离散线性一致性算法噪声问题研究

多智能体一致性问题在传感网、社交网、协同控制等诸多领域有着广泛的实际应用背景, 本文对离散线性一致性算法的噪声问题进行了研究, 证明了离散线性 一致性算法的噪声不可控性; 提出基于抑噪算子ε(t)的噪声控制策略, 指出当ε(t)为t-0.5的高阶无穷小时, 抑噪后的一致性算法噪声可控; 分析了抑噪算子对一致性 算法收敛性的影响, 证明了在无噪声条件下, 当抑噪算子ε(t为t-1的低阶无穷小时, 抑噪后的一致性算法依然可以使Agent收敛至原收敛状态x*.在上述结论基础上进一步指出, 当t→∞ 时, 若抑噪算子ε(t)的阶在t-0.5~t-1之间, 所有Agent 的状态将以原收敛状态x* 为中心呈正态分布. 最后, 以DHA 为例对相应理论结果进行了验证和讨论. 本文为线性一致性算法的噪声控制提供了理论依据, 对抑噪算s子的确定有较强的指导意义.     

Robust estimation without positive real condition

The strictly positive real (SPR) condition on the noise model is necessary for a discrete-time linear stochastic control system with unmodeled dynamics, even so for a time-invariant ARMAX system, in the past robust analysis of parameter estimation. However, this condition is hardly satisfied for a high-order and/or multidimensional system with correlated noise. The main work in this paper is to show that for robust parameter estimation and adaptive tracking, as well as closed-loop system stabilization, the SPR condition is replaced by a stable matrix polynomial. The main method is to design a “two-step” recursive least squares algorithm with or without a weighted factor and with a fixed lag regressive vector and to define an adaptive control with bounded external excitation and with randomly varying truncation     

On-line parallel heuristics, processor scheduling and robot searching under the competitive framework

In this paper we investigate parallel searches on m concurrent rays for a point target t located at some unknown distance along one of the rays. A group of p agents or robots moving at unit speed searches for t. The search succeeds when an agent reaches the point t. Given a strategy S the competitive ratio is the ratio of the time needed by the agents to find t using S and the time needed if the location of t had been known in advance. We provide a strategy with competitive ratio of 1+2(m/p−1)(m/(m−p))^m/p and prove that this is optimal. This problem has applications in multiple heuristic searches in AI as well as robot motion planning. The case p = 1 is known in the literature as the cow path problem.     

A robust model reference adaptive control without strictly positive real condition

This paper considers the model reference adaptive control problem for the system (1) with unmodelled dynamics m n dominated by a small constant k multiplied by a quantity dependent on the input, output and noise. Using bounded external excitation and randomly varying truncation techniques, we give design method of a model reference adaptive controller which is similar to that in Meyn and Brown (1992). It is shown that the closed-loop system is globally stable, the estimation error for parameter contained in the modelled part is of order k, and the closed-loop system under the model reference adaptive control law is suboptimal in the sense of lim sup 1/n n (A m (z) y i+1 - B m (z)z* i - C m (z) w i+1 ) 2 h O( k 2 ) + n 2 b 1 2 while the SPR condition usually used in other papers is replaced by a stability condition.     

Analytic-numerical solutions with a priori error bounds for time-dependent mixed partial differential problems

The aim of this paper is double. First, we point out that the hypothesis D(t₁)D(t₂) = D(t₂)D(t₁) imposed in [1] can be removed. Second, a constructive method for obtaining analytic-numerical solutions with a prefixed accuracy in a bounded domain Ω(t₀,t₁) = [0,p] × [t₀,t₁], for mixed problems of the type u_t(x,t) − D(t)u_xx(x,t) = 0, 0 < x < p, t> 0, subject to u(0,t) = u(p,t) = 0 and u(x,0) = F(x) is proposed. Here, u(x,t) and F(x) are r-component vectors, D(t) is a C^{r × r} valued analytic function and there exists a positive number δ such that every eigenvalue z of (1/2) (D(t) + D(t)^H) is bigger than δ. An illustrative example is included.     

