Foley-Sammon optimal discriminant vectors using kernel approach

A new nonlinear feature extraction method called kernel Foley-Sammon optimal discriminant vectors (KFSODVs) is presented in this paper. This new method extends the well-known Foley-Sammon optimal discriminant vectors (FSODVs) from linear domain to a nonlinear domain via the kernel trick that has been used in support vector machine (SVM) and other commonly used kernel-based learning algorithms. The proposed method also provides an effective technique to solve the so-called small sample size (SSS) problem which exists in many classification problems such as face recognition. We give the derivation of KFSODV and conduct experiments on both simulated and real data sets to confirm that the KFSODV method is superior to the previous commonly used kernel-based learning algorithms in terms of the performance of discrimination.     

Active control of vibration using a neural network

Feedforward control of sound and vibration using a neural network-based control system is considered, with the aim being to derive an architecture/algorithm combination which is capable of supplanting the commonly used finite impulse response filter/filtered-x least mean square (LMS) linear arrangement for certain nonlinear problems. An adaptive algorithm is derived which enables stable adaptation of the neural controller for this purpose, while providing the capacity to maintain causality within the control scheme. The algorithm is shown to be simply a generalization of the linear filtered-x LMS algorithm. Experiments are undertaken which demonstrate the utility of the proposed arrangement, showing that it performs as well as a linear control system for a linear control problem and better for a nonlinear control problem. The experiments also lead to the conclusion that more work is required to improve the predictability and consistency of the performance before the neural network controller becomes a practical alternative to the current linear feedforward systems.     

Application of the harmonic acceleration method for nonlinear dynamic analysis

The application of the harmonic acceleration method, which shows some advantages in linear transient vibration analysis compared to the other commonly used methods, is extended to nonlinear problems employing the mode superposition technique and an iteration procedure. Stability analysis shows that the method is unconditionally stable. A higher degree of accuracy is achieved. The efficiency of the method is illustrated by some simple numerical examples.     

A Robust Solution to the Stereo-Vision-Based Simultaneous Localization and Mapping Problem with Steady and Moving Landmarks

The problem of visual simultaneous localization and mapping (SLAM) is examined in this paper using recently developed ideas and algorithms from modern robust control and estimation theory. A nonlinear model for a stereo-vision-based sensor is derived that leads to nonlinear measurements of the landmark coordinates along with optical flow-based measurements of the relative robot–landmark velocity. Using a novel analytical measurement transformation, the nonlinear SLAM problem is converted into the linear domain and solved using a robust linear filter. Actually, the linear filter is guaranteed stable and the SLAM state estimation error is bounded within an ellipsoidal set. A mathematically rigorous stability proof is given that holds true even when the landmarks move in accordance with an unknown control input. No similar results are available for the commonly employed extended Kalman filter, which is known to exhibit divergence and inconsistency characteristics in practice. A number of illustrative examples are given using both simulated and real vision data that further validate the proposed method.     

Adaptive control with nonlinear filtering

Nonlinear filters are not commonly used in parameter adaptive control. They make, however, a noteworthy alternative especially when state space models and control algorithms are used. In practice most of the well-known nonlinear filters like the extended Kalman filter suffer computational eomplexity and robustness problems. In the paper a filter is introduced which is not new but less well known and suffers less with these problems. Relationship with extended Kalman-type filters is shown. It is further shown how simultaneous state and parameter estimation can be done without using extended state techniques. Adaptive control design is based on a linear state space model and per interval quadratic performance index. A multivariable PI-type controller is obtained, the parameters of which can be easily calculated. As a case example application to servo control of a real life hydraulic manipulator is considered.     

