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This paper proposes an efficient implementation for a continuous terminal algorithm (CTA). Although CTA is a continuous version of the famous twisting algorithm (TA), the conventional implementations of this CTA still suffer from chattering, especially when the gains and the time-step sizes are selected very large. The proposed implementation is based on an implicit Euler method and it totally suppresses the chattering. The proposed implementation is compared with the conventional explicit Euler implementation through simulations. It shows that the proposed implementation is very efficient and the chattering is suppressed both in the control input and output.

This paper derives new results for the analysis of nonlinear systems by extending contraction theory in the framework of vector distances. A new tool, vector contraction analysis utilizing a notion of the vector-valued norm which evidently induces a vector distance between any pair of trajectories of the system, offers an amenable framework as each component of vector-valued norm function satisfies fewer strict conditions as that of standard contraction analysis. Particularly, every element of vector-valued norm derivative need not be strictly negative definite for convergence of any pair of trajectories of the system. Moreover, vector-valued norm derivative satisfies a component-wise inequality employing some comparison system. In fact, the convergence analysis is performed by comparing the relative distances between any pair of the trajectories of the original nonlinear system and the comparison system. Comparison results are derived by utilizing the concepts of quasi-monotonicity property of the function and vector differential inequalities. Moreover, the results are also derived in the framework of the cone ordering instead of utilizing component-wise inequalities between vectors. In addition, the proposed framework is illustrated by examples.

This paper studies the super-twisting algorithm (STA) for adaptive sliding mode design. The proposed method tunes the two gains of STA on line simultaneously such that a second order sliding mode can take place with small rectifying gains. The perturbation magnitude is obtained exactly by employing a third-order sliding mode observer in opposition to the conventional approximations by using a first order low pass filter. While driving the sliding variable to the sliding mode surface, one gain of the STA automatically converges to an adjacent area of the perturbation magnitude in finite time. The other gain is adjusted by the above gain to guarantee the robustness of the STA. This method requires only one parameter to be adjusted. The adjustment is straightforward because it just keeps increasing until it fulfills the convergence constraints. For large values of the parameter, chattering in the update law of the two gains is avoided by employing a geometry based backward Euler integration method. The usefulness is illustrated by an example of designing an equivalent control based sliding mode control (ECBC-SMC) with the proposed adaptive STA for a perturbed LTI system.