A method to construct a family of strict Lyapunov functions, i.e., with negative definite derivative, for the super-twisting algorithm, without or with perturbations, is provided. This second order sliding modes algorithm is widely used to design controllers, observers and exact differentiators. The proposed Lyapunov functions ascertain finite time convergence, provide an estimate of the convergence time, and ensure the robustness of the finite-time or ultimate boundedness for a class of perturbations wider than the classical ones for this algorithm. Since the Lyapunov functions and their derivatives are quadratic forms, the operation with them is as simple as for linear time invariant systems.

Strict Lyapunov Functions for the Super-Twisting Algorithm

Distributed consensus tracking is addressed in this paper for multi-agent systems with Lipschitz-type node dynamics. The main contribution of this work is solving the consensus tracking problem without the assumption that the topology among followers is strongly connected and fixed. By using tools from M-matrix theory, a class of consensus tracking protocols based only on the relative states among neighboring agents is designed. By appropriately constructing Lyapunov function, it is proved that consensus tracking in the closed-loop multi-agent systems with a fixed topology having a directed spanning tree can be achieved if the feedback gain matrix and the coupling strength are suitably selected. Furthermore, with the assumption that each possible topology contains a directed spanning tree, it is theoretically shown that consensus tracking under switching directed topologies can be achieved if the control parameters are suitably selected and the dwell time is larger than a positive threshold. The results are then extended to the case where the communication topology contains a directed spanning tree only frequently as the system evolves with time. Finally, some numerical simulations are given to verify the theoretical analysis.

Consensus Tracking of Multi-Agent Systems With Lipschitz-Type Node Dynamics and Switching Topologies

In this paper, an output feedback stabilization of perturbed double-integrator systems using super-twisting control (STC) is studied. It is shown that when STC is implemented based on super-twisting observer (STO), then it is not possible to achieve second-order sliding mode (SOSM) using continuous control on the chosen sliding surface. Two methodologies are proposed to circumvent the above-mentioned problem. In the first method, control input is discontinuous, which may not be desirable for practical systems. In the second method, continuous STC is proposed based on higher order sliding mode observer (HOSMO) that achieves SOSM on the chosen sliding surface. For simplicity, we are considering here only the perturbed double integrator, which can be generalized for an arbitrary-order systems. Numerical simulations and experimental validation are also presented to show the effectiveness of the proposed method.

/pdf/implementation-of-super-twisting-control-super-twisting-and-5g4fgle0su.pdf

Implementation of Super-Twisting Control: Super-Twisting and Higher Order Sliding-Mode Observer-Based Approaches

This paper presents a new adaptive sliding-mode control (ASMC) scheme that uses the time-delay estimation (TDE) technique, then applies the scheme to robot manipulators. The proposed ASMC uses a new adaptive law to achieve good tracking performance with small chattering effect. The new adaptive law considers an arbitrarily small vicinity of the sliding manifold, in which the derivatives of the adaptive gains are inversely proportional to the sliding variables. Such an adaptive law provides remarkably fast adaptation and chattering reduction near the sliding manifold. To yield the desirable closed-loop poles and simplify a complicated system model by adapting feedback compensation, the proposed ASMC scheme works together with a pole-placement control (PPC) and a TDE technique. It is shown that the tracking errors of the proposed ASMC scheme are guaranteed to be uniformly ultimately bounded (UUB) with arbitrarily small bound. The practical effectiveness and the fast adaptation of the proposed ASMC are illustrated in simulations and experiments with robot manipulators, and compared with those of an existing ASMC.

A New Adaptive Sliding-Mode Control Scheme for Application to Robot Manipulators

This paper presents a comprehensive overview of the application of Variable Structure Systems (VSSs) with Sliding Mode (SM) methods in motion control systems. Our aim is to give implementable sliding mode design solutions for complex motion systems, actuators and supply converters. This paper provides a frame for further study of sliding mode applications in motion control systems.

Variable Structure Systems With Sliding Modes in Motion Control—A Survey

The first sliding mode control application may be found in the papers back in the 1930s in Russia. With its versatile yet simple design procedure the methodology is proven to be one of the most powerful solutions for many practical control designs. For the sake of demonstration this paper is oriented towards application aspects of sliding mode control methodology. First the design approach based on the regularization is generalized for mechanical systems. It is shown that stability of zero dynamics should be taken into account when the regular form consists of blocks of second-order equations. Majority of applications in the paper are related to control and estimation methods of automotive industry. New theoretical methods are developed in the context of these studies: sliding made nonlinear observers, observers with binary 
measurements, parameter estimation in systems with sliding mode control.

/pdf/sliding-mode-control-on-electro-mechanical-systems-1y8q1wge46.pdf

Hao Chi Chang

Papers

Sliding mode control on electro-mechanical systems