some comments on the behavior of sign sign adaptive identifiers abstract the sign sign variant of the lms algorithm with and without leakage is investigated three basic questions are considered through simplistic examples the need to redefine input excitations for parameter convergence the influence of designer selected variables on offset due to leakage and the effect of measurement noise in underexcited systems

robust h control of uncertain fuzzy systems under time varying sampling robust h sampled data control for uncertain fuzzy systems is discussed in many practical situations a system is modeled as a continuous time fuzzy system while the control input is the zero order hold which can be represented as a piecewise continuous delay here we take a delay system approach to the h sampled data control problem the closed loop system with a sampled data state feedback controller becomes a system with time varying delay first h performance conditions for the closed loop system are given in terms of linear matrix inequalities lmis such conditions are derived by using the leibniz newton formula and free weighting matrix method for fuzzy time delay systems under the assumption that sampling time is not greater than some prescribed number then a design method for a robust h sampled data state feedback controller for uncertain fuzzy systems is proposed numerical examples are given to illustrate our robust h sampled data state feedback control design

a feedback approach to the steady state performance of fractionally spaced blind adaptive equalizers this paper proposes a new approach to the analysis of the steady state performance of constant modulus algorithms cma which are among the most popular adaptive schemes for blind equalization a major feature of the proposed feedback approach is that it bypasses the need for working directly with the weight error covariance matrix in so doing approximate expressions for the steady state mean square error of several cm algorithms are derived including cma2 2 cma1 2 normalized cma and a new normalized cma variant with less bias a comparison among the various algorithms is also performed along with several simulation results the conclusions confirm the superior performance of cma2 2

convergence of the sign algorithm for adaptive filtering with correlated data convergence of a decreasing gain sign algorithm sa for adaptive filtering is analyzed the presence of the hard limiter in the algorithm makes a rigorous analysis difficult therefore there are few results available such results normally include restrictive assumptions such as the assumptions that successive observation vectors are independent and the new error signal of the adaptive filter has a time invariant probability density function the former assumption is not valid in the context of adaptive filtering since two successive observation vectors share most of their components while the latter assumption is a restriction on the adaptive weights whose evolution is a priori unknown in lieu of using these assumptions an almost sure convergence of the sa is proved under the assumption that the sequence of observation vectors is m dependent this assumption allows strong correlation between successive observations

on tuning leakage for performance robust adaptive control the behavior of a model reference adaptive control mrac system is examined the effect of the inclusion of leakage in the adaptive algorithm is studied and the leakage mechanism of the sigma modification and the e sub 1 modification are discussed a useful approximation is given for a particular mrac problem for the small step size average parameter estimate motion away from an arbitrary stabilizing adaptive controller parameterization arrow diagrams verify the conclusion that designer selected leakage constants must be well chosen in relation to the unknown plant and environment to avoid bursting by an adaptive controller in the absence of adequate excitation by the external forcing function s

tracking analysis of the sign algorithm in nonstationary environments a tracking analysis of the adaptive filters equipped with the sign algorithm and operating in nonstationary environments is presented under the assumption that the nonstationary can be modeled using a random disturbance it is shown that the long term time average of the mean absolute error is bounded and that there exists an optimal choice of the convergence constant mu which minimizes this quality applying the commonly used independence assumption and under the assumption that the nonstationarity is solely due to the time varying behavior of the optimal coefficients it is shown that the distributions of the successive coefficient misalignment vectors converge to a limiting distribution when the adaptive filter is used in the system identification mode finally under the additional assumption that the signals involved are zero mean and gaussian a set of nonlinear difference equations is derived that characterizes the mean and mean squared behavior of the filter coefficients and the mean squared estimation error during adaptation and tracking results of several experiments that show very good correlation with the theoretical analyses are presented

convergence analysis of self adaptive equalizers a theoretical analysis of self adaptive equalization for data transmission is carried out starting from known convergence results for the corresponding trained adaptive filter the development relies on a suitable ergodicity model for the sequence of observations at the output of the transmission channel thanks to the boundedness of the decision function used for data recovery it can be proved that the algorithm is bounded strong convergence results can be reached when a perfect noiseless equalizer exists the algorithm will converge to it if the eye pattern is initially open otherwise convergence may take place towards certain other stationary points of the algorithm for which domains of attraction have been defined some of them will result in a poor error rate the case of a noisy channel exhibits limit points for the algorithm that differ from those of the classical trained algorithm the stronger the noise the greater the difference is one of the principal results of this study is the proof of the stability of the usual decision feedback algorithms once the learning period is over

transient and steady state analysis of the affine combination of two adaptive filters in this paper we propose an approach to the transient and steady state analysis of the affine combination of one fast and one slow adaptive filters the theoretical models are based on expressions for the excess mean square error emse and cross emse of the component filters which allows their application to different combinations of algorithms such as least mean squares lms normalized lms nlms and constant modulus algorithm cma considering white or colored inputs and stationary or nonstationary environments since the desired universal behavior of the combination depends on the correct estimation of the mixing parameter at every instant its adaptation is also taken into account in the transient analysis furthermore we propose normalized algorithms for the adaptation of the mixing parameter that exhibit good performance good agreement between analysis and simulation results is always observed

brief sufficient conditions for the existence of an unbounded solution readily verifiable conditions under which a dynamical system of the form x f x possesses an unbounded solution are presented the results are illustrated by showing they can be used to infer results about lack of global stabilizability for nonlinear control systems the key observation in the paper is that behaviour at infinity can be studied using local methods applied to an auxiliary system

tracking properties and steady state performance of rls adaptive filter algorithms adaptive signal processing algorithms derived from ls least squares cost functions are known to converge extremely fast and have excellent capabilities to track an unknown parameter vector this paper treats analytically and experimentally the steady state operation of rls recursive least squares adaptive filters with exponential windows for stationary and nonstationary inputs a new formula for the estimation noise has been derived involving second and fourth order statistics of the filter input as well as the exponential windowing factor and filter length furthermore it is shown that the adaptation process associated with lag effects depends solely on the exponential weighting parameter in addition the calculation of the excess mean square error due to the lag for an assumed markov channel provides the necessary information about tradeoffs between speed of adaptation and steady state error it is also the basis for comparison to the simple lms algorithm in a simple case of channel identification it is shown that the lms and rls adaptive filters have the same tracking behavior finally in the last part we present new rls restart procedures applied to transversal structures for mitigating the disastrous results of the third source of noise namely finite precision arithmetic

