analysis of a high order iterative learning control algorithm for uncertain nonlinear systems with state delays key words iterative learning control nonlinear systems convergence analysis high order scheme delay tracking abstract for a better learning transient along the iteration number direction a pid type iterative learning control algo rithm is proposed for a class of delayed uncertain nonlinear systems which perform a given task repeatedly the convergence conditions for the proposed high order learning control have been established the tracking error bound is proven to be a class k function of the bounds of reinitialization errors uncer tainties and disturbances to the systems it is also indicated that the time delays in the system states do not play a significant role in the ilc convergence property the e ectiveness of the high order ilc algorithm is demonstrated by simulation examples 1998 elsevier science ltd all rights reserved

identification of composite local linear state space models using a projected gradient search an identification method is described to determine a weighted combination of local linear state space models from input and output data normalized radial basis functions are used for the weights

learning variable structure control approaches for repeatable tracking control tasks for the finite period tracking control task repeatable over iterations uniformly bounded learning control is synthesized into variable structure control to approximate the equivalent control and to realize perfect tracking

networks with condorcet solutions abstract a condorcet point on a network is a point such that thare is no other point closer to a strict majority of users which are located at the vertices of the network a complete characterization is given of those networks on which no condorcet paradox occurs in the sense that for each distribution of users there exists at least one condorcet point moreover those networks are characterized on which condorcet points and weber points always coincide a polynomial algorithm is obtained which decides whether a given network has condorcet points for all distributions of users on the network

on d type and p type ilc designs and anticipatory approach many schemes of iterative learning control ilc have been developed for continuous time non linear dynamic systems to improve tracking performance two schemes d type and p type have been the bases for many ilc designs recently the anticipatory ilc scheme has been introduced on the basis of a different approach wang 1998a 1999 in this paper these basic schemes are compared from both analysis and implementation view points the anticipatory ilc scheme is designed on the basis of a causal pair of the action taken and its resulting state variables this approach has the anticipatory characteristics of the d type ilc and the simplicity for implementation of p type ilcs a sampled data ilc scheme is presented as another form of this anticipatory ilc scheme furthermore control device saturation is taken into account and tracking error convergence results are established with proofs the convergence results are also provided in the presence of uncertainties disturbances and measurement noises ex

asymptotic learning control for a class of cascaded nonlinear uncertain systems the problem of learning unknown functions in a class of cascaded nonlinear systems is studied the functions to be learned are those functions that are imbedded in the system dynamics and are of known period of time in addition to the unknown periodic time functions nonlinear uncertainties bounded by known functions of the state are also admissible the objective of the paper is to find an iterative learning control under which the class of nonlinear systems are globally stabilized in the sense of being uniform bounded their outputs are asymptotically convergent and a combination of the time functions contained in system dynamics are asymptotically learned to this end a new type of differential difference learning law is utilized to generate the proposed learning control that yields both asymptotic stability of the system output and asymptotic convergence of the learning error the design is carried out by applying the lyapunov direct method and backward recursive design method

brief on the equivalence of causal lti iterative learning control and feedback control the goal of iterative learning control ilc is to improve the accuracy of a system that repeatedly follows a reference trajectory this paper proves that for each causal linear time invariant ilc there is an equivalent feedback that achieves the ultimate ilc error with no iterations remarkably this equivalent feedback depends only on the ilc operators and hence requires no plant knowledge this equivalence is obtained whether or not the ilc includes current cycle feedback if the ilc system is internally stable and converges to zero error there exists an internally stabilizing feedback that approaches zero error at high gain since conventional feedback requires no iterations there is no reason to use causal ilc

iterative learning control for non linear systems described by a blended multiple model representation this paper deals with the design of gain scheduling based iterative learning controllers for continuous time non linear systems described by a blended multiple model representation sufficient conditions guaranteeing the convergence of the infinity norm as well as the u norm of the tracking error are derived the effectiveness of the proposed control scheme is illustrated on an example of a non affine in input system

