Network structure preserving model reduction with weak a priori structural information

Abstract

Abstract-This paper extends a state projection method for structure preserving model reduction to situations where only a weaker notion of system structure is available. This weaker notion of structure, identifying the causal relationship between manifest variables of the system, is especially relevant is settings such as systems biology, where a clear partition of state variables into distinct subsystems may be unknown, or not even exist. The resulting technique, like similar approaches, does not provide theoretical performance guarantees, so an extensive computational study is conducted, and it is observed to work fairly well in practice. Moreover, necessary conditions, characterizing structurally minimal realizations, and sufficient conditions, characterizing edge loss resulting from the reduction process, are presented.

Mathematical relationships between representations of structure in linear interconnected dynamical systems

“…Exploring how to appropriately define the complexity of a subsystem or signal structure is in itself an open research problem. Thus, future work [13] will explore the resulting research problems that arise from the results in this paper, [7], [6] and the framework provided in [9]. At a more general level, future research will also investigate different representations of structure and their relationships within the behavioral framework provided in [4].…”

“…The dynamical structure function (defined in [5] and discussed in [7], [8], [6], [12], [9]) is a representation that describes the direct causal dependence among a subset of state variables; it is the mathematical analogue of signal structure.…”

“…System structure in this context refers to the sharing of variables or the linking of terminals and ports [4]. At the same time, [5], [6], [7], [8] describe system structure in terms of the dependencies between manifest variables of the system. Alternatively, system structure in the context of decentralized control can refer to the location of zero entries in a transfer function matrix.…”

Optimal Distributed Control for Platooning via Sparse Coprime Factorizations

“…It is precisely this particular type of constraints on the coprime factors of the controller that induces the distributed DRAFT implementation of resulted controllers as a network of linear time-invariant subsystems, such that the sub-controller on board each vehicle uses only information from its predecessor in the string. This approach to distributed controllers as linear dynamical networks hinges on the concept of dynamical structure functions, originally introduced in [47], [35] and further developed in [36], [37], [38], [39], [40], [41], [43], [44], [45].…”

A Minimal Realization Technique for the Dynamical Structure Function of a Class of LTI Systems

“…As a result, the DSF is a useful modeling tool for complex networks where some information about the network's global structure is desired without engaging the full complexity of a complete state-space realization [2], [5], [6], [12]- [14]. Examples of applications that have effectively leveraged the DSF as a modeling technology include system biology, in the reconstruction of biochemical reaction networks [4], [10], [17], [18]; computer science, in the vulnerability analysis and design of secure architectures for cyber-physical systems [26], [27]; and distributed systems, in the design of distributed and decentralized control systems [28]- [30] and structure-preserving model-reduction [15]. Underlying all of these applications, however, is the theoretical question relating a Dynamical Structure Function to its minimal state realizations.…”