How to get the output response from a state space equation? - matlab

Question

For a state space equation in which matrix A is dependent on variable t(time), how can I get the step or output response?
This is the code, which doesn't work:
A = [sin(t) 0;0 cos(t)];
B = [0.5; 0.0];
C = [1 0; 0 1];
G = ss(A,B,C,[]);
step(G,t)
x0 = [-1;0;2];
initial(G,x0)
Here are error message:
Error using horzcat Dimensions of matrices being concatenated are not
consistent.
Error in Response (line 11) A = [sin(t) 0;0 cos(t)];

Answer 1

As pointed already, you can only use the ss function to generate LTI systems, but you can discretise your model analytically using methods like forward Euler, backwards Euler, Tustin etc. and simulate your model using a for loop.
For your example, you could run something like this:
consider a sampling period h = 0.01 (or lower if the dynamics are not captured properly);
select N as the number of steps for the simulation (100, 1000 etc.);
declare the time instants vector t = (0:N-1)*h;
create a for loop which computes the system states and outputs, here using the forward Euler method (see https://en.wikipedia.org/wiki/Euler_method):
% A = [sin(t) 0;0 cos(t)];
B = [0.5; 0.0];
C = [1 0; 0 1];
D = [0; 0];
x0 = [-1;1]; % the initial state must have two elements as this is a second-order system
u = 0; % constant zero input, but can be modified
N = 1000;
h = 0.01;
t = (0:N-1)*h;
x_vec = [];
y_vec = [];
xk = x0;
yk = [0;0];
for k=1:N
Ad = eye(2)+h*[sin(t(k)) 0; 0 cos(t(k))];
Bd = h*B; % C and D remain the same
yk = C*xk + D*u;
xk = Ad*xk + Bd*u;
x_vec = [x_vec, xk]; % keep results in memory
y_vec = [y_vec, yk];
end
% Plot output response
plot(t,y_vec);