On Padé-type model order reduction of J-Hermitian linear dynamical systems

A simple, yet powerful approach to model order reduction of large-scale linear dynamical systems is to employ projection onto block Krylov subspaces. The transfer functions of the resulting reduced-order models of such projection methods can be characterized as Padé-type approximants of the transfer function of the original large-scale system. If the original system exhibits certain symmetries, then the reduced-order models are considerably more accurate than the theory for general systems predicts. In this paper, the framework of J-Hermitian linear dynamical systems is used to establish a general result about this higher accuracy. In particular, it is shown that in the case of J-Hermitian linear dynamical systems, the reduced-order transfer functions match twice as many Taylor coefficients of the original transfer function as in the general case. An application to the SPRIM algorithm for order reduction of general RCL electrical networks is discussed.     

Some properties of the Hessian of the logarithmic barrier function

More than twenty years ago, Murray and Lootsma showed that Hessian matrices of the logarithmic barrier function become increasingly ill-conditioned at points on the barrier trajectory as the solution is approached. This paper explores some further characteristics of the barrier Hessian. We first show that, except in two special cases, the barrier Hessian is ill-conditioned in an entire region near the solution. At points in a more restricted region (including the barrier trajectory itself), this ill-conditioning displays a special structure connected with subspaces defined by the Jacobian of the active constraints. We then indicate how a Cholesky factorization with diagonal pivoting can be used to detect numerical rank-deficiency in the barrier Hessian, and to provide information about the underlying subspaces without making an explicit prediction of the active constraints. Using this subspace information, a close approximation to the Newton direction can be calculated by solving linear systems whose condition reflects that of the original problem.     

A geometrical numerical linear algebra approach to residual-enhanced parametric model reduction

The most prominent approaches to model reduction share the general principle, that a given large-scale system is projected onto a suitable subspace spanned by a low-dimensional basis. The projection basis essentially determines the approximation quality of the resulting reduced order system. Nonlinear parametric dependencies may be taken into account by applying interpolation techniques on matrix manifolds for computing projection bases and/or reduced systems at arbitrary parameter conditions, see [1] for a recent survey. In this short note, an original approach to enhance the prediction capabilities of the interpolated reduced-order projection bases for parametrized linear systems is proposed. To this end, a suitable residual-based goal function is introduced, which measures the approximation quality of a given projection subspace. As this function is invariant under changes of basis, it is to be considered as a mapping on the Grassmann-Manifold Gr(n, p) of p-dimensional subspaces of ℝⁿ. As such, it may be optimized using the conjugate gradient method specifically adapted to the manifold structure of the Grassmannian Gr(n, p), [2,3]. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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??In this paper, semiparametric estimation of a regression function in the third order partially linear autoregressive model with first order autoregressive errors is mainly studied. We suppose that the regression function has a parametric framework, and use the conditional least squares method to obtain the parameter estimators. Then semiparametric estimators of the regression function can be given by combining with the nonparametric kernel function adjustment. Furthermore, under certain conditions, the consistency of the estimators is proved. Finally, simulation research is presented to evaluate the effectiveness of the proposed method.     

仿射内点既约投影Hessian算法解非线性约束优化

本文结合非单调内点回代技术,提供了新的仿射信赖域方法解含有非负变量约束和非线性等式约束的优化问题.为求解大规模问题,采用等式约束的Jacobian矩阵的QR分解和两块校正的双边既约Hessian矩阵投影,将问题分解成零空间和值空间两个信赖域子问题.零空间的子问题为通常二次目标函数只带椭球约束的信赖域子问题,而值空间的子问题使用满足信赖域约束参数的值空间投影向量方向.通过引入Fletcher罚函数作为势函数,将由两个子问题结合信赖域策略构成的合成方向,并使用非单调线搜索技术回代于可接受的非负约束内点步长.在合理的条件下,算法具有整体收敛性且两块校正的双边既约Hessian投影法将保持超线性收敛速率.非单调技术将克服高度非线性情况,加快收敛进展.     

