A Necessary and Sufficient Condition for Robust Stability of LTI Discrete-Time Systems using Sum-of-Squares Matrix Polynomials
Bibliographic record
Abstract
This paper deals with the robust stability of discrete-time systems with convex polytopic uncertainties. First, it is proved that the parameter-dependent Lyapunov function can be assumed to be a polynomial with a specific bound on its degree. Then, it is shown that the robust stability of any system is equivalent to the existence of two matrix polynomials with some bounds on their degrees, where these two polynomials and also the corresponding Lyapunov matrix polynomial satisfy a specific relation. Furthermore, a method is presented to convert the problem of existence of such polynomials to a set of linear matrix inequalities and equalities, which is referred to as semidefinite programming (SDP), and can be solved by using a number of available softwares. One of the capabilities of the proposed method is that the bounds obtained for the degrees of the related polynomials can be replaced by any smaller numbers in order to simplify the computations, at the cost of a potentially conservative result. Moreover, in the case when it is desired to accurately solve the robust stability problem while the degrees of the related polynomials are large, a computationally efficient method is proposed to convert the problem to the SDP with a reduced number of variables. The efficacy of this work is demonstrated in two numerical examples
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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 | 0.001 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".