A note on finite-time and fixed-time stability

Arising in a large number of application areas, sparse recovery (SR) has been exhaustively investigated and many algorithms have been proposed. Different from the numerical methods realized by iterative algorithm, the recent continuous approach is realized through analog circuit, which takes advantage of short time-delay and fast convergence faster. However, the existing continuous method for SR still has the space to further improve the convergence speed. Consequently, in this paper, we propose a new dynamical continuous system to solve the sparse signal recovery problem with fixed-time convergence property.

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“…Remark 2. From (26), it can be found that the T max is independent of the initial conditions. In addition, the T max is smaller with smaller α.…”

Section: A A) Tu U U(t)mentioning

confidence: 99%

Dynamical sparse signal recovery with fixed-time convergence

Ren

Lyu

et al. 2019

Signal Processing

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“…Theorem 1: For the second-order multi-variable system (5) with unknown nonlinear dynamics satisfying Assumption 1, the global fixed-time stability is realized at the origin under the proposed controller (12), and the upper bound of the settling time is given by…”

Section: Fixed-time Stability Of Second-order Nonlinear Systemsmentioning

confidence: 99%

“…Then by observing the obtained expression in (23), it can be seen that the control input is always well defined during the sliding motion if 1/2 < q 2 < 1. Therefore, the singularity of controller can be avoided for s = 0 by imposing the requirement 1/2 < q 2 < 1 to the designed controller (12).…”

Section: Fixed-time Stability Of Second-order Nonlinear Systemsmentioning

confidence: 99%

“…For a linear single input system, an implicit Lyapunov function method was applied to establish certain criteria for the fixed-time stability in [11]. In [12], a general approach was provided to reveal the essence of finite-time stability and fixed-time convergence for the considered system, and conditions for finite-time and fixedtime convergence were derived. For a SISO dynamic system, it was shown that any finite-time convergent homogeneous sliding mode controller could be transformed into a fixed-time convergent one, featuring an upper bound of convergence time [13].…”

Section: Introductionmentioning

confidence: 99%

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Fixed-Time Stabilization of Second-Order Systems with Unknown Nonlinear Inherent Dynamics

Yang

Yue

Wang

2018

2018 UKACC 12th International Conference on Control (CONTROL)

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This paper studies the fixed-time stabilization control for a class of second-order systems with unknown nonlinear dynamics and external disturbance. To achieve the fixed-time stability, two polynomial feedbacks, one with fractional exponent and the other with exponent greater than 1, are exploited in the sliding surface design. Then, considering the unknown nonlinear dynamics and the external disturbance, the unit-vector technique, coupled with the upper bounds related to the Lipschitz condition of nonlinear dynamics and the uncertain disturbance, is utilized to the fixed-time stability controller based on the designed sliding surface function. Finally, a forced pendulum system is used as case study example to examine the effectiveness of the developed design framework.

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“…Being a particular case of finite-time stability, this type of stability, known as fixedtime stability, inherits many of the results derived for finite-time stable systems. Although some sufficient conditions for fixed-time stability can be found in [9,11,13,14] and references therein, so far, necessary and sufficient conditions that take into account the regularity of the settling-time function haven't been presented. The aim of this article is to provide these conditions along with a characterization of the fixed-time stability property using Lyapunov theory.…”

Section: Introductionmentioning

confidence: 99%

Conditions for fixed-time stability and stabilization of continuous autonomous systems

Lopez-Ramirez

Efimov

Polyakov

et al. 2019

Systems & Control Letters

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This work presents Lyapunov analysis conditions for fixed-time stability, a property where all the system's trajectories converge exactly to zero in a finite amount of time that is independent of the system's initial condition. Necessary and sufficient conditions for fixed-time stability without taking into account the regularity of the settling-time function are presented first. Next, a characterization for fixed-time stability with continuous settling-time function is introduced. A particular form of the characterizing functions follows, it allows to establish more constructive conditions and in order to obtain a converse result, the concept of complete fixed-time stability is introduced. A set of academic examples and an example of allocation of mobile agents illustrate the given concepts. Finally, a sufficient condition for fixed-time stabilization of nonlinear affine systems is obtained.

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A note on finite-time and fixed-time stability

Cited by 153 publications

References 24 publications

Dynamical sparse signal recovery with fixed-time convergence

Dynamical sparse signal recovery with fixed-time convergence

Fixed-Time Stabilization of Second-Order Systems with Unknown Nonlinear Inherent Dynamics

Conditions for fixed-time stability and stabilization of continuous autonomous systems

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