1. Ryszard Gessing Silesian Technical University Gliwice, Poland
Whether and When the Conventional Controllers Operate Well with Nonlinear Plants
Ryszard Gessing
Silesian Technical University
Gliwice, Poland

2. Outline of the Presentation
Introduction
Magnetic Levitation (narmamaglev.mdl)
Robot Arm (mrefrobotarm.mdl)
Criterion RD1 (Relative Degree = 1)
Stirred Tank Reactor (predcstr.mdl)
Conclusions

3. Introduction
In MATLAB NN Demos the systems with the following nonlinear plants
are considered:
Magnetic Levitation
Robot Arm
Stirred Tank Reactor
It will be shown that applying some conventional controllers –
gives better result;
confirms the usability of the Criterion RD1 (relative Order =1),
which is not commonly known in literature.

4. Outline of the Presentation
Introduction
Magnetic Levitation (narmamaglev.mdl)
Robot Arm (mrefrobotarm.mdl)
Criterion RD1 (Relative Degree = 1)
Stirred Tank Reactor (predcstr.mdl)
Conclusions

5. Magnetic Levitation (narmamaglev.mdl)
Plant:
Balance of forces:

6. Model of the magnetic levitation
Static characteristic:

7. Applied Controllers
1. The controller NARMA-L2 from MATLAB demo which has to illustrate the possibilities of the neural networks;
2. The conventional controller PD with the constraints of the
control the same as in NARMA-L2:
narmamaglev.mdl narmamaglev2,mdl narmamaglev1.mdl
maglev.mdl maglev1.mdl

8. The control waveforms for
The NN controller
NARMA-L2
(after learning).
The conventional
controller PD.

9. The control waveforms for
The conventional
controller PD.
The NN controller
NARMA-L2
(after learning).

10. Outline of the Presentation
Introduction
Magnetic Levitation (narmamaglev.mdl)
Robot Arm (mrefrobotarm.mdl)
Criterion RD1 (Relative Degree = 1)
Stirred Tank Reactor (predcstr.mdl)
Conclusions

11. Robot arm (mrefrobotarm.mdl)
From balance of the torques:

12. Model of the robot arm
Static characteristic:
(exists for )

13. Time responses for u=9*1(t) and u=11*1(t)
For nonlinear dynamic integrator.

14. The applied controllers
1. The „Model Reference Controller” from MATLAB demo which
has to illustrate the possibilities of the NN controllers;
2. The modified controller PD with the same control constraints as
in the NN MRC, implementing the control with model reference.
Compensator PD
mrefrobotarm.mdl robotarm0.mdll

15. The control waveforms -0.7<r<0.7
MRC NN controller
PD controller

16. The control waveforms PD controller -2<r<2 MRC NN controller

17. Outline of the Presentation
Introduction
Magnetic Levitation (narmamaglev.mdl)
Robot Arm (mrefrobotarm.mdl)
Criterion RD1 (Relative Degree = 1)
Stirred Tank Reactor (predcstr.mdl)
Conclusions

18. The Criterion RD1
Compensator PD
Assumptions: The linear or nonlinear plant has minimum phase
zeros and the polynomial Q(s), (Q(0)=1) is stable.
The Criterion RD1:
The relative order of the OL system is equal to 1.
Then the CL system is usually stable
for large values of the gain k and
it has very good properties.
where Z(s) = kQ(s)G(s) E(s)

19. Very simple design Determine the relative degree d of the plant;
2. Apply the polynomial:
3. Choose the time constant T so that
where or somewhat more
phase of the OL TF (linear system), or by means
of trials (nonlinear system).
4. Apply approximation:
where or less.

20. Examples
Magnetic levitation
Robot arm

21. Properties of the considered system
Operates for linear and nonlinear plants;
Is very robust with respect to large and fast parameter changes;
Has very fast transients;
Needs actuators accepting “nervous operation”;
Measurement noises cause some problems.

22. Outline of the Presentation
Introduction
Magnetic Levitation (narmamaglev.mdl)
Robot Arm (mrefrobotarm.mdl)
Criterion RD1 (Relative Degree = 1)
Stirred Tank Reactor (predcstr.mdl)
Conclusions

23. Stirred tank reactor (predcstr.mdl)
h – level
y - concentration

24. Stirred tank reactor (predcstr.mdl) static characteristic
Ymx=21.665

25. Time response for stepwise input u
yo=0, ho=30
yo=22, ho=30

26. Applied controllers:
1. NN predictive controller having to illustrate the possibilities
of neural networks – stepwise change of the reference every
20 time unites in the interval 20-23, control constraints: 0-4;
2. Since d=1 in accordance with Criterion RD1 the proportional P
controller with gain k=200, under changes of the reference
and control constraints as for the system with NN controller.
predcstr.mdl cstr0.mdl

27. Control waveforms (stepwise change of the reference r every 20 time units)
NN predictive
controller
Conventional
P controller

28. Control waveforms (stepwise change of the reference every 200 time units)
NN predictive
controller
Time of simulation
15 min.
Conventional
P controller
Time of simulation
5 sec.

29. Outline of the Presentation
Introduction
Magnetic Levitation (narmamaglev.mdl)
Robot Arm (mrefrobotarm.mdl)
Criterion RD1 (Relative Degree = 1)
Stirred Tank Reactor (predcstr.mdl)
Conclusions

30. Conclusions
It is worthwhile to know the Criterion RD1, though not always it may be applied;
For its applying usually only the knowledge about the relative degree of the plant (linear or nonlinear) is needed;
The common view that for control of nonlinear plants the advanced controllers are needed is not valid (especially in the case of RD smaller than 3);
It is seen that the conventional controllers PD or even P operate significantly better even with strongly nonlinear plants than NN controllers;
Applying of NN controllers to these plants is not justified and probably results from the above mentioned common view;
Some limitation in applying of the controllers with high gain, resulting from Criterion RD1 is the appearance of measurement noises, but this problem goes beyond the scope of the present paper.

31. Example
(nonstationary plant of the third order)
- varying

32. Parameter changes:
Time responses for stepwise changes of the reference:
Control constraints: