In the real world you will never have this :

You will have

1/(s-a+epsilon) and (s-a)/(s+b)

and epsilon will evoluate ...

The plant you want to control is normally dynamic and has sensor and actuators that over time may get their behavior not in line with the mathematical model that the controller was based on. The way to go about this is to get a feedback and compare it to desired response in order to regulate the plant output. Additionally, some plants such as aircraft influenced by the environment which may introduce some disturbance and the sensors may be amalgamated with noise so the plant output will not be what is expected as the ideal environment have changed. Also, some input may excite some modes which need to be suppressed by the controller once detected thru feedback.

Lastly, in modeling plants, always there is some degree of uncertainty and in order to cope with those uncertainties, you will need to have a feedback and address those deviations in the controller during design.

So, in short, open loop system does not guarantee proper control of plant.

Additionally to the previous comments, I put out that some plants also have unstable zeros that must not be cancelled by an unstable zero (cascading a controller). In this case feedback configurations arrives to a better performance. Also, even if a non-realistic (without uncertainties and disturbances) unstable-pole zero cancellation is realized, and this ideal system behaves "well" from an input to output point of view, there are internal unstable dynamics that can produce overheating or saturations or even damage internal components.

Your system will still be internally unstable, so your cascade solution does not work (even in the unlikely case that you get exact matching of the pole and zero). Try it in the lab and see what happens.

Hi

Real world not always behave like you said. Everybody knows that uncertainties of modeling and disturbances like noise in sensors and actuators can unstabilize the system even if you compensate with cascade controller like you propose.

Feedback allows you to establish a kind of “measurement” of the deviation from the set point due to these uncertainties. Your controller should work very well if the system is completely like the transfer function with zero uncertainties.

:)

If we push one step further we could rephrase the question as: why doing feedback control is we can do open-loop control ? Because if we know exactly the unstable pole, let's do open-loop control with the (filtered) inverse of the plant's transfer function ?. The answer is always the same: a marginal error on the estimation of the value of "b" will potentially make it blow up. Open-loop control is nice if you know perfectly the plant, or if you don't really care about statism or stability. If not FB control is clearly much more robust.

I would remind the concept of controllability. It is not recommended to cancel poles with zero. Even if you use exact parameters and put the perfect zero in front of a stable pole, you end with an uncontrollable stable pole. This is not desirable, yet not a tragedy yet.

Now, if you use exact parameters and put the perfect zero in front of the unstable pole, you get an uncontrollable unstable pole and this ends in tragedy.

That's why people take the pain to design closed-loop control systems.

if there is an input disturbance, then the system will go unstable without feedback.

Loss of controllability is another issue due to pole-zero cancellation.

Moreover, perfect pole-zero cancellation might not happen in practical systems

Feedback is required to handle uncertainties and disturbances

Cancelling an unstable pole creates a system that lacks internal stability. This is a serious problem for any system that is to be implemented in practice:

Internal stability. Definition. A system is internally stable if for all initial conditions, and all bounded signals injected at any place in the system, all states remain bounded for all future time.