rw
Sinusoidal Solution: The particular solution is X = (jwI - A)^(-1)BU. This gives us the complex vector X, which tells us the amplitude and phase of the state variables’ sinusoidal responses. The full time-domain solution for the state vector is then x(t) = Xe^(jwt).
Existence Condition: The solution exists only if the matrix (jwI - A) is invertible. A matrix is invertible if and only if its determinant is non-zero. Therefore, the solution is valid when: det(jwI - A) ≠ 0
But this is equivalent to saying that the jw is not an eigenvalue of A. The eigenvalues of A represent the system’s natural frequencies.questions
So If the input frequency w is such that jw equals one of the eigenvalues of A, then det(jwI - A) = 0, the inverse doesn’t exist, and the solution blows up (becomes infinitely large). This is the phenomenon of resonance. The system is being driven at one of its natural frequencies, leading to an unbounded response. Everything goes boom.