nonlinearities enhance parameter convergence in output feedback systems while the parameter convergence properties of standard adaptive algorithms for linear systems are well established there are no similar results on the parameter convergence of adaptive controllers for nonlinear systems which have gained popularity in recent years in this paper we focus on a recently developed class of adaptive schemes for output feedback nonlinear systems and show that parameter convergence is guaranteed if and only if an appropriately defined signal vector which does not depend on closed loop signals is persistently exciting then we develop an analytic procedure which allows us given a specific nonlinear system and a specific reference signal to determine a priori whether or not this vector is persistently exciting pe and hence whether or not the parameter estimates will converge in the process we show that the presence of nonlinearities usually reduces the sufficient richness sr requirements on the reference signals and hence enhances parameter convergence this is the first result on the relationship between pe and sr for adaptive nonlinear control systems

disturbance generated bifurcation in a prototype adaptive system with e sub 1 modification law the e sub 1 modification law has been proposed by k s narendra and a m annaswamy ibid vol 32 no 2 pp 134 5 1987 for robust model reference adaptive control in the presence of bounded disturbances using this law it is shown by the present authors that in the case of regulation a prototype adaptive control system with a single unknown pole and an unknown constant disturbance at its input possesses multiple equilibria and undergoes various bifurcations simulations and local analysis give a complete characterization of the possible dynamics exhibited by the disturbance adaptive system as the constant disturbance is varied saddle node bifurcation subcritical hopf bifurcation and saddle connection bifurcation occur in that order consequently in this case the e sub 1 modification law is qualitatively equivalent to the sigma modification law

a stability instability boundary for disturbance free slow adaptation with unmodeled dynamics the instability of adaptive schemes for small values of parameter adjustment gain has the form of a slow drift of adjustable parameters with an averaging analysis we derive not only a sufficient condition for stability of this drift but also a sufficient condition for it to be unstable a sharp boundary between stability and instability is signal dependent and much less demanding than the usual strict positive realness property the new positivity condition has a simple signal energy interpretation

lyapunov redesign for structural convergence improvement in adaptive control the relative improvement of the convergence rate in adaptive error models is discussed it is shown how to improve convergence positive realness and robustness by modifying the structure of the error model the proposed approaches to designing adaptive control algorithms using the lyapunov method not only improve the rate of convergence but also provide flexibility in choosing nonlinear control inputs boundedness of the nonlinear functions of the modified schemes is shown

persistency of excitation in possible unstable continuous time systems identification and indirect adaptive control the author proves persistency of excitation of the output of an output reachable possibly unstable linear system under certain input conditions and applies this result to adaptive identification and indirect adaptive control in adaptive identification he proves exponential parameter convergence regardless of stability of the identified plant in indirect adaptive control he proves exponential parameter convergence along with asymptotic time invariance and global experimental stability of the controlled closed loop system

reproducing kernel hilbert space approach for the online update of radial bases in neuro adaptive control classical work in model reference adaptive control for uncertain nonlinear dynamical systems with a radial basis function rbf neural network adaptive element does not guarantee that the network weights stay bounded in a compact neighborhood of the ideal weights when the system signals are not persistently exciting pe recent work has shown however that an adaptive controller using specifically recorded data concurrently with instantaneous data guarantees boundedness without pe signals however the work assumes fixed rbf network centers which requires domain knowledge of the uncertainty motivated by reproducing kernel hilbert space theory we propose an online algorithm for updating the rbf centers to remove the assumption in addition to proving boundedness of the resulting neuro adaptive controller a connection is made between pe signals and kernel methods simulation results show improved performance

analysis of adaptive identifiers in the presence of unmodeled dynamics averaging and tuned parameters the authors analyze the behavior of a standard identifier when the plant contains additional dynamics called unmodeled dynamics which invalidate the known order assumption the first result of the analysis is an input richness condition which does not depend on the order of the unmodeled dynamics to guarantee persistency of excitation of the regressor then it is shown that the persistently exciting pe condition leads to a bibo bounded input bounded output stability property for the identifier the method of averaging is used to formally define the notion of tuned parameters as the equilibrium of the identifier averaged system it is shown that the tuned parameters always exist and that the actual parameters converge to some neighborhood of the tuned parameters from the definition of the tuned parameters an explicit expression to calculate and interpret them as the fixed parameter values that minimize the mean squared output error is derived

