controlled invariance for discrete time nonlinear systems with an application to the disturbance decoupling problems invariant distributions are defined for discrete time nonlinear control systems and necessary and sufficient conditions are given for their controlled invariance this extends to discrete time systems the basic tool which has been so important in solving the various synthesis problems for continuous time systems to indicate their utility in the discrete time setting they are used to locally solve the disturbance decoupling problem

some model theory of polish structures we introduce a notion of polish structure and in doing so provide a setting which allows the application of ideas and techniques from model theory descriptive set theory topology and the theory of profinite groups we define a topological notion of independence in polish structures and prove that it has some nice properties using this notion we prove counterparts of some basic results from geometric stability theory in the context of small polish structures then we prove some structural theorems about compact groups regarded as polish structures each small nm stable compact g group is solvable by finite each small compact g group of finite n m rank is nilpotent by finite examples of small polish structures and groups are also given

amenability definable groups and automorphism groups abstract we prove several theorems relating amenability of groups in various categories discrete definable topological automor phism group to model theoretic invariants quotients by connected components lascar galois group g compactness among the main tools which is possibly of independent interest is the adaptation and generalization of theorem 12 of 27 to other settings including amenable topological groups whose topology is generated by a family of subgroups on the side of definable groups we prove that if g is definable in a model m and g is definably amenable then the connected components g m 00 and g m 000 coincide answering positively a question from 22 we also prove some natural counterparts for topological groups using our generalizations of 27 by finding a dictionary relating quotients by connected components and galois groups of categorical theories we conclude that if m is countable and categorical and aut m is amenable as a topological group then t th m is g compact i e the lascar galois group gal l t is compact hausdorff equivalently the natural epimorphism from gal l t to gal k p t is an isomorphism we also take the opportunity to further develop the model theoretic approach to topological dynamics obtaining for example some new invariants for topological groups as well as allowing a uniform approach to the theorems above and the various categories

a new algorithm for constructing approximate transformations for nonlinear systems an algorithm is provided for constructing an approximating transformation for a feedback linearizable system and for a system that is not feedback linearizable instead of approximating the given nonlinear system the authors approximate the transformation in a proper sense they assume that the lead variables of a transformation have a certain form with unknown parameters substitute these variables into the required equations and determine the unknown parameters through linear algebraic equations there is a tremendous savings in terms of number of equations and unknowns and it is sometimes possible to find exact or best possible transformations in a surprisingly small number of steps

strict feedforward control systems linearisability and convergent normal forms this article discusses the feedback equivalence of multi inputs feedforward control systems via smooth resp analytic feedback transformations we first address the state resp feedback linearisation problem and provide easily computable algorithms that yield explicit state resp feedback linearising coordinates for systems in strict feedforward form the application of the algorithms does not require checking the commutativity resp involutivity of the distributions associated with the system and the algorithms fail after few steps if the system is not linearisable in the latter case the algorithms are extended to provide coordinate systems bringing the system into a normal form which is a smooth resp analytic counterpart of kang s formal normal form illustrative examples for both the linearisation and convergent normal form include the vertical take off and landing aircraft the multi vehicle wireless testbed among others

a direct method for the construction of nonlinear discrete time observer with linearizable error dynamics we provide a direct method for the design of nonlinear discrete time observers by a construction of a change of variables an explicit expression of the change of variables is given some simulations for chaotic systems such as lozi system and he spl acute non system are provided to illustrate the proposed method

nonlinear discrete time observer design with linearizable error dynamics a necessary and sufficient condition for the existence of a discrete time nonlinear observer with linearizable error dynamics is provided the result can be applied to any real analytic nonlinear system whose linear part is observable the necessary and sufficient condition is the solvability of a nonlinear functional equation furthermore the well known siegel s theorem on the linearizability of a mapping is naturally reproduced in a corollary the proposed observer design method is constructive and can be applied approximately to any sufficiently smooth linearly observable system yielding a local observer with approximately linear error dynamics

