Abstract
This paper shows that within the space of all LTI systems, equipped with the Zariski topology, the set of impulse controllable systems contains an open dense set of systems; in other words, impulse controllable systems are generic. This genericity persists for many closed subsets of LTI systems of interest, such as the class of singular descriptor systems.
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Notes
By shift invariance, the choice of \(t=1\) above is not important, any \(t \ne 0\) suffices. We use the value \(\left( X_M({\frac{\textsf {d} }{\textsf {dt} }}) f \right) _{t = 1}\) as an initial value of a potential solution f to \(M({\frac{\textsf {d} }{\textsf {dt} }})\), i.e. for \(\lim _{t \nearrow 0} f(t)\).
Indeed, the kernel of the operator \(M({\frac{\textsf {d} }{\textsf {dt} }}): ({\mathcal {D}}')^k \rightarrow ({\mathcal {D}}')^\ell \) is isomorphic to \({\textsf {Hom}}_A(A^k/M, ~{\mathcal {D}}')\), the module of homomorphisms from \(A^k/M\) to the space of distributions \({\mathcal {D}}'\), for instance [9].
We refer to any standard book on Algebra for definitions, for instance, D. Eisenbud: Commutative algebra with a view towards algebraic geometry.
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Acknowledgements
The second author thanks the Department of Electrical Engineering, IIT Bombay, for its hospitality during the time this paper was written.
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Belur, M.N., Shankar, S. The persistence of impulse controllability. Math. Control Signals Syst. 31, 487–501 (2019). https://doi.org/10.1007/s00498-019-00250-x
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DOI: https://doi.org/10.1007/s00498-019-00250-x