MODELING OF DC MOTOR SPEED CONTROL

Control Systems Lab

DC MOTOR SPEED

System Modeling Physical Setup 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 translational motion.

Input of a system is the voltage source (V)

Output is the rotational speed of the shaft dθ/dt.

SYSTEM REQUIREMENTS

we will require that the steady-state error of the motor speed be less than 1%. 

it must accelerate to its steady-state speed as soon as it turns on. In this case, we want it to have a settling time less than 2 seconds. 

 A speed faster than the reference may damage the equipment, we want to have a step response with overshoot of less than 5%.

In short Settling time less than 2 seconds Overshoot less than 5% Steady-state error less than 1%

PHYSICAL PARAMETERS

Physical Parameter Description

J moment of inertia of the rotor 0.01 kg.m^2

b motor viscous friction constant 0.1 N.m.s

Ke electromotive force constant 0.01 V/rad/sec

Kt motor torque constant 0.01 N.m/Amp

R electric resistance 1Ω

L electric inductance 0.5 H

SYSTEM EQUATIONS

Motor torque is proportional to only the armature current Ia by a constant factor Kt 

Τ=KtIa The back emf,  ε, is proportional to the angular velocity of the shaft by a

constant factor Ke.

ε=Kes θ By using Kirchhoff's voltage law

Ia = (V – ε)/(Ls+R) Mechanical torque is sum of inertia and friction

Τ=Js2 θ+Ds θ  Rotational speed is considered the output and the armature voltage is

considered the input.

θ/V = Kt/[(Ls+R)(Js2+Ds)+ KtKes]

MATLAB REPRESENTATION

• OPEN LOOP RESPONSEJ = 0.01; b = 0.1; K = 0.01; R = 1; L = 0.5; s = tf('s'); P_motor = K/((J*s+b)*(L*s+R)+K^2)

P_motor = 0.01 --------------------------- 0.005 s^2 + 0.06 s + 0.1001

Continuous-time transfer function.

CONTROLLER DESIGN

PID Controller Design : Proportional controlFrom the plot we see that both the steady-state error and the overshoot are too large. When we increase the proportional gain Kp , steady-state error will reduce . When we increase Kp results in increased overshoot, therefore, it appears that not all of the design requirements can be met

with a simple proportional controller.

TUNING THE GAINS

Kp = 100; Ki = 200; Kd = 1;As expected, the steady-state error is now eliminated much more quickly

than before. However, the large Ki has greatly increased the overshoot. Let's increase Kd in an attempt to reduce the overshoot.

SIMULINK

COMBINE REPRESENTATION

Modeling Equations

Controller design

Simulink design

PID controller

DESIGN REQUIREMENTS

FLOW DIAGRAM

State space