a high order internal model based iterative learning control scheme for nonlinear systems with time iteration varying parameters in this technical note we propose a new iterative learning control ilc scheme for nonlinear systems with parametric uncertainties that are temporally and iteratively varying the time varying characteristics of the parameters are described by a set of unknown basis functions that can be any continuous functions the iteratively varying characteristics of the parameters are described by a high order internal model hoim that is essentially an auto regression model in the iteration domain the new parametric learning law with hoim is designed to effectively handle the unknown basis functions the method of composite energy function is used to derive convergence properties of the hoim based ilc namely the pointwise convergence along the time axis and asymptotic convergence along the iteration axis comparing with existing ilc schemes the hoim based ilc can deal with nonlinear systems with more generic parametric uncertainties that may not be repeatable along the iteration axis the validity of the hoim based ilc under identical initialization condition i i c and the alignment condition is also explored

iterative learning control for a class of nonlinear systems abstract for precise tracking control of a class of nonlinear systems over a finite time interval an iterative learning control method using the relative degree of a system is proposed the class of nonlinear systems where the iterative learning control is applied can be extended by using the proposed method a sufficient condition is derived for guaranteeing a uniform convergence of the output to the desired output the results obtained for a class of nonlinear systems are shown to be a generalization of the existing results for linear time invariant systems

a survey on iterative learning control for nonlinear systems in this article we review the recent advances in iterative learning control ilc for nonlinear dynamic systems in the research field of ilc two categories of system nonlinearities are considered namely the global lipschitz continuous glc functions and local lipschitz continuous llc functions ilc for glc systems is widely studied and analysed using contraction mapping approach and the focus of recent exploration moves to application problems though a number of theoretical issues remain open ilc for llc systems is currently a hot area and the recent research focuses on ilc design and analysis by means of lyapunov approach the objectives of this article are to introduce recent development and advances in nonlinear ilc schemes highlight their effectiveness and limitations as well as discuss the directions for further exploration of nonlinear ilc

simple measure for complexity a measure of complexity is proposed based on appropriately defined notions of order and disorder which has a considerable degree of flexibility in its dependence on these concepts the possible functional dependencies which result encompass those of many earlier definitions of complexity the proposed measure is in principle easy to calculate and has the property of an intensive thermodynamic quantity with appropriate choices of parameters it behaves similarly to effective measure complexity for the logistic map it is also a generalization of the normalized complexity of lopez ruiz et al but does not suffer from over universality

effective normalization of complexity measurements for epoch length and sampling frequency the algorithmic complexity of a symbol sequence is sensitive to the length of the message additionally in those cases where the sequence is constructed by the symbolic reduction of an experimentally observed wave form the corresponding value of algorithmic complexity is also sensitive to the sampling frequency in this contribution we present definitions of algorithmic redundancy that are sequence sensitive generalizations of shannon s original definition of information redundancy in contrast with algorithmic complexity we demonstrate that algorithmic redundancy is not sensitive to message length or to observation scale sampling frequency when stationary systems are examined

iterative learning control design for linear discrete time systems with multiple high order internal models this work is focused on the iterative learning control ilc design for linear discrete time systems with iteration varying factors including reference initial state and exogenous disturbances first multiple high order internal models hoims are given for various iteration varying factors the reference is generated by an hoim and the initial state and the exogenous disturbances ultimately satisfy hoims but do not strictly follow hoims in finite iteration interval then a new ilc scheme a special high order ilc ho ilc is constructed according to an augmented hoim that is the aggregation of all hoims for simplicity the case with only iteration varying reference is first considered where the asymptotic stability and monotone convergence based ilc design methods are both presented by using the 2 d analysis approach next more iteration varying factors are further considered in this situation the hoim based ilc is transformed into a controller design problem of a 2 d roesser model with non zero boundary states and disturbances where the 2 d h performance is studied in consequence an ilc design criterion is presented to achieve perfect tracking and 2 d h performance for comparison the monotone convergence based ilc design method is extended to the situation with more iteration varying factors utilizing information provided by the multiple hoims it is verified that ho ilc outperforms low order ilc lo ilc in presence of iteration varying factors meanwhile the 2 d h based ilc is shown to be superior to the monotone convergence based ilc finally a microscale robotic deposition system with iteration varying factors is given to illustrate the advantage of the proposed 2 d h based ilc