observability quadratic normal form for discrete time systems this note deals with quadratic observability normal form for nonlinear discrete time single input single output siso system first of all the main concept of quadratic equivalence with respect to the observability property is introduced for discrete time systems subsequently normal form structure for discrete time system is developed for system with unobservable linear approximation in one direction finally the effect of the so called resonant terms on the observer design and synchronization of chaotic systems is pointed out in an illustrative example

nonlinear control via uniform system approximation abstract we introduce the notion of uniform system approximation and use it to solve the following problem given a single input nonlinear system f g x f x g x u that is linearly controllable on a manifold e of equilibrium points find vector fields f g such that f g and f g agree to first order on e and f g is input to state linearizable on a neighborhood of e the solution proceeds by constructing a linearizable uniform system approximation and yields as a by product a feedback control law that can be used to guarantee stable operation of the given system in a neighborhood of e this includes for example smooth transitions between operating points that are not necessarily close to each other a simple example is given to illustrate the approach

normal forms and approximated feedback linearization in discrete time abstract the paper discusses approximated feedback linearization of nonlinear discrete time dynamics which are controllable in first approximation and introduces two types of normal forms the study is set in the context of differential difference representations of discrete time dynamics proposed in monaco normand cyrot in normand cyrot ed perspectives in control a tribute to ioan dore landau springer londres 1998 pp 191 205

sampled data adaptive control of a class of continuous nonlinear systems the problem is addressed of deriving sampled data control laws for a class of nonlinear continuous systems the concept of approximate state feedback linearizing control is defined for sampled data models arising from the sampling of continuous time systems such control laws are then designed for both non adaptive and adaptive cases and a detailed performance analysis is performed

global stabilization of discrete time homogeneous systems abstract the feedback stabilization of continuous time r homogeneous systems has been studied by several authors see for instance dayawansa and martin proceedings of the 28th c d c tampa florida december 1989 hermes j differential equations 92 1991 76 89 hermes systems control lett 24 1995 7 11 kawski proceedings of the 27th c d c texas 1988 texas kawski control theory and advanced technology vol 6 mita press tokyo 1996 pp 497 516 sepulchre and aeyels math control signal systems 9 1996 34 58 in kawski analysis of controlled dynamical systems birkhauser basel 1991 kawski proceedings of ifac nolcos 1995 kawski gave an intrinsic concept of homogeneity with respect to general smooth dilations in this note we define the homogeneity of transition maps w r t continuous dilations and we show how one can construct a global stabilizing feedback control for discrete time homogeneous systems which are locally asymptotically stabilizable

nonlinear discrete time control of systems with a naimark sacker bifurcation in this paper we study the problem of the stabilization of nonlinear control system with one complex uncontrollable mode we find normal forms and quadratic invariants then we compute center manifolds and use bifurcation theory to synthesize quadratic controllers

approximate linearization by state feedback and coordinate change a nonlinear system can always be approximated to first order by linear systems it has been shown by jakubczyk and respondek 4 and hunt and su 3 that certain nonlinear systems are the exact transforms of linear systems under nonlinear state coordinate change and nonlinear state feedback in this paper we give necessary and sufficient conditions for a nonlinear system to be approximated to higher order by the transform of a linear system the use of this technique in the design of nonlinear compensators has been suggested recently by the author 6

the global existence of nonlinear observers with linear error dynamics a topological point of view in this paper we show that under suitable assumptions there exists a global homeomorphism 1 of rn which maps a nonlinear system x f x x 0 x0 y h x onto a linear system with output injection z az y z 0 x0 thus an observer for state x can be directly constructed as x f x y h x which is a generalized version of luenberger observer an important feature of the obtained result is that there is no need to find the corresponding change of coordinates explicitly which is different from current various existing approaches

