stabilization and regulation of nonlinear singularly perturbed systems composite control for a general class of nonlinear singularly perturbed systems a two time scale analysis and design procedure for stabilization and regulation is presented it is shown that under a set of conditions the composite control proposed by chow and kokotovic 5 asymptotically stabilizes the desired equilibrium and produces a finite cost j uc which tends to the cost of the slow regulator as the perturbation parameter e tends to zero thus the well posedness of the full regulator problem is established furthermore under different stability requirements on the fast subsystem explicit upper and lower bounds on j uc are obtained finally for each result in this paper an upper bound on the perturbation parameter e is provided under which the result is valid

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

a new approach to the perfect regulation and the bounded peaking in linear multivariable control systems this paper presents a new approach to the perfect regulation p r and the bounded peaking b p in linear multivariable control systems based on the pole and the eigenvector assignment technique roughly speaking the p r represents an ideal control action which reduces the settling time to 0 while the b p simply means the bounded overshoot in this ideal situation simple frequency domain characterizations of the p r and the b p are derived they reveal some invariance properties of the p r and the b p that provide a powerful tool for achieving the p r and the b p the existence condition for the p r is derived which turns out to be identical to the result obtained in the optimal regulator theory this result is extended to a more general control objective of attaining the p r for one output while keeping the b p for another output a condition on which the p r is realized by an output feedback is also derived a simple design algorithm is proposed for achieving the p r which is essentially the eigenvector assignment procedure finally an application to a real system is discussed

perfect and subperfect regulation in linear multivariable control systems this paper is concerned with the synthesis of a high gain feedback control called perfect regulation p r for linear multivariable systems with external signals the existence condition for p r is derived in which the minimum phase property plays an essential role the use of an observer is discussed for achieving p r under restricted state observation some asymptotically ideal feedback properties are demonstrated such as the complete desensitization the complete servo performance with decoupling and the complete disturbance rejection which formulate the loop tightness for multivariable systems a design method for applying p r to non minimum phase systems called subperfect regulation s p r is proposed based on the factorization of the plant transfer function matrix into the minimum phase part and the totally non minimum phase part it is a multivariable extension of a well known design technique for scalar systems to overcome the difficulty of phase non minimality computational procedures for p r and s p r are discussed an illustrative example is shown

convergence properties of two nonconforming finite elements abstract two nonconforming triangular elements for second order problems are analyzed one element is found to be convergent the other does not yield approximate solutions converging to the solution of the given boundary value problem but to a wrong limit which is the solution of another variational equation this strange convergence behaviour may be typical for a wider class of nonconforming elements violating the basic precondition for convergence the problem of patch subdivisions is also considered

multiparameter singular perturbations of linear systems with multiple time scales a joint multitime scale multiparameter singular perturbation is formulated and resolved in the context of linear time varying systems an interesting feature of the solution procedure is a hierarchial scheme of aggregating and arranging of the groups of small parameters according to their order the scheme provides a suitable framework for establishing qualitative properties of multitime scale systems

model reduction and perturbation structures synthesis of optimal or suboptimal state feedbacks with the help of various perturbation techniques has been handled extensively in the literature for example by kokotovic and by sandell et al one of the main obstacles to applying these formalisms to large linear systems is the difficulty of finding a suitable perturbation structure a systematic way is given to construct regular and singular perturbation links between the original and a good reduced order model with these perturbation structures not only suboptimal state feedback but also suboptimal output feedback can be synthesized

order and the index of singular time invariant linar systems various concepts of order have been proposed in the literature it is shown that many of these are directly related to the index of a system

construction of darboux coordinates and poincare birkhoff normal forms in noncanonical hamiltonian systems abstract we demonstrate a general method to construct darboux coordinates via normal form expansions in noncanonical hamiltonian system obtained from e g a variational approach to quantum systems the procedure serves as a tool to naturally extract canonical coordinates out of the variational parameters and at the same time to transform the energy functional into its poincare birkhoff normal form the method is general in the sense that it is applicable for arbitrary degrees of freedom in arbitrary orders of the local expansion and it is independent of the precise form of the hamilton operator the method presented allows for the general and systematic investigation of quantum systems in the vicinity of fixed points which e g correspond to ground excited or transition states moreover it directly allows to calculate classical and quantum reaction rates by applying transition state theory

