explicit formulas for opuc and popuc associated with measures which are simple modifications of the lebesgue measure we consider nontrivial probability measures obtained as simple modifications of the lebesgue measure which include mass points at z 1 and z i the orthogonal polynomials para orthogonal polynomials and toeplitz matrices associated with these measures are presented

nonlinear control of feedforward systems with bounded signals the stabilization problem for a class of nonlinear feedforward systems is solved using bounded control it is shown that when the lower subsystem of the cascade is input to state stable and the upper subsystem not exponentially unstable global asymptotic stability can be achieved via a simple static feedback having bounded amplitude that requires knowledge of the upper part of the state only this is made possible by invoking the bounded real lemma and a generalization of the small gain theorem thus stabilization is achieved with typical saturation functions saturations of constant sign or quantized control moreover the problem of asymptotic stabilization of a stable linear system with bounded outputs is solved by means of dynamic feedback finally a new class of stabilizing control laws for a chain of integrators with input saturation is proposed some robustness issues are also addressed and the theory is illustrated with examples on the stabilization of physical systems

the discrete zak transform application to time frequency analysis and synthesis of nonstationary signals algorithms to compute coefficients of the finite double sum expansion of time varying nonstationary signals and to synthesize them from a finite number of expansion coefficients are presented the algorithms are based on the computation of the discrete zak transform dzt fast algorithms to compute dzt are presented modifications to the algorithms which increase robustness are given

verblunsky coefficients related with periodic real sequences and associated measures on the unit circle it is known that given a pair of real sequences cn n 1 dn n 1 with dn n 1 a positive chain sequence we can associate a unique nontrivial probability measure on the unit circle precisely the measure is such that the corresponding verblunsky coefficients n n 0 are given by the relation n 1 n 1 1 2mn icn1 icn n 1 where 0 1 0 1 n k 1n 1 ick 1 ick n 1n 1 and mn n 0 is the minimal parameter sequence of dn n 1 in this paper we consider the space denoted by npnp of all nontrivial probability measures such that the associated real sequences cn n 1 and mn n 1 are periodic with period p for p np n by assuming an appropriate metric on the space of all nontrivial probability measures on the unit circle we show that there exists a homeomorphism gpgp between the metric subspaces npnp and vpvp where vpvp denotes the space of nontrivial probability measures with associated p periodic verblunsky coefficients moreover it is shown that the set fpfp of fixed points of gpgp is exactly vp npvp np and this set is characterized by a p 1 p 1 dimensional submanifold of rprp we also prove that the study of probability measures in npnp is equivalent to the study of probability measures in vpvp furthermore it is shown that the pure points of measures in npnp are in fact zeros of associated para orthogonal polynomials of degree p we also look at the essential support of probability measures in the limit periodic case i e when the sequences cn n 1 and mn n 1 are limit periodic with period p finally we give some examples to illustrate the results obtained

para orthogonal polynomials from constant verblunsky coefficients abstract orthogonal polynomials on the unit circle associated with constant verblunsky coefficients are also known as geronimus polynomials we consider the properties of some special sequences of para orthogonal polynomials that follow from the geronimus polynomials and as applications obtain information concerning certain associated quadrature rules and also other related orthogonal polynomials positive chain sequences play an important role in our study

saturated delayed controls for feedforward nonlinear systems the saturated delayed feedback is presented to globally stabilize a class of feedforward nonlinear systems whose nominal dynamics is the cascade of multiple oscillators and multiple integrators key strategies are stated as follows 1 to compensate the arbitrarily large input delay and to carry out the recursive analysis we transform the concerned system into a normal form 2 in the convergence analysis for multiple oscillators that are relatively difficult to treat we combine the saturated delayed terms define the composite lyapunov functions and use the cauchy formula 3 for ease of the analysis on nonlinear terms we employ saturation levels to restrict them as applications the usual feedforward nonlinear systems with multiple integrators as the nominal dynamics and the null controllable linear systems are dealt with in some new manners and a saturated delayed controller is presented for the well known tora system

