On discrete-time port-Hamiltonian (descriptor) systems

Abstract

Port-Hamiltonian (pH) systems have been studied extensively for linear continuous-time dynamical systems. In this manuscript a discrete-time pH descriptor formulation is presented for linear, completely causal, scattering passive dynamical systems that is purely based on the system coefficients. The relation of this formulation to positive and bounded real systems and the characterization via positive semidefinite solutions of Kalman-Yakubovich-Popv inequalities is also studied.

A geometric framework for discrete time port‐Hamiltonian systems

“…An extension of Definition 3.1 in this direction is rather straight forward, but left for future work. Here, one introduces Lagrangian effort variables 𝑒 𝐿,𝑘 which must fulfill (𝑥 𝑘+1 , 𝑒 𝑘,𝐿 ) ∈  ℎ 2 ,1 () for some ℎ > 0 and one has to replace 𝑥 𝑘 in the left hand side expression in (10) by 𝑒 𝑘,𝐿 .…”

“…In [10], a definition of discrete-time scattering pH descriptor systems was presented for the case that the system is causal, that is, the solution 𝑥 𝑘 at index 𝑘 does not depend on future inputs. This has been characterized in [10] by the Kronecker index of the pair (𝐸, 𝐴) being at most one. In this case, the system can be transformed to a reduced standard discrete-time system…”