Optimal finite-dimensional solution for a class of nonlinear observation problems

The filtering problem in a differential system with linear dynamics and observations described by an implicit equation linear in the state is solved in finite-dimensional recursive form. The original problem is posed as a deterministic fixed-interval optimization problem (FIOP) on a finite time interval. No stochastic concepts are used. Via Pontryagin's principle, the FIOP is converted into a linear, two-point boundary-value problem. The boundary-value problem is separated by using a linear Riccati transformation into two initial-value problems which give the equations for the optimal filter and filter gain. The optimal filter is linear in the state, but nonlinear with respect to the observation. Stability of the filter is considered on the basis of a related properly linear system. Three filtering examples are given.     

A Practical Method for Designing Linear Quadratic Regulator for Commensurate Fractional-Order Systems

The methods currently available for designing a linear quadratic regulator for fractional-order systems are either based on sufficient-type conditions for the optimality of functionals or generate very complicated analytical solutions even for simple systems. It follows that the use of such methods is limited to very simple problems. The present paper proposes a practical method for designing a linear quadratic regulator (assuming linear state feedback), Kalman filter, and linear quadratic Gaussian regulator/controller for commensurate fractional-order systems (in Caputo sense). For this purpose, considering the fact that in dealing with fractional-order systems the cost function of linear quadratic regulator has only one extremum, the optimal state feedback gains of linear quadratic regulator and the gains of the Kalman filter are calculated using a gradient-based numerical optimization algorithm. Various fractional-order linear quadratic regulator and Kalman filter design problems are solved using the proposed approach. Specifically, a linear quadratic Gaussian controller capable of tracking step command is designed for a commensurate fractional-order system which is non-minimum phase and unstable and has seven (pseudo) states.     

Numerical experiments on a hybrid WENO5 filter for shock‐capturing

In the present paper, a hybrid filter is introduced for high accurate numerical simulation of shock‐containing flows. The fourth‐order compact finite difference scheme is used for the spatial discretization and the third‐order Runge–Kutta scheme is used for the time integration. After each time‐step, the hybrid filter is applied on the results. The filter is composed of a linear sixth‐order filter and the dissipative part of a fifth‐order weighted essentially nonoscillatory scheme (WENO5). The classic WENO5 scheme and the WENO5 scheme with adaptive order (WENO5‐AO) are used to form the hybrid filter. Using a shock‐detecting sensor, the hybrid filter reduces to the linear sixth‐order filter in smooth regions for damping high frequency waves and reduces to the WENO5 filter at shocks in order to eliminate unwanted oscillations produced by the nondissipative spatial discretization method. The filter performance and accuracy of the results are examined through several test cases including the advection, Euler and Navier–Stokes equations. The results are compared with that of a hybrid second‐order filter and also that of the WENO5 and WENO5‐AO schemes.     

Optimization approach to the design of frequency sampling filters

Many digital signal processing applications require linear phase filtering. For applications that require narrow-band linear phase filtering, frequency sampling filters can implement linear phase filters more efficiently than the commonly used direct convolution filter. In this paper, a technique is developed for designing linear phase frequency sampling filters. A frequency sampling filter approximates a desired frequency response by interpolating a frequency response through a set of frequency samples taken from the desired frequency response. Although the frequency response of a frequency sampling filter passes through the frequency samples, the frequency response may not be well behaved between the specific samples. Linear programming is commonly used to control the interpolation errors between frequency samples. The design method developed in this paper controls the interpolation errors between frequency samples by minimizing the mean square error between the desired and actual frequency responses in the stopband and passband. This design method describes the frequency sampling filter design problem as a constrained optimization problem which is solved using the Lagrange multiplier optimization method. This results in a set of linear equations which when solved determine the filter's coefficients.This work was partially funded by The National Supercomputing Center for Energy and the Environment, University of Nevada Las Vegas, Las Vegas, Nevada and by NSF Grant MIP-9200581.     

中心对称仿酉矩阵的构造及二元具有线性相位正交小波滤波器组的参数化

中心对称仿酉矩阵(简记为CSPM)在线性相位的小波滤波器组的构造中起着重要的作用,本文给出偶数阶CSPM的表达式,矩阵中的元素为二元一次多项式.基于已给出的CSPM,给出具有线性相位的二元正交小波滤波器组的参数化,通过选取不同的参数可以得到的具有线性相位的正交小波滤波器组.最后给出算例.     