immersion of discrete time nonlinear system by regular dynamic state feedback into a linear system necessary and sufficient conditions for local solvability of the title problem around a given trajectory are obtained the proposed conditions are less restrictive than those obtained by lee and marcus for the problem of immersion a nonlinear system into a linear system via static state feedback instrumental in the problem solution is the inversion structure algorithm for a discrete time nonlinear system the solvability conditions are expressed in terms of the inversion algorithm moreover the construction of both the dynamic state feedback and the immersion map relies on this algorithm finally it is shown that for locally right invertible systems the considered problem unlike the problem of immersion via static state feedback is always solvable

feedback classification of multi input nonlinear control systems we study the feedback group action on multi input nonlinear control systems with uncontrollable mode we follow slightly an approach proposed in kang and krener w kang and a j krener siam j control optim 30 1992 pp 1319 1337 which consists of analyzing the system and the feedback group step by step we construct a normal form which generalizes on one hand the results obtained in the single input case and on the other hand those recently obtained by the same author in the controllable case we illustrate our results by studying the caltech multi vehicle wireless testbed mvwt and the prototype of planar vertical takeoff and landing aircraft pvtol we also study the notion of bifurcation of controllability for systems with one nonzero uncontrollable mode we first show that the equilibria for those systems is a p dimensional submanifold p equals number of inputs provided that one term in their normal form is nonzero we show that these systems are linearly controllable hence stabilizable at any nearby equilibrium point of the origin

on input output linearization of discrete time nonlinear systems abstract we consider a nonlinear discrete time system of the form x t 1 f x t u t y t h x t where x r n u r m y r q and f and h are analytic necessary and sufficient conditions for local input output linearizability are given we show that these conditions are also sufficient for a formal solution to the global input output linearization problem finally we show that zeros at infinity of can be obtained by the structure algorithm for locally input output linearizable systems

on the convergence of multi grid methods with transforming smoothers in the present paper we give a convergence theory for multi grid methods with transforming smoothers as introduced in 31 applied to a general system of partial differential equations the theory follows hackbusch s approach for scalar pde and allows a convergence proof for some well known multi grid methods for stokes and navier stokes equations as dgs by brandt dinar 5 tilu from 31 and the simple methods by patankar spalding 23

on weak relative degree and the mcgehee transformation points where jumps in relative degree occur are presented as singularities in the vector field of the internal dynamics of systems that are candidates for input output linearization via nonlinear feedback for a certain class of vector fields it is shown that the point at which the relative degree is weak can be transformed to a saddle equilibrium point at the origin

on dynamic input output linearization of discrete time nonlinear systems this paper studies the problem of linearizing the input output map of an analytic discrete time nonlinear system locally around a given trajectory necessary and sufficient conditions are given for the existence of a regular dynamic state feedback control law under which the input dependent part of the response of a nonlinear system becomes linear in the input and independent of the initial state the proposed conditions are less restrictive than those obtained by lee and marcus for linearizing the input output map via a static state feedback instrumental in the problem solution is the inversion structure algorithm for a discrete time nonlinear system firstly the solvability conditions are expressed in terms of the inversion algorithm secondly the proof of the existence and construction of the dynamic state feedback compensator relies on this algorithm

a class of iterative methods for solving saddle point problems we consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points we are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations we present an iterative method involving two levels of iteration similar in some respects to the uzawa algorithm we relate the rates of convergence of the outer and inner iterations proving that under natural hypotheses the outer iteration achieves the rate of convergence of the inner iteration the technique is applied to finite element approximations of the stokes equations

exact and approximate feedback linearization without the linear controllability assumption the feedback linearization problem of nonlinear control systems has been solved in the literature under the assumption that the nonlinear system is linearly controllable in this paper the assumption of linear controllability is removed necessary and sufficient conditions are given through lie orbital symmetries thus giving a geometric characterization of the problem in the analytic case both the exact and the approximate linearization problems are considered in the analytic case

