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

antiinvariant subspaces of maximum dimension abstract for a linear transformation t we determine the maximum of the dimensions of the subspaces m for which m and t m are disjoint

technical communique adaptive ilc for a class of discrete time systems with iteration varying trajectory and random initial condition in this work we present a discrete time adaptive iterative learning control ailc scheme to deal with systems with time varying parametric uncertainties using the analogy between the discrete time axis and the iterative learning axis the new adaptive ilc can incorporate a recursive least squares rls algorithm hence the learning gain can be tuned iteratively along the learning axis and pointwisely along the time axis when the initial states are random and the reference trajectory is iteration varying the new ailc can achieve the pointwise convergence over a finite time interval asymptotically along the iterative learning axis

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

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

an iterative identification method for linear continuous time systems this paper presents a novel approach for the identification of continuous time systems directly from sampled i o data based on trial iterations the method achieves identification through iterative learning control ilc concepts in the presence of heavy measurement noise the robustness against measurement noise is achieved through 1 projection of continuous time i o signals onto a finite dimensional parameter space and 2 kalman filter type noise reduction in addition an alternative simpler method is given with some robustness analysis the effectiveness of the method is demonstrated through numerical examples for a nonminimum phase plant

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

iterative learning control of hamiltonian systems i o based optimal control approach in this note a novel iterative learning control scheme for a class of hamiltonian control systems is proposed which is applicable to electromechanical systems the proposed method has the following distinguished features this method does not require either the precise knowledge of the model of the target system or the time derivatives of the output signals despite the lack of information the tracking error monotonously decreases in l sub 2 sense and further perfect tracking is achieved when it is applied to mechanical systems the self adjoint related properties of hamiltonian systems proven in this note play the key role in this learning control those properties are also useful for general optimal control furthermore experiments of a robot manipulator demonstrate the effectiveness of the proposed method

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

robust self learning fuzzy controller design for a class of nonlinear mimo systems a new learning paradigm that can be applied for the design of a fuzzy logic based learning controller that is robust to external signals such as disturbances and set point changes is proposed in the paper it is well known that the self organizing fuzzy controller proposed by procyk and mamdani is sensitive to external signals such a phenomenon may be observed in other types of direct fuzzy logic based learning controllers that utilize an adaptation scheme in which the locational information of the current error state vector determines the degree of adaptation that should be made in the learning controller to resolve the problem of sensitivity to external signals it is proposed that learning and modification of the controller be made in consideration of the motional trend of the error state vector as well as its locational information this paradigm is adopted in the design of a new robust self learning fuzzy controller for a class of nonlinear mimo systems the well known techniques of sliding mode control and the fuzzy decision making method are utilized to implement the proposed learning scheme in the fuzzy learning controller via simulation study and experimental results the proposed learning controller is verified to be robust in the presence of external disturbances

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

iterative learning of model reference adaptive controller for uncertain nonlinear systems with only output measurement in this paper a model reference adaptive control strategy is used to design an iterative learning controller for a class of repeatable nonlinear systems with uncertain parameters high relative degree initial output resetting error input disturbance and output noise the class of nonlinear systems should satisfy some differential geometric conditions such that the plant can be transformed via a state transformation into an output feedback canonical form a suitable error model is derived based on signals filtered from plant input and output the learning controller compensates for the unknown parameters uncertainties and nonlinearity via projection type adaptation laws which update control parameters along the iteration domain it is shown that the internal signals remain bounded for all iterations the output tracking error will converge to a profile which can be tuned by design parameters and the learning speed is improved if the learning gain is large

a 2d systems approach to iterative learning control for discrete linear processes with zero markov parameters in this paper a new approach to iterative learning control for the practically relevant case of deterministic discrete linear plants with uniform rank greater than unity is developed the analysis is undertaken in a 2d systems setting that by using a strong form of stability for linear repetitive processes allows simultaneous con sideration of both trial to trial error convergence and along the trial performance resulting in design algorithms that can be computed using linear matrix inequalities lmis finally the control laws are experimentally verified on a gantry robot that replicates a pick and place operation commonly found in a number of applications to which iterative learning control is applicable

