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In this article, the sliding mode control (SMC) of fractional nonlinear systems (FNSs) with disturbances di(t)$$ {d}_i(t) $$ is studied. First, a useful fractional power‐rate inequality DαV(x(t))≤−μVβ(x(t))$$ {D}^{\alpha }V\left(x(t)\right)\le -\mu {V}^{\beta}\left(x(t)\right) $$ for β∈(0,1)$$ \beta \in \left(0,1\right) $$ is extended to a more general form β>0$$ \beta >0 $$. Based on the generalized inequality, the method of a modified fractional integral SMC is completed, in which the parameter of the symbolic function is extended to ς>1/2$$ \varsigma >1/2 $$ and γ>1/2$$ \gamma >1/2 $$. This increases the degree of freedom of the sliding mode surface (SMS). Then, the SMC of FNSs is studied for the cases of known and unknown disturbances. In the case of known disturbance, it is proved by the generalized inequality and quadratic Lyapunov function method that the state of the FNSs can converge asymptotically to zero on the SMS. For the case of unknown disturbance, a fractional disturbance observer is used to estimate the disturbance by introducing auxiliary variables zi(t)=di(t)−δxi(t)$$ {z}_i(t)={d}_i(t)-\delta {x}_i(t) $$. The disturbance estimation error d˜i(t)$$ {\tilde{d}}_i(t) $$ of the proposed FNSs is proved to be bounded. An equivalent global control law is obtained and asymptotic convergence of the FNSs under the action of the controller is proved. Finally, two numerical simulation examples are given to verify the feasibility of the method proposed in this article. [ABSTRACT FROM AUTHOR]

Abstract In this paper, based on the asymptotic expansion of Mittag-Leffler function and the fractional comparison principle, an improved fractional Halanay inequality with time-varying coefficients is proved by introducing parameters λ i and δ. The fractional autonomous Halanay inequality is generalized to the fractional non-autonomous case. Moreover, based on the improved fractional Halanay inequality and Lyapunov functional method, a novel sufficient condition on self synchronization of the fractional non-autonomous Hopfield neural networks with time delay is obtained. Finally, three numerical examples are given to demonstrate the effectiveness of proposed methods. [ABSTRACT FROM AUTHOR]

This paper is concerned with the exponential stability of delayed neural networks. Combined the Legendre polynomials with free-weighting matrices technique, an improved free-matrix-based single integral inequality is given, which includes the general single integral inequality and the free-matrix-based single integral inequality as special cases. Furthermore, a free-matrix-based double integral inequality which improves the existing results is derived. As applications of these novel free-matrix-based integral inequalities, several exponential stability criteria with less conservatism for the delayed neural networks are obtained. The effectiveness of our main results is illustrated by three numerical examples from the literatures. [ABSTRACT FROM AUTHOR]

In this paper, not only some novel sufficient criteria for stability of discrete delay neural networks but also a new discriminant method for self synchronization of Hopfield neural networks are considered. Some novel stability criteria for discrete delay neural networks are obtained if the time average of the difference for two coefficients on some fixed lengths is lower bounded by some positive numbers. Moreover, a new discrete Halanay inequality is given, which extends the existing results. Based on the Halanay inequality, a novel sufficient criterion on self synchronization of Hopfield neural networks with time delay is obtained. Two numerical examples are given to demonstrate the effectiveness of proposed methods. [ABSTRACT FROM AUTHOR]

Auxiliary function-based summation inequalities are addressed in this technical note. By constructing appropriate auxiliary functions, several new summation inequalities are obtained. A novel sufficient criterion for asymptotic stability of discrete-time systems with time-varying delay is obtained in terms of linear matrix inequalities. The advantage of the proposed method is demonstrated by two classical examples from the literature. [ABSTRACT FROM AUTHOR]

In this paper, based on the discrete Wirtinger inequality, a novel summation inequality is established which extends the Jensen inequality. By the technique of the novel inequality, a sufficient criterion on asymptotic stability of discrete neural networks with time-varying delays is obtained. A numerical example is given to demonstrate the effectiveness of the inequality. [ABSTRACT FROM AUTHOR]

This paper focuses on the issue of robust stability analysis for recurrent neural networks (RNNs) with leakage delay. By constructing a novel Lyapunov-Krasovskii functional together with the reciprocally convex approach and the free-weighting matrix technique, some less conservative stability criteria in terms of linear matrix inequalities for RNNs are derived. The new contribution of this paper is that a novel delay-partitioning method is proposed, and some new zero equalities are introduced. Finally, several examples are given to demonstrate the effectiveness of the proposed methods. The simulated results reveal that the leakage delay has great influence on the dynamical systems, and it cannot be neglected. [ABSTRACT FROM AUTHOR]

This paper focuses on the stability and passivity analysis for a class of discrete-time linear systems. A novel general free-matrix-based summation inequality is proposed, and two improved stability criteria in terms of linear matrix inequalities (LMIs) are derived by employing the new summation inequality. Furthermore, when an external disturbance appears, a sufficient condition is established to ensure the considered linear system to be passive. Three numerical examples are provided to demonstrate the improvement of the proposed method. [ABSTRACT FROM AUTHOR]

Self-locking stand-alone cages can achieve satisfactory clinical results and fusion rate. However, there have been no reports on the causes and relationship of different fusion state. This study is to classify the different fusion states of the index level and to explore the potential contributing factors and links of them. From June 2008 to October 2011, 42 patients underwent anterior cervical discectomy and fusion with MC+ cages. More than 5 years' follow-up was reviewed. The fusion state and the relevant clinical and radiologic records were reviewed retrospectively. At the last follow-up, the fusion proportion of type I, II, III, and IV was 11.7%, 16.9%, 26.9%, and 42.9%, respectively. The overall fusion rate was 97.4%. For all the fused types, significant improvement for the visual analog scale, Japanese Orthopaedic Association, and Neck Disability Index scores was found at the last follow-up (P < 0.05). However, there were no significant differences between the 4 types (P > 0.05). For sagittal vertical axis, type IV was significantly larger than that of type I, II, and III (P < 0.05), and for range of motion, type III was significantly larger than that of type II and IV (P < 0.05). For anterior cervical discectomy and fusion with self-locking stand-alone cages, the fusion of the index level seems to be a progressive dynamic process during the mid-term follow-up, which may be influenced by the location of the cage, the aagittal vertical axis of the index level, and the global range of motion of the cervical spine. Satisfactory clinical results could be achieved by all the fused types. [ABSTRACT FROM AUTHOR]