reduced order feedback control of distributed parameter systems via singular perturbation methods abstract implementable feedback control of distributed parameter systems must often be based on reduced order models due to the infinite dimensional nature of the actual open loop system the behavior of controllers designed via reduced order models obtained with singular perturbation techniques is analyzed when such controllers are used in the actual distributed parameter system the closed loop stability is in question the results presented here provide bounds on the smallness of the singular perturbation parameter to ensure stable operation such a priori bounds may be used to evaluate potential reduced order controllers for distributed parameter systems

control of singularly perturbed hybrid stochastic systems we study a class of optimal stochastic control problems involving two different time scales the fast mode of the system is represented by deterministic state equations whereas the slow mode of the system corresponds to a jump disturbance process under a fundamental ergodicity property for a class of infinitesimal control systems associated with the fast mode we show that there exists a limit problem which provides a good approximation to the optimal control of the perturbed system both the finite and infinite discounted horizon cases are considered we show how an approximate optimal control law can be constructed from the solution of the limit control problem in the particular case where the infinitesimal control systems possess the so called turnpike property i e characterized by the existence of global attractors the limit control problem can be given an interpretation related to a decomposition approach

near optimal control of nonstandard singularly perturbed systems abstract near optimal control of nonstandard multiparameter multitime scale singularly perturbed systems is studied using the descriptor variable approach in particular the slow models of the considered systems are allowed to exhibit impulses in the case of multiparameter singular perturbations the near optimal control is a cascade connection of separately designed slow and fast subregulators in the case of multitime scale singular perturbations the near optimal control is hierarchically composed of n 1 subregulators a parallel algorithm is provided for designing the subregulators separately the knowledge of the small singular perturbation parameters is not required in the overall design furthermore the frequency responses of the near optimal systems are analyzed and several design examples are given

almost invariant subspaces an approach to high gain feedback design part ii almost conditionally invariant subspaces in part i of this paper almost controlled invariant subspaces were studied in this part we will consider their duals the almost conditionally invariant subspaces these concepts give immediately by dualization of the almost disturbance decoupling control by state feedback the solution of the almost disturbance decoupled estimation problem finally we consider the problem of approximate disturbance decoupling by measurement feedback and it is shown that this problem is solvable to any arbitrary degree of accuracy if and only if 1 almost disturbance decoupling by state feedback and 2 almost disturbance decoupled estimation of the to be controlled output are both possible

partial cheap control of the time invariant regulator a singular perturbation analysis of the partial cheap control of the linear time invariant regulator is presented in which some but not necessarily all of the control inputs have arbitrarily small cost weighting in the limit the m 2 initial cheap control inputs arc impulsive while the outer solution asymptotically approaches a singular are associated with a dynamical system and matrix riccati equation both of order n m 2 a two time scale decomposition of the regulator problem yields a slow control and a composite control that provide near optimal performance further the partial cheap control results are used to solve the dual almost singular optimal linear filtering problem

a quadratic programming flux correction method for high order dg discretizations of sn transport abstract we present a new flux fixup approach for arbitrarily high order discontinuous galerkin dg discretizations of the s n transport equation and we demonstrate the compatibility of this approach with the variable eddington factor vef method 1 2 the new fixup approach is sweep compatible during a transport sweep block gauss seidel iteration in which the scattering source is lagged a local quadratic programming qp problem is solved in each spatial element to ensure that the solution satisfies certain physical constraints including local particle balance in this paper we describe two choices of physical constraints resulting in two variants of the method qp zero qpz and qp maximum principle qpmp in qpz the finite element coefficients of the solution are constrained to be nonnegative in qpmp they are constrained to adhere to an approximate discrete maximum principle there are two primary takeaways in this paper first when the positive bernstein basis is used for dg discretization the qpmp method eliminates negativities preserves high order accuracy for smooth problems and significantly dampens unphysical oscillations in the solution the latter feature the dampening of unphysical oscillations is an improvement upon standard simpler fixup approaches such as the approach described in 3 denoted as the zero and rescale zr method in this paper this improvement comes at a moderate computational cost but it is not prohibitive our results show that even in an unrealistic worst case scenario where 83 of the spatial elements require a fixup the computational cost of performing a transport sweep with fixup is only 31 greater than performing one without fixup the second takeaway is that the vef method can be used to accelerate the convergence of transport sweeps even when a fixup is applied when optically thick regions are present transport sweeps converge slowly regardless of whether a fixup is applied and acceleration is needed however attempting to apply standard diffusion synthetic acceleration dsa to fixed up transport sweeps results in divergence for optically thick problems our results show that the same is not true for vef when vef is combined with fixed up transport sweeps the result is a scheme that produces a nonnegative solution converges independently of the mean free path and in the case of the qpmp fixup adheres to an approximate discrete maximum principle

