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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 investigate the effect of Polyetheretherketone (PEEK) rod semi-rigid pedicle screw fixation system in lumbar spine non-fusion surgery., Methods: A total of 74 patients with tow-level lumbar degenerative diseases who underwent surgery from March 2017 to December 2019 were divided into PEEK rod group and titanium rod group. In the PEEK rod group, there were 34 patients, including 13 males and 21 females, aged from 51 to 79 years old with an average of (62.4±6.8) years old;There were 1 patient of L 1 -L 3 segments, 7 patients of L 2 -L 4 segments, 20 patients of L 3 -L 5 segments and 6 patients of L 4 -S 1 segments. In the titanium rod group, there were 40 patients, including 17 males and 23 females, aged from 52 to 81 years old with an average of (65.2±7.3) years old;There were 3 patient of L 1 -L 3 segments, 11 patients of L 2 -L 4 segments, 19 patients of L 3 -L 5 segments and 7 patients of L 4 -S 1 segments. The general conditions of operation, such as operation time, intraoperative blood loss, postoperative drainage was recorded. The visual analogue scale (VAS) for low back pain and Oswestry disability index (ODI) were compared in preoperatively and postoperatively(3 months, 12 months and last follow-up) between two groups. The change of range of motion (ROM) was observed by flexion and extension x-ray of lumbar., Results: All patients successfully completed the operation. The follow-up time ranged from 22 to 34 months with an average of(26.8±5.6) months. The operative time (142.2±44.7) min and intraoperative blood loss(166.5±67.4)ml in PEEK group were lower than those in titanium group [(160.7±57.3) min、(212.8±85.4) ml]( P <0.05). There was no significant differences in postoperative drainage between the two groups ( P >0.05). At the final follow-up visit, in PEEK group and titanium group VAS of low back pain[(0.8±0.4) points vs (1.0±0.5) points], VAS for leg pain [ (0.7±0.4) points vs (0.8±0.5) points] and ODI [(9.8±1.6)% vs (12.1±1.5)%] were compared with preoperative [ (5.8±1.1) points vs (6.0±1.1)points], [ (7.2±1.7) points vs (7.0±1.6) points], [(68.5±8.9)% vs(66.3±8.2)%] were significantly different( P <0.05). There was no significant difference in VAS scores between the two groups at each postoperative time point ( P >0.05). At 3 months after surgery, there was no difference in ODI between the two groups ( P >0.05). There were significant differences in ODI between PEEK group and titanium rod group at 12 months [(15.5±2.1)% vs (18.4±2.4)%] and at the last follow-up [(9.8±1.6)% vs (12.1±1.5)%] ( P <0.05). The total range of motion (ROM) of lumbar decreased in both groups after surgery. At 12 months after surgery and the last follow-up, the PEEK group compared with the titanium rod group, the total range of motion of lumbar was statistically significant ( P <0.05). The range of motion (ROM) of the fixed segments decreased in both groups after surgery. The ROM of the fixed segments in PEEK group decreased from (9.5±4.6)° to (4.1±1.9)° at the last follow-up ( P <0.05), which in the titanium rod group was decreased from (9.8±4.3)°to (0.9±0.5)° at the last follow-up ( P <0.05). The range of motion (ROM) of upper adjacent segment increased in both groups, there was statistical significance in the ROM of upper adjacent segment between the two groups at 12 months after surgery and the last follow-up, ( P <0.05). There was no screw loosening and broken rods in both groups during the follow-up period., Conclusion: The PEEK rod semi-rigid pedicle screw internal fixation system used in lumbar non-fusion surgery can retain part of the mobility of the fixed segment, showing comparable short-term clinical efficacy to titanium rod fusion. PEEK rod semi-rigid pedicle screw internal fixation system is a feasible choice for the treatment of lumbar spine degenerative diseases, and its long-term efficacy needs further follow-up observation.

Objective: To observe the early clinical effect of perfusion bone cement screw for lumbar degenerative diseases with osteoporosis., Methods: The clinical data of 28 patients with lumbar degenerative diseases combined with moderate to severe osteoporosis treated by posterior lateral graft fusion with perfusion of bone cement screws from June 2015 to June 2017 were retrospectively analyzed. There were 9 males and 19 females, aged from 55 to 86 years old with an average of 76 years. Anteroposterior, oblique, and dynamic radiography were performed before operation, and the diagnosis was confirmed by CT, MRI and dual energy X-ray absorptionmetry(DXA). All the patients had moderate to severe lumbar spinal stenosis, including 16 cases with degenerative scoliosis, 12 cases with degenerative lumbar spondylolisthesis, and 16 cases with lumbar disc herniation. According to Jikei grade of osteoporosis, 9 cases were grade II and 19 cases were grade III. Visual analogue scale (VAS) and Japanese Orthopaedic Association (JOA) score were used to assess the improvement of lumbar leg pain, and neurological function. Imaging data were used to observe the circumstance of pedicle loosening, prolapse, breakage and bone cement leakage, and comprehensively evaluate the fusion., Results: The hospital stay was from 10 to 14 days with an average of 12 days; the operative time was 100 to 150 min with an average of 120 min;the blood loss was 200 to 600 ml with an average of 350 ml (for operations more than 3 vertebral segments, blood filtration recovery was intraoperatively used);the postoperative drainage volume was 150 to 600 ml with an average of 300 ml, no allogeneic blood was used in all the patients. Bone cement of 2 to 3 ml were injected into each vertebral body, and bone cement leakage occurred in 2 cases during injection, both of which were paravertebral vessel leakage, and there was no evidence of intravertebral leakage. The injection of bone cement was terminated in a timely manner without serious complications such as nerve injury, bone cement toxicity, and vascular embolization and pulmonary embolism. According to the fusion criteria by X-ray, 18 cases achieved strong bone fusion, and 10 cases were inaccurate fusion, but no pseudarthrosis occurred in the fusion segment. No screw loosening, prolapse or fracture were found, and postoperative VAS, JOA scores were significantly improved., Conclusions: Perfusion bone cement screw technique can obtain satisfactory effect in treating lumbar degenerative diseases with osteoporosis., Competing Interests: The authors of this article and the planning committee members and staff have no relevant financial relationships with commercial interests to disclose., (Copyright© 2019 by the China Journal of Orthopaedics and Traumatology Press.)