quiver diagrams and signed adaptive filters a graphical technique for exploring the behavior of deterministic discrete time adaptive algorithms excited by periodic inputs is examined the constraint to periodic inputs is convenient the underlying concepts used also apply to stationary inputs drawn from finite alphabets the resulting graph is based on plotting single period parameter trajectories in the parameter error space and is essentially a discrete version of flow diagrams associated with continuous time systems for the least mean square lms adaptive fir filter with the algorithm step size sufficiently small the change of adapted parameters from a particular initial value is given by the total update over the input period with the parameterization frozen at its initial value this results in the familiar elliptical hyperparaboloid surface steepest descent interpretation this method can be extended to the lms adaptive infinite input response iir filter with suitable approximations based on a small step size

mean square performance of a family of affine projection algorithms affine projection algorithms are useful adaptive filters whose main purpose is to speed the convergence of lms type filters most analytical results on affine projection algorithms assume special regression models or gaussian regression data the available analysis also treat different affine projection filters separately this paper provides a unified treatment of the mean square error tracking and transient performances of a family of affine projection algorithms the treatment relies on energy conservation arguments and does not restrict the regressors to specific models or to a gaussian distribution simulation results illustrate the analysis and the derived performance expressions

stability analysis of sampled data systems via free matrix based time dependent discontinuous lyapunov approach in this paper a new time dependent discontinuous lyapunov functional namely free matrix based time dependent discontinuous fmbtdd lyapunov functional is introduced for stability analysis of sampled data systems first a modified free matrix based integral inequality mfmbii is derived based on the existing free matrix based integral inequality 1 and it is applied to develop a stability criterion for sampled data systems and then inspired by mfmbii fmbtdd term is established that leads to efficient stability conditions four numerical examples are given to demonstrate the effectiveness of the proposed methods

adaptive filtering with binary reinforcement recently there has been increased interest in high speed adaptive filtering where the usual stochastic gradient or least mean square lms algorithm is replaced with the simpler algorithm where adaptation is guided only by the polarity of the error signal in this paper the convergence of this binary reinforcement br algorithm is proved under the usual independence assumption and the surprising observation is made that unlike the lms algorithm convergence occurs for any positive value of the step size parameter while the stochastic gradient algorithm attempts to minimize a mean square error cost function the binary reinforcement algorithm in fact attempts to minimize a mean absolute error cost function it is proved for the binary reinforcement algorithm that the tap weight vector converges in distribution to a random vector that is suitably concentrated about the optimal value based on a least mean absolute error cost function for a sufficiently small step size the expected cost of the asymptotic weight vector can be made as close as desired to thc minimum cost attain ed by the optimum weight vector

comparison of the convergence of two algorithms for adaptive fir digital filters the convergence properties of two different algorithms for the updating of the coefficients of an adaptive fir digital filter are investigated and compared with one another these algorithms are the stochastic iteration algorithm and the sign algorithm in this latter algorithm a one bit gradient estimation is used which makes its implementation very simple the convergence is characterized by the residual echo variance after convergence and a parameter that indicates the speed of the convergence it is shown that the convergence of the sign algorithm can always be assured but is much slower than that of the stochastic iteration algorithm if the same variance of the residual echo is to be obtained

stochastic minimization with constant step size asymptotic laws the behavior of the stationary distribution of a markovian process of the form qquad x n 1 a pi s x n a ay n a as a tends to zero is studied the process is a constant step size constant gain constrained stochastic approximation process with pi s being the projection onto the convex set of constraints s if the gains a are held constant the process x n a does not converge to the point of solution but the stationary distribution of x n a converges to the unit mass at the solution if a to 0 properly normalized the stationary distribution converges to a nondegenerated limit this limit depends on the smoothness of the feasible set s in the neighborhood of the solution in particular on the dimension of the largest linear subspace contained in the tangential cone

improved convergence analysis of stochastic gradient adaptive filters using the sign algorithm convergence analysis of stochastic gradient adaptive filters using the sign algorithm is presented in this paper the methods of analysis currently available in literature assume that the input signals to the filter are white this restriction is removed for gaussian signals in our analysis expressions for the second moment of the coefficient vector and the steady state error power are also derived simulation results are presented and the theoretical and empirical curves show a very good match

an improved result on exponential stabilization of sampled data fuzzy systems this brief paper studies the exponential stabilization problem for the takagi sugeno fuzzy systems with a variable sampling different from previous results the gains of fuzzy state feedback controller adopted in this paper are time varying during two consecutive sampling instants which can contribute to the enlargement of the allowable sampling interval by choosing a suitable design parameter to reduce the design conservativeness a novel fuzzy time dependent lyapunov functional ftdlf is put forward to fully exploit the accessible information about the sampling pattern and the fuzzy basis functions moreover a more relaxed constraint condition is presented to ensure the positive definiteness of the ftdlf on sampling intervals by resorting to the novel ftdlf and the relaxed constraint condition new exponential stabilization criteria dependent on and independent of upper bounds on time derivatives of fuzzy basis functions are established by which a larger sampling interval can be achieved one example is offered to demonstrate the validity and superiorities of the obtained new results

tracking analysis of the sign sign algorithm for nonstationary adaptive filtering with gaussian data this correspondence is concerned with the analysis of the sign sign algorithm ssa when used to track a plant with randomly time varying parameters the input of the plant and the plant noise are assumed stationary and gaussian the paper derives expressions of the steady state excess mean square error spl xi the steady state mean square weight misalignment spl eta and the step sizes that minimize each one of them the paper also presents a comparison among the tracking properties of the ssa the sign algorithm sa and the signed regressor algorithm sra it is found that the three algorithms share the features that 1 spl xi does not depend on the eigenvalue spread of the input covariance matrix 2 spl eta depends on the eigenvalue spread and 3 the step size that minimizes spl xi is different from the one that minimizes spl eta the minimum values of spl xi attained by the sa and the sra are equal to each other and they are 1 db less than the one attained by the ssa the ratio of the minimum value of spl eta of the ssa to the one of the sa is found to be dependent on the input eigenvalue spread for equal eigenvalues this ratio is equal to 1 db the minimum value of spl eta of the ssa is found to be 1 db higher than the one of the sra independently of the input eigenvalue spread it is found that an advantage of the ssa with respect to both the sa and the sra is that the two optimum step sizes of the ssa are independent of the mean square plant input and the mean square plant noise

