A Bernstein Polynomial Approach to Estimating Reachable Set of Periodic Piecewise Polynomial Systems

• Published in: IEEE transactions on automatic control Vol. 66; no. 10; pp. 4812 - 4819
• Main Authors: Xie, Xiaochen, Fan, Chenchen, Kwok, Ka-Wai, Lam, James
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.10.2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Bernstein polynomial; Estimation; Functions (mathematics); Mathematical analysis; periodic piecewise polynomial systems (PPPSs); Polynomial matrices; reachable set estimation; Subsystems; Switches; Symmetric matrices; Technological innovation; Thermal stability; Time-varying systems
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2020.3037041

In this article, a Bernstein polynomial approach is first applied to the estimation of reachable set for a class of periodic piecewise polynomial systems, whose subsystems are time-varying and can be expanded to Bernstein polynomial forms. A lemma on the negativity/positivity for a class of Bernstein polynomial matrix functions is presented, which can provide a feasible set larger than that by the existing method. Based on the integration of the presented lemma and the theory of matrix polynomials, two tractable sufficient conditions are developed. For comparison of conservatism, the reachable set estimation is achieved through optimizing the ellipsoidal bounding region. Four sets of constraints with different conservatism are derived and compared. The effectiveness and superiority of the Bernstein polynomial approach in reachable set estimation are demonstrated via an illustrative example. The results show that the proposed approach enables lower conservatism in reachable set estimation, providing an intuitive route to tackle time-varying parameter products with high powers.