The Lyapunov matrix function and the stability of hybrid systems

1985 ◽  
Vol 21 (4) ◽  
pp. 386-393
Author(s):  
A. A. Martynyuk
Keyword(s):  
Hybrid Systems ◽  
Matrix Function ◽  
Lyapunov Matrix ◽  
The Stability

The Lyapunov matrix-function and stability of hybrid systems

1986 ◽  
Vol 10 (12) ◽  
pp. 1449-1457 ◽  
Author(s):  
A.A. Martynyuk
Keyword(s):  
Hybrid Systems ◽  
Matrix Function ◽  
Lyapunov Matrix

scholarly journals Stabilisation of Discrete-Time Piecewise Homogeneous Markov Jump Linear System with Imperfect Transition Probabilities

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Ding Zhai ◽  
Liwei An ◽  
Jinghao Li ◽  
Qingling Zhang
Keyword(s):  
Linear System ◽  
Discrete Time ◽  
Transition Probabilities ◽  
Feedback Controller ◽  
Sufficient Condition ◽  
Markov Jump ◽  
Stochastically Stable ◽  
Lyapunov Matrix ◽  
The Stability ◽  
Markov Jump Linear System

This paper is devoted to investigating the stability and stabilisation problems for discrete-time piecewise homogeneous Markov jump linear system with imperfect transition probabilities. A sufficient condition is derived to ensure the considered system to be stochastically stable. Moreover, the corresponding sufficient condition on the existence of a mode-dependent and variation-dependent state feedback controller is derived to guarantee the stochastic stability of the closed-loop system, and a new method is further proposed to design a static output feedback controller by introducing additional slack matrix variables to eliminate the equation constraint on Lyapunov matrix. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed methods.

scholarly journals Analysis of stability problems via matrix Lyapunov functions

1990 ◽  
Vol 3 (4) ◽  
pp. 209-226 ◽  
Author(s):  
Anatoly A. Martynyuk
Keyword(s):  
Lyapunov Functions ◽  
Systems Of Nonlinear Equations ◽  
Matrix Functions ◽  
Singularly Perturbed Systems ◽  
Two Component System ◽  
Two Component ◽  
Lyapunov Matrix ◽  
Analysis Of Stability ◽  
The Stability ◽  
The Given

The stability of nonlinear systems is analyzed by the direct Lyapunov's method in terms of Lyapunov matrix functions. The given paper surveys the main theorems on stability, asymptotic stability and nonstability. They are applied to systems of nonlinear equations, singularly-perturbed systems and hybrid systems. The results are demonstrated by an example of a two-component system.

scholarly journals An Approach for the Global Stability of Mathematical Model of an Infectious Disease

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1778
Author(s):  
Mojtaba Masoumnezhad ◽  
Maziar Rajabi ◽  
Amirahmad Chapnevis ◽  
Aleksei Dorofeev ◽  
Stanford Shateyi ◽  
...  
Keyword(s):  
Infectious Disease ◽  
Mathematical Model ◽  
Global Stability ◽  
Incidence Rates ◽  
Endemic Equilibrium ◽  
Symmetry Properties ◽  
Nonlinear Incidence Rates ◽  
Lyapunov Matrix ◽  
The Stability ◽  
The Mathematical Model

The global stability analysis for the mathematical model of an infectious disease is discussed here. The endemic equilibrium is shown to be globally stable by using a modification of the Volterra–Lyapunov matrix method. The basis of the method is the combination of Lyapunov functions and the Volterra–Lyapunov matrices. By reducing the dimensions of the matrices and under some conditions, we can easily show the global stability of the endemic equilibrium. To prove the stability based on Volterra–Lyapunov matrices, we use matrices with the symmetry properties (symmetric positive definite). The results developed in this paper can be applied in more complex systems with nonlinear incidence rates. Numerical simulations are presented to illustrate the analytical results.

scholarly journals A Feasibility Analysis of Controlling a Hybrid Power System over Short Time Intervals

Energies ◽  
2020 ◽  
Vol 13 (21) ◽  
pp. 5682
Author(s):  
Tianyao Zhang ◽  
Diyi Chen ◽  
Jing Liu ◽  
Beibei Xu ◽  
Venkateshkumar M
Keyword(s):  
Solar Radiation ◽  
Hybrid Systems ◽  
Feasibility Analysis ◽  
Energy Sources ◽  
Systematic Change ◽  
Short Term ◽  
Time Intervals ◽  
Hybrid Power ◽  
The Stability ◽  
Short Time