two novel iterative learning control schemes for systems with randomly varying trial lengths abstract this paper proposes two novel improved iterative learning control ilc schemes for systems with randomly varying trial lengths different from the existing works on ilc with variable trial lengths that advocate to replace the missing control information by zero the proposed learning algorithms are equipped with a searching mechanism to collect useful but avoid redundant past tracking information which could expedite the learning speed the searching mechanism is realized by the newly defined stochastic variables and an iteratively moving average operator the convergence of the proposed learning schemes is strictly proved based on the contraction mapping methodology two illustrative examples are provided to show the superiorities of the proposed approaches

on initial conditions in iterative learning control initial conditions or initial resetting conditions play a fundamental role in all kinds of iterative learning control methods in this note we study five different initial conditions disclose the inherent relationship between each initial condition and corresponding learning convergence or boundedness property the iterative learning control method under consideration is based on lyapunov theory which is suitable for plants with time varying parametric uncertainties and local lipschitz nonlinearities

stability of learning control with disturbances and uncertain initial conditions the authors investigate the effects of state disturbances output noise and errors in initial conditions on a class of learning control algorithms they present a simple learning algorithm and exhibit via a concise proof bounds on the asymptotic trajectory errors for the learned input and the corresponding state and output trajectories furthermore these bounds are continuous functions of the bounds on the initial condition errors state disturbances and output noise and the bounds are zero in the absence of these disturbances

a delay parabolic cauchy dirichlet problem in interpolation space on etudie la solution d une equation parabolique non lineaire a retard avec conditions de cauchy et de dirichlet

on the p type learning control sufficient conditions for the robustness and convergence of p type learning control algorithms for a class of time varying nonlinear systems are presented the authors prove the uniform boundedness of the system state and the input control with respect to the existence of errors of initialization measurement noises and fluctuations of system dynamics furthermore the system output converges uniformly to the desired one in absence of all disturbances finally specialization of the results to linear systems are presented

initial shift issues on discrete time iterative learning control with system relative degree this paper deals with the initial shift problem that arises from discrete time iterative learning control a unified learning scheme is considered for a class of nonlinear systems with well defined relative degree which adopts the error data with anticipation in time and provides wider freedom for the updating law formation the sufficient convergence condition is derived to enable the system to possess asymptotic tracking capability and the converged output trajectory can be assessed by the initial condition the tracking performance is improved further by the introduction of initial rectifying action and the complete tracking is achieved over a specified interval

prediction and entropy of nonlinear dynamical systems and symbolic sequences with lro abstract following shannon we introduce higher order entropies and derive dynamic entropies the nth order dynamic entropy conditional entropy is a measure of the uncertainty of the next state which follows after the observation of n foregoing states the asymptotic behaviour of the dynamic entropies at large n is studied for several nonlinear model systems and for symbolic sequences with long range order lro for example we investigate 1d maps texts dna strings and time series it is shown that the existence of long correlations improves the possibility of predictions characteristic scaling laws for the higher order shannon entropies and the conditional entropies are derived and a new interpolation formula is tested finally instead of the dynamic entropies which yield mean values of the uncertainty predictability we investigate the local values of the uncertainty predictability

robust iterative learning control for a class of nonlinear systems a novel nonlinear control scheme robust iterative learning control rilc is developed in this paper the new robust ilc system provides a general framework targeting at synthesizing learning control and robust control methods with the help of lyapunov s direct method thereafter being able to handle more general classes of nonlinear uncertain systems in the proposed control scheme learning control and variable structure control are made to function in a complementary manner the nonlinear learning control strategy is applied directly to the structured system uncertainties which can be separated and expressed as products of unknown state independent functions and known state dependent functions for non structured system uncertainties associated with known bounding functions as the only a priori knowledge variable structure control vsc strategy is applied to ensure the global asymptotic stability in addition important issues regarding the objective trajectory categories resetting condition derivative signal requirement and their relationships have been made clear in this paper

brief iterative learning control with initial rectifying action this paper addresses the initial shift problem in iterative learning control with system relative degree the tracking error caused by nonzero initial shift is detected when applying a conventional learning algorithm finite initial rectifying action is introduced in the learning algorithm and is shown effective in the improvement of tracking performance in particular robustness with respect to variable initial shifts the uniform convergence of the output trajectory to a desired one jointed smoothly with a specified transient trajectory from the starting position is ensured in the presence of fixed initial shift

