A graph-theoretic method to find decentralized fixed modes of LTI systems
ABSTRACT This paper deals with the decentralized pole assignability of interconnected systems by means of linear time-invariant (LTI) controllers. A simple graph-theoretic approach is proposed to identify the distinct decentralized fixed modes (DFMs) of the system, i.e., the unrepeated modes which cannot be moved by means of a LTI decentralized controller. The state-space representation of the system is transformed to the decoupled form using a proper change of coordinates. For any unrepeated mode, a matrix is then computed which resembles the transfer function matrix of the system at some point in the complex plane. A bipartite graph is constructed accordingly in terms of the computed matrix. Now, the problem of verifying if this mode is a DFM of the system reduces to checking if the constructed graph has a complete bipartite subgraph with a certain property. The sole restriction of this work is that it is only capable of identifying the distinct DFMs of a system. However, it is axiomatic that most of the modes of the real-world systems are normally distinct. The primary advantage of the present paper is its simplicity, compared to the existing ones which often require evaluating the rank of several matrices.
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- "Several results are reported in the literature for structurally constrained control of large-scale systems in the contexts of decentralized and overlapping control, to address the shortcomings of the traditional control techniques , , , , . The concepts of controllability and observability were introduced in the literature, and it was shown that they play a key role in various feedback control analysis and design problems such as model reduction, optimal control, state estimation, etc. . "
Conference Paper: Controllability and Observability of Uncertain Systems: A Robust Measure[Show abstract] [Hide abstract]
ABSTRACT: This paper deals with the class of polynomially uncertain continuous-time linear time-invariant (LTI) systems whose uncertainties belong to a semi-algebraic set. The objective is to determine the minimum of the smallest singular value of the controllability or observability Gramian over the uncertainty region. This provides a quantitative measure for the robust controllability or observability degree of the system. To this end, it is shown that the problem can be recast as a sum-of-squares (SOS) problem. In the special case when the uncertainty region is polytopic, the corresponding SOS formulation can be simplified significantly. One can apply the proposed method to any large-scale interconnected system to identify those inputs and outputs that are more effective in controlling the system. This enables the designer to simplify the control structure by ignoring those inputs and outputs whose contribution to the overall control operation is relatively weak. A numerical example is presented to demonstrate the efficacy of the results.Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on; 01/2010
- "Several methods have been proposed in the literature to find the DFMs of a system , , , . So long as there exist no unstable DFMs, one can design a stabilizing LTI decentralized controller. "
Conference Paper: A New Decentralization Technique for Interconnected Systems[Show abstract] [Hide abstract]
ABSTRACT: This paper deals with LTI interconnected systems whose subsystems have coupled dynamics. The objective is to decentralize a given centralized controller satisfying some prescribed design specifications. More precisely, a parameterized decentralized controller is to be designed such that the state and the input of the system under the obtained decentralized controller can become arbitrarily close to those of the system under the given centralized controller, by tuning its free parameters. To this end, a two-level decentralized controller is designed, where the top level captures the dynamics of the centralized control system. The notion of structural initial value observability is introduced to design the bottom level of the decentralized controller. This method can decentralize every generic centralized controller, provided the interconnected system satisfies very mild conditions.Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on; 01/2010
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- "Another issue is that a generic static decentralized controller may not be able to sufficiently displace a mode so that it is recognized as a non DFM. A graph-theoretic method was proposed in the recent work , which constructs a bipartite graph corresponding to each unrepeated mode of the system. This work states that the mode is a DFM if and only if the graph contains a bipartite subgraph satisfying two specific properties. "
ABSTRACT: Given an interconnected system, this paper is concerned with the time complexity of verifying if any given unrepeated mode of the system is a decentralized fixed mode (DFM). It is shown that checking the decentralized fixedness of any distinct mode is tantamount to testing the strong connectivity of a digraph formed based on the system. It is subsequently proved that the time complexity of this decision problem using the proposed approach is the same as the complexity of matrix multiplication. This work concludes that the identification of distinct decentralized fixed modes (by means of a deterministic algorithm, rather than a randomized one) is computationally very easy, although the existing algorithms for solving this problem would wrongly imply that it is cumbersome. This paper provides not only a complexity analysis, but also an efficient algorithm for tackling the underlying problem.American Control Conference, 2009. ACC '09.; 07/2009