steering control of singularly perturbed systems a composite control approach abstract this paper presents a composite control approach to the problem of steering the state of a singularly perturbed system from a given initial state to a given final state while minimizing a cost functional the problem is treated for a class of nonlinear systems whose fast dynamics are weakly nonlinear in the fast variables and control inputs the composite control comprises three components a reduced control a feedback boundary layer stabilizing control and an open loop right boundary layer control it is shown that application of the composite control results in a final state that is o close to the desired state and a value of the cost that is o close to the optimal cost of the reduced problem

unique normal forms for vector fields and hamiltonians on construit des formes canoniques pour tous les champs vectoriels resp hamiltoniens en equilibre a 0 r n relativement au groupe des diffeomorphismes formels resp des transformations canoniques tangentiels a l identite

direct construction of a dividing surface of minimal flux for multi degree of freedom systems that cannot be recrossed the fundamental assumption of transition state theory is the existence of a dividing surface having the property that trajectories originating in reactants resp products must cross the surface only once and then proceed to products resp reactants recently it has been shown wiggins et al 2001 phys rev lett 86 5478 uzer et al 2002 nonlinearity 15 957 how to construct a dividing surface in phase space for hamiltonian systems with an arbitrary finite number of degrees of freedom having the property that trajectories only cross once locally in this letter we provide an argument showing that the flux across this dividing surface is a minimum with respect to certain types of variations of the dividing surface

a singular perturbation canonical form of invertible systems determination of multivariate root loci a new canonical form of multivariable invertible systems is given this form is suitable for the singular perturbation analysis of such singular problems as multi variable asymptotic root loci under high gain feedback cheap control state variable estimation with weak measurement noise etc appropriate selection and decomposition of state variables into slow and fast changing groups is a key feature that loads to well defined singular perturbation models the selection of state variables is accomplished without explicitly calculating any eigenvalues of the given system but by knowing the direct relationship of the output and its differential coefficients with the input the use of the canonical form to characterize the multivariable root loci under high gain feedback is demonstrated

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

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

further reductions of normal forms for dynamical systems abstract we propose in this paper a method for obtaining a significant refinement of normal forms for dynamical systems or vector fields with concrete and interesting applications we use lower order nonlinear terms in the normal form for the simplifications of higher order terms our approach is applicable for both the non nilpotent and the nilpotent cases for dynamical systems of dimensions 2 and 3 we give an algorithm that leads to interesting finite order normal forms which are optimal or unique with respect to equivalence by formal near identity transformations we can compute at the same time a formal diffeormorphism that realizes the normalization comparisons with other methods are given for several examples

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

bifurcation in the presence of small noise the methods of deterministic bifurcation arc sensitive to the addition of small amounts of white noise thus for example in systems whose macroscopic deterministic description arises from an aggregation of microscopically fluctuating dynamics the predictions of deterministic bifurcation may be incorrect here we use laplace s method of steepest descent to study bifurcation in the presence of small noise motivation for this study arises from maxwell s equal area rule for phase transitions in van der waals gases the theory is then applied to the study of the dynamics of noisy constrained or implicitly defined dynamical systems

linear grading function and further reduction of normal forms abstract in this note an idea of quasi homogeneous normal form theory using new grading functions is introduced the definition ofnth order normal form is given and some sufficient conditions for the uniqueness of normal forms are derived a special case of the unsolved problem in a paper of baider and sanders for the unique normal form of bogdanov takens singularities is solved

on the solutions of linear differential equations with singular coefficients abstract equations of the form e ax u with e and a square matrices and e singular are considered the controversy that exists in the literature concerning the solutions of such equations is investigated solutions are arrived at through an application of singular perturbation theory

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

hypernormal form theory foundations and algorithms abstract hypernormal forms unique normal forms simplest normal forms are investigated both from the standpoint of foundational theory and algorithms suitable for use with computer algebra the baider theory of the campbell hausdorff group is refined by a study of its subgroups to determine the smallest substages into which the hypernormalization process can be divided this leads to a linear algebra algorithm to compute the generators needed for each substage with the least amount of work a concrete interpretation of jan sanders spectral sequence for hypernormal forms is presented examples are given and a proof is given for a little known theorem of belitskii expressing the hypernormal form space in the inner product style as the kernel of a higher order differential operator

on the existence of positive diagonal p such that pa a t p l 0 given a real square matrix a we address the question does there exist a positive diagonal matrix p such that pa a t p is negative definite we present an algorithm which when applied to a gives a definite yes or no answer to this question if the answer is yes the algorithm will provide one such p

