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Homework Statement
I have calculated the following mechanical system
[tex]
\left( \begin{array}{c}\dot{\mathbf{x_1}}(t) & \dot{\mathbf{x_2}}(t) \end{array} \right) = \left( \begin{array}{cc}0.5 & 0 & 0 & 1\end{array} \right) \cdot \left( \begin{array}{c}x_{1}(t) & x_{2}(t)\end{array} \right) + \left( \begin{array}{c}0.5 & 1\end{array} \right)\cdot u(t)
[/tex]
[tex]
\left( \begin{array}{c}y_{1}(t) & y_{2}(t)\end{array} \right) = \left( \begin{array}{cc}1 & 0\end{array} \right) \cdot \left( \begin{array}{c}x_{1}(t) & x_{2}(t)\end{array} \right)
[/tex]
The question is to find the expression for u(t) that brings the system to its restposition in 2 seconds. Afterwards i have to simulate this time response in matlab with the function lsim.
The initial conditions of the system are:
[tex]
x_1(0) = 10
[/tex]
[tex]
x_2(0) = 1
[/tex]
Homework Equations
[tex]
u(t) = B^T\exp^{A^T(t_1t)}W_c^{1}(t_1)[\exp^{At_1}x_0x_1]
[/tex]
[tex]
W_c(t_1) = \int_0^{t_1} \exp^{A\gamma}BB^T\exp^{A^T\gamma} d\gamma
[/tex]
The Attempt at a Solution
I calculated u(t) with matlab with the following code
Code:
A = [0.5 0;0 1];
B = [0.5; 1];
C = [1 0];
D = 0;
syms t;
Wc = int(expm(A*t)*B*transpose(B)*expm(transpose(A)*t),0,2);
u = transpose(B)*expm(transpose(A)*(2t))*inv(Wc)*(expm(A*2)*[10;1][0;0]);
sys = ss(A,B,C,D);
t = 0:0.1:2;
u = subs(u,t);
lsim(sys,u,t)
The result i get is given in the lsimresults.bmp.
While i expect it to curve a bit down/up the original time responses (1st and 2nd thumbnails) where u(t) = 0, so that the amplitude of the system becomes 0 at 2 seconds.
So, if you there is anybody who can give me some help where it goes wrong. Or if i interpretate something wrong maybe?
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