BIBO stability

We also see another common definition—bounded input bounded output (BIBO) stability A system is BIBO stable if the output response is bounded for any bounded input. One illustration of this definition is to consider a hypothetical situation with a closed-loop pole at the origin. In such a case, we know that if we apply an impulse input or a rectangular pulse input, the response remains bounded. However, if we apply a step input, which is bounded, the response is a ramp, which has no upper bound. For this reason, we cannot accept any control system that has closed-loop poles lying on the imaginary axis. They must be in the LHP. 1... 

The complete Routh array analysis allows us to find, for example, the number of poles on the imaginary axis. Since BIBO stability requires that all poles lie in the left-hand plane, we will not bother with these details (which are still in many control texts). Consider the fact that we can calculate easily the exact roots of a polynomial with MATLAB, we use the Routh criterion to the extent that it serves its purpose.1 That would be to derive inequality criteria for proper selection of controller gains of relatively simple systems. The technique loses its attractiveness when the algebra becomes too messy. Now the simplified Routh-Hurwitz recipe without proof follows. 

If any one of the coefficients is negative, at least one root has a positive real part (i.e., in the right hand plane). If any of the coefficients is zero, not all of the roots are in the left hand plane it is likely that some of them are on the imaginary axis. Either way, stop. This test is a necessary condition for BIBO stability. There is no point in doing more other than to redesign the controller. 

Definition 7—Bounded-input-bounded-output (BIBO) stability. A... 

The advantage of FG stability over BIBO stability is that systems such as in Example 3 no longer have to be characterized as stable, a conclusion that agrees with intuition. Indeed, for Example 3 we have that pulses of infinitesimally small amplitude drive the output y to 2 consequently. 

Furthermore, conjugate poles on the imaginary axis are BIBO stable—a step input leads to a sustained oscillation that is bounded in time. But we do not consider this oscillatory steady state as stable, and hence we exclude the entire imaginary axis. In an advanced class, you should find more mathematical definitions of stability. 

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