On the end of Zentralblatt fur Mathematik und ihre Grenzgebiete there is the section Controllability, Observability and System Structure.

I have one problem. In the countries where cybernetics is not accepted as important area of research, the mathematicians say: controllability is not important, it is plagiarism, the technical experts say: controllability is not important for engineering because it is part of specialization in mathematics. What to do?

Dear Danilo,

I tend to agree with Harald. If a system is not controllable or close to being not controllable, then it will be a total waste of time to try to come up with a controller in practice. The concepts of controllability and observability can have important practical consequences even when the system is controllable and observable. If you are curious, see some of the references in Article Structural, dynamical and symbolic observability: From dynam...

Ok, having said all that, it is worth remembering that such concepts were developed in the 1950s well after the 2nd World War where a lot of control techniques were developed and used. Hence, it is possible to do some control without such concepts. As always, the issue is how much of the problem do we want to understand and what type of solutions are we aiming at.

Regards.

Luis

Luis, it is OK, your examples are finite dimensional. What do you say for the case of infinite dimensional SISO and MIMO systems ( Hilbert space and Banach space case, see for example the book of Balakrishnan ( Applied functional analysis, The methods of optimization in Hilbert space)). It is developed 20 years later, 1970s. I hope that you will agree with me that knowledges of much more mathematics are needed. The papers of Roberto Triggiani was very popular at these times. Regards, Danilo.

For single input single output infinite dimensional systems (SISO), controllability is actually the case of operators with cyclic vectors. When I was in Leningrad 1978 I could not find in the libraries the paper of Domingo Herrero about multicyclic operators. I was very happy to see that only Nikolai Kapitonovich Nikolski ( LOMI) had the preprint of this paper.

From theoretical point of view, ordinary differential equations are mainly finite dimensional case, partial differential equations are infinite dimensional case.

I believe that checking the contralibility is the first and important step. After all we can only control a controlable system.

This question is very interesting for me, from now on I will pay more attention to this and specially the opinion and point of view of my colleagues.

Controllability is the criteria under which the whole system is controllable. For example, if we have unstable and stable part of the system, then for stabilization of the system is necessary that uncontrollable part is stable and controllable unstable part can be made stable ( at linear systems exponentially stabilizable,( strong, uniform etc)) . This approach holds true at the state space systems.

Observability is dual property of controllability, i.e. calculation with "adjungierte Operatoren".

Dear Danilo,

I just saw your answers and additional points of view. You are right concerning dimensionality. In the system is infinite dimensional because of a time delay, then we may avoid the problem by taking a discrete-time representation and using standard controllability tools. Now, as you pointed out, if the system is governed by partial differential equations the problem is different and much harder, I guess. Having said that, let me raise the following *practical* question: in the case of high-dimentional (not to mention infinte-dimensional) systems, doest it make sense, or better, is it really necessary to be able to place the system state anywhere in such a high-dimensional space? Perhaps the region of interest is a low-dimensional manifold where the controlled system should stay...

Regards.