L/sup p/ approximation of Sigma-Pi neural networks

A feedforward Sigma-Pi neural network with a single hidden layer of m neurons is given by /sup m//spl Sigma//sub j=1/c/sub j/g(n/spl Pi//sub k=1/x/sub k/-/spl theta//sub k//sup j///spl lambda//sub k//sup j/) where c/sub j/, /spl theta//sub k//sup j/, /spl lambda//sub k//spl isin/R. We investigate the approximation of arbitrary functions f: R/sup n//spl rarr/R by a Sigma-Pi neural network in the L/sup p/ norm. An L/sup p/ locally integrable function g(t) can approximate any given function, if and only if g(t) can not be written in the form /spl Sigma//sub j=1//sup n//spl Sigma//sub k=0//sup m//spl alpha//sub jk/(ln|t|)/sup j-1/t/sub k/.     

Remarks on the L/sup p/-input converging-state property

Let X /spl sub/ /spl Ropf//sup N/ and consider a system x/spl dot/ = f(x,u), f : X /spl times/ /spl Ropf//sup M/ /spl rarr/ /spl Ropf//sup N/, with the property that the associated autonomous system x/spl dot/ = f (x,0) has an asymptotically stable compactum C with region of attraction A. Assume that x is a solution of the former, defined on [0,/spl infin/), corresponding to an input function u. Assume further that, for each compact K /spl sub/ X, there exists k > 0 such that |f(z,v) - f(z,0)| /spl les/ k|v| for all (z,v) /spl isin/ /spl times/ /spl Ropf//sup M/. A simple proof is given of the following L/sup p/-input converging-state property: if u /spl isin/ L/sup p/ for some p /spl isin/ [1,/spl infin/) and x has an /spl omega/-limit point in A, then x approaches C.     

Optimal quantization of periodic task requests on multiple identical processors

We simplify the periodic tasks scheduling problem by making a trade off between processor load and computational complexity. A set N of periodic tasks, each characterized by its density /spl rho//sub i/, contains n possibly unique values of /spl rho//sub i/. We transform N through a process called quantization, in which each /spl rho/i /spl isin/ N is mapped onto a service level s/sub j/ /spl isin/ L, where |L| = l /spl Lt/ n and /spl rho//sub i/ /spl les/ s/sub j/, (this second condition differentiates this problem from the p-median problem on the real line). We define the periodic task quantization problem with deterministic input (PTQ-D) and present an optimal polynomial time dynamic programming solution. We also introduce the problem PTQ-S (with stochastic input) and present an optimal solution. We examine, in a simulation study, the trade off penalty of excess processor load needed to service the set of quantized tasks over the original set, and find that, through quantization onto as few as 15 or 20 service levels, no more than 5 percent processor load is required above the amount requested. Finally, we demonstrate that the scheduling of a set of periodic tasks is greatly simplified through quantization and we present a fast online algorithm that schedules quantized periodic tasks.     

Computation of the frequency response of multilinear affine systems

Deals with the problem of computing the frequency response of an uncertain transfer function whose numerator and denominator polynomials are multiples of independent uncertain polynomials of the form P(s, q) = l/sub o/ (q) + l/sub 1/ (q) s + /spl middot//spl middot//spl middot/ + l/sub n/, (q) s/sup n/ whose coefficients depend linearly on q = [q/sub 1/, q/sub 2/, ..., q/sub q/]/sup T/ and the uncertainty box is Q = {q: q/sub i/ /spl epsiv/ [q/sub i/, q/sub i/], i = 1, 2,..., q}. Using the geometric structure of the value set of P(s, q), a powerful edge elimination procedure is proposed for computing the Bode, Nyquist, and Nichols envelopes of these uncertain systems. A numerical example is included to illustrate the benefit of the method presented.     

A necessary algebraic condition for controllability and observability of linear time-varying systems

In this note, we give an algebraic condition which is necessary for the system x'(t)=A(t)x(t)+B(t)u(t), y(t)=C(t)x(t), either to be totally controllable or to be totally observable, where x/spl isin//spl Ropf//sup d/, u/spl isin//spl Ropf//sup p/, y/spl isin//spl Ropf//sup q/, and the matrix functions A, B and C are (d-2), (d-1) and (d-1) times continuously differentiable, respectively. All conditions presented here are in terms of known quantities and therefore easily verified. Our conditions can be used to rule out large classes of time-varying systems which cannot be controlled and/or observed no matter what the nonzero time-varying coefficients are. This work is motivated by the deep result of Silverman and Meadows.     

