When is a linear convolution system stabilizable?

Lack of operator closedness and closability arguments are used to discuss the impossibility of finite ℓ_p/L_p gain stabilization of unstable finite-dimensional linear (convolution operator) systems in the doubly infinite time-axis case for any 1p∞. The presented analysis generalizes and refines a result due to Georgiou and Smith (IEEE Trans. Automat. Control 40 (1995) 516) for the L₂(−∞,∞) setting.     

A small gain theorem for linear stochastic systems

A well-known result in linear control theory is the so-called “small gain” theorem stating that if given two plants with transfer matrix functions T₁ and T₂ in H^∞ such that T₁ < γ and T₂ < 1/γ, when coupling T₂ to T₁ such that u₂ = y₁ and u₁ = y₂ one obtains an internally stable closed system. The aim of the present paper is to describe a corresponding result for stochastic systems with state-dependent white noise.     

Robust H∞ control design for systems with structured parameter uncertainty

In a recent paper a unification of the H₂ (LQG) and H_∞ control-design problems was obtained in terms of modified algebraic Riccati equations. In the present paper these results are extended to guarantee robust H₂ and H_∞ performance in the presence of structured real-valued parameter variiations (ΔA, ΔB, ΔC) in the state space model. For design flexibility the paper considers two distinct types of uncertainty bounds for both full- and reduced-order dynamic compensation. An important special case of these results generates H₂/H_∞ controller designs with guaranteed gain margins.     

A comment on “L∞ optimal control of SISO continuous-time systems”

The paper “L_∞ optimal control of SISO continuous-time systems” by Wang and Sznaier (Wang & Sznaier (1997). Automatica, 33 (1), 85–90) studies the problem of designing a controller that optimally minimizes the peak absolute value of system output, due to a fixed input signal. With a newly defined function space A_∞, it was claimed that the set of all L_∞-bounded outputs could be parameterized and that the problem could be transformed to a minimal distance problem on L_∞ space. We believe, however, their formulation has essential flaws.     

Simultaneous H control of a finite collection of linear plants with a single nonlinear digital controller

This paper considers the problem of simultaneous H^∞ control of a finite collection of linear time-invariant systems via a nonlinear digital output feedback controller. The main result is given in terms of the existence of suitable solutions to Riccati algebraic equations and a dynamic programming equation. Our main result shows that if the simultaneous H^∞ control problem for k linear time-invariant plants of orders n₁,n₂,…,n_k can be solved, then this problem can be solved via a nonlinear time-invariant controller of order nn₁+n₂++n_k.     

Simultaneous H2/H∞ optimal control The state feedback case

A simultaneous H₂/H_∞ control problem is considered. This problem seeks to minimize the H₂ norm of a closed-loop transfer matrix while simultaneously satisfying a prescribed H_∞ norm bound on some other closed-loop transfer matrix by utilizing dynamic state feedback controllers. Such a problem was formulated earlier by Rotea and Khargonekar (Automatica, 27, 307–316, 1991) who considered only so called regular problems. Here, for a class of singular problems, necessary and sufficient conditions are established so that the posed simultaneous H₂/H_∞ problem is solvable by using state feedback controllers. The class of singular problems considered have a left invertible transfer function matrix from the control input to the controlled output which is used for the H₂ norm performance measure. This class of problems subsumes the class of regular problems.     

基于核技巧和超图正则的稀疏非负矩阵分解

针对传统的非负矩阵分解（NMF）应用于聚类时,没有同时考虑到鲁棒性和稀疏性,导致聚类性能较低的问题,提出了基于核技巧和超图正则的稀疏非负矩阵分解算法（KHGNMF）。首先,在继承核技巧的良好性能的基础上,用L_2,1范数改进标准非负矩阵分解中的F范数,并添加超图正则项以尽可能多地保留原始数据间的内在几何结构信息;其次,引入L_2,1/2伪范数和L_1/2正则项作为稀疏约束合并到NMF模型中;最后,提出新算法并将新算法应用于图像聚类。在6个标准的数据集上进行验证,实验结果表明,相对于非线性正交图正则非负矩阵分解方法,KHGNMF使聚类性能（精度和归一化互信息）成功地提升了39%~54%,有效地改善和提高了算法的稀疏性和鲁棒性,聚类效果更好。     

基于Hadoop平台的并行DHP数据分析方法

由候选项集C₂生成频繁2-项集L₂是关联规则Apriori算法的一个瓶颈。直接哈希修剪（DHP）算法利用一个生成的Hash表H₂删减C₂中无用的候选项集,以此提高L₂的生成效率。但传统DHP算法是一个串行算法,不能有效处理较大规模数据。针对这一问题,提出DHP的并行化算法——H_DHP。首先,对DHP算法并行化策略的可行性进行了理论分析与证明;其次,基于Hadoop平台,把Hash表H₂的生成以及频繁项集L₁、L₃~L_k的生成方法进行了并行实现,并借助Hbase数据库生成关联规则。仿真实验结果表明：与传统DHP算法相比,H_DHP算法在数据的处理时间效率、处理数据集的规模大小,以及加速比和可扩展性等方面都有较好的性能。     

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.     

