Stability Of Homogeneous Systems Research Articles

Stability analysis of homogeneous cooperative positive differential systems with time-varying delays and its generalization

This article is devoted to studying the asymptotic behavior of differential systems with bounded delays. We first focus on the analysis of homogeneous cooperative positive systems. Under the assumption that the vector field is homogeneous of a degree greater than 1, we show that the non-trivial solutions of the system converge to the origin at a polynomial rate. In the case when the degree of homogeneity equals 1, we prove that the solutions will decay at an exponential rate. As a generalization of these results, we consider nonlinear non-autonomous differential systems with time-varying delays that are bounded above by stable homogeneous positive systems. By some additional imposed conditions, in light of the comparison principle, we obtain the locally exponential stability and polynomial stability of the equilibrium point to these systems. Finally, specific examples and discussions are provided to illustrate the validity of the proposed theoretical results.

On Stability of Homogeneous Systems in Presence of Parasitic Dynamics

In this paper, the effect of a homogeneous parasitic dynamics on the stability of a homogeneous system, when homogeneity degrees are possibly different, is studied via ISS approach in the framework of singular perturbations. Thus, the possibilities to reduce the order of the interconnected system considering only the reduced-order dynamics and neglecting the parasitic ones are examined. Proposed analysis discovers three kinds of stability in the behavior of such an interconnection by assuming that both, reduced-order and unforced parasitic dynamics, are globally asymptotically stable. In the first case, global asymptotic stability of the interconnection can be concluded, when the homogeneity degrees of both systems coincide and the singular perturbation parameter is small enough. In the second case, only local stability of interconnection can be ensured, when the homogeneity degree of reduced-order dynamics is greater than the homogeneity degrees of the parasitic ones. Helpfully, the proposed approach allows to estimate the domain of attraction for the system's trajectories as a function of the singular perturbation parameter. In the third case, only practical stability can be guaranteed, when the homogeneity degree of reduced-order dynamics is smaller than the homogeneity degrees of the parasitic ones. Additionally, the proposed approach provides an estimation of the asymptotic bound of the system's trajectories in terms of the singular perturbation parameter.

Stability of homogeneous systems with distributed delay and time-varying perturbations

For a class of nonlinear systems with homogeneous right-hand sides of non-zero degree and distributed delays, the problem of stability robustness of the zero solution with respect to time-varying perturbations multiplied by a nonlinear functional gain is studied. It is assumed that the disturbance-free and delay-free system (that results after substitution of non-delayed state for the delayed one) is globally asymptotically stable. First, it is demonstrated that in the disturbance-free case the zero solution is either locally asymptotically stable or practically globally asymptotically stable, depending on the homogeneity degree of the delay-free counterpart. Second, using averaging tools several variants of the time-varying perturbations are considered and the respective conditions are derived evaluating the stability margins in the system. The results are obtained by a careful choice and comparison of Lyapunov–Krasovskii and Lyapunov–Razumikhin approaches. Finally, the obtained theoretical findings are illustrated on two mechanical systems.

Stability of homogeneous positive systems with time-varying delays

Stability analysis of positive linear systems has been well-developed, while attention to positive homogeneous nonlinear systems is notably less paid. This note deepens and extends some previous results on stability of homogeneous systems with bounded time-varying delays. Based on positivity property, exponential stability of a class of delayed homogeneous system with degree confined to an interval is studied, and its state boundary is further investigated in any fixed time, which is more precise than the former result in some cases. Additionally, finite-time stability is obtained for a delay-free homogeneous system and an explicit boundary is presented.

Homogeneous Lp Stability for Homogeneous Systems

The motivation of this paper comes from the fact that <i>L_p</i>&#x2014;stability and <i>L_p</i>&#x2014;gain, using the classical signal norms, is not well-defined for arbitrary continuous weighted homogeneous systems. However, using homogeneous signal norms it is possible to show that every internally stable homogeneous system has a globally defined finite homogeneous <i>L_p</i>&#x2014;gain, for <i>p</i> sufficiently large. If the system has a homogeneous approximation, the homogeneous <i>L_p</i>&#x2014;gain is inherited locally. Homogeneous <i>L_p</i>&#x2014;stability can be characterized by a homogeneous dissipation inequality, which in the input affine case can be transformed to a homogeneous Hamilton-Jacobi inequality. An estimation of an upper bound for the homogeneous <i>L_p</i>&#x2014;gain can be derived from these inequalities. Homogeneous <i>L</i>_&#x221E;&#x2014;stability is also considered and its strong relationship to Input-to-State stability is studied. These results are extensions to arbitrary homogeneous systems of the well-known situation for linear time-invariant systems, where the Hamilton-Jacobi inequality reduces to an algebraic Riccati inequality. A natural application of finite-gain homogeneous <i>L_p</i>&#x2014;stability is in the study of stability for interconnected systems. An extension of the small-gain theorem for negative feedback systems and results for systems in cascade are derived for different homogeneous norms. Previous results in the literature use classical signal norms, hence, they can only be applied to a restricted class of homogeneous systems. The results are illustrated by several examples.