Transient response control via characteristic ratio assignment

This note develops an approach to directly control the transient response of linear time-invariant control systems. We begin by considering all-pole transfer functions of order n for which we introduce a set of parameters /spl alpha//sub i/, i=1,...n called the characteristic ratios. We also introduce a generalized time constant /spl tau/. We prove that /spl alpha//sub 1/ and /spl tau/ can be used to characterize the system overshoot to a step input and the speed of response, respectively. By independently adjusting /spl alpha//sub 1/ and /spl tau/ in all-pole systems, arbitrarily small or no overshoot as well as arbitrarily fast speed of response can be achieved. These formulas are used to develop a procedure to design feedback controllers with feedforward or two parameter output feedback type for achieving time response specifications. For a minimum phase plant we show that arbitrary transient response specifications, namely one with independently specified overshoot and specified rise time or speed of response can be exactly attained.     

Robust H/sub /spl infin// static output feedback control of fuzzy systems: an ILMI approach

This paper examines the problem of robust H/sub /spl infin// static output feedback control of a Takagi-Sugeno fuzzy system. The proposed robust H/sub /spl infin// static output feedback controller guarantees the L/sub 2/ gain of the mapping from the exogenous disturbances to the regulated output to be less than or equal to a prescribed level. The existence of a robust H/sub /spl infin// static output feedback control is given in terms of the solvability of bilinear matrix inequalities. An iterative algorithm based on the linear matrix inequality is developed to compute robust H/sub /spl infin// static output feedback gains. To reduce the conservatism of the design, the structural information of membership function characteristics is incorporated. A numerical example is used to illustrate the validity of the design methodologies.     

Channel assignment with separation for interference avoidance in wireless networks

Given an integer /spl sigma/>1, a vector (/spl delta//sub 1/, /spl delta//sub 2/,..., /spl delta//sub /spl sigma/-1/), of nonnegative integers, and an undirected graph G=(V, E), an L(/spl delta//sub 1/, /spl delta//sub 2/,..., /spl delta//sub /spl sigma/-1/)-coloring of G is a function f from the vertex set V to a set of nonnegative integers, such that |f(u)-f(v)|/spl ges//spl delta//sub i/, if d(u,v)=i, for 1相似文献   

Learning capability and storage capacity of two-hidden-layer feedforward networks

The problem of the necessary complexity of neural networks is of interest in applications. In this paper, learning capability and storage capacity of feedforward neural networks are considered. We markedly improve the recent results by introducing neural-network modularity logically. This paper rigorously proves in a constructive method that two-hidden-layer feedforward networks (TLFNs) with 2/spl radic/(m+2)N (/spl Lt/N) hidden neurons can learn any N distinct samples (x/sub i/, t/sub i/) with any arbitrarily small error, where m is the required number of output neurons. It implies that the required number of hidden neurons needed in feedforward networks can be decreased significantly, comparing with previous results. Conversely, a TLFN with Q hidden neurons can store at least Q/sup 2//4(m+2) any distinct data (x/sub i/, t/sub i/) with any desired precision.     

Fuzzy H/sub /spl infin// output feedback control design for singularly perturbed systems with pole placement constraints: an LMI approach

This paper examines the problem of designing an H/sub /spl infin// output feedback controller with pole placement constraints for singular perturbed Takagi-Sugeno (TS) fuzzy models. We propose a fuzzy H/sub /spl infin// output feedback controller that not only guarantees the /spl Lscr//sub 2/-gain of the mapping from the exogenous input noise to the regulated output to be less than some prescribed value, but also ensures closed-loop poles of each subsystem are in a prespecified linear matrix inequality (LMI) region. In order to alleviate the numerical stiffness caused by the singular perturbation /spl epsiv/, the design technique is formulated in terms of a family of /spl epsiv/-independent linear matrix inequalities. The proposed approach can be applied both standard and nonstandard singularly perturbed nonlinear systems. A numerical example is provided to illustrate the design developed in this paper.     