kernel decomposition of time frequency distributions bilinear time frequency distributions tfds offer improved time frequency resolution over linear representations but suffer from difficult interpretation higher implementation cost and the lack of associated low cost signal synthesis algorithms in the paper the authors introduce some new tools for the interpretation and quantitative comparison of high resolution tfds these tools are used in related work to define low cost high resolution tfds and to define linear low cost signal synthesis algorithms associated with high resolution tfds first each real valued tfd is associated with a self adjoint linear operator spl psi the spectral representation of spl psi expresses the tfd as a weighted sum of spectrograms sps it is shown that the sp decomposition and weyl correspondence do not yield useful interpretations for high resolution tfds due to the fact that spl psi is not positive

integrator forwarding a new recursive nonlinear robust design we consider the global stabilization of nonlinear systems in strict feedforward form we show that these systems while not feedback linearizable can be globally transformed by feedback and diffeomorphism into lower triangular form the transformation is explicit and recursive we employ this transformation to design a globally stabilizing controller with optimality properties and an input to state stability property with respect to matched uncertainties c 1997 elsevier science ltd

an explicit formula of bounded feedback stabilizers for feedforward systems abstract this work constitutes continuation of a recent result of the same authors which provides an explicit formula for bounded feedback stabilizers for a wide class of triangular nonlinear systems in the present paper the previous result is extended for nonlinear systems having feedforward structure and whose linearization at the equilibrium is in general uncontrollable

discrete gabor transform a feasible algorithm for implementing the gabor expansion the coefficients of which are computed by the discrete gabor transform dgt is presented for a given synthesis window and sampling pattern computing the auxiliary biorthogonal function of the dgt is nothing more than solving a linear system the dgt presented applies for both finite as well as infinite sequences by exploiting the nonuniqueness of the auxiliary biorthogonal function at oversampling an orthogonal like dgt is obtained as the discrete fourier transform dft is a discrete realization of the continuous time fourier transform similarly the dgt introduced provides a feasible vehicle to implement the useful gabor expansion

robust wigner distribution with application to the instantaneous frequency estimation the wigner distribution wd produces highly concentrated time frequency tf representation of nonstationary signals it may be used as an efficient signal analysis tool including the cases of frequency modulated signals corrupted with the gaussian noise in some applications a significant amount of impulse noise is present then the wd fails to produce satisfactory results the robust periodogram has been introduced for spectral estimation of this kind of noisy signals it can produce good concentration for pure harmonic signals however it is not so efficient in the cases of signals with rapidly varying frequency this is the motivation for introducing the robust wd it is a reliable tf representation tool for wide class of nonstationary signals corrupted with impulse noise this distribution produces good accuracy of the instantaneous frequency if estimation using the huber 1981 loss function a generalization of the wd is presented it includes both the standard and the robust wd as special cases this distribution can be used for tf analysis of signals corrupted with a mixture of impulse and gaussian noise the presented theory is illustrated on examples including applications on the if estimation and time varying filtering of signals corrupted with a mixture of the gaussian and impulse noise the case study analysis of the if estimators accuracy based on the standard and the robust wd forms is performed in order to improve the if estimation a median filter is applied on the obtained if estimate

saturated stabilization for an uncertain cascaded system subject to an oscillator abstract this paper investigates saturated stabilization of an uncertain cascaded system that consists of an uncertain double integrator and an uncertain oscillator the saturation reduction analysis is carried out by using the time interval based contradiction approach while the exponential stability analysis of the corresponding reduced system is conducted by using the hurwitz lemma in doing the saturation reduction analysis an intractable problem is how to estimate the states of the oscillator fully using the property of saturation functions as well as the technique of lyapunov inequalities we can first estimate the states of the oscillator and then achieve the time derivative bounds that some saturated terms possess in a finite time this has helped to carry out the saturation reduction analysis as the applications of the suggested algorithm the stabilizing controllers have been provided for two benchmark systems the uncertain two mass spring and the translational oscillator with a rotational actuator tora

