How to insert innitial condition of x(t) with relative of time

Now I'm generating DC motor code to get some value from a state space equation plant.

But I have 2 question from this.

1. I need to know how can I insert the innitial condition of x(t) with realative of time since 0-1 second
2. If x(t) is matrix 3x1 and it have 3 variable x1 = supply current, x2 = angular displacement, x3 = angular velocity. How can I get the value of x2 at 1 second

This is my code

function [xt,x1,x2,x3] = motor1(t,xt)
xt = zeros(3,1) 

L = 0.3; %H inductance
R = 2.5; %ohm resistance
Kv = 0.0195; %Vs/rad electromotive constant 
Kt = 0.0195; %Nm/A torque constant
J = 17.2*10^-7; %kgm^2 inertia
D = 1*10^-6; %N.m.rad/s friction constant
A_motor = [-R/L 0 -Kv/L; 0 0 1; Kt/J 0 D/J];
B_motor = [1/L;0;0];
c = [1 1 1];
C_motor = diag(c);
D_motor = [0;0;0];
x1 = xt(1);
x2 = xt(2);
x3 = xt(3);

%y = lsim(A_motor,B_motor,C_motor,D_motor,t)
%x0 = [s1;s2;s3] 
% s1 = is
% s2 = ceta
% s3 = omega

% Create state space 
motor = ss(A_motor,B_motor,C_motor,D_motor)
% Check eigenvalues
%pole=eig(A_motor)
% Tranfer function
%TF_motor = tf(motor)\
e_At = expm(A_motor);
S = @(t) (e_At*(B_motor)*12);

x = ((e_At)*(xt))+(integral(S,0,1,'ArrayValued',true))
dx = ((A_motor)*(x))+((B_motor)*12)
y = ((C_motor)*x)+((D_motor)*12)

CM=ctrb(A_motor,B_motor)
step(motor)

Answer 1

[TED MOSBY]

Hi,

Initial Condition: You can specify the initial condition 'x0' as a 3x1 matrix. This matrix represents the initial values of your state variables: supply current, angular displacement, and angular velocity. You can then simulate the system over a specified time range.

Simulating the System: You can use MATLAB's "lsim" function to simulate the response of your system to a specified input over time. This function allows you to specify the initial condition and will provide the state trajectory over time.

Have a look at the below revised code:

function [xt, x1, x2, x3] = motor1(t, x0) 
% Define system parameters 
L = 0.3; % H inductance 
R = 2.5; % ohm resistance 
Kv = 0.0195; % Vs/rad electromotive constant  
Kt = 0.0195; % Nm/A torque constant 
J = 17.2e-7; % kgm^2 inertia 
D = 1e-6; % N.m.rad/s friction constant 

% State-space matrices 
A_motor = [-R/L 0 -Kv/L; 0 0 1; Kt/J 0 -D/J]; 
B_motor = [1/L; 0; 0]; 
C_motor = eye(3); % Output all states 
D_motor = [0; 0; 0]; 

% Create state-space system 
motor = ss(A_motor, B_motor, C_motor, D_motor); 

% Define input voltage as a step input (e.g., 12V) 
u = 12 * ones(size(t)); 

% Simulate the system 
[y, t_out, x] = lsim(motor, u, t, x0); 

% Extract state variables at t = 1 second 
[~, idx] = min(abs(t_out - 1)); % Find index closest to t = 1 
x1 = x(idx, 1); % Supply current 
x2 = x(idx, 2); % Angular displacement 
x3 = x(idx, 3); % Angular velocity 

% Output the state at each time step 
xt = x; 
end 
t = linspace(0, 5, 1000); % Time vector from 0 to 5 seconds 
x0 = [0; 0; 0]; % Initial condition: all states start at zero 
[xt, x1, x2, x3] = motor1(t, x0); 
disp(['Angular displacement at 1 second: ', num2str(x2)]); 

www.mathworks.com/help/control/ref/dynamicsystem.lsim.html?searchHighlight=lsim&s_tid=srchtitle_support_results_1_lsim

Hope this helps!