brief nonlinear learning control for a class of nonlinear systems based on the lyapunov s direct method a new learning control design is proposed the proposed technique can be applied in two ways it is either the standard backward recursive design or its extension in the first case the design yields a class of learning control with a difference learning law under which the class of nonlinear systems is guaranteed to be asymptotically stable with respect to the number of trials in performing repeated tasks however implementation of the difference learning control requires derivative measurement of the state for guaranteed stability and performance as required by most of the existing linear learning control laws to overcome this difficulty the proposed design extends the recursive design by employing a new state transformation and a new lyapunov function and it yields a class of learning control with a difference differential learning law compared with the existing design methods most of which are based on linear analysis and design the extension not only guarantees global stability and good performance but also removes such limitations as derivative measurement lipschitz condition and resetting of initial conditions in addition the proposed design does not rely on the property of a system under consideration such as the input output passivity

some complete lie algebras abstract we give the derivation algebra der h and the holomorph h h of the finite dimensional heisenberg algebra h over the complex field c we also give the derivation algebra der h h of h h we prove that der h is a simple complete lie algebra h h is not a complete lie algebra but its derivation algebra der h h is a simple complete lie algebra

on a discrete time stochastic learning control algorithm the learning gain for a selected learning algorithm is derived based on minimizing the trace of the input error covariance matrix for linear time varying systems it is shown that if the product of the input output coupling matrices is a full column rank then the input error covariance matrix converges uniformly to zero in the presence of uncorrelated random disturbances however the state error covariance matrix converges uniformly to zero in presence of measurement noise moreover it is shown that if a certain condition is met then the knowledge of the state coupling matrix is not needed to apply the proposed stochastic algorithm the proposed algorithm is shown to suppress a class of nonlinear and repetitive state disturbance the application of this algorithm to a class of nonlinear systems is also considered a numerical example is included to illustrate the performance of the algorithm

brief paper analysis of two particular iterative learning control schemes in frequency and time domains this paper deals with iterative learning control design for multiple input multiple output mimo linear time invariant lti systems two particular ilc schemes are considered and analyzed in both frequency and time domains some remarks on the convergence implementation robustness with respect to disturbances and reinitialization errors as well as positive realness issues related to both schemes are provided

robust optimal design and convergence properties analysis of iterative learning control approaches in this paper we address four major issues in the field of iterative learning control ilc theory and design the first issue is concerned with ilc design in the presence of system interval uncertainties targeting at time optimal fastest convergence and robustness properties concurrently we formulate the ilc design into a min max optimization problem and provide a systematic solution for linear type ilc consisting of the first order and higher order ilc schemes inherently relating to the first issue the second issue is concerned with the performance evaluation of various ilc schemes convergence speed is one of the most important factors in ilc a learning performance index q factor is introduced which provides a rigorous and quantified evaluation criterion for comparing the convergence speed of various ilc schemes we further explore a key issue how does the system dynamics affect the learning performance by associating the time weighted norm with the supreme norm we disclose the dynamics impact in ilc which can be assessed by global uniform bound and monotonicity in iteration domain finally we address a rather controversial issue in ilc can the higher order ilc outperform the lower order ilc in terms of convergence speed and robustness by applying the min max design which is robust and optimal and conducting rigorous analysis we reach the conclusion that the q factor of ilc sequences of lower order ilc is lower than that of higher order ilc in terms of the time weighted norm

an iterative learning controller with initial state learning in iterative learning control ilc a common assumption is that the initial states in each repetitive operation should be inside a given ball centred at the desired initial states which may be unknown this assumption is critical to the stability analysis and the size of the ball will directly affect the final output trajectory tracking errors in this paper this assumption is removed by using an initial state learning scheme together with the traditional d type ilc updating law both linear and nonlinear time varying uncertain systems are investigated uniform bounds for the final tracking errors are obtained and these bounds are only dependent on the system uncertainties and disturbances yet independent of the initial errors furthermore the desired initial states can be identified through learning iterations

