A graph-theoretic method to find decentralized fixed modes of LTI systems

Department of Electrical and Computer Engineering, Concordia University, Montréal, Que., Canada H3G 1M8
Automatica (Impact Factor: 3.13). 12/2007; DOI: 10.1016/j.automatica.2007.04.019
Source: DBLP

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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    ABSTRACT: This paper presents a novel sequential technique for pole-assignment in linear time-invariant (LTI) decentralized control systems. Generalized sampled-data hold functions (GSHF) are used as local controllers to place the modes of the equivalent discrete-time closed-loop system in the desired locations in the z-plane. These locations are assumed to be obtained by using a proper mapping from the continuous-time domain. The GSHFs are obtained one at a time, in a sequential fashion. In other words, each local controller is designed for the equivalent discrete-time closed-loop model associated with the previously designed controllers. While no bound is provided on the intersample ripple, the convergence of the samples to zero ensures that the intersample values will also approach zero as time increases. The main characteristic of the proposed method is that unlike conventional pole-placement algorithms, the design complexity here does not increase after each local controller is obtained. A numerical example is provided which confirms the efficacy of the proposed pole-placement technique.
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