decomposition of generalized frequency response functions for nonlinear systems using symbolic computation a symbolic manipulation procedure which both computes and automatically decomposes the generalized frequency response functions of nonlinear rational polynomial and integrodifferential equation models into a set of closed form nth order transfer functions is presented the symbolic representation exposes the explicit relationship between the model parameters and the nonlinear transfer functions in the frequency domain and leads to important insights into the characterization of nonlinear systems examples for each model type are included to demonstrate the symbolic transfer function approach

dynamic feedback stabilization of an unstable wave equation abstract in this paper we consider the stabilization of an unstable wave equation through a dynamic boundary compensator the measurement which is one internal point of the state of the wave equation is fluxed into the compensator while the output of the compensator is forced into the boundary of the wave equation the well posedness of the closed loop system is proved by using the semigroup method by choosing appropriate controller parameters and adopting nyquist criterion for distributed parameter systems the eigenvalues of the closed loop systems are shown to be inside the left half complex plane then the closed loop system is proved to be exponentially stable by using the riesz basis approach numerical simulations are presented to verify the effectiveness of the proposed compensator

an analysis and design method for a class of nonlinear systems with nested saturations abstractthis paper is concerned with the analysis and design of a class of nonlinear systems subject to nested saturations the proposed controller incorporates both an extended state observer eso which is utilised to estimate the nonlinear dynamics of the plant as well as a set of observer based feedbacks we first present analysis results for systems with nonlinear esos and show that local stabilisation can be achieved in a region including the origin then in the case that the eso is in linear form leso the conditions for determining the estimate of the domain of attraction of the resulting closed loop system are formulated as a convex optimisation problem a linear matrix inequality based algorithm is then established to design the feedback gains and the leso gain an illustrative example is given to show the effectiveness of the proposed approach

discrete time instantaneous frequency and its computation classical results indicate that the means of the fourier and instantaneous frequencies are exactly equal and the weighted variance of the fourier frequency is greater than or equal to that of the instantaneous frequency these results are investigated for the discrete case in which the signal and its spectrum are both assumed to be periodic specifically the cases in which the discrete time instantaneous frequency is defined as the forward backward and central differences are discussed a simple algorithm for computing the discrete time instantaneous frequency is presented whereby descriptors for the characterization of the signal are constructed the application of this algorithm is discussed through examples

design of observers for linear systems with unknown inputs with a straightforward treatment a design of reduced order observers is presented for linear systems with unknown inputs the observers are derived with a physical meaning some explanations are given within the framework of descriptor system observer design principles the conditions for the existence of the observers are presented some illustrative examples are included

the active disturbance rejection control to stabilization for multi dimensional wave equation with boundary control matched disturbance in this paper we consider boundary stabilization for a multi dimensional wave equation with boundary control matched disturbance that depends on both time and spatial variables the active disturbance rejection control adrc approach is adopted in investigation an extended state observer is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multi dimensional system by infinitely many test functions the disturbance is canceled in the feedback loop together with a collocated stabilizing controller all subsystems in the closed loop are shown to be asymptotically stable in particular the time varying high gain is first time applied to a system described by the partial differential equation for complete disturbance rejection purpose and the peaking value reduction caused by the constant high gain in literature the overall picture of the adrc in dealing with the disturbance for multi dimensional partial differential equation is presented through this system the numerical experiments are carried out to illustrate the convergence and effect of peaking value reduction

on the conditions of exponential stability in active disturbance rejection control based on singular perturbation analysis abstractstability of active disturbance rejection control adrc is analysed in the presence of unknown nonlinear and time varying dynamics in the framework of singular perturbations the closed loop error dynamics are semi decoupled into a relatively slow subsystem the feedback loop and a relatively fast subsystem the extended state observer respectively it is shown analytically and geometrically that there exists a unique exponential stable solution if the size of the initial observer error is sufficiently small i e in the same order of the inverse of the observer bandwidth the process of developing the uniformly asymptotic solution of the system reveals the condition on the stability of the adrc and the relationship between the rate of change in the total disturbance and the size of the estimation error the differentiability of the total disturbance is the only assumption made