Generalized Hopf bifurcation emerged from a corner in general planar piecewise smooth systems

We investigate a generalized Hopf bifurcation emerged from a corner located at the origin which is the intersection of n$n$ discontinuity boundaries in planar piecewise smooth dynamical systems with the Jacobian matrix of each smooth subsystem having either two different nonzero real eigenvalues or a pair of complex conjugate eigenvalues. We obtain a novel result that the generalized Hopf bifurcation can occur even when the Jacobian matrix of each smooth subsystem has two different nonzero real eigenvalues. According to the eigenvalues of the Jacobian matrices and the number of smooth subsystems, we provide a general method and prove some generalized Hopf bifurcation theorems by studying the associated Poincaré map.     

Bifurcations and Exact Solutions of the Raman Soliton Model in Nanoscale Optical Waveguides with Metamaterials

In this paper, we study Raman soliton model in nanoscale optical waveguides with metamaterials, having polynomial law non-linearity. By using the bifurcation theory method of dynamical systems to the equations of $\phi(\xi)$, under 24 different parameter conditions, we obtain bifurcations of phase portraits and different traveling wave solutions including periodic solutions, homoclinic and heteroclinic solutions for planar dynamical system of the Raman soliton model. Under different parameter conditions, 24 exact explicit parametric representations of the traveling wave solutions are derived. The dynamic behavior of these traveling wave solutions are meaningful and helpful for us to understand the physical structures of the model.     

随机截尾下污染数据部分线性模型的强相合估计

考虑一类新的污染数据部分线性模型,当受污染后的因变量被随机右截断时,就截断分布已知的情形,利用所获得截断观测数据构造了模型中的参数分量,非参数分量及污染系数的估计量,并在适当的条件下,证明了这些估计量的强相合性.     

Equi-surjective systems of linear operators and applications

In this paper we study a system of linear operators between finite-dimensional Euclidean spaces. Emphasis is made on unbounded systems and sufficient conditions are established for their equi-surjectivity. An application is presented in which a system of approximate Jacobian matrices is used to obtain a parametric interior mapping theorem. A multiplier rule for vector problems is also derived.     

Estimates for the accuracy of the projection-difference method for an abstract hyperbolic-parabolic system like systems of thermoelasticity equations

We obtain energy estimates for the accuracy of a three-layer scheme of the projection- difference method for a system of abstract differential equations in a Hilbert space, which generalizes a number of linear systems of coupled thermoelasticity equations in the absence of any special conditions on the projection subspaces.     

纵向数据半参数回归模型的估计方法

本文考虑纵向数据半参数回归模型,通过考虑纵向数据的协方差结构,基于Profile最小二乘法和局部线性拟合的方法建立了模型中参数分量、回归函数和误差方差的估计量,来提高估计的有效性,在适当条件下给出了这些估计量的相合性.并通过模拟研究将该方法与最小二乘局部线性拟合估计方法进行了比较,表明了Profile最小二乘局部线性拟合方法在有限样本情况下具有良好的性质.     

Transforms from differential equations to difference equations and vice-versa applied to computer control systems

This letter derives the transform relationship between differential equations to difference equations and vice-versa, applied to computer control systems. The key is to obtain the rational fraction transfer function model of a time-invariant linear differential equation system, using the Laplace transform, and to obtain the impulse transfer function model of a time-invariant linear difference equation, using the shift operator. Finally, we find the discrete-time models of the first-order, second-order and third-order systems from their continuous-time models and vice-versa and find the mapping relationship between the coefficients of discrete-time models and the continuous-time models using the bilinear transform. An example is provided to demonstrate the proposed model transform methods.     

