A Comparison of conventional and nonconventional methods of DC motor speed control

15th Workshop on International Stability, Technology, and Culture The International Federation of Automatic Control June 6-8, 2013. Prishtina, Kosovo

A Comparison of conventional and nonconventional methods of DC motor speed control Ermira Buzi*, Petraq Marango** * Electrical Engineering Faculty, Polytechnic University of Tirana Tirana, Albania (e-mail: [email protected]). ** Electrical Engineering Faculty, Polytechnic University of Tirana (e-mail: [email protected]) Abstract: In this paper we study the speed control of a DC motor with conventional (PID) and nonconventional (Neural Network – NN) methods. DC motor model and performance indicators criteria are defined. PID controller is tuned through Matlab scripting and tools and relevant results of closed loops system are received. For nonconventional method inverse model control schema is used, where a three layer NN presents the inverse model, and it is trained by back propagation algorithm. Simulation results are presented to compare the two control methods and conclude the benefits of NN usage in control. Considering the broad range of DC motor usage, where robotics is included, the study is presented as a way for improving the closed system’s dynamics. Keywords: Neural networks, back propagation algorithm, DC motor, neural control, PID controller. D

1. INTRODUCTION

Va(s)

NN covers different areas like data mining, function approximation, pattern recognition. In this paper we will study their function approximation capability for speed control of a DC motor with nonconventional (Neural Network – NN) method. The NN parameters will be adjusted for having the NN model’s response the same as DC motor model’s ones (et al Hagan 1996). Back propagation algorithm is used for NN parameters tuning. The approximated model will be used in inverse model control schema. PID controller will be used on conventional control method. Simulation results will be presented to compare the two control methods.

+

1 b  Js

+

ω(s)

Ke

Fig. 1. Block diagram of DC motor

G( s) 

Kt  ( s)  Va ( s) Ra  La s b  Js   Kt K e

(1)

Motor’s transfer function will be:

G( s) 

2. DC MOTOR MODEL AND PERFORMANCE INDICATORS CRITERIA

2.256  10 5 s 2 + 1059 s + 1.973  10 4

(2)

Step response of model G(s) is given on Fig. 2. Discrete transfer function of the motor for sampling period T = 0.01sec is: (et al Veisllari 1997)

2.1 The DC motor model We have taken under consideration the speed control of DC motor with permanent magnet of module G14 EV(et al Elettronica Veneta 2004). Its technical parameters are: rotor’s inertia J = 350*10-7 kg m2; mechanical time constant Tm = 0.17s, back-emf constant (K=Ke=Kt) = 0.06 Nm/A; armature resistance Ra = 8 ohm; armature inductance La = 7.6 mH

G( z ) 

1.801 z -1  0.1758z-2 1 - 0.8272 z -1  2.529  10-5 z  2

(3)

This gives the difference equation: y(k )  0.82 y(k -1) - 2.5210-5 y(k  2)  1.8 u(k -1)  0.17 u(k - 2)

(4)

Block diagram of the model is given in Fig.1. and motor’s model on eq. (1), (et al Marango 2001). Model’s input is armature voltage Va and the output is speed ω.

978-3-902823-34-2/2013 © IFAC

-

-

Kt Ra  La s

where y(k) and u(k) are the speed and the control signal on sampling time kT, for k = 1, 2, 3, ... N.

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10.3182/20130606-3-XK-4037.00054

IFAC SWIIS 2013 June 6-8, 2013. Prishtina, Kosovo

H ( z) 

3.326 10 -5 z 3  0.106 z 2  0.054 z  0.006 z 3  1.721z 2  0.773z  0.006

(6)

Fig. 2. Step response of opened loop system. 2.2 Performance Indicators For control design we define the performance indicators criteria as:

Fig. 4. PID tuning through Matlab’s pidtool System response towards stair signal will be as per Fig. 5

Steady state error: e = 0; Overshoot in %: < 15% Settling time: tv ≤ 0.25 sek Response of closed loop system for both cases, with PID control and with Inverse Neural Model control, should meet these criteria. 3. PID CONTROL Block diagram of closed loop system with PID control is on Fig. 3, where G(s) is the DC motor transfer function and Gc(s) is transfer function of PID controller of the form:

GC ( s)  K p (1 

1  Td  s) Ti  s Fig. 5. Response of closed loop system with PID controller

R(s) +

Gc(s)

G(s)

ω(s)

From the results, the tuned PID fulfils the performance indicators criteria.

-

4. CONTROL WITH INVERSE NEURAL NETWORK MODEL

Fig. 3. Block diagram of closed loop system with PID controller

We replace the PID controller on Fig. 3 with Inverse Neural Model and compare the results for both methods. Closed loop system will be as per Fig. 6.

