Design and Analysis of DC Motor With PID Controller - IRD India

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ISSN (PRINT) : 2320 – 8945, Volume -1, Issue -3, 2013

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Design and Analysis of DC Motor With PID Controller - A State Space Approach

K. Venu Ch. Rushikesh V. Rajasekhar Department of Electrical and Electronics Engineering

VNR Vignana Jyothi Institute of Engineering and Technology, Hyderabad-500090, India [email protected], [email protected] & [email protected]

Abstract - DC motors, because of their simplicity, ease of application, reliability and favourable cost have been a backbone of industrial applications. In these applications, the motor should be precisely controlled to give the desired performance. Modelling of any system is an important task in control applications because the electrical and mechanical components should be represented in mathematical form. Although several methods are available for representing DC Motor model, the state space approach is unique in the sense, accurate and precise control is possible. This paper presents the state space approach in modelling of DC Motor. Mathematical models are developed for simple closed loop control of DC Motor and closed loop control with proportional Integral and Derivative (PID) controller. This approach of modelling is useful for designing intelligent control techniques like Fuzzy control, neural networks etc.

I. INTRODUCTION

In most of the advanced control algorithms DC Motors are used because of stable and linear characteristics associated with it. Also various speed control methods are available for DC Motor to meet the desired performance. Hence modelling should be done in such a way that every control algorithm available can be implemented. Generally modelling of any system is to represent mechanical, electrical or any physical systems or components in mathematical form. The DC Motor can be modelled by using four basic equations.

Regularly every physical system is represented in the form of transfer function which is a relation between input and output, but absence of initial conditions limits this form of modelling. DC Motor can also be modelled by using state space equations. The state space representation is relation between state variables, their derivatives, input and output. Feasibility of defining initial conditions and being simple first order differential equations the state space approach finds its application in modelling various physical systems.

II. DC MOTOR MODELLING

The DC motor modeling is done summing the torques acting on the rotor inertia and integrating the acceleration to the velocity and also Kirchhoff‟s laws to armature circuit.

Figure1: Dynamic Motor Model

The mathematical model of DC motor can be constructed by suing four basic equations of motor.

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(2)

(3)

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Assuming physical parameters of the motor J: moment of inertia of the rotor 0.068 kg/m²

B:motor viscousfriction constant 0.03475NM/rad.sec K_b: electromotive force constant 3.475V/rad/sec Kt: motor torque constant 3.475 N.m/Amp Ra: electric resistance 7.56 Ohm La: Motor Inductance 0.055 H T_f: Friction Torque 0.212N-m T = Time constant 40msec

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ITSI Transactions on Electrical and Electronics Engineering (ITSI-TEEE)

ISSN (PRINT) :2320 – 8945, Volume -1, Issue -3, 2013

12 III. TRANSFER FUNCTION OF DC MOTOR

The transfer function of DC Motor is derived by simplifying four basic equations and applying laplace transform. Here we will take that the input of the system is the reference voltage (u_a) applied to the motor's armature, while the output is the rotational speed of the shaft (w_r).

STATE SPACE FORM

In general state space representation has the form

y = Cx +Du

x is a column vector of dimension „n‟ called State Vector.

u is the input of the system y is the output of the system A is system matrix

B is input matrix C is Output matrix D is feedback matrix

IV. STATE SPACE EQUATIONS FOR OPEN LOOP CONTROL OF DC MOTOR

Simple open loop network of DC Motor can be represented in state space form by taking current and speed as state variables.

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Let the friction torque T_f = 0 then we can write state space matrix as

= +

Y = + 0

Substituting motor parameters in the above matrix then we get:

system matrix A =

input matrix B = output matrix c ₌

and the feedback matrix is zero matrix_.

The State space equations after substituting motor parameters are:

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V. STATE SPACE EQUATIONS FOR CLOSED LOOP CONTROL OF DC MOTOR In closed loop network the difference between the reference input and feedback input is fed to the armature of DC Motor.

(9) u is the reference input

Equations (5) (6) and (9) constitute state space equations for closed loop control of DC Motor and the resultant state space matrices are given below.

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Y = + 0

Substituting physical parameters of the motor we get state space equations as

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The state space matrix after substituting physical parameters of the motor is:

VI. STATE SPACE EQUTATIONS FOR PID CONTROLLED DC MOTOR

The speed of the DC Motor can be controlled by installing a PID controller in the closed loop network of DC motor. With PID controller we get three state space variables and the single input single output DC Motor network is converted into five input and single output network.

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Let

Therefore equations (5) (6) and (15) form the state space equations for PID Controlled DC Motor. From the equation (15) it is observed that the input to the DC Motor or the armature voltage is becoming a state variable in the presence of the PID controller.

The state space equations can be represented in the matrix form given below:

BU

A=

B =

After substuting the physical parameters of the motor the state space matrices are modified as:

A=

– –

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14 B=

VII. SIMULATIONS AND RESULTS

The state space equations are solved in the interval of 0 to 10seconds with zero initial conditions in MATLAB. Step load is applied and a reference speed of 100rad/sec is set. The results are presented for open loop control, closed loop control and with PID controller.

The proportional gain (K_p), proportional integral gain (Ki) , proportional derivative gain (Kd) are chosen such that the performance of the system is improved

Figure 1 Open loop response of DC Motor

Figure 2 closed loop response of DC Motor

Figure 3 Closed loop response with PID controller VIII. DISCUSSIONS AND CONCLUSIONS Technically the simulink model drawn from transfer function of DC motor and MATLAB code generated from state space equations are same. Hence the results obtained from transfer function analysis and the state space analysis coinciding with each other is observed.

Feasibility in expressing initial conditions and time variant complex inputs to the DC motor are the unique features possible with state space approach of DC Motor modeling. Manipulation in any of the parameters is also possible while solving differential equation which is useful for implementing artificial intelligent control algorithms like Fuzzy control.

IX. REFERENCES

[1] Farzan Rashidi', Mehran Rashidi: and Arash Hashemi-Hoseinit ,“Speed Regulation of DC Motors Using Intelligent Controllers”, 0-7803- 7729-X/03/ $17.00 02003 IEEE, pp.925-930 [2] Waleed Abd El-Meged El-badry, “Design and

implementation of Real Time DC motor speedcontrolusing Fuzzy logic”

[3] Fuzzy Logic Controller for contolling dc motor speed using MATLAB applications Nur Azliza Ali

[4] DC Motors dynamic model and control techniques by Luca Zaccarian

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