Stability of fractional order systems

The theory and applications of fractional calculus (FC) had a considerable progress during the last years. Dynamical systems and control are one of the most active areas, and several authors focused on the stability of fractional order systems. Nevertheless, due to the multitude of efforts in a short period of time, contributions are scattered along the literature, and it becomes difficult for researchers to have a complete and systematic picture of the present day knowledge. This paper is an attempt to overcome this situation by reviewing the state of the art and putting this topic in a systematic form. While the problem is formulated with rigour, from the mathematical point of view, the exposition intends to be easy to read by the applied researchers. Different types of systems are considered, namely, linear/nonlinear, positive, with delay, distributed, and continuous/discrete. Several possible routes of future progress that emerge are also tackled

About Robust Stability of Caputo Linear Fractional Dynamic Systems with Time Delays through Fixed Point Theory

This paper investigates the global stability and the global asymptotic stability independent of the sizes of the delays of linear time-varying Caputo fractional dynamic systems of real fractional order possessing internal point delays. The investigation is performed via fixed point theory in a complete metric space by defining appropriate nonexpansive or contractive self-mappings from initial conditions to points of the state-trajectory solution. The existence of a unique fixed point leading to a globally asymptotically stable equilibrium point is investigated, in particular, under easily testable sufficiency-type stability conditions. The study is performed for both the uncontrolled case and the controlled case under a wide class of state feedback laws.Ministerio de Educación (DPI2009-07197) y Gobierno Vasco (IT378-10 SAIOTEK S-PE09UN12

Laminar chaos in systems with quasiperiodic delay

A new type of chaos called laminar chaos was found in singularly perturbed dynamical systems with periodic time-varying delay [Phys. Rev. Lett. 120, 084102 (2018)]. It is characterized by nearly constant laminar phases, which are periodically interrupted by irregular bursts, where the intensity levels of the laminar phases vary chaotically from phase to phase. In this paper, we demonstrate that laminar chaos can also be observed in systems with quasiperiodic delay, where we generalize the concept of conservative and dissipative delays to such systems. It turns out that the durations of the laminar phases vary quasiperiodically and follow the dynamics of a torus map in contrast to the periodic variation observed for periodic delay. Theoretical and numerical results indicate that introducing a quasiperiodic delay modulation into a time-delay system can lead to a giant reduction of the dimension of the chaotic attractors. By varying the mean delay and keeping other parameters fixed, we found that the Kaplan-Yorke dimension is modulated quasiperiodically over several orders of magnitudes, where the dynamics switches quasiperiodically between different types of high- and low-dimensional types of chaos.Comment: 17 pages, 9 figure

Stability Analysis and Decentralized Control of Coupled Oscillators with Feedback Delays

Most dynamic systems do not react instantaneously to actuation signals. The temporal evolution of some others is based on retarded communications or depends on information from the past. In such cases, the mathematical models used to describe these systems must include information about the past dynamics of the states. These models are often referred to as delay or retarded systems. Delays could channel energy in and out of a system at incorrect time intervals producing instabilities and rendering controllers\u27 performance ineffective. The purpose of this research is two folds. The first investigates the effect of inherent system delays on the stability of coupled oscillators subjected to decentralized control and the second studies the prospectus of augmenting the delay into a larger delay period that could actually stabilize the coupled system and enhance its damping characteristics. Towards these ends, a system of two linearly-coupled oscillators with decentralized delayed-proportional feedback is considered. A comprehensive linear stability analysis is utilized to generate maps that divide the controllers\u27 gain and delay domain into regions of stability for different coupling values. These maps are then used to draw definite conclusions about the effect of coupling on the stability of the closed-loop in the presence of delay. Once the stability maps are generated, the Lambert-W function approach is utilized to find the stability exponents of the coupled system which, in turn, is used to generate damping contours within the pockets of stability. These contours are used to choose gain-delay combinations that could augment the inherent feedback delays into a larger delay period which can enhance the damping characteristics and reduce the system settling time significantly. An experimental plant comprised of two mass-spring-damper trios coupled with a spring is installed to validate the theoretical results and the proposed control hypothesis. Different scenarios consisting of different gains and delays are considered and compared with theoretical findings demonstrating very good agreement. Furthermore, the proposed delayed-proportional feedback decentralized controller is tested and its ability to dampen external oscillations is verified through different experiments. Such a research endeavor could prove very beneficial to many vital areas in our life. A good example is that of the coupled system of the natural and artificial cardiac pacemakers where the natural pacemaker represents a rhythmic oscillating system and the coupled artificial pacemaker provides a stabilizing signal through a feedback mechanism that senses the loss in rhythm. In this system, even the minute amount of delay in the sensing-actuating could prove very detrimental. The result of this research contributes to the solution of this and similar problems

Spectrum analysis of LTI continuous-time systems with constant delays: A literature overview of some recent results

In recent decades, increasingly intensive research attention has been given to dynamical systems containing delays and those affected by the after-effect phenomenon. Such research covers a wide range of human activities and the solutions of related engineering problems often require interdisciplinary cooperation. The knowledge of the spectrum of these so-called time-delay systems (TDSs) is very crucial for the analysis of their dynamical properties, especially stability, periodicity, and dumping effect. A great volume of mathematical methods and techniques to analyze the spectrum of the TDSs have been developed and further applied in the most recent times. Although a broad family of nonlinear, stochastic, sampled-data, time-variant or time-varying-delay systems has been considered, the study of the most fundamental continuous linear time-invariant (LTI) TDSs with fixed delays is still the dominant research direction with ever-increasing new results and novel applications. This paper is primarily aimed at a (systematic) literature overview of recent (mostly published between 2013 to 2017) advances regarding the spectrum analysis of the LTI-TDSs. Specifically, a total of 137 collected articles-which are most closely related to the research area-are eventually reviewed. There are two main objectives of this review paper: First, to provide the reader with a detailed literature survey on the selected recent results on the topic and Second, to suggest possible future research directions to be tackled by scientists and engineers in the field. © 2013 IEEE.MSMT-7778/2014, FEDER, European Regional Development Fund; LO1303, FEDER, European Regional Development Fund; CZ.1.05/2.1.00/19.0376, FEDER, European Regional Development FundEuropean Regional Development Fund through the Project CEBIA-Tech Instrumentation [CZ.1.05/2.1.00/19.0376]; National Sustainability Program Project [LO1303 (MSMT-7778/2014)

Positivity and Stability of the Solutions of Caputo Fractional Linear Time-Invariant Systems of Any Order with Internal Point Delays

This paper is devoted to the investigation of the nonnegative solutions and the stability and asymptotic properties of the solutions of fractional differential dynamic systems involving delayed dynamics with point delays. The obtained results are independent of the sizes of the delays