weak convergence and local stability properties of fixed step size recursive algorithms a recursive equation that subsumes several common adaptive filtering algorithms is analyzed for general stochastic inputs and disturbances by relating the motion of the parameter estimate errors to the behavior of an unforced deterministic ordinary differential equation ode the odes describing the motion of several common adaptive filters are examined in some simple settings including the least mean square lms algorithm and all three of its signed variants the signed regressor the signed error and the sign sign algorithms stability and instability results are presented in terms of the eigenvalues of a correlation like matrix this generalizes known results for lms signed regressor lms and signed error lms and gives new stability criteria for the sign sign algorithm the ability of the algorithms to track moving parameterizations can be analyzed in a similar manner by relating the time varying system to a forced ode the asymptotic distribution about the forced ode is an ornstein uhlenbeck process the properties of which can be described in a straightforward manner

robust h stabilization for t s fuzzy systems with time varying delays and memory sampled data control abstract in this paper we address the robust h stabilization for t s fuzzy systems with time varying delays based on memory sampled data control by developing some new terms an improved piecewise lyapunov krasovskii functional lkf is constructed to take full advantage of characteristic about real sampling pattern furthermore some relaxed matrices proposed in the lkf are not necessarily positive definite by using the lkf and free matrix based fmb integral inequality some sufficient criteria are established to ensure the stability of fuzzy systems and reduce the influence of external disturbance with an h norm bound then the memory sampled data controller can be derived by solving a group of linear matrix inequalities lmis with the maximal sampling period finally a numerical example is given to demonstrate the benefits and the superiority of the approach proposed

adaptive filter performance with nonlinearities in the correlation multiplier it is well known that the correlation multiplier used to form the update signal in adaptive transversal filters need not be a true multiplier by using an approximate multiplier such as a sign bit only multiplier implementation can be made less costly in this paper we derive expressions for the convergence rate and misalignment noise of adaptive transversal filters with arbitrary correlation multiplier non linearities the analysis shows any such nonlinearities to always harm performance and quantifies the penalty suffered sign bit only multiplication suffers a significant penalty and power of two multiplication is suggested as a better alternative its performance is almost equal to that of a true multiplier and in a digital implementation at least it is only slightly more difficult to realize than a sign bit only multiplier the theory is developed in a way emphasizing physical understanding and has been verified to be in good agreement with simulation

an improved input delay approach to stabilization of fuzzy systems under variable sampling in this paper we investigate the problem of stabilization for sampled data fuzzy systems under variable sampling a novel lyapunov krasovskii functional lkf is defined to capture the characteristic of sampled data systems and an improved input delay approach is proposed by the use of an appropriate enlargement scheme new stability and stabilization criteria are obtained in terms of linear matrix inequalities lmis compared with the existing results the newly obtained ones contain less conservatism some illustrative examples are given to show the effectiveness of the proposed method and the significant improvement on the existing results

steady state mse performance analysis of mixture approaches to adaptive filtering in this paper we consider mixture approaches that adaptively combine outputs of several parallel running adaptive algorithms these parallel units can be considered as diversity branches that can be exploited to improve the overall performance we study various mixture structures where the final output is constructed as the weighted linear combination of the outputs of several constituent filters although the mixture structure is linear the combination weights can be updated in a highly nonlinear manner to minimize the final estimation error such as in singer and feder 1999 arenas garcia figueiras vidal and sayed 2006 lopes satorius and sayed 2006 bershad bermudez and tourneret 2008 and silva and nascimento 2008 we distinguish mixture approaches that are convex combinations where the linear mixture weights are constrained to be nonnegative and sum up to one singer and feder 1999 arenas garcia figueiras vidal and sayed 2006 affine combinations where the linear mixture weights are constrained to sum up to one bershad bermudez and tourneret 2008 and finally unconstrained linear combinations of constituent filters kozat and singer 2000 we investigate mixture structures with respect to their final mean square error mse and tracking performance in the steady state for stationary and certain nonstationary data respectively we demonstrate that these mixture approaches can greatly improve over the performance of the constituent filters our analysis is also generic such that it can be applied to inhomogeneous mixtures of constituent adaptive branches with possibly different structures adaptation methods or having different filter lengths

asymptotically efficient parameter estimation using quantized output observations this paper studies identification of systems in which only quantized output observations are available an identification algorithm for system gains is introduced that employs empirical measures from multiple sensor thresholds and optimizes their convex combinations strong convergence of the algorithm is first derived the algorithm is then extended to a scenario of system identification with communication constraints in which the sensor output is transmitted through a noisy communication channel and observed after transmission the main results of this paper demonstrate that these algorithms achieve the cramer rao lower bounds asymptotically and hence are asymptotically efficient algorithms furthermore under some mild regularity conditions these optimal algorithms achieve error bounds that approach optimal error bounds of linear sensors when the number of thresholds becomes large these results are further extended to finite impulse response and rational transfer function models when the inputs are designed to be periodic and full rank

improving the tracking capability of adaptive filters via convex combination as is well known hessian based adaptive filters such as the recursive least squares algorithm rls for supervised adaptive filtering or the shalvi weinstein algorithm swa for blind equalization converge much faster than gradient based algorithms such as the least mean squares algorithm lms or the constant modulus algorithm cma however when the problem is tracking a time variant filter the issue is not so clear cut there are environments for which each family presents better performance given this we propose the use of a convex combination of algorithms of different families to obtain an algorithm with superior tracking capability we show the potential of this combination and provide a unified theoretical model for the steady state excess mean square error for convex combinations of gradient and hessian based algorithms assuming a random walk model for the parameter variations the proposed model is valid for algorithms of the same or different families and for supervised lms and rls or blind cma and swa algorithms

a remark on an example by teel hespanha with applications to cascaded systems the properties of a system proposed by teel and hespanha which is globally exponentially stable but with state that can be driven to infinity by an arbitrarily small exponentially decaying disturbance are discussed in detail these are used to propose a family of systems with a similar property and to argue that unstable behavior may be nongeneric and not detected by means of simulations finally sufficient conditions for the existence of unbounded trajectories in cascaded systems are given

