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

QUESTION1- mathematical model

state-space model of the DC motor

d/dt[theta I]=[-b/J kt/J ; -km/La -Ra/La]*[theta ; I]+[0;1/La]*Vs

note-W(s)/Ia(s)=kt/J(s)+b

Ia(s)=V(s)-Vb(s)/La(s)+Ra

OLTF-W(s)/V(s)=kt/J*La*s^2+(J*Ra+b*La)*s+(b*Ra+kt*km)

SIMULINK MODEL

modelling

% motor parameters
J=0.85; % (kgm^2) moment of inertia of the rotor
b=0.1; % (Nms) motor viscous friction constant
km=0.01;% (v/rad/sec) back emf
kt=0.01;% (Nm/amp) motor torque constant
Ra=1;% (ohm) armature resistance
La=0.5;% (H) armature inductance

% equivalent wn
fn=1; %hz
wn=2*pi*fn; %rad/sec

% design parameters
zeta=0.9;% underdamped

% desired poles
p1=-zeta*wn-wn*sqrt(zeta^2-1);
p2=-zeta*wn+wn*sqrt(zeta^2-1);
P=[p1;p2];

% A & B matrices
A=[-b/J kt/J;-km/La -Ra/La];
B=[0;1/La];
c=[1 0];
D=0;

% poles placement design
K=place(A,B,P);

% gains
kp=38.3133;
kd=-2.8584;
ki=10;

plotting

subplot(321);plot(out,w,'r-');grid;
subplot(321);title('speed');
subplot(322);plot(out,tq,'m-');grid;title('torque');
subplot(323);plot(out,ia,'k-');grid;title('armature current');
subplot(324);plot(out,Vs),grid;title('supply voltage');
subplot(325);plot(out,Vb,'g');grid;title('back emf');

results

conclusion

speed of the dc motor has been maintained under different conditions.