How does feedback make a system stable even in the presence of an inherently unstable element in that?

Question

Consider the transfer function of a block being $$\frac{1}{s-0.5}$$ which is unstable in open loop. If we go for a feedback, magically it becomes stable even with the presence of the inherently unstable block. My thoughts on these are

1. When an element is unstable , it doesn't mean it is unstable for all inputs. For certain inputs the block can still produce stable outputs.
2. During feedback, the input given to the the block gets modified such that output doesn't blow up.

My questions are

1. Are the above observations correct?
2. The impulse response of the block is $$ e^{0.5t}u(t) $$ What the block does is, take the area under the signal at present instant and blow it up for subsequent instants. If so, a positive valued input should blowup the output exponentially. Then why is the output not blown up (exponentially) in following case, shown in figure?

The signal shown as input is the actual error signal when we use the block in a closed loop format with unit step input.

Answer 1

"When an element is unstable , it doesn't mean it is unstable for all inputs. For certain inputs the block can still produce stable outputs."

At first, an "element" (a part) cannot be unstable. It is better to use the term "active block" or "amplifier". Hence, an amplifier can be unstable if it has feedback, which produces a closed-loop pole (pair) with a positive real part. This is true for all inputs because it is the feedback loop which produces self-excitement of the system - not the input signal.

"During feedback, the input given to the the block gets modified such that output doesn't blow up."

An unstable system can be stabilized if the overall negative feedback overrides (dominates) the positive feedback which would cause instability (without negative feedback). In your case (example), the gain block has an inherent positive feedback (causing a pole at +0.5). This gain block ist stabilized using - as shown - 100% negative feedback causing a total closed-loop gain of "2 (and a pole at "-0.5")

Answer 2

Not necessarily all feedbacks stabilize the system. In your pic, if it were positive feedback, the system would be unstable.

When an element is unstable , it doesn't mean it is unstable for all inputs. For certain inputs, the block can still produce stable outputs.

No. When an LTI system(like the one you have) is stable, it stable for all stable inputs. When an LTI system is unstable, the response is unstable for all stable and unstable inputs. Only for non-linear systems, a certain input range may stabilize and few will lead to an unstable response.

During feedback, the input given to the the block gets modified such that output doesn't blow up.

Yes. The reason why it is stable in this case is that for some input say x when given just to your unstable system, it generates a response(high value). This high value is given back to your unstable system but with reversed polarity/effect. So this feedback signal again when given back to the system, it generates high value(this time in opposite polarity) and the process continues and so on. So for any signal, your unstable system generates a response which is fed back negatively/counteractively to the system thereby stabilizing it.

Though the system is overall stable, internally it is unstable

Answer 3

\$V_{out} = \dfrac{1}{s-0.5}\times (V_{in} - V_{out}\$)

Rearranging: -

\$V_{out}(s-0.5+1) = V_{in}\$

and: -

\$\dfrac{V_{out}}{V_{in}} = \dfrac{1}{s+0.5}\$ i.e. a previously unstable system becomes stable with negative feedback.

Should the minus 0.5 term have been minus 1 (or more negative) then using the simple negative feedback in the question would not have produced stability.