MODELING OF SPEED CONTROL IN A DC MOTOR USING PROPORTIONAL INTEGRAL DERIVATIVES CONTROLLER

MODELING OF SPEED CONTROL IN A DC MOTOR USING PROPORTIONAL INTEGRAL DERIVATIVES CONTROLLER

                                                               ¹ James Agajo ² Avazi Isaac    

           ¹ Dept. of Electrical and Electronics Engineering, Federal Polytechnic, Auchi, Edo state Nigeria                                                

                      ²Dept. of Electrical and Electronics Engineering, University of  Abuja, Nigeria

                                 Phone: +2348053312732 , agajojul@yahoo.com

ABSTRACT: This project aims at designing, analyzing, and modeling of a dc motor speed control using Proportional Integral Derivatives Controller(PID).It entails how a PID controller can be used to achieve a desired speed response in a dc motor speed control. It presents a detailed simulation of how this PID controller and technology is added to the dc motor speed control with their varying parameters for desired response. Using the Dynast Shell Simulator shows the hardware and it sub system in  their simulated platform without the use of complex equations to achieve your aim. This project relates the place of PID technology in the present control discipline, how it reduces and eliminates errors in the control processes using Dynast Shell Simulator.

1.0              BACKGROUND

In spite of the development of power electronics resources, the direct current machines are becoming more useful as they have found wide application, that is, automobile industry (electric vehicle), weak power using battery system (motor of toy), the electric traction in the multi-machine systems, etc.

The speed of DC motor can be adjusted to a great extent so as to provide easy control and high performance. There are several conventional and numeric controller types intended for controling the DC motor speed at its executing various tasks.[3]

PID which simply stand for Proportional Integral Derivative Controller which is used in control processes to eliminate and reduce errors.This errors represents the difference between where you want to go and where you are actually at.

Errors is thus defined as the difference between the set point and measurement.

PID controllers are widely used in industrial plants because it is simple and robust.

D.C. motors are motors that run on Direct Current from a battery or D.C. power supply. Direct Current is the term used to describe electricity at a constant voltage.

When a battery or D.C. power supply is connected between a D.C motor's electrical leads, the motor converts electrical energy to mechanical work as the output shaft rotates.

There are different kinds of D.C. motors, but they all work on the same principles.
"The electric motor is the most convenient of all sources of motive power. It is clean and silent, starts instantly, and can be built large enough to drive the world's fastest trains or small enough to work in a wrist.

PID Controllers are designed to eliminate the need for continuous operator attention. Cruise control in a car and a house thermostat are common examples of how controllers are used to automatically adjust some variable to hold the measurement (or process variable) at the set-point. The set-point is where you would like the measurement to be. Error is defined as the difference between set-point and measurement. Error = (set-point) - (measurement). The variable being adjusted is called the manipulated variable which usually is equal to the output of the controller. The output of PID controllers will change in response to a change in measurement or set-point.

This project will enhance the reduction of overshoot, reduction of settling time and the steady state trasient response of the control signal.

PID controller came to existence because it eliminate these error or reduce them to beareable extent.

These error reduction led to the design of dc motor speed control for effective plant control.

2.0 METHODOLOGY

The design and implementation of dc motor speed control using PID technology is achieved by dynast shell simulator.

When using Dynast shell simulator, you need not to deal with any equations at all. You can easily set up the system model in graphical Form from a kit of dynamic elements.

2.1 SECTION OF DYNAST SHELL SOLVER

The DYNAST simulation system consists of two separate parts. DYNAST  Solver and a DYNAST working environment. DYNAST Solver is composed of several sections sharing common data. The section SYSTEM reads in the system-model description in the form of a set of algebro-differential equations, a block diagram, a multipole diagram, or in a form combining freely these approaches.

2.2 INPUT LANGUAGE OF DYNAST SHELL SOLVER

DYNAST input-language is composed of statements coded in ASCII characters and both upper and lower case letters may be used in them. DYNAST, however, is not case sensitive and converts all the letters into the upper-case ones. Each of the statements is terminated by the semicolon character '

;

'. A statement may continue on several lines, and there may be several statements placed in one line. The maximum number of characters in one line including spaces is 80.