strong d stability abstract a square matrix f is said to be d stable if the eigenvalues of df have negative real parts for any diagonal matrix d with positive diagonal elements the perturbational properties of d stability and block d stability a generalization due to khalil and kokotovic are examined notions of strong d stability and strong block d stability are introduced and a general class of strongly block d stable matrices is identified briefly a matrix is strongly d stable if is is d stable and if every sufficiently small perturbation of the matrix is also d stable the utility of this new concept is illustrated by proving a stability theorem for time invariant multiparameter singular perturbation problems the novelty of this theorem is that is applies to two time scale as well as multiple time scale systems indeed regardless of the relative magnitudes of the singular perturbation parameters the crucial assumption of the theorem is the strong block d stability of an associated boundary layer system

optimal control of linear systems with balanced reduced order models perturbation approximations abstract in this article we study balanced model reduction of linear systems for feedback control problems specifically we focus on linear quadratic regulators with collocated inputs and outputs and we consider perturbative approximations of the dynamics in the case that the hankel singular values corresponding to the hardly controllable and observable states go to zero to this end we consider different perturbative scenarios that depend on how the negligible states scale with the small hankel singular values and derive the corresponding limit systems as well as approximate expressions for the optimal feedback controls our approach that is based on a formal asymptotic expansion of an algebraic riccati equations associated with the pontryagin maximum principle and that is validated numerically shows that model reduction based on open loop balancing can also give good closed loop performance

singularly perturbed difference equations in optimal control problems a linear regular problem involving singularly perturbed difference equations is considered the basic features of singularly perturbed systems order reduction separation of time scales boundary layers etc are highlighted application to the numerical treatment of control problems involving stiff differential equations is discussed

generalization of tychonov s theorem with applications to adaptive control of siso delay systems abstract this paper deals with single input single output delay systems with unknown transfer function parameters including unknown non commensurate real delays a generalization of tychonov s theorem is obtained this generalization is used to show the stability of a smooth adaptive controller when applied to a large class of delay systems this paper generalizes the results in 1 2

h control of linear singularly perturbed systems with small state delay an infinite horizon h state feedback control problem for singularly perturbed linear systems with a small state delay is considered an asymptotic solution of the hybrid system of riccati type algebraic ordinary differential and partial differential equations with deviating arguments associated with this problem is constructed based on this asymptotic solution conditions for the existence of a solution of the original h problem independent of the singular perturbation parameter are derived a simplified controller with parameter independent gain matrices solving the original problem for all sufficiently small values of this parameter is obtained an illustrative example is presented

control of a hybrid stochastic system abstract we consider in this paper a continuous time stochastic hybrid control system with a finite time horizon the objective is to minimize a linear function of the expected state trajectory the state evolves according to a linear dynamics however the parameters of the state evolution equation may change at discrete times according to a controlled markov chain which has finite state and action spaces we use a procedure similar in form to the maximum principle this determines a control strategy which is asymptotically optimal as the number of transitions during the finite time horizon grows to infinity

robust redesign of adaptive control effects of unmodeled high frequency dynamics on stability and performance of adaptive control schemes are analyzed in the regulation problem global stability properties are no longer guaranteed but a region of attraction exists for exact adaptive regulation the dependence of the region of attraction on unmodeled parasitics is examined first then the general case of model reference adaptive control is considered in which parasitics can destroy stability and boundedness properties a modified adaptive law is proposed guaranteeing the existence of a region of attraction from which all signals converge to a residual set which contains the equilibrium for exact tracking the size of this set depends on design parameters the frequency range of parasitics and the reference input signal characteristics

on a representation of the limit occupational measures set of a control system with applications to singularly perturbed control systems a representation of the limit occupational measures set of a control system in terms of the vector function defining the system s dynamics is established applications in averaging of singularly perturbed control systems are demonstrated

application of vibrational control to linear quadratic control problems abstract the application of the theory of vibrational control to linear quadratic control problems is developed the solution of the matrix riccati differential equation rde and the optimal trajectory are found approximately as power series of a small parameter the minimizing effect of vibrations on the maximal solution of rde and the cost functional is studied the region of attraction of the maximal solution of rde for the case of hamiltonian matrix with imaginary axis eigenvalues is found special attention is paid to the application of vibrations to the linear quadratic problem of stabilization with respect to a part of variables and transfer of the other variables to a given position a problem of vibrational stabilization and optimal control of a carriage with an inverted pendulum is solved as an example