comment ii on simple measure for complexity the measure of complexity recently proposed by shiner davidson and landsberg phys rev e 59 1459 1999 does not adequately describe the transition from regular to indexed languages observed at the period doubling accumulation points of quadratic maps this comment points to a generic inadequacy of that measure

on non local stability properties of extremum seeking control in this paper we consider several extremum seeking schemes and show under appropriate conditions that these schemes achieve extremum seeking from an arbitrarily large domain of initial conditions if the parameters in the controller are appropriately adjusted this non local stability result is proved by showing semi global practical stability of the closed loop system with respect to the design parameters we show that reducing the size of the parameters typically slows down the convergence rate of the extremum seeking controllers and enlarges the domain of the attraction our results provide guidelines on how to tune the controller parameters in order to achieve extremum seeking simulation examples illustrate our results

convergence of iterative learning control for siso nonrepetitive systems subject to iteration dependent uncertainties abstract this paper studies the robust convergence properties of iterative learning control ilc for single input single output siso nonrepetitive systems subject to iteration dependent uncertainties that arise in not only initial states and external disturbances but also plant models given an extended relative degree condition it is possible to propose necessary and sufficient nas conditions for robust ilc convergence the tracking error bound is shown to depend continuously on the bounds of the iteration dependent uncertainties when the iteration dependent uncertainties are bounded nas conditions exist to guarantee bounded system trajectories and output tracking error if the iteration dependent uncertainties converge then nas conditions ensure bounded system trajectories and zero output tracking error the results are also extended to a class of affine nonlinear systems satisfying a lipschitz condition simulation tests on a representative batch process demonstrate the validity of the obtained robust ilc convergence results

oscillations of solutions of vector differential equations of parabolic type with functional arguments vector parabolic differential equations with functional arguments are studied and the oscillations of solutions of boundary value problems are investigated our approach is to reduce the oscillation problems to the nonexistence of positive solutions of scalar differential inequalities by employing the concept of h oscillation introduced by domslak see r courant d hilbert methods of mathematical physics vol i interscience new york 1996 where h denotes a unit vector

d type anticipatory iterative learning control for a class of inhomogeneous heat equations in this paper a d type anticipatory iterative learning control ilc scheme is applied to the boundary control of a class of inhomogeneous heat equations where the heat flux at one side is the control input while the temperature measurement at the other side is the control output by transforming the inhomogeneous heat equation into its integral form and exploiting the properties of the embedded jacobi theta functions the learning convergence of ilc is guaranteed through rigorous analysis without any simplification or discretization of the 3d dynamics in the time space as well as iteration domains the adopted ilc scheme makes full use of the process repetition and deals with state independent or state dependent uncertainties meanwhile due to the feedforward characteristic of ilc the proposed scheme not only makes anticipatory compensation possible to overcome the heat conduction delay in boundary output tracking but also eliminates the gain margin limitation encountered in feedback control in the end an illustrative example is presented to demonstrate the performance of the proposed ilc scheme

easily adaptable complexity measure for finite time series we present a complexity measure for any finite time series this measure has invariance under any monotonic transformation of the time series has a degree of robustness against noise and has the adaptability of satisfying almost all the widely accepted but conflicting criteria for complexity measurements surprisingly the measure is developed from kolmogorov complexity which is traditionally believed to represent only randomness and to satisfy one criterion to the exclusion of the others for familiar iterative systems our treatment may imply a heuristic approach to transforming symbolic dynamics into permutation dynamics and vice versa

technical communique iterative learning control design without a priori knowledge of the control direction in this work we explore the possibility of designing a new iterative learning control scheme for systems without a priori knowledge of the control direction by incorporating a nussbaum type function a new learning control mechanism is constructed with both differential and difference updating laws the new learning control mechanism can warrant a l t 2 convergence of the tracking error sequence along the iteration axis in the presence of time varying parametric uncertainties and local lipschitz nonlinearities

a statistical measure of complexity with nonextensive entropy abstract a statistical measure of complexity utilising the concept of entropy or information is proposed our way in this study is to use a nonextensive entropy instead of an extensive additive shannon entropy in the definition but can be characterised as a difference between the qth order renyi entropy and the second one furthermore we devise a conditional joint and mutual complexity measure as a coherent possibility the behavior of the measure for the logistic map shows that it is more sensitive to nonextensivity at the transition point ac 3 8284 than any other values when 0