decentralized sampled data h fuzzy filter for nonlinear large scale systems this paper presents a decentralized sampled data h fuzzy filter design method for nonlinear large scale systems which are represented by a takagi sugeno t s fuzzy model based on the t s fuzzy model the error system between the nonlinear large scale system and the filter is obtained the discretization process of the error system is accomplished with the exact discrete time approach to eliminate the exact approximate mismatch by using the discrete time lyapunov sense the sufficient condition of the asymptotic stability for the error system is given and a prescribed level of the h norm is ensured to guarantee the h fuzzy filter performance finally numerical examples are given to show the effectiveness of the proposed methods

convergence analysis of an adaptive filter equipped with the sign sign algorithm convergence with probability 1 of the sign sign algorithm with a decreasing step size is proven when it is used in adaptive plant identification the plant input and plant noise are assumed stationary and m dependent the assumptions do not pose any upper bound on the memory m of the plant input and plant noise and hence they allow arbitrarily strong correlation in these signals a main interest of the paper is that it provides a condition of persistent excitation of the algorithm examples satisfying this condition are provided

low complexity variable forgetting factor mechanism for blind adaptive constrained constant modulus algorithms in this paper we propose a novel low complexity variable forgetting factor vff mechanism for blind adaptive constrained constant modulus ccm recursive least squares rls algorithms applied to linear interference suppression in direct sequence code division multiple access ds cdma systems the proposed variable forgetting factor mechanism employs an updated component related to the time average of the constant modulus cm cost function to automatically adjust the forgetting factor in order to ensure good tracking of the interference and the channel convergence and tracking analyses are carried out analytical expressions for predicting the mean squared error of the proposed adaptation technique are obtained simulation results are presented for nonstationary environments and show that the proposed variable forgetting factor mechanism achieves superior performance to previously reported methods at a reduced complexity

analysis of stochastic gradient algorithms for linear regression problems parameter estimation problems that can be formulated as linear regressions are quite common in many applications recursive on line sequential estimation of such parameters can be performed using the recursive least squares rls algorithm or a stochastic gradient version of this algorithm in this paper the convergence properties of the gradient algorithm are analyzed under the assumption that the gain tends to zero the technique is the same as the so called ordinary differential equation approach but the treatment here is self contained and includes a proof of the boundedness of the estimates a main result is that the convergence conditions for the gradient algorithm are the same as those for the recursive least squares algorithm

sampled data robust h control for t s fuzzy systems with time delay and uncertainties abstract this paper investigates sampled data robust h control for t s fuzzy systems with time delay and uncertainties by introducing the free weighting matrices to deal with the integral items and converting the coupling time varying matrix inequalities into a group of decoupling matrix inequalities an innovative delay dependent stabilization criterion is first presented then a sampled data h controller is designed to stabilize the t s fuzzy system to achieve the prescribed disturbance attenuation level which is dependent on both the size of state delay and upper bound of allowable sampling period since 1 the sawtooth structure of delay is fully considered 2 neither model transformation nor bounding techniques are employed in deriving the delay dependent results and 3 none of the integral items are arbitrarily ignored and magnified the less conservative results can be expected a practical application example is given to demonstrate the effectiveness of the proposed methods

characterizing persistent excitation for the sign sign equation error identifier abstract the sign sign ss algorithm is a computationally efficient adaptive identifier often used in signal processing tasks it is obtained by introducing signum functions on both the regressor and the prediction error multiplicands in the update kernel of the well known lms algorithm this paper gives a deterministic persistent excitation condition on the regressor sequence which guarantees ss convergence it also gives conditions under which ss may diverge and discusses how these persistent excitation conditions may be verified through a finite amount of computations

convergence of an adaptive linear estimation algorithm in this work we prove the almost sure convergence of an adaptive linear estimator governed by a stochastic gradient algorithm with decreasing step size in the presence of correlated observations two complementary contributions are added to the famous 1977 ljung theorem first we drop the condition of nondivergence of the algorithm assumed by ljung while that condition can be ensured by adding a barrier the convergence of the suitably bounded algorithm itself is not established even on the basis of ljung theorem here the barrier problem is overcome by proving that it is not necessary for the convergence our second contribution is to generalize the model describing the correlated observations no state space model is used and no linear relationship between the observations and the signal to be estimated needs to be assumed instead we use a decreasing covariance model that agrees with a very wide class of practical applications

a unified approach to the steady state and tracking analyses of adaptive filters most adaptive filters are inherently nonlinear and time variant systems the nonlinearities in the update equations tend to lead to difficulties in the study of their steady state performance as a limiting case of their transient performance this paper develops a unified approach to the steady state and tracking analyses of adaptive algorithms that bypasses many of these difficulties the approach is based on the study of the energy flow through each iteration of an adaptive filter and it relies on a fundamental error variance relation

joint identification of plant rational models and noise distribution functions using binary valued observations system identification of plants with binary valued output observations is of importance in understanding modeling capability and limitations for systems with limited sensor information establishing relationships between communication resource limitations and identification complexity and studying sensor networks this paper resolves two issues arising in such system identification problems first regression structures for identifying a rational model contain non smooth nonlinearities leading to a difficult nonlinear filtering problem by introducing a two step identification procedure that employs periodic signals empirical measures and identifiability features rational models can be identified without resorting to complicated nonlinear searching algorithms second by formulating a joint identification problem we are able to accommodate scenarios in which noise distribution functions are unknown convergence of parameter estimates is established recursive algorithms for joint identification and their key properties are further developed

stability and stabilization for singular switching semi markovian jump systems with generally uncertain transition rates this paper studies the problems of stability and stabilization for a class of singular switching semi markovian jump systems the general transition rates in the semi markov process cover completely unknown and uncertain bounded as two special cases first sufficient conditions are developed to ensure the unforced system to be regular impulse free and exponentially mean square stable then by proposing a state feedback controller sufficient conditions in terms of strict linear matrix inequalities are derived to guarantee the closed loop system to be stochastically stabilziable finally a numerical example is provided to show the effectiveness of the obtained results

sampled data synchronization of chaotic lur e systems via input delay dependent free matrix zero equality approach abstract this paper focuses on the sampled data synchronization problem of chaotic lur e systems clss by using sampled output of the systems with variable sampling rates one novel approach input delay dependent free matrix zero equality iddfmze approach is proposed for the first time the iddfmze approach can not only fully capture the available information on the actual sampling pattern but deploy more system information at the dynamic partitioning point a new lyapunov krasovskii functional lkf with some new terms is constructed which can use more information of the activation function at the dynamic partitioning point based on the presented iddfmze approach and the constructed lkf developed synchronization criterion is obtained in the form of linear matrix inequalities lmis the desired sampled data controller is designed under larger sampling period finally the superiority of proposed results is shown by two numerical examples