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

near optimum regulators for stochastic linear singularly perturbed systems this paper presents a new approach to the decomposition and approximation of linear quadratic gaussian estimation and control problems for singularly perturbed systems the kalman filter is decomposed into separate slow mode and fast mode filters via the use of a decoupling transformation a near optimal control law is derived by approximating the coefficients of the optimal control law the order of approximation of the optimal performance is 0 mu n where n is the order of approximation of the coefficients

some results in nonlinear output regulation and feedback stabilization the problems of nonlinear output regulation and feedback stabilization on compacta are studied high gain control laws are given to solve output regulation on compacta for a class of nonlinear systems for which exact tracking with internal stability may be impossible with some modification this result is applied to feedback stabilization on compacta for nonlinear systems whose zero dynamics may be globally unstable

subsystems time scales and multimodeling through a couple of naive examples the control theorists are invited to re examine the role of modeling in the study of large scale dynamic systems instead of assuming the existence of n diagonally dominant blocks it is possible to justify one strongly coupled slow core and n weakly coupled fast subsystems this structure is exhibited with a physically meaningful choice of state variables the controls are introduced following the recent concept of multimodeling

uniform asymptotic stability of non linear singularly perturbed general and large scale systems new general advantageous criteria are established for uniform asymptotic stability of singularly perturbed general and large scale systems possibly containing non differentiable non linearities which often appear in control systems for systems with multiple time scales caused by several small parameters or for large scale systems an essential order reduction of aggregation matrices is achieved the results are conceptually and numerically applied to the absolute stability analysis of singularly perturbed lurie postnikov systems

the existence of a unique and bounded solution of the algebraic riccati equation of multimodel estimation and control problems abstract the existence of a unique and bounded solution of the algebraic matrix riccati equation arising in multimodel estimation and control problems is established for sufficiently small values of two parameters this is obtained under control oriented assumptions imposed on the slow and fast subsystems and by applying the implicit function theorem

brief paper on applications of singular perturbation techniques in nonlinear optimal control the technique of singular perturbations spt has been applied with considerable success in several nonlinear optimal control problems in many cases the zero order approximation of the optimal control function has been expressed in a feedback form this paper deals with topics involved in such closed loop application which seem to merit further discussion it is formally demonstrated that a forced singular perturbation model obtained by artificial insertion of the perturbation parameter results in the same zero order composite feedback control solution as a classical singularly perturbed model where a small parameter of physical significance appears as a consequence of a scaling transformation the accuracy of the zero order feedback approximation depends in both cases on the actual time scale separation of the variables two inherent limitations of the feedback solution are also pointed out 1 first and higher order correction terms of the zero order approximation have to be computed by a predictive or off line integration 2 on line implementation of spt control strategy in a terminal boundary layer requires iterative computations a simple pursuit problem serves as an illustrative example

the equivalence of time scale decomposition techniques used in the analysis and design of linear systems linear system models can be decomposed into reduced order fast and slow subsystems using spectrum separating solutions to dual riccati type equations eigenspace power iterations provide globally convergent iterative schemes for obtaining these solutions from this basis an equivalence of previous decomposition techniques is then achieved including the classic two time scale asymptotic series decomposition of singularly perturbed systems the result is a unified theory for the decomposition of time scales in linear systems applicable to both continuous and discrete time models

on the construction of asymptoticallly optimal controls for singularly perturbed systems we construct asymptotically optimal controls for singularly perturbed nonlinear systems where both the slow and the fast motions are controlled for this purpose we give necessary optimality conditions for an averaged system for the slow motion

coherency based decomposition and aggregation the relationship between coherency area decomposition and aggregation is investigated for a class of models of electromechanical networks including power system models it is established that coherency conditions and a physically meaningful area aggregate reproducing a prescribed set of system modes depend on external connections only coherency with respect to a set of slowest modes slow coherency and an earlier developed grouping algorithm are used to identify weakly coupled areas as demonstrated on a 48 machine model of the united states north east power system