unified iterative learning control schemes for nonlinear dynamic systems with nonlinear input uncertainties in many ilc algorithms nonlinear input uncertainties such as saturation dead zone and hysteresis which do exist due to practical implementations are always ignored although various ilc algorithms have been proposed to compensate various nonlinear input uncertainties a systematic design framework is still missing this note presents a unified design framework to deal with very general nonlinear input uncertainties the concept of a dual loop ilc is introduced one ilc loop ilc loop 1 is designed for the nominal model without nonlinear input uncertainties the other ilc loop ilc loop 2 uses some iterative algorithms to handle nonlinear input uncertainties two ilc loops can be designed independently and are connected by a proper time scale separation our first result shows that by using time scale separation the overall system semi globally practically converges to the desired trajectory if ilc loop 2 uniformly converges furthermore if ilc loop 2 converges almost monotonically ilc loop 1 and ilc loop 2 can update simultaneously to achieve uniform convergence of the overall system

analysis of iterative learning control for a class of nonlinear discrete time systems abstract an output feedback iterative learning control scheme is proposed for a class of nonlinear discrete time systems in the presence of direct transmission from system inputs to outputs the convergence condition of the iterative learning control scheme is analyzed the effectiveness of suggested discrete time iterative learning control algorithm is demonstrated with a single link robotic manipulator as the illustrative example 1997 elsevier science ltd

a generalized iterative learning controller against initial state error in this paper the previous results that the performance of iterative learning control ilc algorithm can be improved by adding a proportional term and or an integral term of error in d type ilc a

an iterative learning control approach for linear systems with randomly varying trial lengths this technical note addresses an iterative learning control ilc design problem for discrete time linear systems where the trial lengths could be randomly varying in the iteration domain an ilc scheme with an iteration average operator is introduced for tracking tasks with non uniform trial lengths which thus mitigates the requirement on classic ilc that all trial lengths must be identical in addition the identical initialization condition can be absolutely removed the learning convergence condition of ilc in mathematical expectation is derived through rigorous analysis as a result the proposed ilc scheme is applicable to more practical systems in the end two illustrative examples are presented to demonstrate the performance and the effectiveness of the averaging ilc scheme for both time invariant and time varying linear systems

an average operator based pd type iterative learning control for variable initial state error this note studies the effect of variable initial state error in iterative learning control ilc systems and proposes a new ilc algorithm based on an average operator then it is shown that when the proposed algorithm is applied to linear time invariant lti systems the effect of the initial state error can be exactly estimated under a specific condition while the existing algorithms guarantee only the boundness of the error or the convergence from stochastic point of view to show the validity of the proposed algorithm a numerical example is given

optimal iterative learning control design for multi agent systems consensus tracking abstract under a repeatable operation environment this paper proposes an iterative learning control scheme that can be applied to multi agent systems to perform consensus tracking under the fixed communication topology the agent dynamics are modeled by time varying nonlinear equations which satisfy the global lipschitz continuous condition in addition the desired consensus trajectory is only accessible to a subset of the followers by using the concept of the graph dependent matrix norm the convergence conditions can be specified at the agent level which depend on a set of eigenvalues that are associated with the communication topology the results are first derived for homogeneous agent systems and then extended to heterogeneous systems next optimal controller gain design methods are proposed in the sense that the norm of tracking error converges at the fastest rate which imposes a tightest bounding function for the actual tracking error in the norm analysis framework in the end an illustrative example of a group of heterogeneous agents is provided to demonstrate the effectiveness of the proposed design methods

analyticity of the maximal solution of an abstract nonlinear parabolic equation where u t x 0 7 j x r r satisfies other suitable conditions problem p is studied under the assumption that af au generates an analytic semigroup in e and has domain f which corresponds to a parabolicity condition on equation p problems of this kind have been studied by monotonicity methods see brezis 111 crandall ligget 3 and crandall pazy 4 to obtain global but weak solutions of p here we find a maximally defined strict solution by means of a linearization method and by virtue of maximal regularity results for the linear case this method has been used by da prato grisvard in 5 to obtain strict solutions of p with arguments which seem to be more simple and suitable to our purpose we get also the analyticity of u with respect to 1 w when f is analytic in section 1 some notation and assumptions are given problem p is studied in section 2 where we prove existence and uniqueness of the strict solution of p which is defined on a maximal time interval contained in 0 t and it is continuous with respect to t 0 in section 3 we give conditions to get global existence of the solution in section 4 we study analyticity with respect to t ug of the maximally defined solution of p and in section 5 a simple example is given in a subsequent paper we will show how this theory applies to the study of the analyttctty of the solutions of nonlinear parabol ic partial differential equations in spaces of holder continuous functions

