Finite-Time Stability and High Relative Degrees in Sliding-Mode Control

Abstract

Abstract. Establishing and exactly keeping constraints of high relative degrees is a central problem of the modern sliding-mode control. Its solution in finite-time is based on so-called high-order sliding modes, and is reduced to finite-time stabilization of an auxiliary uncertain system. Such stabilization is mostly based on the homogeneity approach. Robust exact differentiators are also developed in this way and are used to produce robust output-feedback controllers. The resulting controllers feature high accuracy in the presence of sampling noises and delays, ultimate robustness to the presence of unaccounted-for fast stable dynamics of actuators and sensors, and to small model uncertainties affecting the relative degrees. The dangerous types of the chattering effect are removed artificially increasing the relative degree. Parameters of the controllers and differentiators can be adjusted to provide for the needed convergence rate, and can be also adapted in real time. Simulation results and applications are presented in the fields of control, signal and image processing.

Monitoring function based extremum seeking control for uncertain relative degrees with light source seeking experiments

“…For example any actuator or sensor can always be integrated in the plant drastically changing the relative degree [30], which corroborates with the relevance of the proposed control strategy. A known weak point of the standard sliding mode based control approaches is the requirement that the relative degree of the sliding variable be well-defined, constant and known.…”

“…A known weak point of the standard sliding mode based control approaches is the requirement that the relative degree of the sliding variable be well-defined, constant and known. Any small perturbation or model inaccuracy can lead to the decrease of the relative degree, or even to its disappearance [30], [31].…”

Finite-time control of robotic manipulators

“…In general, in that case, only local uncertainty suppression is available in the literature for multi-input systems. In such a context, a class of gain-function robust controllers with single input and adjustable amplitude was recently proposed in works Levant (2011) and Levant and Livne (2012) to overcome globally unbounded uncertainty problem.…”

“…Pydynowski, 1996;Ferrara & Capisani, 2011;Levant, 1998Levant, , 2003Levant, , 2005Levant, , 2011Levant & Michael, 2009;Mondal & Mahanta, 2014;Shtessel, Shkolnikov, & Brown, 2003;Sira-Ramírez, 1992). Nevertheless, the approaches from Bartolini, Ferrara, Punta (2000), Bartolini, Ferrara, Usai et al (2000), Bartolini et al (2003), Bartolini and Pydynowski (1996), Ferrara and Capisani (2011), Mondal and Mahanta (2014), Shtessel et al (2003) and Sira-Ramírez (1992) are able to steer a tracking error to zero asymptotically and those from Levant (1998Levant ( , 2003Levant ( , 2005Levant ( , 2011 and Levant and Michael (2009) are only applicable to single input dynamic systems. In order to both increase tracking accuracy and accelerate a convergence process to the stable equilibrium, terminal sliding mode (TSM) control techniques have been offered as a particularly useful tool for high precision control of robotic manipulators.…”

Robust Task Space Finite-Time Chattering-Free Control of Robotic Manipulators

“…In general, in that case, only local uncertainty suppression is available in the literature for multi-input systems. In such a context, a class of gain-function robust controllers with single input and adjustable amplitude was recently proposed in works [9,11] to overcome globally unbounded uncertainty problem. -Remark 3.…”