direct singular perturbation analysis of high gain and cheap control problems a new and direct approach to the analysis and design of high gain feedback and cheap control problems is developed transformation of such singular problems to singular perturbation models of conventional type early at the onset of analysis or design allows a direct application of the readily available singular perturbation literature the approach permits a simple characterization of a multivariable root loci under a high gain feedback and b asymptotic behavior of optimal closed loop poles state and control trajectories performance index and optimal transfer function as the control cost coefficient in the performance index goes to zero

weak connections time scales and aggregation of nonlinear systems if nonlinear subsystems with continuous equilibria are weakly connected their local behavior is fast compared with the system wide behavior caused by the connections the slow behavior is described by an aggregate model which appears as a slow subsystem in the singular perturbation form of the model in this way earlier linear aggregation results by simon et al in economics and slow coherency results in power systems are extended to nonlinear systems and related to singular perturbations

paper on coherency based decomposition algorithms a new algorithm of coherency based decomposition is presented and applied contrary to avramovic and coworkers algorithm it does not necessarily imply in practice the knowledge of the eigenvalues and eigenvectors of the system or an a priori knowledge of the number of coherent areas as long as two classes of dynamics slow and fast exist its applicability to the case of a 48 machine system is demonstrated

convergence in the boundary layer for singularly perturbed equations a singularly perturbed linear finite dimensional ordinary differential equation is considered on the half line 0 the reduced system is assumed to have a unique solution in the sense of distributions it is proved that under reasonable conditions the solution of the full system converges to that of the reduced in the distributional sense and the order of the limiting distribution is computed an application to the optimal linear quadratic regulator with cheap control is described

characterization and computation for the bound e in linear time variantsingularly perturbed systems abstract this paper gives a complete solution to the bound problem in a linear time invariant singularly perturbed system a criterion is presented to determine the existence of the bound

paper optimal control of markov chains admitting strong and weak interactions this paper is devoted to the study of an optimal control problem for a markov chain with generator b ea where e is a small parameter it is shown that an approximate solution can be calculated by a policy improvement algorithm involving computations relative to an aggregated problem the dimension of which is given by n the number of ergodic sets for the b matrix together with a family of decentralized problems the dimensions of which are given by the number of elements in each ergodic set for the b matrix

on a class of optimal control problems with an almost cost free solution for a class of linear singular optimal control problems with a nonunique singular arc the solution of the corresponding nearly singular problem is analyzed and a limit solution based on formal singular perturbations is derived a rigorous proof of the correctness of the result is given by constructing a convergent power series satisfying the riccati equation of the nearly singular problem

stability bounds of singularly perturbed discrete systems the authors propose a systematic approach to determine the exact stability bound of singularly perturbed discrete time systems by combining three conditions of the critical stability criterion with a bialternate product for discrete time systems the stability problems of two types of singularly perturbed discrete time systems can be solved via the derived scheme several examples are given to illustrate the proposed results

optimal control problems over large time intervals abstract the paper focuses on problems of optimal control over a large time interval with end states prescribed for a wide class of systems and performance indices an approximate solution is characterized by piecing together two solutions for two infinite time problems these solutions exhibit similar transient behaviour

perturbation analysis of a system of quasi variational inequalities for optimal stochastic scheduling optimal scheduling of continuously evolving dynamical systems involves the analysis of certain partial differential inequalities on solution dependent domains these are the quasi variatlonai inequalities qvi s introduced for such problems by a bensoussan and j l lions they arise naturally in such applications as inventory control and unit commitment scheduling in electric energy systems qvi s are in these contexts the analog of the dynamic programming partial differential equations of r bellman for continuous control problems analytical treatment of qvi s involves not only the solution of partial differential inequalities but also the simultaneous analysis of the free boundaries of the domain of the solution in effect the optimal return function is the solution of the inequalities and the optimal control switching policy is defined by the free boundaries this paper considers a system of two singularly perturbed second order quasi variational inequalities with a turning point in a bounded domain in r1 asymptotic approximations for the free boundaries are derived using standard perturbation analysis for boundary value problems also local regularity properties of the limit solution are obtained the results are illustrated by a parametric study of a simple model of commitment scheduling for a single unit in a power generation system

