SYSTEM OF DIFFERENTIAL EQUATIONS

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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 x : State variable matrix (nx1) A : System matrix (nxn) * Non-derivative terms are on the right hand side u : Input matrix (mx1) B : Matrix with dimension (nxm)

s I X(s) – x0 = A X(s) + B U (s) Laplace Transformation: s X(s) - x0 = A X(s) + B U(s) s I X(s) – x0 = A X(s) + B U (s) [s I – A] X(s) = x0 + B U (s) X(s) = [s I – A]-1 x0 + [s I – A]-1 B U (s) Homogenous part Particular part

Example: At t=0 u=-2 , v=3 ; F(s) = -4/s D(s)=det[sI-A]=s2+3s+120 : Eigenvalue equation X(s) = [s I – A]-1 x0 + [s I – A]-1 B U (s)

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

Example: f(t):Input x(t) and y(t) : Outputs to be found and Let’s use the variables