H∞ estimation for fuzzy membership function optimization

Given a fuzzy logic system, how can we determine the membership functions that will result in the best performance? If we constrain the membership functions to a specific shape (e.g., triangles or trapezoids) then each membership function can be parameterized by a few variables and the membership optimization problem can be reduced to a parameter optimization problem. The parameter optimization problem can then be formulated as a nonlinear filtering problem. In this paper we solve the nonlinear filtering problem using H_∞ state estimation theory. However, the membership functions that result from this approach are not (in general) sum normal. That is, the membership function values do not add up to one at each point in the domain. We therefore modify the H_∞ filter with the addition of state constraints so that the resulting membership functions are sum normal. Sum normality may be desirable not only for its intuitive appeal but also for computational reasons in the real time implementation of fuzzy logic systems. The methods proposed in this paper are illustrated on a fuzzy automotive cruise controller and compared to Kalman filtering based optimization.     

Synchronization of FitzHugh–Nagumo systems in EES via H∞ variable universe adaptive fuzzy control

In this paper, H_∞ variable universe adaptive fuzzy control law is derived and applied to synchronize two FitzHugh—Nagumo neurons. Firstly, the different dynamical behavior of the nonlinear cable model based on the FitzHugh–Nagumo model responding to the various external electrical simulations (EES) is studied. Next, using the result of the analysis, a robust adaptive variable universe fuzzy control scheme is designed to synchronize two neurons. The computer simulations is provided for verify the efficiency of the designed synchronization scheme.     

A new formulation of some discrete-time stochastic-parameter state estimation problems

Linear unbiased full-order state estimation problem for discrete-time models with stochastic parameters and additive finite energy type disturbance signals is reformulated in terms of linear matrix inequalities. Two estimation problems that are considered are the design for mean-square bounded estimation error and the design for the mean-square stochastic version of the suboptimal H_∞ estimator. These two designs are shown to apply to both the estimation with random sensor delay and estimation under observation uncertainty.     

Reduced-order controllers for unstable multivariable linear systems that can be balanced

This paper considers the problem of H_∞-stabilization of unstable multivariable linear systems. The major features of the approach are: (1) a reduced-order model is obtained using low-frequency balancing, the approximant will have the same number of unstable poles as the original system, (2) the controller design is accomplished by dynamic output feedback, and (3) sufficient conditions in the form of two algebraic Riccati equations and an upper bound explicitly characterize a H_∞-controller of lower dimensions. At the end, an illustrative example is given to show the simplicity of the procedure.     

The smallest degree sum that yields potentially kCℓ-graphic sequences

Gould et al. (Combinatorics, Graph Theory and Algorithms, Vol. 1, 1999, pp. 387–400) considered a variation of the classical Turán-type extremal problems as follows: For a given graph H, determine the smallest even integer σ(H,n) such that every n-term graphic sequence π=(d₁,d₂,…,d_n) with term sum σ(π)=d₁+d₂++d_nσ(H,n) has a realization G containing H as a subgraph. In this paper, for given integers k and ℓ, ℓ7 and 3kℓ, we completely determine the smallest even integer σ(_kC_ℓ,n) such that each n-term graphic sequence π=(d₁,d₂,…,d_n) with term sum σ(π)=d₁+d₂++d_nσ(_kC_ℓ,n) has a realization G containing a cycle of length r for each r, krℓ.     

Relationships between discrete-time and continuous-time algebraic riccati inequalities

The bilinear transformation is used to establish a direct relationship between a discrete-time algebraic Riccati inequality (DARI) and an associated continuous-time algebraic Riccati inequality (CARI). It is shown that under mild conditions, the DARI is solvable if and only if the corresponding CARI is solvable. The relationship between the DARI and the CARI is then used to translate the general solvability conditions for a CARI given by Scherer into analogous conditions for the DARI. It is shown how such conditions can be applied to determine the solvability of a discrete-time H_∞ control problem whose solution set is characterized by two DARIs.     

On an explicit formula for the distribution of the supremum

Let M_∞ be the supremum of a random walk drifting to -∞ which is generated by the partial sums of a sequence of independent identically distributed random variables with a common distribution F. We prove that the moment generating function E exp(sM_∞) is a rational function if and only if the function ∫₀^∞ exp(sx)F(dx) is rational.     

