robust control of cascaded and individually feedback linearizable nonlinear systems abstract feedback linearization technique is applied to design robust controllers for a class of nonlinear uncertain systems every system in the class is a series connection of finite number of nonlinear subsystems which are individually feedback linearizable and have significant but matched nonlinear functional uncertainties the key difference between this work and existing robust results is that the matching conditions and the existence of feedback linearization transformation are only required individually for each subsystem a constructive robust control design procedure is proposed for the unique control in the overall system the procedure is conceptually simple and contains three steps first perform input output feedback linearization for all individual subsystems second design fictitious robust controllers for inputs of each subsystem as if the subsystem had an independent control finally a mapping is developed which maps the individual fictitious controls recursively into the unique control of the overall system since neither the matching conditions nor feedback linearization are required for the overall system this becomes the broadest class of nonlinear uncertain or known systems for which a robust regular tracking control can be constructed using feedback linearization method global stability of the overall system is always guaranteed

on vanishing stability regions in nonlinear systems with high gain feedback when local stabilization of nonlinear systems is achieved by linear feedback the resulting stability region may vanish as the feedback gains increase this is demonstrated by examples in which neglected nonlinearities create an unstable limit cycle around an asymptotically stable equilibrium as the feedback gains tend to infinity the unstable limit cycle shrinks to the equilibrium if a feedback linearization design is applied the same instability mechanism may occur when the nonlinearities are not precisely known

induced i l fuzzy topological spaces abstract in this paper we introduce the notion of induced i l fuzzy topological spaces by using the i l valued lower semicontinuous mappings and investigate some of their basic properties in particular we give a necessary and sufficient condition for an i l fuzzy topological space to be induced and prove that this kind of induced space preserves the cartesian product and the n compactness as well

a model reference adaptive control scheme for pure feedback nonlinear systems a model reference adaptive control mrac scheme is presented for nonlinear systems in a pure feedback canonical form with unknown parameters the present of parameter uncertainty in the system causes imperfect linearization i e it introduces nonlinear additive terms in the transformed coordinates under some mild technical assumptions global convergence of the output error is established for all initial estimates of the parameter vector lying in an open neighborhood of the true parameters in the parameter space

an insertion theorem for continuous valued functions and its consequences spaces in which open sets take values in the l interval i l are investigated we prove a theorem concerning the insertion of a continuous function with values in i l with l a complete lattice we establish certain factorizations of the functors i l and i l with l a hypercontinuous lattice the latter result and the insertion theorem provide a partial answer to a recent open question related to insertion of lattice valued functions as another application of the insertion theorem we establish a relation between complete regularity in the categories of l topological and i l topological spaces with l a frame

singular pdes and the single step formulation of feedback linearization with pole placement the present work proposes a new formulation and approach to the problem of feedback linearization with pole placement the problem under consideration is not treated within the context of geometric exact feedback linearization where restrictive conditions arise from a two step design method transformation of the original nonlinear system into a linear one in controllable canonical form with an external reference input and the subsequent employment of linear pole placement techniques in the present work the problem is formulated in a single step using tools from singular pde theory in particular the mathematical formulation of the problem is realized via a system of first order quasi linear singular pdes and a rather general set of necessary and sufficient conditions for solvability is derived by using lyapunov s auxiliary theorem the solution to the system of singular pdes is locally analytic and this enables the development of a series solution method that is easily programmable with the aid of a symbolic software package under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear state feedback law computed through the solution of the system of singular pdes both feedback linearization and pole placement design objectives may be accomplished in a single step effectively overcoming the restrictions of the other approaches by bypassing the intermediate step of transforming the original system into a linear controllable one with an external reference input

connectedness and local connectedness in lowen spaces a fuzzy connectedness axiom and a matching fuzzy local connectedness are defined in this paper for lowen spaces it is shown that such lowen spaces have properties corresponding to those of traditional connected and locally connected topological spaces it is also proved that an l topological space x d is fuzzy connected resp fuzzy locally connected in our sense if and only if its induced i l topological space x w d is too some related categorical results are also obtained and a fuzzy version of topological intersection theorems is given by using our notion of connectedness

