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On Existence of Stabilizing Switching Laws within a Class of Unstable Linear Systems

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  • Sendren Sheng-Dong Xu
  • Chih-Chiang Chen

Abstract

The equivalence of two conditions, condition (3) and condition (4) stated in Problem Statement section, regarding the existence of stabilizing switching laws between two unstable linear systems first appeared in (Feron 1996). Although Feron never published this result, it has been referenced in almost every survey on switched systems; see, for example, (Liberzon and Morse 1999). This paper proposes another way to prove the equivalence of two conditions regarding the existence of stabilizing switching laws between two unstable linear systems. One is effective for theoretical derivation, while the other is implementable, and a class of stabilizing switching laws have been explicitly constructed by Wicks et al. (1994). With the help of the equivalent relation, a condition for the existence of controllers and stabilizing switching laws between two unstabilizable linear control systems is then proposed. Then, the study is further extended to the issue concerning the construction of quadratically stabilizing switching laws among N unstable linear systems and N unstabilizable linear control systems. The obtained results are employed to study the existence of control laws and quadratically stabilizing switching laws within a class of unstabilizable linear control systems. The numerical examples are illustrated and simulated to show the feasibility and effectiveness of the proposed methods.

Suggested Citation

  • Sendren Sheng-Dong Xu & Chih-Chiang Chen, 2013. "On Existence of Stabilizing Switching Laws within a Class of Unstable Linear Systems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:681523
    DOI: 10.1155/2013/681523
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    References listed on IDEAS

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    1. Manlika Rajchakit & Grienggrai Rajchakit, 2012. "Mean Square Exponential Stability of Stochastic Switched System with Interval Time‐Varying Delays," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    2. Manlika Rajchakit & Grienggrai Rajchakit, 2012. "Mean Square Exponential Stability of Stochastic Switched System with Interval Time-Varying Delays," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-12, June.
    3. Shipei Huang & Zhengrong Xiang & Hamid Reza Karimi, 2013. "Stabilization and Controller Design of 2D Discrete Switched Systems with State Delays under Asynchronous Switching," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    4. Shipei Huang & Zhengrong Xiang & Hamid Reza Karimi, 2013. "Stabilization and Controller Design of 2D Discrete Switched Systems with State Delays under Asynchronous Switching," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-12, May.
    5. Xia Li & Hamid Reza Karimi & Zhengrong Xiang, 2013. "Output Feedback Control of Discrete Impulsive Switched Systems with State Delays and Missing Measurements," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-10, October.
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