new results on composite control of singularly perturbed uncertain linear systems abstract we consider the problem of stabilizing uncertain singularly perturbed linear time varying systems whose fast dynamics can be unstable we propose a class of nonlinear composite controllers which assure global uniform ultimate boundedness of the trajectories of the closed loop system provided the singular perturbation parameter is sufficiently small these controllers consist of a fast controller which stabilizes the fast dynamics and a slow controller which yields the desired stability properties for the slow dynamics we consider the structure of the fast controller to be simpler than structures previously proposed in the literature to obtain these controllers we first develop some new results for singularly perturbed systems under output feedback in particular it is shown that the matching assumption which deals with the manner in which the uncertainties enter the system and is made on the reduced order system is invariant under linear feedback of the fast variable

balancing for a class of non linear singularly perturbed systems in this work a balancing method is investigated for a class of non linear singularly perturbed systems the main result presented here shows that the well known two stages strategy used with singular perturbations in control theory can be extended to compute a balancing form of non linear singularly perturbed systems so an approximate balancing form is derived from the balancing forms of the slow and fast subsystems both computed separately this two stage method avoids the difficult task of solving high dimensional and ill conditioned hamilton jacobi equations due to the presence of the small singular perturbation parameter

multivariable circle criteria for multiparameter singularly perturbed systems a direct approach to the lur e problem for singularly perturbed systems sps is proposed in contrast to previous results the feedback connection between the linear and nonlinear parts of sps is allowed to depend essentially on both the slow and the fast variables the lur e problem for multiparameter sps is studied by the same framework

coupling an eulerian fluid calculation to a lagrangian solid calculation with the ghost fluid method we propose a numerical method for modeling multimaterial flows where the domain is decomposed into separate eulerian and lagrangian subdomains that is the equations are written in eulerian form in one subdomain and in lagrangian form in the other subdomain this is of interest for example when considering the effect of underwater explosions on the hull of a ship or the impact of a low speed projectile on a soft explosive target on the one hand high speed fluid flows are traditionally modeled by applying shock capturing schemes to the compressible euler equations to avoid problems associated with tangling of a lagrangian mesh on the other hand solid dynamics calculations are traditionally carried out using lagrangian numerical methods to avoid problems associated with numerical smearing in eulerian calculations we use the ghost fluid method to create accurate discretizations across the eulerian lagrangian interface the numerical method is presented in both one and two spatial dimensions three dimensional extensions to the interface coupling method are straightforward

robust stabilization of singularly perturbed systems abstract we discuss the problem of designing stabilizing controllers for singularly perturbed systems on the basis of simplified models in 1 it was shown that a constant gain output feedback controller designed on the basis of the simplified model need not stabilize the true system containing both fast and slow modes this phenomenon was then expanded to include the case where the simplified system is strictly proper in 2 the objectives of this note are threefold i to show that given any proper system and any stabilizing controller for it that is proper but not strictly proper there exists a singular perturbation of the system that is destabilized by that controller ii to show that any strictly proper controller for a singularly perturbed system designed on the basis of a reduced order model will stabilize the true system for sufficiently small values of the fast dynamics parameter and iii to provide a characterization in the same spirit as 3 4 of the set of all strictly proper controllers that stabilize a given proper plant by combining these results it is possible to generate the class of all robustly stabilizing controllers for a given singularly perturbed system

a further note on the robustness of output feedback control methods to modeling errors in an earlier note it was shown that designing a nonstrictly proper output feedback compensator using a model that neglects parasitic elements might lead to an unstable closed loop system even when the neglected modes are asymptotically stable the purpose of this note is to show that this instability phenomenon might take place even when the reduced order design model is strictly proper

the guaranteed cost control problem of uncertain singularly perturbed systems abstract in this paper we study the algebraic riccati equation corresponding to the guaranteed cost control theory for an uncertain singularly perturbed system the construction of the controller involves solving the full order algebraic riccati equation with small parameter e under control oriented assumptions we first provide the sufficient conditions such that the full order algebraic riccati equation has a positive semi definite stabilizing solution next we propose an iterative algorithm based on the kleinman algorithm to solve the algebraic riccati equation which depends on the parameter e our new idea is to use the solutions of the reduced order algebraic riccati equations for the initial condition by using the iterative algorithm we can easily obtain a required solution of the algebraic riccati equation moreover using the initial conditions without e we show that there exists an e such that the proposed algorithm has quadratic convergence finally in order to show the effectiveness of the proposed algorithm numerical examples are included

