An SOS method for the design of continuous and discontinuous differentiators

201

155

Summary In this paper, we provide a new nonconservative upper bound for the settling time of a class of fixed‐time stable systems. To expose the value and the applicability of this result, we present four main contributions. First, we revisit the well‐known class of fixed‐time stable systems, to show the conservatism of the classical upper estimate of its settling time. Second, we provide the smallest constant that the uniformly upper bounds the settling time of any trajectory of the system under consideration. Third, introducing a slight modification of the previous class of fixed‐time systems, we propose a new predefined‐time convergent algorithm where the least upper bound of the settling time is set a priori as a parameter of the system. At last, we design a class of predefined‐time controllers for first‐ and second‐order systems based on the exposed stability analysis. Simulation results highlight the performance of the proposed scheme regarding settling time estimation compared to existing methods.

Section: Resultsmentioning

confidence: 99%

Enhancing the settling time estimation of a class of fixed‐time stable systems

Aldana‐López

Gómez‐Gutiérrez

Jiménez‐Rodríguez

et al. 2019

201

155

“…An important feature of the method is that it is also useful to design some parameters of the system; however, this problem becomes bilinear and harder to solve. Although the assumption of the existence of LFs in the class of GFs seems to be strong, the method has been successfully applied on some significant continuous and discontinuous systems, as can be seen in the examples of this paper and in the works of Sanchez et al, where the existence of LF in the class of GFs was proven for the family of systems considered there. Nonetheless, the general existence result remains as a fundamental problem that must be solved.…”

Section: Discussionmentioning

confidence: 95%

“…In this section, we continue the development of some examples initiated in Section 2. Two additional examples can be found in the works of Sanchez et al and Torres‐González et al…”

Section: Examplesmentioning

confidence: 95%

“…Among the high‐order sliding mode algorithms, an important one is the arbitrary order robust and exact differentiator . The undisturbed dynamics of the differentiation error constitutes a GF system . In the case of the first‐order differentiator, such a dynamics corresponds to the well‐known super‐twisting algorithm .…”

Section: Motivational Examplesmentioning

confidence: 99%

“…This LF constitutes a GF, and remarkably, it can be written as a quadratic form. In the case of the second‐order differentiator, the error dynamics is

ė_{1} = - λ_{1} {⌈ e_{1} ⌋}^{\frac{2}{3}} + e_{2}, ė_{2} = - λ_{2} {⌈ e_{1} ⌋}^{\frac{1}{3}} + e_{3}, ė_{3} = - λ_{3} {⌈ e_{1} ⌋}^{0} .

In other works, some GFs have been provided as LFs for the differential inclusion associated to , for example, the GF

V : R^{3} \to double-struckR

given by

V false(e false) = β_{1} {false| e_{1} false|}^{\frac{5}{3}} - β_{12} e_{1} e_{2} + β_{2} {false| e_{2} false|}^{\frac{5}{2}} - β_{23} e_{2} e_{3}^{3} + β_{3} {false| e_{3} false|}^{5}

.…”

Section: Motivational Examplesmentioning

confidence: 99%

See 2 more Smart Citations

Design of Lyapunov functions for a class of homogeneous systems: Generalized forms approach

Sánchez

Moreno

2018

Self Cite

Summary In this paper, we provide a method to design Lyapunov functions (LFs) for a class of homogeneous systems described by functions that we call generalized forms (GFs). Homogeneous polynomial systems and several high‐order sliding modes are included in the class. The LF candidate is chosen from the same class of functions and it is parameterized in its coefficients. Since the derivative of the LF candidate along the system's trajectories is also a GF, the problem is reduced to verify positive definiteness of two GFs. We establish a procedure to represent a GF with a finite set of polynomials. Thus, the problem is changed to determine positive definiteness of a set of polynomials. Such a problem can be solved by means of Pólya's theorem or the sum of squares representation.

Homogeneous output‐feedback control with disturbance‐observer for a class of nonlinear systems

Sánchez

Moreno²

2020

Self Cite

Summary In this article, we propose a finite‐time output‐feedback control scheme for a class of nonlinear systems. The dynamic part of the controller consists of an extended order observer, which is based on the higher order sliding‐mode exact differentiator. With such an observer, the states of the system are exactly estimated in finite‐time; moreover, the additional state in the observer estimates exactly and in finite‐time Lipschitz disturbances in the system. Such an estimation is used by the static part of the controller to compensate the disturbances. The static part of the controller can be chosen from a class of homogeneous controllers. The whole control scheme allows to recover in finite‐time several useful properties of homogeneous systems despite the additional and uncertain nonlinear terms. The effect of the noise in the measurement is also studied.