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研究了二阶线性系统族共同二次Lyapunov函数的存在性问题。系统模型为∑Ai:x(t)=Aix(t),其中Ai∈R2×2为Hurwitz常矩阵。借助Lyapunov稳定性理论和矩阵理论,对子系统矩阵包含有限个对角阵和有限个具有复数特征值矩阵的二阶系统族,分3种情况证明了其存在共同二次Lyapunov函数,将存在共同二次Lyapunov函数的充分条件转化为若干个代数不等式,并基于定理的证明过程给出了一个共同二次Lyapunov函数的求法。验证该充分条件容易在计算机上编程实现,从而具有较强的工程实用性。最后通过数值算例来验证了该充分条件的有效性以及更低的保守性。
We study the existence of the common quadratic Lyapunov functions for families of second-order linear systems. The system model is ΣAi: x (t) = Aix (t), where Ai∈R2 × 2 is the Hurwitz constant matrix. By means of Lyapunov stability theory and matrix theory, the subsystems matrix is composed of a finite number of diagonal arrays and a finite number of second-order system families with complex eigenvalue matrices. The existence of common quadratic Lyapunov functions is proved in three cases The sufficient condition of the quadratic Lyapunov function is transformed into several algebraic inequalities, and a common quadratic Lyapunov function is given based on the proof of the theorem. Verify that the sufficient conditions are easy to program on the computer, which has strong engineering practicability. Finally, a numerical example is given to verify the validity of this sufficient condition and lower conservativeness.