near optimal output feedback regulation of ill conditioned linear systems a two time scale approximation for the linear quadratic optimal output feedback regulator program is examined necessary conditions for optimality as well as an algorithm for computing locally near optimal gains are derived if it is assumed that the slow and fast subsystem initial conditions are uniformly distributed optimal gains for the two time scale problem provide a second order approximation to optimal closed loop performance in the unperturbed system this is verified with a numerical example

composite control of a singularly perturbed uncertain system with slow non linearities a practical stability controller for a class of non linear singularly perturbed uncertain dynamical systems is designed using the composite control technique the class of systems considered has non linearities that depend upon the slow dynamics of the system the system is uncertain in the slow dynamics as only a nominal slow behaviour is assumed to be known together with some bound on the non linearities the controller guarantees practical stability in the sense that it drives the state of the system in a sufficiently small neighbourhood of the origin

robust model tracking control for a class of nonlinear plants we propose a new model following control scheme for a class of nonlinear plants the feedback control signal is a continuous function of all its arguments it is shown that this scheme guarantees that tracking error remains bounded and tends to a neighborhood of the origin with a rate not inferior to an exponential one furthermore it allows the designer to arbitrarily prescribe the rate of convergence and the size of the set of ultimate boundedness

on the robustness of output feedback control methods to modeling errors singular perturbation analysis is employed to study the robustness of output feedback control schemes for linear time invariant systems the effect of designing the control strategy using a model that neglects unknown parasitic elements is considered it is shown that such a strategy may in general be destabilizing even when the neglected modes are asymptotically stable special structures that will not permit this situation and will guarantee asymptotic stability are illustrated

feedback control of two time scale nonlinear systems this article concerns a class of two time scale single input single output nonlinear systems for such systems combination of singular perturbation and geometric methods is employed to synthesize well conditioned static state feedback laws that induce a well characterized input output behaviour and guarantee stability of the closed loop system the developed control laws are applied to two nonlinear chemical processes with time scale multiplicity and their performance is evaluated through simulations

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

singular perturbations and input to state stability this paper establishes a type of total stability for the input to state stability property with respect to singular perturbations in particular if the boundary layer system is uniformly globally asymptotically stable and the reduced system is input to state stable with respect to disturbances then these properties continue to hold up to an arbitrarily small offset for initial conditions disturbances and their derivatives in an arbitrarily large compact set as long as the singular perturbation parameter is sufficiently small

robust semiglobally practical stabilization for nonlinear singularly perturbed systems in this work the problem of semiglobally practical stabilization is considered for nonlinear singularly perturbed systems with unknown parameters the composite lyapunov function for the full systems is established by both that of the slow subsystem and the boundary layer system a state feedback control law for the linear part of the slow subsystem and boundary layer system is proposed which renders the whole closed loop system semiglobally stable the upper bound expression of e is given to obtain the condition of asymptotic stability for the system a simulation example is given to demonstrate the effectiveness and feasibility of the controller

on output feedback control of singularly perturbed systems we treat the problem of robustness of output feedback controllers with respect to singular perturbations given a singularly perturbed control system whose boundary layer system is exponentially stable and whose reduced order system is exponentially stabilizable via a possibly dynamical output feedback controller we present a sufficient condition which ensures that the system obtained by applying the same controller to the original full order singularly perturbed control system is exponentially stable for sufficiently small values of the perturbation parameter this condition which is less restrictive than those previously given in the literature is shown to be always satisfied when the singular perturbation is due to the presence of fast actuators and or sensors furthermore we show explicitly that in the linear time invariant case if this condition is not satisfied then there exists an output feedback controller which stabilizes the reduced order system but destabilizes the full order system

h sub 2 guaranteed cost control for singularly perturbed uncertain systems the h sub 2 guaranteed cost control problem for a singularly perturbed norm bounded uncertain system is addressed using the quadratic stabilizability framework after defining the corresponding slow and fast uncertain subsystems the set of quadratic stabilizing composite controls is characterized two riccati equations have to be solved one for the slow subsystem and the other for the fast subsystem choosing appropriately the weighting matrices it is shown how to pick up in the set of quadratic stabilizing composite controls a control minimizing an upper bound on the h sub 2 norm of a certain transfer matrix the case of the guaranteed cost control problem for the reduced system is also investigated

steady and unsteady shock waves on overlapping grids the interaction between steady and unsteady shock waves and interfaces of overlapping grids is studied conditions permitting shock penetration through an interface and ensuring solution uniqueness are derived easily usable corrections ensuring shock interface penetration and convergence are devised