state feedback control of non linear systems a design method for state feedback controllers for single input non linear systems is proposed the method makes use of the transformations of the non linear system into controllable like canonical forms the resulting non linear state feedback is designed in such a way that the eigenvalues of the linearized closed loop model are invariant with respect to any constant operating point the method constitutes an alternative approach to the design methodology recently proposed by baumann and rugh also a review of different transformation methods for non linear systems is presented an example and simulation results of different control strategies are provided to illustrate the design technique

linearization of discrete time nonlinear systems and a canonical structure linearizable nonlinear discrete time systems are described as a composition of diffeomorphisms linearizability of a general nonlinear system is obtained as an extension of linearizability of the system expressed in terms of the diffeomorphisms

an input output characterization for linearization by feedback for a stationary initially relaxed single input single output bilinear realizable system that is described by a sequence of regular transfer functions the relative degrees of the individual transfer functions are defined in a manner similar to the linear case these relative degrees are used to characterize the region where linear input output behavior can be obtained by state feedback in a bilinear realization of the system

structure of algebras of commutative matrices commutative spaces of matrices a nk 1 akjk are studied where jk is a set of 0 1 matrices with prescribed sum s and the ak s are complex parameters the great number of commutative spaces a obeying simple prescriptions on the first two rows of s can all be described in terms of two basic algebras of matrices denoted by tn and n a relationship between the structure of spaces a and the multiplicative complexity of the bilinear forms defined by the jk s is discussed

a new method for linearizing non linear systems the pseudolinearization consider the single input non linear system x f x u it has been shown that under mild conditions there exist transformations z t x v s x u such that in the z space the linear tangent model is independent of the operating point a constructive procedure to obtain the above transformations is described this new method is computationally attractive and effective in many practical cases

global state and feedback equivalence of nonlinear systems the problem of finding global state space transformations and global feedback of the form u t x t to transform a given nonlinear system xdot f x g x u to a controllable linear system on rn or on an open subset of rn is considered here we give a complete set of differential geometric conditions which are equivalent to the existence of a solution to the above problem

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

local convexity and local boundedness of induced i l topological vector spaces in this paper we prove that an l topological vector space is locally convex locally bounded iff its induced i l topological vector space is locally convex locally bounded where i l is the l fuzzy unit interval also we give a necessary and sufficient condition for the induced i l topological vector space to be i l fuzzy normable

a functional equations approach to nonlinear discrete time feedback stabilization through pole placement abstract the present work proposes a new approach to the nonlinear discrete time feedback stabilization problem with pole placement the problem s formulation is realized through a system of nonlinear functional equations and a rather general set of necessary and sufficient conditions for solvability is derived using tools from functional equations theory one can prove that the solution to the above system of nonlinear functional equations is locally analytic and an easily programmable series solution method can be developed under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear discrete time state feedback control law that are both computed through the solution of the system of nonlinear functional equations the feedback stabilization with pole placement design objective can be attained under rather general conditions the key idea of the proposed single step design approach is to bypass the intermediate step of transforming the original system into a linear controllable one with an external reference input associated with the classical exact feedback linearization approach however since the proposed method does not involve an external reference input it cannot meet other control objectives such as trajectory tracking and model matching

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

on some sets of permutation matrices abstract in 5 a class of p p circulant matrices was studied where p is a prime and necessary and sufficient conditions were presented for the matrices in to be commutative and to be closed with respect to matrix multiplication here we show that these properties also hold for n n circulant matrices where n is a positive integer with an additional condition namely contains an n cycle

feedforward and feedback linearization of non linear systems with disturbances the problem of transforming non linear systems with input disturbances to their linear equivalents is investigated a characterization of the class of non linear systems with disturbances that are equivalent to controllable quasi linear systems with disturbances z az bv z d is given the necessary transformations state space coordinate changes and control input space coordinate changes by non linear feedforward and feedback are presented and their implications for control system synthesis are discussed finally a numerical example is given

feedback equivalence of control systems abstract cartan s method of equivalence is applied to the problem of equivalence of 2 state systems with scalar control under feedback the differential invariants produced by the method completely characterize equivalence classes the phenomenon of linearizability is associated with the presence of infinite lie pseudogroups the generic non linearizable case has a geometrically defined variational problem which yields a time optimal closed loop control

