It is well known that controllability and observability are mathematical duals. So my question is, can algorithms that are proposed to check controllability be used to check observability as well?
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Sure. But more precisely, reachability and observability are duals. So any algorithm that is designed to check reachability of $(A, B)$ can be used to check the observability of $(A,C)$ by just checking the reachability of $(A^T,C^T)$ and vice versa.
obareey
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1It seems in nonlinear dynamical systems, controllability and observablity are not duals anymore, right? – m.taheri Aug 30 '19 at 09:08
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After a quick search I believe it is the case for certain systems, see for example: https://www.researchgate.net/publication/266366723_Duality_of_observability_and_controllability_properties_of_nonlinear_dynamic_systems. If you happen to construct a function that converts two problems to each other, then you could use the same algorithm for both problems. – obareey Aug 30 '19 at 14:05
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Thank you @obareey. Here : https://math.stackexchange.com/questions/3030305/what-is-the-difference-between-controllability-and-reachability you have said that reachability and controllability is not equivalent in the case of discrete time linear systems. So how I can use duality between reachability and observability to infer duality between controllability and observability? – m.taheri Aug 30 '19 at 15:31
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In general, you cannot. You have to check the specific conditions that the duality exists. For example, if $A$ does not have any zero eigenvalues, then duality exists between controllability and observability (in discrete time systems). Controllability is actually the dual of constructability in every case for linear systems.. – obareey Aug 31 '19 at 12:09