Plot step response without using step function - matlab

Question

I want to plot the step response. I know that I can use step function with state space equations, but I try to get same results using plot function. Here is my sample of code:
for i=1:201
u(i) = 1;
x1(i+1) = (-(b/J)*x1(i) + (K/J)*x2(i));
x2(i+1) = (-(K/L)*x1(i) - (R/L)*x2(i) + (1/L)*u(i));
y(i) = x1(i);
end
and this is the state space equations:
A = [-b/J K/J
-K/L -R/L];
B = [0
1/L];
C = [1 0];
D = 0;
If i do:
t = 0:1:200;
plot(t, y)
it is not working and I want to have the same results like the step function below:
sys = ss(A,B,C,D);
step(sys)
You can find my state space equation here.

Answer 1

The reason for the mismatch is that sys is a continuous time model, whereas the computation of y treats it as a discrete-time system.
The following is a way of estimating the step-response of a continuous-time system in the discrete-time domain:
% Given from the problem statement
A = [-b/J K/J
-K/L -R/L];
B = [0
1/L];
C = [1 0];
D = 0;
% this is your continuous-time model
sys = ss(A,B,C,D);
% define the sample rate of the equivalent discrete-time model
Ts = 1/10;
% this needs to be something smaller than the time-constants in your model,
% so that you have enough resolution to represent the continuous-time
% signal.
% convert the system to the equivalent discrete-time model
sysd = c2d(sys,Ts);
% define how long a step response you'd like to compute
T = 7;
% this should be long enough to cover the length of the step response
t = 0:Ts:T; % time-grid for the plot
nSmp = length(t); % total number of samples to be computed
% initializations
y = NaN(1, nSmp); % output vector
u = ones(1, nSmp); % unit step input
X = [0; 0]; % state vector, initialized to 0
% compute the samples of the step-response
% (i prefer to use vectorized form to keep the code concise)
for i=1:nSmp
y(i) = sysd.C * X + sysd.D * u(i);
X = sysd.A * X + sysd.B * u(i);
end
% plot continous-time step response
figure;
step(sys);
% plot simulated discrete-time step response
figure;
plot(t, y, 'r')
xlabel('Time (s)');
ylabel('Amplitude');
title('Simulated Step Response');