global convergence of nonlinear cascade flows with morse bott zero dynamics abstract we derive a sufficient condition that a flow captures the dynamics on an invariant submanifold this leads to a refinement of the lasalle invariance principle as a consequence we generalize a well known global asymptotic stability result of nonlinear cascade systems to show global convergence to a compact invariant set this includes the case where a globally asymptotically stable system is coupled to a morse bott flow

a homotopy algorithm for approximating geometric distributions by integrable systems in the geometric theory of nonlinear control systems the notion of a distribution and the dual notion of codistribution play a central role many results in nonlinear control theory require certain distributions to be integrable distributions and codistributions are not generically integrable and moreover the integrability property is not likely to persist under small perturbations of the system therefore it is natural to consider the problem of approximating a given codistribution by an integrable codistribution and to determine to what extent such an approximation may be used for obtaining approximate solutions to various problems in control theory in this note we concentrate on the purely mathematical problem of approximating a given codistribution by an integrable codistribution we present an algorithm for approximating an m dimensional nonintegrable codistribution by an integrable one using a homotopy approach the method yields an approximating codistribution that agrees with the original codistribution on an m dimensional submanifold e 0 of r n

asymptotic stabilization of a generic class of three dimensional homogeneous quadratic systems abstract we give necessary and sufficient conditions for the asymptotic stabilizability of a generic class of single input three dimensional homogeneous quadratic systems

nonlinear observer design in the siegel domain we extend the method of kazantzis and kravaris systems control lett 34 1998 pp 241 247 for the design of an observer to a larger class of nonlinear systems the extended method is applicable to any real analytic observable nonlinear system it is based on the solution of a first order singular nonlinear pde this solution yields a change of state coordinates which linearizes the error dynamics under very general conditions the existence and uniqueness of the solution is proved lyapunov s auxiliary theorem and siegel s theorem are obtained as corollaries the technique is constructive and yields a method for constructing approximate solutions

optimization of large scale truss structures using sparse sand formulations sparsity features of simultaneous analysis and design sand formulations are studied and exploited for optimization of large scale truss structures three formulations are described and implemented with an existing analysis code sand formulations have large number of variables however gradients of the functions and hessian of the lagrangian are quite sparsely populated therefore this structure of the problem is exploited and an optimization algorithm with sparse matrix capability is used to solve large scale problems an existing analysis software is integrated with an optimizer based on a sparse sequential quadratic programming sqp algorithm to solve sample problems the formulations and algorithms work quite well for the sample problems and their performances are compared and discussed for all the cases considered the sand formulations out perform the conventional formulation except one case further research is suggested to fully study and utilize sparse features of the alternative sand formulations and to develop more efficient sparse solvers for large scale and more complex applications

state space newton s method for topology optimization abstract we introduce a new algorithm for solving certain classes of topology optimization problems which enjoys fast local convergence normally achieved by the full space methods while working in a smaller reduced space the computational complexity of newton s direction finding subproblem in the algorithm is comparable with that of finding the steepest descent direction in the traditional first order nested reduced space algorithms for topology optimization that is the space reduction is computationally inexpensive and more importantly it does not ruin the sparsity of the full space system of optimality conditions the fast local convergence of the algorithm allows one to efficiently solve a sequence of optimization problems for varying parameters numerical continuation this can be utilized for eliminating the errors introduced by the approximate enforcement of the boundary conditions or 0 1 type constraints on the design field through penalties in many topology optimization approaches we test the algorithm on the benchmark problems of dissipated power minimization for stokes flows and in all cases the algorithm outperforms the traditional first order reduced space nested approaches by a factor varying from two to almost twenty in terms of the number of iterations while attaining an almost unprecedented accuracy in solving the discretized topology optimization problem finally we present a few extensions to the algorithm one involving computations on adaptively refined meshes and another related to solving topology optimization problems for non newtonian fluids

