singular perturbation method
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The motion control of a surface ship based on a four degrees of freedom (4-DoF) (surge, sway, roll, and yaw) maneuvering motion model is studied in this paper. A time-scale decomposition method is introduced to solve the path-following problem, implementing Rudder Roll Stabilization (RRS) at the same time. The control objectives are to let the ship to track a predefined curve path under environmental disturbances, and to reduce the roll motion at the same time. A singular perturbation method is used to decouple the whole system into two subsystems of different time scales: the slow path-following subsystem and the fast roll reduction subsystem. The coupling effect of the two subsystems is also considered in this framework of analysis. RRS control is only possible when there is the so-called bandwidth separation characteristic in the ship motion system, which requires a large bandwidth separation gap between the two subsystems. To avoid the slow subsystem being affected by the wave disturbances of high frequency and large system uncertainties, the adaptive control is introduced in the slow subsystem, while a Proportion-Differentiation (PD) control law is adopted in the fast roll reduction subsystem. Simulation results show the effectiveness and robustness of the proposed control strategy.

A linear discrete stable control system is considered. The Power Factor Correction (PFC) converter to allow independent control of current and voltage. It converter are fast and slow states to inheres sty present small parameters inductor and capacitor its computes stiffness and to include switching ripple effects. As an alternative a Singular Perturbation Method (SPM) is presented Initial Value Problem (IVP) and Boundary Value Problem (BVP). It is applied to two state switching power converters to provide rigorous justification of the time scale separation. It is modeled as a one parameter singularly perturbed system. SPM consists of an outer series solution and one boundary layer correction (BLC) solution. A boundary layer correction is required to recover the initial conditions lost in the process of degeneration and to improve the solution. SPM is carried out up to second-order approximate solution for the PFC converter model for IVP and BVP. The results are compared with the exact solution (between with and without parameters). The results substantiate the application.

<pre>Classical approximations in chemical kinetics, the quasi-steady-state approximation (QSSA) and the partial-equilibrium approximation (PEA), are used to reduce rate equations for the concentrations and the extents of the reaction steps, respectively. We make precise two conditions on the rate constants necessary and sufficient to eliminate a well-chosen variable in the vicinity of a steady state. The first condition expresses that dynamics admits a small characteristic time associated with a fast variable. The second condition ensures that the fast variable is a concentration for QSSA and an extent for PEA. Both approximations exploit the zeroth order of a singular perturbation method. Eliminating a fast variable does not mean that it has reached a steady state. The fast evolution is considered over and the slow evolution of the eliminated variable is governed by the slow variables. The evolution of the slow variables occurs on a slow manifold in the space of the concentrations or the extents. In some cases the dynamics of the slow variables can be associated with a reduced chemical scheme. QSSA and PEA are applied to three chemical schemes associated with linear and nonlinear dynamics. We find that QSSA cannot be identified with the elimination of a reactive intermediate. The nonlinearities of the rate equations induce a more complex behavior.</pre>

This paper investigates the controllability of discrete-time leader-follower multiagent systems (MASs) with two-time-scale and heterogeneous features, motivated by the fact that many real systems are operating in discrete-time. In this study, singularly perturbed difference systems are used to model the two-time-scale heterogeneous discrete-time MASs. To avoid the ill-posedness problem caused by the singular perturbation parameter when using the classical control theory to study the model, the singular perturbation method was first applied to decompose the system into two subsystems with slow-time-scale and fast-time-scale feature. Then, from the perspective of algebra and graph theory, several easier-to-use controllability criteria for the related MASs are proposed. Finally, the effectiveness of the main results is verified by simulation.

<pre>Classical approximations in chemical kinetics, the quasi-steady-state approximation (QSSA) and the partial-equilibrium approximation (PEA), are used to reduce rate equations for the concentrations and the extents of the reaction steps, respectively. We make precise two conditions on the rate constants necessary and sufficient to eliminate a well-chosen variable in the vicinity of a steady state. The first condition expresses that dynamics admits a small characteristic time associated with a fast variable. The second condition ensures that the fast variable is a concentration for QSSA and an extent for PEA. Both approximations exploit the zeroth order of a singular perturbation method. Eliminating a fast variable does not mean that it has reached a steady state. The fast evolution is considered over and the slow evolution of the eliminated variable is governed by the slow variables. The evolution of the slow variables occurs on a slow manifold in the space of the concentrations or the extents. In some cases the dynamics of the slow variables can be associated with a reduced chemical scheme. QSSA and PEA are applied to three chemical schemes associated with linear and nonlinear dynamics. We find that QSSA cannot be identified with the elimination of a reactive intermediate. The nonlinearities of the rate equations induce a more complex behavior.</pre>

<pre>Classical approximations in chemical kinetics, the quasi-steady-state approximation (QSSA) and the partial-equilibrium approximation (PEA), are used to reduce rate equations for the concentrations and the extents of the reaction steps, respectively. We make precise two conditions on the rate constants necessary and sufficient to eliminate a well-chosen variable in the vicinity of a steady state. The first condition expresses that dynamics admits a small characteristic time associated with a fast variable. The second condition ensures that the fast variable is a concentration for QSSA and an extent for PEA. Both approximations exploit the zeroth order of a singular perturbation method. Eliminating a fast variable does not mean that it has reached a steady state. The fast evolution is considered over and the slow evolution of the eliminated variable is governed by the slow variables. The evolution of the slow variables occurs on a slow manifold in the space of the concentrations or the extents. In some cases the dynamics of the slow variables can be associated with a reduced chemical scheme. QSSA and PEA are applied to three chemical schemes associated with linear and nonlinear dynamics. We find that QSSA cannot be identified with the elimination of a reactive intermediate. The nonlinearities of the rate equations induce a more complex behavior.</pre>