on the statistical efficiency of the lms algorithm with nonstationary inputs a fundamental relationship exists between the quality of an adaptive solution and the amount of data used in obtaining it quality is defined here in terms of misadjustment the ratio of the excess mean square error mse in an adaptive solution to the minimum possible mse the higher the misadjustment the lower the quality is the quality of the exact least squares solution is compared with the quality of the solutions obtained by the orthogonalized and the conventional least mean square lms algorithms with stationary and nonstationary input data when adapting with noisy observations a filter trained with a finite data sample using an exact least squares algorithms will have a misadjustment given by m frac n n frac number of weights number of training samples if the same adaptive filter were trained with a steady flow of data using an ideal orthogonalized lms algorithm the misadjustment would be m frac n 4 tau mse frac number of weights number of training samples thus for a given time constant tau mse of the learning process the ideal orthogonalized lms algorithm will have about as low a misadjustment as can be achieved since this algorithm performs essentially as an exact least squares algorithm with exponential data weighting it is well known that when rapid convergence with stationary data is required exact least squares algorithms can in certain cases outperform the conventional widrow hoff lms algorithm it is shown here however that for an important class of nonstationary problems the misadjustment of conventional lms is the same as that of orthogonalized lms which in the stationary case is shown to perform essentially as an exact least squares algorithm

a tight upper bound of the average absolute error in a constant step size sign algorithm a direct performance index of the adaptive filtering sign algorithm sa is the average absolute error aae at the output of the filter adopting this performance index an easy analysis of sa is achieved under a weak assumption it is proved for both deterministic and random inputs to the filter that the aae has a tight upper bound that exceeds the minimum aae by half the product of the step size and power of the filter input the assumption used is existence of average squared and average absolute values of filter input signals a practical interest of the result is that it provides a formula for the biggest step size as a function of tolerable adaptation noise to desired signal ratio

convergence analysis of the sign algorithm without the independence and gaussian assumptions the paper is concerned with rigorous convergence analysis of the sign algorithm sa in the context of adaptive plant identification asymptotic time averaged convergence for the mean absolute weight misalignment is proved for all values of the algorithm step size and initial weight vector the paper has three main contributions with respect to available convergence results of the sa the first is the deletion of the gaussian assumption which is important when covering the case of discrete valued data no assumption about the distribution of the regressor sequence is used except for the usual assumption of positive definite covariance matrix the assumptions used about the noise allow nonexistence unboundedness and vanishing of the noise probability density function for arguments strictly different from zero the second contribution is the deletion of the assumption of independent successive regressors this deletion is important since in applications two successive regressors usually share all their components except two hence they are strongly dependent even for white plant input the case of colored noise is also analyzed finally the third contribution is the extension of the above results to the nonstationary case the used assumptions allow nonstationarity of the plant input plant noise and plant parameters

new result on reliable h performance state estimation for memory static neural networks with stochastic sampled data communication abstract this work examines the h performance state estimation problem for memory static neural networks msnns with reliable state feedback stochastic sampled data control ssdc the purpose of presenting this study is to determine whether the h performance and criteria with less conservatism for stability could be gained by ssdc for msnns or not firstly we suppose that the sampling interval values follow bernoulli distribution and the probability of occurrence are teadfast constant then generalize it to a more universal form secondly on basis of considering the sampling input delay and its sawtooth structure characteristics a modified augmented lyapunov krasovskii functional lkf is constructed on account of the free matrix based integral inequality fmbii together with generalized free weighting matrix gfwm inequality which can reduce the conservatism of h performance criteria thirdly the expected estimator gain matrix can be designed in the light of the solution to linear matrix inequalities lmis finally an numerical example is given to check the superiority of the proposed msnns control design technique

event triggered sampling control for stability and stabilization of memristive neural networks with communication delays this paper investigates the global asymptotic stability and stabilization of memristive neural networks mnns with communication delays via event triggered sampling control first based on the novel approach in lemma 1 the concerned mnns are converted into traditional neural networks with uncertain parameters next a discrete event triggered sampling control scheme which only needs supervision of the system state at discrete instants is designed for mnns for the first time thanks to this controller the number of control updates could greatly reduce then by getting utmost out of the usable information on the actual sampling pattern a newly augmented lyapunov krasovskii functional lkf is constructed to formulate stability and stabilization criteria it should be mentioned that the lkf is positive definite only at endpoints of each subinterval of the holding intervals but not necessarily positive definite inside the holding intervals finally the feasibility and effectiveness of the proposed results are tested by two numerical examples

transient and tracking performance bounds of the sign sign algorithm the paper provides a rigorous tracking analysis of the sign sign algorithm when used in the identification of a time varying plant with a white gaussian input the plant parameters vary according to a random walk model the assumptions allow nonstationarity of the plant input plant noise and increments of the plant parameters upper bounds are derived for the long term averages of the mean of the weight misalignment norm mean absolute error mean square weight misalignment and mean square error these bounds hold for all values of the algorithm step size all initial filter weight settings and all degrees of nonstationarity of the plant input plant noise and plant parameter increments lower bounds of the mean square weight misalignment and mean square error are also derived the step sizes that minimize the above bounds are derived a transient analysis of the algorithm is done in the case of a time invariant plant a tight lower bound of the convergence time is derived the above analytical results are supported by computer simulations

comparison of rls lms and sign algorithms for tracking randomly time varying channels the performance of adaptive fir filters governed by the recursive least squares rls algorithm the least mean square lms algorithm and the sign algorithm sa are compared when the optimal filtering vector is randomly time varying the comparison is done in terms of the steady state excess mean square estimation error spl xi and the steady state mean square weight deviation spl eta it is shown that spl xi does not depend on the spread of eigenvalues of the input covariance matrix r in the cases of the lms algorithm and the sa while it does in the case of the rls algorithm in the three algorithms spl eta is found to be increasing with the eigenvalue spread the value of the adaptation parameter that minimizes spl xi is different from the one that minimizes spl eta it is shown that the minimum values of spl xi and spl eta attained by the rls algorithm are equal to the ones attained by the lms algorithm in any one of the three following cases 1 if r has equal eigenvalues 2 if the fluctuations of the individual elements of the optimal vector are mutually uncorrelated and have the same mean square value or 3 if r is diagonal and the fluctuations of the individual elements of the optimal vector have the same mean square value conditions that make the values of spl xi and spl eta of the lms algorithm smaller or greater than the ones of the rls algorithm are derived for gaussian input data the minimum values of spl xi and spl eta attained by the sa are found to exceed the ones attained by the lms algorithm by 1 db independently of r and the mutual correlation between the elements of the optimal vector

