Small Singular Perturbation Parameter Research Articles

Finite-Time Event-Triggered Stabilization for Discrete-Time Fuzzy Markov Jump Singularly Perturbed Systems.

The finite-time event-triggered stabilization is studied for a class of discrete-time nonlinear Markov jump singularly perturbed models with partially unknown transition probabilities (TPs). T-S fuzzy strategy is adopted to characterize the related nonlinear Markov jump singularly perturbed models. The control objective is to make sure that the system states remain within a bounded domain during a fixed-time interval. First, a mode-dependent event-triggered scheme is constructed to reduce the communication burden and save the network bandwidth. On that basis, by using a new Lyapunov function, a developed finite-time stability criterion is derived for the corresponding system to avoid an ill-conditioned issue due to a small singular perturbation parameter. Moreover, the mode-dependent fuzzy controller gain and the event-triggered parameter are co-designed under the framework of partially unknown TPs. Finally, the feasibility of the main results is provided to verify the finite-time event-triggered control strategy.

Designing Distributed Impulsive Controller for Networked Singularly Perturbed Systems

This paper develops a novel synthesis approach for synchronization of a network of singularly perturbed systems (SPSs) with a small singular perturbation parameter (SPP) <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> via distributed impulsive control. First, a decoupling method in the setting of directed networks is employed to decompose networked SPSs related to complex eigenvalues of the Laplacian matrix. Then, based on an improved piecewise continuous Lyapunov function, an <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula>-dependent synchronization criterion is established. The relationship among the impulse interval, the impulse gain matrix and <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> is revealed. By employing the newly-obtained synchronization criterion, sufficient conditions on the existence of an <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula>-dependent impulse gain matrix are derived. Finally, an example is simulated to verify the effectiveness of the theoretical results.

LMI-based Control of Singularly Perturbed Switched Affine Systems

In this paper, we tackle the problem of stabilizing singularly perturbed switched affine systems. Even though efficient control strategies based on the solution of Linear Matrix Inequalities (LMIs) have been presented in the switched affine systems literature, the presence of the small singular perturbation parameter may introduce numerical difficulties in the solution of these LMIs due to ill conditioning. Moreover, this parameter may be uncertain or its use in the control law may not be practical. Here, we propose an LMI-based control design strategy that avoids the conditioning issues and that also provides an estimation of the domain of attraction. Furthermore, the resulting controller does not depend on the singular perturbation parameter. The proposed method is illustrated on a numerical example.

Finite-Time Stabilization of Markov Switching Singularly Perturbed Models

This brief is concerned with the issue of finite-time stabilization of discrete-time stochastic singularly perturbed models, in which the stochastic process is regulated by a Markov chain with partially unknown transition probabilities (TPs). The slow-state and fast-state variable are considered in the modeling, and the corresponding Markov switching model with a singularly perturbed parameter is obtained in a unified framework. Ill-conditioned problems caused by a small singular perturbation parameter are prevented by developing a finite-time stability criterion for the resultant system. Furthermore, feasible conditions are derived for the desired finite-time state feedback controller by using matrix inequalities that are independent of the singularly perturbed parameter. Finally, a gear-driven DC motor model is applied to illustrate the effectiveness of the described control strategy.

Synchronization for singularity-perturbed complex networks via event-triggered impulsive control

&lt;p style='text-indent:20px;'&gt;This paper studies synchronization of singularity-perturbed complex networks (SPCNs) with a small singular perturbation parameter (SPP) via event-triggered impulsive control (ETIC). A novel dynamic event-triggered mechanism is proposed where an auxiliary impulse parameter is introduced to regulate the triggering threshold dynamically for saving the network resource. Based on SPP-dependent Lyapunov function, some sufficient conditions involving the impulsive gain, triggering parameters and singular perturbation parameter (SPP) are obtained to synchronize the SPCNs, and the upper bound of SPP is also determined. Moreover, it proves that the Zeno behavior can be excluded. Finally, two simulations are provided to demonstrate the validity of the obtained results.&lt;/p&gt;

Numerical analysis for a class of coupled system of singularly perturbed time‐dependent convection‐diffusion equations with a discontinuous source term

AbstractIn this article, we present a parameters‐uniform numerical method for a time‐dependent weakly coupled system of two convection‐diffusion equations that has a discontinuity, along the line in the source term. The second order term of each equation is multiplied by a small singular perturbation parameter and these parameters are assumed to be different in magnitude. On an appropriate Shishkin mesh, the considered problem is discretized using an upwind central difference scheme away from the line of discontinuity, and a particular upwind central difference scheme along the line of discontinuity. The numerical approximations produced by this scheme are uniformly convergent of first‐order in time and almost first‐order in space with respect to both perturbation parameters. Numerical results are presented to validate the theoretical results.