persistency of excitation results for structured nonminimal models it is frequently convenient to use specially structured nonminimal models for parameter estimation such as in the case of direct adaptive control where the model is parameterized in terms of the desired control law parameters this is done for example in direct model reference adaptive control and in direct pole assignment adaptive control algorithms it is shown that the parameters appearing in these nonminimal models can be uniquely estimated if and only if a certain design identity has a unique solution the result is used to develop persistency of excitation results for these models

parameter convergence and uniqueness in nonminimal parameterizations for multivariable adaptive control the issue of parameter convergence in multivariable adaptive control is addressed in a general framework parameter convergence is guaranteed if a certain design identity has a unique solution and if the inputs satisfy persistency of excitation conditions the uniqueness of the solution of the design identity can be obtained in general by using parameterizations that although nonminimal are structured so as to guarantee uniqueness this concept is illustrated with a direct adaptive pole placement algorithm which is modified to guarantee uniqueness and it is shown how the results can be used to establish stability and convergence properties of the algorithm

frequency domain conditions for parameter convergence in multivariable recursive identification abstract a multivariable adaptive identifier is designed on the basis of a canonical form leading to a unique set of parameters to identify frequency domain conditions for parameter convergence are derived the sufficient condition to guarantee convergence in all cases is that n v distinct frequency components be present in each input of the plant where v is the maximum of the observability indices and n the order of the plant on the other hand the necessary condition requires n mv distinct frequency components distributed over the inputs where m is the number of inputs and at least v frequency components are in each separate input through examples it is shown that convergence may or may not occur for intermediate numbers of frequency components depending on their location and on the plant parameters

bayesian nonparametric adaptive control using gaussian processes most current model reference adaptive control mrac methods rely on parametric adaptive elements in which the number of parameters of the adaptive element are fixed a priori often through expert judgment an example of such an adaptive element is radial basis function networks rbfns with rbf centers preallocated based on the expected operating domain if the system operates outside of the expected operating domain this adaptive element can become noneffective in capturing and canceling the uncertainty thus rendering the adaptive controller only semiglobal in nature this paper investigates a gaussian process based bayesian mrac architecture gp mrac which leverages the power and flexibility of gp bayesian nonparametric models of uncertainty the gp mrac does not require the centers to be preallocated can inherently handle measurement noise and enables mrac to handle a broader set of uncertainties including those that are defined as distributions over functions we use stochastic stability arguments to show that gp mrac guarantees good closed loop performance with no prior domain knowledge of the uncertainty online implementable gp inference methods are compared in numerical simulations against rbfn mrac with preallocated centers and are shown to provide better tracking and improved long term learning

multivariable adaptive control identifiable parameterizations and parameter convergence the paper presents a direct model reference adaptive control algorithm for multivariable systems that has guaranteed parameter convergence properties the algorithm requires fewer parameters than existing schemes and the parameters converge to their nominal values under persistency of excitation conditions simple frequency domain conditions on the external inputs are derived to satisfy these conditions the convergence of the controller parameters to their nominal values guarantees a certain degree of robustness of the algorithms and ensures that the model matching objective is satisfied asymptotically

collocation and iterated collocation methods for a class of weakly singular volterra integral equations we discuss the convergence properties of spline collocation and iterated collocation methods for a weakly singular volterra integral equation associated with certain heat conduction problems this work completes the previous studies of numerical methods for this type of equations with noncompact kernel in particular a global convergence result is obtained and it is shown that discrete superconvergence can be achieved with the iterated collocation if the exact solution belongs to some appropriate spaces some numerical examples illustrate the theoretical results

brief interval excitation through impulse sequences a technical lemma we present a technical lemma that establishes in a general framework an interval excitation property using a specific class of impulse sequences according to such an interval excitation property an impulse applied within a time interval generates excitation within the same interval this property does not hold for arbitrary exciting sequences the lemma is a unified statement of previous results corresponding to particular choices of the exciting sequence