a new approach to robust guaranteed cost control for uncertain multimodeling systems the guaranteed cost control problem for multimodeling systems with norm bounded uncertainty is investigated the main contribution in this paper is that a new independent controller is derived by solving the reduced order slow and fast algebraic riccati equations ares whose dimension is smaller than the dimension of full order multiparameter algebraic riccati equation mare it is shown that if these ares have a positive definite stabilizing solution then the closed loop system is quadratically stable and has the cost bound

robust semi global output tracking for nonlinear singularly perturbed systems in this work a robust controller design methodology for a broad class of singularly perturbed nonlinear systems with time varying uncertainties is developed when the fast subsystem is stabilizable and the slow subsystem is input output linearizable with input to state stable iss inverse dynamics the developed state feedback controller guarantees boundedness of the trajectories of the closed loop system and robust output tracking with arbitrary degree of asymptotic attenuation of the effect of uncertainty on the output for initial conditions and uncertainties in an arbitrarily large compact set so long as the singular perturbation parameter is sufficiently small the controller is a continuous function of the state of the system and its construction requires the knowledge of bounding functions on the size of uncertainty the developed method is applied to a non isothermal continuous stirred tank reactor with fast jacket dynamics

wronskians geometry and some general solutions to the nonlinear liouville like pde s the formal properties of n dimensional wronskians and their geometric interpretation enable one to construct explicit analytic solutions to some nonlinear partial differential equations pde s that generalize the liouville equation xy e2 the studied pde s are a lmf const b lmf g lmg f and c lm lmf const where lm is the nonlinear differential operation lmf det kx lyf with k l 0 1 m some nontrivial formal properties of the composition of the lm s are established

a robust and accurate numerical method for transcritical turbulent flows at supercritical pressure with an arbitrary equation of state this paper addresses issues in high fidelity numerical simulations of transcritical turbulent flows at supercritical pressure the proposed strategy builds on a tabulated look up table method based on refprop database for an accurate estimation of non linear behaviors of thermodynamic and fluid transport properties at the transcritical conditions based on the look up table method we propose a numerical method that satisfies high order spatial accuracy spurious oscillation free property and capability of capturing the abrupt variation in thermodynamic properties across the transcritical contact surface the method introduces artificial mass diffusivity to the continuity and momentum equations in a physically consistent manner in order to capture the steep transcritical thermodynamic variations robustly while maintaining spurious oscillation free property in the velocity field the pressure evolution equation is derived from the full compressible navier stokes equations and solved instead of solving the total energy equation to achieve the spurious pressure oscillation free property with an arbitrary equation of state including the present look up table method flow problems with and without physical diffusion are employed for the numerical tests to validate the robustness accuracy and consistency of the proposed approach

bibo stabilization of time delayed system with nonlinear perturbation time delayed control system with the nonlinear perturbation is considered novel bibo stabilization criteria are established by the lyapunov functional and formulated in terms of existence of a positive definite solution to an auxiliary algebraic riccati matrix equation the robust quadratic stability for such systems is also discussed a numerical example is given to illustrate the effectiveness of our result

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

compensation of measurable disturbances for two time scale nonlinear systems this paper deals with a class of two time scale nonlinear systems with time varying disturbances modeled within the framework of singular perturbations systems in both standard and nonstandard form are considered for these systems we synthesize well conditioned control laws that utilize feedback of the full state vector and feedforward compensation of disturbances to exponentially stabilize the fast dynamics and enforce a pre specified input output behavior independently of the disturbances in the closed loop slow subsystem singular perturbation methods are employed to establish that the discrepancy between the output of the closed loop full order system and the output of the closed loop slow subsystem is proportional to the value of the singular perturbation parameter the stability of the closed loop system is analyzed using lyapunov s direct method and precise conditions that guarantee boundedness of the trajectories for sufficiently small values of the singular perturbation parameter are derived the application of the developed methodology is illustrated through a catalytic continuous stirred tank reactor example modeled as a singularly perturbed system in nonstandard form

a five equation model for the simulation of miscible and viscous compressible fluids abstract typical multispecies compressible navier stokes computations employ conservative equations for mass fraction transport upwind discretisations of these governing equations produce spurious pressure oscillations at diffuse contact surfaces between gases of differing ratio of specific heat capacities which degrade the convergence rate of the algorithm adding quasi conservative equations for volume fraction can solve this error however this approach has been derived only for immiscible fluids here a five equation quasi conservative model is proposed that includes the effects of species diffusion viscosity and thermal conductivity the derivation of the model is presented along with a numerical method to solve the governing equations at second order accuracy in space and time formal convergence studies demonstrate the expected order of accuracy is achieved for three benchmark problems cross validated against two standard mass fraction models in these test cases the new model has between 2 and 10 times lower error for a given grid size simulations of a two dimensional air sf6 richtmyer meshkov instability demonstrate that the new model converges to the solution with four times fewer points in each direction when compared to the mass fraction model in an identical numerical framework this represents an 40 times lower computational cost for an equivalent error in two dimensional computations the proposed model is thus very suitable for direct numerical simulation and large eddy simulation of compressible mixing

