A new class of fast finite-time discontinuous controllers Emmanuel Cruz-Zavala 1 and Jaime A. Moreno 2 Abstract— We introduce a new class of high-order sliding mode (HOSM) controllers with improved convergence rate for a single-input uncertain system. The controllers are designed by means of Control Lyapunov Functions (CLF’s) and properties of homogeneous systems in the bi-limit. Therefore, the proposed controllers in feedback with the system make the close-loop sys- tem homogeneous in the bi-limit. Robustness and convergence rate are improved in comparison with the reported HOSM. I. INTRODUCTION Sliding mode control (SMC) is considered to be effective to deal with uncertainties and disturbances in nonlinear dynamical systems.The classical SMC design is based on finding a discontinuous control input to drive the state trajec- tories to reach the so called sliding surface in finite time [18]. The sliding surface is a manifold previously designed to at- tain some dynamic performance requirements. Unfortunately, standard SMC has two drawbacks: it is restricted to systems having relative degree equal to one and it yields the well- known chattering effect, a phenomenon of high-frequency and finite-amplitude oscillations of the controlled system provided by the use of a first-order sliding mode (FOSM) which degrades the performance in practical applications. Higher-order sliding modes HOSM’s [11] preserve the main robustness properties of standard SMC but additional advantages are obtained. For instance, HOSM’s can alleviate the chattering effect, finite time convergence to the origin is ensured and better accuracy with respect to discrete sampling time is attained [12]. In contrast to the classical SMC design, the HOSM controllers are constructed by means of homogeneity tools. However, HOSM can deal only with bounded perturbations and the convergence rate is slower when the trajectories are far away from the origin. Fast Terminal Sliding Mode based Control (FTSMC) can deal with these problems. However, it has a stronger restriction for applicability purposes as the Terminal Slid- ing Mode based Control (TSMC), [14], [19]. Since the control signal becomes unbounded for some sets, the state trajectories need to start in a prescribed sector of the state space in order to avoid an unbounded control signal. To overcome this drawback, the Nonsingular Fast Terminal Sliding Mode based Control (NFTSMC) rewrites the well- known fast terminal sliding surface (FTSS) to obtain non singular control inputs,[13]. The terminal controllers are based on the standard SMC design and it has been suitable *Financial support from CONACyT CVU 267513, PAPIIT, UNAM, grant IN113614, and Fondo de Colaboraci´ on del II-FI,UNAM, IISGBAS-109- 2013, is gratefully acknowledged. E. Cruz-Zavala (emitacz@yahoo.com.mx) and J.A. Moreno (JMorenoP@ii.unam.mx) are with the Instituto de Ingenier´ ıa, UNAM. for second order systems. However, extensions to obtain fast nonsingular control inputs for arbitrary order systems seem to be difficult under this approaches. New strategies need to be taken into account in order to develop a new class discontinuous controllers with fast finite-time convergence. It is a fact that, Lyapunov’s methods together with homogeneity theory led to a recursive design of homogeneous finite-time continuous state feedback con- trollers for a class of nonlinear dynamical system [8], [9]. Both methodologies play a main role in analysis and design for nonlinear systems. The paper is focused on designing a new class of HOSM controllers for a nonlinear perturbed dynamical system with a single control input. An improvement of the convergence rate and robustness properties to certain class of perturbations are attained by the proposed family of controllers. This new class of discontinuous controllers is, in general, nonhomogeneous and has new features. The controllers are discontinuous on continuous manifolds, which are complectly different from those described in HOSM, [12]. The controllers are obtained by means of Control Lyapunov Functions (CLF). This functions are designed by using the homogeneous in the bi-limit properties, [1]. A Lyapunov’s design provides a way to estimate the convergence time and leads to find sufficient condition for the tuning of the controller’s gain. With an appropriated selection on the parameters, we can recover homogeneous finite-time controllers and homoge- neous controllers with exponential convergence from the same control law structure. II. PRELIMINARIES In this section, we present some important definitions. Let R ≥0 := [0, ∞) and ⌈·⌋ m := |·| m sign (·), for any m ≥ 0. This operator preserves the sign of the value of the functions. According to this ⌈x⌋ 0 = sign (x), d dx ⌈x⌋ m = m|x| m−1 and d dx |x| m = m|x| m−1 sign (x)= m⌈x⌋ m−1 . Note that ⌈x⌋ m = x m , for any odd integer m. A. Standard Homogeneity Properties We recall the concepts of continuous homogeneous func- tions and vector fields from [2]. The latter concept has been extended to (Filippov) Differential Inclusions in [12]. Definition 1: Let Δ r ǫ x := (ǫ r 1 x 1 , ..., ǫ r n x n ) = diag(ǫ ri )x be the dilation operator for any ǫ > 0 and ∀x ∈ R n , where the weights of the coordinates r i > 0 can be grouped in the vector of weights r =(r 1 , ..., r n ). 1) A function V : R n → R is called r-homogeneous of degree m ∈ R with respect to (w.r.t.) Δ r ǫ x, if the identity V (Δ r ǫ x)= ǫ m V (x),∀x ∈ R n , holds. 13th IEEE Workshop on Variable Structure Systems, VSS’14, June 29 -July 2, 2014, Nantes, France. 978-1-4799-5566-4/14/$31.00@2014 IEEE