Recursive identification for EIV ARMAX systems

The input u _k and output y _k of the multivariate ARMAX system A(z)y _k = B(z)u _k + C(z)w _k are observed with noises: u _k ^ob ≜ u _k + ε _k ^u and y _k ^ob ≜ y _k + ε _k ^y , where ε _k ^u and ε _k ^y denote the observation noises. Such kind of systems are called errors-in-variables (EIV) systems. In the paper, recursive algorithms based on observations are proposed for estimating coefficients of A(z), B(z), C(z), and the covariance matrix Rw of w _k without requiring higher than the second order statistics. The algorithms are convenient for computation and are proved to converge to the system coefficients under reasonable conditions. An illustrative example is provided, and the simulation results are shown to be consistent with the theoretical analysis.     

Generalized dilations and the stabilization of uncertain systems via measurement feedback

We study the problem of semiglobally stabilizing uncertain nonlinear system

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, with (A,B) in Brunowski form. We prove that if p₁(z,u,t)u and p₂(z,u,t)u are of order greater than 1 and 0, respectively, with “generalized” dilation δ_l(z,u)=(l¹⁻ⁿz₁,…,l⁻¹z_n−1,z_n,lu) and uniformly with respect to t, where z_i is the ith component of z, then we can achieve semiglobal stabilization via arbitrarily bounded linear measurement feedback.     

Asymptotic behavior of nonoscillatory solutions of neutral difference equations

The authors consider the m^th-order neutral difference equation D_m(y(n) + p(n)y(n − k) + q(n)f(y(σ(n))) = e(n), where m ≥ 1, {p(n)}, {q(n)}, {e(n)}, and {a₁(n)}, {a₂(n)}, …, {a_m−1(n)} are real sequences, a_i(n) > 0 for i = 1,2,…, m−1, a_m(n) ≡ 1, D₀z(n) = y(n)+p(n)y(n − k), D_iz(n) = a_i(n)ΔD_i−1z(n) for i = 1,2, …, m, k is a positive integer, {σ(n)} → ∞ is a sequence of positive integers, and R → R is continuous with u f(u) > 0 for u ≠ 0. In the case where {q(n)} is allowed to oscillate, they obtain sufficient conditions for all bounded nonoscillatory solutions to converge to zero, and if {q(n)} is a nonnegative sequence, they establish sufficient conditions for all nonoscillatory solutions to converge to zero. Examples illustrating the results are included throughout the paper.     

Tensor and border rank of certain classes of matrices and the fast evaluation of determinant inverse matrix and eigenvalues

The tensor rankA of the linear spaceA generated by the set of linearly independent matricesA ₁, A₂, …, A_p, is the least integert for wich there existt diadsu ^(r) v ^(r)τ, τ=1,2,...,t, such that . IfA=n+k,k≪n then some computational problems concerning matricesA∈A can be solyed fast. For example the parallel inversion of almost any nonsingular matrixA∈A costs 3 logn+0(log² k) steps with max(n ²+p (n+k), k² n+nk) processors, the evaluation of the determinant ofA can be performed by a parallel algorithm in logp+logn+0 (log² k) parallel steps and by a sequential algorithm inn(1+k ²)+p (n+k)+0 (k ³) multiplications. Analogous results hold to accomplish one step of bisection method, Newton's iterations method and shifted inverse power method applied toA−λB in order to compute the (generalized) eigenvalues provided thatA, B∈A. The same results hold if tensor rank is replaced by border rank. Applications to the case of banded Toeplitz matrices are shown. Dedicated to Professor S. Faedo on his 70th birthday Part of the results of this paper has been presented at the Oberwolfach Conference on Komplexitatstheorie, November 1983     

Oscillation criteria of certain nonlinear partial difference equations

This paper is concerned with the nonlinear partial difference equation with continuous variables

,where a, σ_i, τ_i are positive numbers, h_i(x, y, u) ε C(R⁺ × R⁺ × R, R), uh_i(x, y, u) > 0 for u ≠ 0, h_i is nondecreasing in u, i = 1, …, m. Some oscillation criteria of this equation are obtained.     