an iterative learning control law for dynamical systems abstract this paper is concerned with an iterative learning control law which enables us to find a control input that generates the desired output exactly over a finite time interval through the repetition of trials we derive a sufficient condition for nonlinear systems to achieve the desired output by the iterative learning control based on this result we show that the direct transmission term of the plant plays a crucial role in the error convergence of the learning process further we identify the class of plants to which the learning control law is applicable

fuzzy system based adaptive iterative learning control for nonlinear plants with initial state errors in this paper a fuzzy system based adaptive iterative learning controller is proposed for a class of non lipschitz nonlinear plants which can repeat a given task over a finite time interval the variable initial resetting state errors at the beginning of each trial is considered to overcome the initial errors a time varying boundary layer is introduced to design an error function based on the error function the main structure of this controller is constructed by a fuzzy iterative learning component and a feedback stabilization component the fuzzy system is used as an approximator to compensate for the plant unknown nonlinearity since the optimal parameters for a good fuzzy approximation are in general unavailable the adaptive algorithms are derived along the iteration axis to search for suitable parameter values and then guarantee the closed loop stability and learning convergence it is shown that all the adjustable parameters as well as internal signals remain bounded for all iterations there even exist initial state errors the norm of tracking error vector will asymptotically converge to a tunable residual set as iteration goes to infinity and the learning speed can be easily improved if the learning gain is large

a combined adaptive law for fuzzy iterative learning control of nonlinear systems with varying control tasks to deal with the iterative control of uncertain nonlinear systems with varying control tasks nonzero initial resetting state errors and nonrepeatable mismatched input disturbance a new adaptive fuzzy iterative learning controller is proposed in this paper the main structure of this learning controller is constructed by a fuzzy learning component and a robust learning component for the fuzzy learning component a fuzzy system used as an approximator is designed to compensate for the plant nonlinearity for the robust learning component a sliding mode like strategy is applied to overcome the nonlinear input gain input disturbance and fuzzy approximation error both designs are based on a time varying boundary layer which is introduced not only to solve the problem of initial state errors but also to eliminate the possible undesirable chattering behavior a new adaptive law combining time and iteration domain adaptation is derived to search for suitable values of control parameters and then guarantee the closed loop stability and error convergence this adaptive algorithm is designed without using projection or deadzone mechanism with a suitable choice of the weighting gain the memory size for the storage of parameter profiles can be greatly reduced it is shown that all the adjustable parameters as well as internal signals remain bounded for all iterations moreover the norm of tracking state error vector will asymptotically converge to a tunable residual set even when the desired tracking trajectory is varying between successive iterations

iterative learning control design based on composite energy function with input saturation in this work an iterative learning control scheme is designed for a class of nonlinear uncertain systems with input saturation the analysis of convergence in the iteration domain is based on composite energy function which consists of both input and state information along the time and iteration axes through rigorous analysis the learning convergence in the iteration domain can be guaranteed under the input saturation provided the desired trajectory is realizable within the saturation bound

complexity fractals disease time and cancer despite many years of research a method to precisely and quantitatively determine cancer disease state remains elusive current practice for characterizing solid tumors involves the use of varying systems of tumor grading and staging and thus leaves diagnosis and clinical staging dependent on the experience and skill of the physicians involved although numerous disease markers have been identified no combination of them has yet been found that produces a quantifiable and reliable measure of disease state newly developed genomic markers and other measures based on the developing sciences of complexity offer promise that this situation may soon be changed for the better in this paper we examine the potential of two measures of complexity fractal dimension and percolation for use as components of a yet to be determined disease time vector that more accurately quantifies disease state the measures are applied to a set of micrographs of progressive rat hepatoma and analyzed in terms of their correlation with cell differentiation ratio of tumor weight to rat body weight and tumor growth time the results provide some support for the idea that measures of complexity could be important elements of any future cancer disease time vector