The moment Lyapunov exponent for a three-dimensional stochastic system

This paper presents a method, through which the pth moment stability of a linear multiplicative stochastic system, that is a linear part of a co-dimension two-bifurcation system upon a three-dimensional center manifold and is subjected to a parametric excitation by an ergodic real noise, is obtained. The excitation included is assumed to be an integrable function of an n-dimensional Ornstein–Uhlenbeck vector process that is the output of a linear filter system and both the strong mixing condition, which is the sufficient condition for the stochastic averaging method, and the delicate balance condition are removed in the present study. By using a perturbation method and the spectrum representations of both the Fokker Planck operator and its adjoint one of the linear filter system, the asymptotic expressions of the moment Lyapunov exponent are obtained, which match the numerical results well.     

A Kalman filter as a minimax estimator

A minimax terminal state estimation problem is posed for a linear plant and a generalized quadratic loss function. Sufficient conditions are developed to insure that a Kalman filter will provide a minimax estimate for the terminal state of the plant. It is further shown that this Kalman filter will not generally be a minimax estimate for the terminal state if the observation interval is arbitrarily long. Consequently, a subminimax estimate is defined, subject to a particular existence condition. This subminimax estimate is related to the Kalman filter, and it may provide a useful estimate for the terminal state when the performance of the Kalman filter is no longer satisfactory.     

Kalman滤波的自适应算法

1 引 言 本文,我们讨论时不变线性随机系统 这里A、Γ和C分别是已知的n×n,n×p和q×n阶常数矩阵,1≤p,q≤n,且{ξ_k}{η_k}是均 值为零的高斯白噪声序列,有     

Generalized H ∞-optimal filtering under external and initial perturbations

We consider the problem of optimal filtering of unmeasured variables of a linear dynamical system by linear stationary filters. The filtering performance functional to be minimized is given by the maximum relative integral filtering error over all external perturbations as well as initial perturbations caused by the unknown initial conditions of the system. We show that the optimal filter implements a trade-off between an H _∞-optimal filter and a minimax observer.     

Factorizations and construction of linear phase paraunitary filter banks and higher multiplicity wavelets

It is known that paraunitary matrices can be factorized into shift products of orthogonal matrices or linear factors. When the number of rows of such a matrix (i.e. the number of channels of a paraunitary filter bank) is even, the symmetry constraints corresponding to the linear phase property of the filter bank can be expressed as restrictions on factors — except the very first one, all must be centrosymmetric. For odd numbers of rows the situation is more complicated. It turns out that paraunitary matrices comprising an even number of square blocks do not exist and quadratic centrosymmetric factors have to be used in the 0-shift product factorization. The centrosymmetric linear and quadratic factors can be easily obtained from partitions of centrosymmetric orthogonal matrices. Their parameterizations are also described.The characterizations of paraunitary matrices obtained from these factorizations are complete; the question of the number of free parameters is discussed. Furthermore, the proposed factorizations also allow us to derive lattice structures for linear phase paraunitary filter banks and, since the basic regularity conditions can be incorporated as a constraint on the first factor, they can be used also for the construction of symmetric higher multiplicity wavelets.and Cooperative Research Centre for Sensor Signal and Information ProcessingThe author is an Overseas Postgraduate Research Scholar supported by the Australian Government.     

具有一定消失矩的优化线性相位滤波器

利用消失矩特性和编码误差最小,给出了一类有限长线性相位的双正交小波滤波器组BNVF的构造方法,BNVF的综合低通和分解高通的系数为二进分数,将其用于图像的分解与重构时,可避免一些乘法运算．三组BNVF用于图像变换编码时,其压缩性能不亚于Cohen,Daubechies等构造的9／7步滤波器组CDF-9／7,且计算复杂度更低．     

An SQP-filter method for inequality constrained optimization and its global convergence

In this paper, we combine the filter technique with a modified sequential quadratic programming (SQP) method. The optimization solution is obtained by reducing step length, which is obtained by an exact linear search. Furthermore, this method can start with an infeasible initial point. The method uses a filter to promote global convergence.     

Filtering and Control with Information Increasing

In this paper, we consider a filtering problem where the observation filtration is enlarged with a future information. In the linear case, we obtain the filter equations and study the associated linear regulator problem.     

