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

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

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.

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

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

Model of the magnetic levitation Static characteristic:

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

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

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

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

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

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

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

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

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

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

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

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)

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.

Examples Magnetic levitation Robot arm

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.

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

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

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

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

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

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

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.

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

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.

Example (nonstationary plant of the third order) - varying

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