a confluence matrix condition for exponential error convergence in overparametrized adaptive systems many practical adaptive feedforward systems are overparametrized and for this reason will not satisfy persistent excitation pe conditions for these systems a weaker pe condition is proposed under which it is shown that the while parameters may not converge the cancellation error the error between the desired and estimated outputs still converges exponentially bounds are given on the exponential rate of convergence useful for understanding the various tradeoffs and for systematic optimization and design purposes the convergence rate is determined by properties of the confluence matrix defined herein that plays a role similar to that played by the autocorrelation matrix for fully pe systems as a case study the structure of the confluence matrix is examined in detail for adaptive systems with a tap delay line tdl regressor and sinusoid excitation

a result on exponential tracking error convergence and persistent excitation conditions are investigated for exponential convergence of the tracking error in feedforward adaptive systems having insufficient excitation particular attention is paid to the adaptive gradient algorithm with periodic excitation in the overparameterized case a main result is that for a bounded periodic regressor the tracking error converges exponentially without regard to parameter convergence or to the degree of overparameterization these results weaken the persistent excitation pe conditions and parameter convergence conditions generally considered necessary to ensure exponential tracking convergence in this class of systems

randomized algorithms for robust controller synthesis using statistical learning theory by now it is known that several problems in the robustness analysis and synthesis of control systems are np complete or np hard these negative results force us to modify our notion of solving a given problem an approach that is recently gaining popularity is that of using randomized algorithms which can be used to solve a problem approximately most of the time we begin with the premise that many problems in robustness analysis and synthesis can be formulated as the minimization of an objective function with respect to the controller parameters it is argued that in order to assess the performance of a controller as the plant varies over a prespecified family it is better to use the average performance of the controller as the objective function to be minimized rather than its worst case performance as the worst case objective function usually leads to rather conservative designs then it is shown that a property from statistical learning theory known as uniform convergence of empirical means ucem plays an important role in allowing us to construct efficient randomized algorithms for a wide variety of controller synthesis problems in particular whenever the ucem property holds there exists an efficient i e polynomial time randomized algorithm using very recent results in statistical learning theory it is shown that the ucem property holds in any problem in which the satisfaction of a performance constraint can be expressed in terms of a finite number of polynomial inequalities in particular several problems such as robust stabilization and weighted h 2 h norm minimization are amenable to the randomized approach

adaptive cancellation of periodic synchronously sampled interference a special case of the adaptive noise canceller is considered in which the interference is harmonic and synchronously sampled the implementation of the lms algorithm in this case can be made very efficient and its behavior can be represented exactly by that of a linear time invariant comb filter

sinusoidal interference rejection analysis of an lms adaptive feedforward controller with a noisy periodic reference this paper applies a new vector subspace model to determine the non wiener solutions of the lms algorithm when the reference input is an arbitrary noisy periodic signal the lms weights are modeled as a deterministic time varying mean plus a zero mean fluctuating part for the mean weight each harmonic component of the periodic reference is shown to excite only a two dimensional 2 d subspace of the n dimensional tap weight space the fluctuating part of the weight is due to the reference noise that excites the full n dimensional space the power spectrum of the error is computed using the correlation function of the weight fluctuations the error is shown to be primarily the sum of a white noise and the canceled periodic residual for an independent noise vector model when the effects of the tapped delay line are incorporated in the model the noise becomes colored with a spectrum centered about the canceled sinusoidal frequency this weight model significantly simplifies the understanding of the non wiener behavior and can be applied to active noise cancellation problems when filters appear in the cancellation loop

criterion for the convergence of the solution of the riccati differential equation the optimal control problem for a linear system with a quadratic cost function leads to the matrix riccati differential equation the convergence of the solution of this equation for increasing time interval is investigated as a function of the final state penalty matrix a necessary and sufficient condition for convergence is derived for stabilizable systems even if the output in the cost function is not detectable an algorithm is developed to determine the limiting value of the solution which is one of the symmetric positive semidefinite solutions of the algebraic riccati equation examples for convergence and nonconvergence are given a discussion is also included of the convergence properties of the solution of the riccati differential equation to any real symmetric not necessarily positive semidefinite solution of the algebraic riccati equation

dissipative lqg control systems an indefinite lyapunov test matrix for closed loop dissipativeness using lqg linear quadratic gaussian design is presented the application of this test to relaxation systems lossless systems hamiltonian systems positive real systems and second order systems with collocated rate sensors and actuators is explored the lyapunov test also helps clarify the role of collocated sensors and actuators in lqg design a set of duality theorems based on the indefinite lyapunov test matrix is derived

non wiener solutions for the lms algorithm a time domain approach a time domain analysis of the lms algorithm is presented for a sinusoidal deterministic reference input for the sinusoidal reference input only the n dimensional time varying linear matrix recursion for the weight vector is solved using a 2 d orthogonal subspace decomposition using this weight vector solution it is shown that there exists a linear time invariant relationship between the desired input and the filter output