3.0 SYSTEM DESIGN AND ANALYSIS

3.1  DC MOTOR SPEED CONTROL

The striped thing between the motor and the dc generator is a shaft which couples them together mechanically. The motor drives the generator (tachometer) via this shaft.

The set speed control provides a dc voltage, say 12 volts for maximum speed and zero for stationary. This could be a potentiometer providing any voltage in a range from zero to +12 volts. The differential amplifier will amplify any difference between its two input voltages.

If the motor is stationary and the speed control is moved from zero to half speed then, since the tachometer is not rotating and not producing an output voltage,  there will a differential in voltages at the two inputs of the difference amplifier. Therefore there will be an output voltage from the amplifier.

Since this voltage is not high enough in value to drive the motor, it is increased in amplitude by the dc amplifier. A dc amplifier is a special type of amplifier which can increase dc voltages.[9]

4.2 MODEL OF A DC MOTOR

DC machines are characterized by their versatility. By means of various combinations of shunt, series, and separately-excited field windings they can be designed to display a wide variety of volt-ampere or speed-torque characteristics for both dynamic and steady-state operation. Because of the ease with which they can be controlled systems of DC machines have been frequently used in many applications requiring a wide range of motor speeds and a precise output motor control .[11]

In this paper, the separated excitation DC motor model is chosen according to its good electrical and mechanical performances more than other DC motor models. The DC motor is driven by applied voltage.

4.3 DC MOTOR SPEED MODELLING

A common actuator in control systems is the DC motor. It directly provides rotary motion and, coupled with wheels or drums and cables, can provide transitional motion.

For this example, we will assume the following values for the physical parameters. These values were derived by experiment from an actual motor in Carnegie Mellon's undergraduate controls lab.

* moment of inertia of the rotor (J) = 0.01 kg.m²/s² * damping ratio of the mechanical system (b) = 0.1 Nms * electromotive force constant (K=Ke=Kt) = 0.01 Nm/Amp * electric resistance (R) = 1 ohm * electric inductance (L) = 0.5 H * input (V): Source Voltage * output (theta): position of shaft * The rotor and shaft are assumed to be rigid

The motor torque, T, is related to the armature current, i, by a constant factor Kt. The back emf, e, is related to the rotational velocity by the following equations:

T=K_τi      ,  T=KeθIn     SI units (which we will use), Kt (armature constant) is equal to Ke (motor constant).

From the figure above we can write the following equations based on Newton's law combined with Kirchhoff's law:

Jθ+bθ= Ki

Ldi/dt + Ri = ∇-Kθ

Transfer Function

Using Laplace Transforms, the above modeling equations can be expressed in terms of s.

S(Js+b) θ(s)=KI(s)

(Ls+R)I(s)=V-Ks θ(s)

By eliminating I(s) we can get the following open-loop transfer function, where the rotational speed is the output and the voltage is the input.

θ/V = K/(Js+b)(Ls+R)+K²

 State-Space

In the state-space form, the equations above can be expressed by choosing the rotational speed and electric current as the state variables and the voltage as an input. The output is chosen to be the rotational speed.

4.4 Design requirements

First, our uncompensated motor can only rotate at 0.1 rad/sec with an input voltage of 1 Volt (this will be demonstrated later when the open-loop response is simulated). Since the most basic requirement of a motor is that it should rotate at the desired speed, the steady-state error of the motor speed should be less than 1%. The other performance requirement is that the motor must accelerate to its steady-state speed as soon as it turns on. In this case, we want it to have a settling time of 2 seconds. Since a speed faster than the reference may damage the equipment, we want to have an overshoot of less than 5%.

If we simulate the reference input (r) by an unit step input, then the motor speed output should have:

Ø       Settling time less than 2 seconds

Ø       Overshoot less than 5%

Ø       Steady-state error less than 1%

4.5 PID CONTROLLER

PID defines th three terms proportionality namely : P -Proportional, I - Integral, D – Derivative.

4.5 TUNING PID

Tuning a system means adjusting three multipliers Kp, Ki and Kd adding in various amounts of these functions to get the system to behave the way you want. The table summarizes the PID terms and their effect on a control system.[5]

TABLE 4.1: PID TERM AND EFFECT ON CONTROL SYSTEM

TERM                             MATH FUNCTION                 EFFECT ON CONTROL SYSTEM

PROPORTIONAL       KP x Verror                                Typically the main drive in a control          

                                                                                          loop, KP reduces a large part of the  

                                                                                          overall error

INTEGRAL               KI x  ∫ Verror dt                           Reduces the final error in a system.   

