SYSTEM STABILITY

2.6 SYSTEM STABILITY

To understand the intuitive basis for the BIBO (bounded-input/bounded-output) stability of a system introduced in Section 1.7 , let us examine the stability concept as applied to a right circular cone. Such a cone can be made to stand forever on its circular base, on its apex, or on its side. For this reason, these three states of the cone are said to be equilibrium states. Qualitatively, however, the three states show very different behavior. If the cone, standing on its circular base, were to be disturbed slightly and then left to itself, it would eventually return to its original equilibrium position. In such a case, the cone is said to be in stable equilibrium. In contrast, if the cone stands on its apex, then the slightest disturbance will cause the cone to move farther and farther away from its equilibrium state. The cone in this case is said to be in an unstable equilibrium. The cone lying on its side, if disturbed, will neither go back to the original state nor continue to move farther away from the original state. Thus it is said to be in a neutral equilibrium. Clearly, when a system is in stable equilibrium, application of a small disturbance (input) produces a small response. In contrast, when the system is in unstable equilibrium, even a minuscule disturbance (input) produces an unbounded response. BIBO stability definition can be understood in the light of this concept. If every bounded input produces bounded output, the system is (BIBO) stable. [ † ] In contrast, if even one bounded input results in unbounded response, the system is (BIBO) unstable.

For an LTIC system

Therefore

Moreover, if x(t) is bounded, then |(x − τ)t < K 1 < ∞, and

Hence for BIBO stability

This is a sufficient condition for BIBO stability. We can show that this is also a necessary condition (see Prob. 2.6-4). Therefore, for an LTIC system, if its impulse response h(t) is absolutely integrale, the system is (BIBO) stable. Otherwise it is (BIBO) unstable. In addition, we shall show in Chapter

4 that a necessary (but not sufficient) condition for an LTIC system described by Eq. (2.1) to be BIBO stable is M ≤ N. If M > N, the system is unstable. This is one of the reasons to avoid systems with M > N.

Because the BIBO stability of a system can be ascertained by measurements at the external terminals (input and output), this is an external stability criterion. It is no coincidence that the BIBO criterion in (2.64) is in terms of the impulse response, which is an external description of the system.

As observed in Section 1.9 , the internal behavior of a system is not always ascertainable from the external terminals. Therefore, the external (BIBO) stability may not be correct indication of the internal stability. Indeed, some systems that appear stable by the BIBO criterion may be internally unstable. This is like a room inside a house on fire: no trace of fire is visible from outside, but the entire house will be burned to ashes.

The BIBO stability is meaningful only for systems in which the internal and the external description are equivalent (controllable and observable systems). Fortunately, most practical systems fall into this category, and whenever we apply this criterion, we implicitly assume that the system in fact belongs to this category. The internal stability is all-inclusive, and the external stability can always be determined from the internal stability. For this The BIBO stability is meaningful only for systems in which the internal and the external description are equivalent (controllable and observable systems). Fortunately, most practical systems fall into this category, and whenever we apply this criterion, we implicitly assume that the system in fact belongs to this category. The internal stability is all-inclusive, and the external stability can always be determined from the internal stability. For this