Analysis of Steady State Behavior of Second Order Sliding Mode Algorithms I.Boiko, L. Fridman, R. Iriarte Abstract—An analysis of a second order sliding mode algorithm known as the super-twisting algorithm is carried out in the frequency domain with the use of the describing function method. It is shown that in the presence of an actuator, the transient process converges to a periodic motion. Parameters of this periodic motion are analyzed. A comparison between the periodic solutions in the systems with higher order sliding mode controllers and the oscillations that occur in classical sliding mode systems with actuators is done. I. INTRODUCTION H IGHER order sliding modes (SM) have received a lot of attention from the control research community over the last decade (see bibliography in [1-11]. The main reasons for the use of the higher order sliding mode algorithms are: a higher accuracy of resulting motions; the possibility of using a continuous control law (super twisting or twisting as a filter); the possibility of utilizing the Coulomb friction in the control algorithm [7]; the finite time convergence for the systems with arbitrary relative degree [1]. It is known that the first order SM in systems with actuators of relative degree two or more is realized as chattering [10,11]. For the same reason, it would be logical to expect a similar behavior from a real second order SM, as the second order SM algorithms contain the sign function or the infinite gain. The modes that occur in a relay feedback system with the plant being the order 1, 2, 3, etc. dynamics were studied in publications [12,13]. It has been proven in those works that for the plant of order 3 and higher the point of the origin cannot be a stable equilibrium point. A similar behavior, therefore, can be expected from a system with a second order SM algorithm. Thus, the objective of this paper is to analyze the motions that occur in a system with the super-twisting algorithm, to show the existence of the periodic motions, to assess the parameters of those motions to be able to generate requirements to the actuator dynamics, and to compare those parameters with the parameters of chattering in the corresponding first order SM [14] algorithms. I. Boiko is with SNC-Lavalin, 909 5th avenue SW, Calgary, Alberta, T2P 3G5, Canada (email: iboiko@ieee.org). L. Fridman and R. Iriarte are with Universidad Nacional Autónoma de México, UNAM, Facultad de Ingeniería, Ciudad Universitaria, CP 04510 Mexico City, Mexico (emails: lfridman@verona.fi-p.unam.mx ririarte@dctrl.fi-b.unam.mx). Given the objective of the outlined analysis and the facts that the introduction of an actuator increases the order of the system, and at least two nonlinearities are present in a second order SM algorithm, the analysis of corresponding Poincare maps becomes very complicated. In this case the describing function (DF) method [15] seems to be a good choice as a method of analysis, as it provides a relatively simple and efficient solution of the problem. The paper is organized as follows. At first the model of the system involving the super-twisting algorithm suitable for the frequency domain analysis is obtained. Then the DF model of the algorithm is obtained. After that it is shown that a periodic motion occurs and the problem of finding the parameters of this periodic motion is considered. Finally, a number of examples are considered and a comparison is done page limits. II. SUPER-TWISTING ALGORITHM AND ITS DF ANALYSIS The super-twisting algorithm is one of the popular algorithms among the second order sliding mode algorithms. It is used for the plants with relative degree one. Let the plant (or plant plus actuator) be given by the following differential equations: Cx y Bu Ax x = + = & (1) where A and B are matrices of respective dimensions, y can be treated as either the sliding variable or the output of the plant. We shall also use the plant description in the form of a transfer function W(s), which can be obtained from the formulas (1) as follows: B A Is C s W 1 ) ( ) ( − − = The control u for the super-twisting algorithm is given as Proceeding of the 2004 American Control Conference Boston, Massachusetts June 30 - July 2, 2004 0-7803-8335-4/04/$17.00 ©2004 AACC WeA19.4 632