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

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

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

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

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

neural network control of a rotating elastic manipulator abstract nonminimum phase property of a rotating elastic manipulator causes difficulties for both classical and neural network inverse model control while most of the neural network methods for control of elastic manipulators do not appear to converge to a solution when the system is lightly damped in this paper an appropriate cost function for a neural controller is proposed in the designed neural control system there are only three layer feedforward networks consisting of an input layer with two nodes one hidden layer and output layer with one node the number of units in the hidden layer and the value of the learning rate are robust to this designed network algorithm in order to simulate the transient response of the rotating elastic manipulator system a single input single output state space representation is presented for the system nonlinear model it can be seen from the simulation results the designed neural controller can not only achieve very good tracking performance zero steady state errors and strong robustness to system parameter uncertainty but also reject the effects of the input torque disturbance

decentralized iterative learning control for a class of large scale interconnected dynamical systems the problem of decentralized iterative learning control for a class of large scale interconnected dynamical systems is considered in this paper it is assumed that the considered large scale dynamical systems are linear time varying and the interconnections between each subsystem are unknown for such a class of uncertain large scale interconnected dynamical systems a method is presented whereby a class of decentralized local iterative learning control schemes is constructed it is also shown that under some given conditions the constructed decentralized local iterative learning controllers can guarantee the asymptotic convergence of the local output error between the given desired local output and the actual local output of each subsystem through the iterative learning process finally as a numerical example the system coupled by two inverted pendulums is given to illustrate the application of the proposed decentralized iterative learning control schemes

brief paper noise tolerant iterative learning control for a class of continuous time systems the paper proposes a noise tolerant iterative learning control ilc for a class of linear continuous time systems which achieves high precision tracking for uncertain plants by iteration of trials in the presence of heavy measurement noise the robustness against measurement noise is achieved through i projection of continuous time i o signals onto a finite dimensional parameter space ii using error data of all past iterations via an integral operation in the learning law and iii noise reduction by h 2 optimization subject to a specified convergence speed of the ilc

state steering by learning for a class of nonlinear control systems abstract the problem of steering the state of nonlinearly perturbed linear systems by learning is investigated a family of algorithms which compute the steering control by means of successive trials on the real plant is presented convergence in the face of a class of nonlinear plant perturbations is proven state feedback linearizable systems are shown to be addressable by the presented algorithms two examples illustrate the applicability of algorithms

brief paper convergence and robustness of discrete time nonlinear systems with iterative learning control in this paper iterative learning control design of a class of discrete time nonlinear dynamic systems with disturbances are considered an iterative learning control law is proposed to overcome the uncertainties in system parameters and disturbances it is shown that the system outputs states and control inputs can be guaranteed to converge to desired trajectories in the absence of state output disturbances and repeatability uncertainty in the presence of these disturbances and initial state uncertainty the tracking errors will be bounded experiment is carried out to verify the theory and results are presented

an output based adaptive iterative learning controller for high relative degree uncertain linear systems in this paper we derive an output tracking error model based on signals filtered from plant input and output and then present a new output based adaptive iterative learning controller for repeatable linear systems with unknown parameters high relative degree initial resetting error input disturbance and output noise the proposed controller solves the important robustness issues without assuming the bounds of uncertainties to be sufficiently small and can be applied to high relative degree plants without using output differentiation control parameters are updated between successive iterations so as to compensate for unknown system parameters and uncertainties it is shown that the internal signals inside closed loop learning system remain bounded and the output tracking error will asymptotically converge to a profile tunable by some design parameters furthermore the learning speed is easily improved if the learning gain is increased

