scholarly journals A new H2-norm Lyapunov function for the stability of a singularly perturbed system of two conservation laws

2013 ◽  
Author(s):  
Ying Tang ◽  
Christophe Prieur ◽  
Antoine Girard
Keyword(s):  
Lyapunov Function ◽  
Conservation Laws ◽  
Singularly Perturbed ◽  
Singularly Perturbed System ◽  
Perturbed System ◽  
The Stability

scholarly journals Decomposition and stability of linear singularly perturbed systems with two small parameters

2021 ◽  
Vol 13 (1) ◽  
pp. 15-21
Author(s):  
O.V. Osypova ◽  
A.S. Pertsov ◽  
I.M. Cherevko
Keyword(s):  
Original System ◽  
Zero Solution ◽  
Singularly Perturbed ◽  
Reduction Principle ◽  
Singularly Perturbed Systems ◽  
Singularly Perturbed System ◽  
Perturbed System ◽  
Integral Manifolds ◽  
Small Parameters ◽  
The Stability

In the domain $\Omega =\left\{\left(t,\varepsilon _{1}, \varepsilon _{2} \right): t\in {\mathbb R},\varepsilon _{1}>0, \varepsilon _{2} >0\right\}$, we consider a linear singularly perturbed system with two small parameters \[ \left\{ \begin{array}{l} {\dot{x}_{0} =A_{00} x_{0} +A_{01} x_{1} +A_{02} x_{2},} \\ {\varepsilon _{1} \dot{x}_{1} =A_{10} x_{0} +A_{11} x_{1} +A_{12} x_{2},} \\ {\varepsilon _{1} \varepsilon _{2} \dot{x}_{2} =A_{20} x_{0} +A_{21} x_{1} +A_{22} x_{2},} \end{array}\right. \] where $x_{0} \in {\mathbb R}^{n_{0}}$, $x_{1} \in {\mathbb R}^{n_{1}}$, $x_{2} \in {\mathbb R}^{n_{2}}$. In this paper, schemes of decomposition and splitting of the system into independent subsystems by using the integral manifolds method of fast and slow variables are investigated. We give the conditions under which the reduction principle is truthful to study the stability of zero solution of the original system.

Multivariable PID Using Singularly Perturbed System

2014 ◽  
Vol 67 (5) ◽  
Author(s):  
Mashitah Che Razali ◽  
Norhaliza Abdul Wahab ◽  
Sharatul Izah Samsudin
Keyword(s):  
Controller Design ◽  
Treatment Plant ◽  
Original System ◽  
Singularly Perturbed ◽  
Order System ◽  
Singularly Perturbed System ◽  
Modeling And Control ◽  
Dynamic Matrix ◽  
Perturbed System ◽  
The Stability

The paper investigates the possibilities of using the singularly perturbation method in a multivariable proportional-integral-derivative (MPID) controller design. The MPID methods of Davison, Penttinen-Koivo and Maciejowski are implemented and the effective of each method is tested on wastewater treatment plant (WWTP). Basically, this work involves modeling and control. In the modeling part, the original full order system of the WWTP was decomposed to a singularly perturbed system. Approximated slow and fast models of the system were realized based on eigenvalue of the identified system. The estimated models are then used for controller design. Mostly, the conventional MPID considered static inverse matrix, but this singularly perturbed MPID considers dynamic matrix inverse. The stability of the singularly perturbed system is established by using Bode analysis, whereby the bode plot of the model system is compared to the original system. The simulation results showed that the singularly perturbed method can be applied into MPID. The three methods of MPID have been compared and the Maciejowski shows the best closed loop performance.

Output Feedback Controller Design Based on Fuzzy Singularly Perturbed Model

2012 ◽  
Vol 466-467 ◽  
pp. 1402-1406 ◽  
Author(s):  
Li Li ◽  
Fu Chun Sun
Keyword(s):  
Closed Loop ◽  
Controller Design ◽  
Reference Model ◽  
Singularly Perturbed ◽  
Singularly Perturbed System ◽  
Tracking Errors ◽  
Perturbed System ◽  
Closed Loop Systems ◽  
The Stability ◽  
Perturbed Model

We will propose a kind of new controller for the nonlinear singularly perturbed system with immeasurable states on the basis of the Fuzzy Singularly Perturbed Model (FSPM). An observer is designed to approximate the immeasurable tracking errors. Two-step approach and Schur theory are used to solve Bilinear Matrix Inequation(BMI) to achieve the gains of the controller and the observer. It can make the states of the plant to follow those of the stable reference model. Lyapunov constitute techniques can be used to prove the stability of the closed-loop systems. Finally the simulation is offered to illustrate the effectiveness of the proposed approach.

scholarly journals Complete Controllability Conditions for Linear Singularly Perturbed Time-Invariant Systems with Multiple Delays via Chang-Type Transformation

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 71 ◽  
Author(s):  
Olga Tsekhan
Keyword(s):  
Sufficient Conditions ◽  
Singularly Perturbed ◽  
Degenerate System ◽  
Multiple Delays ◽  
Complete Controllability ◽  
Singularly Perturbed System ◽  
System With Delay ◽  
Perturbed System ◽  
Time Invariant ◽  
Controllability Conditions

The problem of complete controllability of a linear time-invariant singularly-perturbed system with multiple commensurate non-small delays in the slow state variables is considered. An approach to the time-scale separation of the original singularly-perturbed system by means of Chang-type non-degenerate transformation, generalized for the system with delay, is used. Sufficient conditions for complete controllability of the singularly-perturbed system with delay are obtained. The conditions do not depend on a singularity parameter and are valid for all its sufficiently small values. The conditions have a parametric rank form and are expressed in terms of the controllability conditions of two systems of a lower dimension than the original one: the degenerate system and the boundary layer system.

NUMERICAL COMPUTATION OF CANARDS

2000 ◽  
Vol 10 (12) ◽  
pp. 2669-2687 ◽  
Author(s):  
JOHN GUCKENHEIMER ◽  
KATHLEEN HOFFMAN ◽  
WARREN WECKESSER
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Electrical Potential ◽  
Slow Manifold ◽  
Singularly Perturbed ◽  
Singularly Perturbed Systems ◽  
Singularly Perturbed System ◽  
Chemical Systems ◽  
Perturbed System ◽  
Physical And Chemical

Singularly perturbed systems of ordinary differential equations arise in many biological, physical and chemical systems. We present an example of a singularly perturbed system of ordinary differential equations that arises as a model of the electrical potential across the cell membrane of a neuron. We describe two periodic solutions of this example that were numerically computed using continuation of solutions of boundary value problems. One of these periodic orbits contains canards, trajectory segments that follow unstable portions of a slow manifold. We identify several mechanisms that lead to the formation of these and other canards in this example.

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