connected components of definable groups and o minimality i abstract we give examples of definable groups g in a saturated model sometimes o minimal such that g 00 g 000 yielding also new examples of non g compact theories we also prove that for g definable in a saturated o minimal structure g has a bounded orbit i e there is a type of g whose stabilizer has bounded index if and only if g is definably amenable giving a positive answer to a conjecture of newelski and petrykowski in this special case of groups definable in o minimal structures

stabilization of single input nonlinear systems using piecewise constant controllers the purpose of this note is to present a new approach for stabilizing single input control affine systems for which the robust relative degree equals the dimension of the state space in contrast to the method used in approximate feedback linearization we employ piecewise constant controllers which allows us to account for higher order terms

quadratic forms and approximate feed back linearization in discrete time quadratic feedback linearization is discussed for discrete time systems i e approximate feedback linearization up to the second order with respect to input and state variables after introducing in discrete time two equivalence classes invariant under quadratic static state feedback and diffeomorphism two canonical quadratic representations are proposed then it is shown that any nonlinear discrete time dynamics which are controllable in their first approximations can be linearized up to the second order under dynamic feedback the results stated here can be considered as the general discrete time analogues of the continuous time ones kang and krener 1992

reversible linear and nonlinear discrete time dynamics it is shown that any causal discrete time system can be realized with a reversible or invertible state variable representation this bridges the gap with continuous time where flows always satisfy this property and settles an old debate on the relevance of invertibility in nonlinear discrete dynamics as the result applies to the linear case it is shown how the classic approach should be slightly modified and several examples of the benefits that can be obtained are given difference algebra is employed for nonlinear systems it leads to a most elegant characterization of causality which enables reversibility to be demonstrated module theory suffices in the linear case which is supplemented by elementary matrix manipulations

alternative formulations for structural optimization an evaluation by using trusses the concept of simultaneous analysis and design sand is revisited with two objectives in mind 1 to study and evaluate three alternative formulations for structural optimization using trusses as examples and 2 to implement the formulations with a structural analysis program and study their implementation aspects the idea is to gain insights for use of the alternative formulations for structural optimization three alternative formulations are defined using displacements and forces or stresses as optimization variables all of the problem functions in the formulations become explicit in terms of the variables gradients of the functions can be evaluated more easily compared to the conventional approach where special design sensitivity analysis methods are needed existing analysis software is integrated with an optimizer based on a sparse sequential quadratic programming method to solve examples with known solutions with the sand type formulations only the pre and postprocessing capabilities of the analysis software are used to evaluate the problem functions however the conventional formulation also needs the equation solving capabilities for analysis as well as design sensitivity analysis performance of the formulations is evaluated with extensive numerical experiments and their advantages and disadvantages are discussed based on the present research it is concluded that the alternative sand formulations are more efficient compared to the conventional approach and have potential for further development for practical applications particularly the formulations that do not require calculation of the global stiffness matrix

right coprime factorizations and stabilization for nonlinear systems the problem of constructing right coprime factorizations which are based on the graph of an input output map instead of the bezout identity of nonlinear input output maps is considered the map is assumed to arise from a state variable realization with a fixed initial state the main result is that the existence of a stabilizing state feedback implies the existence of a right coprime factorization for the map the technique is illustrated by application to nonlinear systems which are affine in the control and have a controllable linear past and nonlinear systems which are feedback linearizable a notion of input output stability that requires a bound on the magnitude of the input signals is introduced methods of constructing such bounds are developed for a locally feedback linearizable system the problem of input bounds is transferred to the equivalent linear system this leads to a technique that allows state and input constraints for a feedback linearizable system to be mapped to the equivalent linear system

on the linear equivalents of nonlinear systems this paper studies feedback equivalence of nonlinear systems an equivalence relation is established for the systems x f x u a complete characterization of the class of nonlinear systems which are equivalent to controllable linear systems is obtained