Literature about the importance of renewable energy resources, including wind and solar energy, is becoming increasingly important; however, these energy sources are unstable and volatile in nature, and are usually integrated with conventional energy sources, such as hydropower, forming hybrid power generation systems that maintain a stable grid. Short-term changes in wind speed or solar radiation intensity have a great impact on the stability of hybrid systems, and have been reported in the literature. However, reliable models to manage such systems are lacking, and previous studies have regarded the hour scale as the minimum baseline for systematic change. In this article, hybrid power systems are proposed that are controlled on very short time intervals. The results of a feasibility analysis of the proposed model indicate the viability of complementary hybrid systems in controlling and maintaining the stability, which are subjected to short durations of fluctuations in wind or solar radiation. The simulation results indicate that the influence of the shutdown of the wind turbine, with the regulation effect of the hydro power, is 3–5 times greater than that of the short-term wind turbulence fluctuation. When the hydro turbine is adopted to adjust the short-term fluctuation of solar radiation, the effect on the system was suppressed to 0.02–0.2 times of the former.

Perspectives and results on the stability and stabilizability of hybrid systems

2000 ◽  
Vol 88 (7) ◽  
pp. 1069-1082 ◽  
Author(s):  
R.A. Decarlo ◽  
M.S. Branicky ◽  
S. Pettersson ◽  
B. Lennartson
Keyword(s):  
Hybrid Systems ◽  
The Stability

scholarly journals TheZ-Transform Method and Delta Type Fractional Difference Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Dorota Mozyrska ◽  
Małgorzata Wyrwas
Keyword(s):  
Matrix Function ◽  
Initial Value Problems ◽  
Stability Problem ◽  
Difference Operators ◽  
Fractional Order Systems ◽  
Initial Value ◽  
Fractional Difference ◽  
Transform Method ◽  
Delta Type ◽  
The Stability

The Caputo-, Riemann-Liouville-, and Grünwald-Letnikov-type difference initial value problems for linear fractional-order systems are discussed. We take under our consideration the possible solutions via the classicalZ-transform method. We stress the formula for the image of the discrete Mittag-Leffler matrix function in theZ-transform. We also prove forms of images in theZ-transform of the expressed fractional difference summation and operators. Additionally, the stability problem of the considered systems is studied.

scholarly journals Nonlinear Voltage Control for Three-Phase DC-AC Converters in Hybrid Systems: An Application of the PI-PBC Method

Electronics ◽  
2020 ◽  
Vol 9 (5) ◽  
pp. 847 ◽  
Author(s):  
Federico M. Serra ◽  
Lucas M. Fernández ◽  
Oscar D. Montoya ◽  
Walter Gil-González ◽  
Jesus C. Hernández
Keyword(s):  
Hybrid Systems ◽  
Direct Current ◽  
Output Voltage ◽  
Lyapunov Theory ◽  
Experimental Results ◽  
Proportional Integral ◽  
Three Phase ◽  
The Stability ◽  
Ac Converters ◽  
Lc Filter

In this paper, a proportional-integral passivity-based controller (PI-PBC) is proposed to regulate the amplitude and frequency of the three-phase output voltage in a direct-current alternating-current (DC-AC) converter with an LC filter. This converter is used to supply energy to AC loads in hybrid renewable based systems. The proposed strategy uses the well-known proportional-integral (PI) actions and guarantees the stability of the system by means of the Lyapunov theory. The proposed controller continues to maintain the simplicity and robustness of the PI controls using the Hamiltonian representation of the system, thereby ensuring stability and producing improvements in the performance. The performance of the proposed controller was validated based on simulation and experimental results after considering parametric variations and comparing them with classical approaches.

Stability analysis on a hybrid PDE–ODE system describing intermittent hormonal therapy of prostate cancer

2018 ◽  
Vol 28 (03) ◽  
pp. 487-523 ◽  
Author(s):  
Kurumi Hiruko ◽  
Shinya Okabe
Keyword(s):  
Prostate Cancer ◽  
Optimal Control ◽  
Hybrid Systems ◽  
Hormonal Therapy ◽  
Sufficient Condition ◽  
Systems Modeling ◽  
Ode System ◽  
The Stability ◽  
Stable Control ◽  
Ode Systems

We consider the stability of control prescribed by hybrid PDE–ODE systems modeling intermittent hormonal therapy of prostate cancer. Hybrid systems can be regarded as a generalization of optimal control. However, since the purpose of hybrid systems is not only minimization or maximization of a corresponding functional, it is not clear what is optimal in hybrid systems. In this paper, we shall give a concept of stability of the control prescribed by the hybrid PDE–ODE systems. Moreover, we show a sufficient condition on initial data for the existence of the stable control. Finally, we apply the main result to several mathematical models describing intermittent hormonal therapy of prostate cancer.

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