algebraic multidimensional phase unwrapping and zero distribution of complex polynomials characterization of multivariate stable polynomials we define the multidimensional unwrapped phase for any finite extent multidimensional signal that may have its zero on the distinguished boundary of the unit polydisc by using this definition we deduce that multivariate stable polynomials can be simply characterized in terms of the proposed unwrapped phase a rigorous symbolic algebraic solution to the exact phase unwrapping problem for multidimensional finite extent signals is also proposed this solution is based on a newly developed general sturm sequence and does not need any numerical root finding or numerical integration technique furthermore it is shown that the proposed algebraic phase unwrapping algorithm can be used to determine the exact zero distribution of any univariate complex polynomial without suffering the so called singular case problem

on near controllability nearly controllable subspaces and near controllability index of a class of discrete time bilinear systems a root locus approach this paper studies near controllability of a class of discrete time bilinear systems via a root locus approach a necessary and sufficient criterion for the systems to be nearly controllable is given in particular by using the root locus approach the control inputs which achieve the state transition for the nearly controllable systems can be computed furthermore for the non nearly controllable systems nearly controllable subspaces are derived and near controllability index is defined accordingly the controllability properties of such a class of discrete time bilinear systems are fully characterized finally examples are provided to demonstrate the results of the paper

control of uncertain nonlinear systems based on observers and estimators in this paper we consider a class of nonlinear dynamical systems with zero dynamics which is subject to both unknown nonparametric dynamics and external disturbances and is required to track a given reference signal by using the output feedback our controller is designed based on both the extended state observer eso and the projected gradient estimator while the eso is used to estimate the total uncertainties the projected gradient algorithm is used to estimate the nonparametric uncertainties treated as time varying parameters this method overcomes the difficulties that the traditional active disturbance rejection control adrc technique needs to have a good prior estimate for the uncertainties in the input channel the closed loop system is shown to be semi globally stable and at the same time the tracking error can be made arbitrarily small

stabilisation of discrete time bilinear systems by constant control inputs abstractin this paper the stabilisation problem of discrete time bilinear systems by using constant control inputs is considered it is shown that the spectrum of a matrix derived from the system

on controllability near controllability controllable subspaces and nearly controllable subspaces of a class of discrete time multi input bilinear systems abstract this paper considers a class of discrete time multi input inhomogeneous bilinear systems the structure of such systems is most close to linear time invariant systems but they own a strong property that is if the systems are uncontrollable they can still be nearly controllable necessary and sufficient conditions for controllability and near controllability of the systems are established by using a classical decomposition furthermore a geometric characterization is given for the systems such that controllable subspaces and nearly controllable subspaces are derived and characterized similar results on controllability are also obtained for the continuous time counterparts of the systems finally examples are provided to demonstrate the conceptions and results of this paper

homogeneous lyapunov function for homogeneous continuous vector field the goal of this article is to provide a construction of a homogeneous lyapunov function p associated with a system of differential equations j f x x r n 1 under the hypotheses 1 f c r n vanishes at x 0 and is homogeneous 2 the zero solution of this system is locally asymptotically stable moreover the lyapunov function v x tends to infinity with 1 x i and belongs to c r 0 r n cp with p e as large as wanted as application to the theory of homogeneous systems we present two well known results of robustness in a slightly extended form and with simpler proofs

controllability and near controllability of a class of discrete time inhomogeneous bilinear systems in this paper a class of discrete time inhomogeneous bilinear systems whose controllability cannot be checked by the classical results is considered by using an algebraic technique it is shown t

on convergence of active disturbance rejection control for a class of uncertain stochastic nonlinear systems in this paper the practical mean square convergence of active disturbance rejection control for a class of uncertain stochastic nonlinear systems modelled by the ito type stochastic differential e

on controllability of discrete time bilinear systems by near controllability abstract in this paper controllability of discrete time bilinear systems is studied by applying a recent result on near controllability a new sufficient condition for controllability of the systems is presented where controllability is proved by approximation with near controllability the new condition is algebraically verifiable and is hence easy to apply compared with a classical result on controllability of discrete time bilinear systems which can be effective even when the classical result does not work furthermore the control inputs to achieve the transition of the systems between any given pair of states are approximately computable according to near controllability therefore near controllability can be used to not only better characterize the system properties but also prove controllability with computable control inputs the new condition is then generalized to derive similar results on controllability and near controllability of the systems finally examples are given to illustrate the results of this paper