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

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

the validity of the perturbation approximation for rough surface scattering using a gaussian roughness spectrum the validity of the perturbation approximation for rough surface scattering is examined 1 by comparison with exact results obtained by solving an integral equation and 2 through comparison of low order perturbation predictions with higher order predictions the pressure release boundary condition is assumed and the field quantity calculated is the bistatic scattering cross section a gaussian roughness spectrum is used and the surfaces have height variations in only one direction it is found in general that the condition kh 1 k is the acoustic wavenumber h is the rms surface height is insufficient to guarantee the accuracy of first order or higher order perturbation theory when the surface correlation length l becomes too large or too small with h held fixed higher order perturbation terms can make larger contributions to the scattering cross section than lower order terms an explanation for this result is given the regions of validity for low order perturbation theory are also given th

electromagnetic scattering by rough conduction circular cylinders i angular corrugation the coherent and incoherent responses of perfectly conducting angularly corrugated surfaces which are infinite circular cylinders on the average are calculated using a second order perturbation method the general approach used is to apply the extinction theorem to write an integral equation for the surface current density and then to use perturbation theory to solve it these results are recast by using the only nontrivial pade approximant for this approximation this recasting yield a considerable qualitative improvement which at least in one special case is shown to be quantitative as well

k fold anti invariant subspaces of a linear mapping abstract let be a linear bijection on a finite dimensional vector space and k 0 an integer a subspace t is called a k fold anti invariant subspace if t t t k t t t k we find the maximal dimension of such subspaces t

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

covering gl v and products of blockcyclic matrices let v be an n dimensional vector space over a field k where n n n 3 k 4 a linear transformation v v is called blockcyclic if the minimum rank satisfies n 2 we study products of blockcyclic conjugacy classes in gl v example if and are products of blockcyclic conjugacy classes in gl v then contains a cyclic transformation the results are used in our study on products of singular similarity classes knuppel and nielsen in press 8 a further application of our methods is the following theorem let k n n 1 a non central conjugacy class of gl v and gl v such that det det k then k a similar result was proved in lev s article lev 1996 10

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 eigenvalue correction due to scattering by a rough wall of an acoustic waveguide in this paper a derivation of the attenuation factor in a waveguide with stochastic walls is presented the perturbation method and fourier analysis are employed to derive asymptotically consistent boundary value problems at each asymptotic order the derived approximation predicts the attenuation of the propagating mode in a rough waveguide through a correction to the eigenvalue corresponding to smooth walls the proposed approach can be used to derive results that are consistent with those obtained by bass et al ieee trans antennas propag 22 278 288 1974 the novelty of the method is that it does not involve the integral dyson type equation and as a result the large number of statistical moments included in the equation in the form of the mass operator of the volume scattering theory the derived eigenvalue correction is described by the correlation function of the randomly rough surface the averaged solution in the plane wave regime is approximated by the exponential function dependent on the derived eigenvalue correction the approximations are compared with numerical results obtained using the finite element method fem an approach to retrieve the correct deviation in roughness height and correlation length from multiple numerical realizations of the stochastic surface is proposed to account for the oversampling of the rough surface occurring in the fem meshing procedure

exponential stability of nonlinear differential repetitive processes with applications to iterative learning control abstract this paper studies exponential stability properties of a class of two dimensional 2d systems called differential repetitive processes drps since a distinguishing feature of drps is that the problem domain is bounded in the time direction the notion of stability to be evaluated does not require the nonlinear system defining a drp to be stable in the typical sense in particular we study a notion of exponential stability along the discrete iteration dimension of the 2d dynamics which requires the boundary data for the differential pass dynamics to converge to zero as the iterations evolve our main contribution is to show under standard regularity assumptions that exponential stability of a drp is equivalent to that of its linearized dynamics in turn exponential stability of this linearization can be readily verified by a spectral radius condition the application of this result to iterative learning control ilc is discussed theoretical findings are supported by a numerical simulation of an ilc algorithm

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

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

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

a note on convergence property of iterative learning controller with respect to sup norm we discuss some undesirable phenomena associated with the norm that is used in the proof of convergence of the iterative learning controller ilc specifically we first report that huge overshoot may be observed in the output trajectory error even though the applied learning algorithm has been proved to be exponentially convergent with respect to the norm this is because of the way in which the norm converges differs from that of the sup norm we then investigate the relation between the norm and the sup norm finally we present a simple method to remedy the undesirable feature of the conventional ilc

brief paper novel iterative learning controls for linear discrete time systems based on a performance index over iterations an optimal iterative learning control ilc is proposed to optimize an accumulative quadratic performance index in the iteration domain for the nominal dynamics of linear discrete time systems properties of stability convergence robustness and optimality are investigated and demonstrated in the case that the system under consideration contains uncertain dynamics the proposed ilc design can be applied to yield a guaranteed cost ilc whose solution can be found using the linear matrix inequality lmi technique simulation examples are included to demonstrate feasibility and effectiveness of the proposed learning controls