robust iterative learning control design is straightforward for uncertain lti systems satisfying the robust performance condition this note demonstrates that the design of a robust iterative learning control is straightforward for uncertain linear time invariant systems satisfying the robust performance condition it is shown that once a controller is designed to satisfy the well known robust performance condition a convergent updating rule involving the performance weighting function can be directly obtained it is also shown that for a particular choice of this weighting function one can achieve a perfect tracking in the case where this choice is not allowable a sufficient condition ensuring that the least upper bound of the spl lscr sub 2 norm of the final tracking error is less than the least upper bound of the spl lscr sub 2 norm of the initial tracking error is provided this sufficient condition also guarantees that the infinity norm of the final tracking error is less than the infinity norm of the initial tracking error

technical communique a robust approach to iterative learning control design for uncertain systems this note proposes a frequency domain design method to achieve an iterative learning controller for uncertain feedback control systems for a linear time invariant plant with multiplicative perturbations and a feedback controller we derive a sufficient condition for the iterative process to converge for all admissible plant uncertainties based on the derived sufficient condition we show that the iterative learning control design problem can be reformulated as a general robust control problem setup consequently the iterative learning controller under plant uncertainty can be systematically designed by wellestablished robust control design procedures

adaptive learning control of nonlinear systems by output error feedback this paper addresses the problem of designing an output error feedback control for single input single output nonlinear systems with uncertain smooth output dependent nonlinearities whose local lipschitz constants are known the considered systems are required to be observable minimum phase with known relative degree and known high frequency gain sign linear systems are included the reference output signal is assumed to be smooth and periodic with known period by developing in fourier series expansion a suitable periodic input reference signal an output error feedback adaptive learning control is designed which ldquolearnsrdquo the input reference signal by identifying its fourier coefficients bounded closed loop signals and exponential tracking of both input and output reference signals are obtained when the fourier series expansion is finite while arbitrary small tracking errors are exponentially achieved otherwise the resulting control is not model based is independent of the system order and depends only on the relative degree the reference signal period and the high frequency gain sign

a robust high order p type iterative learning controller using current iteration tracking error a high order p type updating law in which the current iteration tracking error cite is employed is proposed for iterative learning control ilc of a class of uncertain nonlinear repetitive systems uniform boundedness of the tracking error is established in the presence of uncertainty disturbance and initialization error the bound and the ilc convergence rate can be adjusted by tuning the learning gain of the cite

derivation algebras of centerless perfect lie algebras are complete abstract it is proved that the derivation algebra of a centerless perfect lie algebra of arbitrary dimension over any field of arbitrary characteristic is complete and that the holomorph of a centerless perfect lie algebra is complete if and only if its outer derivation algebra is centerless

fixed point theorem based iterative learning control for ltv systems with input singularity in this paper we address a challenging and open problem how to design a suitable iterative learning control ilc system in the presence of input singularity which is incurred by the singularities of the system direct feed through term considering two typical types of input singularities we first revise the ilc operators accordingly by adding a forgetting factor and incorporating a time varying learning gain in the sequel guarantee ilc operators to be contractible next using the banach fixed point theorem we demonstrate that the output sequence can either enter and remains ultimately in a designated neighborhood of the target trajectory or is bounded by a class k function finally an illustrative example is presented

brief on the p type and newton type ilc schemes for dynamic systems with non affine in input factors in this paper p type learning scheme and newton type learning scheme are proposed for quite general nonlinear dynamic systems with non affine in input factors using the contraction mapping method it is shown that both schemes can achieve asymptotic convergence along learning repetition horizon in order to quantify and evaluate the learning performance new indices q factor and q order are introduced in particular to evaluate the learning convergence speed it is shown that the p type iterative learning scheme has a linear convergence order with limited learning convergence speed under system uncertainties on the other hand if more of system information such as the input jacobian is available newton type iterative learning scheme which is originated from numerical analysis can greatly speed up the learning convergence speed the effectiveness of the two learning control methods are demonstrated through a switched reluctance motor system

