1. SYSTEM OF DIFFERENTIAL EQUATIONS f(t) : Input u(t) and v(t) : Outputs to be found System of constant coefficient differential equations with two unknowns * First order derivative terms are on the left hand side * Non-derivative terms are on the right hand side x : State variable matrix (nx1) A : System matrix (nxn) u : Input matrix (mx1) B : Matrix with dimension (nxm)

2. s X(s) - x 0 = A X(s) + B U(s) s I X(s) – x 0 = A X(s) + B U (s) [s I – A] X(s) = x 0 + B U (s) X(s) = [s I – A] -1 x 0 + [s I – A] -1 B U (s) Homogenous partParticular part Laplace Transformation:

3. D(s)=det[sI-A]=s 2 +3s+120 : Eigenvalue equation At t=0 u=-2, v=3 ; F(s) = -4/s X(s) = [s I – A] -1 x 0 + [s I – A] -1 B U (s) Example:

4. With Matlab : Calculation of eigenvalues: a=[0,1 ; -120,-3] ; eig(a) syms s; a=[0,1;-120,-3] ; i1=eye(2); a1=inv(s*i1-a); pretty(a1) Determination of matrix [sI-A]  1 :

5. Example: f(t):Inputx(t) and y(t) : Outputs to be found and Let’s use the variables