on the stability of multiple time scale systems the stability of time invariant multiparameter singular perturbation problems is considered and the implications of two time scale stability results for multiple time scale systems are clarified an example shows that the asymptotic stability of a multiparameter singular perturbation problem under the bounded mutual ratios assumption for arbitrary bounds on the ratios of the small parameters does not imply asymptotic stability under the multiple time scales assumption for any ordering of the smallness of the parameters however this conclusion does apply when only two small parameters are present and the fast variables are scalar valued a multiparameter singularly perturbed system may be asymptotically stable for all sufficiently small and positive values of the perturbation parameters even though the boundary layer system does not satisfy the d stability criterion these examples are discussed in the light of the strong d stability condition which must be imposed to obtain results that are robus

a new parameter estimate in singular perturbations abstract a new upper bound is obtained for the singular perturbation parameter of an asymptotically stable singularly perturbed system general time invariant systems with a single small parameter are considered the paper employs a riccati equation whose solution is known to facilitate the exact decoupling of fast and slow dynamics an application of the brouwer fixed point theorem to the riccati equation and of liapunov s direct method to the fast and slow subsystems results in the desired upper bound computation of the estimate requires only the solution of two liapunov matrix equations

on the connection between nonlinear differential algebraic equations and singularly perturbed control systems in nonstandard form considers a class of control systems represented by nonlinear differential equations depending on a small parameter the systems are not in the standard singularly perturbed form and therefore one of the challenges is to show that the control systems do represent singularly perturbed two time scale systems assumptions are introduced which guarantee that an equivalent representation for the systems can be obtained in the standard singularly perturbed form thereby justifying the two time scale property the equations for the slow dynamics are characterized by a set of differential algebraic equations which have been studied previosly in the literature the fast dynamics are characterized by differential equations both the slow and the fast dynamics are easily derived and are defined in terms of variables that define the original control system control design for the class of systems being considered is studied using the composite control approach

composite controls for singularly perturbed systems the controllability and the stabilizability invariance of the reduced system slow subsystem of the nonlinear singularly perturbed system due to a class of fast feedback controls is shown the general type composite control for the nonlinear singularly perturbed system is proposed and its application for the synthesis problem is considered

an asymptotic error analysis of identifiers and adaptive observers in the presence of parasitics the behavior of identifiers and adaptive observers in the presence of parasitics is analyzed bounds for the state parameter error are established the adaptive schemes considered are shown to be robust provided the input signal is rich for the dominant modes but does not contain high frequencies in the parasitic range the bounds indicate possibilities for reducing the error by a proper choice of the input signal

singular perturbation in mayer s problem for linear systems the well posedness of mayer s problem for a linear system with a small parameter in the derivatives and constrained controls is studied the convergence of the optimal control when the parameter tends to zero is analyzed the state constrained case is discussed

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

asymptotic stability of nonlinear multiparameter singularly perturbed systems sufficient conditions are derived to guarantee the asymptotic stability of a class of nonlinear singularly perturbed systems with several perturbation parameters of the same order estimates of the region of attraction and bounds on the small parameters are obtained

decomposition and stability for multiparameter singular perturbation problems time scale separation and stability of linear time varying and time invariant multiparameter singular perturbation problems are analyzed the first problem considered in the paper is that of deriving upper bounds on the small parasitic parameters ensuring the existence of an invertible bounded transformation exactly separating fast and slow dynamics this problem is most interesting for the time varying case the analysis of this problem in the time varying case requires the two time scale setting introduced by h k khalil and p v kokotovic 16 this entails that the mutual ratios of the small parameters are bounded by known positive constants the second problem considered is to derive parameter bounds ensuring that the system in question is uniformly asymptotically stable the results on decomposition are used to facilitate the derivation of these latter bounds fortunately the analysis of decomposition and stability questions for time invariant multiparameter singular perturbation problems requires no restriction on the relative magnitudes of the small parameters a concept of strong d stability is introduced and shown to greatly simplify the stability analysis of time invariant multiparameter problems

balanced model reduction of linear systems with nonzero initial conditions singular perturbation approximation abstract in this article we study balanced model reduction of linear control systems using the singular perturbation approximation balanced model reduction techniques have been successfully applied to systems with homogeneous initial conditions with one of their most important features being a priori l2 and h bounds for the approximation error the main focus of this work is to derive an l2 error bound for the singular perturbation approximation for system with inhomogeneous initial conditions extending related work on balanced truncation this l2 error bound measures the difference between the input output maps of the original and of the reduced initial value systems the advantages and flexibility of this approach are demonstrated with a variety of numerical examples