                                                                                          Summing even a small error over  

                                                                                          time produces a drive signal large  

                                                                                          enough to move the system.

DERIVATIVE          KD x dVerror / dt                          The output changes quickly. This  

                                                                                          helps reduce overshoot and ringing. It  

                                                                                          has no effect on final error   

Here's a straight forward approach to get you up and soloing quickly.

1. SET KP. Starting with KP=0, KI=0 and KD=0, increase KP until the output starts overshooting and ringing significantly.

2. SET KD. Increase KD until the overshoot is reduced to an acceptable level.

3. SET KI. Increase KI until the final error is equal to zero.

4.6 The characteristics of P, I, and D controllers

A proportional controller (Kp) will have the effect of reducing the rise time and will reduce ,but never eliminate. An integral control (Ki) will have the effect of eliminating the steady-state error, but it may make the transient response worse. A derivative control (Kd) will have the effect of increasing the stability of the system, reducing the overshoot, and improving the transient response. Effects of each of controllers Kp, Kd, and Ki on a closed-loop system are summarized in the table shown below. [2]

TABLE 4.2 PID CLOSED LOOP RESPONSE

CL RESPONSE  RISE TIME    OVERSHOOT   SETTLING TIME    S-S ERROR

Kp                        Decrease          Increase               Small Change                 Decrease

Ki                         Decrease          Increase               Increase                          Eliminate

Kd                      Small Change    Decrease              Decrease                      Small Change 

Note that these correlations may not be exactly accurate, because Kp, Ki, and Kd are dependent of each other. In fact, changing one of these variables can change the effect of the other two. For this reason, the table should only be used as a reference when you are determining the values for Ki, Kp and Kd.

From the main problem, the dynamic equations and the open-loop transfer function of the DC Motor are:

S(Js+b) θ(s)=KI(s)

(Ls+R)I(s)=V - Ks θ(s)

θ/V = K/(Js+b)(Ls+R)+K²

FIG 4.3 IMPLEMENTATION OF PID CONTROLLER

With a 1 rad/sec step input, the design criteria are:

Ø       Settling time less than 2 seconds

Ø       Overshoot less than 5%

Ø       Steady-stage error less than 1%

First, let's take a look at how the PID controller works in a closed-loop system. The variable (e) represents the tracking error, the difference between the desired input value (R) and the actual output (Y). This error signal (e) will be sent to the PID controller, and the controller computes both the derivative and the integral of this error signal. The signal (u) just past the controller is now equal to the proportional gain (Kp) times the magnitude of the error plus the integral gain (Ki) times the integral of the error plus the derivative gain (Kd) times the derivative of the error.[7]

U=ke + Ki∫edt +K_Dde/dt

This signal (u) will be sent to the plant, and the new output (Y) will be obtained. This new output (Y) will be sent back to the sensor again to find the new error signal (e). The controller takes this new error signal and computes its derivative and its integral again. This process goes on and on.

4.4 RUNNING SIMULATION AND DESIGN USING DYNAST SHELL

This is the final step as regards the problem set up, it is done after the dc motor has been modelled, after the pid controller has been added to the design.

Also,  having inputed the correct parameters of the pid controller to ensure the desired response.

4.5 HARDWARE SUB SYSTEM

A Permanent magnet DC machine

Ø      Interface

             A

[V]

armature-circuit +terminal

             B

[V]

armature-circuit –terminal

             S

[rad/s]

machine shaft

             H

[rad/s]

machine housing

        External Parameters

          Kem = 1

[V.s/rad]

machine electro-mechanical constant

          La = 0

[H]

armature-circuit inductance

         Ra = 0.1

[Ω]

armature-circuit resistance

         Jm = 0

[kg.m²]

armature moment of inertia

         Bm = 0

[N.m.s/rad]

armature-to-housing damping factor

4.8.2 PID CONTROLLER

Ø      Interface

Up             positive input,   Un             negative input

Ø      Y               OutputExternal Parameters

        Kp= 1[–]  proportional gain

       Ki=  0 [s^–1] integral gain

     Kd=  0 [s]  derivative gain

                Tau = 0 [s]  time constant of derivative controller

4.6 Integrator block

       Outputs the integral of scaled input signal

         where t₀ is the starting time of transient analysis.

