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

nonlinear decoupling via feedback a differential geometric approach the paper deals with the nonlinear decoupling and noninteracting control problems a complete solution to those problems is made possible via a suitable nonlinear generalization of several powerful geometric concepts already introduced in studying linear multivariable control systems the paper also includes algorithms concerned with the actual construction of the appropriate control laws

dynamics of piecewise linear maps and sets of nonnegative matrices abstract we consider maps f k v min a k av and g k v max a k av where k is a finite set of nonnegative matrices and by min and max we mean component wise minimum and maximum we transfer known results about properties of g k to f k in particular we show existence of nonnegative generalized eigenvectors of f k give necessary and sufficient conditions for existence of strictly positive eigenvector of f k study dynamics of f k on the positive cone we show the existence and construct matrices a and b possibly not in k such that f k n v a n v and g k n v b n v for any strictly positive vector v

robust tracking in nonlinear systems a nonlinear feedback multivariable controller is used to implement multivariable tracking in a nonlinear system the tracking error is measured by a general function of system state and the input command the controller is robust in the sense that the tracking error is ultimately bounded in the presence of modeling errors free parameters which affect the form of the controller allow flexibility in determining such factors as the size of the ultimate bound the rate of error decay excursion of the control conditions on the class of modeling errors and the level of system gain restrictive assumptions on the structure of the model and the modeling errors are required but they are treated in a transformation framework which allows the generalization of similar conditions which appear in the prior literature these assumptions hold for robotic manipulators this application is investigated at some length and it appears that the resulting control scheme may have advantages over others which have been proposed in the robotics literature

some new classes of complex symmetric operators we say that an operator t e b h is complex symmetric if there exists a conjugate linear isometric involution c so that t ct c we prove that binormal operators operators that are algebraic of degree two including all idempotents and large classes of rank one perturbations of normal operators are complex symmetric from an abstract viewpoint these results explain why the compressed shift and volterra integration operator are complex symmetric finally we attempt to describe all complex symmetric partial isometries obtaining the sharpest possible statement given only the data dim ker t dim ker t

local stabilization of minimum phase nonlinear systems abstract in this note we briefly review the notion of a nonlinear minimum phase system i e a system which when constrained in such a way as to produce zero output evolves with an asymptotically stable dynamics the main purpose of the paper is to show that any minimum phase nonlinear system can always be locally stabilized by smooth state feedback

robust output tracking for nonlinear uncertain systems abstract the problem of robust output tracking for nonlinear systems in the presence of uncertainties is further studied by extending the results of behtash 1990 to systems with mismatched uncertainties a new design method which is called the stable combined variable perturbation method scvpm is proposed and a robust controller is presented using high gain feedback our results are compatible with those of behtash 1990 in a sense that the new controller will degenerate to the one of behtash 1990 if the so called generalized matching condition is satisfied simulation results verified the effectiveness of the newly proposed method and controller

complex symmetric generators of c algebras abstract this paper gives necessary and sufficient conditions for an essentially normal operator t to have its c algebra c t generated by a complex symmetric operator also it is completely determined when c t is isomorphic to a c algebra singly generated by complex symmetric operators these both depend only on the singular part of t

almost disturbance decoupling for single input single output nonlinear systems the almost disturbance decoupling problem for nonlinear single input single output systems is addressed by using singular perturbation methods and high gain feedback sufficient conditions and the explicit high gain nonlinear state feedback in solvable cases are given they generalize both previous almost results for linear systems and exact ones for nonlinear systems the necessity of the conditions is discussed in particular an example is given where the main structural condition is not satisfied and the high gain control designed on the basis of linear approximations fails to achieve almost disturbance decoupling for the original system

