946 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 2004 Technical Notes and Correspondence_______________________________ Analysis of Second-Order Sliding-Mode Algorithms in the Frequency Domain I. Boiko, L. Fridman, and M. I. Castellanos Abstract—A frequency domain analysis of the second-order sliding-mode algorithms, particularly of the twisting algorithm is carried out in the frequency domain with the use of the describing function method and Tsypkin’s approach. It is shown that in the presence of an actuator, the transient process may converge to a periodic motion. Param- eters of this periodic motion are analyzed. A comparison of 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. Index Terms—Chattering, relay control, sliding-mode (SM) , variable structure systems. I. INTRODUCTION Higher order sliding modes (SM) have received a lot of attention from the control research community over the last decade (see the bibliographies in [1]–[9]). The main advantages of the higher order sliding-mode algorithms are: A higher accuracy of resulting motions; the possibility of using continuous control laws (super twisting or twisting as a filter); the possibility of utilizing the Coulomb friction in the control algorithm [7]; and finite time convergence for the systems with arbitrary relative degree. It is known that the first order SM in systems with actuators of rel- ative degree two or higher is realized as chattering [2], [9]. For the same reason, it would be logical to expect a similar behavior from a second-order SM, as the aforementioned algorithms contain the sign function. The modes that occur in a relay feedback system with the plant being the order 1, 2, 3, etc. dynamics were studied in publica- tions [10], [11]. 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 equilib- rium point. Similar behavior, therefore, can be expected from a system with a second-order SM algorithm. Thus, the objective of this note is to analyze the motions that occur in a system with one of the most pop- ular second-order SM algorithm—the twisting algorithm, to show the existence of periodic motions, to estimate the parameters of those mo- tions, and to compare the latter with the parameters of chattering in the systems with asymptotic second-order relay control [6], [10], [11] and first-order SM control [12]. Given the objective of the outlined analysis and the fact that the in- troduction of an actuator increases the order of the system, the analysis of corresponding Poincare maps becomes complicated. In this case, the describing function (DF) method [13] seems to be a good choice. The DF method provides a simple and efficient solution of the problem. However, the DF method provides only an approximate solution and Manuscript received June 14, 2003; revised October 19, 2003 and February 19, 2004. Recommended by Associate Editor M. Reyhanoglu. I. Boiko is with the SNC-Lavalin, Calgary, AB T2P 3G5, Canada (e-mail: i.boiko@ieee.org). L. Fridman and M. I. Castellanos are with the División de Posgrado, Facultad de Ingeniería, the National University of Mexico, Edificio A, Ciudad Universitaria, C.P. 70-256 México City, México (e-mail: lfridman@verona.fi-p.unam.mx). Digital Object Identifier 10.1109/TAC.2004.829615 for that reason the Tsypkin’s method [10] modified to accommodate the analyzed problem is used for the analysis as well. The latter does not require involvement of the filter hypothesis and provides an exact solution of the problem of finding the parameters of self-excited oscil- lations in a relay feedback system. However, the twisting algorithm is not equivalent to the relay feedback control and some modifications to the Tsypkin’s method need to be done to accommodate this method to the analyzed algorithm. The note is organized as follows. At first the model of the system in- volving the twisting algorithm suitable for the frequency domain anal- ysis is obtained. Then, DF model and the model suitable for deriving the Tsypkin’s locus of the system are built. After that the DF analysis and the exact analysis of the system with the twisting algorithm are considered. Finally, a number of examples are considered and a com- parison is done. II. TWISTING ALGORITHM AND ITS DF ANALYSIS The twisting algorithm is one of the simplest and most popular algo- rithmsamongthesecond-ordersliding-modealgorithms.Therearetwo ways to use the twisting algorithm [8]: to apply it to a plant of relative degree two, or to apply it to a plant of relative degree one and intro- duce an integrator in series with the plant (twisting-as-a-filter). For the plants of relative degree two it can be formulated as follows. Let the plant (or the plant plus actuator) be given by the following differential equations: (1) where is an -dimenaionsl state vector, is a scalar control, and are matrices of respective dimensions, and is scalar and can be treated as either the sliding variable or the output of the plant. Also, let the control of the twisting algorithm be given as follows [5], [8]: (2) where and are positive values, . Assume that a peri- odic motion occurs in the system with the twisting algorithm. Then the system can be analyzed with the use of the DF method. As normally accepted in the DF analysis, we assume that the plant has a magnitude characteristic of a low-pass filter. Find the DF of the twisting algo- rithm as the first harmonic of the periodic control signal divided by the amplitude of —in accordance with the definition of the DF [13] where istheamplitudeoftheinputtothenonlinearity(of inour case) and is the frequency of . However, the twisting algorithm can be analyzed as the parallel connection of two ideal relays where the input to the first relay is the sliding variable and the input to the second relay is the derivative of the sliding variable. The DF for those nonlin- earities are known [13]. For the first relay the DF is: , and for the second relay it is: , where is the ampli- tude of . Also, take into account the relationship between and in the Laplace domain, which gives the relationship between the amplitudes and , where is the frequency of the 0018-9286/04$20.00 © 2004 IEEE