Question

MATLAB - Simulink Solution

U 005) Esin - M = 2kg; m= 0.1 kg; l= 0.5m I cos e M 8:Pendulum Angle 8:Angular velocity of pendulum x:Position of the car *:Velocity of the car The linear model of the system is introduced by following equations 0 = 20.6010 - u x = 0.5u - 0.49050 Suppose states of system are x = [o o 0 1 001 20.601 0 0 0 A ;C = [0 0 1 0] 0 0 0 1 0 1-0.4905 0 0 0 Output the system is position of the car. Design a controller such that the poles of closed loop system are $1,2 = -1 7 3i; S3 = -5;54 = -5; If you want to increase the type of the system, you need to choose a suitable pole for that. 0 ;B=

Answer 1

%%%%%%%%%%. MATLAB CODE %%%%%%%%%%%%%%

% state space equation matrices

A = [0 1 0 0;

20.601 0 0 0;

0 0 0 1;

-0.4905 0 0 0];

B = [0;-1;0;0.5];

C = eye(4); % for getting output of all states

% eigen values of A = poles of open loop

E = eig(A);

% desired pole location

last_pole = -0.5; % you can change last pole location a/c to your need

p = [-1+3i; -1-3i; -5; last_pole];

% determining K matrix (gain matrix) for pole placement

K = place(A,B,p);

% integrating solution

X0 = [0.1 0 0 0]';

X = X0;

storeY = [];

t0 = 0;

dt = 0.1;

tf = 10;

T = t0:dt:tf;

for t = t0:dt:tf

if t <=10

r = 1;

else

r = 0;

end

U = r-K*X;

XDOT = A*X + B*U;

X = X + XDOT*dt;

Y = C*X;

storeY = [storeY Y];

end

figure

subplot(411)

plot(T,storeY(1,:))

legend('\theta')

subplot(412)

plot(T,storeY(2,:))

legend('theta\_dot')

subplot(413)

plot(T,storeY(3,:))

legend('x')

subplot(414)

plot(T,storeY(4,:))

legend('x\_dot')

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(I) With r = 0 && x = [0 0 0 0]'

1 0 -1 0 4 6 8 10 12 14 16 18 20 N 1 theta_dot 0 -1 0 4 6 8 10 12 14 16 18 20 N 1 X 0 2 4 6 8 10 12 14 16 18 20 1 X_dot 0 -1 0 2 4 6 8 10 12 14 16 18 20

(I) With r = 1 && x = [0 0 0 0]'

0.05 0 -0.05 0 1 2. 3 4 On (o 7 8 9 10 0.2 theta_dot 0 -0.2 0 1 3 4 On 6 7 8 9 10 N 0.5 X 0 -0.5 0 1 2 3 4 5 6 7 8 9 10 0.5 X_dot 0 -0.5 0 1 2 3 4 5 6 7 8 9 10

(I) With r = 0 && x = [0.1 0 0 0]'

(I) With r = 1 && x = [0.1 0 0 0]'

0.1 0 -0.1 0 1 2. 3 4 On (o 7 8 9 10 0.5 theta_dot 0 -0.5 0 1 2 3 4 On 6 7 8 9 10 0.4 X 0.2 0 0 1 2 3 4 5 6 7 8 9 10 0.5 X_dot 0 -0.5 0 1 2 3 4 5 6 7 00 9 10

0.2 0 -0.2 0 1 2. 3 4 On co 7 8 9 10 1 theta_dot 0 -1 0 1 2 3 4 On 6 7 8 9 10 0.5 X -0.5 0 1 2 3 4 5 6 7 8 9 10 1 X_dot 0 -1 0 1 2 3 4 5 6 7 8 9 10

(I) With r = 1 && x = [0 0.05 0 0]'

0.1 0 -0.1 0 0.2 1 2. 3 4 On co 7 8 9 10 theta_dot 0 -0.2 0 1 2 3 4 On 6 7 8 9 10 0.5 X -0.5 0 1 2 3 4 5 6 7 8 9 10 0.5 X_dot 0 -0.5 0 1 2 3 4 5 6 7 8 9 10

note: I've attached an entirely MatLab solution to this problem. Coz sharing solution of SIMULINK doesn't worth it here as it will be consisting of only block diagrams.

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I’ve attached the solution above. I hope you will find it useful.

If any doubt, FEEL FREE to comment anytime.

Thank You!

1 0 -1 0 4 6 8 10 12 14 16 18 20 N 1 theta_dot 0 -1 0 4 6 8 10 12 14 16 18 20 N 1 X 0 2 4 6 8 10 12 14 16 18 20 1 X_dot 0 -1 0 2 4 6 8 10 12 14 16 18 20

We were unable to transcribe this image

We were unable to transcribe this image

0.2 0 -0.2 0 1 2. 3 4 On co 7 8 9 10 1 theta_dot 0 -1 0 1 2 3 4 On 6 7 8 9 10 0.5 X -0.5 0 1 2 3 4 5 6 7 8 9 10 1 X_dot 0 -1 0 1 2 3 4 5 6 7 8 9 10

0.1 0 -0.1 0 0.2 1 2. 3 4 On co 7 8 9 10 theta_dot 0 -0.2 0 1 2 3 4 On 6 7 8 9 10 0.5 X -0.5 0 1 2 3 4 5 6 7 8 9 10 0.5 X_dot 0 -0.5 0 1 2 3 4 5 6 7 8 9 10