transition kernels for bilinear time frequency distributions the author proposes a sequence of time frequency signal representations which offer a continuous transition from time smoothed pseudo wigner ville distributions pwvd to a specified class of positive representations this allows the selection of a representation which offers the desired tradeoff between the inherent localization properties of the pwvd and the lack of interference terms present in the class of positive representations a cause of some criticism of the method the sequence converges uniformly to the specified positive distribution

oversampling in the gabor scheme a method for calculating the coefficients of the gabor expansion in the case of oversampling is presented the method is based on the concept of frames and utilizes the zak transform as such the zak transform highlights the meaning and importance of frames in the context of oversampling and other aspects of signal representation by means of nonorthogonal bases

alias free generalized discrete time time frequency distributions a definition of generalized discrete time time frequency distribution that utilizes all of the outer product terms from a data sequence so that one can avoid aliasing is introduced the new approach provides 1 proper implementation of the discrete time spectrogram 2 correct evaluation of the instantaneous frequency of the underlying continuous time signal and 3 correct frequency marginal the formulation provides a unified framework for implementing members of cohen s class which was formulated in the continuous time domain some requirements for the discrete time kernel in the new approach are discussed in association with desirable distribution properties some experimental results are provided to illustrate the features of the proposed method

regularity and unitarity of bilinear time frequency signal representations two structural properties of bilinear time frequency representations btfrs of signals are introduced and studied the definition of these properties is based on a linear operator description of btfrs the first property termed regularity has important implications with respect to the recovery of signals from the btfr outcome the derivation of other bilinear signal representations from a btfr the btfrs reaction to linear signal transformations and the construction of bases of induced btfr domain spaces the second property called unitarity is equivalent to validity of moyal s formula 1949 unitarity is thus necessary and sufficient for a closed form solution of optimal signal synthesis and for a btfr formulation or optimal detection estimation methods unitarity also allows the systematic construction of btfr product relations like the wigner distribution s interference formula and the ambiguity function s self transformation property unitarity permits the construction of induced orthogonal projection operators and guarantees the orthonormality of induced basis functions

robust global stabilization of a class of uncertain feedforward nonlinear systems abstract the problem of global asymptotically stabilizing a certain class of uncertain feedforward nonlinear systems is considered the control law is obtained by nesting saturation functions whose amplitude can be rendered arbitrarily small with respect to previous works on the subject the design procedure is able to deal with uncertain possibly time varying parameters ranging within the prescribed compact sets which can affect also the linear approximation of the system the small gain theorem for nonlinear systems which are input to state stable with restrictions is shown to be a key tool for designing a state feedback saturated control law

simple saturated designs for ancbc systems and extension to feedforward nonlinear systems this article re examines the robust stabilisation of the asymptotically null controllable with bounded controls ancbc systems and extends the established algorithm to a wide class of feedforward nonlinear systems whose nominal dynamics contains both multiple integrators and multiple oscillators based on the notion of converging input bounded state cibs rather than small input small state siss the computation burden in sussmann et al sussmann h j sontag e d and yang y 1994 a general result on the stabilization of linear systems using bounded controls ieee transactions on automatic control 39 2411 2425 is reduced and a class of simple saturated control laws is presented for the cibs stabilisation of ancbc systems then by combining the technique of dealing with higher order terms the algorithm for ancbc systems is extended to feedforward nonlinear systems