state and feedback linearizations of single input control systems abstract in this paper we address the problem of state resp feedback linearization of nonlinear single input control systems using state resp feedback coordinate transformations although necessary and sufficient geometric conditions have been provided in the early eighties the problems of finding the state resp feedback linearizing coordinates are subject to solving systems of partial differential equations we will provide here a solution to the two problems by defining algorithms allowing to compute explicitly the linearizing state resp feedback coordinates for any nonlinear control system that is indeed linearizable resp feedback linearizable each algorithm is performed using a maximum of n 1 steps n being the dimension of the system and they are made possible by explicitly solving the flow box or straightening theorem we illustrate with several examples borrowed from the literature

some numerical aspects of approximate linearization of single input non linear systems we characterize approximate linearization to order m of a single input non linear system in terms of the existence of an integrating factor to order m 2 which is equivalent to the well known condition that a system of vector fields be order m 1 involutive we present a constructive method to obtain the required transformation for approximate linearization which results in substantial computational savings

on the synthesis of linear input output responses for nonlinear systems necessary and sufficient conditions are given for the existence of a feedback control law under which the input dependent part of the response of a given nonlinear system becomes linear in the input and independent of the initial state

groups measures and the nip we discuss measures invariant measures on definable groups and genericity often in an nip failure of the independence property environment we complete the proof of the third author s conjectures relating definably compact groups g in saturated o minimal structures to compact lie groups we also prove some other structural results about such g for example the existence of a left invariant finitely additive probability measure on definable subsets of g we finally introduce a new notion compact domination domination of a definable set by a compact space and raise some new conjectures in the o minimal case

immersion and immersion by nonsingular feedback of a discrete time nonlinear system into a linear system an analytic discrete time nonlinear system is considered necessary and sufficient conditions for immersion of the system into a linear system are given local immersion is defined by nonsingular feedback into a linear system and given necessary and sufficient conditions for this problem it is shown that a similar approach can also be applied to continuous time affine nonlinear systems

a neural approach for control of nonlinear systems with feedback linearization several schemes for feedback linearization using neural networks have been investigated and compared then an approach to design a neurocontroller in the sense of feedback linearization is introduced the contents include 1 full input output linearization when a system has relative degree n 2 partial input output linearization when a system has relative degree r r n and 3 approximate linearization when the involutivity condition does not hold corresponding programs and examples are given to illustrate the proposed methodology

an interpolation strategy for discrete time bilinear mpc problems input output i o feedback linearization suffers from a number of restrictions which have limited its use in model based predictive control some of these restrictions do not apply to the case of bilinear systems but problems with input constraints and unstable zero dynamics persist this paper overcomes these difficulties by means of an interpolation strategy involved in this interpolation is a feasible and stabilizing trajectory which is computed through the use of invariant feasible sets and a more aggressive trajectory which can be chosen to be either the unconstrained optimal trajectory or any alternative one

feedback classification of nonlinear single input control systems with controllable linearization normal forms canonical forms and invariants we study the feedback group action on single input nonlinear control systems we follow an approach of kang and krener based on analyzing step by step the action of homogeneous transformations on the homogeneous part of the same degree of the system we construct a dual normal form and dual invariants with respect to those obtained by kang we also propose a canonical form and a dual canonical form and show that two systems are equivalent via a formal feedback if and only if their canonical forms resp their dual canonical forms coincide we give an explicit construction of transformations bringing the system to its normal dual normal canonical and dual canonical forms we illustrate our results by simple examples on mathbb r 3 and mathbb r 4

on approximate feedback equivalence of affine control systems as a generalization of some ideas of krener 1984 a b we introduce problems of k l and k r feedback equivalence of affine systems and relate them to the action of so called k l and k r feedback groups then using some elementary properties of jets of smooth maps we establish necessary conditions under which the groups act on a manifold of systems with open orbits this result allows us to determine a range of dimensions n m k r where n is the dimension of the state space and m the number of input within which the problems are ill posed in particular the result applies to problems of k r reduction of affine systems i e of finding a feedback transforming the kth order taylor polynomial of an affine system into the rth order one r k

