Talk:Control Systems/Stability

I would say instability is also non-infinite, as in simple undesirable oscillationMstachowsky (discuss • contribs) 13:49, 10 April 2014 (UTC)

Some points: In response to the comment above, instability is a test that is either/or. A system can't be "a little bit unstable", and therefore the presence (or absence) of oscillations is not indicative of instability. The confusion comes because you are mixing up a signal with a system. *systems* are unstable, while *signals* do not have a criterion by which to judge stability. Consider the following:

let H(s) = 1/s. This system is unstable. To prove this, we just need to give it a unit step input. Although the input is bounded, the output is unbounded, thus there is at least one bounded input for which the output is unbounded, and we violate the rule of BIBO stability right off the bat. However, if you consider the system's response to a "boxcar" input (which is zero everywhere but some finite time interval, during which it is constant), the system output will not oscillate and will not go to infinity. All that this means is that there is *some* bounded input for which the output is bounded, but since it is not true for *every* bounded input, the system is not BIBO stable.

Second, the claim that an LTI system is stable if all of its poles lie in the OLHP is not sufficient, but it is necessary. A second condition is that the system is proper. Thus saying that an LTI system is stable *if and only if* its poles are in the OLHP is not correct.

Consider:

H(s) = s^2/(s+1)

All if its poles are in the OLHP, but we can re-write as:

H(s) = (s-1) + 1/(s+1)

The step response of this function is:

y(t) = ilaplace(H(s)/s) = delta(t) - e^-t

which is unbounded. Thus the system has all poles in the OLHP but is not BIBO stable. All improper tfs have a proof for instability that is similar, and thus all improper TFs are unstable.

What are peoples' thoughts?Mstachowsky (discuss • contribs) 13:49, 10 April 2014 (UTC)

On Marginal stability: a marginally stable system is unstable since it fails the BIBO stability test. A system with two poles at +/- jw will produce an unbounded output when the input is u(t) = sin(wt). However, in many introductory linear systems courses the concept of "marginal stability" is talked about. Since we use BIBO stability as the most coarse definition of stability, I suggest that we change the section to read something like:

A system with poles on the imaginary axis is often referred to as "marginally stable". This means that there are many bounded inputs for which the output will also be bounded. However (etc...)

Thoughts?Mstachowsky (discuss • contribs) 13:27, 11 April 2014 (UTC)

Calling the angular frequency (ω) as complex frequency (s)?
In the section of Marginal Stability it reads "When the poles of the system in the complex S-Domain exist on the complex frequency axis (the vertical axis)." It's the first time I hear the name "complex frequency" for the variable in the vertical axis. That's wrong. It's not called like that, but as "angular frequency". What is called "complex frequency" is s, not ω. Ι will edit the section to reflect this. Alej27 (discuss • contribs) 00:49, 28 April 2020 (UTC)