a nonlinear small gain theorem for the analysis of control systems with saturation a nonlinear small gain theorem is presented that provides a formalism for analyzing the behavior of certain control systems that contain or utilize saturation the theorem is used to show that an iterative procedure can be derived for controlling systems in a general nonlinear feedforward form this result in turn is applied to the control of 1 linear systems stable and unstable with inputs subject to magnitude and rate saturation and time delays 2 the cascade of globally asymptotically stable nonlinear systems with certain linear systems those that are stabilizable right invertible and such that all of their invariant zeros have nonpositive real part 3 the inverted pendulum on a cart and 4 the planar vertical takeoff and landing aircraft

global decentralised stabilisation for a class of uncertain large scale feedforward nonlinear systems in this paper the problem of global decentralised stabilisation for a class of uncertain large scale feedforward nonlinear systems is investigated the system under consideration is allowed to contain unknown non lipschitz continuous nonlinear terms the design of the global decentralised controllers takes a two steps procedure first of all based on the adding a power integrator technique and the homogeneous domination approach a local homogeneous decentralised controller is proposed for each subsystem of the large scale feedforward nonlinear system then we integrate a series of nested saturation functions with the homogeneous decentralised controllers and adjust the saturation levels to ensure globally asymptotic stability of the closed loop system simulation studies are conducted to illustrate the effectiveness of the proposed control method

new results in decentralized adaptive non linear stabilization using output feedback this paper proposes new results in decentralized adaptive stabilization for a class of large scale non linear systems with output measurements like previous work the parametric uncertainty does not satisfy any matching conditions and the uncertain interconnections may be highly non linear unlike most previous results in the literature of decentralized control the development of our systematic approach does not rely on the crucial assumption that the unmeasured zero dynamics of each individual local system are linear and enter the system additively and linearly simulation results based on a practical example of two inverted pendulums on carts demonstrate the effectiveness of the proposed decentralized adaptive stabilization methodology

the zak transform and decimated time frequency distributions in this paper the interrelation between the zak transform and the generalized discrete time frequency distribution gdtfd is examined starting with the discrete zak transform its definition is broadened to include arbitrary windows creating the windowed zak transform wzt the wzt is then combined with the spectrogram it is demonstrated that the spectrogram based upon the wzt called the zak spectrogram zs is a generalization of the standard spectrogram next building upon the idea of the weighted spectrogram the weighted zs is used to produce a new class of gdtfd called the decimated gdtfd d gdtfd the decimated gdtfd is similar to the gdtfd except it trades bandwidth for computational speed and requires significantly less storage in order to be implemented the reduction in discrete bandwidth is from 2 spl pi for the gdtfd to 2 spl pi m for the d gdtfd an important attribute of the dgdtfd is that it requires significantly less storage than the gdtfd the d gdtfd requires only 1 m sup 2 of the storage of the gdtfd an example using the binomial distribution is given to illustrate the connection between the d gdtfd and the gdtfd throughout the paper examples of multirate systems are given which could be used to implement the building blocks of the dgdtfd this makes the d gdtfd practical to implement using slower and less expensive elements

output feedback control of large scale nonlinear time delay systems in lower triangular form this paper is concerned with the stabilization problem for a class of large scale nonlinear time delay systems in lower triangular form the uncertain nonlinearities are assumed to be bounded by continuous functions of the outputs or delayed outputs multiplied by unmeasured states or delayed states an observer based output feedback control scheme is proposed using the dynamic gain control design approach based on lyapunov stability theory global asymptotic stability of the closed loop control system is proved contrary to many existing control designs for lower triangular nonlinear systems the celebrated backstepping method is not utilized here an example is finally given to demonstrate the effectiveness of the proposed design procedure

decentralized robust stabilization for large scale feedforward non linear systems in this paper we consider the decentralized robust stabilization problem for large scale feedforward non linear systems which possess an upper triangular structure while in the decentralized control area the commonly studied large scale feedback non linear systems possess a lower triangular structure our decentralized robust stabilizer for each subsystem takes a nested saturation feedback with the saturation levels satisfying a set of equations and inequalities global asymptotic stability of the closed loop large scale system has been proved and a simulation example is given