l uniform spaces versus i l uniform spaces this paper studies relationships between the categories of i l uniform spaces l uniform spaces and uniform spaces we construct two adjunctions f l j l between the category of i l uniform spaces and the category of l uniform spaces and f l j l between the category of l uniform spaces and the category of uniform spaces with l a complete lattice with an order reversing involution in both cases which with l 0 1 and l i 0 1 respectively reduce to the adjunction from the category of i uniform spaces to the category of uniform spaces investigated by katsaras if l is any complete lattice not necessarily with an order reversing involution then so is the l unit interval i l and we have another adjunction viz f i l j i l from the category of i l uniform spaces to the category of uniform spaces we show that the following two factorizations hold f i l f l f l and j i l j l j l when l is a meet continuous lattice with an order reversing involution there is also a natural link between f l j l and the existing adjunction w l i l from the category of i l topological spaces to the one of l topological spaces via the forgetful functors an essential tool in this paper is the theory of galois connections it is emphasized that no kind of distributivity is assumed on the lattice l in this paper

dynamic compensation of measurable disturbances in non linear multivariable systems this paper concerns multivariable nonlinear systems with measurable disturbance inputs for such systems we address the problem of eliminating the effect of the disturbance inputs on the outputs and achieving a well characterized linear input output behaviour in the closed loop system necessary and sufficient conditions are derived under which dynamic feedforward static state feedback laws solve the above problem and explicit synthesis formulas are developed

on time scaling for nonlinear systems application to linearization in this note we propose a method to analyze systems in a time scale which is varied depending on the state such as dt d tau s x where t and are the actual time scale and that of new one respectively and s x is the function which we call time scaling function analysis of the system in the new time scale enables us to investigate the intrinsic structure of the system a linearization problem in the new time scale is formulated as wide sense feedback equivalence and is solved it is also shown that the time scaling function which makes the system linear is derived as the solution of differential equations

on l tychonoff spaces topics of this paper include 1 basic terminology and auxiliary observations 2 pointfree characterizations of completely l regular spaces 3 embedding spaces of prescribed weight applications to l tychonoff and l to spaces 4 remarks on liu s characterizations of complete l regularity involving points 5 extending continuous l real functions from h compact subspaces 6 conditional completeness of c x r l 7 remarks on the compactification of liu and luo 8 l creg l creg and l tych l tych 9 lower order axioms for the l line a new coherent system of separation axioms for l topological spaces 10 open questions

some comments on global linearization of nonlinear systems it is shown that even in the case of a single input system on r 2 it will be rather difficult to find conditions sufficient to guarantee existence of a global linearization and that local conditions alone will definitely not be enough a class of examples which is fairly typical and it is hoped useful is discussed

on the largest feedback linearizable subsystem abstract a feedback invariant set of integers is associated with any nonlinear multivariable system which is linear with respect to the inputs it is shown to be the set of controllability indices of the largest feedback linearizable subsystem i e the largest subsystem which can be made locally linear and controllable by means of nonsingular feedback transformations

state observers and state feedback controllers for a class of non linear systems a design methodology for observers and controllers for a class of single input single output non linear systems is developed a non linear observer form is defined and a corresponding observer using non linear observer gains is specified the resulting error dynamics which are non linear are asymptotically stable for a proper choice of those gains and bounded inputs and outputs a non linear controller form is defined and a corresponding controller which results in linear closed loop dynamics with arbitrary eigenvalue placement is devised transformations from the class of interest to the defined forms are derived in the observer case the transformation to the observer form and the non linear entries in the state matrix of that form can be determined separately leading to a system of linear partial differential equations a major simplification in the controller case a simple well known system of linear partial differential equations is obtained the efficacy of the developed methodology is sho

linearization and input output decoupling for general nonlinear systems necessary and sufficient conditions are obtained for the linearization and input decoupling by state feedback of general nonlinear systems it is shown how these conditions can be derived from the already known conditions for affine nonlinear systems thereby also elucidating the existing theory for affine systems

the differential geometric structure of general mechanical systems from the lagrangian point of view several differential geometric points of view on analytical mechanics of systems with a finite number of degrees of freedom are developed in generality emphasizing cartan s calculus of differential forms and ehresmann s theory of jet spaces the classical theory of lagrange s equations with external forces and constraints holonomic or nonholonomic is put into an invariant and coordinate free form the relation between this lagrangian and the hamiltonian symplectic approach which is that most extensively used in the contemporary mathematical physics literature is also developed

permutation matrix groups with positive sum abstract explicit forms are given for all commutative sets of permutation matrices which sum to a positive matrix and for nonabelian groups of permutation matrices of order twice an odd prime which sum to the matrix of all ones a relationship to circulants of level k is indicated

