Graph-theoretic approach to symbolic analysis of linear descriptor systems

AbstractContinuous descriptor systems EẋAx+Bu, yCx, where E is a possibly singular matrix, are symbolically analyzed by means of digraphs. Starting with four different digraph characterizations of square matrices and determinants, the author favors the Cauchy-Coates interpretation. Then, an appropriate digraph representation of the matrix pencil (sE−A) is given, which is followed by a digraph interpretation of det(sE−A) and the transfer-function matrix C(sE−A)−1B. Next, a graph-theoretic procedure is derived to reveal a possibly hidden factorizability of the determinant det(sE−A). This is very important for large-scale systems. Finally, as an application of the derived results, an electrical network is analyzed symbolically

Digraph based determination of Jordan block size structure of singular matrix pencils

AbstractThe generic Jordan block sizes corresponding to multiple characteristic roots at zero and at infinity of a singular matrix pencil will be determined graph-theoretically. An application of this technique to detect certain controllability properties of linear time-invariant differential algebraic equations is discussed

When is the discretization of a spatially distributed system good enough for control?

Accepted versio

A model structure-driven hierarchical decentralized stabilizing control structure for process networks

Based on the structure of process models a hierarchically structured state-space model has been proposed for process networks with controlled mass convection and constant physico-chemical properties. Using the theory of cascade-connected nonlinear systems and the properties of Metzler and Hurwitz matrices it is shown that process systems with controlled mass convection and without sources or with stabilizing linear source terms are globally asymptotically stable. The hierarchically structured model gives rise to a distributed controller structure that is in agreement with the traditional hierarchical process control system structure where local controllers are used for mass inventory control and coordinating controllers are used for optimizing the system dynamics. The proposed distributed controller is illustrated on a simple non-isotherm jacketed chemical reactor

Behavior Modes, Pathways and Overall Trajectories: Eigenvector and Eigenvalue Analysis of Dynamic Systems

One of the most fundamental principles in system dynamics is the premise that the structure of the system will generate its behavior. Such philosophical position has fostered the development of a number of formal methods aimed at understanding the causes of model behavior. To most in the field of system dynamics, behavior is commonly understood as modes of behavior (e.g., exponential growth, exponential decay, and oscillation) because of their direct association with the feedback loops (e.g., reinforcing, balancing, and balancing with delays, respectively) that generate them. Hence, traditional research on formal model analysis has emphasized which loops cause a particular “mode” of behavior, with eigenvalues representing the most important link between structure and behavior. The main contribution of this work arises from a choice to focus our analysis in the overall trajectory of a state variable – a broader definition of behavior than that of a specific behavior mode. When we consider overall behavior trajectories, contributions from eigenvectors are just as central as those from eigenvalues. Our approach to understanding model behavior derives an equation describing overall behavior trajectories in terms of both eigenvalues and eigenvectors. We then use the derivatives of both eigenvalues and eigenvectors with respect to link (or loop) gains to measure how they affect overall behavior trajectories over time. The direct consequence of focusing on behavior trajectories is that system dynamics researchers' reliance on eigenvalue elasticities can be seen as too-narrow a focus on model behavior – a focus that has excluded the short term impact of a change in loop (or link) gain in its analysis