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

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 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

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

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

selection of the learning gain matrix of an iterative learning control algorithm in presence of measurement noise arbitrary high precision output tracking is one of the most desirable control objectives found in industrial applications regardless of measurement errors the main purpose of this paper is to supply to the iterative learning control ilc designer guidelines to select the corresponding learning gain in order to achieve this control objective for example if certain conditions are met then it is necessary for the learning gain to converge to zero in the learning iterative domain in particular this paper presents necessary and sufficient conditions for boundedness of trajectories and uniform tracking in presence of measurement noise and a class of random reinitialization errors for a simple ilc algorithm the system under consideration is a class of discrete time affine nonlinear systems with arbitrary relative degree and arbitrary number of system inputs and outputs the state function does not need to satisfy a lipschitz condition this work also provides a recursive algorithm that generates the appropriate learning gain functions that meet the arbitrary high precision output tracking objective the resulting tracking output error is shown to converge to zero at a rate inversely proportional to square root of the number of learning iterations in presence of measurement noise and a class of reinitialization errors two illustrative numerical examples are presented

the extended covering number of sln is n 1 abstract for a special linear group sln k k a field we want to find the minimal integer k extended covering number such that 1 c1 ck for arbitrary non central conjugacy classes ci of sln k using methods from chevalley groups length theorems on products of simple mappings and theorems on products of cyclic mappings we find that k n 1 provided n and the field are not too small

a discrete time learning control algorithm for a class of linear time invariant systems a discretized version of the d type learning control algorithm is presented for a mimo linear discrete time system a necessary and sufficient condition for uniform convergence of the proposed learning algorithm is presented then we prove that the same condition is sufficient for the global robustness of the proposed learning algorithm to state disturbances measurement noise at the output and reinitialization error are present at each iteration a numerical example is given to illustrate the results

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

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 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

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

optimality of first order ilc among higher order ilc higher order iterative learning control ho ilc algorithms use past system control information from more than one past iterative cycle this class of ilc algorithms have been proposed aiming at improving the learning efficiency and performance this paper addresses the optimality of ho ilc in the sense of minimizing the trace of the control error covariance matrix in the presence of a class of uncorrelated random disturbances it is shown that the optimal weighting matrices corresponding to the control information associated with more than one cycle preceding the current cycle are zero that is an optimal ho ilc does not add to the optimality of standard first order ilc in the sense of minimizing the trace of the control error covariance matrix the system under consideration is a linear discrete time varying systems with different relative degree between the input and each output

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

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

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

on functional approximation of the equivalent control using learning variable structure control a learning variable structure control lvsc approach is originated to obtain the equivalent control of a general class of multiple input multiple output mimo variable structure systems under repeatable control tasks lvsc synthesizes variable structure control vsc as the robust part which stabilizes the system and learning control lc as the plug in intelligent part which completely nullifies the effects of the matched uncertainties on tracking error rigorous proof based on energy function and functional analysis shows that the tracking error sequence converges uniformly to zero and that the bounded lc sequence converges to the equivalent control almost everywhere

a stochastic iterative learning control algorithm with application to an induction motor a recursive optimal algorithm based on minimizing the input error covariance matrix is derived to generate the learning gain matrix of a p type ilc for linear discrete time varying systems with arbitrary relative degree it is shown that in the case where the number of inputs is not greater than the number of outputs the input error covariance matrix converges to zero at a rate inversely proportional to the number of iterations in the presence of uncorrelated random state disturbance reinitialization errors and measurement noise the state error covariance matrix converges to zero at a rate inversely proportional to the number of iterations in the presence of measurement noise in the case where the number of inputs is greater than the number of outputs then the system output error converges to zero at a rate inversely proportional to the number of iterations in presence of measurement noise another suboptimal recursive algorithm is also proposed based on unknown system dynamics and unknown disturb

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

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