This article investigates exact robust stability bounds of output feedback controlled fractional-order systems with the commensurate order ∈ [1, 2) and single parameter perturbations in all system coefficient matrices. First, a sufficient and necessary condition for robust asymptotical stability of such systems is obtained by using the Kronecker product. Then the maximal upper bounds and minimum lower bounds for robust asymptotical stability are established, respectively, without conservatism by transforming such problems into checking whether the matrix with single parameter perturbations is nonsingular or not. Finally, two numerical examples are given to show the effectiveness of the proposed results.

BACKGROUND Lumbar disc herniation (LDH) has emerged as one of the most common causes of low back pain. The routine treatment approach involves chemonucleolysis therapy, discectomy by percutaneous endoscopy, and percutaneous laser disc decompression. Unfortunately, all of these methods carry inherent risk of causing harm to the patient and, as such, there is an unmet but urgent need for an effective and safe noninvasive treatment for LDH. The purpose of this report is to describe a non-invasive method for re-absorption of LDH. CASE SUMMARY A 34-year-old woman was admitted with a complaint of waist pain that she reported as having become acutely aggravated over the past 3 d and accompanied by discomfort in the right lower limb. Her self-reported medical history included persistent postpartum low back pain from 7 years prior. Physical exam showed positivity for neck flexion test (Lindner sign) and supine abdomen test; the straight leg-raising test showed right 60(+) and left 80(-). Findings from standard imaging (magnetic resonance) and collective physical examinations indicated a L5/S1 herniated lumbar disc. Treatment consisted of three-dimensional (balanced regulating) spinal manipulation and acupuncture, upon which the LDH resolved by retraction. CONCLUSION Following L5/S1 herniated lumbar disc diagnosis, three-dimensional (balanced regulating) spinal manipulation combined with acupuncture therapy is an effective treatment.