State Matrix and Output of LTI System

Hello Dear All, My matrix is as of the following:

%xdot=A*x+B*u
%y=C*x+D*u

A=[0 1 0 ; 0 0 1; -2 -4 -3];
B=[1 0;0 1; -1 1];
C=[0 1 -1; 1 2 1];
D=[0 0;0 0];
x=[1;0;0]; %initial condition

I want to evaluate x and y for t=0:1:10 and u=[1;1] as of below but i could not succeed.

T=1
maxt=10
for t=1:maxt/T
  u(t)=[1;1];
 x(:,t+1)=(eye(n)+T*A)*x(:,t)+T*B*u(t); 
end
Would you be kind to help me?

Ps. I was asked to determine the state x(t) from 0 to 10 and the output y(t) of the LTI system

Answer 1

[Rick Rosson]

Please format your code.

[ali veysel]

sorry, done.

[Rick Rosson]

Thanks!

Answer 2

[Rick Rosson]

Do you have access to the Control System Toolbox, or just core MATLAB itself?

Answer 3

[Rick Rosson]

Please try:

   %%Time specifications

   stopTime = 10;
   Fs = 1;
   dt = 1/Fs;
   t = (0:dt:stopTime)';

   N = size(t,1);

   %%State space model

   A=[0 1 0 ; 0 0 1; -2 -4 -3];
   B=[1 0;0 1; -1 1];
   C=[0 1 -1; 1 2 1];
   D=[0 0;0 0];

   %%Pre-allocation

   u = zeros(2,N);
   x = zeros(3,N);
   y = zeros(2,N);

   %%Initial conditions

   u(:,1) = [ 1 ; 1 ];
   x(:,1) = [ 1 ; 0 ; 0 ];
   y(:,1) = ...

   %%Simulation - main loop

   for k = 2:N
       u(:,k) = [ 1 ; 1 ];
       x(:,k) = x(:,k-1) + dt*( ... );
       y(:,k) = ...

   end

   %%Plot results

   figure;

   ax(1) = subplot(2,1,1);
   plot(t,x);

   ax(2) = subplot(2,1,2);
   plot(t,y);

   linkaxes(ax,'x');

HTH.

Rick

[ali veysel]

Rick,

Thank you a lot for your help on this!

The formulas are like this as you konw already:

xdot=Ax+Bu

y=Cx+Du

For the simulation main loop, i believe i got it wrong:

u(:,k) = [ 1 ; 1 ];

x(:,k) = x(:,k-1) + dt*u(:,k);

y(:,k) = y(:,k)*x(:,k)

Answer 4

[Rick Rosson]

Close, but not quite correct. In the main loop, you need to use the state space model to compute the updated state variables x and the value of the output variables y. So x(:,k) should depend on A and B. Likewise, y(:,k) should depend on C and D.

Take the equations as given, and then write them out in the form of MATLAB code using the appropriate variables. Double check to make sure that the inner dimensions match on each matrix multiplication.

Hint:

xdot = dx/dt

dx = x_k - x_k-1

Therefore

x_k = . . . ?

[ali veysel]

I got your point I guess:

u(:,k)=[1;1];

x(:,k+1)=(eye(n1)+T*A)*x(:,k)+(T*B*u(:,k));

y(:,k)=(C)*x(:,k);

changed my code like this now i think i got the correct results.

[Rick Rosson]

It looks much better now. You may need to change the subscripting on the 2nd line to avoid an out of bounds error. Try x(:,k) = ... x(:,k-1) ... u(:k-1)