Question 4.8: Finding a bounded excitation that produces an unbounded resp......
Finding a bounded excitation that produces an unbounded response
Consider an integrator for which y(t)=\int\limits_{−∞}^{t} x(τ)dτ. Find the eigenvalues of the solution of this equation and find a bounded excitation that will produce an unbounded response.
By applying Leibniz’s formula for the derivative of an integral of this type, we can differentiate both sides and form the differential equation y'(t) = x(t). This is a very simple differential equation with one eigenvalue and the homogeneous solution is a constant because the eigenvalue is zero. Therefore this system should be BIBO unstable. A bounded excitation that has the same functional form as the homogeneous solution produces an unbounded response. In this case, a constant excitation produces an unbounded response. Since the response is the integral of the excitation, it should be clear that as time t passes, the magnitude of the response to a constant excitation grows linearly without a finite upper bound.