robust stabilization with positive real uncertainty beyond the small gain theory in many applications of feedback control phase information is available concerning the plant uncertainty for example lightly damped flexible structures with colocated rate sensors and force actuators give rise to positive real transfer functions closed loop stability is thus guaranteed by means of negative feedback with strictly positive real compensators in this paper the properties of positive real transfer functions are used to guarantee robust stability in the presence of positive real but otherwise unknown plant uncertainty these results are then used for controller synthesis to address the problem of robust stabilization in the presence of positive real uncertainty one of the principal motivations for these results is to utilize phase information in guaranteeing robust stability in this these sense results go beyond the usual limitations the small gain theorem and quadratic lyapunov functions which may be conservative when phase information is available the results of this paper are based upon a riccati equation formulation of the positive real lemma and thus are in the spirit of recent riccati based approaches to bounded real h control

the boundedness of all products of a pair of matrices is undecidable we show that the boundedness of the set of all products of a given pair sigma of rational matrices is undecidable furthermore we show that the joint or generalized spectral radius rho sigma is not computable because testing whether rho sigma less than or equal to1 is an undecidable problem as a consequence the robust stability of linear systems under time varying perturbations is undecidable and the same is true for the stability of a simple class of hybrid systems we also discuss some connections with the so called finiteness conjecture our results are based on a simple reduction from the emptiness problem for probabilistic finite automata which is known to be undecidable c 2000 elsevier science b v all rights reserved

continuity properties of the real complex structured singular value the structured singular value function mu is defined with respect to a given uncertainty set this function is continuous if the uncertainties are allowed to be complex however if some uncertainties are required to be real then it can be discontinuous it is shown that mu is always upper semi continuous and conditions are derived under which it is also lower semi continuous with these results the real parameter robustness problem is reexamined a related although not equivalent problem is formulated which is always continuous and the relationship between the new problem and the original real mu m problem is made explicit a numerical example and results obtained via this related problem are presented

computational complexity of real structured singular value in l sub p setting this paper studies a generalized real structured singular value spl mu problem where uncertain parameters are bounded by an l sub p norm two results are presented the first one shows that this generalized spl mu problem is np hard for any given rational number p spl isin 1 spl infin the np hardness holds as long as le the size of the largest repeated block exceeds one this result generalizes the known np hardness result for the conventional spl mu problem with p spl infin the second result which strengthens the first one considers the approximability problem of the generalized spl mu we show that the problem of obtaining an estimate for the generalized spl mu with some guaranteed bounds on the relative error remains to be np hard regardless how large this bound is

controllability and observability criteria for multivariable linear second order models criteria are discussed for the determination of controllability stabilizability observability or detectability of linear second order multivariable models of for example large space structures an initial modal transformation is not required and the criteria are thus applicable to models with arbitrary damping coefficients moreover the criteria are modal in the sense that some or all of the modes may be tested for controllability or observability this aspect has advantages if not all the modes are known or easily computable the criteria are further illustrated for a number of important special cases in a series of corollaries

on the lti properties of adaptive feedforward systems with tap delay line regressors it is shown that an adaptive system whose regressor is formed by tap delay line tdl filtering of a multitone sinusoidal signal is representable as a parallel connection of a linear time invariant lti block and a linear time varying ltv block a norm bound induced 2 norm is computed explicitly on the ltv block and is shown to decrease as n sup 1 where n is the number of taps hence the adaptive system becomes lti in the limit as the number of taps goes to infinity in the more realistic case where the number of taps n is finite the new lti plus norm bounded perturbation representation renders for the first time the adaptive system analyzable by standard robust control methods

a general theory of linear time invariant adaptive feedforward systems with harmonic regressors establishes necessary and sufficient conditions for an adaptive system with a harmonic regressor i e a regressor comprised exclusively of sinusoidal signals to admit an exact linear time invariant lti representation these conditions are important because a large number of adaptive systems used in practice have sinusoidal regressors and the stability and performance of such systems having lti representations can be completely analyzed by well known methods the theory is extended to applications where the lti conditions do not hold in which case the harmonic adaptive system can be written as the parallel connection of a purely lti subsystem and a linear time varying ltv subsystem an explicit upper bound is established on the induced two norm of the ltv block which allows systematic treatment using emerging robust control methods applicable to lti systems with norm bounded ltv perturbations

persistence of excitation conditions and the convergence of adaptive schemes in the study of the behavior of adaptive filtering algorithms persistence of excitation of the input process arises as a sufficient condition for convergence and perhaps more importantly for convergence rate of the parameter estimates in this paper the underlying nature of the persistence requirement is presented and discussed relating its normal specification in terms of moment conditions with covariance decays etc to sample path properties deterministic and stochastic persistence conditions and persistence measures are treated as well as persistence conditions for output error equation error and adaptive control schemes