gradient based variable forgetting factor rls algorithm in time varying environments in this paper a new control mechanism for the variable forgetting factor vff of the recursive least square rls adaptive algorithm is presented the control algorithm is basically a gradient based method of which the gradient is derived from an improved mean square error analysis of rls the new mean square error analysis exploits the correlation of the inverse of the correlation matrix with itself that yields improved theoretical results especially in the transient and steady state mean square error it is shown that the theoretical analysis is close to simulation results for different forgetting factors and different model orders the analysis yields a dynamic equation of mean square error that can be used to derive a dynamic equation of the gradient of mean square error to control the forgetting factor the dynamic equation can produce a positive gradient when the error is large and a negative gradient when the error is in the steady state compared with other variable forgetting factor algorithms the new control algorithm gives fast tracking and small mean square model error for different signal to noise ratios snrs

sign sign lms convergence with independent stochastic inputs the sign sign adaptive least mean square lms identifier filter is a computationally efficient variant of the lms identifier filter it involves the introduction of signum functions in the traditional lms update term consideration is given to global convergence of parameter estimates offered by this algorithm to a ball with radius proportional to the algorithm step size for white input sequences specially from gaussian and uniform distributions

almost sure convergence of a decreasing gain sign algorithm for adaptive filtering the convergence analysis of a sign algorithm with decreasing gain when governing the weights of an adaptive filter is presented the analysis is done in a noiseless case a rigorous proof of almost sure convergence to the optimal filter is attained under a weak ergodicity assumption that includes the case important in practice of correlated observations

optimization of adaptive identification for time varying filters adaptive identification in a time varying context is studied when controlled by the lms algorithm with constant gain under the assumption of correlated successive input vectors it is well known by experience that the tracking mean square error mse epsilon mu results from the tradeoff between the gradient part which is increasing and the lag contribution which is decreasing in this note we clarify the relative roles of the gradient and lag errors by proving their decoupled character this property relies upon independence between the additive noise at the output of the plant to be identified and the information vector at the plant input convergence of the mse is established rather than assumed quantitative evaluations of upper and lower bounds allow an approximate optimization of the gain in two important cases the optimum is exact one of these cases is slow variations it is defined in a quantitative manner thanks to the ratio of the variation noise to the output additive noise

a novel approach to stability and stabilization of fuzzy sampled data markovian chaotic systems abstract this paper studies the problems of stability and stabilization of a class of markovian chaotic systems via fuzzy sampled data control different from some existing results the markovian jumping with general uncertain transition rates is considered for such systems a novel approach i e input delay dependent vector approach is proposed for the first time based on this approach an input delay dependent lyapunov krasovskii functional lkf with cubic function of input delay is successfully constructed and the convex combination technique is used with ease to derive stability and stabilization criteria compared with some existing stability and stabilization criteria for chaotic systems our results are not only less conservative but also with reduced calculation complexity finally the superiorities of proposed results are illustrated by two numerical examples

further refinements in stability conditions for time varying delay systems abstract this paper addresses the problem of assessing the stability of linear time invariant lti systems with time varying delay the first contribution is a new stability criterion specified as a negativity condition for a quadratic function parameterized by the delay this result mainly follows from an augmented affine parameter dependent lyapunov krasovskii functional which in turn takes advantages of convexity properties then as a second contribution we invoke a result from robust control literature to show how the proposed stability condition can be checked exactly in terms of linear matrix inequality lmi conditions the improvements obtained by the proposed refinements are illustrated via numerical examples drawn from the literature

stability and stabilization of takagi sugeno fuzzy systems via sampled data and state quantized controller in this paper we investigate the problem of stability and stabilization for sampled data fuzzy systems with state quantization by using an input delay approach the sampled data fuzzy systems with state quantization are transformed into a continuous time system with a delay in the state the transformed system contains nondifferentiable time varying state delay based on some integral techniques some new stability and stabilization criteria are first proposed by a modified lyapunov functional furthermore in the case of no quantization some new stability and stabilization criteria are also obtained it is shown that the new stability and stabilization criteria can provide a larger upper bound of the sampling interval than some existing ones in the literature two simulation examples are given to show the effectiveness of the proposed design method

transient analysis of adaptive filters with error nonlinearities the paper develops a unified approach to the transient analysis of adaptive filters with error nonlinearities in addition to deriving earlier results in a unified manner the approach also leads to new performance results without restricting the regression data to being gaussian or white the framework is based on energy conservation arguments and avoids the need for explicit recursions for the covariance matrix of the weight error vector

krylov proportionate adaptive filtering techniques not limited to sparse systems this paper proposes a novel adaptive filtering scheme named the krylov proportionate normalized least mean square kpnlms algorithm kpnlms exploits the benefits i e fast convergence for sparse unknown systems of the proportionate nlms algorithm but its applications are not limited to sparse unknown systems a set of orthonormal basis vectors is generated from a certain krylov sequence it is proven that the unknown system is sparse with respect to the basis vectors in case of fairly uncorrelated input data different adaptation gain is allocated to a coefficient of each basis vector and the gain is roughly proportional to the absolute value of the corresponding coefficient of the current estimate kpnlms enjoys i fast convergence ii linear complexity per iteration and iii no use of any a priori information numerical examples demonstrate significant advantages of the proposed scheme over the reduced rank method based on the multistage wiener filter mwf and the transform domain adaptive filter tdaf both in noisy and silent situations

slow state variables feedback stabilization for semi markov jump systems with singular perturbations the slow state variables feedback stabilization problem for semi markov jump discrete time systems with slow sampling singular perturbations is discussed in this work a new fairly comprehensive system model semi markov jump system with singular perturbations which is more general than markov jump model is employed to describe the phenomena of random abrupt changes in structure and parameters of the systems based on a slow state variables feedback control scheme a novel technique to design the desired controller is presented and the allowed maximum of singular perturbation parameter can be calculated with the help of the discrete time semi markov kernel approach some sojourn time dependent and less conservative sufficient conditions are established via a novel matrix decoupling technique to ensure the solvability of the problem to be addressed finally an illustrative example is given to show the superiority and usefulness of the proposed method

an invariant measure approach to the convergence of stochastic approximations with state dependent noise a new method is presented for quickly getting the ode ordinary differential equation associated with the asymptotic properties of the stochastic approximation x n 1 x n a n f x n zeta n or the projected algorithm for the constrained problem either a n to 0 or a n can be constant in which case the analysis is on the sequence obtained when a to 0 the method requires that x n zeta n 1 be markov with a feller transition function but little else the simplest result requires that if x n equiv x the corresponding noise process zeta n x n geqq 0 have a unique invariant measure but the nonunique case can also be treated no mixing condition is required nor the construction of averaged test functions and f cdot cdot need not be continuous a detailed analysis of the way that zeta n varies with x n is not required for the class of sequences treated the conditions seem easier to verify than for other methods there

stability and stabilization of t s fuzzy systems with time varying delays via augmented lyapunov krasovskii functionals this paper is concerned with the problem of stability and stabilization for takagi sugeno t s fuzzy systems with time varying delays by constructing a suitable lyapunov krasovskii functional sufficient conditions for ensuring the asymptotic stability and stabilization of the concerned fuzzy systems have been derived within the framework of linear matrix inequalities lmis the advantage and effectiveness of the proposed criteria will be shown through the comparison of maximum delay bounds with some results obtained by recently published papers via four numerical examples

robust h control of t s fuzzy systems with input time varying delays a delay partitioning method abstract addressed in this paper is the robust h control issue of t s fuzzy systems with input time varying delays by means of the delay partitioning method the delay interval is partitioned into multiple unequal subintervals whose lengths satisfy a geometric sequence on this basis a modified lyapunov krasovskii functional is presented to analyze asymptotic stability of the open loop system then a state feedback controller that ensures a prescribed h performance level for the closed loop system is proposed in linear matrix inequality format finally two numerical examples are given to illustrate the effectiveness and advantages of the obtained results

a robust direct adaptive controller this paper proposes a new direct adaptive control algorithm which is robust with respect to additive and multiplicative plant unmodeled dynamics the algorithm is designed based on the reduced order plant which is assumed to be minimum phase and of known order and relative degree but is analyzed with respect to the overall plant which due to the unmodeled dynamics may be nonminimum phase and of unknown order and relative degree it is shown that if the unmodeled dynamics are sufficiently small in the low frequency range then the algorithm guarantees boundedness of all signals in the adaptive loop and small residual tracking errors for any bounded initial conditions in the absence of unmodeled dynamics the residual tracking error is shown to be zero

analysis of a sign algorithm with delayed data an adaptive filter whose weights are adapted using a sign algorithm with a delayed error signal is analyzed for stationary environments it is proved that the excess average absolute estimation error is bounded for all values of the error signal delay and the algorithm step size for the nonstationary case when the optimal filter weights are time varying the optimum step size which minimizes the excess average absolute error is derived it is shown that the optimum step size does not depend on the additive noise power the analytical results are supported by computer simulations

a tight small gain theorem for not necessarily iss systems a new version of the small gain theorem is presented for nonlinear finite dimensional systems the result provides conditions for global asymptotic stability under relaxed assumptions in particular the two interconnected subsystems need not be input to state stable in open loop

homogeneous approximation recursive observer design and output feedback we introduce two new tools that can be useful in nonlinear observer and output feedback design the first one is a simple extension of the notion of homogeneous approximation to make it valid both at the origin and at infinity homogeneity in the bi limit exploiting this extension we give several results concerning stability and robustness for a homogeneous in the bi limit vector field the second tool is a new recursive observer design procedure for a chain of integrator combining these two tools we propose a new global asymptotic stabilization result by output feedback for feedback and feedforward systems

reduced complexity self tuning adaptive algorithms in application to channel estimation in this letter reduced complexity self tuning algorithms are proposed using simplified parameter updating procedures convergence analysis based on the independence assumption and the ordinary differential equation ode method shows that the tuning parameter of the proposed algorithm attains the same limit as the conventional self tuning adaptive algorithm simulations are carried out for channel estimation to support the analysis and performance of the proposed algorithms

applications of a kushner and clark lemma to general classes of stochastic algorithms two general classes of stochastic algorithms are considered including algorithms considered by ljung as well as algorithms of the form theta n 1 theta n gamma n 1 v n 1 theta n z where z is a stationary ergodic process it is shown how one can apply a lemma of kushner and clark to obtain properties of these algorithms this is done by using in particular martingale arguments in the generalized ljung case in these various situations the convergence is obtained by the method of the associated ordinary differential equation under the classical boundedness assumptions in the case of linear algorithms the boundedness assumptions are dropped

further results on stabilization of chaotic systems based on fuzzy memory sampled data control this note investigates sampled data control for chaotic systems a memory sampled data control scheme that involves a constant signal transmission delay is employed for the first time to tackle the stabilization problem for takagi sugeno fuzzy systems the advantage of the constructed lyapunov functional lies in the fact that it is neither necessarily positive on sampling intervals nor necessarily continuous at sampling instants by introducing a modified lyapunov functional that involves the state of a constant signal transmission delay a delay dependent stability criterion is derived so that the closed loop system is asymptotically stable the desired sampled data controller can be achieved by solving a set of linear matrix inequalities compared with the existing results a larger sampling period is obtained by this new approach a simulation example is presented to illustrate the effectiveness and conservatism reduction of the proposed scheme

convergence of the signed output error adaptive identifier this paper considers an adaptive output error identifier with a signum function in its update kernel it is shown that the classical strict positive real spr condition required for the convergence of traditional adaptive identifiers does not suffice for the convergence of this signed identifier instead what is needed is a stronger operator condition called the strict dominant passive sdp condition we give an analog of the kalman yakubovic popov lemma for the sdp conditions and using it give a convergence proof under the assumptions of persistent excitation and sdp

reciprocally convex approach to stability of systems with time varying delays whereas the upper bound lemma for matrix cross product introduced by park 1999 and modified by moon park kwon and lee 2001 plays a key role in guiding various delay dependent criteria for delayed systems the jensen inequality has become an alternative as a way of reducing the number of decision variables it directly relaxes the integral term of quadratic quantities into the quadratic term of the integral quantities resulting in a linear combination of positive functions weighted by the inverses of convex parameters this paper suggests the lower bound lemma for such a combination which achieves performance behavior identical to approaches based on the integral inequality lemma but with much less decision variables comparable to those based on the jensen inequality lemma

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

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

analytical performance of the lms algorithm on the estimation of wide sense stationary channels the performance of the least mean square lms algorithm on the estimation of time varying channels is analytically evaluated using the estimation error correlation matrix the mean square weight error mswe and the mean square estimation error mse as parameters expressions for those parameters are obtained from a set of hypotheses usually adopted in the communication systems context the channel is modeled as a wide sense stationary wss discrete time stochastic field with known autocorrelation the expressions for the steady state mswe and mse are particularized for the class of wss channel models and an original analysis of the optimum lms step size parameter for usual channel models is addressed for the sake of comparison with other works the analytical step size optimization for random walk models is also considered several estimates of mswe curves obtained by computer simulation are compared with analytical results for validation purposes a very good agreement between simulated and analytical results for both the mswe expressions and the optimum value of the lms step size parameter is shown

convergence analysis of adaptive filtering algorithms with singular data covariance matrix the paper provides a rigorous analysis of the behavior of adaptive filtering algorithms when the covariance matrix of the filter input is singular the analysis is done in the context of adaptive plant identification the considered algorithms are lms rls sign sa and signed regressor sra algorithms both the signal and weight behavior of the algorithms are considered the signal behavior is evaluated in terms of the moments of the excess output error of the filter the weight behavior is evaluated in terms of the moments of the filter weight misalignment vector it is found that the rls and sra diverge when the input covariance matrix is singular the steady state signal behavior of the lms and sa can be made arbitrarily fine by using sufficiently small step sizes of the algorithms indeed the long term average of the mean square excess error of the lms is proportional to the algorithm step size the long term average of the mean absolute excess error of the sa is proportional to the square root of the algorithm step size on the other hand the steady state weight behavior of both the lms and sa have biases that depend on the weight initialization the analytical results of the paper are supported by simulations

shrinkage linear and widely linear complex valued least mean squares algorithms for adaptive beamforming in this paper shrinkage linear complex valued least mean squares sl clms and shrinkage widely linear complex valued least mean squares swl clms algorithms are devised for adaptive beamforming by exploiting the relationship between the noise free a posteriori and a priori error signals the sl clms method is able to provide a variable step size to update the weight vector for the adaptive beamformer significantly enhancing the convergence speed and decreasing the steady state misadjustment on the other hand besides adopting a variable step size determined by minimizing the square of the augmented noise free a posteriori errors the swl clms approach exploits the noncircular properties of the signal of interest which considerably improves the steady state performance simulation results are presented to illustrate their superiority over the clms complex valued normalized lms variable step size recursive least squares rls algorithms and their corresponding widely linear based schemes additionally our proposed algorithms are more computationally efficient than the rls solutions though they may have a slightly slower convergence rate

sharf convergence properties a class of stable algorithms for adapting infinite impulse response iir digital filters based on the concepts of nonlinear stability theory prominent in the control literature is emerging while this class of adaptive filters offers much promise in practical applications little has been done toward providing a characterization that would guide selection of design parameters such as adaptation constants and error smoothing coefficients this paper focuses on the simplest well behaved member of this class of adaptive recursive filters sharf progression from a local linearization of the nonlinear parameter estimate convergence behavior through an idealized eigenvalue eigenvector analysis of the parameter estimate time varying recursion to lyapunov function establishment for the full output and parameter error system reveals the exponential local nongradient descent convergence character of sharf and provides initial insight into the effects of adaptation constants and error smoothing coefficients on these characteristics

adaptive tracking of linear time variant systems by extended rls algorithms we exploit the one to one correspondences between the recursive least squares rls and kalman variables to formulate extended forms of the rls algorithm two particular forms of the extended rls algorithm are considered one pertaining to a system identification problem and the other pertaining to the tracking of a chirped sinusoid in additive noise for both of these applications experiments are presented that demonstrate the tracking superiority of the extended rls algorithms compared with the standard rls and least mean squares lms algorithms

convergence of an adaptive filter with signed filtered error the need for high speed adaptive filters has prompted the search for alternatives to the popular lms algorithm one modification is the replacement of the prediction error term in the lms update kernel by its signum function at the same time in noise and echo cancellation problems reduced residual noise variance often requires error filtering in the paper the authors consider the situation where the signum function of such a filtered error appears in the update kernel without the signum function the error model of the filtered algorithms is similar to those in output error identifiers and a strictly positive real spr error filter suffices for convergence the authors analyze the signed filtered error algorithm to show that because of the sign operator a spr error filter no longer guarantees convergence a new concept in systems theory namely that of strictly dominant passivity sdp is introduced instead a system is sdp if the average product of its output and the signum of the input is positive the authors conduct a systems theoretic investigation of the sdp condition and prove that the algorithm under study is convergent if the inverse of the error filter is sdp

optimum step size of sign algorithm for nonstationary adaptive filtering the adaptive filtering sign algorithm is analyzed in the case of nonstationary and correlated data the excess time average absolute estimation error eaae is used as a performance index of the algorithm it is proved that the eaae is the sum of two terms one proportional to the algorithm step size mu and the other proportional to 1 mu the optimum step size that minimizes the eaae is found equal to square root q p with q being the power of fluctuations of the optimal filter weights and p being the power of filter input this formula is derived under the assumption that increments of optimal filter weights are white computer simulations are used to examine the validity of the formula for other models of variation of the optimal filter simulation results show a good fit with the obtained optimum step size formula

identification input design for consistent parameter estimation of linear systems with binary valued output observations input design is of essential importance in system identification for providing sufficient probing capabilities to guarantee convergence of parameter estimates to their true values this paper presents conditions on input signals that characterize their probing richness for strongly consistent parameter estimation of linear systems with binary valued output observations necessary and sufficient conditions on periodic signals are derived for sufficient richness these conditions are further studied under different system configurations including open loop and feedback systems and different scenarios of noises including actuator noise input measurement noise and output measurement noise in addition to system parameter estimation essential properties of identifiability and input conditions are also derived when sensor thresholds or noise distribution functions are unknown the findings of this paper provide a foundation to study identification of systems that either use binary valued or quantized sensors or involve communication channels which mandate quantization of signals