Improving the predictions of the circle criterion by combining quadratic forms
HENRY M. POWER, Ah Chung Tsoi · IEEE Transactions on Automatic Control · 1973
By using a Lyapunov function which consists of different quadratic forms in various sectors of the (u, (du/d\tau)) plane, the prediction of the circle criterion that the null solution of(d^{2}u/d\tau^{2}) + 2(du/d\tau) + f(\tau, u, (du/d\tau))\cdotp u = 0is asymptotically stable for0 \leq \alpha < f(\cdotp) < \beta, with\beta = (\sqrt{alpha} + 2)^{2}, is improved to\beta = \[\{\frac{(\sqrt{\alpha} + 1)^{2} + 1 + \sqrt{(\sqrt{\alpha} + 1)^{4} + 2 (\sqrt{\alpha} + 1)^{2} + 5}}{2}\}^{\frac{1}{2}} + 1 \]^{2}.