adaptive control of arma plants using worst case design by semi infinite optimization this paper presents a new approach to on line control system tuning based on worst case design using semi infinite optimlzation together with a plant uncertainty identification scheme which this approach requires

richness and excitation on an interval with application to continuous time adaptive control richness and excitation on an interval is presented as a useful tool which offers flexibility and mathematical simplification of the excitation issue feedback laws are considered which when they are applied to any controllable time invariant linear system result in a closed loop system such that the feedback signal the measure of excitation which it produces and the state vector of the system are all of the same order of magnitude on an interval such feedback laws generate excitation of time varying linear systems provided these systems are almost time invariant and controllable in the interval of interest using these results a smooth adaptive controller which controls and adapts simultaneously all the time and stabilizes and asymptotically regulates an arbitrary time invariant controllable linear system of known order is obtained

superconvergence of collocation methods for volterra and abel integral equations of the second kind this paper deals with the question of the attainable order of convergence in the numerical solution of volterra and abel integral equations by collocation methods in certain piecewise polynomial spaces and which are based on suitable interpolatory quadrature for the resulting moment integrals the use of a nonlinear variation of constants formula for the representation of the error function in terms of the defect allows for a unified treatment of equations with continuous and weakly singular kernels

on parameter convergence in adaptive control it is well known that the parameter error as well as the model plant mismatch error in a model reference adaptive scheme tends exponentially to zero iff a certain sufficient richness condition holds for signals inside the time varying plant control loop in this paper we give conditions on the reference signal the exogenous input to the adaptive loop namely that it have as many spectral lines as there are unknown parameters in order to guarantee parameter convergence

mathematical programming methods in the numerical solution of volterra integral and integro differential equations with weakly singular kernel abstract nonlinear volterra integral and integro differential equations with weakly singular kernel are considered and solved numerically using nonlinear mathematical programming methods based on minimax approximations in both cases polynomial and multiquadric approximation are used

order of convergence of one step methods for volterra integral equations of the second kind nice proofs of convergence and asymptotic expansions are known for one step methods for ordinary differential equations it is shown that these proofs can be generalized in a natural way to extended one step methods for volterra integral equations of the second kind furthermore the convergence of mixed one step methods is investigated for both types general volterra runge kutta methods are considered as examples

natural continuous extensions of runge kutta methods for volterra integral equations of the second kind and their applications we consider a very general class of runge kutta methods for the numerical solution of volterra integral equations of the second kind which includes as special cases all the more important methods which have been considered in the literature the main purpose of this paper is to define and prove the existence of the natural continuous extensions nce s of runge kutta methods i e piecewise polynomial functions which extend the approximation at the grid points to the whole interval of integration the particular properties required of the nce s allow us to construct the tail approximations which are quite efficient in terms of kernel evaluations

h optimization and optimal rejection of persistent disturbances abstract in this note it is shown that in a certain well known normed space of persistent signals a natural optimal disturbance rejection problem is equivalent to h optimization thus clarifying certain misconceptions on h optimization in the literature these conclusions are further strengthened using techniques of wiener s generalized harmonic analysis

a survey of recent advances in the numerical treatment of volterra integral and integro differential equations abstract the present survey paper samples recent advances in the numerical analysis of volterra integral equations of the first and second kind and of integro differential equations including equations with weakly singular kernels except for some important earlier references the discussion focuses on the development which has taken place during the last dozen years a fairly extensive bibliography selected to be representative rather than comprehensive complements the paper

exponential convergence of adaptive identification and control algorithms output and equation error adaptive identification algorithms are shown to be exponentially convergent under a deterministic or stochastic persistently exciting or spanning condition on the system inputs together with several other standard conditions an adaptive control algorithm is shown to be exponentially convergent under a deterministic or stochastic persistently exciting condition on the reference trajectory together with some standard conditions

a reliable technique for solving the weakly singular second kind volterra type integral equations abstract in this paper the adomian decomposition method and the phenomenon of the self canceling noise terms are used for solving the weakly singular volterra type linear and nonlinear integral equations the solution obtained is in the form of a convergent power series with elegantly computable terms comparing this scheme with many collocation type methods that use the nonpolynomial basis shows that the present approach is effective and powerful many test modeling problems from mathematical physics linear and nonlinear are discussed to illustrate the effectiveness and the performance of the decomposition method