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

global transformations of nonlinear systems recent results have established necessary and sufficient conditions for a nonlinear system of the form dot x t f x t u t g x t with f 0 0 to be locally equivalent in a neighborhood of the origin in rnto a controllable linear system we combine these results with several versions of the global inverse function theorem to prove sufficient conditions for the transformation of a nonlinear system to a linear system in doing so we introduce a technique for constructing a transformation under the assumptions that g ldot f dotg ad n 1 f ldotg span an n dimensional space and that g ldot f ldot g ad n 2 f ldotg is an involutive set

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

l fuzzy normal spaces and tietze extension theorem abstract in this paper we prove a characterization theorem for normal l fuzzy topological spaces l is an infinitely distributive lattice with an order reversing involution in the particular case l 0 1 this theorem reduces to a known result of kat tov fund math 38 1951 85 91 and tong duke math j 19 1952 289 292 as an important application we obtain a fuzzy version of tietze extension theorem this yields an affirmative answer to a question raised in a series of papers by rodabaugh fuzzy sets and systems 11 1983 163 183

higher separation axioms in l topologically generated i l topological spaces the main result of this paper is a theorem about inserting a pair of semicontinuous l real valued functions which extends the insertion theorem of kubiak comment math univ carolinae 34 1993 357 362 from l 0 1 to an arbitrary meet continuous lattice l endowed with an order reversing involution with this result it is shown that the normality type separation axioms in top l are preserved by the functor which takes an l topological space x to the i l topological space l x obtained by providing the set x with the i l topology consisting of all lower semicontinuous functions from x to i l the same is proved for the case of the regularity axiom

an extended direct scheme for robust adaptive nonlinear control abstract the proposed adaptive scheme achieves regulation for a class of nonlinear systems with unknown constant parameters and unmodeled dynamics the scheme does not employ overparametrization and does not restrict the class of nonlinearities by any growth conditions instead the dependence on the unknown parameters is restricted by an extended matching condition which however is satisfied in many systems of practical importance such as most types of electric motors

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

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

method for the global analysis of non linear parametrized systems a method for the global analysis of non linear systems is presented which provides a framework for the analysis of both continuous and discrete time non linear systems for the diverse range of model types by extending a numerical approach to enable the analysis of non linear parametrized non linear systems a global perspective is achieved that is difficult to obtain using analytical methods alone simple examples are used to illustrate how the method characterizes the system stability at the same time as detecting typical qualitative bifurcation phenomena a non linear feedback system is analysed in order to show how the approach provides both qualitative and quantitative information over a defined region of the system parameter space the method proves to be a powerful tool which can be used to probe the non linear characteristics of a system

method for the control of a special class of non linear systems abstract a method for the numerical synthesis of compensators for non linear control systems is proposed the non linear systems investigated are the type in which the system can be represented by two separate parts a linear element and a non linear element as shown in fig 1 the approach adopted is called the method of inequalities in which the problem is formulated as a set of inequalities which represent the specified closed loop performance of the system as well as constraints of an engineering and financial kind the non linear part of the system is linearized using an exponential input describing function an inequality is then formulated concerning the non linear element to ensure that every point in the admissible space represents a closed loop system which does not exhibit autonomous limit cycle oscillation this inequality is defined by using an enhanced sinusoidal describing function which takes account of the effects of higher harmonic signal components as well as of a fundamental sinusoi

galois connections between categories of l topological spaces a galois connection between two concrete categories a and b is a pair of concrete functors f email protected b g email protected a such that id y fg y y email protected b is a natural transformation from the functor email protected g to the identity functor on b and id x email protected gf x email protected a is a natural transformation from the identity functor on a email protected f in this paper it is demonstrated that for any complete lattices l 1 l 2 every galois connection between the category l 1 top of l 1 topological spaces and the category l 2 top of l 2 topologically spaces is determined by an l 1 topology d on l 2 several examples of such galois connections are given under a mild assumption it is showed that every element of the l 1 topology d is order preserving

