A Framework for Structural Input/Output and Control Configuration Selection in Large-Scale Systems (1309.5868v3)

Abstract: This paper addresses problems on the structural design of control systems taking explicitly into consideration the possible application to large-scale systems. We provide an efficient and unified framework to solve the following major minimization problems: (i) selection of the minimum number of manipulated/measured variables to achieve structural controllability/observability of the system, and (ii) selection of the minimum number of feedback interconnections between measured and manipulated variables such that the closed-loop system has no structurally fixed modes. Contrary to what would be expected, we show that it is possible to obtain a global solution for each of the aforementioned minimization problems using polynomial complexity algorithms in the number of the state variables of the system. In addition, we provide several new graph-theoretic characterizations of structural systems concepts, which, in turn, enable us to characterize all possible solutions to the above problems.

• The paper introduces a novel framework that jointly addresses structural I/O selection and control configuration to ensure decentralized system controllability and observability.
• It employs graph-theoretical techniques and maximum matchings to systematically determine the minimal set of sensors and actuators for robust system performance.
• The approach provides efficient polynomial-time algorithms that reduce complexity in large-scale system design and enhance overall control capability.

Structural Input/Output and Control Configuration Selection in Large-Scale Systems

The paper "A Framework for Structural Input/Output and Control Configuration Selection in Large-Scale Systems" presents a comprehensive methodology for tackling the intricate problem of structural design in complex large-scale systems, such as power grids, industrial processes, wireless networks, and biological networks. The paper is motivated by the need for scalable techniques that integrate design and decision-making within a single framework, focusing explicitly on the selection of input/output (I/O) variables and control configuration (CC) for decentralized systems.

Overview and Problem Definition

The paper addresses two foundational questions in systems control theory: which variables should be measured and manipulated (I/O selection), and which feedback connections should be established between these variables (CC selection). These problems, particularly the latter, are vital in decentralized control where understanding the necessary sensor-controller connections dramatically impacts system stability, performance, cost, and complexity.

A salient feature of the approach is its reliance on structural systems theory, which emphasizes properties deducible from the sparsity patterns of system matrices, rather than specific numerical values. This approach offers flexibility in scenarios where exact parameter values are unknown or difficult to determine. The research defines two specific computational problems: the sparsest I/O selection problem (formulated as problem $\mathcal{P}_1$) and the joint I/O and CC selection problem ($\mathcal{P}_2$).

Methodological Contributions

Key methodological contributions of the paper include:

1. Feasible Dedicated Input/Output Configurations: The research provides criteria for determining the minimal set of state variables that need to be actuated or monitored to achieve structural controllability or observability. This is achieved through a systematic exploration of maximum matchings within bipartite graphs representing system states.
2. New Characterization of Structural Controllability: A novel graph-theoretical characterization allows for the investigation of maximal matching properties in bipartite graphs to ensure controllability and observability through minimal actuation and sensing setups.
3. Joint I/O and Control Configuration Design: The paper elaborates on designing systems devoid of structurally fixed modes by integrating the I/O design with feedback structures. This unified approach ensures the existence of feedback matrices facilitating pole placement, thereby achieving desired dynamic behaviors.
4. Algorithmic Solutions: The paper presents efficient polynomial-time algorithms for solving the presented problems. This is a notable achievement given the inherent combinatorial nature of these problems.

Implications and Future Directions

The implications of this research are profound for both theoretical advancements and practical applications in the control community. The structural approach to I/O and CC selection bridges the gap between theoretical control design and practical implementability in complex systems with uncertain parameters. The polynomial-time algorithms make this approach viable for large-scale systems encountered in industrial applications.

The theoretical insights provided could be further explored to address configurations with heterogeneous costs, where inputs, outputs, and feedback links have varying costs, thereby broadening the applicability of these results. Additionally, extensions to dynamic and nonlinear systems control could further solidify the framework’s robustness and adaptability.

Overall, the paper contributes a substantial step forward in the structured design of large-scale systems, mitigating complexity while enhancing control efficacy, and opening avenues for continued exploration and refinement in this critical area of systems engineering.