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

on m complex symmetric weighted shift operators on cn abstract in this paper we study m complex symmetric weighted shift operators on c n let t be the backward weighted shift on c n for some n 2 we consider when t and t a the matrix of entries the moduli of those of t are both m complex symmetric with the same standard conjugation c give as well some unitary operators useful in the study and generalize to upper triangular matrices also we show that if t is 2k complex symmetric with the standard conjugation c for some k n with k n then t is 2 k 1 complex symmetric with the conjugation c

globally stable robust tracking of nonlinear systems using variable structure control and with an application to a robotic manipulator the globally stable robust output tracking for a class of nonlinear systems is considered based only on the knowledge of the bounds on the uncertainties a variable structure control vsc law is developed under the structure matching assumption it is shown that the outputs of the closed loop system asymptotically track given output trajectories despite the uncertainties while maintaining the boundedness of all signals inside the loop all signals inside the loop are shown to be bounded for all time to illustrate the efficiency of the controller the approach is applied to the case of a two degree of freedom dof robotic manipulator with variable payload numerical simulation results are also provided

weyl s theorem for complex symmetric operators abstract an operator on hilbert space is complex symmetric if it can be represented as a symmetric matrix relative to some orthonormal basis of the space it is proved in this paper that each complex symmetric operator on a complex separable hilbert space has a compact perturbation being complex symmetric and satisfying weyl s theorem where the compact can be chosen with arbitrarily small norm

on the stabilization and observation of nonlinear uncertain dynamic systems the problem of stabilization and observation of uncertain and or nonlinear dynamic systems for which matching conditions are not satisfied is investigated no statistical description of uncertain elements is assumed the uncertain and or nonlinear quantities are described only in terms of bounds on their possible sizes stabilizing feedback controllers and observers are proposed whose design is based on the constructive use of lyapunov functions and the bellman gronwall lemma

robustness of uncertain systems in the absence of matching assumptions abstract we consider a class of uncertain dynamical systems containing uncertain elements these may be due to model parameter uncertainty input disturbance and measurement noise we require no a priori information on the uncertain elements except that their restraint sets are assumed to be known and compact this paper describes a general approach for designing controls to assure practical stability in the absence of matching assumptions which most previous results have needed the result is applicable to non linear systems a specialization to linear systems is then given and a comparison is made with previous work

direct approach to almost disturbance and almost input output decoupling a direct solution to the almost input output and disturbance decoupling problem is given the approach is based on a new canonical form for linear systems which is obtained in a fully constructive way together with some elementary approximation results the solution is given in the form of an explicit feedback law parameterized by a scalar e by letting e go to zero the accuracy of the resulting approximate solution can be improved to any desired degree

complex symmetric monomial toeplitz operators on the unit ball abstract monomial toeplitz operators t z p z q on the weighted bergman space of the unit ball in c n are natural extension of the weighted shift operators in higher dimensions dim ker t z p z q may occur for non trivial case thus complex symmetric phenomena will appear in this paper we completely characterize when the monomial toeplitz operator t z p z q on the weighted bergman space is j a symmetric where j a f z f a z with the symmetric unitary n n matrix a moreover we also determine all possible forms of a such that t z p z q is j a symmetric as an application we construct the weakly closed unital commutative algebra generated by complex symmetric toeplitz operators

output tracking control of nonlinear systems with mismatched uncertainties abstract this paper addresses the output tracking problem of a class of nonlinear systems given input output linearizability of the class of nonlinear systems a robust output tracking controller derived via a lyapunov based approach is proposed here the model uncertainties do not need to satisfy the conventional matching condition but states and tracking error remain bounded furthermore the magnitude of the tracking error will bear dominant dependence on the magnitude of the mismatched uncertainties

multiple surface sliding control of a class of uncertain nonlinear systemsm we develop a multiple surface sliding control for a class of single input single output nonlinear systems whose uncertainties do not satisfy the standard matching condition a coordinate dependence condition on the uncertainty bound is assumed a computed normal form is defined to handle such uncertainties instead of the normal form used in conventional input output linearization a control algorithm which can be applied to global tracking problems has been developed and evaluated for a benchmark example

