Relative genericity of controllablity and stabilizability for differential-algebraic systems

Abstract

The present note is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021) on generic controllability and stabilizability of linear differential-algebraic equations. We resolve the drawback that genericity is considered in the unrestricted set of system matrices $$(E,A,B)\in \mathbb {R}^{\ell \times }\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}$$ ( E , A , B ) ∈ R ℓ × × R ℓ × n × R ℓ × m , while for relative genericity we allow the restricted set $$\Sigma _{\ell ,n,m}^{\le r} := \{(E,A,B)\in \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m} \,\big \vert \,\textrm{rk}\,_{\mathbb {R}} E \le r\}$$ Σ ℓ , n , m ≤ r : = { ( E , A , B ) ∈ R ℓ × n × R ℓ × n × R ℓ × m | rk R E ≤ r } , where $$ r\in {\mathbb {N}}$$ r ∈ N . Our main results are characterizations of generic controllability and generic stabilizability in $$\Sigma _{\ell ,n,m}^{\le r}$$ Σ ℓ , n , m ≤ r in terms of the numbers $$\ell , n, m, r$$ ℓ , n , m , r .

Differential-algebraic systems are generically controllable and stabilizable

“…The relative Zariski topology on V has, in general, not the same favourable properties regarding the Euclidean topology (and the Lebesgue measure, if there is some reasonable way of defining such a measure) which allows to study genericity in the sense of Definition 2.5 as a reasonable concept; the simplest example is a discrete set with at least two points. To overcome this issue, Kirchhoff used in [23] an adapted concept for genericity in some reference se, which was refined in [19] and is defined as follows. ).…”

“…However, the concept of genericity of controllability for DAE-systems has the drawback that the set Σ ℓ,n,m is too "large": if ℓ = n, then in each arbitrarily small neighbourhood of (E, A, B) ∈ Σ ℓ,n,m there is some invertible E ′ ∈ K n×n such that (E ′ , A, B) is an ordinary differential equation. To resolve this drawback, Kirchhoff [23] introduced the concept of relative genericity 2 , and Ilchmann and Kirchhoff [19,Thm. 3.2] showed in 2022 that a similar characterization to (1.4) holds for relative genericity of controllability for various reference sets such as (E, A, B) ∈ Σ ℓ,n,m rk R E ≤ r for r ∈ N. 1 A set S ⊆ R n is called generic if, and only if, there exist a proper algebraic variety V ∈ V prop n (R) so that S c ⊆ V.…”

“…This class is a generalization of (1.5) and a restriction of (1.3). Due to the particular structure in (1.1), it is not readily possible to apply the results of [18,Theorem 2.3] and of [19,Thm. 3.2].…”

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The present work is a successor of [18] on generic controllability and of [19] on relative generic controllability of linear differential-algebraic equations. We extend the result from general, unstructured differential-algebraic equations to differential-algebraic equations of port-Hamiltonian type.

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“…if V is an analytic submanifold with countable atlas), then the properties of the relative Zariski topology can be invoked to conclude that every relative generic set is conull w.r.t. a Lebesgue-type measure, see [19,Proposition 2.8]. Thus, the authors prefer this concept of relative genericity over an adaptation of the purely topological concept of definition 2.3 w.r.t.…”