convergence properties of adaptive systems and the definition of exponential stability the convergence properties of adaptive systems in terms of excitation conditions on the regressor vector are well known with persistent excitation of the regressor vector in model reference adaptive control the state error and the adaptation error are globally exponentially stable or equivalently exponentially stable in the large when the excitation condition however is imposed on the reference input or the reference model state it is often incorrectly concluded that the persistent excitation in those signals also implies exponential stability in the large the definition of persistent excitation is revisited so as to address some possible confusion in the adaptive control literature it is then shown that persistent excitation of the reference model only implies local persistent excitation weak persistent excitation weak persistent excitation of the regressor is still sufficient for uniform asymptotic stability in the large but not exponential stability in the large we show that there exists an infinite region in the state space of adaptive systems where the state rate is bounded this infinite region with finite rate of convergence is shown to exist not only in classic open loop reference model adaptive systems but also in a new class of closed loop reference model adaptive systems

on the local stability properties of adaptive parameter estimators with composite errors and split algorithms the stability characteristics of a generalized error system structure are examined for adaptive parameter estimation systems this error system form encompasses the structure of a number of particular applications of adaptive parameter estimation theory which fall outside the framework of more familiar error system models the consequences for error system stability which derive from the added generality are analyzed it is demonstrated that when using averaging theory techniques to analyze the error system stability conditions must be augmented for the basic error system in order to ensure stability for the generalized error system the inadequacy of regressor spectral restrictions and persistent spanning conditions to guarantee local error system stability of the generalized structure is rigorously established alternative conditions which will yield such local stability are demonstrated to illustrate the concepts involved the recursive identification of parameters is examined in a parallel form realization of a linear system

persistency excitation in continuous time systems for the class of linear time invariant single input continuous systems we find conditions on the input under which the state is persistently exciting for adaptive identification purposes these conditions are expressed through time domain properties of the filtered input they are both necessary and sufficient and no prior constraints are placed on the structure of the input wave or its boundedness

integrability for solutions to some anisotropic elliptic equations abstract we consider the boundary value problem i 1 n d i a i x d u x 0 x u x u x x we show that higher integrability of the boundary datum u forces solutions u to have higher integrability as well assumptions on a i x z are suggested by the euler equation of the anisotropic functional d 1 u p 1 d 2 u p 2 d n u p n d x

nonlinear averaging analysis of the incremental pole placement adaptive controller abstract incremental pole placement is an indirect adaptive control algorithm aimed at allocating the closed loop poles at some desired positions by incrementally tuning the controller parameters it differs from the classical algorithm in that the closed loop poles are estimated directly together with the plant poles and zeros the deviation between the obtained closed loop poles and the desired ones is then used in the update mechanism cautious control is implemented naturally as the control update and estimation algorithm are highly decoupled this paper addresses the performance of the incremental pole placement a nonlinear averaging analysis is presented the stability conditions describe the achievable performance in terms of model plant mismatch and noise characteristics the analysis indicates how robustness with respect to the unmodelled unparameterized dynamics can be improved

composite model reference adaptive control with parameter convergence under finite excitation a new parameter estimation method is proposed in the framework of composite model reference adaptive control for improved parameter convergence without persistent excitation the regressor filtering scheme is adopted to perform the parameter estimation with signals that can be obtained easily a new framework for residual signal construction is proposed the incoming data are first accumulated to build the information matrix and then its quality is evaluated with respect to a chosen measure to select and store the best one the information matrix is built to have full rank after sufficient but not persistent excitation in this way the exponential convergence of both tracking error and parameter estimation error can be guaranteed without persistent oscillation in the external command which drives the system numerical simulations are performed to verify the theoretical findings and to demonstrate the advantages of the proposed adaptation law over the standard direct adaptation law