the omega limit sets of ray contractive operators abstract in this paper we study the omega limit sets of a ray contractive map t without the compactness assumption on the orbits if there exists a compact subset of k the interior of a closed convex cone k such that t and t k 0 k then we are able to prove that x either consists of a single point or forms a cycle 2 periodic orbit for any x k

global robust stabilization of minimum phase nonlinear systems with uncertainty the problem of global stabilization via state feedback is studied for a class of minimum phase nonlinear systems with unknown unmodelled time varying parameters or disturbances belonging to a given compact set sufficient conditions are presented for the existence of a robust stabilizing state feedback controller the controller is constructed explicitly by using a lyapunov based recursive design procedure the proposed feedback control law is a smooth function of the state and requires information of bounding functions on the size of uncertainties in the case where the disturbance signals represent only an unknown constant vector parameter and appear linearly in the system an adaptive controller is developed based on similar arguments achieving global stability with state regulation the main results of this paper incorporate and generalize a number of robust and adaptive control schemes for global stabilization of uncertain feedback linearizable systems without zero dynamics

complex symmetric operators and applications ii a bounded linear operator t on a complex hilbert space h is called complex symmetric if t ct c where c is a conjugation an isometric antilinear involution of h we prove that t cj t where j is an auxiliary conjugation commuting with t t t we consider numerous examples including the poincare neumann singular integral bounded operator and the jordan model operator compressed shift the decomposition t cj t also extends to the class of unbounded c selfadjoint operators originally introduced by glazman in this context it provides a method for estimating the norms of the resolvents of certain unbounded operators

on linear maps preserving complex symmetry abstract an operator t on a complex hilbert space h is said to be complex symmetric if there exists a conjugate linear isometric involution c h h so that c t c t this paper is devoted to describing which linear maps leave the class of complex symmetric operators invariant complete characterizations are obtained for several classes of linear maps including similarity transformations surjective linear isometries multiplication operators and certain completely positive maps

a new systematic design of high gain feedback for nonlinear systems with unmatched uncertainties the systematic design of a robust output tracking algorithm for a feedback linearizable system with unmatched uncertainties is presented the proposed technique is based on a backstepping procedure to construct a time varying singularly perturbed system and the underlying theoretical approach by the lyapunov theory the main result is that a dynamically parametrized state feedback is developed to cancel the output tracking error finally an illustrative example through a chemical reactor system is given to demonstrate the utilization of the approach developed in this paper

complex symmetric toeplitz operators on the dirichlet space abstract we study when a toeplitz operator t on the dirichlet space of the unit disk is complex symmetric with respect to a class of conjugations and find surprisingly that the case of complex symmetries of toeplitz operators according to these conjugations is very few we also show that if t is complex symmetric then the curve t t must be nowhere winding furthermore the spectrum and invertibility of complex symmetric toeplitz operators are described

complex symmetric partial isometries an operator t b h is complex symmetric if there exists a conjugate linear isometric involution c h h so that t ct c we provide a concrete description of all complex symmetric partial isometries in particular we prove that any partial isometry on a hilbert space of dimension 4 is complex symmetric

characterizations of linear volterra integral equations with nonnegative kernels abstract we first introduce the notion of positive linear volterra integral equations then we offer a criterion for positive equations in terms of the resolvent in particular equations with nonnegative kernels are positive next we obtain a variant of the paley wiener theorem for equations of this class and its extension to perturbed equations furthermore we get a perron frobenius type theorem for linear volterra integral equations with nonnegative kernels finally we give a criterion for positivity of the initial function semigroup of linear volterra integral equations and provide a necessary and sufficient condition for exponential stability of the semigroups

robust output tracking for nonlinear systems with weakly non minimum phase this paper is concerned with the problem of designing a robust output tracking controller for mimo nonlinear systems with weakly non minimum phase based on our system formulation control plants with uncertainties and or with actuator dynamics fall into the class under consideration the controller design here is divided into two phases fast feedback control and slow feedback control so that a final composite control is obtained the former is chosen to stabilize the boundary layer system whereas the latter essentially handles the mismatched uncertainties after the system is reformulated under some mild assumptions it is shown that the overall states are bounded and the tracking errors converge to a residual set whose size is a class k function of e as e 0 the residual set shrinks to the origin an interesting application to a simplified aircraft model with fast actuator dynamics which turns out to be weakly non minimum phase is given the computer simulation has verified the expected satisfac

