Article Transactions of the Institute of Measurement and Control 1–13 Ó The Author(s) 2018 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0142331218785694 journals.sagepub.com/home/tim Non-smooth convex Lyapunov functions for stability analysis of fractional-order systems Aldo-Jonathan Mun ˜ oz-Va ´zquez 1 , Vicente Parra-Vega 2 and Anand Sa ´nchez-Orta 2 Abstract Based on proximal subdifferentials and subgradients, and instrumented with an extended Caputo differintegral operator, the stability analysis of a gen- eral class of fractional-order nonlinear systems is considered by means of non-smooth but convex Lyapunov functions. This facilitates concluding the Mittag–Leffler stability for fractional-order systems whose solutions are not necessarily differentiable in any integer-order sense. As a solution to the problem of robust command of fractional-order systems subject to unknown but Lebesgue-measurable and bounded disturbances, a unit-vector-like integral sliding mode controller is proposed. Numerical simulations are conducted to highlight the reliability of the proposed method in the analysis and design of fractional-order systems closed by non-smooth robust controllers. Keywords convex Lyapunov functions, fractional sliding mode control, fractional-order systems, Mittag–Leffler stability Introduction Fractional calculus foundation is as old as the conventional integer-order calculus (Podlubny, 1999); moreover, despite there having been reported recent impressive advances in the- ory and applications (Chen K et al., 2017; Efe, 2011; Stamova et al., 2017), there remains some fundamental open problems in modelling, control and stability analysis for fractional-order nonlinear systems. The importance of frac- tional calculus becomes evident when studying some charac- teristics of advanced physical phenomena, which are better and more deeply understood by considering inherent features of fractional operators, such as non-locality, memory and heritage. Nevertheless, the most popular differintegral opera- tors still consider integer-order derivatives in their definitions (Podlubny, 1999), thus limiting and compromising some applications of the fractional-order techniques. On stability of fractional-order systems Stability analysis in fractional-order systems involves model- ling and control, and is one of the most important issues. Internal stability in fractional-order time-invariant systems has been addressed by Matignon (1996), with the analysis for rational-order systems by Petra´ sˇ (2009). In addition, the case of fractional-order nonlinear systems was studied by Li et al. (2009, 2010) and Zhang et al. (2011) by extending the Lyapunov direct method – more precisely, by extending the particular case of exponential stability to the more general notion of Mittag–Leffler stability, but without designing the Lyapunov function and therefore making the method unsui- table for several applications. A big step was taken by Alikhanov (2011), Aguila- Camacho et al. (2014) and Duarte-Mermoud et al. (2015), by demonstrating a fractional-order differential inequality for the Caputo operator, later extended to the Riemann-Liouville case by Liu et al. (2016), based on quadratic Lyapunov func- tions to prove asymptotic stability of fractional-order non- linear systems equilibria. However, this explicitly required the differentiability of the pseudo-state. Moreover, the method of Aguila-Camacho et al. (2014) and Duarte-Mermoud et al. (2015) for analysing Mittag– Leffler stability using quadratic Lyapunov functions, based on the work of Li et al. (2009), Li et al. (2010) and Zhang et al. (2011), is too conservative, and stands for a particular case of Ding et al. (2015) that extends the inequality demon- strated by Alikhanov (2011), Aguila-Camacho et al. (2014) and Duarte-Mermoud et al. (2015) to the case of more gen- eral Lyapounov functions of even degrees. In this direction, a great and promising advance on stabi- lity analysis of fractional-order systems, based on the classical 1 Mechatronic Engineering, Polytechnic University of Victoria, Ciudad Victoria, Tamaulipas, Mexico 2 Robotics and Advanced Manufacturing, Research Center for Advanced Studies, Saltillo, Coahuila, Mexico Corresponding author: Aldo-Jonathan Mun ˜oz-Va ´zquez, Mechatronic Engineering, Polytechnic University of Victoria, Ciudad Victoria, Tamaulipas, Mexico Email: aldo.munoz.vazquez@gmail.com