On the complex zeros of Hμ(z), J(z), J(z) for real or complex order

Propositions about the nonexistence of complex zeros of the functions H_μ(z)=J_μ(z)+zJ^′_μ(z),J^′_μ(z),J^″_μ(z), where J^′_μ(z) and J^″_μ(z) are the first two derivatives of the Bessel functions J_μ(z), for μ in general complex are proved. Bounds for the purely imaginary zeros of the above functions assuming their existence are given. Thus for the range of values for which these bounds are violated there are no purely imaginary zeros of the above functions. Finally, some known results from previous work are generalized in the present paper.     

Intersections of nest algebras in finite dimensions

If are maximal nests on a finite-dimensional Hilbert space H, the dimension of the intersection of the corresponding nest algebras is at least dim H. On the other hand, there are three maximal nests whose nest algebras intersect in the scalar operators. The dimension of the intersection of two nest algebras (corresponding to maximal nests) can be of any integer value from n to n(n+1)/2, where n=dim H. For any two maximal nests there exists a basis {f₁,f₂,…,f_n} of H and a permutation π such that and where M_i=  span{f₁,f₂,…,f_i} and N_i= span{f_π(1),f_π(2),…,f_π(i)}. The intersection of the corresponding nest algebras has minimum dimension, namely dim H, precisely when π(j)=n−j+1,1jn. Those algebras which are upper-triangular matrix incidence algebras, relative to some basis, can be characterised as intersections of certain nest algebras.     

Analysis and numerics for a parabolic equation with impulsive forcing

The paper considers a one-dimensional particle-continuum model, with impulsive interaction between the fluid and a number of pointwise particles. A simplification results in a system of ODEs coupled with a parabolic PDE forced by a nonlinear term involving a sum of Dirac delta functions. The existence of a mild solution is proved using a combination of energy estimates and semigroup theory. However, the regularity of these solutions is shown to be limited to C^0,1 by the impulsive terms. The convergence of a Galerkin method is established simultaneously with a proof of continuous dependence, and thus uniqueness, of solutions for the underlying system. The peculiarities of the system imply this analysis must be performed in L_∞. The C^0,1 regularity of the solution determines a suboptimal rate of convergence for the Galerkin method. The theoretical results are verified by MATLAB computations.     

Dynamic programming in problems of identifying distributed parameter systems

The problem of identifying the input of a system governed by a “semi-linear” evolution equation of parabolic type, based on the results of observations subject to undefined disturbances, is investigated. Estimates of the input, optimal in the sense of the so-called H_∞-criterion, are obtained. The information function of the system—the value function in an appropriate optimization problem—is evaluated. The relations between the information function and information sets are indicated. Optimality principles adequate to the proposed formulations of the problem are formulated and the corresponding dynamic programming equations are derived. Procedures for regularizing the problem, based on evolution equations of the input estimates, are proposed for the heat-conduction equation.     

On the stationary Lp-approximation power to derivatives by radial basis function interpolation

In this paper, we consider approximation to derivatives of a function by using radial basis function interpolation. Most of well-known theories for this problem provide error analysis in terms of the so-called native space, say C_φ. However, if a basis function φ is smooth, the space C_φ is extremely small. Thus, the purpose of this study is to extend this result to functions in the homogenous Sobolev space.     

Tensor product of operads and iterated loop spaces

A algebraic characterization of an n-fold loop space in terms of its n different 1-fold loop structures is established. This amounts to describing the higher homotopy commutativity for such a space as a strict partial commutativity of the 1-fold loop structures. The tensor product of operads (a special case of the construction for algebraic theories) is ideally suited for this. In particular we show that the operad of little n-cubes C_n is homotopy equivalent to the n-fold tensor product Cⁿ₁, i.e., ‘tensoring these A_∞-structures yields an iterated loop structure’. This is not true for arbitrary A_∞-operads.     

Spectra of multilevel toeplitz matrices: Advanced theory via simple matrix relationships

We consider the eigenvalue and singular-value distributions for m-level Toeplitz matrices generated by a complex-valued periodic function ƒ of m real variables. We show that familiar formulations for ƒ L_∞ (due to Szegő and others) can be preserved so long as f L₁, and what is more, with G. Weyl's definitions just a bit changed. In contrast to other approaches, the one we follow is based on simple matrix relationships.     