riemann problem and level set approaches for homentropic two fluid flow computations a finite volume method is presented for the computation of compressible flows of two immiscible fluids at very different densities a novel ingredient in the method is a linearized two fluid osher scheme allowing for flux computations in the case of different fluids e g water and air left and right of a cell face a level set technique is employed to distinguish between the two fluids the level set equation is incorporated into the system of hyperbolic conservation laws fixes are presented for the solution errors pressure oscillations that may occur near two fluid interfaces when applying a capturing method the fixes are analyzed and tested for two fluid flows with arbitrarily large density ratios a simple variant of the ghost fluid method appears to be a perfect remedy computations for compressible water air flows yield perfectly sharp pressure oscillation free interfaces the masses of the separate fluids appear to be conserved up to first order accuracy

accurate computation of contact discontinuities in flows with general equations of state abstract the evolution of a contact discontinuity in a real fluid flow is particularly difficult to compute with a conservative scheme indeed godunov type schemes involve an averaging step which may create an intermediate state with an incorrect pressure spurious acoustic waves are then generated these waves affect transient simulations as well as implicit time marching procedures for steady state computations however a formulation involving the pressure as a dependent variable can be used to restore a fast convergence for steady state computations

hybrid multifluid algorithms extensions of many successful single component schemes to compute multicomponent gas dynamics suffer from oscillations and other computational inaccuracies near material interfaces that are caused by the failure of the schemes to maintain pressure equilibrium between the fluid components a new algorithm based on the compressible euler equations for multicomponent fluids augmented by the pressure evolution equation is presented the extended set of equations offers two alternative ways to update the pressure field i using the equation of state or ii using the pressure evolution equation in a numerical implementation these two procedures generally yield different answers the former is a standard conservative update but may produce oscillations near material interfaces the latter is nonconservative but becomes exact near interfaces and automatically maintains pressure equilibrium a hybrid scheme which selects from the two pressure update procedures is presented the scheme perfectly conserves total mass and momentum and conserves total energy everywhere except at a finite very small number of grid cells computed solutions exhibit oscillation free interfaces and have em negligible relative conservation errors in total energy even for very strong shocks the proposed hybrid approach and switching strategies are independent of the numerical implementation and may provide a simple framework within which to extend one s favourite scheme to solve multifluid dynamics

robust stability of two time scale systems with nonlinear uncertainties this note is concerned with the robust stability problem of uncertain two time scale systems where the only information available for system uncertainties is their norm upper bounds the stability bound spl epsiv sup is derived via a state transformation and the constructive use of a lyapunov function finally examples are given to illustrate the applications of the proposed results

a model and numerical method for compressible flows with capillary effects a new model for interface problems with capillary effects in compressible fluids is presented together with a specific numerical method to treat capillary flows and pressure waves propagation this new multiphase model is in agreement with physical principles of conservation and respects the second law of thermodynamics a new numerical method is also proposed where the global system of equations is split into several submodels each submodel is hyperbolic or weakly hyperbolic and can be solved with an adequate numerical method this method is tested and validated thanks to comparisons with analytical solutions laplace law and with experimental results on droplet breakup induced by a shock wave

an adaptive ghost fluid finite volume method for compressible gas water simulations an adaptive ghost fluid finite volume method is developed for one and two dimensional compressible multi medium flows in this work it couples the real ghost fluid method gfm c w wang t g liu b c khoo a real ghost fluid method for the simulation of multi medium compressible flow siam j sci comput 28 2006 278 302 and the adaptive moving mesh method h z tang t tang moving mesh methods for one and two dimensional hyperbolic conservation laws siam j numer anal 41 2003 487 515 h z tang t tang p w zhang an adaptive mesh redistribution method for non linear hamilton jacobi equations in two and three dimensions j comput phys 188 2003 543 572 and thus combines their advantages this work shows that the local mesh clustering in the vicinity of the material interface can effectively reduce both numerical and conservative errors caused by the gfm around the material interface and other discontinuities besides the improvement of flow field resolution the adaptive gfm also largely increases the computational efficiency several numerical experiments are conducted to demonstrate robustness and efficiency of the current method they include several 1d and 2d gas water flow problems involving a large density gradient at the material interface and strong shock interface interactions the results show that our algorithm can capture the shock waves and the material interface accurately and is stable and robust even for solutions with large density and pressure gradients