Linear System Theory and Design
主讲:朱芳来
minimalPart 3
Chapter 5 Stability
5.1 Introduction
System are designed to perform some task or to process signals. If a system is not stable, the system may burn out, disintegrate, or saturate when a signal , no matter how small, is applied. Therefore an unstable system is useless in practice and stability is a basic requirement for all system.
The response of linear systems can always be decomposed as the zero-state response and the zero-input response. We will introduce the BIBO stability for the zero-state response and marginal and asymptotic stability for the zero-input response.
5.2 Input-Output Stability of LTI Systems
Consider a SISO linear time-invariant (LTI) system described by
(5.1)
where g(t) is the impulse response and the system is linear time-invariant, cause, and relaxed at t=0.
∫∫−=−=t t d t u g d u t g t y 00)()()()()(ττττττ
Definition: A system is said to be BIBO stable (bounded-input bounded-output stable) if every bounded input excites a bounded output.
Theorem 5.1A SISO system described by (5.1) is BIBO stable if and only if g(t) is absolutely integrable in [0,∞) , i.e. there exists a constant M such that
holds.
∞
<
≤
∫M
dt
t
g t
)(
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