Stability and Stabilization

Qualitative dynamics of three different loop quantizations of spatially flat isotropic and homogeneous models is studied using effective spacetime description of the underlying quantum geometry. These include the standard loop quantum cosmology (LQC), its recently revived modification (referred to as mLQC-I), and another related modification of LQC (mLQC-II) whose dynamics is studied in detail for the first time. Various features of LQC, including quantum bounce and preinflationary dynamics, are found to be shared with the mLQC-I and mLQC-II models. We study universal properties of dynamics for chaotic inflation, fractional monodromy inflation, Starobinsky potential, non-minimal Higgs inflation, and an exponential potential. We find various critical points and study their stability, which reveal various qualitative similarities in the post-bounce phase for all these models. The pre-bounce qualitative dynamics of LQC and mLQC-II turns out to be very similar, but is strikingly different from that of mLQC-I. In the dynamical analysis, some of the fixed points turn out to be degenerate for which center manifold theory is used. For all these potentials, non-perturbative quantum gravitational effects always result in a non-singular inflationary scenario with a phase of super-inflation succeeded by the conventional inflation. We show the existence of inflationary attractors, and obtain scaling solutions in the case of the exponential potential. Since all of the models agree with general relativity at late times, our results are also of use in classical theory where qualitative dynamics of some of the potentials has not been studied earlier. 1 Here "loop cosmology," refers collectively to various possible loop quantizations of cosmological spacetimes in the framework of LQG and should not be confused with LQC, here defined by the model developed in Refs. [6-8], which is an example of such a quantization.arXiv:1807.05236v2 [gr-qc]

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“…(3.10)- (3.11). To show that the origin is indeed stable in the expanding phase, one need appeal to the Lyapunov theorem [52] which states:…”

Section: Appendix A: Stability Analysis Of the Equilibriummentioning

confidence: 99%

Qualitative dynamics and inflationary attractors in loop cosmology

Singh

Wang

2018

Phys. Rev. D

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“…which has a solution in _ Xð0Þ ¼ 0, where X ¼ ½X 1 X 2 � T . To demonstrate Lyapunov local Asymptotic Stability by the direct method, a candidate function V must have the following properties [34,35]:…”

Section: Outer Loopmentioning

confidence: 99%

“…{V}_{S}\left(\omega Lcos{X}_{2}-Rsin{X}_{2}\right)-\omega L{V}_{S}\right)$ which has a solution in

\dot{X}(0)=0

, where

X={\left[{X}_{1}{X}_{2}\right]}^{T}

. To demonstrate Lyapunov local Asymptotic Stability by the direct method, a candidate function V must have the following properties [34, 35]:

V(0)=0

V(X) > 0,X\ne 0,X\in {D}_{X}

V(X)\to \infty ,\Vert X\Vert \to \infty

\dot{V}(0)=0

\dot{V}(X)< 0,X\ne 0,X\in {D}_{X}

…”

Section: Resiliency and Stability In A Distributed Control Environmentmentioning

confidence: 99%

Real‐time resilient microgrid power management based on multi‐agent systems with price forecast

2022

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Microgrids have emerged to diversify conventional electric generation using small-scale distributed generation. Large efforts have been put into designing control strategies to optimise the power schedules of microgrids, however, verification that such control systems also are reliable in terms of stability during normal operation and fault conditions is needed. This study presents a hierarchical distributed control system that fulfils these conditions for an AC microgrid. The stability maintained by proposed controller, considering the large signal model, is analysed with the use of Lyapunov's direct method. Resilient control distribution is achieved by the implementation of suitable forecast models and fault-tolerance mechanisms to avoid single points of failure. The resilience of the control system is verified with the use of graph theory. The stable and resilient operation of the proposed control system is tested by a real-time microgrid model implemented with an OPAL-RT real-time simulator, combined with a communication network built with Raspberry Pis, testing the control system presented under normal and faulty conditions. Simulation results show a stable operation in terms of voltage and frequency in both conditions, resilient operation is shown for the faulty condition case. Additionally, cost minimisation performance is included to validate optimal power management capabilities.

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“…An important property of the Kalman model to be valid is that the pair (A,C) be detectable [2]. We use the Popov-Belevitch-Hautus (PBH) Lemma [20] to easily discard unacceptable A b values. We then use the Kalman model to obtain an expression of the Kalman filter.…”

Section: Kalman Modelmentioning

confidence: 99%

Flexible Launch Vehicle Control Using Robust Observer-Based Controller Obtained through Structured H ∞ Synthesis

Chambon¹,

Apkarian²,

Burlion³

2015

Advances in Aerospace Guidance, Navigation and Control

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Abstract. Control of a flexible launch vehicle in the atmospheric ascent phase is highly challenging as it involves multiple concurrent design requirements. This ranges from reduction of the angle of attack in face of wind, minimum gain-phase and parametric margins as well as flexible modes attenuation. In this work, we discuss recently available non-smooth optimization techniques as a central tool to solve this problem. We consider designing an observer-based controller based on a Kalman filter suitably augmented with Dryden wind dynamics. We suggest a nonconservative approach to handle model uncertainties based upon multiple models of the launcher. This preliminary work aims at testing the potential of non-smooth controller tuning on a generic launcher model. It will serve as a stepping stone for a more in-depth study of the benchmark developed by M. Ganet at Airbus Defence and Space.

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Stability and Stabilization

Cited by 51 publications

References 0 publications

Qualitative dynamics and inflationary attractors in loop cosmology

Qualitative dynamics and inflationary attractors in loop cosmology

Real‐time resilient microgrid power management based on multi‐agent systems with price forecast

Flexible Launch Vehicle Control Using Robust Observer-Based Controller Obtained through Structured H ∞ Synthesis

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