comments on solutions to the combined sensitivity and complementary sensitivity problem in control systems in the above mentioned paper by lim et al 1996 a numerical example is given to show that the nevanlinna pick interpolation formulas appearing in khargonekar and tannenbaum 1985 and krein and nudel man 1977 are incorrect by comparing the paper of krein and nudel man with adamjan 1971 we illustrate that the correct formulas are obtained by simply taking the transpose of the pick matrix given in the first two above mentioned papers

stochastic p type d type iterative learning control algorithms this paper presents stochastic algorithms that compute optimal and sub optimal learning gains for a p type iterative learning control algorithm ilc for a class of discrete time varying linear systems the optimal algorithm is based on minimizing the trace of the input error covariance matrix the state disturbance reinitialization errors and measurement errors are considered to be zero mean white processes it is shown that if the product of the input output coupling matrices c t 1 b t is full column rank then the input error covariance matrix converges to zero in presence of uncorrelated disturbances another sub optimal p type algorithm which does not require the knowledge of the state matrix is also presented it is shown that the convergence of the input error covariance matrices corresponding to the optimal and sub optimal p type and d type algorithms are equivalent and all converge to zero at a rate inversely proportional to the number of learning iterations a transient response pe

an iterative learning control theory for a class of nonlinear dynamic systems abstract an iterative learning control scheme is presented for a class of nonlinear dynamic systems which includes holonomic systems as its subset the control scheme is composed of two types of control methodology a linear feedback mechanism and a feedforward learning strategy at each iteration the linear feedback provides stability of the system and keeps its state errors within uniform bounds the iterative learning rule on the other hand tracks the entire span of a reference input over a sequence of iterations the proposed learning control scheme takes into account the dominant system dynamics in its update algorithm in the form of scaled feedback errors in contrast to many other learning control techniques the proposed learning algorithm neither uses derivative terms of feedback errors nor assumes external input perturbations as a prerequisite the convergence proof of the proposed learning scheme is given under minor conditions on the system parameters

brief sampled data iterative learning control for nonlinear systems with arbitrary relative degree in this paper a sampled daata iterative learning control method is proposed for nonlinear systems without restriction on system relative degree the learning algorithm does not require numerical differentiations of any order from the tracking error a sufficient condition is derived to guarantee the convergence of the system output at each sampling instant to the desired trajectory numerical simulation is conducted to demonstrate the theoretical result

a composite energy function based learning control approach for nonlinear systems with time varying parametric uncertainties a new learning control approach is developed in this note to address a class of nonlinear systems with time varying parametric uncertainties the concept of composite energy function cef which provides the system information along both time and learning repetition horizons is introduced in the analysis of learning control cef consists of two parts the first part is a standard lyapunov function which is used to access system behavior along time horizon during each learning cycle the second part is an l sup 2 norm of parametric learning errors which reflects the variation of the system status when the control system is updated on the basis of learning cycles the proposed learning control algorithm achieves asymptotical convergence along a learning repetition horizon at the same time the boundedness and pointwise convergence of the tracking error along time horizon is guaranteed the proposed learning control strategy is applicable to quite general classes of nonlinear systems without requiring the global lipschitz continuity condition and zero relative degree condition

on lie algebras with only inner derivations it is the purpose of this article to give a characterization of abelian by nilpotent lie algebras which have only inner derivations the motivation has been provided by two recent papers 3 41 which deal with a similar problem in group theory the lie algebra case allows some modifications and simplifications as compared to the group theory case we begin with a brief summary about lie algebras which have only inner derivations and also about lie algebras which must possess outer derivations if the algebra has nondegenerate killing form then all derivations are inner 6 p 741 while if the algebra is nilpotent then there exist outer derivations s this latter result has been extended in a variety of ways see 131 and the bibliography given there in particular we men tion that if the algebra is solvable and has no outer derivations then its center must be 0 2 121 on the other hand there are examples of lie algebras with nonzero center and only inner derivations see 9 also a general method for constructing algebras with only inner derivations is described in 7 in another direction schenkman s famous derivation tower theorem lo asserts that an algebra with zero center can be sub invariantly embedded in an algebra with only inner derivations schenkman s paper also provides bounds on this embedding which are actually met as is shown in 8 all lie algebras considered here are finite dimensional over a field let