Interface    In    Input           

     Out             Output

Ø      External Parameters:         

Ø       c = 1  coefficient

4.7 MODELLING USING DYNAST SHELL

When using Dynast shell, you need not to deal with any equations at all.You can easily set up

 Step Response Computation

the system model in graphical form from a kit of dynamic elements.

To compute the open-loop step response of the system using DYNCAD

Ø      pull-down the menu DYNAST

Ø      click Compute analysis

4.9 TRANSFER FUNCTION COMPUTATION

DYNAST can also provide linearization and semisymbolic analysis of the system model. To see the following list of poles and zeros of the system transfer function(s)

Ø      pull down the menu Results in DYNCAD

Ø      click Textual

You can even find in the listing the following semisymbolic expression for the step response

Here the PID control loop is added to the main system design, having tunned to the desired parameters to effect the desired response.

4.11 SIMULATION RESULT

Having set the proportional, integral and derivatives parameter to the desired value, which are;

Kp= 100

With proportional band, the controller output is proportional to the error or a change in measurement (depending on the controller).

(controller output) = (error)*100/(proportional band)

Ki=200

With integral action, the controller output is proportional to the amount of time the error is present. Integral action eliminates offset.

CONTROLLER OUTPUT = (1/INTEGRAL) (Integral of) e(t) d(t)  Therefore the Integral gain reduces the settling time. Integral

Term  will eliminate the steady-state error

Kd=10

With derivative action, the controller output is proportional to the rate of change of the measurement or error. The controller output is calculated by the rate of change of the measurement with time.

                                      dm

      CONTROLLER OUTPUT = DERIVATIVE ----

                                      dt

Where m is the measurement at time t.

Derivative at this parameter reduces overshoot.

5.0 CONCLUSION AND RECOMMENDATION

I will conclude this project having laid a detailed algorithm that shows the design and analysis of a dc motor speed control using pid controller to correct errors.

This simply entails that pid controller proposes that it gives the desire response when added to a dc motor speed control.
In this project i only showed the reason why a pid controller should be used to design a dc motor speed control.

Having studied both designs of dc motor speed control with and without pid, i conclude by writing that it is the preferred means for controlling the speed of a dc motor control.

With the aid of dynast shell simulator, a comprehensive design of the model was carried out with ease, that is eliminating the need for complex equations and calculations to achieve your aim and require standard.

Without no doubt, i believe, having gone through this project you are convinced that pid technology is the preferred means of speed control especailly in dc motor and other control system

Also proved that dynast shell is a better means of achieving your desire design requirement,simulation and analysis of your control design.

REFERENCE

[1] Boumediène ALLAOUA^*, and Brahim MEBARKI  Setting Up PID DC Motor Speed Control Alteration Parameters Using Particle. Swarm Optimization StrategyBrahim GASBAOUI  2007.

  [2] Kamaraj Tuning Algorithms for PID Controller Using

      Soft Computing Techniques by 2001.

[3] lancet MitIntro to D.C. Motors User conference,Austin texas     Vol 11 2004.       

 [4] Rick Bickley Dc Motor Control System For Robot Applications Rick Bickley Publisher 2003.

[5]  K. SamsaiOptimal PI Controller Design Simulation, Robotic and AutomationResearchUnitSchool of Electrical Engineering 2007.

[6] Y.P Wang , a Genetic Algorithm to design PI Controller, Int. Conference Power Sytem Technology 2007

[7] Matt Krass, PID Control Theory, Matt Krass 2^nd Edition 2006.

[8] SAFFET AYASUNDC Motor Speed Control Methods Using MATLAB/Simulink Department of Electrical and Electronics Engineering, Nigde University, Nigde 51100, Turkey 2006.

[9] S. Li and R. Challoo, Restructuring an electric machinery course      with an integrative approach and computer-assisted teaching methodology, IEEE Trans Educ 49 (2006)

[10] M.Y. Chow and Y. Tipsuwan, Gain adaptation of networked DC motor controllers based on QOS variations, IEEE Trans Ind Electron 50 (2003).

[11] S. J. Chapman, Electric machinery fundamentals, 3rd

         edition, WCB/McGraw-Hill, New York, 1999.