Differential-algebraic systems are generically controllable and stabilizable

Self Cite

The present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021. https://doi.org/10.1007/s00498-021-00287-x), Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x) on (relative) generic controllability of unstructured linear differential-algebraic systems and of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) on (relative) generic controllability of port-Hamiltonian descriptor systems. We extend their results to (relative) genericity of observability. For unstructured differential-algebraic systems, criteria for (relative) generic observability are derived from Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x) using duality. This is not possible for port-Hamiltonian systems. Hence, we tweak the results of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) and derive similar criteria as for the unstructured case. Additionally, we consider certain rank constraints on the system matrices.

“…The following lemma is a slight improvement of [8,Lemma 3.6] and characterizes the matrices E and Q that may appear in a port-Hamiltonian system (1.5).…”

Section: Auxiliary Results: Relative Generic Setsmentioning

confidence: 99%

Section: Gl(r K )mentioning

confidence: 99%

Section: Remark 44 Instead Of Viewing H Rmentioning

confidence: 99%

Section: Remark 44 Instead Of Viewing H Rmentioning

confidence: 99%

Section: Remark 44 Instead Of Viewing H Rmentioning

confidence: 99%

See 3 more Smart Citations

Generic observability for port-Hamiltonian descriptor systems

Kirchhoff

2024

Self Cite

Relative genericity of controllablity and stabilizability for differential-algebraic systems

Ilchmann

Kirchhoff

2022

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 .

Port-Hamiltonian descriptor systems are relative generically controllable and stabilizable

Ilchmann,

Kirchhoff,

Schaller

2024

Self Cite

The present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021) on generic controllability and of Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2022) 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. We derive results on relative genericity. These findings are the basis for characterizing relative generic controllability of port-Hamiltonian systems in terms of dimensions. A similar result is proved for relative generic stabilizability.