A new unconditionally stable ADI compact scheme for the two-space-dimensional linear hyperbolic equation

We formulate a new alternating direction implicit compact scheme of O(τ²+h ⁴) for the linear hyperbolic equation u _tt +2α u _t +β² u=u _xx +u _yy +f(x, y, t), 0<x, y<1, 0<t≤T, subject to appropriate initial and Dirichlet boundary conditions, where α>0 and β≥0 are real numbers. In this article, we show the method is unconditionally stable by the Von Neumann method. At last, numerical demonstrations are given to illustrate our result.     

A sixth-order extrapolation method for special nonlinear fourth-order boundary value problems

A sixth-order convergent finite difference method is developed for the numerical solution of the special nonlinear fourth-order boundary value problem y^(iv)(x) = f(x, y), a < x < b, y(a) = A₀, y″(a) = B₀, y(b) = A₁ y′(b) = B₁, the simple-simple beam problem.The method is based on a second-order convergent method which is used on three grids, sixth-order convergence being obtained by taking a linear combination of the (second-order) numerical results calculated using the three individual grids.Special formulas are proposed for application to points of the discretization adjacent to the boundaries x = a and x= b, the first two terms of the local truncation errors of these formulas being the same as those of the second-order method used at the other points of each grid.Modifications to these two formulas are obtained for problems with boundary conditions of the form y(a) = A₀, y′(a) = C₀, y(b) = A₁, y′(b) = C₁, the clamped-clamped beam problem.The general boundary value problem, for which the differential equation is y^(iv)(x) = f(x, y, y′, y″, y‴), is also considered.     

The construction of mutually independent Hamiltonian cycles in bubble-sort graphs

A Hamiltonian cycle C=? u ₁, u ₂, …, u _n(G), u ₁ ? with n(G)=number of vertices of G, is a cycle C(u ₁; G), where u ₁ is the beginning and ending vertex and u _i is the ith vertex in C and u _i ≠u _j for any i≠j, 1≤i, j≤n(G). A set of Hamiltonian cycles {C ₁, C ₂, …, C _k } of G is mutually independent if any two different Hamiltonian cycles are independent. For a hamiltonian graph G, the mutually independent Hamiltonianicity number of G, denoted by h(G), is the maximum integer k such that for any vertex u of G there exist k-mutually independent Hamiltonian cycles of G starting at u. In this paper, we prove that h(B _n )=n?1 if n≥4, where B _n is the n-dimensional bubble-sort graph.     

数据仓库系统中层次式Cube存储结构

区域查询是数据仓库上支持联机分析处理(on-line analytical processing,简称OLAP)的重要操作.近几年,人们提出了一些支持区域查询和数据更新的Cube存储结构.然而这些存储结构的空间复杂性和时间复杂性都很高,难以在实际中使用.为此,提出了一种层次式Cube存储结构HDC(hierarchical data cube)及其上的相关算法.HDC上区域查询的代价和数据更新代价均为O(log^dn),综合性能为O((logn)^2d)(使用C_qC_u模型)或O(K(logn)^d)(使用C_qn_q+C_un_u模型).理论分析与实验表明,HDC的区域查询代价、数据更新代价、空间代价以及综合性能都优于目前所有的Cube存储结构.     

Constrained controllability of perturbed discrete-time systems

In this paper we study the global null controllability with bounded controls of perturbed linear discrete-time systems in Rⁿ . More explicitly, a globally null-controllable autonomous discrete-time system is perturbed by ‘suitably small’ terms V(k) and B(k) to obtain a system of the form x(k + 1) = [A + V(k)]x(k) + [B + B(k)]u(k) and sufficient conditions are given to ensure the global null controllability of the resulting perturbed system.     

