BIBO Stability

A BIBO (bounded-input bounded-output) stable system is a system for which the outputs will remain bounded for all time, for any finite initial condition and input. A continuous-time linear time-invariant system is BIBO stable if and only if all the poles of the system have real parts less than 0.

For example, consider the following system:

The transfer function of this sytem is 1 / (s -  1). The transfer function has a pole with a real part that is not less than zero (the pole is at s = 1). That means that this system is BIBO unstable. The solution of this system can be derived as follows:

So if u(t ) = 1 and x(0) = - 1, we get y(t) = - 1. In other words, y(t) is bounded for all time. However, this does not mean that the system is BIBO stable! BIBO stability requires that the output remain bounded for all time, for all initial conditions and inputs - not just for some specific initial condition and input. In other words, if we can find even one initial condition and input that causes one of the outputs to approach infinity with time, then the system is BIBO unstable. For the above system, we can choose (for example) u(t ) = 0 and x(0) = 1. In this case y(t) = et. In other words, y(t) approaches infinity with time, which proves that the system is BIBO unstable.

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Last Revised: August 5, 2002