Tonametl Sanchez

Design of Continuous Twisting Algorithm

Abstract For the double integrator with matched Lipschitz disturbances we propose a continuous homogeneous controller providing finite-time stability of the origin. The disturbance is compensated exactly in finite time using a discontinuous function through an integral action. Since the controller is dynamic, the closed loop is a third order system that achieves a third order sliding mode in the steady state. The stability and robustness properties of the controller are proven using a smooth and homogeneous strict Lyapunov function (LF). In a first stage, the gains of the controller and the LF are designed using a method based on Polya’s Theorem. In a second stage the controller’s gains are adjusted through a sum of squares representation of the LF.

Construction of Lyapunov Functions for a Class of Higher Order Sliding Modes algorithms

A method to construct (strict) Lyapunov Functions for a class of Higher Order Sliding Modes (HOSM) algorithms, that are homogeneous and piecewise state affine is presented. It is shown first that several HOSM algorithms presented in the literature posses these properties. The basic idea of the construction method is borrowed from the constructive proofs of the Lyapunovs Converse Theorems. It is shown, by means of some concrete examples of second and third order, that the construction of the Lyapunov Function can be done for this class of systems. The obtained Lyapunov functions allow the estimation of the convergence time, the values of the gains that render the origin finite time stable, and the robustness of the algorithms to bounded perturbations.

A constructive Lyapunov function design method for a class of homogeneous systems

In this paper a method to design homogeneous Lyapunov functions for a class of homogeneous dynamical systems is proposed. The method is constructive and simple. The procedure exploits homogeneity and polynomial-like properties of the class of functions considered for the vector fields and for the Lyapunov function candidates. It also makes strong use of an algebraic result known as Pólyas Theorem to establish the required positive and negative definite properties of the Lyapunov function and its derivative. We illustrate the method by designing smooth Lyapunov functions for two systems. The first one is a Second Order Sliding Mode algorithm known as Super Twisting. We propose here for the first time a smooth Lyapunov function for it. The second example corresponds to a double integrator controlled by a continuous homogeneous state feedback, whose origin is finite time stable.

Smooth Lyapunov function and gain design for a Second Order Differentiator

We design a smooth Lyapunov function for the Levants Second Order Differentiator. The Lyapunov function construction method takes advantage of the structure of the system vector field to choose a candidate function. Both, the vector field and the candidate function belong to a special class of homogeneous functions. The problem of proving the positiveness of the function and the negativeness of its derivative is reduced, by using Pólyas Theorem, to the problem of solving a system of inequalities. Such inequalities are linear in the coefficients of the candidate function and also linear in the system parameters, but bilinear in both. The gains of the differentiator are designed during the construction process, and through the Lyapunov function, convergence time is estimated.

An SOS method for the design of continuous and discontinuous differentiators

ABSTRACT Given a (differentiable) signal, it is an important task for many applications to estimate on line its derivatives. Some well-known algorithms to solve this problem include the (continuous) high-gain observers and (discontinuous) Levants exact differentiators. With exception of the linear high-gain observers, a systematic design of the gains of nonlinear differentiators is, in general, a difficult task and an open research field. In this work, we propose a novel method for the gain-tuning of a family of homogeneous differentiators which estimate the time-derivatives of a signal in finite-time. We show that the stability analysis and the gain-tuning of such differentiators can be done under a unified Lyapunov function framework, and it is converted to a sum of squares problem, that can be solved using LMIs, much in the same spirit of the linear systems.

On a sign controller for the triple integrator

It is well known that for a double integrator a controller using only the signs of the state variables, the Twisting controller, is able to bring all trajectories to the origin in finite time and robustly with respect to bounded matched perturbations. In this paper we analyze a triple integrator with a controller consisting of the sum of the signs of the state variables multiplied by constant gains. We show that, in contrast to the twisting controller, there is no set of gains for which the origin is a globally asymptotically stable equilibrium point. There is however a set of gains such that for almost all initial conditions the trajectories converge in finite time to the origin.

Construction of a Smooth Lyapunov Function for the Robust and Exact Second-Order Differentiator

Differentiators play an important role in (continuous) feedback control systems. In particular, the robust and exact second-order differentiator has shown some very interesting properties and it has been used successfully in sliding mode control, in spite of the lack of a Lyapunov based procedure to design its gains. As contribution of this paper, we provide a constructive method to determine a differentiable Lyapunov function for such a differentiator. Moreover, the Lyapunov function is used to provide a procedure to design the differentiator’s parameters. Also, some sets of such parameters are provided. The determination of the positive definiteness of the Lyapunov function and negative definiteness of its derivative is converted to the problem of solving a system of inequalities linear in the parameters of the Lyapunov function candidate and also linear in the gains of the differentiator, but bilinear in both.

Lyapunov functions for Twisting and Terminal controllers

In this work, homogeneous Lyapunov functions for the Second Order Sliding Mode algorithms “Twisting” and “Terminal” are designed. Such functions are obtained using a systematic and constructive method. The designing method and the Lyapunov functions provide sets of gains that guarantee finite time stability of the systems origin in the disturbed case. It is also proved that the conditions for the controllers gains are even necessary and sufficient.

Output feedback Continuous Twisting Algorithm

Abstract Two output feedback controllers based on the Continuous Twisting Algorithm are provided. In those controllers, the state observers are based on the first and the second order Robust Exact Differentiators. The stability of the closed loops is proven through input-to-state stability properties. In the case of the second order differentiator, the conservation of homogeneity allows the output feedback scheme to preserve the robustness and accuracy properties of the state feedback Continuous Twisting Algorithm. In the same case, a smooth homogeneous Lyapunov function is constructed for the closed loop. A separation principle in the design of the controller and the observers is established. A qualitative analysis of the performance of the controllers in the presence of noise in the measurement is carried out. One of the schemes is used for output feedback control of a class of nonlinear systems.