scholarly journals Stability of fractional positive nonlinear systems

2015 ◽  
Vol 25 (4) ◽  
pp. 491-496 ◽  
Author(s):  
Tadeusz Kaczorek
Keyword(s):  
Nonlinear Systems ◽  
Continuous Time ◽  
Lyapunov Method ◽  
Linear Part ◽  
Stability Conditions ◽  
Continuous Time Systems ◽  
Nonlinear Vector ◽  
The Stability ◽  
Metzler Matrix ◽  
Time Systems

AbstractThe conditions for positivity and stability of a class of fractional nonlinear continuous-time systems are established. It is assumed that the nonlinear vector function is continuous, satisfies the Lipschitz condition and the linear part is described by a Metzler matrix. The stability conditions are established by the use of an extension of the Lyapunov method to fractional positive nonlinear systems.

scholarly journals Lyapunov Functions for State Observers of Dynamic Systems Using Hamilton–Jacobi Inequalities

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 202 ◽  
Author(s):  
Angelo Alessandri
Keyword(s):  
Nonlinear Systems ◽  
Dynamic Systems ◽  
Continuous Time ◽  
Lyapunov Functions ◽  
Estimation Error ◽  
State Observers ◽  
Continuous Time Systems ◽  
Estimation Problems ◽  
The Stability ◽  
Time Systems

Lyapunov functions enable analyzing the stability of dynamic systems described by ordinary differential equations without finding the solution of such equations. For nonlinear systems, devising a Lyapunov function is not an easy task to solve in general. In this paper, we present an approach to the construction of Lyapunov funtions to prove stability in estimation problems. To this end, we motivate the adoption of input-to-state stability (ISS) to deal with the estimation error involved by state observers in performing state estimation for nonlinear continuous-time systems. Such stability properties are ensured by means of ISS Lyapunov functions that satisfy Hamilton–Jacobi inequalities. Based on this general framework, we focus on observers for polynomial nonlinear systems and the sum-of-squares paradigm to find such Lyapunov functions.

scholarly journals Metzler cyclic electric systems

2017 ◽  
pp. 109-118
Author(s):  
Wojciech Mitkowski
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Continuous Time ◽  
Dynamic Properties ◽  
Electrical Circuit ◽  
Continuous Time Systems ◽  
Linear Differential ◽  
Metzler Matrix ◽  
Time Systems

In applications we meet with positive (nonnegative) systems. A system generated by a linear differential equation with a Metzler-matrix is called a nonnegative system. The paper shows the basic dynamic properties of nonnegative continuous-time systems. Considered was the example of an oscillating electrical circuit with a cyclic Metzler-matrix, enabling the correct physical interpretations. Cykliczne obwody elektryczne Metzlera Streszczenie: Układy dodatnie (nieujemne) można spotkać w wielu zastosowaniach. Układem nieujemnym jest układ generowany równaniem różniczkowym liniowym z macierzą Metzlera. W artykule przedstawiono podstawowe własności dynamiczne układów nieujemnych z czasem ciągłym. Rozważono przykład elektrycznego obwodu oscylacyjnego z cykliczną macierzą Metzlera umożliwiający odpowiednie interpretacje fizyczne. Słowa kluczowe: układy dodatnie, układy Metzlera, układy cykliczne, obwody elektryczne

scholarly journals Positive stable realizations with system Metzler matrices

2011 ◽  
Vol 21 (2) ◽  
pp. 167-188 ◽  
Author(s):  
Tadeusz Kaczorek
Keyword(s):  
Transfer Matrix ◽  
Continuous Time ◽  
Similarity Transformation ◽  
Transfer Matrices ◽  
Numerical Examples ◽  
Continuous Time Systems ◽  
Metzler Matrix ◽  
Time Systems

Positive stable realizations with system Metzler matricesConditions for the existence of positive stable realizations with system Metzler matrices for linear continuous-time systems are established. A procedure for finding a positive stable realization with system Metzler matrix based on similarity transformation of proper transfer matrices is proposed and demonstrated on numerical examples. It is shown that if the poles of stable transfer matrix are real then the classical Gilbert method can be used to find the positive stable realization.

Positivity and Stability of Standard and Fractional Descriptor Continuous-Time Linear and Nonlinear Systems

2018 ◽  
Vol 19 (3-4) ◽  
pp. 299-307 ◽  
Author(s):  
Tadeusz Kaczorek
Keyword(s):  
Nonlinear Systems ◽  
Continuous Time ◽  
Lyapunov Method ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear And Nonlinear ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Linear And Nonlinear Systems ◽  
Time Linear

AbstractThe positivity and stability of standard and fractional descriptor continuous-time linear and nonlinear systems are addressed. Necessary and sufficient conditions for the positivity of descriptor linear and sufficient conditions for nonlinear systems are established. Using an extension of Lyapunov method sufficient conditions for the stability of positive nonlinear systems are given. The considerations are extended to fractional nonlinear systems.

scholarly journals Positivity and Linearization of a Class of Nonlinear Continuous–Time Systems by State Feedbacks

2015 ◽  
Vol 25 (4) ◽  
pp. 827-831 ◽  
Author(s):  
Tadeusz Kaczorek
Keyword(s):  
Nonlinear Systems ◽  
Continuous Time ◽  
State Feedback ◽  
Sufficient Conditions ◽  
Closed Loop System ◽  
Continuous Time Systems ◽  
Necessary And Sufficient ◽  
Time Systems ◽  
Nonlinear State ◽  
Continuous Time System

Abstract The positivity and linearization of a class of nonlinear continuous-time system by nonlinear state feedbacks are addressed. Necessary and sufficient conditions for the positivity of the class of nonlinear systems are established. A method for linearization of nonlinear systems by nonlinear state feedbacks is presented. It is shown that by a suitable choice of the state feedback it is possible to obtain an asymptotically stable and controllable linear system, and if the closed-loop system is positive then it is unstable.

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