Robust regional eigenvalue‐clustering analysis for linear perturbed discrete time‐delay systems

Regional Eigenvalue-Clustering Robustness Analysis for Linear Discrete Time-Delay Systems with Both Structured and Unstructured Parameter Uncertainties

“…It will be shown later in Examples 1 and 2 given below that the asymmetric sufficient condition (15) gives less conservative results than the symmetric sufficient condition of Hsieh et al (7), (8) , and the reason why the proposed asymmetric sufficient condition (15) is less conservative will be also explained in Remark 4 given below.…”

“…Lemma 1 (7) : If all the eigenvalues of the nominal system (8) lie inside a specified region D of the z-plane, then…”

“…Lemma 2 (7) : All the eigenvalues of the uncertain system (1) lie inside the specified region D, if the nominal system (8) has all its eigenvalues located within a specified region D, and if…”

“…Thus, in order to get around this difficulty, all the researchers (2) -(9) directly adopted the linear discrete time-delay system (1) instead of the linear augmented discrete system (3) to develop computationally simple conditions to analyze the robustness problem of eigenvalue-clustering inside a specified region for the linear discrete time-delay system with structured (elemental) parameter uncertainties. Note that the results proposed by Hsieh et al (7), (8) can be applied to analyze the robustness problem of eigenvalue-clustering inside a specified region which may be allowed to be any shape, whereas the results given by Wang and Wang (2), (3) , Trinh and Aldeen (4) , Shyu and Liaw (5) , Chang (6) , and Chen et al (9) can only be applied to analyze the robustness problem of eigenvalueclustering within a specified circular region. For the case of eigenvalue-clustering in a circular region, the sufficient condition proposed by Hsieh et al (7), (8) is shown to be less conservative than those proposed by Wang and Wang (2), (3) , Trinh and Aldeen (4) , Shyu and Liaw (5) , and Chang (6) ; besides, it can be seen that the sufficient condition proposed by Hsieh et al (7), (8) is the generalized version of the results given by Chen et al (9) On the other hand, it is well known that an approximate system model is always used in practice and sometimes the approximation error should be covered by introducing both structured (elemental) and unstructured (norm-bounded) parameter uncertainties in control system analysis and design (10) .…”

“…Three examples are given in section 3 for illustration. For the case of only considering structured (elemental) parameter uncertainties, a comparison of conservatism between the criteria, which are, respectively, proposed in this paper and by Hsieh et al (7), (8) , is also given in this section. Finally, section 4 offers some conclusions.…”