Open AccessJournal Article10.1016/J.SYSCONLE.2021.104914

Structurally quotient fixed modes

- 01 May 2021

- Vol. 151, pp 104914

TL;DR: In this article, a necessary and sufficient graph-theoretical characterization of quotient fixed modes occurring in parametric decentralized control systems is provided, and an efficient polynomial-time algorithm for the verification of this graph theoretical condition is presented.

About: This article is published in Systems & Control Letters. The article was published on 01 May 2021. and is currently open access. The article focuses on the topics: Quotient & Approximation algorithm.

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The authors provide a necessary and sufficient graph-theoretical characterization of quotient fixes modes occurring in parametric decentralized control systems.. Specifically, the authors introduce the notion of structurally quotient fixed modes ( SQFMs ) that generically captures the quotient fixed modes and only depends on the system ’ s structure.. Additionally, the authors provide an efficient polynomial-time algorithm for the verification of this graph-theoretical condition.. The authors show that this algorithm can be parallelized, and linear-time computational complexity approximation algorithms can be considered to attain a sub-optimal solution.. Lastly, the authors discuss the implications of the actuation–sensing–communication capabilities and the systems ’ interconnections on the existence of SQFM.

Fig. 1. In (a), we have a digraph representation of the structural plant (Ā, B̄, C̄), with four states, two subsystems (one with input u1 and output y1 , and the other with input u2 and output y2), with quotient subsystems Q1 and Q2 , delimited by the gray boxes. The obtained quotient system digraph by Lemma 1 is depicted in (b), where the red edge in (b) corresponds to the red edge in (a) connecting the subsystem Q1 with the subsystem Q1 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. Decomposition in paths and cycles of each subsystem’s digraph of the structural plant (Ā, B̄, C̄, K̄ ), i.e., D(Q1) and D(Q2), obtained with Algorithm 1. n red, we have this decomposition for D(Q1) with only once cycle. In blue, we ave this decomposition for D(Q1) with only two cycles. (For interpretation of he references to color in this figure legend, the reader is referred to the web ersion of this article.)

Fig. 3. In (a), we have a digraph representation of the structural plant (Ā, B̄, C̄), with five states, two subsystems (one with inputs u1, u2 and outputs y1, y2 , and he other with inputs u3, u4, u5 and output y3, y4, y5), with quotient subsystems Q1 and Q2 , delimited by the gray boxes. The obtained quotient system digraph by Lemma 1 is depicted in (b).

Fig. 2. Example of how to add edges between two subsystems’ subsystems digraph representations to remove SQFMs, illustrating the case where one subsystem has SFMs and the other subsystems does not. To simplify the visualization and the analysis, we use meta-nodes that represent an SCC of the system’s digraph representation given as follows: the source SCCs of the structural plant digraph representation are denoted by N⊤1 ,N ⊤ 2 , and N ⊤ 3 ; the arget SCCs are denoted by N⊥6 N ⊥ 7 ,N ⊥ 8 , and N ⊥ 9 ; and the other SCCs are denoted by N4 , and N5 . For simplicity, here we consider that both subsystems are such that all the states belong to a disjoint union of cycles. Thus, the dashed red edges indicate that there is one edge from and to the state variables in different SCCS. Notice that after considering these edges, the structurally quotient system digraph changes into a single partition that is now structurally controllable and observable, and therefore, the subsystem does not have SFM, which implies that the original information pattern does lead to the existence of SQFM. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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References

Linear Multivariable Control: A Geometric Approach

On the stabilization of decentralized control systems

Large-scale dynamic systems: stability and structure : D.D. Siljak 416 pages, diagrams, 6 x 9 in., Elsevier/North-Holland, New York, 1978. Price, $25.00.

Control structure design for complete chemical plants

19th IEEE conference on decision and control

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