applications of singular perturbation techniques to control problems this paper discusses typical applications of singular perturbation techniques to control problems in the last fifteen years the first three sections are devoted to the standard model and its time scale stability and controllability properties the next two sections deal with linear quadratic optimal control and one with cheap near singular control then the composite control and trajectory optimization are considered in two sections and stochastic control in one section the last section returns to the problem of modeling this time in the context of large scale systems the bibliography contains more than 250 titles

linear compensator designs based exclusively on input output information are never robust with respect to unmodeled dynamics the effects of unmodeled higher order dynamics or parasitics on the stability of linear control systems are investigated a description is first given of a class of perturbations of a given state equation that cannot be distinguished from the original on the basis of input output measurements alone then it is shown that given any plant compensator pair perturbations of each system of this type can always be found that destabilize the closed loop configuration the effect of destabilizing perturbations on output behavior is explored

robustness of absolute stability the stability of a class of single input single output singularly perturbed systems formed by a linear time invariant feedforward block with a sector bounded time varying feedback is considered it is shown that if the reduced order slow subsystem is absolutely stable and the parasitics are asymptotically stable and sufficiently fast then the full system is absolutely stable bounds on the singular perturbation parameter for uniform asymptotic stability and absolute stability are obtained

robustness of exponential stability to singular perturbations and delays singularly perturbed nonlinear differential equations with small time delays in the slow variables are considered averages of the fast variables are used in order to obtain a sufficient condition under which the exponential stability of the slow subsystem is robust to singular perturbations and delays

actor critic type learning algorithms for markov decision processes algorithms for learning the optimal policy of a markov decision process mdp based on simulated transitions are formulated and analyzed these are variants of the well known actor critic or adaptive critic algorithm in the artificial intelligence literature distributed asynchronous implementations are considered the analysis involves two time scale stochastic approximations

non standard singularly perturbed control systems and differential algebraic equations we consider a class of linear control systems represented by equations depending on a small parameter but which are not in the standard singularly perturbed form one of the challenging system theoretic problem is to obtain an equivalent representation for the control system which if possible is in the standard singularly perturbed form assumptions are introduced which guarantee that an equivalent representation can be obtained which is in the standard singularly perturbed form thereby justifying the two time scale property the equations for the slow dynamics are characterized by a set of differential algebraic equations which are easily derived from the original equations by setting the parameter to zero the original assumptions guarantee the existence of solutions of the obtained differential algebraic equations the fast dynamics are characterized by different equations in terms of matrices that define the original control system control design for the system being considered is studied using th

a reduction process for perturbed markov chains given a perturbed markov chain with generator a varepsilon a 0 varepsilon a 1 varepsilon 2 a 2 cdots it is shown that to each time scale t t varepsilon t varepsilon 2 cdots is associated a reduced markov chain this process of reduction has only finitely many steps it is given in a recursive way and each reduced chain admits as state space the recurrent classes of the reduced chain defined at the previous stage it is shown that if lambda is an eigenvalue different from zero of the generator of the kth reduced chain with multiplicity p then a varepsilon admits p eigenvalues lambda varepsilon such that lambda varepsilon varepsilon k lambda o varepsilon k a basis of ker a 0 resp ker a 0 which is robust to the perturbation is constructed

invariant slow manifold approach for exact dynamic inversion of singularly perturbed linear mechanical systems with admissible output constraints abstract we propose an approach for the exact dynamic inversion of singularly perturbed second order linear systems through asymptotic expansion in a singular parameter we show that the inversion solution corresponding to the invariant slow manifold can be expressed as a converging infinite series under desired output constraints composed of exponential support functions in the complex domain we provide systematic mathematical procedures to obtain the closed form invariant slow manifold along with required admissible boundary conditions numerical examples are given to validate the proposed approach

brief paper a multimodel approach to stochastic team problems this paper develops decentralized team strategies for decision makers using different models of the same large scale system multiparameter singular perturbations are employed to capture the multimodel nature of the fast dynamic subsystems interconnected through slow dynamic variables the small parameters are appropriately scaled so that the variables in both time scales are well defined the system considered is linear and the cost criterion is quadratic first a multimodel solution is obtained when the decision makers make different noisy linear observations of the random initial state only then the solution to this static team problem is utilized to obtain a multimodel solution to the dynamic team problem with sampled observations under the one step delay observation sharing pattern in both cases the well posedness of the multimodel solution is demonstrated