Singular Limit in Hopf Bifurcation for Doubly Diffusive Convection Equations II: Bifurcation and Stability

A singular perturbation problem from the artificial compressible system to the incompressible system is considered for a doubly diffusive convection when a Hopf bifurcation from the motionless state occurs in the incompressible system. It is proved that the Hopf bifurcation also occurs in the artificial compressible system for small singular perturbation parameter, called the artificial Mach number. The time periodic solution branch of the artificial compressible system is shown to converge to the corresponding bifurcating branch of the incompressible system in the singular limit of vanishing artificial Mach number.

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

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.

A Posteriori Error Estimates for Hughes Stabilized SUPG Technique and Adaptive Refinement for a Convection-Diffusion Problem

The motive of the present work is to propose an adaptive numerical technique for singularly perturbed convection-diffusion problem in two dimensions. It has been observed that for small singular perturbation parameter, the problem under consideration displays sharp interior or boundary layers in the solution which cannot be captured by standard numerical techniques. In the present work, Hughes stabilization strategy along with the streamline upwind/Petrov-Galerkin (SUPG) method has been proposed to capture these boundary layers. Reliable a posteriori error estimates in energy norm on anisotropic meshes have been developed for the proposed scheme. But these estimates prove to be dependent on the singular perturbation parameter. Therefore, to overcome the difficulty of oscillations in the solution, an efficient adaptive mesh refinement algorithm has been proposed. Numerical experiments have been performed to test the efficiency of the proposed algorithm.

Robin boundary value problems for a singularly perturbed weakly coupled system of convection–diffusion equations having discontinuous source term

In this paper, we consider a weakly coupled system of convection–diffusion equations subject to Robin boundary conditions, and having boundary and interior layers. The diffusion term of each equation is multiplied by a small singular perturbation parameter, but these parameters are assumed to be different in magnitude, and the source term is having a discontinuity at a point in the interior of the domain. An upwind scheme is used for the considered problem in conjunction with piecewise uniform Shishkin mesh. It is proved that the numerical approximations produced by this method are almost first order uniformly convergent with respect to both small parameters. Numerical results are presented to validate the theoretical results.

Parameter-uniform convergence of a numerical method for a coupled system of singularly perturbed semilinear reaction–diffusion equations with boundary and interior layers

In this paper, we consider a coupled system of m ( ≥ 2 ) singularly perturbed semilinear reaction–diffusion equations with a discontinuous source term having a discontinuity at a point in the interior of the domain. The diffusion term of each equation is multiplied by small singular perturbation parameter, but these parameters are assumed to be different in magnitude. A numerical method is constructed on a variant of Shishkin mesh. The approximations generated by this method are shown to be almost second order uniformly convergent with respect to all perturbation parameters. Numerical results are in support of the theoretical results. • A coupled system of m ( ≥ 2 ) singularly perturbed semilinear reaction-diffusion equations is considered. • The source term is having a discontinuity at a point in the interior of the domain. • The diffusion term of each equation is multiplied by a small perturbation parameter. • All the perturbation parameters are different in magnitude. • The constructed numerical method has almost second parameter uniform convergence.

New Designs of Linear Observers and Observer-Based Controllers for Singularly Perturbed Linear Systems

This paper considers the design of observers for systems with slow and fast modes. The existing methods are able to design independent slow and fast observers with $O(\epsilon)$ accuracy only, where $\epsilon$ is a small positive singular perturbation parameter that indicates separation of system state-space variables into slow and fast. In this paper, the design of independent slow and fast reduced-order observers was performed with very high accuracy $O(\epsilon ^{i}), i=2,3,....$ . The results obtained are extended to the design of corresponding observer driven controllers. The design allows complete time-scale separation for both the observer and controller through the decomposition into slow and fast time scales. This method reduces both offline and online computations. The effectiveness of the new methods is demonstrated through both theoretical and simulation results.

Optimal uniform-convergence results for convection–diffusion problems in one dimension using preconditioning

A linear one-dimensional convection–diffusion problem with a small singular perturbation parameter ε is considered. The problem is discretized using finite-difference schemes on the Shishkin mesh. Generally speaking, such discretizations are not consistent uniformly in ε , so ε -uniform convergence cannot be proved by the classical approach based on ε -uniform stability and ε -uniform consistency. This is why previous proofs of convergence have introduced non-classical techniques (e.g., specially chosen barrier functions). In the present paper, we show for the first time that one can prove optimal convergence inside the classical framework: a suitable preconditioning of the discrete system is shown to yield a method that, uniformly in ε , is both consistent and stable. Using this technique, optimal error bounds are obtained for the upwind and hybrid finite-difference schemes.

