Project 2 - Speed Control of a Direct Current (DC) motor

Parameters of DC Motor:

J = Moment of inertia

Bm = Viscous friction

R = Armature resistance

L = Armature inductance

Kb = Backemf constant

Kt = Torque constant

Vs = Supply voltage

Vb = Back emf voltage = Kb*w

I = Armature current

w=wn=speed

T = Motor Torque = Kt*I

State space representation of the DC motor:

X_dot = Ax +Bu ...................(1)

Y = Cx + Du ....................(2)

[w_dot; I_dot] = [-Bm/J Kt/J; -Kb/L -R/L][w; I] + [0; 1/L][V] .............(3)

Y(t) = [1 0][w; I] + [0][V] ..........................(4)

Comparing (1) and (2) with (3) and (4), we get,

A=[-Bm/J Kt/J; -Kb/L -R/L]

B=[0; 1/L]

Pole placement design:

K=place(A,B,p)

K=[K1,K2]

K1 = Kp % Proportional gain

K2 = Kd % Derivative gain

Choose Ki = 0 initially % Integral gain

p=[p1;p2]

p1 = -z*wn + wn*sqrt(z^2-1)

p2 = -z*wn - wn*sqrt(z^2-1)

Choose z=0.7 for under damped system

fn=1

wn=2*pi*fn

PID controller design:

u(t) = Kp(xref - x1) + Ki*int(xref - x1)dt + Kd*d/dt*(xref - x1);

where

u(t) = control input

Kp=Proportional gain

Ki=Integral gain

Kd=Derivative gain

xref - x1 = error signal

xref = speed reference = wref

x1 = speed =w

Hence,

u(t) = Kp(wref - w) + Ki*int(wref -w)dt + Kd*d/dt*(wref-w);

1. Simulink model to simulate the design, Kp = 25.7469, Kd = -1.6018, Ki=0, Set speed wref = 30:

Simulink output:

From the above output, PID design fails to track the set speed of 30 and had the actual speed of 22.

2. Simulink model to simulate the design, Kp = 25.7469, Kd = -1.6018, Ki designed to 55 from 0, Set speed wref = 30:

Simulink output:

From the above output, PID design track the set speed of 30 with the actual speed and the design objectives like settling time, overshoot and steady state error are met.

3. Simulink model to simulate the design, Kp = 25.7469, Kd = -1.6018, Ki = 55, Set speed wref = 100:

Simulink output:

From the above output, PID design track the set speed of 100 with the actual speed and the design objectives like settling time, overshoot and steady state error are met.

If we change any one of the gain in the above PID gain set (Kp = 25.7469, Ki = 55, Kd = -1.6018), then the design will fail to track the set speed.