Dear Zahoor Ahmed,

Here is a add about this subject.

A system is said to be controllable if starting from a state x0 at time t0 and a command u (t) we can take the latter to a state x (t1) at time t1> t0 (the trajectory can be any , the state is invariant at time t1).

A system is said to be observable if from a state x (t1) t1> t0 and a command u (t) we can reconstruct the initial state x0 (same remark as for controllable).

We can test these two properties with the rank criterion or linear independence and linear dependence, we also notice that these two properties are dual and that there is an extension of these properties taking into account the stability that is 'reachability and detectability deduced from the Hautus criterion.

All of his claims are made for state representation.

Now with regard to invertibility and inversion of the system (used for transfert function).

A system is called invertible if it produces distinct output signals for distinct input signals.

If an invertible system produces the output y(t) for the input x(t) then its inverse produces the output x(t) for the input y(t).

Best regards

Dear Harald Kirchsteiger, Mohamed-Mourad Lafifi , Janusz Milek , thanks to all.

Hi, during my thesis (92 ...97), I have had some works (see some of my works in researchgate) related to observability and controlability. The two properties are related to the testability of protocols.

I agree all, but let's remind that the Hautus criterion it's correct: Popov–Belevitch–Hautus criterion.

The first that write this was Vasile Mihai POPOV in Hyperstability of automation systems 1965

Thanks to all, if we consider controller parametrization (internal stability analysis), then how we can understand the above parameters, Harald Kirchsteiger ,Janusz Milek Mohamed-Mourad Lafifi @Harald Kirchsteiger

Dear Zahoor Ahmed,

If I understood correctly the extension of the question, the answer is as follows in case of linear system there is a conservation of these properties (controllability, observability) and reinforcement of the stability during the change premeditated choice of the system's spectrum by introduction of the controller parametrization.

For nonlinear or fractional systems the question is treated in a singular way.

Also i suggest you to see links and attached files on topic.

https://hal.archives-ouvertes.fr/tel-02437362/document

https://hal.inria.fr/hal-00866648/document

Conference Paper Observer-Based Controllers for Fractional Differential Systems

Best regards

Thank you very much dear @Mohamed-Mourad Lafifi , My concern is for a continuous time linear system in frequency domain. If you have any experience or literature, please share with me. Thank you very much for your interest and quick response.

Dear Zahoor Ahmed ,

Here are the references in the field in question.

Article An Input–Output Parametrization of Stabilizing Controllers: ...

https://ethz.ch/content/dam/ethz/special-interest/mavt/dynamic-systems-n-control/idsc-dam/Lectures/Digital-Control-Systems/Slides_DigReg_2013.pdf

http://www.cse.psu.edu/~gik2/teach/Linear_Systems-4.pdf

Thesis Repetitive Control Systems: Stability and Periodic Tracking ...

Best regards