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I have two questions:

  • Why we study stability in open loop rather than closed loop (which is the system that will be implemented).
  • and is there a relation between response time in closed loop & open loop (I know that this depends on the system, but some general points will be appreciated)

So as you would guess from my questions I have a difficulty in understanding the performance's study of a feedback control system. For me, if we're going to implement a closed loop system (process+ controller+ feedback loop) we should study its stability not the open loop's (process+ controller) because it's the one that we are interested in and the one we're going to use.

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  • \$\begingroup\$ Where's this "you must only study open-loop stability in your study of closed-loop systems" directive? \$\endgroup\$ – Phil Frost Nov 22 '13 at 2:36
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    \$\begingroup\$ @PhilFrost I mean when we want to study the stability of a closed loop system we draw the open loop's Bode plot and we decide from it if the closed loop system is stable or not (from the phase margin for example). \$\endgroup\$ – Heisenberg Nov 22 '13 at 2:52
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You can't implement a stable closed-loop system without knowing what the open-loop response looks like.

A simple example might be controlling the brightness of an incandescent lamp using a photo-diode (very fast) as feedback of brightness.

At rest, the system has no-problem then you set a demand that wants to see X watts per square metre produced at 1m. The photodiode will tell you the watts per square metre hitting it but if you don't take into account the time-lag (or inertia) of the lamp your control system will ramp up to maximum power before your lamp has started to glow.

The photo-diode, at some point later registers the correct amount of light and the driving system instantly "levels-out" because it believes the lamp has hit the demand but, the lamp will glow a bit brighter because of thermal lag (or inertia) and then the control loop will switch off and what you'll get is possibly a self-oscillating system and it may take ages before the system settles down.

Along the way you may even destroy the lamp.

What about other control loops for things like linear actuators - you set a demand position and an amplifier starts driving the motor to the correct position but you get overshoot because the motor and mechanism have inertia.

Basically, if you don't respect the open-loop response you have a recipe for disaster.

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When studying a control system, one wants to know how a the controller's choice of output stimulus at any given step will affect the ranges of system states at future moments in time; generally, one will try to work things so that the ranges of possible system states are quasi-centered upon the expected desired states, but always include any states that may turn out to be necessary to avoid disaster. If one assumes that one will generally want to avoid needlessly changing the output stimulus too fast, one may estimate what future system states would be if one started with a particular output stimulus and ramped it up at the desired rate, and then figure what the states would be if one started with that stimulus and ramped it down. One should choose a stimulus where the envelope of future states fits with desired future behavior.

The reason one uses open-loop behavior when performing such analyses is that it will be much easier to solve for what the system should do now if one can compute an envelope of future behavior using only the present system state and the presently-generated output stimulus, versus needing to consider the effects of one's solution upon future behavior. Open-loop analysis will often give an envelope of possible future behaviors rather than a particular exact expected behavior, but in many cases having an envelope may be just as useful and in some cases even more so.

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