Feedback Stabilization Over a First Order Moving Average Gaussian Noise Channel

Recent developments in information theory by Y.-H. Kim have established the feedback capacity of a first order moving average additive Gaussian noise channel. Separate developments in control theory have examined linear time invariant feedback control stabilization under signal to noise ratio (SNR) constraints, including colored noise channels. This note considers the particular case of a minimum phase plant with relative degree one and a single unstable pole at z=phi (with |phi| > 1) over a first order moving average Gaussian channel. SNR constrained stabilization in this case is possible precisely when the feedback capacity of the channel satisfies CFB ges log₂ |phi|. Furthermore, using the results of Kim we show that there exist linear encoding and decoding schemes that achieve stabilization within the SNR constraint precisely when CFB ges log₂ |phi|.     

Nonnegative solutions of singular boundary value problems with sign changing nonlinearities

The paper presents sufficient conditions for the existence of positive solutions of the equation x″(t) + q(t)f(t,x(t),x′(t)) = 0 with the Dirichlet conditions x(0) = 0, x(1) = 0 and of the equation (p(t)x′(t))′ + p(t)q(t)f(t,x(t),p(t)x′(t)) = 0 with the boundary conditions lim_t→o⁺ p(t)x′(t) = 0, x(1) = 0. Our nonlinearity f is allowed to change sign and f may be singular at x = 0. The proofs are based on a combination of the regularity and sequential techniques and the method of lower and upper functions.     

Tree spanners on chordal graphs: complexity and algorithms

A tree t-spanner T in a graph G is a spanning tree of G such that the distance in T between every pair of vertices is at most t times their distance in G. The T t-S problem asks whether a graph admits a tree t-spanner, given t. We substantially strengthen the hardness result of Cai and Corneil (SIAM J. Discrete Math. 8 (1995) 359–387) by showing that, for any t4, T t-S is NP-complete even on chordal graphs of diameter at most t+1 (if t is even), respectively, at most t+2 (if t is odd). Then we point out that every chordal graph of diameter at most t−1 (respectively, t−2) admits a tree t-spanner whenever t2 is even (respectively, t3 is odd), and such a tree spanner can be constructed in linear time.

The complexity status of T 3-S still remains open for chordal graphs, even on the subclass of undirected path graphs that are strongly chordal as well. For other important subclasses of chordal graphs, such as very strongly chordal graphs (containing all interval graphs), 1-split graphs (containing all split graphs) and chordal graphs of diameter at most 2, we are able to decide T 3-S efficiently.     

Constructible functions in cellular automata and their applications to hierarchy results

We investigate time-constructible functions in one-dimensional cellular automata (CA). It is shown that (i) if a function t(n) is computable by an O(t(n)−n)-time Turing machine, then t(n) is time constructible by CA and (ii) if two functions are time constructible by CA, then the sum, product, and exponential functions of them are time constructible by CA. As an application, it is shown that if t₁(n) and t₂(n) are time constructible functions such that lim_n→∞ t₁(n)/t₂(n) = 0 and t₁(n)n, then there is a language which can be recognized by a CA in t₂(n) time but not by any CA in t₁(n) time.     

New results on oscillation for delay differential equations with piecewise constant argument

We introduce a new technique to obtain some new oscillation criteria for the oscillating coefficients delay differential equation with piecewise constant argument of the form x′(t) + a(t)x(t) + b(t)x({t − k}) = 0, where a(t) and b(t) are right continuous functions on [−k, ∞), k is a positive integer, and [·] denotes the greatest integer function. Our results improve and generalize the known results in the literature. Some examples are also given to demonstrate the advantage of our results.     

Stability of nonlinear time-varying feedback systems

In this paper, the stability of nonlinear time-varying feedback systems is studied using a “passive operator” technique. The feedback system is assumed to consist of a linear time-invariant operator G(s) in the forward path and a nonlinear time-varying gain function f(·)K(t) in the feedback path. The stability condition indicates that the bound on the time derivative [dK(t)/(dt)] depends both on the nonlinearity f(·) and the multiplier Z(s) chosen to make G(s)Z(s) positive real. It is also shown that the main result in this paper can be specialized to yield many of the results obtained so far for nonlinear time-invariant systems and linear time-varying systems.     