Lyapunov‐based consistent discretization of stable homogeneous systems

SummaryIn this article, we propose a discretization scheme for asymptotically stable homogeneous systems. This scheme exploits the information provided by a homogeneous Lyapunov function of the system. The main features of the scheme are: (1) the discretization method is explicit and; (2) the discrete‐time system preserves the asymptotic stability, the convergence rate, and the Lyapunov function of the original continuous‐time system.

A Consistent Discretisation method for Stable Homogeneous Systems based on Lyapunov Function

In this paper we propose a discretisation scheme for continuous and asymptotically stable homogeneous systems. This method is based on the dynamics of the system projected on a level surface of a homogeneous Lyapunov function. The discretisation method is explicit and preserves the convergence rate of the continuous-time system.

Discretization of homogeneous systems using Euler method with a state-dependent step

Numeric approximations to the solutions of asymptotically stable homogeneous systems by Euler method, with a step of discretization scaled by the state norm, are investigated (for the explicit and implicit integration schemes). It is proven that for a sufficiently small discretization step the convergence of the approximating solutions to zero can be guaranteed globally in a finite or a fixed time depending on the degree of homogeneity of the system, but in an infinite number of discretization iterations. The maximal admissible step can be estimated by analyzing the system properties on the sphere. It is shown that the absolute and relative errors of the discretizations are globally bounded functions, thus the approximations approaching the solutions with the step converging to zero. In addition, it is established that the proposed discretization approach preserves robustness with respect to exogenous perturbations. Efficiency of the designed discretization algorithms is demonstrated in simulations.

New insight into experimental and computational studies of Choline chloride-based ‘green’ ternary deep eutectic solvent (TDES)

In this study, we have examined the influence of ternary component (n-butanol, iso-butanol, and butandiol) as hydrogen bond donor (HBD) on binary Deep eutectic solvents (BDES) as hydrogen bond acceptor (HBA) which is composed of Choline chloride (ChCl) and Malonic acid (MA) in equimolar ratio (1:1) also called as maline to understand the hydrogen bonding interactions involved therein. The physicochemical properties of such as density, sound velocity, refractive index, conductivity, and viscosity of maline and TDES were evaluated at 303.15 K. Fourier-transform infrared spectroscopy (FT-IR) validated the molecular structures of the components and the involvement of extensive hydrogen bonding. Here the combination of maline (ChCl: MA) with the mentioned alcohols was simulated using Gauss View 5.0.9 package. Study showed that maline and butandiol exhibited a well-established and high relative hydrogen bond networks between them leading to a larger melting point depression in comparison to n-butanol and iso-butanol. Such a response could be related to the stable homogeneous system formed by butandiol and not by n-butanol and iso-butanol. These findings are further supported and significantly validated by the evaluation of Total energy (T.E) which accounts for the involvement of molecular orbitals. Additionally, the 3D-molecular electrostatic potential (3D-MEP) description was performed to predict the coordination of electrophilic and nucleophilic sites between the HBA and HBD.

Homogeneous Lyapunov Functions: From Converse Design to Numerical Implementation

The problem of the synthesis of a homogeneous Lyapunov function for an asymptotically stable homogeneous system is studied. First, for systems with nonnegative degree of homogeneity, several expressions of homogeneous Lyapunov functions are derived, which depend explicitly on the supremum or the integral (over finite or infinite intervals of time) of the system solutions. Second, a numeric procedure is proposed, which ensures the construction of a homogeneous Lyapunov function. The analytical results are illustrated by simulations.

Acceleration of finite‐time stable homogeneous systems

SummaryStabilization rates of power‐integrator chains are easily regulated. It provides a framework for acceleration of uncertain multiple‐input–multiple‐output dynamic systems of known relative degrees (RDs). The desired rate of the output stabilization (sliding‐mode control) is ensured for an uncertain system if its RD is known, and a rough approximation of the high‐frequency gain matrix is available. The uniformly bounded convergence time (fixed‐time stability) is obtained as a particular case. The control can be kept continuous everywhere except the sliding‐mode set if the partial RDs are equal. Similarly, uncertain smooth systems of complete multiple‐input–multiple‐output RDs (ie, lacking zero dynamics) are stabilized by continuous control at their equilibria in finite time and are accelerated. Output‐feedback controllers are constructed. Computer simulation demonstrates the efficiency of the proposed approach.