Linearization of nonlinear filter generators and its application to cryptanalysis of stream ciphers

Nonlinear filter generators are commonly used as keystream generators in stream ciphers. A nonlinear filter generator utilizes a nonlinear filtering function to combine the outputs of a linear feedback shift register (LFSR) to improve the linear complexity of keystream sequences. However, the LFSR-based stream ciphers are still potentially vulnerable to algebraic attacks that recover the key from some keystream bits. Although the known algebraic attacks only require polynomial time complexity of computations, all have their own constraints. This paper uses the linearization of nonlinear filter generators to cryptanalyze LFSR-based stream ciphers. Such a method works for any nonlinear filter generators. Viewing a nonlinear filter generator as a Boolean network that evolves as an automaton through Boolean functions, we first give its linearization representation. Compared to the linearization representation in Limniotis et al. (2008), this representation requires lower spatial complexity of computations in most cases. Based on the representation, the key recoverability is analyzed via the observability of Boolean networks. An algorithm for key recovery is given as well. Compared to the exhaustive search to recover the key, using this linearization representation requires lower time complexity of computations, though it leads to exponential time complexity.     

一种非单调滤子信赖域算法解线性不等式约束优化

本文给出了一种新的多维滤子算法结合非单调信赖域策略解线性约束优化.目标函数及其投影梯度的分量组成了新的多维滤子，并且与信赖域半径有关.当信赖域半径充分小时，新的滤子能接受试探点，避免算法无限循环.非单调信赖域策略保证了新算法的整体收敛性.目前为止，多维滤子算法局部收敛性分析仍然没有解决，在合理假设下，我们分析了新算法的局部超线性收敛性.数值结果验证了算法的有效性.     

Linear Matrix Inequality Optimization Approach to Exponential Robust Filtering for Switched Hopfield Neural Networks

This paper is concerned with the delay-dependent exponential robust filtering problem for switched Hopfield neural networks with time-delay. A new delay-dependent switched exponential robust filter is proposed that results in an exponentially stable filtering error system with a guaranteed robust performance. The design of the switched exponential robust filter for these types of neural networks can be achieved by solving a linear matrix inequality (LMI), which can be easily facilitated using standard numerical packages. An illustrative example is given to demonstrate the effectiveness of the proposed filter.     

Lanczos-Based Exponential Filtering for Discrete Ill-Posed Problems

We describe regularizing iterative methods for the solution of large ill-conditioned linear systems of equations that arise from the discretization of linear ill-posed problems. The regularization is specified by a filter function of Gaussian type. A parameter determines the amount of regularization applied. The iterative methods are based on a truncated Lanczos decomposition and the filter function is approximated by a linear combination of Lanczos polynomials. A suitable value of the regularization parameter is determined by an L-curve criterion. Computed examples that illustrate the performance of the methods are presented.     

An Improved Method for Designing Quadrature Mirror Filter Banks via Unconstrained Optimization

This paper proposes an algorithm to design a two-channel linear phase quadrature mirror filter (QMF) bank. The design problem is presented systematically as an unconstrained optimization that minimizes the weighted sum of error of transfer function of the filter bank at quadrature frequency, stopband energy and the passband error of a prototype filter (PF). A new method is developed for the design of a low pass prototype filter for QMF banks. For solving given optimization problem, Quasi-Newton optimization technique is used. Numerical examples and comparisons with several existing methods are included to show the performances and effectiveness of this method. An application of the proposed method is considered in the area of subband coding of the images.     

ON ENERGY CONSERVATION BY TRIGONOMETRIC INTEGRATORS IN THE LINEAR CASE WITH APPLICATION TO WAVE EQUATIONS

Trigonometric integrators for oscillatory linear Hamiltonian differential equations are considered. Under a condition of Hairer & Lubich on the filter functions in the method, a modified energy is derived that is exactly preserved by trigonometric integrators. This implies and extends a known result on all-time near-conservation of energy. The extension can be applied to linear wave equations.     

具有Markov跳变参数的广义系统鲁棒故障检测

针对一类广义不确定时滞Markov跳变系统,应用鲁棒控制理论分析重构的增广系统,获取了系统的故障检测滤波器.利用构造的随机Lyapunov-Krasovskii函数和线性矩阵不等式,证明并给出了故障检测滤波器有解的充分条件和优化设计方法,同时使得残差体现其对扰动的抑制作用和对故障的灵敏性.所设计的滤波器使系统具有随机稳定性,抑制干扰能力强,满足所给的范数指标.仿真算例对结果进行了验证,在故障信息发生时,故障检测滤波器可以很灵敏地检测出故障,并能有效地抑制未知扰动对残差的影响在给定的范围内.