systematic design of robust controllers for nonlinear uncertain systems the design problem of robust stabilizing controllers for nonlinear uncertain systems is considered in this paper the nominal system is input output linearizable and the uncertainty is assumed to be either generalized matching or mismatched the control design combines differential geometric feedback linearization with the deterministic approach in which no statistical property of the uncertainty is assumed and used the only information on the uncertainty which is needed for the design is its possible bound two robust stabilizing controllers for generalized matching uncertain systems are given for mismatched uncertain systems a systematic design procedure is first presented then four robust stabilizing controllers are given the robust controllers which are given in forms of explicit feedback laws based on the nominal system and are parametrized by the robust design parameters related to the bounds of uncertainties are more practicable than the existing ones in the literature numerical example

disturbance decoupling by measurement feedback with stability or pole placement in this paper we solve the disturbance decoupling problem by measurement feedback and requiring stability or pole placement on the closed loop system the problem is attacked using the geometric approach through the concepts of a bmod mathcal b invariant and controllability subspaces and their duals a mid k invariant and complementary observability subspaces the solution of this problem has an interesting structure consisting of a feedback processor which decomposes into i a disturbance decoupling loop ii a disturbance input stabilization or pole placement loop and iii a controlled output stabilization or pole placement loop

robust output tracking for non linear systems the problem of output tracking for a single input single output non linear system in the presence of uncertainties is studied the notions relative degree and minimum phase for non linear systems are reviewed given a bounded desired tracking signal with bounded derivatives a control law is designed for minimum phase non linear systems which results in tracking of this signal by the output this control law is modified in the presence of uncertainties associated with the model vector fields to reduce the effects of these uncertainties on the tracking errors two types of uncertainties are considered those satisfying a generalized matching condition but otherwise unstructured and linear parametric uncertainties it is shown that for systems with the first type of uncertainty high gain control laws can result in small tracking errors of o where e is a small design parameter an alternative scheme based on variable structure control strategy is shown to yield zero tracking errors adaptive control te

robust stabilization of mimo nonlinear time varying mismatched uncertain systems abstract this paper considers the problem of robust stabilization of multi input multi output mimo nonlinear systems in the presence of mismatched time varying uncertainties combining the differential geometric feedback linearization with the deterministic approach it derives two robust stabilizing controllers i e the high gain feedback controller and the variable structure controller these two controllers only require the knowledge of the nominal system and the bounds of the uncertainties therefore they are more feasible to be implemented both controllers reduce the effects of arbitrarily large bounded mismatched uncertainties successfully to achieve uniform ultimate boundedness stabilization of the mimo nonlinear timevarying uncertain systems

on stability and robust stability of positive linear volterra equations we first introduce the notion of positive linear volterra integrodifferential equations then we give some characterizations of these positive equations an explicit criterion and a perron frobenius type theorem for positive linear volterra integrodifferential equations are given then we offer a new criterion for uniformly asymptotic stability of positive equations finally we study stability radii of positive linear volterra integrodifferential equations it is proved that complex real and positive stability radii of positive linear volterra equations under structured perturbations or affine perturbations coincide and can be computed by explicit formulae to the best of our knowledge most of the results of this paper are new