a direct relation between a signal time series and its unwrapped phase a noniterative method for phase unwrapping a real finite length discrete time signal is described we use an operator which counts sign changes in a sturm sequence generated from the real and imaginary parts of the dft the number of sign changes is related to the number of multiples of which must be added to the principal value arctan to produce unwrapped phase except for the evaluation of trigonometric and inverse trigonometric functions the unwrapped phase at any frequency can be computed in a finite number of steps the approach is illustrated with an example and a fortran program implementation of the algorithm is included in the appendix

on controllability of homogeneous and inhomogeneous discrete time multi input bilinear systems in dimension two abstractin this paper the controllability problem of two dimensional discrete time multi input bilinear systems is completely solved the homogeneous and the inhomogeneous cases are studied separately and necessary and sufficient conditions for controllability are established by using a linear algebraic method which are easy to apply moreover for the uncontrollable systems near controllability is considered and similar necessary and sufficient conditions are also obtained finally examples are provided to demonstrate the results of this paper

minimum seeking for clfs universal semiglobally stabilizing feedback under unknown control directions employing extremum seeking es for seeking minima of control lyapunov function clf candidates we develop 1 the first systematic design of es controllers for unstable plants 2 a simple non model based universal feedback law that emulates in an average sense the l g v controllers for stabilization with inverse optimality and 3 a new strategy for stabilization of systems with unknown control directions as an alternative to nussbaum gain controllers that lack exponential stability lack transient performance guarantees and lack robustness to changes in the control direction the stability analysis that underlies our designs is inspired by an analysis approach synthesized in a recent work by durr stankovic and johansson which combines a lie bracket averaging result of gurvits and li with a semiglobal practical stability result under small parametric perturbations by moreau and aeyels

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

state observation of nonlinear uncertain dynamical systems this note proposes new types of observers for nonlinear dynamical systems subjected to bounded nonlinearities or uncertainties the design of these observers utilizes techniques related to variable structure systems theory a measure for the rate at which the estimates converge to the actual states is derived

on convergence of the nonlinear active disturbance rejection control for mimo systems in this paper the global and semiglobal convergence of the nonlinear active distur bance rejection control adrc for a class of multi input multi output nonlinear systems with large uncertainty that comes from both dynamical modeling and external disturbance are proved as a result a class of linear systems with external disturbance that can be dealt with by the adrc is classified the adrc is then compared both analytically and numerically to the well known in ternal model principle a number of illustrative examples are presented to show the efficiency and advantage of the adrc in dealing with unknown dynamics and in achieving fast tracking with lower overstriking

phase unwrapping of digital signals in order to unwrap the phase of a digital signal the signal s z plane zeros which lie close to the unit circle are located by repeatedly expanding the z plane in small steps and detecting a zero s crossing of the unit circle this detection is accomplished by determining the step which gives rise to a maximum change in the phase at any frequency sampling point and across any segment of the unit circle this method and its associated computer program have been rigorously tested and the phase was successfully unwrapped in most cases with the few exceptions involving sequences generated from high density clusters of zeros

on convergence of tracking differentiator the tracking differentiator was first proposed by han in 1989 and the proof of convergence was presented the first time in han and wang han j q and wang w 1994 nonlinear tracking differentiator journal of systems science and mathematical science 14 177 183 in chinese unfortunately the proof there is incomplete this problem has been open for over two decades in this article we give a rigorous proof under some additional conditions an application for online estimation of the unknown frequencies for the finite sum of the sinusoidal signals is presented the numerical simulations illustrate the effectiveness of the estimation for both linear and nonlinear tracking differentiators

periodic input estimation for linear periodic systems automotive engine applications in this paper we consider periodic linear systems driven by t 0 periodic signals that we desire to reconstruct the systems under consideration are of the form email protected a t x a 0 t w t y c t x email protected r n email protected r m email protected r p m