Through pidtool of Matlab (Fig. 4) we tune the PID parameters as: Kp = 0.0358; Ti = 0.0154; Td = 5.16∙10-6 The tuned PID controller gives the performance indicators of closed loop system: Rise time (seconds) tr = 0.074; Settling time (seconds) ts = 0.247; Overshoot in %: 11

r(k) +

e(k) -

Inverse Neural Model

u(k)

DC Model

y(k)

Transfer function of closed loop system is:

H ( s) 

1.108 10 -8 s 2  0.002s  0.139 -7 3

2

2.26 10 s  0.0003s  0.007s  0.139

Fig. 6 Block diagram of control with inverse neural model (5)

The inverse model of the DC motor is to be identified. Prior identification, we simulate the set of input {u(k)} – output {y(k)} data (Fig.8) and define the NN topology as a three layer neural network. Number of neurons in input layer is defined referring to differential model of DC motor, eq. 4.

Discrete transfer function of the system with sampling time Ts = 0.01 is:

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IFAC SWIIS 2013 June 6-8, 2013. Prishtina, Kosovo

Output layer present the model output, and number of neurons in hidden layer is defined based on simulations results. The selected topology is 6×6×1 (et al Deng 2011, Hagan 1996, Shahin 2008). The training of Inverse Neural Model is done offline, through serial-parallel model, depicted on Fig. 7, (et al Bose 2002 p654). The neural network model is trained with the help of back propagation algorithm. The error received as difference between the target control signal {u(k)} and neural network output {uNN(k)}, is back propagated to the network for tuning the weights of the network’s layers. Minimization of square error e2 is done by gradient descent method, (et al Bose 2002 p637-643).

DC model Diff. eq.

u

y

yN

Z-m(m = -1, 0, 1, 2)

N e

N

+

Inverse Neural Model

uNN

Fig.9. evolution of input layer weights of the neural model

yN(k+1) yN(k) yN(k-1) yN(k-2) uN(k-1) uN(k-2)

. .. Backpropagation algorithm

Fig. 7. Block diagram of identification of inverse neural model Fig.10. evolution of hidden layer weights of the neural model

Fig. 8. Input-output data set used for NN training Fig.11. evolution of training error of the neural model

The simulation results for identification (training) of inverse neural model are shown in Fig 9-12. Evolution of input and hidden layers weights are given in Fig. 9-10. Output layer present the NN output and is given in Fig. 12. Received maximal error is 0.007 and error variance e2 = 0.016.

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IFAC SWIIS 2013 June 6-8, 2013. Prishtina, Kosovo

giving a steady state error = 0.4%, and for reference amplitude r = 2 is y = 2 giving a steady state error = 0%. For having a better comparison of the two control methods we plot the output of closed loop system of both cases as per Fig. 14. It is obvious that control with NN is better than PID control. 6. CONCLUSIONS The speed of DC motor has been controlled using two methods, PID control and Inverse Neural Model control. The NN was trained to identify the inverse model of the motor without knowing the motor’s model, or what is called black box identification method. From the simulations results it can be seen that using NN, there is no need to know the model parameters when designing the system control if inputoutput data sets of the model are available through measurements or simulations.

Fig.12. Target response and output of inverse neural model The offline trained inverse neural model is placed in the closed loop system (Fig. 6). Simulation on Matlab of this system for stair reference signal is given on Fig. 13.

For this example NN control satisfies better than PID controller the performance indicators criteria. Thus NN can be seen as an alternative way for models identification and control for improving the systems dynamics. REFERENCES Bose, Bimal K, (2002) Modern power electronics and AC drivers, p654,637-643, Prentice Hall PTR, ISBN 0-13016743-6 Deng, Jiamei, Stobart, Richard and Maass, Bastian (2011). The Applications of Artificial Neural Networks to Engines, Artificial Neural Networks - Industrial and Control Engineering Applications, Prof. Kenji Suzuki (Ed.), ISBN: 978-953-307-220-3, InTech, Available from: http://www.intechopen.com/books/artificialneural-networksindustrial-and-control-engineeringapplications/the-applications-of-artificial-neuralnetworks-to-engines ElettronicaVeneta, (2004) Controllo di velocita per motore d.c. MODULO G14/EV, Student-Trainer MPT/EV. Treviso, Italy Hagan M. T, Demuth H. B, Beale M. H, 1996, Neural Network Design, PWS Publishing Company, ISBN 0534943322 Marango P, (2001) Kontrolli i Proceseve, SHBLU, ISBN 99927-0-138-2, Tirana, p.246-248 Shahin, M. A., Maier, H. R., and Jaksa, M. B., 2008, State of the Art of Artificial Neural Networks in Geotechnical Engineering, State of the Art Geotechnical Engineering, EJGE Special Volume Bouquet 08, http://www.ejge.com/Bouquet08/Shahin/Shahin_ppr.pdf Veisllari K, (1997) Kontrolli Numerik 1, SHBLU 1997, Tirana, p.52, p.83-84

Fig.13. Reference signal and closed loop system response for control with inverse neural model

Fig. 14 Comparison of output signals of closed loop system for both control methods. From the results, (Fig 13) the settling time of closed loop system with Inverse Neural Model is 0.1 sec and the steady state output value for reference amplitude r = 1 is y = 1.004, 53