Linear quadratic Gaussian (LQG) control design for port-Hamiltonian systems is studied.
The recently proposed method from [42] is reviewed and modified such that the resulting controllers have a port-Hamiltonian (pH) realization. Based on this new modification, a reduced-order controller is obtained by truncation of a balanced system. The approach is shown to be closely related to classical LQG balanced truncation and shares a similar a priori error bound with respect to the gap metric. With regard to this error bound, a theoretically optimal pH-representation is derived. Consequences for pH-preserving balanced truncation model reduction are discussed and shown to yield two different classical H∞ -error bounds. Numerical examples illustrate the main theoretical findings.
We present a novel model-order reduction (MOR) method for linear time-invariant systems that preserves passivity and is thus suited for structure-preserving MOR for port-Hamiltonian (pH) systems. Our algorithm exploits the well-known spectral factorization of the Popov function by a solution of the Kalman-Yakubovich-Popov (KYP) inequality. It performs MOR directly on the spectral factor inheriting the original system’s sparsity enabling MOR in a large-scale context. Our analysis reveals that the spectral factorization corresponding to the minimal solution of an associated algebraic Riccati equation is preferable from a model reduction perspective and benefits pH-preserving MOR methods such as a modified version of the iterative rational Krylov algorithm (IRKA). Numerical examples demonstrate that our approach can produce high-fidelity reduced-order models close to (unstructured) H2 -optimal reduced-order models.
We present a new balancing-based structure-preserving model reduc-
tion technique for linear port-Hamiltonian descriptor systems. The pro-
posed method relies on a modification of a set of two dual generalized
algebraic Riccati equations that arise in the context of linear quadratic
Gaussian balanced truncation for differential algebraic systems. We de-
rive an a priori error bound with respect to a right coprime factorization
of the underlying transfer function thereby allowing for an estimate with
respect to the gap metric. We further theoretically and numerically ana-
lyze the influence of the Hamiltonian and a change thereof, respectively.
With regard to this change of the Hamiltonian, we provide a novel proce-
dure that is based on a recently introduced Kalman–Yakubovich–Popov
inequality for descriptor systems. Numerical examples demonstrate how
the quality of reduced-order models can significantly be improved by first
computing an extremal solution to this inequality.
With this overview we want to provide a compilation of different models for
the description of gas flow in networks in order to facilitate the introduction
to the topic. Special attention is paid to the hierarchical structure inherent
to the modeling, and the detailed description of individual components such
as valves and compressors. Also included are network model classes based
on purely algebraic relations, and energy-based port-Hamiltonian models. A
short overview of basic numerical methods and concepts for the treatment
of hyperbolic balance equations is also given. We do not claim completeness
and refer in many places to the existing literature.