ill conditioning and computational error in interior methods for nonlinear programming ill conditioning has long been regarded as a plague on interior methods but its damaging effects have rarely been documented in fact implementors of interior methods who ignore warnings about the dire consequences of ill conditioning usually manage to compute accurate solutions we offer some insight into this seeming contradiction by analyzing ill conditioning within a primal dual method in which the full usually well conditioned primal dual matrix is transformed to a condensed inherently ill conditioned matrix mpd we show that ill conditioning in the exact condensed matrix closely resembles that known for the primal barrier hessian and then examine the influence of cancellation in the computed constraints using the structure of mpd various bounds are obtained on the absolute accuracy of the computed primal dual steps without cancellation the portion of the computed x step in the small space of mpd a subspace close to the null space of the jacobian of the active constraints has an absolute error bound comparable to machine precision and its large space component has a much smaller error bound with cancellation the usual case the absolute error bounds for both the small and large space components of the computed x step are comparable to machine precision in either case the absolute error bound for the computed multiplier steps associated with active constraints is comparable to machine precision the computed multiplier steps for inactive constraints although converging to zero retain approximately full relative precision because of errors in forming the right hand side the absolute error in the computed solution of the full well conditioned primal dual system is shown to be comparable to machine precision thus under quite general conditions ill conditioning in mpd does not noticeably impair the accuracy of the computed primal dual steps a similar analysis applies to search directions obtained by direct solution of the primal newton equations

eberlein oligomorphic groups we study the fourier stieltjes algebra of roelcke precompact non archimedean polish groups and give a model theoretic description of the hilbert compactification of these groups we characterize the family of such groups whose fourier stieltjes algebra is dense in the algebra of weakly almost periodic functions those are exactly the automorphism groups of 0 stable 0 categorical structures this analysis is then extended to all semitopological semigroup compactifica tions s of such a group s is hilbert representable if and only if it is an inverse semigroup we also show that every factor of the hilbert compactification is hilbert representable

analysis and control of hopf bifurcations in this paper control systems with two uncontrollable modes on the imaginary axis are studied the main contributions include the local orientation control of periodic solutions and center manifolds the quadratic normal form of systems with two imaginary uncontrollable modes the stabilization of the hopf bifurcation by state feedback and the quadratic invariants that characterize the nonlinearity of a system and its hopf bifurcation

simultaneous solution approaches for large optimization problems in this paper efficient simultaneous strategies are presented for the optimization of practical problems involving pde models in particular reduced sequential quadratic programming methods for problems with only few influence variables and simultaneous quadratic programming iterations are discussed as a result we obtain algorithms whose overall computational complexity is reduced considerably in comparison to a black box approach

controllability distributions and systems approximations a geometric approach given a nonlinear system we determine a relation at an equilibrium between controllability distributions defined for a nonlinear system and a taylor series approximation of it the value of such a relation is appreciated if we recall that the solvability conditions as well as the solutions to some control synthesis problems can be stated in terms of geometric concepts like controlled invariant controllability distributions the relation between these distributions at the equilibrium will help us to decide when the solvability conditions of this kind of problems are equivalent for the nonlinear system and its approximation

survey approximate linearization via feedback an overview fostered by a growing interest in nonlinear control theory and catalyzed by the discovery in the early 1980s of the exact conditions under which a nonlinear plant can be linearized by static state feedback and coordinate transformation in the last decades there has been a rapid increase of interest in the search for approximate solutions to the problem of linearizing nonlinear systems by state or output feedback main reason for that is the limited applicability of the rigorous methods and the complexity sensitivity and design difficulties of the exact linearizing compensators if any in the present paper the literature on the subject is reviewed and organized in what is believed to be a new and consistent perspective recent works especially in the area of data based techniques are in fact described and related whenever possible to fundamental results previously obtained by model based differential geometric methods this is expected to bring modern system linearization methods closer to the needs of practicing control engineers and to stimulate further research eventually able to fill visible gaps in this direction