Krylov Subspaces Associated with Higher-Order Linear Dynamical Systems

A standard approach to model reduction of large-scale higher-order linear dynamical systems is to rewrite the system as an equivalent first-order system and then employ Krylov-subspace techniques for model reduction of first-order systems. This paper presents some results about the structure of the block-Krylov subspaces induced by the matrices of such equivalent first-order formulations of higher-order systems. Two general classes of matrices, which exhibit the key structures of the matrices of first-order formulations of higher-order systems, are introduced. It is proved that for both classes, the block-Krylov subspaces induced by the matrices in these classes can be viewed as multiple copies of certain subspaces of the state space of the original higher-order system. AMS subject classification (2000) 65F30, 15A57, 65P99, 41A21     

Asymptotic properties of wavelet estimators in partially linear errors-in-variables models with long-memory errors

While the random errors are a function of Gaussian random variables that are stationary and long dependent, we investigate a partially linear errors-in-variables (EV) model by the wavelet method. Under general conditions, we obtain asymptotic representation of the parametric estimator, and asymptotic distributions and weak convergence rates of the parametric and nonparametric estimators. At last, the validity of the wavelet method is illuminated by a simulation example and a real example.     

Bias-corrected statistical inference for partially linear varying coefficient errors-in-variables models with restricted condition

In this paper, we consider the statistical inference for the partially liner varying coefficient model with measurement error in the nonparametric part when some prior information about the parametric part is available. The prior information is expressed in the form of exact linear restrictions. Two types of local bias-corrected restricted profile least squares estimators of the parametric component and nonparametric component are conducted, and their asymptotic properties are also studied under some regularity conditions. Moreover, we compare the efficiency of the two kinds of parameter estimators under the criterion of Lo?ner ordering. Finally, we develop a linear hypothesis test for the parametric component. Some simulation studies are conducted to examine the finite sample performance for the proposed method. A real dataset is analyzed for illustration.     

Predictive control of nonlinear dynamic processes

Predictive control can be applied if the reference value of the process is known in advance and the deterministic disturbances can be predicted. A cost function defined in the future horizon is minimized. The control signal is calculated for a control horizon, but only the first one is applied and the procedure is repeated (receding horizon strategy). Processes with mild analytical nonlinear characteristics are considered. The possible process models are either nonparametric (linear, Hammerstein, and Volterra weighting function series) or parametric ones (generalized Hammerstein, parametric Volterra, and bilinear models). The algorithms of the optimal and suboptimal predictive control based on the nonparametric and the parametric models mentioned are derived. Several simulations present how effective these methods are. The adaptive case is dealt with as well.     

Decentralized iterative learning control for a class of large scale interconnected dynamical systems

The problem of decentralized iterative learning control for a class of large scale interconnected dynamical systems is considered. In this paper, it is assumed that the considered large scale dynamical systems are linear time-varying, and the interconnections between each subsystem are unknown. For such a class of uncertain large scale interconnected dynamical systems, a method is presented whereby a class of decentralized local iterative learning control schemes is constructed. It is also shown that under some given conditions, the constructed decentralized local iterative learning controllers can guarantee the asymptotic convergence of the local output error between the given desired local output and the actual local output of each subsystem through the iterative learning process. Finally, as a numerical example, the system coupled by two inverted pendulums is given to illustrate the application of the proposed decentralized iterative learning control schemes.     

Multivalued linear projections

A multivalued linear projection operator P defined on linear space X is a multivalued linear operator which is idempotent and has invariant domain. We show that a multivalued projection can be characterised in terms of a pair of subspaces and then establish that the class of multivalued linear projections is closed under taking adjoints and closures. We apply the characterisations of the adjoint and completion of a projection together with the closed graph and closed range theorems to give criteria for the continuity of a projection defined on a normed linear space. A new proof of the theorem on closed sums of closed subspaces in a Banach space (cf. Mennicken and Sagraloff [9, 10]) follows as a simple corollary. We then show that the topological decomposition of a space may be expressed in terms of multivalued projections. The paper is concluded with an application to multivalued semi-Fredholm relations with generalised inverses.