A limit characterization for the number of spanning trees of graphs

In this paper we propose a limit characterization of the behaviour of classes of graphs with respect to their number of spanning trees. Let {G_n} be a sequence of graphs G₀,G₁,G₂,… that belong to a particular class. We consider graphs of the form K_n−G_n that result from the complete graph K_n after removing a set of edges that span G_n. We study the spanning tree behaviour of the sequence {K_n−G_n} when n→∞ and the number of edges of G_n scales according to n. More specifically, we define the spanning tree indicator ({G_n}), a quantity that characterizes the spanning tree behaviour of {K_n−G_n}. We derive closed formulas for the spanning tree indicators for certain well-known classes of graphs. Finally, we demonstrate that the indicator can be used to compare the spanning tree behaviour of different classes of graphs (even when their members never happen to have the same number of edges).     

Discrete-time system modelling in Lp with orthonormal basis functions

In this paper, model sets for linear time-invariant systems spanned by fixed pole orthonormal bases are investigated. The obtained model sets are shown to be complete in L_p(T) (1<p<∞), the Lebesque spaces of functions on the unit circle T, and in C(T), the space of periodic continuous functions on T. The L_p norm error bounds for estimating systems in L_p(T) by the partial sums of the Fourier series formed by the orthonormal functions are computed for the case 1<p<∞. Some inequalities on the mean growth of the Fourier series are also derived. These results have application in estimation and model reduction.     

Integral averages and oscillation of second-order nonlinear differential equations

We present some criteria for the oscillation of the second-order nonlinear differential equation

Full-size image

where a ε C¹([t₀, ∞)) is a nonnegative function, q ε C ([t₀, ∞)) are allowed to change sign on [t₀, ∞), ψ, f ε C¹ , ψ(x) > 0, xf(x) > 0, f′(x) ≥ 0 for x ≠ 0. These criteria are obtained by using a general class of the parameter functions H(t,s) in the averaging techniques and represent extension, as well as improvement of known oscillation criteria of Philos and Purnaras for the generalized Emden-Fowler equation.     

A simpler formula for the singular values of a certain Hankel operator

This paper deals with the problem of computing the singular values and vectors of a Hankel operator with symbol m^*W where m ε H^∞ is arbitrary inner and W ε H^∞ is rational. A simplified version of the formula given in [6] is obtained for computing the singular values of the Hankel operator. This result is applied to the (one-block) H^∞ optimal control of SISO stable infinite dimensional plants and rational weights. Using this new formula a simple expression is derived for the H^∞ optimal controller whose structure was observed in [9].     

L approximation of a parabolic system regarded as a delay system

We present an approach to the problem of finding an L^∞ approximant of the infinite-dimensional system describing the diffusion of heat in a wall. We show that this system can be regarded as a delay system with Laplace variable √s. We are able to get results, established in Zwart et al. (1988), about partial fraction expansion for delay systems, achieved by some adjustment to the specificities of our particular case. The determination of an L^∞ approximant is realised in two steps, using the optimal Hankel-norm approximation.     

Computational complexity of a problem arising in fixed order output feedback design

This paper is concerned with a matrix inequality problem which arises in fixed order output feedback control design. This problem involves finding two symmetric and positive definitive matrices X and Y such that each satisfies a linear matrix inequality and that XY=I. It is well-known that many control problems such as fixed order output feedback stabilization, H_∞ control, guaranteed H₂ control, and mixed H₂/H_∞ control can all be converted into the matrix inequality problem above, including static output feedback problems as a special case. We show, however, that this matrix inequality problem is NP-hard.     

First-order spectra with one binary predicate

The spectrum, Sp(), of a sentence is the set of cardinalities of finite structures which satisfy . We prove that any set of integers which is in Func₁^∞, i.e. in the class of spectra of first-order sentences of type containing only unary function symbols, is also in BIN¹, i.e. in the class of spectra of first-order sentences of type involving only a single binary relation.

We give similar results for generalized spectra and some corollaries: in particular, from the fact that the large complexity class _cNTIME_RAM(cn) is included in Func₁^∞ for unary languages (n denotes the input integer), we deduce that the set of primes and many “natural” sets belong to BIN¹.

We also give some consequences for the image of spectra under polynomials of .     

On the type of a polynomial f, when f(0)f−f(0)f is a monomial

A polynomial is said to be of type (p₁, p₂, p₃) relative to a directed line in the complex plane if, counting multiplicities, it has p₁ zeros to the left of, p₂ zeros on, and p₃ zeros to the right of the line. In this paper we determine explicitly the types of all polynomials belonging to a very restricted (but infinite) family of polynomials. A polynomial ƒ belongs to this family if and only if its coefficients are such that the polynomial ƒ^*(0)ƒ(z)−ƒ(0) ƒ^*(z) is a monomial; here ƒ^* denotes the reflection of ƒ in the directed line.

A special case of the present result appeared in an earlier publication.     

Asymptotic expansions of Hankel transforms of functions with logarithmic singularities

Asymptotic expansions as λ → +∞ are obtained for the Hankel transform

whereJ_v(t) is the Bessel function of the first kind and v is a fixed complex number. The function \tf(t) is allowed to have an asymptotic expansion near the origin of the form

Here, Re _n ↑ +∞ and β_n is an arbitrary complex number.