euclidean space controllability of singularly perturbed linear systems with state delay abstract a singularly perturbed linear time dependent control system with small point and distributed delays in state variables is considered connections between the properties of controllability of the reduced order and boundary layer systems associated with the original one and such a property of the original system itself are established

infinite time regulators for singularly perturbed difference equations linear quadratic regulators for singularly perturbed difference equations are considered asymptotic behaviour of the optimal solution as the perturbation parameter tends to zero is studied the composite feedback control strategy of continuous time singularly perturbed systems is extended to the new problem

singular perturbation results for a class of stochastic control problems composite control originally proposed in a deterministic context is generalized to the problem with white noise inputs however the approach used here is radically different from the deterministic approach presence of noise smoothes the system behavior and allows a more complete solution than in the deterministic case

analysis of a system of singularly perturbed convection diffusion equations with strong coupling a system of singularly perturbed convection diffusion equations with strong coupling via the convection term is studied the system is discretized by an upwinded first order finite difference scheme for which both an a priori and an a posteriori error analysis in the maximum norm are conducted based on the a priori error bound we show that there exists a mesh with n subintervals that gives optimal error bounds of mathcal o left n 1 right which are robust with respect to the diffusion parameters the a posteriori error bounds are used to design an adaptive method numerical experiments complement the theoretical findings

on singular perturbations due to fast actuators in hybrid control systems a stability result is given for hybrid control systems singularly perturbed by fast but continuous actuators if a hybrid control system has a compact set globally asymptotically stable when the actuator dynamics are omitted or equivalently are infinitely fast then the same compact set is semiglobally practically asymptotically stable in the finite speed of the actuator dynamics this result which generalizes classical results for differential equations justifies using a simplified plant model that ignores fast but continuous actuator dynamics even when using a hybrid feedback control algorithm

a singular perturbation approach to modeling and control of markov chains finite state continuous time markov processes with weak interactions are modeled as singularly perturbed systems aggregate states are obtained using a grouping algorithm two time scale expansions simplify cost equations and lead to decentralized optimization algorithms

a multimodel approach to stochastic nash games a decentralized filtering and control scheme is presented for obtaining low order nash equilibrium strategies for decision makers using different models of the same large scale system multiparameter singular perturbations are employed to capture the multimodel nature of the fast dynamic subsystems interconnected through slow dynamic variables the small parameters are appropriately scaled so that the variables in both time scales are well defined the decision makers have decentralized information structures and are constrained to use only finite dimensional compensators of a specified dimension it is shown that the proposed scheme is in fact the asymptotic limit of the exact solution as the small parameters go to zero

adaptive boundary control of a flexible manipulator with input saturation abstractin this study we consider the anti windup design as one of the approaches for the boundary control problem of a flexible manipulator in the presence of system parametric uncertainties external disturbances and bounded inputs the dynamics of the system are represented by partial differential equations pdes using the singular perturbation approach the pde model is divided into two simpler subsystems with the lyapunov s direct method an adaptive boundary control scheme is developed to regulate the angular position and suppress the elastic vibration simultaneously and the adaptive laws are designed to compensate for the system parametric uncertainties and the disturbances the proposed control scheme allows the application of smooth hyperbolic functions which satisfy physical conditions and input restrictions be easily realised numerical simulations demonstrate the effectiveness of the proposed scheme

singular perturbation of the popov kaiman yakubovich lemma absolute stability of linear time invariant single input single output lti siso singularly perturbed systems with a time varying feedback element is examined via the popov kalman yakubovich pky lemma if the reduced order slow subsystem is absolutely stable then so is the full system provided the parasitics are sufficiently fast and stable

disturbance decoupling by high gain feedback asymptotic disturbance decoupling add a concept which is suitable for the study of the effects of high gain feedback on disturbance decoupling is introduced in this note we show that the conditions for add are the conditions for exact disturbance decoupling due to wonham and morse for a reduced order system

correspondence comments on performance evaluation of ukf based nonlinear filtering this note points out that the main result of the paper xiong k zhang h y chan c w 2006 performance evaluation of ukf based nonlinear filtering automatica 42 2 261 270 can be extended to a class of so called gaussian filters it justifies a practical countermeasure of the divergence i e adding small quantities to the noise covariance matrix to stabilize the filter