Bounded-Input Bounded-Output


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Bounded-Input Bounded-Output
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Classical Control Stability
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1 Introduction to BIBO Stability

A system is defined to be BIBO Stable if every bounded input to the system results in a bounded output over the time interval LaTeX: [t_0, \infty). This must hold for all initial times to. So long as we don't input infinity to our system, we won't get infinity output.

A system is defined to be uniformly BIBO Stable if there exists a positive constant k that is independent of t0 such that for all t0 the following conditions:

LaTeX: \|u(t)\| \le 1
LaTeX:  t \ge t_0

implies that

LaTeX: \| y(t) \| \le k

2 Determining BIBO Stability

Mathematically a system f is BIBO stable if an arbitrary input x is bounded by two finite but large arbitrary constants M and -M:

LaTeX: -M < x \le M

Apply the input x, and the arbitrary boundries M and -M to the system to produce three outputs:

LaTeX: y_x = f(x)
LaTeX: y_M = f(M)
LaTeX: y_{-M} = f(-M)

Now, all three outputs should be finite for all possible values of M and x, and they should satisfy the following relationship:

LaTeX: y_{-M} \le y_x \le y_M

If this condition is satisfied, then the system is BIBO stable.