on l tychonoff spaces ii this paper is a sequel to the 1995 paper on l tychonoff spaces the embedding theorem for l topological spaces is shown to hold true for l an arbitrary complete lattice without imposing any order reversing involution on l some results on completely l regular spaces and on l tychonoff spaces which have previously been known to hold true for l a frame are exhibited as ones holding for l a meet continuous lattice for such a lattice an insertion theorem for completely l regular spaces is given some weak forms of separating families of maps are discussed we also clarify the dependence between the sub t0 separation axiom of liu and the l t0 separation axiom of rodabaugh

representations for the drazin inverses of 2 2 block matrices abstract let m denote a 2 2 block complex matrix a b c d where a and d are square matrices not necessarily with the same orders in this paper explicit representations for the drazin inverse of m are presented under the condition that bd i c 0 for i 0 1 n 1 where n is the order of d

on compactness of induced i l fuzzy topological spaces abstract the induced i l fuzzy topological spaces for l fuzzy topological spaces introduced by wang geping is a kind of important fuzzy topological space in this paper the author studies the fuzzy compactness of induced i l fuzzy topological spaces some available relations between molecular nets of an l fuzzy topological space and that of its induced i l fuzzy topological space are firstly presented by using these relations the author obtains that the induced i l fuzzy topological space is strong fuzzy compact resp fuzzy compact n compact if and only if the l fuzzy topological space is strong fuzzy compact resp fuzzy compact n compact the author also proves that the induced i l fuzzy topological space is paracompact if and only if the l fuzzy topological space is paracompact

survey of local connectedness axioms and their properties in l topological spaces based on the notions of fuzzy connectedness defined by lu and li fuzzy connectedness new definitions and comparisons fuzzy sets and systems 157 2006 1928 1940 some new fuzzy local connectedness such as local connectedness local ultra f 1 connectedness local strong f 1 connectedness local ultra f 2 connectedness local strong f 2 connectedness ultra f 1 local connectedness strong f 1 local connectedness ultra f 2 local connectedness and strong f 2 local connectedness of l topological spaces or l cotopological spaces are defined in this paper apart from maintaining some frequently used properties that classical local connected topological spaces possess l topological spaces or l cotopological spaces which satisfy our new fuzzy local connectedness axioms also have some interesting properties for example local connectedness local ultra f 1 connectedness local ultra f 2 connectedness local strong f 1 connectedness and local strong f 2 connectedness are all i l inducible if l is a complete and meet continuous demorgan algebra they are l extension of local connectedness of topological spaces under some moderate restrictions to l it is proved that all l intervals are locally connected if copr l is a generating set of l some categorical results reflecting the differences between classical topology and l topology are obtained the relationships between these notions of connectedness are also investigated in detail

canonical forms of decouplable and controllable linear systems in this paper we present the canonical forms of decouplable and controllable linear systems some of the non zero elements in gilbert s canonically decoupled structure which remain undetermined are explicitly determined by a complete set of invariances our result constitutes the key step to characterization of the whole class of nonlinear systems that are feedback equivalent to decouplable and controllable linear systems

deterministic feedback linearization girsanov transformations and finite dimensional filters the problem of extending state feedback linearization methods of deterministic control theory to stochastic systems is addressed for stratonovitch stochastic differential equations with smooth vector fields necessary and geffucient geometric conditions for local and global linearization by diffeomorphism and absolutely continuous change of probability law are obtained using the interpretation of girsanov transformations as state feedback on brownian motions for stochastic systems with single with single input or onde dimensional brownian motion and single output or one dimensional observation process necessary and sufficient geometric conditions to transform the duncan mortensen zakai dmz equation of filtering into that of an affine prime system are obtained as well as interpretation of gauge transformation as girsanov change of probability law

a categorical accommodation of various notions of fuzzy topology different definitions of fuzzy topology have been stated and developed in the literature but have not been satisfactorily related to each other and to ordinary topology to remedy this a new fuzzy topological category fuzz is defined it is a significant generalization of previous definitions it is the simplest setting yet constructed into which ordinary topology and these previous definitions may be placed by identifying each with a different subcategory of fuzz it therefore allows us to conveniently analyze the relative merits of previous definitions give coherence to known results and indicate appropriate directions for future development and it generates and may be viewed as both including and included in categories of fuzzy topological spaces which exhibit higher order fuzziness