The wonham filter with random parameters: rate of convergence and error bounds

Let /spl alpha/(t) be a finite-state continuous-time Markov chain with generator Q=(q/sup ij/)/spl isin/R/sup m/spl times/m/ and state space M={z/sup i/,...,z/sup m/}, where z/sup 1/ for i/sub /spl middot//spl middot//spl middot//spl middot//m are distinct real numbers. When the state-space and the generator are known a priori, the best estimator of /spl alpha/(t) (in terms of mean square error) under noisy observation is the classical Wonham filter. This note addresses the estimation issue when values of the state-space or values of the generator are unknown a priori. In each case, we propose a (suboptimal) filter and prove its convergence to the desired Wonham filter under simple conditions. Moreover, we obtain the rate of convergence using both the mean square and the higher moment error bounds.     

Quasi-continuous high-order sliding-mode controllers

A universal finite-time-convergent controller is developed capable to control the output of any uncertain single-input-single-output system with a known permanent relative degree r. The tracking error /spl sigma/ is steered to zero by means of a control dependent only on /spl sigma/, /spl sigma//spl dot/, ..., /spl sigma//sup (r-1)/ and continuous everywhere except the set /spl sigma/=/spl sigma//spl dot/=/spl middot//spl middot//spl middot/=/spl sigma//sup (r-1)/=0. A robust output-feedback controller version provides for the tracking accuracy proportional to the sampling noise magnitude.     

Robust H/sub /spl infin// control for uncertain discrete-time-delay fuzzy systems via output feedback controllers

This work investigates the problem of robust output feedback H/sub /spl infin// control for a class of uncertain discrete-time fuzzy systems with time delays. The state-space Takagi-Sugeno fuzzy model with time delays and norm-bounded parameter uncertainties is adopted. The purpose is the design of a full-order fuzzy dynamic output feedback controller which ensures the robust asymptotic stability of the closed-loop system and guarantees an H/sub /spl infin// norm bound constraint on disturbance attenuation for all admissible uncertainties. In terms of linear matrix inequalities (LMIs), a sufficient condition for the solvability of this problem is presented. Explicit expressions of a desired output feedback controller are proposed when the given LMIs are feasible. The effectiveness and the applicability of the proposed design approach are demonstrated by applying this to the problem of robust H/sub /spl infin// control for a class of uncertain nonlinear discrete delay systems.     

A min-max approach to fuzzy clustering, estimation, and identification

This study, for any unknown physical process y=f(x/sub 1/,...,x/sub n/), is concerned with the: 1) fuzzy partition of n-dimensional input space X=X/sub 1//spl times//spl middot//spl middot//spl middot//spl times/X/sub n/ into K different clusters, 2) estimating the process behavior y/spl circ/=f(x/spl circ/) for a given input x/spl circ/=(x/spl circ//sub 1/,/spl middot//spl middot//spl middot/,x/spl circ//sub n/)/spl isin/X, and 3) fuzzy approximation of the process, with uncertain input-output identification data {(x(k)/spl plusmn//spl delta/x/sub k/),(y(k)/spl plusmn/v/sub k/)}/sub k=1,.../, using a Sugeno type fuzzy inference system. A unified min-max approach (that attempts to minimize the worst-case effect of data uncertainties and modeling errors on estimation performance), is suggested to provide robustness against data uncertainties and modeling errors. The proposed method of min-max fuzzy parameters estimation does not make any assumption and does not require a priori knowledge of upper bounds, statistics, and distribution of data uncertainties and modeling errors. To show the feasibility of the approach, simulation studies and a real-world application of physical fitness classification based on the fuzzy interpretation of physiological parameters, have been provided.     

