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For Hammerstein and Wiener systems observed with additive noises the adaptive regulation control is produced by a truncated stochastic approximation (SA) algorithm with truncation regions expanding with a prescribed rate. It is proved that the stochastic adaptive control given in this note is optimal in the sense that it minimizes the long run average of regulation errors a.s. Index Terms--Adaptive regulation, expanding truncations, Hammerstein system, stochastic approximation, Wiener system. Digital Object Identifier 10.1109/TAC.2007.894536

Identification is considered for the Hammerstein system consisting of a static nonlinear block f(*) followed by an ARX subsystem, when the system output is observed with noise. No assumption is made on the structure of f(*). Recursive estimates are given for coefficients of the ARX subsystem and for the value of f (u) at any u. All estimates are proved to converge to the true values with probability one. Numerical examples are provided justifying the theoretical analysis. Index Terms--ARX, Hammerstein system, nonparametric nonlinearity, recursive estimate, strong consistency, stochastic approximation.

This paper deals with identification of Wiener systems with nonlinearity being a discontinuous piece-wise linear function. Recursive estimation algorithms are proposed to estimate six unknown parameters contained in the nonlinearity and all unknown coefficients of the linear subsystem by using the iid Gaussian inputs. The estimates are proved to converge to the corresponding true values with probability one. A numerical example is given to justify the obtained theoretical results. Index Terms--Identification, kernel function, strong consistency, Wiener system.

This paper gives estimates for: 1) coefficients contained in the linear part of the Hammerstein system; 2) the value of the nonlinear function f(u) in the Hammerstein system at any u; 3) Ef([u.sub.k]) and E[u.sub.k]f([u.sub.k]) with [u.sub.k] denoting the system input. No assumption is made on structure of f(*). The estimates given by the stochastic approximation algorithms with expanding truncations are recursive and convergent to the true values with probability one. Two numerical examples are given. Index Terms--Hammerstein system, nonparametric nonlinearity, recursive estimate, stochastic approximation, strong consistency.

An iterative learning control (ILC) algorithm, which in essence is a stochastic approximation algorithm, is proposed for output tracking for nonlinear stochastic systems with unknown dynamics and unknown noise statistics. The nonlinear function of the system dynamics is allowed to grow up as fast as a polynomial of any degree, but the system is linear with respect to control. It is proved that the ILC generated by the algorithm a.s. converges to the optimal one at each time t [EPSILON] [0, 1, ..., N] and the output tracking error is asymptotically minimized in the mean square sense as the number of iterates tends to infinity, although the convergence rate is rather slow. The only information used in the algorithm is the noisy observation of the system output and the reference signal [y.sub.d](t). When the system state equation is free of noise and the system output is realizable, then the exact state tracking is asymptotically achieved and the tracking error is purely due to the observation noise. Index Terms--Almost sure (a.s.) convergence, iterative learning control, nonlinear stochastic system, output tracking, stochastic approximation.

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. Index Terms--Almost sure convergence, diffusion limit, ergodicity, expanding truncation, rate of convergence, sign algorithm.

In this correspondence, we develop adaptive algorithms for multichannel (single-input-multiple-output, or SIMO) blind identification with both statistic and deterministic models. In these algorithms, the estimates are continuously improved while receiving new signals. Therefore, the algorithms can track the channel continuously and thus are amenable to real applications such as wireless communications. At each step, only a small amount of computation is involved. The algorithms are based on stochastic approximation methods. Convergence properties of these algorithms are proved. Simulation examples are presented to show the performance of the algorithms. Index Terms--Stochastic approximation, wireless communication.

Stochastic adaptive stabilization usually leads to the boundedness of the average of squared output of the stabilized system, but gives no conclusion on stability of the resulting system. This note proves that adaptively stabilized stochastic system in its steady state is truly stable in the conventional sense. Index Terms--Adaptive stabilization, ARMA, stability.

A direct adaptive regulator for nonlinear nonparametric systems with measurement corrupted by noise is proposed. Under reasonable conditions the state of the closed-loop system is adaptively regulated so that it converges to zero as time tends to infinity. An illustrative example, being an affine nonlinear system, with all imposed conditions satisfied is given. The method of proof is based on stochastic approximation techniques. Index Terms--Adaptive regulator, nonlinear nonparametric systems, stochastic approximation.

In this paper the Perturbation-Analysis-Robbins-Monro-Single-Run algorithm is applied to estimating the optimal parameter of a performance measure for the GI/G/1 queueing systems, where the algorithm is updated after every fixed-length observation period. Our aim is to analyze the limiting behavior of the algorithm. The almost sure convergence rate of the algorithm is established. It is shown that the convergence rate depends on the second derivative of the performance measure at the optimal point. Index Terms - Convergence rates, perturbation analysis, queueing systems, stochastic approximation, stochastic discrete-event systems.

Two fundamental concepts and quantities, realization factors and performance potentials, are introduced for Markov processes. The relations among these two quantities and the group inverse of the infinitesimal generator are studied. It is shown that the sensitivity of the steady-state performance with respect to the change of the infinitesimal generator can be easily calculated by using either of these three quantities and that these quantities can be estimated by analyzing a single sample path of a Markov process. Based on these results, algorithms for estimating performance sensitivities on a single sample path of a Markov process can be proposed. The potentials in this paper are defined through realization factors and are shown to be the same as those defined by Poisson equations. The results provide a uniform framework of perturbation realization for infinitesimal perturbation analysis (IPA) and non-IPA approaches to the sensitivity analysis of steady-state performance; they also provide a theoretical background for the PA algorithms developed in recent years. Index Terms - Perturbation analysis, Poisson equations, sample-path analysis.

The key issue for adaptive pole-placement control of linear time-invariant systems is the possible singularity of the Sylvester matrix corresponding to the coefficient estimate. However, to overcome the difficulty, the estimate is modified by several methods which are either nonrecursive and with high computational load or recursive but with random search involved. All of the previous works are done under the assumption that the system is controllable. This paper gives the necessary and sufficient condition, which is weaker than controllability, for the system to be adaptively stabilizable. First, a nonrecursive algorithm is proposed to modify the estimates, and the algorithm is proved to terminate in finitely many steps. Then, with the help of stochastic approximation, a recursive algorithm is proposed for obtaining the modification parameters; it is proved that these modification parameters turn out to be a constant vector in a finite number of steps. This leads to the convergence of the modified coefficient estimates. For both algorithms the Sylvester matrices corresponding to the modified coefficient estimates are asymptotically uniformly nonsingular; thus, the adaptive pole-placement control problem can be solved, i.e., the system can be adaptively stabilized. Index Terms - Adaptive pole-placement control, controllability, parameter modification, stochastic approximation.

This paper studies the robustness properties of a basic adaptive control algorithm with respect to plant parameter variation as well as modeling errors and bounded disturbances. The algorithm consists of a projected gradient estimator and a pole assignment controller. A unified method of proof is presented for robust stability of both discrete and continuous-time adaptively controlled time-varying systems. The robust performance of an adaptive pole assignment controller is also discussed.

It is shown that the limit points of a stochastic approximation (SA) algorithm compose a connected set. Conditions are given to guarantee the uniqueness of the limit point for a given initial value. Examples are provided wherein {[x.sub.n]} of SA algorithm converges to a limit [bar] x independent of initial values, but [bar] x is unstable for the differential equation x = f(x) with a nonnegative Lyapunov function. Finally, sufficient conditions are given for stability of x = f(x) at [bar] x if {[x.sub.n]} tends to [bar] x for any initial values. Index Terms--Pathwise convergent, stable/instable limit point, stochastic approximation.