unwrapped phase inversion with an exponential damping abstractfull waveform inversion fwi suffers from the phase wrapping cycle skipping problem when the frequency of data is not low enough unless we obtain a good initial velocity model the phase wrapping problem in fwi causes a result corresponding to a local minimum usually far away from the true solution especially at depth thus we have developed an inversion algorithm based on a space domain unwrapped phase and we also used exponential damping to mitigate the nonlinearity associated with the reflections we construct the 2d phase residual map which usually contains the wrapping discontinuities especially if the model is complex and the frequency is high we then unwrap the phase map and remove these cycle based jumps however if the phase map has several residues the unwrapping process becomes very complicated we apply a strong exponential damping to the wavefield to eliminate much of the residues in the phase map thus making the unwrapping process simple we finally invert the unwrapped

active disturbance rejection control for rejecting boundary disturbance from multidimensional kirchhoff plate via boundary control in this paper an algorithm is developed to reject time and spatially varying boundary disturbances from a multidimensional kirchhoff plate via boundary control the disturbance and control input are assumed to be matched the active disturbance rejection control approach is adopted for developing the algorithm a state feedback scheme is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multidimensional system using infinitely many time dependent test functions the proposed control law cancels the disturbance using its estimated value all subsystems in the closed loop are shown to be asymptotically stable simulation results are presented to validate the theoretical conclusions and to exhibit the reduction in the peaking phenomenon due to the use of time varying gains instead of constant high gains

performance recovery of a class of uncertain non affine systems with unmodelled dynamics an indirect dynamic inversion method abstractthis paper presents an output feedback indirect dynamic inversion idi approach for a class of uncertain nonaffine systems with input unmodelled dynamics compared with previous approaches to achieve performance recovery the proposed method aims at dealing with a broader class of nonaffine in control systems with triangular structure an idi state feedback law is designed first in which less knowledge of the model plant is needed compared to earlier approximate dynamic inversion methods thus yielding more robust performance after that an extended high gain observer is designed to accomplish the task with output feedback finally we prove that the designed idi controller is equivalent to an adaptive proportional integral pi controller with respect to both time response equivalence and robustness equivalence the conclusion implies that for the studied strict feedback non affine systems with unmodelled dynamics there always exits a pi controller to stabilise the systems the effectiveness

phase unwrapping of mr phase images using poisson equation the authors have developed a technique based on a solution of the poisson equation to unwrap the phase in magnetic resonance mr phase images the method is based on the assumption that the magnitude of the inter pixel phase change is less than spl pi per pixel therefore the authors obtain an estimate of the phase gradient by wrapping the gradient of the original phase image the problem is then to obtain the absolute phase given the estimate of the phase gradient the least squares ls solution to this problem is shown to be a solution of the poisson equation allowing the use of fast poisson solvers the absolute phase is then obtained by mapping the ls phase to the nearest multiple of 2 k from the measured phase the proposed technique is evaluated using mr phase images and is proven to be robust in the presence of noise an application of the proposed method to the 3 point dixon technique for water and fat separation is demonstrated

a nonlinear extended state observer based on fractional power functions abstract in this paper we investigate a nonlinear extended state observer eso constructed from piece wise smooth functions consisted of linear and fractional power functions this structure of eso was first proposed in the 1990 s and has been widely used in active disturbance rejection control for engineering controls its convergence however has remained an open problem up to this day the main objective of this paper is to provide a convergence theory with explicit error estimation the performances of this type eso are studied by numerical simulation and compared with linear eso the numerical results show that the eso proposed in this paper enjoys the advantages of smaller peaking value and better measurement noise tolerance

on the convergence of an extended state observer for nonlinear systems with uncertainty abstract the extended state observer first proposed by jingqing han in j q han a class of extended state observers for uncertain systems control decis 10 1 1995 85 88 in chinese is the key link toward the active disturbance rejection control that is taking off as a technology after numerous successful applications in engineering unfortunately there is no rigorous proof of convergence to date in this paper we attempt to tackle this long unsolved extraordinary problem the main idea is to transform the error equation of objective system with its extended state observer into a asymptotical stable system with a small disturbance for which the effect of total disturbance error is eliminated by the high gain

