Adaptive Super-Twist Control with Minimal Chattering Effect Vadim I. Utkin, Alex S. Poznyak and Patricio Ordaz Abstract— The, so-called, adaptive super-twist controller is considered here. It contains the adaptive gain-parameter which adjusts the level of a scalar control action on-line based on direct measurements the "equivalent control" obtained by a first- order low-pass filter. It is shown that the problems of the finite time convergence and keeping a small chattering amplitude can be handled simultaneously if the gain magnitude is reduced to a minimal admissible level defined by the conditions for sliding mode to exist. The suggested methodology is compared nu- merically with another schemes of a gain-adaptation (including σ-adaptation) showing a high effectiveness under a significant level of uncertainties and external disturbances. I. I NTRODUCTION A. Brief survey The basic idea of Adaptive Control Approach consists in designing the systems exhibiting the same dynamic prop- erties under uncertainty conditions based on utilization of current information. It involves modifying the control law used by a controller to cope with the fact that the parameters of the system being controlled are slowly time-varying or uncertain. Even more, adaptive control implies improving dynamic characteristics while properties of a controlled plant or environment are varying [1], [20]. Without adaptation the original Sliding Mode Control (SMC) demonstrate robustness properties with respect to parameter variations and disturbances [22]. The first attempts to apply ideas of adaptation in SMC were made in the 60’s [6], [7] and [8]: the control efficiency was improved by changing the position or equation of the discontinuity surfaces without any information on a plant parameters. The design idea might be formulated as follows: if sliding mode exists, then the coefficients of switching plane can be varied to improve the system dynamics. However those early publications did not take in to account the main obstacle for SMC application - the chattering phenomenon which is inherent in sliding motions (see, for example, [2], [3] and [4]). The phenomenon is well-known from literature on power converters and re- ferred to as “ripple” [16]. Then the efforts of the researchers were oriented to the application of adaptivity principles to reduce the effect of chattering. Since the amplitude of chattering is proportional to discontinuity magnitude in control, one of possible adaptation methods is related to reducing this magnitude to the minimum admissible value dictated by the conditions for SM to exist. So, in [17] the Vadim I. Utkin. is with Departament of Electrical and computer Engineering Ohio State Unviversity, Columbus, ohio, 43210, USA utkin@ece.osu.edu Alex S. Poznyak and Patricio Ordaz are with Automatic Control Departament, CINVESTAV-IPN, A.P. 14-740, C.P. 07360 Mexico D.F apoznyak@ctrl.cinvestav.mx control gain depended on the distance of system state to a discontinuity surface. The tracks of adaptivity can be found in the first publications about variable structure systems with SM (see [22], [23]) with the control gain proportional the system state. Similar ideas were developed later in [11] with different algorithms of tuning a control gain. There exist also adaptive SMC (ASMC) algorithms that allow adjusting dynamically the control gains without knowledge of uncertainties/perturbations bounds. In particular, several adaptive fuzzy SMC algorithms were proposed. However, they do not guarantee the tracking performance (see [15], [19] or overestimate the switching control gains as in [10]). Of course, another efficient tool to suppress chattering is the application of state observers [5], but for this method the plant parameters are assumed to be known. B. Motivation and the design idea In [17] and [21] the adaptation process with the varying magnitude of the control gain terminates at the moment when the sliding mode starts. In [11] the authors tried to con- tinue the adaptation process during sliding mode estimating equivalent control. However, none of the above algorithms resulted in minimum possible value of the discontinuous con- trol. Finding the solution of this problem under uncertainty conditions is the objective of this paper. This leads to the minimization of chattering effect. C. Primitive example We start with a simple example. It is evident that for the first-order system ˙ x (t)= a + u u = −ksign (x (t)) ,k> 0 (1) with known range only 0 < |a|≤ a + of a constant parameter a. If the value of a is unknown, the magnitude of control is selected such that sliding mode exists for the all values of unknown parameter k>a + . However if parameter a is varying, the gain k can be decreased and as a result chattering amplitude can be reduced. The objective of adaptation is decreasing k to the minimal value preserving sliding mode, if parameter a is unknown. If the condition k>a + holds, then sliding mode with x (t) ≡ 0 occurs and control in (1) should be replaced by the, so-called, equivalent control [22] u eq for which the right- hand side in (1) is equal to zero, namely, ˙ x (t)=0= a + u eq (2) that leads to k [sign (x (t))] eq = a (3) 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) Orlando, FL, USA, December 12-15, 2011 978-1-61284-799-3/11/$26.00 ©2011 IEEE 7009