On the computation of the fundamental subspaces for descriptor systems

In this paper, we investigate several theoretical and computational aspects of fundamental subspaces for linear time-invariant descriptor systems, which appear in the solution of many control and estimation problems. Different types of reachability and controllability for descriptor systems are described and discussed. The Rosenbrock system matrix pencil is employed for the computation of supremal output-nulling subspaces and supremal output-nulling reachability subspaces for descriptor systems

Qualitative sign stability of linear time invariant descriptor systems

This article discusses assessing the instability of a continuous linear homogeneous timeinvariant descriptor system. Some necessary conditions and sufficient conditions are derived to establish the stability of a matrix pair by the fundamentals of qualitative ecological principles. The proposed conditions are derived using only the qualitative (sign) information of the matrix pair elements. Based on these conditions, the instability of a matrix pair can easily be determined, without any magnitude information of the matrix pair elements and without numerical eigenvalues calculations. With the proposed theory, Magnitude Dependent Stable, Magnitude Dependent Unstable, and Qualitative Sign Stable matrix pairs can be distinguished. The consequences of the proposed conditions and some illustrative examples are discussed

Lyapunov equations and Riccati equations for descriptor systems

In this paper, two new types of Lyapunov and Riccati equations are presented for linear time-invariant descriptor systems. The two equations play key roles in asymptotic stability analysis and control synthesis for this class of systems. Fundamental properties of the two equations are investigated and interesting results are obtained.published_or_final_versio

Kalman filtering and Riccati equations for descriptor systems

Projet META2The theory of Kalman filtering is extended to the case of systems with descriptor dynamics. Explicit expressions are obtained for this descriptor Kalman filter allowing for the possible singularity of the observation noise covariance. Asymptotic behavior of the filter in the time-invariant case is studied ; in particular, a method for constructing the solution of the algebraic descriptor Riccati equation is presented

Finding NHIM: Identifying High Dimensional Phase Space Structures in Reaction Dynamics using Lagrangian Descriptors

Phase space structures such as dividing surfaces, normally hyperbolic invariant manifolds, their stable and unstable manifolds have been an integral part of computing quantitative results such as transition fraction, stability erosion in multi-stable mechanical systems, and reaction rates in chemical reaction dynamics. Thus, methods that can reveal their geometry in high dimensional phase space (4 or more dimensions) need to be benchmarked by comparing with known results. In this study, we assess the capability of one such method called Lagrangian descriptor for revealing the types of high dimensional phase space structures associated with index-1 saddle in Hamiltonian systems. The Lagrangian descriptor based approach is applied to two and three degree-of-freedom quadratic Hamiltonian systems where the high dimensional phase space structures are known, that is as closed-form analytical expressions. This leads to a direct comparison of features in the Lagrangian descriptor plots and the phase space structures' intersection with an isoenergetic two-dimensional surface and hence provides a validation of the approach.Comment: 39 pages, 7 figures, Submitted to Communications in Nonlinear Science and Numerical Simulatio