University:
Massachusetts Institute of Technology
Course:
16.01 | Unified Engineering I, II, III, & IV
Academic year:

2022
Views:

317

Pages:

8
Author:

germanokrt

Deﬁnition of Stability Stability is an important concept in linear systems — we all want to ﬂy in airplanes with stable control systems! Although many of us have an intuitive feel for the idea of “stability,” we need a working deﬁnition that will allow us to classify systems as either stable or unstable. Working with a partner, draft a deﬁnition of stability. The deﬁnition should be speciﬁc enough that you can test whether a system with impulse response g(t) (or equivalently, transfer function G(s)) is stable or not. Note: There are dozens, and maybe hundreds, of deﬁnitions of stability. There is no “wrong” answer! When you are ﬁnished, press “1” on your PRS remote. Deﬁnition of Stability Solution u(t) G y(t) The LTI system G is Bounded Input / Bounded Output (BIBO) stable if every bounded input u(t) produces a bounded output y(t). Basically, this deﬁnition says that every “nice” input produces a “nice” output. BIBO Stability I Consider the systems F , G, and H, with impulse responses given by f (t) = σ(t)e−2t g(t) = σ(t) h(t) = σ(t)et Which of the systems are BIBO stable? 1. F , G, and H 2. F and G 3. F only 4. G and H 5. H only 6. none of the above BIBO Stability I Consider the systems F , G, and H, with impulse responses given by f (t) = σ(t)e−2t g(t) = σ(t) h(t) = σ(t)et Which of the systems are BIBO stable? The correct answer is: 1. F , G, and H 2. F and G 3. ♥ F only 4. G and H 5. H only 6. none of the above BIBO Stability II Consider the system G with impulse response given by 1 g(t) = σ(t) 1+t Is the system G BIBO stable? 1. Yes 2. No 3. Don’t know BIBO Stability II Consider the system G with impulse response given by 1 σ(t) g(t) = 1+t Is the system G BIBO stable? The correct answer is: 1. Yes 2. ♥ No 3. Don’t know BIBO Stability III Consider the system G with impulse response given by 1 g(t) = √ σ(t) t Is the system G BIBO stable? 1. Yes 2. No 3. Don’t know BIBO Stability III Consider the system G with impulse response given by 1 g(t) = √ σ(t) t Is the system G BIBO stable? The correct answer is: 1. Yes 2. ♥ No 3. Don’t know