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

time reversible hamiltonian systems it is shown that transfer matrices satisfying g s g s g t t s have a minimal hamiltonian realization with an energy which is the sum of potential and kinetic energy yielding the time reversibility of the equations furthermore connections are made with an associated gradient system the theory is applied to electrical networks resulting in a characterization of lct networks

on computational complexity of invalidating structured uncertainty models abstract in this paper we study a class of uncertainty model validation invalidation problems for linear discrete time systems we consider models described by a linear fractional transformation lft of modeling uncertainties which are structured time invariant or time varying and are bounded in h or l 1 norms the experimental data available for use in invalidation are either input output time series observations or frequency response measurements we analyze the computational complexity associated with these problems while earlier work showed that for lft models with an unstructured uncertainty the invalidation problems can be reduced to one of solving linear matrix inequalities and hence may be tackled readily using well developed numerical methods in this paper we provide a simple proof to show that in the presence of structured uncertainties they are all np hard with respect to the number of uncertainty blocks suggesting that the problems are inherently intractable from a computational standpoint additionally we also demonstrate that the problems do become tractable when the lft model description is changed to an additive one leading to lmi based invalidation tests similar to those obtained elsewhere these results indicate that the main source of the computational difficulty lies in the lft description together with the structured nature of the uncertainties

computational complexity of spl mu calculation the structured singular value spl mu measures the robustness of uncertain systems numerous researchers over the last decade have worked on developing efficient methods for computing spl mu this paper considers the complexity of calculating spl mu with general mixed real complex uncertainty in the framework of combinatorial complexity theory in particular it is proved that the spl mu recognition problem with either pure real or mixed real complex uncertainty is np hard this strongly suggests that it is futile to pursue exact methods for calculating spl mu of general systems with pure real or mixed uncertainty for other than small problems

efficient computation of a guaranteed lower bound on the robust stability margin for a class of uncertain systems sufficient conditions for the robust stability of a class of uncertain systems with several different assumptions on the structure and nature of the uncertainties can be derived in a unified manner in the framework of integral quadratic constraints these sufficient conditions in turn can be used to derive lower bounds on the robust stability margin for such systems the lower bound is typically computed with a bisection scheme with each iteration requiring the solution of a linear matrix inequality feasibility problem we show how this bisection can be avoided altogether by reformulating the lower bound computation problem as a single generalized eigenvalue minimization problem which can be solved very efficiently using standard algorithms we illustrate this with several important commonly encountered special cases diagonal nonlinear uncertainties diagonal memoryless time invariant sector bounded popov uncertainties structured dynamic uncertainties and structured parametric uncertainties we also present a numerical example that demonstrates the computational savings that can be obtained with our approach

structured singular values and stability analysis of uncertain polynomials part 2 a missing link abstract in part 1 of this paper a generalized notion of structured singular value is introduced and studied specifically in the case when a certain matrix is of rank one in part 2 of this series we demonstrate that the framework developed in part 1 is particularly suitable for a class of stability problems and its unifies several frequency based stability conditions obtained via alternative approaches generally these problems correspond to a class of uncertain polynomials whose coefficients are perturbed in an affine fashion several stability conditions are derived readily from solution of the structured singular value presented in part 1 which either extend or coincide with previously known results in particular for family of interval and diamond polynomials we further show that the stability conditions based on the structured singular value and the corresponding results in the spirit of kharitonov theorem lead to one another thus establishing a link between the two types of drastically different results and accomplishing one of our main goals in this series a related robust stabilization problem is also investigated in this framework

harmonic generation in adaptive feedforward cancellation schemes the paper investigates the generation of harmonics in adaptive feedforward cancellation schemes specifically it is shown that an algorithm designed to reject a certain number of frequency components may in fact be capable of reducing higher order harmonics as well this effect originates in the time variation of the adaptive parameters a consequence is that the adaptive system may perform better than the algorithm with the fixed ideal parameters surprisingly the response to higher order harmonics can be calculated precisely using a laplace transform analysis the origin of the numerical procedure is in the equivalence between the adaptive feedforward cancellation scheme and a control scheme based on the internal model principle the implication of the results on design and implementation issues is discussed

a new set of invariants for linear systems application to reduced order compensator design a new set of invariants for linear systems weighting the contribution of each state component to the inherent closed loop lqg behavior of the system is presented together with applications to model order reduction and reduced order compensator design

a control theory approach to the stability and transient analysis of the filtered x lms adaptive notch filter a method is presented for analyzing the stability and transient response of the filtered x lms adaptive notch filter by formulating the problem in the complex weight domain and applying standard control theory examples are given for pure delay and second order low pass cancellation path transfer functions the method is also extended to the multichannel case where the eigenvalues of the equivalent open loop transfer function matrix characterize the behavior

on the dissipativity of nonlinear systems fuzzy control approach a controller with large dissipativity is designed and implemented the stability margin is increased in order to tolerate the uncertainties and the disturbance in the nonlinear system in this study the dissipativity of a nonlinear system is facilitated via fuzzy control method first the nonlinear system is represented by an takagi and sugeno fuzzy model if the state variables of the nonlinear system are available then a state feedback fuzzy control is developed to guarantee that the nonlinear control system is dissipative this design problem becomes a linear matrix inequality lmi problem the maximum dissipativity design is obtained by solving an eigenvalue problem evp if the state variables are not available a fuzzy observer based state feedback controller will be used to solve the dissipative problem of the nonlinear system simulation examples are given to illustrate the design procedures and performances of the proposed method