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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]

Summary: This paper focuses on the stability analysis for fractional‐order neural networks with time‐varying delay. A novel Lyapunov's asymptotic stability determination theorem is proved, which can be used for fractional‐order systems directly. Different from the classical Lyapunov stability theorem, constraint condition on the derivative of Lyapunov function is revised as an uniformly continuous class‐K function in the fractional‐order case. Based on this novel Lyapunov stability theorem and free weight matrix method, a new sufficient condition on Lyapunov asymptotic stability of fractional‐order Hopfield neural networks is derived by constructing a suitable Lyapunov function. Moreover, two numerical examples are provided to illustrate the effectiveness of these criteria. [ABSTRACT FROM AUTHOR]

Objective: To provide the cobweb classification system (CCS) for the precise digital location and description of the neurological compression in cervical degenerative disease (CDD), and the reliability and the clinical subgroup analysis of the system were tested and analyzed. Methods: The CCS consisted of three parts: compression zones (1‐12), degrees (a, b) and ossification (s, m, h). Computerized tomography (CT) and magnetic resonance imaging (MRI) images from 238 CDD patients were reviewed. All compression cases were classified by five independent reviewers with varied clinical experience in spine surgery. The reliability of the CCS was tested by calculating the kappa (κ) statistics value. Finally, 74 patients with anterior cervical surgery treatment were enrolled for the clinical subgroup analysis. Results: For the small compression, including single and double compression zones, there was a good interobserver reliability between the reviewers (κ coefficient =.855, P <.001). For the large compression with three or more involved zones, there was a fair reliability between the reviewers (κ coefficient =.696, P <.001). The whole intraobserver reliability was good (κ coefficient =.923, P <.001). For clinical practice, the operative time in the large compression and the m/h group was significantly longer than the small compression and the s group, respectively (P <.05), and the blood loss in the m/h group was significantly increased as well (P <.01). Though the preoperative Japanese Orthopedic Association score in Group b was lower than Group a (P <.05), all patients had achieved significant clinical improvement at last follow‐up. Conclusions: The CCS can be used to provide detailed and objective descriptions of the location, extent, and severity of neurological compressions in CDD with satisfactory reliability. Surgeons should pay more attention to the patient with large zone, degree b, and ossification compression, because the operation may be more challenging. [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]

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]