How to improve mamdani's approach to fuzzy control

Fuzzy control is a methodology that translates “if”-“then” rules, A_ji (x₁) &…& A_jn(x_n) → B_j(u), formulated in terms of a natural language, into an actual control strategy u(x). Implication of uncertain statements is much more difficult to understand than “and,” “or,” and “not.” So, the fuzzy control methodologies usually start with translating “if”-“then” rules into statements that contain only “and,” “not,” and “or.” the first such translation was proposed by Mamdani in his pioneer article on fuzzy control. According to this article, a fuzzy control is reasonable iff one of the rules is applicable, i.e., either the first rule is applicable (A₁₁(x₁) &…& A_1n(x_n) & B₁(u)), or the second one is applicable, etc. This approach turned out to be very successful, and it is still used in the majority of fuzzy control applications. However, as R. Yager noticed, in some cases, this approach is not ideal: Namely, if for some x, we know what u(x) should be, and add this crisp rule to our rules, then the resulting fuzzy control for this x may be different from the desired value u(x). to overcome this drawback, Yager proposed to assign priorities to the rules, so that crisp rules get the highest priority, and use these priorities while translating the rules into a control strategy u(x). In this article, we show that a natural modification of Mamdani's approach can solve this problem without adding any ad hoc priorities. © 1995 John Wiley & Sons, Inc.     

Nonlinear eigenvalue problems for quasilinear systems

The paper deals with the existence of positive solutions for the quasilinear system (Φ(u'))' + λh(t)f(u) = 0,0 < t < 1 with the boundary condition u(0) = u(1) = 0. The vector-valued function Φ is defined by Φ(u) = (q(t)(p(t)u₁), …, q(t)(p(t)u_n)), where u = (u₁, …, u_n), andcovers the two important cases (u) = u and (u) = up > 1, h(t) = diag[h₁(t), …, h_n(t)] and f(u) = (f¹(u), …, fⁿ (u)). Assume that fⁱ and h_i are nonnegative continuous. For u = (u₁, …, u_n), let

, f₀ = maxf¹₀, …, fⁿ₀ and f_∞ = maxf¹_∞, …, fⁿ_∞. We prove that the boundary value problem has a positive solution, for certain finite intervals of λ, if one of f₀ and f_∞ is large enough and the other one is small enough. Our methods employ fixed-point theorem in a cone.     

On a language without star

We solve a problem raised by Boasson [5] concerning the language B^*₁ = s{;xⁿ(y⁺x⁺)^*yⁿ vb; n ? 1s};^*. We prove that t he rational cone generated by B^*₁ which is closed under product but not under the star operation [5] does not contain any non-rational AFL.     

Embedding of Cycles in Twisted Cubes with Edge-Pancyclic

In this paper, we study the embedding of cycles in twisted cubes. It has been proven in the literature that, for any integer l, 4≤l≤2 ⁿ , a cycle of length l can be embedded with dilation 1 in an n-dimensional twisted cube, n≥3. We obtain a stronger result of embedding of cycles with edge-pancyclic. We prove that, for any integer l, 4≤l≤2 ⁿ , and a given edge (x,y) in an n-dimensional twisted cube, n≥3, a cycle C of length l can be embedded with dilation 1 in the n-dimensional twisted cube such that (x,y) is in C in the twisted cube. Based on the proof of the edge-pancyclicity of twisted cubes, we further provide an O(llog l+n ²+nl) algorithm to find a cycle C of length l that contains (u,v) in TQ _n for any (u,v)∈E(TQ _n ) and any integer l with 4≤l≤2 ⁿ .     

Universal stabilization of a class of nonlinear systems with homogeneous vector fields

Under relative-degree-one and minimum-phase assumptions, it is well known that the class of finite-dimensional, linear, single-input (u), single-output (y) systems (A,b,c) is universally stabilized by the feedback strategy u = Λ(λ)y, λ = y², where Λ is a function of Nussbaum type (the terminology “universal stabilization” being used in the sense of rendering /s(0/s) a global attractor for each member of the underlying class whilst assuring boundedness of the function λ(·)). A natural generalization of this result to a class ^k of nonlinear control systems (a,b,c), with positively homogeneous (of degree k 1) drift vector field a, is described. Specifically, under the relative-degree-one (cb ≠ 0) and minimum-phase hypotheses (the latter being interpreted as that of asymptotic stability of the equilibrium of the “zero dynamics”), it is shown that the strategy u = Λ(λ)/vby/vb^k−1y, assures ^k-universal stabilization. More generally, the strategy u = Λ(λ)exp(/vby/vb)y, assures -universal stabilization, where = _{k 1} ^k.     