Planar and axially symmetric configurations which are circumvented with the maximum critical mach number

The structure of planar and axially symmetric configurations which, by satisfying a number of geometrical constraints, are circumvented in a boundless space or in a cylindrical channel by an ideal (non-viscous and non-thermally conducting) gas with a maximal critical Mach number M* is found. The analysis is carried out using the “rectilinearity property” of a sonic line in “subsonic” flows (SF), the “principle of a maximum” for an SF and “comparison theorems” which are either taken from /1/ or serve as a generalization of the corresponding assertions from /1/. Following /1/, configurations are considered which have a plane or axis of symmetry parallel to the velocity V_∞ of the approach stream, while flows in which (including the boundary) the Mach number M 1 are said to be “subsonic”. As usual, by M* we mean a value of M_∞ such that the inequality M1, which is satisfied in the whole stream when M_∞ M*, is violated when M_∞>M*.

The configurations investigated include closed bodies and the leading (trailing) parts of a semi-infinite plate or a circular cylinder in an unbounded flow and in a channel as well as lattices of symmetric profiles. Both in /1/, where the structure of closed planar and axially symmetric bodies was found, as well as in /2/, where such bodies were constructed numerically, the generatrices of all the configurations investigated contain the end planes or the segments replacing them of the maximum permissible slope (in modulus) and the “free” streamlines with M 1. Now, however, unlike in /1, 2/, segments of the horizontals are added to it in the general case. Furthermore, in the case of flows in channels and lattices, the configurations which have been found can be circumvented with the development of finite domains of advancing sonic flow.     

AHP多路径排序度量转换中的非线性问题

由于不同测量条件下的测量结果不是线性可加,AHP用矩阵乘法实现多路径序转换值得商榷.自隶属度从只取"1或0"两个值扩展到可取[0,1]区间上一切实数,可表征界于"是"与"不是"之间所有可能"部分是"模糊状态时起,对二值逻辑的研究已拓展到研究近似推理的模糊逻辑.这是逻辑的一个新的研究方向,目的是在隶属度转换过程中,通过对人类近似推理本领进行规范,使得到的目标值是"真值"在当前条件下的最优近似.模糊逻辑的量化方法是数值计算;推理依据是区分权滤波的冗余理论;实质性计算是由冗余理论导出的、实现隶属度转换的非线性去冗算法;所建的隶属度转换模型也是不同测量条件下高维状态空间上测量结果的非线性可加模型.将一维测量数据映射到高维状态空间上表为隶属度向量,可借助隶属度转换模型解决AHP多路径序转换的非线性计算.     

A note on the representability of compact sets by real sequences

Consider the set K of all nonempty compact subsets of a compact metric space (M, d), endowed with the Hausdorff metric. In this paper, we prove that K is isometric to a subset of l_∞( ). An approximation result is also proved.     

Full quadrature sums for pth powers of polynomials with Freud weights

In this paper, we provide a solution of the quadrature sum problem of R. Askey for a class of Freud weights. Let r> 0, b (− ∞, 2]. We establish a full quadrature sum estimate

1 p < ∞, for every polynomial P of degree at most n + rn^1/3, where W² is a Freud weight such as exp(−¦x¦), > 1, λ_jn are the Christoffel numbers, x_jn are the zeros of the orthonormal polynomials for the weight W², and C is independent of n and P. We also prove a generalisation, and that such an estimate is not possible for polynomials P of degree M = m(n) if m(n) = n + ξ_nn^1/3, where ξ_n → ∞ as n → ∞. Previous estimates could sum only over those x_jn with ¦x_jn¦ σx_1n, some fixed 0 < σ < 1.     

路的字典积的邻和可区别边染色

图G的正常[k]-边染色σ是指颜色集合为[k]={1,2,…,k}的G的一个正常边染色.用w_σ（x）表示顶点x关联边的颜色之和,即w_σ（x）=∑_e∋x σ（e）,并称w_σ（x）关于σ的权.图G的k-邻和可区别边染色是指相邻顶点具有不同权的正常[k]-边染色,最小的k值称为G的邻和可区别边色数,记为χ'_∑（G）.现得到了路P_n与简单连通图H的字典积P_n[H]的邻和可区别边色数的精确值,其中H分别为正则第一类图、路、完全图的补图.