State-Space Method: Impulse Response and Stability

The impulse response h(t) is the response of the system to the delta function, a pulse of vanishingly narrow width. The impulse response is the inverse Lagrangian of the transfer function, and can be expressed in terms of the poles of the system p_i (the entries of matrix A).

Bounded-In, Bounded-Out (BIBO) stability specifies that as long as the input is bounded, the output must be bounded. For BIBO stability, the exponential factor e^pt must decay rather than increase. Thus, all the poles p_i must have negative real parts. For BIBO stability, the norm of the impulse response must always be finite: