On Containment for Linear Systems With Switching Topologies: A Novel State Transition Matrix Perspective
Bibliographic record
Abstract
This article studies the containment control problem for a group of linear systems, consisting of more than one leader, over switching topologies. The input matrices of these linear systems are not required to have full-row rank and the switching can be arbitrary, making the problem quite general and challenging. We propose a novel analysis framework from the viewpoint of a state transition matrix. Specifically, according to the inherent linearity, we successfully establish a connection between state transition matrices of the above multileader system and a virtual leader-following system obtained by combining those leaders. This enlightening result relates the containment problem to a consensus one. Then, by analyzing the property of the state transition matrix, we uncover that each component of any follower's state converges to the convex hull spanned by the corresponding components of the leaders', provided some mild conditions are satisfied. These conditions are derived in terms of the concept of a positive linear system. A special case of the second-order linear system is further discussed to illustrate these conditions. Moreover, two different design methods of the feedback gain matrix are provided, which additionally require that the network topology contains a united spanning tree all the time.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".