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

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

singular perturbational model reduction in the frequency domain singular perturbational approximations for linear continuous time and discrete time systems are developed in the frequency domain it is shown that the familiar singular perturbational result is an approximation at the origin in the complex plane however if the system has multiple time scale effects other approximations can be obtained at different locations on the negative real axis to emphasize such behavior the relationship between singular perturbational approximations and direct subsystem elimination is also investigated

brief paper singular perturbation analysis of a receding horizon controller a new receding horizon approach to the linear regulator problem is presented at each instant of time the control strategy is based on the minimization of a quadratic performance index extending over an interval of fixed length into the future connections with standard results are discussed and under general assumptions the closed loop system is shown to be asymptotically stable a potential advantage of our method lies in the fact that a singular perturbation analysis is well posed this indicates that for plants containing slow and fast modes the design procedure can be split up into two stages the resulting composite control exhibits attractive engineering properties use and effectiveness of the proposed techniques are demonstrated by investigating a practical example

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

stabilization of singularly perturbed linear time invariant systems using low order observers singularly perturbed linear time invariant systems with inadequately modeled but stable fast dynamics are considered conditions are formulated under which a reduced order observer can be designed for estimating the slow states from a knowledge of only the degenerate model a design methodology which is robust with respect to the neglected fast dynamics is presented for stabilizing systems satisfying these conditions finally an illustrative example for the autopilot actuator system design for controlling the lateral motion of an aircraft is given

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

singularly perturbed boundary value problems in discrete systems various types of singularly perturbed boundary value problems arising in discrete systems are formulated for the general type it is seen that the degenerate problem is of reduced order and hence shows its inability to accommodate all the given boundary conditions those left over boundary conditions are readmitted through appropriate corrections for the initial and final boundary layers a perturbation method is developed to obtain an approximate solution composed of an outer series initial boundary layer correction series and final boundary layer correction series an important feature of the method is that the corrections for the original boundary value problem are solved as an initial value problem and a final value problem facilitating a considerable reduction in computation an example is given to demonstrate the method

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

normal form theory and spectral sequences abstract the concept of unique normal form is formulated in terms of a spectral sequence as an illustration of this technique some results of baider and churchill concerning the normal form of the anharmonic oscillator are reproduced the aim of this paper is to show that spectral sequences give us a natural framework in which to formulate normal form theory

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

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

computable error bounds for aggregated markov chains abstract this paper describes a method for computing the steady state probability vector of a nearly completely decomposable markov chain the method is closely related to one proposed by simon and ando and developed by courtois however the method described here does not require the determination of a completely decomposable stochastic approximation to the transition matrix and hence it is applicable to matrices other than stochastic an error analysis of the procedure is given which results in effectively computable error bounds author

state decomposition for singular perturbation order reduction a modal approach singular perturbation techniques for order reduction are based on decomposing the high order system into two parts approximation of the high order system dynamics by the low order model is mainly dependent on this partition this paper introduces a measure for the degree of dependence of the state variables on the high order dominant eigenvalues by means of this measure it is not only possible to find an appropriate decomposition yielding a good low order model but also to get an indication of the order of the reduced model two illustrative examples demonstrate the main results

the patch test a condition for assessing fem convergence this paper discusses various forms of the irons patch test and shows again that its satisfaction is a necessary convergence condition being equivalent to consistency further it is shown that the test can be applied for verification of stability requirements thus providing also a sufficient condition of convergence the patch test is useful in testing the order of asymptotic convergence and the satisfaction of the test gives a guide for development of certain incompatible element forms

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

sub optimal control of discrete regulator problems via time scale decomposition the paper investigates an infinite time discrete regulator problem with a two time scale property sufficient conditions are given for the existence of a suboptimal solution and a near optimum design method is proposed via the time scale decomposition approach the method reduces the system order and has a mode decoupling property

numerical stability and constitution analysis of λ type incompatible elements the weak connection conditions and an independence principle for internal displacements which can guarantee the numerical stability of the incompatible element solution are presented in this paper a number of problems such as the uncoupling of the two types of displacement functions in the type element zero energy modes etc have been investigated some simple and practical guidelines for establishing trial functions of non compatible elements are given

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods