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Adaptive filtering algorithms are investigated when system models are subject to model structure errors and regressor signal perturbations. System models for practical applications are often approximations of high-order or nonlinear systems, introducing model structure uncertainties. Measurement and actuation errors cause signal perturbations, which in turn lead to uncertainties in regressors of adaptive filtering algorithms. Employing ordinary differential equation (ODE) methodologies, we show that convergence properties and estimation bias can be characterized by certain differential inclusions. Conditions to ensure algorithm convergence and bounds on estimation bias are derived. These findings yield better understanding of the robustness of adaptive algorithms against structural and signal uncertainties.

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This paper demonstrates high-performance adaptive control for a laser-beam steering system, which exhibits high-order unknown jitter dynamics. The adaptive controller, which is based on a recursive least-squares finite-impulse-response lattice filter, has the distinguishing feature that variable and high-order adaptive filters can be realized in the real-time implementation. Varying the order of the adaptive controller produces both fast adaptation and optimal steady-state performance in the experiment, without the large transients often produced by fixed-order recursive least-squares adaptive controllers. The steady-state performance of the high-order adaptive controller approximates closely the theoretically achievable minimum-variance steady-state performance, which is derived from the identified plant and jitter dynamics. Experimental results also illustrate the capability of the adaptive controller to adapt rapidly to changing jitter characteristics.

Motivated by the developments on iterate averaging of recursive stochastic approximation algorithms and asymptotic analysis of sign-error algorithms for adaptive filtering, this work develops two-stage sign algorithms for adaptive filtering. The proposed algorithms are based on constructions of a sequence of estimates using large step sizes followed by iterate averaging. Our main effort is devoted to improving the performance of the algorithms by establishing asymptotic normality of a suitably scaled sequence of the estimation errors. The asymptotic covariance is calculated and shown to be the smallest possible. Hence, the asymptotic efficiency or asymptotic optimality is obtained. Then variants of the algorithm including sign-regressor procedures and constant-step algorithms are studied. The minimal window width of averaging is also dealt with. Finally, iterate-averaging algorithms for blind multiuser detection in direct sequence/code-division multiple-access (DS/CDMA) systems are proposed and developed, and numerical examples are examined.

Bounds for Coefficients The Model with an Error in the Equation The Model with No Error in the Equation Instrumental Variable Estimation Modifications to Improve Moment Properties

We present an adaptive control scheme for laser-beam steer- ing by a two-axis microelectromechanical systems MEMS fast steering mirror. Disturbances in the laser beam are rejected by a -synthesis feedback controller augmented by the adaptive control loop, which de- termines control gains that are optimal for the current disturbance acting on the laser beam. The variable-order adaptive controller is based on an adaptive lattice filter that implicitly identifies the disturbance statistics from real-time sensor data. Experimental results demonstrate that the adaptive controller significantly extends the disturbance-rejection band- width achieved by the feedback controller alone. The experimental re- sults also illustrate the value of the variable-order capability of the adap- tive controller. © 2006 Society of Photo-Optical Instrumentation

This paper presents an adaptive control scheme for laser-beam steering by a two-axis MEMS fast steering mirror. Disturbances in the laser beam are rejected by a µ- synthesis feedback controller augmented by the adaptive con- trol loop, which determines control gains that are optimal for the current disturbance acting on the laser beam. The variable- order adaptive controller is based on an adaptive lattice filter that implicitly identifies the disturbance statistics from real-time sensor data. Experimental results demonstrate that the adaptive controller significantly extends the disturbance- rejection bandwidth achieved by the feedback controller alone. The experimental results also illustrate the value of the variable-order capability of the adaptive controller. This paper employs a recursive least-squares lattice filter in the adaptive controller, and introduces a variable-order adaptive control scheme that exploits the order-recursive structure of the lattice filter. The capability to vary the order of the filter in the adaptive controller is important because optimal gains can be identified faster for lower-order filters while higher-order filters are required for optimal steady- state rejection of broadband disturbance. Thus, low filter orders can be used initially for fast adaptation without undesirable transient responses, and the filter order can be increased incrementally to achieve optimal steady-state jitter rejection. Section II describes the experimental hardware and con- figuration. Section III describes the system identification of the mirror dynamics and transfer functions require for control system design. Then Section IV describes the design of the control system, which consists of a linear time- invariant (LTI) feedback control loop augmented by the adaptive control loop. Experimental results for two sets of experiments, each with multiple jitter bandwidths, are presented in Section V.

The Wright-Fisher Model Applications of the Diffusion Approximation Genotypic-Frequency Models Infinitely-Many-Allele Models Problems Notes

The use of adaptive algorithms is now very widespread across such varied applications as system identification, adaptive control, transmission systems, adaptive filtering for signal processing, and several aspects of pattern recognition. Numerous, very different examples of applications are given in the text. The success of adaptive algorithms has inspired an abundance of literature, and more recently a number of significant works such as the books of Ljung and Soderström (1983) and of Goodwin and Sin (1984).

A general introduction to adaptive systems is presented with emphasis
given to: open and closed loop adaptation; the adaptive linear combiner;
and alternative expressions of the gradient. Some theoretical
considerations in adaptive processing of stationary signals are
discussed including: the properties of the quadratic performance
surface; methods for searching the performance surface; and the effect
of gradient estimation on adaptation. Finally, the basic adaptive
algorithm and structures are described, with attention given to LMS
algorithms; the z-transform; the sequential regression algorithm; and
adaptive recursive filters. Some of the applications of adaptive signal
processing are also considered, including adaptive control systems;
adaptive interference canceling; and adaptive beam forming.

Overview.- System Settings.- Stochastic Methods for Linear Systems.- Empirical-Measure-Based Identification: Binary-Valued Observations.- Estimation Error Bounds: Including Unmodeled Dynamics.- Rational Systems.- Quantized Identification and Asymptotic Efficiency.- Input Design for Identification in Connected Systems.- Identification of Sensor Thresholds and Noise Distribution Functions.- Deterministic Methods for Linear Systems.- Worst-Case Identification under Binary-Valued Observations.- Worst-Case Identification Using Quantized Observations.- Identification of Nonlinear and Switching Systems.- Identification of Wiener Systems with Binary-Valued Observations.- Identification of Hammerstein Systems with Quantized Observations.- Systems with Markovian Parameters.- Complexity Analysis.- Space and Time Complexities, Threshold Selection, Adaptation.- Impact of Communication Channels on System Identification.

A new algorithm, which is a variant of the sign algorithm, is proposed for the adaptive adjustment of an FIR digital filter with an aim of improving the original convergence characteristics, yet retaining the advantage of hardware simplicity. Based on a recently proposed theory for the sign algorithm, a practical design method is derived for the new algorithm, and it is shown by computer simulation that the new algorithm in fact performs significantly better than the original algorithm.

System identification is studied in which the system output is quantized, transmitted through a digital communication channel, and observed afterwards. This paper explores strong convergence, efficiency, and complexity of identification algorithms under colored noise and dependent communication channels. It first presents algorithms for certain core identification problems using quantized observations on the basis of empirical measures and nonlinear mappings. Strong consistency (with-probability-one convergence) is established under ??-mixing noises. Furthermore, with pre-quantization signal processing, it is shown that certain modified algorithms can achieve asymptotic efficiency under correlated noises. To improve convergence speeds, quantization threshold adaptation algorithms are introduced. These results are then used to study the impact of communication channels on system identification under dependent channels. The concept of fisher information ratio is introduced to characterize such impact. It is shown that the fisher information ratio can be calculated from certain channel characteristic matrices. The relationship between the fisher information ratio and Shannon's channel capacity is discussed from the angle of time and space information. The methods of identification input designs that link general system parameters to core identification problems are reviewed.

In this paper the Robbins-Monro (RM) algorithm with step-size and truncated at randomly varying bounds is considered under mild conditions imposed on the regression function. It is proved that for its a.s. convergence to the zero of a regression function the necessary and sufficient condition is where ξi denotes the measurement error. It is also shown that the algorithm is robust with respect to the measurement error in the sense that the estimation error for the sought-for zero is bounded by a function g(ε) such that

The sections in this article are1The Problem2Background and Literature3Outline4Displaying the Basic Ideas: Arx Models and the Linear Least Squares Method5Model Structures I: Linear Models6Model Structures Ii: Nonlinear Black-Box Models7General Parameter Estimation Techniques8Special Estimation Techniques for Linear Black-Box Models9Data Quality10Model Validation and Model Selection11Back to Data: The Practical Side of Identification

This work is concerned with robustness, convergence, and stability of adaptive filtering (AF) type algorithms in the presence of model mismatch. The algorithms under consideration are recursive and have inherent multiscale structure. They can be considered as dynamic systems, in which the “state” changes much more slowly than the perturbing noise. Beyond the existing results on adaptive algorithms, model mismatch significantly affects convergence properties of AF algorithms, raising issues of algorithm robustness. Weak convergence and weak stability (i.e., recurrence) under model mismatch are derived. Based on the limiting stochastic differential equations of suitably scaled iterates, stability in distribution is established. Then algorithms with decreasing step sizes and their convergence properties are examined. When input signals are large, identification bias due to model mismatch will become large and unacceptable. Methods for reducing such bias are introduced when the identified models are used in regulation problems.

Algorithms for system identification, estimation, and adaptive control in stochastic systems rely mostly on different types of signal averaging to achieve uncertainty reduction, convergence, stability, and performance enhancement. The core of such algorithms is various types of laws of large numbers that reduce the effect of noises when they are averaged. Many of the noise sequences encountered are often correlated and nonwhite. In the case of state estimation using quantized information such as in networked systems, convergence must be analyzed on double-indexed and randomly weighted sums of mixing-type stochastic processes, which are correlated with the remote past and distant future being asymptotically independent. This paper presents new results on convergence analysis of such processes. Strong laws of large numbers and convergence rates for such problems are established. These results resolve some fundamental issues in state observer designs with random sampling times, quantized information processing, and other applications.

The basic adaptive filtering algorithm X_{n+1}^{epsilon} = X_{n}^{epsilon} - epsilon Y_{n}(Y_{n}^{'}X_{n}^{epsilon} - psi_{n}) is analyzed using the theory of weak convergence. Apart from some very special cases, the analysis is hard when done for each fixed epsilon > 0 . But the weak convergence techniques are set up to provide much information for small epsilon . The relevant facts from the theory are given. Define x^{epsilon}(cdot) by x^{epsilon}(t) = X_{n}^{epsilon} on [nepsilon, nepsilon + epsilon) . Then weak (distributional) convergence of {x^{epsilon}(cdot)} and of {x^{epsilon}(cdot + t_{epsilon})} is proved under very weak assumptions, where t_{epsilon} rightarrow infty as epsilon rightarrow 0 . The normalized errors {(X_{n}^{epsilon} - theta ) / sqrt{epsilon} } are analyzed, where theta is a "stable" point for the "mean" algorithm. The asymptotic properties of a projection algorithm are developed, where the X_{n}^{epsilon} are truncated at each iteration, if they fall outside of a given set.

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.

The asymptotic properties of a recursive adaptive beam former
algorithm are studied. Both decreasing-gain and constant-gain cases are
treated. For the case of decreasing gain the mean square convergence
result is obtained, whereas for constant gain a sharp bound is derived,
and asymptotic analysis for the normalized error is carried out. The
analysis provides a clear picture of the local behaviour of the iterates
near the optimal value. A sequence of scale deviations or normalized
errors is shown to converge to a Gauss-Markov diffusion process which
satisfies a stochastic differential equation

This paper develops asymptotic properties of a class of sign-error algorithms with expanding truncation bounds for adaptive filtering. Under merely stationary ergodicity and finite second moments of the reference and output signals, and using trajectory-subsequence (TS) method, it is proved that the algorithm convergers almost surely. Then, a mean squares estimate is derived for the estimation error and a suitably scaled sequence of the estimation error is shown to converge to a diffusion process. The scaling factor together with the stationary covariance gives the rate of convergence result. Moreover, an algorithm under mean squares criterion with expanding truncation bounds is also examined. Compared with the existing results in the literature, sufficient conditions for almost sure convergence are much relaxed. A simple example is provided for demonstration purpose.

Variable-order adaptive control of a microelectromechanical steering mirror for suppression of laser beam jitter

- N O Pérez
- N Chen
- S Gibson
- N. O. Pérez

Recursive Estimation and Control for Stochastic Systems

- H Chen
- H. Chen