Approximation of stability radii for large-scale dissipative Hamiltonian systems
Linear dissipative Hamiltonian (DH) systems of the form \[ \dot{x} = (J-R)Qx \] with a skew-Hermitian matrix \(J\), a Hermitian positive semidefinite matrix \(R\), and a Hermitian positive definite matrix \(Q\), arise as homogeneous parts of port-Hamiltonian systems without inputs and outputs. These systems are Lyapunov stable and under additional weak conditions even asymptotically stable. However, the matrix \((J - R)Q\) may have purely imaginary eigenvalues, implying that DH systems are not necessarily asymptotically stable. Small perturbations may cause the system to become unstable. Asymptotic stability can only be guaranteed when the stability radius of the system is reasonably large. The authors propose subspace frameworks for the estimation of the stability radii for large scale dissipative Hamiltonian systems. At every iteration, the DH system is projected to small subspaces by preserving its structure. Then the corresponding stability radii are computed for the reduced system by exploiting its eigenvalue optimization characterization. The subspaces used in the projections are expanded to attain the Hermite interpolation properties between the objective eigenvalue function of the full and the reduced problems. The authors show that the stability radii are accurately and efficiently estimated for the DH systems of small and moderate size and for large scale finite-element model of an industrial disk brake. MATLAB implementations of the algorithms and subspace frameworks, as well as some of the data, are here presented.
scientific article; zbMATH DE number 7189692
scientific article