state and input estimation for a class of uncertain systems we consider here the problem of state and input estimation for a class of uncertain systems which consist of a nominally linear part and an uncertain nonlinear time varying part which can be regarded as a state dependent time varying input we obtain a lyapunov type characterization of the assumptions commonly found in the literature on this problem using this characterization we obtain estimators which using only a measured output can asymptotically estimate to any desired accuracy the system state and the input

lyapunov stability theory of nonsmooth systems this paper develops nonsmooth lyapunov stability theory and lasalle s invariance principle for a class of nonsmooth lipschitz continuous lyapunov functions and absolutely continuous state trajectories computable tests based on filipov s differential inclusion and clarke s generalized gradient are derived the primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right hand side such as in nonsmooth dynamic systems or variable structure control

on the existence and uniqueness of solutions in adaptive control systems the issue of existence and uniqueness of solutions which arises in the analysis of adaptive control systems is investigated sufficient conditions that guarantee existence and uniqueness in positive time of solutions to differential equations describing a wide class of adaptive schemes are given the concept of filippov s solution is introduced as a method of dealing with discontinuous adaptive systems

active disturbance rejection control for uncertain nonaffine in control nonlinear systems in this technical note the active disturbance rejection control adrc is generalized to uncertain nonaffine in control nonlinear systems the proposed controller incorporates both an extended state observer eso as well as a dynamic inversion the eso is designed to estimate system state and total uncertainty which includes the uncertain internal dynamics and the external disturbance and is nonaffine in control based on the output of the eso the dynamic inversion is used to deal with the nonaffine in control problem the proposed control has a multi time scale structure in which the eso is the fastest time scale the dynamic inversion is the second and the considered nonlinear system is the slowest the practical convergence of the resulting closed loop system is obtained an example is presented to illustrate the efficiency of the proposed method

exact computation of the unwrapped phase of finite length time series r mcgowan and r kuc recently ieee trans acoust speech signal processing vol assp 30 p 719 1982 showed that a direct relationship between a time series and its unwrapped phase exists they proposed an algorithm for computing the unwrapped phase by counting the number of sign changes in a sturm sequence generated from the real and imaginary parts of the discrete fourier transform their algorithm is limited to relatively short sequences by numerical accuracy an extension of their algorithm is proposed which by using all integer arithmetic permits exact computation of the number of multiples of pi required to determine the unwrapped phase for rational valued time sequences of arbitrary length since the computation is exact the extended numerical algorithm should be of interest when accurate phase unwrapping is required

comparative study of non linear state observation techniques abstract this paper contains a comparative study of four techniques for observing the slates of non linear systems the first technique examined is inspired by bestle and zeitz and icrener and respondek in this method a non linear transformation is found that brings the system into a canonical form from where observer design is facilitated the second technique is attributable to thau in this method the error between the system s true state and the output of the observer is shown to be asymptotically convergent to zero provided that an additional assumption is valid the third technique is due to baumann and rugh in this method extended or pseudolinearization of the error differential equation about a family of equilibrium points results in an observer design such that the eigenvalues of the linearized error equation are locally invariant finally techniques from variable structure systems are utilized to design an observer that yields an exponentially decaying error like thau s observer but unl

full order observers for linear systems with unknown inputs this note presents a simple method to design a full order observer for linear systems with unknown inputs the necessary and sufficient conditions for the existence of the observer are given

output feedback stabilization of an unstable wave equation with general corrupted boundary observation we consider boundary output feedback stabilization for an unstable wave equation with boundary observation subject to a general disturbance we adopt for the first time the active disturbance rejection control approach to stabilization for a system described by the partial differential equation with corrupted output feedback by the approach the disturbance is first estimated by a relatively independent estimator it is then canceled in the feedback loop as a result the control law can be designed almost as that for the system without disturbance we show that with a time varying gain properly designed the observer driven by the disturbance estimator is convergent and that all subsystems in the closed loop are asymptotically stable in the energy state space we also provide numerical simulations which demonstrate the convergence results and underline the effect of the time varying gain estimator on peaking value reduction