback
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State Feedback
A State Feedback approach to control the ABB
Robot
State Feedback
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Approaches

• The state feedback different kind of regulators
  for a single way of regulating
• Pole Placement design
• not a state feedback approach
• a quick try to discover the problem
• LQR method
• how to weight the states ?
• H2 method

State Feedback
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H2 - method
Theory

• Minimisation of a criterion

State Feedback
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H2 - method
Theory

• How to weight the states ?
• Reccati solution fits the optimized regulator.
• State feedback based on an augmented state
• With a constant disturbances model prediction
• With a sinusoidal disturbances model prediction

State Feedback
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H2 - method
Theory

• First approach order 8 system
• Second approach order 4 system
• More general regulator
• Best way to obtain a stable regulator for all the
  sets
• Results 2 regulators
• 1 regulator fitting the linear model
• 1 regulator fitting the non linear model
• gt An highlighting of the compromise between
  performance and robustness

State Feedback
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Results H2 method
Linear model

State Feedback
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Results H2 method
Linear model

Total cost 58,67
State Feedback
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Results H2 method
Global regulator (fitting all the sets)

State Feedback
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Results H2 method
Global sets results

State Feedback
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Results H2 method
Global sets results

Total cost 312,98
State Feedback
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Conclusion

• Several methods were used
• Gave us a good overview of state feedback
  regulation
• Several types of models were designed
• The accuracy of the modeling was improved
• Two different predictive models were designed for
  the disturbances

State Feedback
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Conclusion

• H2 method was finally chosen
• a  procedural  method
• a few parameters to tune
• Results
• Designed a specific regulator for the nominal
  model
• Designed a general regulator working on all model
• The models were too different to obtain a good
  general regulator but stability was achieved
• Conclusion
• Learned a lot
• Very valuable project

State Feedback
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Robust Control
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The H mixed sensitivity design
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Standard Problem
W input signals e signal to be controlled y
mesured signals u control signal
Robust Control Design
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The H mixed sensitivity design
8
Robust Control Design
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The H mixed sensitivity design
8
R
Robust Control Design
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The H mixed sensitivity design
8

• Design with Matlab
• augss, augtf transform the weighted
• system to the standard form.
• hinfopt computes the optimal gamma
• and the corresponding controller

Robust Control Design
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The H mixed sensitivity design
8
mixsyn transforms the weighted system to the
standard form computes the controller
Robust Control Design
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Our first goal
A controller that is stable, and produces a cost
for all models.
Robust Control Design
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Working method

• A script that uses mixsyn to produce a
  controller
• Adjust parameters and weight-functions
• Observer nyquist curves and step responses
• When it looks good - try running checkstability
  and simulations.

Robust Control Design
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Working method
Mixsyn does not produce a fully useful controller
by itself. We need to edit it.
Robust Control Design
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Working method
Moved a pole to zero - fixed stationary error in
step-response
... but step response still shows a slow mode
Robust Control Design
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Working method
Finally...
...by canceling slow pole and zero we obtain a
useful controller.
Robust Control Design
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Move parameters around - trying to optimize....
But
Robust Control Design
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Problem with the models
Stability for all models barely possible
Model no 5 and 9 in set 2
Robust Control Design
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So we make a new controller based on...
A process without interesting dynamics.
Robust Control Design
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A controller that works
slow but stable
Robust Control Design
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A controller that works

• Total Cost 211
• High step response
• OK settling time
• Good noise rejections
• Robustness criterions not met! (but stable for
  all models)

Robust Control Design
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Our second goal
A controller that produces the best possible cost
for the nominal model
Robust Control Design
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The combination test algorithm

• System very sensitive to slight changes
• Script to test automatically different
  combinations.
• Real-time S-functions generated to accelerate
  significantly the linear model.

Robust Control Design
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The combination test algorithm
First, W1 is fixed ? Then W2 is fixed and
W3 is increased until maxIterW3 is reached
? Then W2 is changed and W3 is increased
until maxIterW3 is reached and so on
? until maxIterW2 is reached. Then
W1 is changed and the operations above are
resumed. At each change of a weighting function,
hinfopt is used. When 0.75ltgammalt1.25 the
validation script is called to test if the
controller is good or not.
Robust Control Design
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The combination test algorithm
Results about 1,000 different controllers for
the linear system
One of our result
Total cost 64.78 Module margin 0.69
Robust Control Design
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Tuning method

• Problems
• ABB simulation time consuming
• ? Simpler model
• Mathematical representation
• Graphical user interface Tool

Robust Control Design
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Tuning method
Robust Control Design
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Our best controller so far

• Total Cost 49.68
• Good noise rejections
• Good module margin gt1

Robust Control Design
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Our best controller so far
Zreg 1.0e002   -0.4065 3.4105i
-0.4065 - 3.4105i -0.0565 1.2001i
-0.0565 - 1.2001i -0.0301 0.7787i
-0.0301 - 0.7787i -0.0816 - 0.0613i -0.0019
-0.0001   Preg 1.0e003
  -4.2332 -0.0388 0.3370i
-0.0388 - 0.3370i -0.2348
-0.0043 0.0985i -0.0043 - 0.0985i
0.0088 0.0454i 0.0088 - 0.0454i -0.0000
0.0000i -0.0000 - 0.0000i -0.0000
Robust Control Design
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RESULTS
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Comparison of the controllers
Group
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Comparison of the Sensitivity functions
Group