Analytic numerical solution of coupled semi-infinite diffusion problems

In this paper, we consider coupled semi-infinite diffusion problems of the form u_t(x, t)− A² u_xx(x,t) = 0, x> 0, t> 0, subject to u(0,t)=B and u(x,0)=0, where A is a matrix in , and u(x,t), and B are vectors in . Using the Fourier sine transform, an explicit exact solution of the problem is proposed. Given an admissible error and a domain D(x₀,t₀)={(x,t);0≤x≤x₀, t≥t₀ > 0, an analytic approximate solution is constructed so that the error with respect to the exact solution is uniformly upper bounded by in D(x₀, t₀).     

Existence of multiple positive solutions for functional differential equations

In this paper, the existence of at least three positive solutions for the boundary value problem (BVP) of second-order functional differential equation with the form y″(t) + f(t, y^t) = 0, for t ε [0,1], y(t) -βy′(t) =η(t), for t ε [−τ,0], −γy(t) + Δy′(t) = ζ(t), for t ε [1, 1 + a], is studied. Moreover, we investigate the existence of at least three partially symmetric positive solutions for the above BVP with Δ = βγ.     

强噪声背景下钢丝绳损伤信号降噪方法

钢丝绳损伤信号是一种非平稳无周期性的冲击信号,其特征信号的降噪处理和特征提取成为亟待解决的难题。小波变换方法若小波基或者分解层数不适合,会在信号降噪的同时引入其他噪声干扰,影响信号处理与特征提取的效果。相较于小波变换方法,移位平均法只需要选择一定的移位窗宽即可实现对信号的有效降噪,但移位窗宽需要人为选择,盲目性大。针对上述问题,提出一种强噪声背景下钢丝绳损伤信号降噪方法。利用钢丝绳漏磁检测传感器采集不同类型的断丝数据,向信号中加入强高斯白噪声,以模拟强噪声背景;采用自适应移位平均法对钢丝绳损伤信号进行降噪,利用量子粒子群优化(QPSO)算法优化移位平均法的窗宽;将损伤信号的信噪比(SNR)作为适应度函数,通过QPSO算法使得损伤特征信号SNR最大化,从而实现最优信号降噪效果。实验结果表明,对于强噪声背景下的钢丝绳平稳和波动信号,相较于小波变换,自适应移位平均法的降噪效果更明显,信噪比更高,信号更为平滑。实测结果表明,对于现场采集的噪声相对弱一些的钢丝绳损伤信号,自适应移位平均法的降噪效果也比小波变换好,验证了自适应移位平均法具有较好的通用性。     

Encoding Bandpass Signals Using Zero/Level Crossings: A Model-Based Approach

A new approach to representing a time-limited, and essentially bandpass signal x(t) , by a set of discrete frequency values is proposed. The set of discrete frequency values is the set of locations along the frequency axis at which (real and/or imaginary parts of) the Fourier transform of the signal x(t) cross certain levels (especially zero level). Analogously, invoking time-frequency duality, a set of time instants denoting the zero/level crossings of a waveform x(t) can be used to represent a bandlimited spectrum X(f) . The proposed signal representation is based on a simple bandpass signal model that exploits our prior knowledge of the bandwidth/timewidth of the signal. We call it a Sum-of-Sincs (SOS) model, where Sinc stands for the familiar sinx/x function. Given the discrete fequency/time locations, we can accurately reconstruct the signal x(t) or the spectrum X(f) by solving a simple eigenvalue or a least squares problem. Using this approach as the basis, we propose an analysis/synthesis algorithm to decompose and represent complex multicomponent signals like speech over the entire time-frequency region. The proposed signal representation is an alternative to standard analog to discrete conversion based on the sampling theorem, and in principle, possesses some of the desirable attributes of signal representation in natural sensory systems.     