On the Attainable Order of Collocation Methods for the Neutral Functional-Differential Equations with Proportional Delays

In this paper, we extend the recent results of H. Brunner in BIT (1997) for the DDE y′(t)= by(qt), y(0)=1 and the DVIE y(t)=1+∫₀ ^t by(qs)ds with proportional delay qt, 0<q≤1, to the neutral functional-differential equation (NFDE): and the delay Volterra integro-differential equation (DVIDE) : with proportional delays p _i t and q _i t, 0<p _i ,q _i ≤1 and complex numbers a,b _i and c _i . We analyze the attainable order of m-stage implicit (collocation-based) Runge-Kutta methods at the first mesh point t=h for the collocation solution v(t) of the NFDE and the `iterated collocation solution u _it (t)' of the DVIDE to the solution y(t), and investigate the existence of the collocation polynomials M _m (t) of v(th) or M^ _m (t) of u _it (th), t∈[0,1] such that the rational approximant v(h) or u _it (h) is the (m,m)-Padé approximant to y(h) and satisfies |v(h)−y(h)|=O(h ^{2 m +1}). If they exist, then we actually give the conditions of M _m (t) and M^ _m (t), respectively. Received September 17, 1998; revised September 30, 1999     

Adaptive suboptimal control of a linear system with bounded disturbance

An adaptive suboptimal control of a linear discrete system with unknown parameters is proposed. An additive disturbance v_t acting on the system is supposed to be uniformly bounded. The criterion is sup_vtI(y^∞₁, u^∞₁), where y_t is the output, u_t is the control. The adaptive control law gives almost the same guaranteed value of the criterion as the optimal linear feedback does for a system with known parameters.     

Feedback representations of critical controls for well-posed linear systems

This is the first part in a three part study of the suboptimal full information H^∞ problem for a well-posed linear system with input space U, state space H, and output space Y. We define a cost function Q(x₀,u)=∫〈y(s),Jy(s)〉_Yds, where y∈L²_loc( R ⁺; Y) is the output of the system with initial state x₀∈H and control u∈L²_loc( R ⁺; U), and J is a self-adjoint operator on Y. The cost function Qis quadratic in x₀ and u, and we suppose (in the stable case) that the second derivative of Q(x₀, u) with respect to u is non-singular. This implies that, for each x₀∈H, there is a unique critical control u^crit such that the derivative of Q(x₀, u) with respect to u vanishes at u=u^crit. We show that u^crit can be written in feedback form whenever the input/output map of the system has a coprime factorization with a (J, S)-inner numerator; here S is a particular self-adjoint operator on U. A number of properties of this feedback representation are established, such as the equivalence of the (J, S)-losslessness of the factorization and the positivity of the Riccati operator on the reachable subspace. © 1998 John Wiley & Sons, Ltd.     

Stable reduction of linear discrete-time systems via multipoint squared-magnitude CFE

This paper presents an elegant and computer-oriented procedure for the stable reduction of a linear discrete-time system via a multipoint continued-fraction expansion of its z-transfer function G(z). The proposed procedure involves the following three steps: (a) transform the z-domain squared-magnitude function P(z) = G(z)G(z ^?1) of the system frequency response to P(u) by the transformation u = z + z ^?1; (b) obtain an mth-order multipoint continued-fraction approximant P_m (u) to P(u); (c) factorize P_m (u) to yield a reduced z-transfer function G_m (z). The main feature of the procedure is that it guarantees the stability as well as the minimum-phase characteristics of the system and it also gives good overall approximations to both frequency and time responses.     

Stabilization and decay estimate for distributed bilinear systems

In this paper, we consider the problem of feedback stabilization for the distributed bilinear system y′(t)=Ay(t)+u(t)By(t). Here A is the infinitesimal generator of a linear C⁰ semigroup of contractions on a Hilbert space H and B:H→H is a linear bounded operator. A sufficient condition for feedback stabilization is given and explicit decay estimate is established. Applications to vibrating systems are presented.