Realization and Discretization of Asymptotically Stable Homogeneous Systems

Sufficient conditions for the existence and convergence to zero of numeric approximations to solutions of asymptotically stable homogeneous systems are obtained for the explicit and implicit Euler integration schemes. It is shown that the explicit Euler method has certain drawbacks for the global approximation of homogeneous systems with nonzero degrees, whereas the implicit Euler scheme ensures convergence of the approximating solutions to zero. Properties of absolute and relative errors of the respective discretizations are investigated.

Supervisory acceleration of convergence for homogeneous systems

ABSTRACTThe rate of convergence to the origin for a chain of integrators stabilised by homogeneous feedback is accelerated by a supervisory switching of control parameters. The proposed acceleration algorithm ensures a fixed-time convergence for otherwise exponentially or finite-time stable homogeneous closed-loop systems. Bounded disturbances are taken into account. The results are especially useful in the case of exponentially stable systems widespread in practice. The proposed switching strategy is illustrated by computer simulation.

Acceleration of finite-time stable homogeneous systems

Acceleration of finite-time stable homogeneous systems

ChemInform Abstract: (BeDABCO)2Pd2Cl6 as an Efficient Homogeneous Catalyst for Copper‐Free Sonogashira Cross‐Coupling Reaction.

AbstractThe stable homogeneous catalytic system is nonsensitive to air and moisture.

(BeDABCO)2Pd2Cl6 as an efficient homogeneous catalyst for copper‐free Sonogashira cross‐coupling reaction

An efficient catalytic system using 1‐benzyl‐4‐aza‐1‐azoniabicyclo[2.2.2]octane chloride and palladium chloride ((BeDABCO)2Pd2Cl6) was developed for the Sonogashira reaction. In the presence of a catalytic amount of this efficient, stable homogeneous catalytic system that is non‐sensitive to air and moisture, various aryl halides were efficiently coupled with phenylacetylene in good yields in H2O at 50°C under copper‐free conditions. Benzyl dabco as an efficient ligand and also a quaternary ammonium salt had an efficient stabilizing effect on the Pd(0) species. Copyright © 2014 John Wiley &amp; Sons, Ltd.

Stable and Efficient Homogeneous Photocatalytic H2 Evolution Based on Water Soluble Pyrenetetrasulfonic Acid Functionalized Platinum Nanocomposites

Dressed up for the photoshoot: A stable and efficient homogeneous catalytic system involving 1,3,6,8-pyrenetetrasulfonic acid (PTSA) functionalized Pt nanocomposites as photocatalysts and ethylenediaminetetraacetic acid (EDTA) as a sacrificial reductant in the absence of an electron mediator was prepared.

Settling Time Estimate for a Second Order Sliding Mode Controller: A Homogeneity Approach

AbstractA novel homogeneity approach to obtain an upper bound on the settling time for a robust second order sliding mode controller is presented. The stability analysis is substantiated by a global non-smooth Lyapunov function. The proposed method is applied to the ‘twisting’ controller and is based on a combination of global exponential stability and global finite time stability of switched systems. The homogeneity regions are established and are graphically illustrated. Recommended tuning rules for the twisting controller are presented. The proposed framework does not depend on the differential inequality of the Lyapunov function and hence provides a new methodology for obtaining the upper bound on the settling time for exponentially stable homogeneous systems.

Electrogeneration of Ferrate(VI) in Low-concentration NaOH Solution for Capillary Electrophoresis–Chemiluminescence Detection

Abstract A novel, eco-friendly, and stable homogeneous luminescent system for capillary zone electrophoresis (CE) detection was established, based on the chemiluminescence (CL) flash caused by mixing ferrate(VI) with luminol solution. It was found that the stable ferrate(VI) could be freshly electrogenerated in low-concentration NaOH solution for direct analytical application. To test this CL system, 8-hydroxyquinoline (HQ) was chosen as a model sample by CE detection.

Two Global Stabilizability Results for Homogeneous Systems From a Local Stabilizability Assumption

In this technical note, two problems have been investigated. The first one concerns the global asymptotic feedback stabilization of homogeneous systems. For these systems, it is shown that local asymptotic stabilization implies the global dynamic feedback stabilization one. The second problem consists to show that under a weak assumption on the observability, if a state affine system is locally asymptotically stabilizable then it becomes globally asymptotically stabilizable by using a discontinuous output feedback.