It is known that exact pole placement is NP-hard.
On the other hand it is known that regional pole placement in LMI regions can be solved in polynomial time.
The question is what is the complexity of regional pole placement in non-convex or unconnected regions ?
In this context, does the problem becomes harder if we are retricted to use static output feedback to place the eigenvalues instead of using dynamic feedback ?
Yet in this context, does the problem for discrete-time systems harder than for continuous-time systems ?
The question is not fully clear to me. But I cam give some simple and standard response as follows:
For instance , if you have a controlled plant linear transfer function G = B/A with polynomials B , A and n=degree A>=m=degree B ( for realizability), B and A without cancellations ( for controllability of state space realizations of G of dimension n , minimal state dimension) then you can find polynomials S and R such that H (state space realizable feedback compensator)=S/R ( for instance with degree R=degree Am- degree A > degree S) such that
AR+BS=Am
Am is a prefixed stable polynomial ( defined by the designer) which contains the suited closed -loop poles . The above one is a diophantine equation in the ring of polynomials : for given data polynomials , you can find solution polynomials.
In that way , the closed- loop poles are assigned in ( stable: unstable also could be but non sense in applications) arbitrary positions for a linear controllable plant and this is not related to continuous or discrete( the polynomials can be in s ( Laplace op.) or D=d/dt ( time- derivative op. ) for "continuous-time " or in z ( z- transform) or q ( adveance discrete operator) for "discrete-time") . It is not either related to the domain where they belong ( domain being connected or not) since Am can have zeros ( closed-loop designed poles) in any chosen allocation.
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