Stabilization of Discrete Singularly Perturbed Systems Under Composite Observer-Based Control

1999 ◽  
Vol 123 (1) ◽  
pp. 132-139 ◽  
Author(s):  
Feng-Hsiag Hsiao ◽  
Jiing-Dong Hwang ◽  
Shing-Tai Pan
Keyword(s):  
Singular Perturbation ◽  
Stability Criterion ◽  
Original System ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Stability Conditions ◽  
Singularly Perturbed Systems ◽  
Singularly Perturbed System ◽  
Perturbed System ◽  
Perturbed Systems

New stability conditions for discrete singularly perturbed systems are presented in this study. The corresponding slow and fast subsystems of the original discrete singularly perturbed system are first derived. The observer-based controllers for the slow and the fast subsystems are then separately designed and a composite observer-based controller for the original system is subsequently synthesized from these observer-based controllers. Finally, a frequency domain ε-dependent stability criterion for the original discrete singularly perturbed system under the composite observer-based controller is proposed. If any one condition of this criterion is fulfilled, stability of the original system by establishing that of its corresponding slow and fast subsystems is thus investigated. An illustrative example is given to demonstrate that the upper bound of the singular perturbation parameter ε can be obtained by examining this criterion.

Discrete-Time Nonlinear Singularly Perturbed System Identification Using Coupled Multimodel Representation

2020 ◽  
Vol 143 (2) ◽  
Author(s):  
Asma Ben Rajab ◽  
Nesrine Bahri ◽  
Majda Ltaief
Keyword(s):  
Discrete Time ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
System Model ◽  
Model Parameters ◽  
Singularly Perturbed Systems ◽  
Perturbed System ◽  
Marquardt Algorithm ◽  
Perturbed Systems ◽  
Full Knowledge

Abstract Many control and observability theories for singularly perturbed systems require the full knowledge of system model parameters exceptionally if the system is considered as black box. To overcome this problem and to obtain an accurate and faithful model, this paper describes a new identification method for discrete-time nonlinear singularly perturbed systems (NLSPS) using the coupled state multimodel representation. The Levenberg–Marquardt algorithm is used to identify not only the submodels parameters but also the perturbation parameter ε. Two cases are considered to identify these systems. The first one supposes that the perturbation parameter ε of the real system is known and thus only the submodels parameters are identified. The second case supposes that this perturbation parameter is unknown and has to be identified with the other submodels parameters. The simulation example demonstrates the effectiveness of the proposed identification.

scholarly journals Passive Control andε-Bound Estimation of Singularly Perturbed Systems with Nonlinear Nonlinearities

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Linna Zhou ◽  
Chunyu Yang
Keyword(s):  
Singular Perturbation ◽  
Passive Control ◽  
Controller Design ◽  
Design Method ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Singularly Perturbed Systems ◽  
Storage Function ◽  
Rlc Circuit ◽  
Perturbed Systems

This paper considers the problems of passivity analysis and synthesis of singularly perturbed systems with nonlinear uncertainties. By a novel storage function depending on the singular perturbation parameterε, a new method is proposed to estimate theε-bound, such that the system is passive when the singular perturbation parameter is lower than theε-bound. Furthermore, a controller design method is proposed to achieve a predefinedε-bound. The proposed results are shown to be less conservative than the existing ones because the adopted storage function is more general. Finally, an RLC circuit is presented to illustrate the advantages and effectiveness of the proposed methods.

Duck Trajectories of Three-Dimensional Singularly Perturbed Systems

2007 ◽  
Vol 14 (2) ◽  
pp. 341-350
Author(s):  
Nikolai Kh. Rozov
Keyword(s):  
Three Dimensional ◽  
Slow Motion ◽  
Singularly Perturbed ◽  
Fast Variable ◽  
Singularly Perturbed Systems ◽  
Singularly Perturbed System ◽  
Motion Trajectories ◽  
Perturbed System ◽  
General Manner ◽  
Perturbed Systems

Abstract For the singularly perturbed system of three equations with one fast variable and two slow ones the problem of the emergence of duck trajectories is considered in the case with two different slow motion trajectories intersecting in a general manner.