New Designs of Reduced-Order Observer-Based Controllers for Singularly Perturbed Linear Systems

The slow and fast reduced-order observers and reduced-order observer-based controllers are designed by using the two-stage feedback design technique for slow and fast subsystems. The new designs produce an arbitrary order of accuracy, while the previously known designs produce the accuracy of O(ϵ) only where ϵ is a small singular perturbation parameter. Several cases of reduced-order observer designs are considered depending on the measured state space variables: only all slow variables are measured, only all fast variables are measured, and some combinations of the slow and fast variables are measured. Since the two-stage methods have been used to overcome the numerical ill-conditioning problem for Cases (III)–(V), they have similar procedures. The numerical ill-conditioning problem is avoided so that independent feedback controllers can be applied to each subsystem. The design allows complete time-scale separation for both the reduced-order observer and controller through the complete and exact decomposition into slow and fast time scales. This method reduces both offline and online computations.

Recursive Reduced-Order Algorithm for Singularly Perturbed Cross Grammian Algebraic Sylvester Equation

A new recursive algorithm is developed for solving the algebraic Sylvester equation that defines the cross Grammian of singularly perturbed linear systems. The cross Grammian matrix provides aggregate information about controllability and observability of a linear system. The solution is obtained in terms of reduced-order algebraic Sylvester equations that correspond to slow and fast subsystems of a singularly perturbed system. The rate of convergence of the proposed algorithm isOε, whereεis a small singular perturbation parameter that indicates separation of slow and fast state variables. Several real physical system examples are solved to demonstrate efficiency of the proposed algorithm.

Approximating Dynamics of a Singularly Perturbed Stochastic Wave Equation with a Random Dynamical Boundary Condition

This work is concerned with a singularly perturbed stochastic nonlinear wave equation with a random dynamical boundary condition. A splitting is used to establish the approximating equation of the system for a sufficiently small singular perturbation parameter. The approximating equation is a stochastic parabolic equation when the power exponent of the singular perturbation parameter is in $[1/2, 1)$ but is a deterministic wave equation when the power exponent is in $(1, +\infty)$. Moreover, if the power exponent of a singular perturbation parameter is bigger than or equal to $1/2$, the same limiting equation of the system is derived in the sense of distribution, as the perturbation parameter tends to zero. This limiting equation is a deterministic parabolic equation.

Robust Stability of Singularly Perturbed Impulsive Systems Under Nonlinear Perturbation

The robust stability problem for singularly perturbed impulsive systems under nonlinear perturbation is considered. The uncertainties are assumed to be limited by their upper norm bound. By applying the vector Lyapunov function method and a two-time scale comparison principle, a sufficient condition that ensures robust exponential stability for sufficiently small singular perturbation parameter is derived. Moreover, the stability bound of the singular perturbation parameter can be obtained by solving a set of matrix inequalities. Finally, two numerical examples are given to illustrate the effectiveness of the proposed results.

Exponential stability of a class of singularly perturbed stochastic time-delay systems with impulse effect

In this paper, the exponential stability of a class of singularly perturbed stochastic time-delay systems with impulse effect is considered. By using the comparison principle and time scale decomposition, a Lyapunov-based sufficient condition for exponential stability in mean square for a sufficiently small singular perturbation parameter ε is derived. Some known results are generalized and improved.

Efficient numerical procedures for solving closed-loop Stackelberg strategies with small singular perturbation parameter

In this paper, the computation of the linear closed-loop Stackelberg strategies with small singular perturbation parameter that characterizes singularly perturbed systems (SPS) are studied. The attention is focused on a new numerical algorithm for solving a set of cross-coupled algebraic Lyapunov and Riccati equations (CALRE). It is proven that the new algorithm guarantees the local quadratic convergence. A numerical example is solved to show a reduction of the average CPU time compared with the existing algorithm.

A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations

A Dirichlet boundary value problem for a delay parabolic differential equation is studied on a rectangular domain in the x - t plane. The second-order space derivative is multiplied by a small singular perturbation parameter, which gives rise to parabolic boundary layers on the two lateral sides of the rectangle. A numerical method comprising a standard finite difference operator (centred in space, implicit in time) on a rectangular piecewise uniform fitted mesh of N x × N t elements condensing in the boundary layers is proved to be robust with respect to the small parameter, or parameter-uniform, in the sense that its numerical solutions converge in the maximum norm to the exact solution uniformly well for all values of the parameter in the half-open interval ( 0 , 1 ] . More specifically, it is shown that the errors are bounded in the maximum norm by C ( N x - 2 ln 2 N x + N t - 1 ) , where C is a constant independent not only of N x and N t but also of the small parameter. Numerical results are presented, which validate numerically this theoretical result and show that a numerical method consisting of the standard finite difference operator on a uniform mesh of N x × N t elements is not parameter-robust.