global stability proofs for continuous time indirect adaptive control schemes this note presents a general stability proof for continuous time adaptive control with very general assumptions on the identifier and controller we show that if the exogenous input signal is rich enough then both the identifier and the controller converge to their true values applications of the proof to pole placement design and design based on the factorization approach are discussed

on the necessity of block invariance for the convergence of adaptive pole placement algorithm with persistently exciting input global convergence of a number of discrete time adaptive control algorithms with persistently exciting inputs has been established in the literature a key element of the convergence proof is the block invariance feature of the algorithm while the parameters are estimated at every sample the controller parameters remain invariant throughout a block of samples and are updated only at the end of each such block the necessity of this block invariance feature for guaranteeing convergence is established here by a counterexample

parameter information content of measurable signals in direct adaptive control it is postulated that available parameter information is a functional of all past history of measurable signals from this point of view direct model reference adaptive controllers become functional differential equations a modified model reference adaptive control structure and functional adaptation algorithm are presented it is shown that the system is well defined stable and achieves asymptotic model transfer function matching under weaker excitation conditions than those required by earlier model reference adaptive systems it is claimed that the information content perspective developed in this paper has implications in robust adaptive control theory

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations

persistency of excitation and overparametrization in model reference adaptive control the problem of overparameterization of the plant in model reference adaptive control mrac is analyzed overparametrization may arise when an upper bound for the order of the plant is used in the design of the mrac or when the transfer function of the modeled part of the plant has common zeros and poles it is shown that under parametrization the parameters for model plant transfer function matching lie in a linear variety if the reference input signal has enough frequencies the estimated parameters will converge to this variety in the presence of modeling errors such as bounded disturbances the parameters may drift to infinity along the linear variety independently of the number of frequencies in the reference input signal in the case of exact parameterization but close zero pole cancellation the signal vector is shown to be persistently exciting with a low level of excitation such a low level of excitation may lead to poor rate of convergence and to loss of robustness with respect to modeling errors

numerical solution of a class of integro differential equations by the tau method with an error estimation the tau method by construction produces approximate polynomial solutions of differential equations the purpose of this paper is to extend the tau method to the integro differential equations an efficient error estimation for the tau method is also introduced details of this method are presented and some numerical results along with estimated errors are given to clarify the method and its error estimator

necessary and sufficient conditions for parameter convergence in adaptive control a complete description of parameter convergence in model reference adaptive control may be given in terms of the spectrum of the exogenous reference input signal abstract using generalized harmonic analysis a complete description of parameter convergence in model reference adaptive control mrac is given in terms of the spectrum of the exogenous reference input signal roughly speaking if the reference signal contains enough frequencies then the parameter vector converges to its correct value if not it converges to an easily characterizable subspace in parameter space

global estimation of n unknown frequencies given a signal which is available for measurements and consists of the sum of a bias and n sinusoids with unknown amplitudes frequencies and phases a dynamic algorithm is designed to recover online the unknown values of the frequencies for any initial condition and any value of frequencies amplitudes and phases

dominant richness and improvement of performance of robust adaptive control abstract several recent robust adaptive control schemes guarantee stability and residual tracking errors which are small in the mean for any bounded initial conditions independent of any persistence of excitation condition since smallness in the mean does not always guarantee small bounds for the tracking error at steady state the tracking performance of these robust schemes may not be acceptable in this paper we show that the convergence and the tracking performance of these globally stable adaptive control schemes can be considerably improved if the reference input signal is chosen to be dominantly rich we show that dominantly rich signals maintain a high level of persistence of excitation relative to the level of the modeling error which guarantees exponential convergence and small bounds for the tracking and parameter error at steady state

superconvergence of collocation methods for a class of weakly singular volterra integral equations we discuss the application of spline collocation methods to a certain class of weakly singular volterra integral equations it will be shown that by a special choice of the collocation parameters superconvergence properties can be obtained if the exact solution satisfies certain conditions this is in contrast with the theory of collocation methods for abel type equations several numerical examples are given which illustrate the theoretical results

product integration methods for second kind abel integral equations the construction and convergence of high order product integration methods for the second kind abel equation are discussed and the results of de hoog and weiss are generalised backward difference methods are introduced and numerical results are presented which verify the theoretical rates of convergence