A Product–Delay algorithm for graphic design

A Product–Delay algorithm is presented for creating graphic designs on a computer. In this algorithm two functions u(t) and v(t) are multiplied yielding a function x(t). Another function y(t) is formed by delaying or advancing x(t) by a fixed amount of time t. These functions are evaluated over a suitable time interval and the results are plotted in the x–y plane. For appropriate choices of the functions and parameters, the x–y displays exhibit interesting geometric patterns. In this paper the algorithm is illustrated with a pair of sine and square waves. It is shown that a wide variety of graphic designs can be created with these simple waveforms. By virtue of its simplicity this algorithm can be programmed easily and quickly using general purpose software such as Maple, Matlab or Mathematica. It can be executed on standard platforms such as IBM PC compatibles, Macintosh computers or workstations. Some results in polar coordinates are also given.     

Blossoms are polar forms

Consider the functions H(t):=t² and h(u,v):=uv. The identity H(t)=h(t,t) shows that h is the restriction of h to the diagonal u=v in the uv-plane. Yet, in many ways, a bilinear function like h is simpler than a homogeneous quadratic function like H. More generally, if F(t) is some n-ic polynomial function, it is often helpful to study the polar form of F, which is the unique symmetric, multiaffine function ƒ(u_1,…u_n) satisfying the identity F(t)=f(t,…,t). The mathematical theory underlying splines is one area where polar forms can be particularly helpful, because two pieces F and G of an n-ic spline meet at a point r with C^k parametric continuity if and only if their polar forms ƒ and g agree on all sequences of n arguments that contain at least n-k copies of r.

The polar approach to the theory of splines emerged in rather different guises in three independent research efforts: Paul de Faget Casteljau called it ‘shapes through poles’; Carl de Boor called it ‘B-splines without divided differences’; and Lyle Ramshaw called it ‘blossoming’. This paper reviews the work of de Casteljau, de Boor, and Ramshaw in an attempt to clarify the basic principles that underly the polar approach. It also proposes a consistent system of nomenclature as a possible standard.     

Bisector curves of planar rational curves

This paper presents a simple and robust method for computing the bisector of two planar rational curves. We represent the correspondence between the foot points on two planar rational curves C₁(t) and C₂(r) as an implicit curve (t,r)=0, where (t,r) is a bivariate polynomial B-spline function. Given two rational curves of degree m in the xy-plane, the curve (t,r)=0 has degree 4m−2, which is considerably lower than that of the corresponding bisector curve in the xy-plane.     

On the error in Padé approximations for functions defined by Stieltjes integrals

Consdier I(z) = ∫^b_a w(t)f(t, z) dt, f(t, z) = (1 + t/z)⁻¹. It is known that generalized Gaussian quadrature of I(z) leads to approximations which occupy the (n, n + r − 1) positions of the Padé matrix table for I(z). Here r is a positive integer or zero. In a previous paper the author developed a series representation for the error in Gaussian quadrature. This approach is now used to study the error in the Padé approximations noted. Three important examples are treated. Two of the examples are generalized to the case where f(t, z) = (1 + t/z)^−v.     

A system theoretical approach to the optimization of cancer treatment

The paper describes an extremely concise mathematical model specifying quantitatively the influence exerted on teh age-dependent patient survival probability z(t), or output function, by the time distribution of the cytostatic treatment ƒ = dF/dt or input functions, F being the total dose administered.

The model consists of a set of only four simultaneous linear differential equations of first order where a realistic representation is provided for all main causal connections occurring within the host-cancer system throughout its lifespan.

The peculiar circumstances where (i) no cancer disease is present, or (ii) no cytostatic treatment is administered (i.e. F = 0 for any t) or (iii) neither the cancer is present, nor the treatment is given, become simply special mathematical cases of the general differential problem outlined. Following the model described here, the choice of the time distribution of the cytostatic treatment becomes mathematically defined as an optimal control problem.