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

on a class of positive linear differential equations with infinite delay abstract we give an explicit criterion for positivity of the solution semigroup of linear differential equations with infinite delay and a perron frobenius type theorem for positive equations furthermore a novel criterion for the exponential asymptotic stability of positive equations is presented finally we provide a sufficient condition for the exponential asymptotic stability of positive equations subjected to structured perturbations a simple example is given to illustrate the obtained results

asymptotic stabilization of minimum phase nonlinear systems how a class of multivariable nonlinear systems can be stabilized about an equilibrium via smooth state feedback is shown more precisely conditions under which for every compact set of initial states it is possible to design a feedback law which drives to the equilibrium all initial states in this compact set are described the theory includes the development of globally defined transformations of the system equations to their global normal form

a new class of stabilizing controllers for uncertain dynamical systems this paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems the vector of uncertain parameters q cdot is time varying and its values q t lie within a prespecified bounding set q in r p furthermore no statistical description of q cdot is assumed and the controller is shown to render the closed loop system practically stable in a so called guaranteed sense that is the desired stability properties are assured no matter what admissible uncertainty q cdot is realized within the perspective of previous research in this area this paper contains one salient feature the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time invariant in contrast in much of the previous literature see for example 1 2 7 and 9 a linear system is stabilized via nonlinear control another feature of this paper is the fact that the

a mathcal b invariant distributions and disturbance decoupling of nonlinear systems the concept of a b invariant subspaces has resulted in a unified approach to many of the basic structural properties of time invariant linear systems w m wonham lecture notes in economics and mathematical systems vol 101 springer verlag new york 1974 the purpose of this paper is to introduce the more general notion of a mathcal b invariant distributions on differentiable manifolds and to use this idea to study the disturbance decoupling problem for a class of nonlinear systems which evolve on real analytic manifolds

on the closure of the complex symmetric operators compact operators and weighted shifts abstract we study the closure cso of the set cso of all complex symmetric operators on a separable infinite dimensional complex hilbert space among other things we prove that every compact operator in cso is complex symmetric using a construction of kakutani as motivation we also describe many properties of weighted shifts in cso cso in particular we show that weighted shifts which demonstrate a type of approximate self similarity belong to cso cso as a byproduct of our treatment of weighted shifts we explain several ways in which our result on compact operators is optimal

operator based robust control design for nonlinear plants with perturbation in this article operator based robust control design for nonlinear plants with perturbation is considered by using robust right coprime factorisation approach in detail the property of unimodular operator is studied and a control design structure is proposed based on the proposed design scheme the designed system is robustly stable also output tracking performance can be realised simultaneously finally the effectiveness of the proposed design scheme is demonstrated by a simulation example

approximation of complex symmetric operators an operator t on a complex hilbert space mathcal h is called a complex symmetric operator if there exists a conjugate linear isometric involution c mathcal h rightarrow mathcal h so that ctc t in this paper we study the approximation of complex symmetric operators by virtue of an intensive analysis of compact operators in singly generated c algebras we obtain a complete characterization of norm limits of complex symmetric operators and provide a classification of complex symmetric operators up to approximate unitary equivalence this gives a general solution to the norm closure problem for complex symmetric operators as an application we provide a concrete description of partial isometries which are norm limits of complex symmetric operators

nonlinear perron frobenius theory in finite dimensions abstract a nonlinear perron frobenius theory in finite dimensions is described which applies to positively homogeneous and increasing maps sufficient conditions for the existence and uniqueness of eigenvectors in the interior of a cone are developed even when eigenvectors at the boundary of the cone exist several ways to benefit from nonlinearities are pointed out in order to show that the classical linear theory truly deserves nonlinear generalization the main technical novelties are a link between collatz wielandt numbers and g teaux derivatives and the almost one dimensional dynamics of the nonlinear map

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

on ultimate boundedness control of uncertain systems in the absence of matching assumptions this paper describes an approach to ultimate boundedness control under hypotheses less stringent than heretofore required instead of imposing the so called matching conditions of previous authors a certain decomposition of the dynamics is performed the resulting decomposed system has two parts a matched portion and a mismatched portion a certain measure of mismatch m is defined and it is shown that effective control is possible as long as m does not exceed some critical mismatch threshold m it is seen that this threshold depends on the choice of feedback hence it becomes possible to investigate the tradeoffs between the size of the mismatch threshold and the sizes of the feedback gains which achieve the desired ultimate boundedness of solutions