BIBO Stability

A signal x(t) is bounded if there is a constant C, so that |x(t)| ≤ C for all t.

Examples of bounded signals include:

• cos(3t)

• 7e−2t u(t)

• e^(2t) u(1 − t).

Example of unbounded signals include:

• t2

• e^(2t)

• u(t),

• e−t

• 1/t

BIBO Stable

A system is BIBO (bounded input/bounded output) stable if every bounded input x(t) results in a bounded output y(t), as depicted by the simple example shown in the figure below. It does not require that an unbounded input result in a bounded output (this would be unreasonable). Stability is obviously a desirable property for a system.

Example1:

Proving Stability using poles

In general, a system transfer function H(s) is given in the form of a ratio of two polynomials,

H(s) = N(s)/D(s)

Suppose N(s) is of degree m (highest power of s) and D(s) is of degree n. It then follows that H(s) has m zeros and n poles. The zeros and poles may be real or complex, also distinct or repeated.

• We know h(t) (the impulse response must be bounded).

• H(s) the transfer function can only have a certain amount of cases

  1. real distinct poles - Overdamp

  2. repeated poles - critically damp

  3. complex poles - critically damp

  4. Combination of the above poles

• We can see that in all of these cases the poles must lie in the left hand plane of the s-domain.

Pole/Zero Plots of Transfer functions

Poles are represented with x's and Zeros are represented with o's

Stability

A system whose transfer function H(s) is a strictly proper rational function is BIBO stable if and only if its distinct and repeated poles, whether real or complex, reside in the OLHP(Open Left Hand Plane) of the s-domain, which excludes the imaginary axis. Furthermore, the locations of the zeros of H(s) have no bearing on the system’s stability.