Designing asymmetric Hopfield-type associative memory with higher order hamming stability

The problem of optimal asymmetric Hopfield-type associative memory (HAM) design based on perceptron-type learning algorithms is considered. It is found that most of the existing methods considered the design problem as either 1) finding optimal hyperplanes according to normal distance from the prototype vectors to the hyperplane surface or 2) obtaining weight matrix W=[w/sub ij/] by solving a constraint optimization problem. In this paper, we show that since the state space of the HAM consists of only bipolar patterns, i.e., V=(v/sub 1/,v/sub 2/,...,v/sub N/)/sup T//spl isin/{-1,+1}/sup N/, the basins of attraction around each prototype (training) vector should be expanded by using Hamming distance measure. For this reason, in this paper, the design problem is considered from a different point of view. Our idea is to systematically increase the size of the training set according to the desired basin of attraction around each prototype vector. We name this concept the higher order Hamming stability and show that conventional minimum-overlap algorithm can be modified to incorporate this concept. Experimental results show that the recall capability as well as the number of spurious memories are all improved by using the proposed method. Moreover, it is well known that setting all self-connections w/sub ii//spl forall/i to zero has the effect of reducing the number of spurious memories in state space. From the experimental results, we find that the basin width around each prototype vector can be enlarged by allowing nonzero diagonal elements on learning of the weight matrix W. If the magnitude of w/sub ii/ is small for all i, then the condition w/sub ii/=0/spl forall/i can be relaxed without seriously affecting the number of spurious memories in the state space. Therefore, the method proposed in this paper can be used to increase the basin width around each prototype vector with the cost of slightly increasing the number of spurious memories in the state space.     

Periodic orbits from /spl Delta/-modulation of stable linear systems

The /spl Delta/-modulated control of a single input, discrete time, linear stable system is investigated. The modulation direction is given by c/sup T/x where c /spl isin/R/sup n//spl bsol/{0} is a given, otherwise arbitrary, vector. We obtain necessary and sufficient conditions for the existence of periodic points of a finite order. Some concrete results about the existence of a certain order of periodic points are also derived. We also study the relationship between certain polyhedra and the periodicity of the /spl Delta/-modulated orbit.     

Sparse component analysis and blind source separation of underdetermined mixtures

In this letter, we solve the problem of identifying matrices S /spl isin/ /spl Ropf//sup n/spl times/N/ and A /spl isin/ /spl Ropf//sup m/spl times/n/ knowing only their multiplication X = AS, under some conditions, expressed either in terms of A and sparsity of S (identifiability conditions), or in terms of X (sparse component analysis (SCA) conditions). We present algorithms for such identification and illustrate them by examples.     

Adaptive optimization of least-squares tracking algorithms: with applications to adaptive antenna arrays for randomly time-varying mobile communications systems

Adaptive antenna arrays are used for reducing the effects of interference and increasing capacity in mobile communications systems. Typical algorithms recursively compute the antenna weights that minimize the weighted error function (at discrete times kh, k=1,2,..., for a sampling interval h) /spl sigma//sub l=1//sup k//spl alpha//sup k-l/[e/sub l/(W)]/sup 2/, where e/sub l/(W) is a measure of the reception error at time lh with antenna weight vector W, and /spl alpha/<1. The forgetting factor /spl alpha/<1 allows tracking as conditions change and the minimization is used only to get the weights. The average detection error rate depends heavily on the chosen value of /spl alpha/, whose optimal value can change rapidly in time, perhaps significantly in seconds. We add another adaptive loop that tracks the optimal value of /spl alpha/ and greatly improves the operation when the environment is randomly time-varying. The additional adaptive loop is based on an approximation to a natural "gradient descent" method. The algorithm is practical and can improve the performance considerably. In terms of average detection error rates and for all of the scenarios tested, the new system tracks the optimal value of /spl alpha/ well, and always performs better (sometimes much better) than the original algorithm that uses any fixed value of /spl alpha/. Although the initial motivation arises in adaptive antennas, the method can be used to improve algorithms for tracking parameters of time-varying nonlinear systems, where similar issues are involved.     

Reduced-order solutions for the singular H/sup /spl infin// filtering problem

We consider the filtering H/sup /spl infin// problem for linear time-varying systems. This problem has been already solved in the case when the direct feedthrough matrix F between the disturbance and the output vectors is full-row rank. Here, we consider the case when the problem is singular, that is when this feedthrough matrix is not full-row rank. We show that in this case a reduced-order observer can be designed to meet the desired performance. Moreover, we apply the obtained results to solve a particular H/sup /spl infin// singular output feedback problem.