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.

scholarly journals Additive fault tolerant control of nonlinear singularly perturbed systems against actuator fault

2017 ◽  
Vol 68 (1) ◽  
pp. 68-73 ◽  
Author(s):  
Adel Tellili ◽  
Aymen Elghoul ◽  
Mohamed Naceur Abdelkrim
Keyword(s):  
Singular Perturbation ◽  
Fault Tolerant ◽  
Fault Tolerant Control ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Actuator Faults ◽  
Singularly Perturbed Systems ◽  
Singular Perturbation Method ◽  
Numerical Application ◽  
Perturbed Systems

AbstractThis paper presents the design of an additive fault tolerant control for nonlinear time-invariant singularly perturbed systems against actuator faults based on Lyapunov redesign principle. The overall system is reduced into subsystems with fast and slow dynamic behavior using singular perturbation method. The time scale reduction is carried out when the singular perturbation parameter is set to zero, thus avoiding the numerical stiffness due to the interaction of two different dynamics. The fault tolerant controller is computed in two steps. First, a nominal composite controller is designed using the reduced subsystems. Secondly, an additive part is combined with the basic controller to overcome the fault effect. The derived ε - independent controller guarantees asymptotic stability despite the presence of actuator faults. The Lyapunov stability theory is used to prove the stability provided the singular perturbation parameter is sufficiently small. The theoretical results are simulated using a numerical application.

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.

scholarly journals Diagnosis of Discrete-Time Singularly Perturbed Systems Based on Slow Subsystem

2014 ◽  
Vol 8 (4) ◽  
pp. 175-180 ◽  
Author(s):  
Adel Tellili ◽  
Nouceyba Abdelkrim ◽  
Bahaa Jaouadi ◽  
Mohamed Naceur Abdelkrim
Keyword(s):  
Discrete Time ◽  
Diagnostic Algorithm ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Order System ◽  
Singularly Perturbed Systems ◽  
Perturbed Systems ◽  
Parity Space ◽  
The One ◽  
Slow Subsystem

Abstract This paper deals with the diagnosis of discrete-time singularly perturbed systems presenting two time scales property. Parity space method is considered to generate the fault detection residual. The focus is in two directions. First, we discuss the residual illconditioning caused by the singular perturbation parameter. Then, the use of the slow subsystem is considered to make the fault diagnosis easier. It is shown that the designed diagnostic algorithm based on reduced order model is close to the one synthesized using the full order system. The developed approach aims at reducing the computational load and the ill-conditioning for stiff residual generation problem. Two examples of application are used to demonstrate the efficiency of the proposed method.

scholarly journals On Output Feedback Multiobjective Control for Singularly Perturbed Systems

2011 ◽  
Vol 2011 ◽  
pp. 1-28
Author(s):  
Mehdi Ghasem Moghadam ◽  
Mohammad Taghi Hamidi Beheshti
Keyword(s):  
Numerical Simulations ◽  
Output Feedback ◽  
Design Procedure ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Dynamic Output Feedback ◽  
Singularly Perturbed Systems ◽  
Dynamic Output ◽  
Perturbed Systems ◽  
And Performance

A new design procedure for a robust and control of continuous-time singularly perturbed systems via dynamic output feedback is presented. By formulating all objectives in terms of a common Lyapunov function, the controller will be designed through solving a set of inequalities. Therefore, a dynamic output feedback controller is developed such that and performance of the resulting closed-loop system is less than or equal to some prescribed value. Also, and performance for a given upperbound of singular perturbation parameter are guaranteed. It is shown that the -dependent controller is well defined for any and can be reduced to an -independent one so long as is sufficiently small. Finally, numerical simulations are provided to validate the proposed controller. Numerical simulations coincide with the theoretical analysis.

scholarly journals Reduced-Order Algorithm for Eigenvalue Assignment of Singularly Perturbed Linear Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Heonjong Yoo ◽  
Zoran Gajic ◽  
Kyeong-Hwan Lee
Keyword(s):  
Singular Perturbation ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Sylvester Equation ◽  
Singularly Perturbed Systems ◽  
Reduced Order ◽  
Recursive Solution ◽  
System Variables ◽  
Perturbed Linear Systems ◽  
Eigenvalue Assignment

In this paper, we present an algorithm for eigenvalue assignment of linear singularly perturbed systems in terms of reduced-order slow and fast subproblem matrices. No similar algorithm exists in the literature. First, we present an algorithm for the recursive solution of the singularly perturbed algebraic Sylvester equation used for eigenvalue assignment. Due to the presence of a small singular perturbation parameter that indicates separation of the system variables into slow and fast, the corresponding algebraic Sylvester equation is numerically ill-conditioned. The proposed method for the recursive reduced-order solution of the algebraic Sylvester equations removes ill-conditioning and iteratively obtains the solution in terms of four reduced-order numerically well-conditioned algebraic Sylvester equations corresponding to slow and fast variables. The convergence rate of the proposed algorithm is Oε, where ε is a small positive singular perturbation parameter.

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.

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