{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:29:12Z","timestamp":1760149752616,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,5]],"date-time":"2023-09-05T00:00:00Z","timestamp":1693872000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, stability is studied for a class of switched singular systems containing both stable and unstable modes. By introducing a time-varying piecewise Lyapunov function (TVPLF) and a mode-dependent average dwell time (ADT) switching rule, the computable sufficient conditions for system stability are derived. The time-varying piecewise Lyapunov functions are piecewise continuously differentiable on every mode (but may not be differentiable at the interpolating points of the dwell time). This Lyapunov function method is particularly advantageous in overcoming the limitations of traditional multiple Lyapunov function (MLF) methods, which may not have a feasible solution when dealing with switched systems containing only unstable modes. As such, the TVPLF offers greater flexibility in application. Compared with the conventional ADT switching rule, the mode-dependent ADT switching rule not only enables each mode to have its own ADT but also allows for its own switching strategy. Specifically, the stable mode adopts a slow switching strategy while the unstable mode adopts a fast one, thereby reducing the conservatism of the ADT switching rule. Furthermore, based on the stability analysis, the time-varying controllers are proposed to stabilize the switched singular system, which can be expressed as the sequential linear combination of a series of linear state feedback on each mode. The proposed controllers are continuous for each mode, which are different from the controllers designed through the traditional MLF and MDLF methods, where the controllers designed by traditional MLF are the time-invariant linear state feedback in each mode while the controllers designed by the MDLF are piecewise continuous for each mode.<\/jats:p>","DOI":"10.3390\/e25091300","type":"journal-article","created":{"date-parts":[[2023,9,5]],"date-time":"2023-09-05T09:46:20Z","timestamp":1693907180000},"page":"1300","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Stability Analysis of Switched Linear Singular Systems with Unstable and Stable Modes"],"prefix":"10.3390","volume":"25","author":[{"given":"Jiandong","family":"Xiong","sequence":"first","affiliation":[{"name":"College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yidian","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yanfang","family":"Huo","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongpeng","family":"Zhao","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"618","DOI":"10.1109\/TAC.2003.809795","article-title":"On common quadratic Lyapunov functions for pairs of stable LTI systems whose system matrices are in companion form","volume":"4","author":"Shorten","year":"2003","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"885","DOI":"10.1109\/TAC.2003.811274","article-title":"On quadratic Lyapunov functions","volume":"5","author":"Cheng","year":"2003","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1016\/j.sysconle.2005.04.011","article-title":"Lie-algebraic stability conditions for nonlinear switched systems and differential inclusions","volume":"1","author":"Margaliot","year":"2006","journal-title":"Syst. Control Lett."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Sun, Z., and Ge, S. (2005). Switched Linear System: Control and Design, Springer.","DOI":"10.1007\/1-84628-131-8"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Liberzon, D. (2003). Switching in Systems and Control: Foundations and Applications, Birkhauser.","DOI":"10.1007\/978-1-4612-0017-8"},{"key":"ref_6","first-page":"1615","article-title":"Asymptotic stability, L1 gain and time-varying H\u221e control of switched positive systems","volume":"10","author":"Xiong","year":"2019","journal-title":"Control Theory Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1132","DOI":"10.1016\/j.automatica.2012.03.008","article-title":"Stability of switched positive linear systems with average dwell time switching","volume":"6","author":"Zhao","year":"2012","journal-title":"Automatica"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3502","DOI":"10.1109\/TAC.2016.2614911","article-title":"New results on stability of slowly switched systems: A multiple discontinuous Lyapunov function approach","volume":"7","author":"Zhao","year":"2017","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"4197","DOI":"10.1177\/0142331219852461","article-title":"Stability and L2-gain analysis of linear switched singular systems by using multiple discontinuous Lyapunov function approach","volume":"15","author":"Wei","year":"2019","journal-title":"Trans. Inst. Meas. Control"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1109\/TAC.2010.2097351","article-title":"Robust stability and stabilization of linear switched systems with dwell time","volume":"2","author":"Allerh","year":"2011","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"940","DOI":"10.1016\/j.automatica.2013.12.028","article-title":"Stabilization of switched continuous-time systems with all modes unstable via dwell time switching","volume":"3","author":"Xiang","year":"2014","journal-title":"Automatica"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"7685","DOI":"10.1109\/TSMC.2020.2982010","article-title":"Dwell-time-based exponential stabilization of switched positive systems with actuator saturation","volume":"12","author":"Ma","year":"2021","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1080\/00207177408932729","article-title":"Structural properties of linear dynamical systems","volume":"2","author":"Rosenbrock","year":"1974","journal-title":"Int. J. Control"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Dai, L. (1989). Singular Control Systems, Springer.","DOI":"10.1007\/BFb0002475"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Duan, G. (2010). Analysis and Design of Descriptor Linear Systems, Springer.","DOI":"10.1007\/978-1-4419-6397-0"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Aplevich, J.D. (1991). Implicit Linear Systems, Springer.","DOI":"10.1007\/BFb0044363"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1080\/002077299292155","article-title":"Bifurcation on stability of singular systems with delay","volume":"6","author":"Li","year":"1999","journal-title":"Int. J. Syst. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/S0167-6911(02)00204-9","article-title":"On controllability and observability for a class of impulsive systems","volume":"3","author":"Guan","year":"2002","journal-title":"Syst. Control Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2075","DOI":"10.1016\/S0005-1098(02)00104-8","article-title":"Robust decentralized stabilization for a class of large-scale time-delay uncertain impulsive dynamical systems","volume":"38","author":"Guan","year":"2002","journal-title":"Automatica"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1016\/j.sysconle.2006.05.002","article-title":"Stability robust stabilization and H\u221e control of singular-implusive systems","volume":"55","author":"Yao","year":"2006","journal-title":"Syst. Control Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1109\/TCSII.2005.848959","article-title":"Robust H\u221e control of singular impulsive systems with uncertain perturbations","volume":"6","author":"Guan","year":"2005","journal-title":"IEEE Trans. Circ. Syst.-II Express Briefs"},{"key":"ref_22","unstructured":"Daniel, L., and Stephan, T. (2009, January 16\u201318). On stability of linear switched differential algebraic equations. Proceedings of the Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, China."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1481","DOI":"10.1016\/j.automatica.2013.02.002","article-title":"Stability analysis of switched linear singular systems","volume":"5","author":"Zhou","year":"2013","journal-title":"Automatica"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1007\/s11768-008-6140-0","article-title":"H\u221e control for switched and impulsive singular system","volume":"1","author":"Yin","year":"2008","journal-title":"J. Control Theory Appl."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Su, X., and Wu, W. (2016, January 7\u20138). The study of exponential stbility of singular switched systems. Proceedings of the 10th International Conference on Intelligent Systems and Control, Coimbatore, India.","DOI":"10.1109\/ISCO.2016.7726947"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"110264","DOI":"10.1016\/j.automatica.2022.110264","article-title":"A unified framework of convex stability conditions for 2-D switched systems with stable or unstable modes","volume":"141","author":"Zhuang","year":"2022","journal-title":"Automatica"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"8073","DOI":"10.1002\/rnc.6265","article-title":"Exponential stability of totally positive switched linear systems with both stable and unstable subsystems","volume":"14","author":"Pan","year":"2022","journal-title":"Int. J. Robust Nonlinear Control"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/9\/1300\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:46:44Z","timestamp":1760129204000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/9\/1300"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,5]]},"references-count":27,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["e25091300"],"URL":"https:\/\/doi.org\/10.3390\/e25091300","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2023,9,5]]}}}