Time discretized operators. Part 2: towards the theoretical design of a new generation of a generalized family of unconditionally stable implicit and explicit representations of arbitrary order for computational dynamics

The new generation of a generalized family of time discretized operators encompassing implicit and explicit representations that are unconditionally stable and which theoretically inherit Nth-order time accurate features developed in Part 1 are restricted here in Part 2 of the exposition to second-order time accurate operators. As such, unconditionally stable implicit representations are first described followed by unconditionally stable explicit representations. The theoretical design leading to computational algorithms with excellent algorithmic attributes for applicability to practical situations are also addressed for both the implicit and explicit unconditionally stable representations of time discretized operators. Attention is first focused on linear problems and extensions to nonlinear situations are subsequently briefly addressed.     

一类非线性切换系统的观测器设计

针对在实际工程系统中量测设备在经济性与使用性上存在限制,系统状态通常不能完全观测的问题,考虑了含李普希兹非线性干扰项的切换系统观测器设计问题,通过设计适当的切换策略,提出了一种全局二次稳定的观测器设计方案。该方案适用于有限和无限切换系统,其主要结果以线性矩阵不等式(LMI)的形式表达。该方案简单、易实现,仿真算例验证了其有效性。     

Recursive Iterative Algorithms for Adaptive Identification of Nonlinear Concentrated Dynamic Systems

Identification algorithms for concentrated dynamic systems are studied. The apparatus of pseudo-inversion of block matrices underlies adaptive identification. It is used to construct a recursive algorithm for linear least-squares problems and a recursive iterative algorithm for nonlinear least-squares problems.     

Adaptive estimation of discrete-time systems with nonlinear parameterization

This paper concerns adaptive estimation of dynamic systems which are nonlinearly parameterized. A majority of adaptive algorithms employ a gradient approach to determine the direction of adjustment, which ensures stable estimation when parameters occur linearly. These algorithms, however, do not suffice for estimation in systems with nonlinear parameterization. We introduce in this paper a new algorithm for such systems and show that it leads to globally stable estimation by employing a different regression vector and selecting a suitable step size. Both concave/convex parameterizations as well as general nonlinear parameterizations are considered. Stable estimation in the presence of both nonlinear parameters and linear parameters which may appear multiplicatively is established. For the case of concave/convex parameterizations, parameter convergence is shown to result under certain conditions of persistent excitation.     

Variable projection algorithms with sparse constraint for separable nonlinear models

Separable nonlinear models are widely used in various fields such as time series analysis, system modeling, and machine learning, due to their flexible structures and ability to capture nonlinear behavior of data. However, identifying the parameters of these models is challenging, especially when sparse models with better interpretability are desired by practitioners. Previous theoretical and practical studies have shown that variable projection (VP) is an efficient method for identifying separable nonlinear models, but these are based on L2 penalty of model parameters, which cannot be directly extended to deal with sparse constraint. Based on the exploration of the structural characteristics of separable models, this paper proposes gradient-based and trust-region-based variable projection algorithms, which mainly solve two key problems: how to eliminate linear parameters under sparse constraint; and how to deal with the coupling relationship between linear and nonlinear parameters in the model. Finally, numerical experiments on synthetic data and real time series data are conducted to verify the effectiveness of the proposed algorithms.     

Optimal control problems treated with algorithms of linear and nonlinear programming

Algorithms solving optimal control problems for linear discrete systems and linear continuous systems (without discretization) are discussed. The algorithms are based on a new approach to solving linear programming problems worked out in Minsk (USSR). A new method for solving nonlinear programming problems is justified. It uses the network interpretation of nonlinear functions and special network operations. Results of numerical experiment (on geometric programming problems) are given. In conclusion an algorithm of solving optimal control problem for the system with nonlinear input is described.     

经典线性算法的非线性核形式

经典线性算法的非线性核形式是近10年发展起来的一类非线性机器学习技术．它们最显著的特点是利用满足Mercer条件的核函数巧妙地推导出线性算法的非线性形式。并表述为与样本数目有关、与维数无关的优化问题．为了提高数值计算的稳定性、控制算法的推广能力以及改善迭代过程的收敛性。部分算法还采用了正则化技术．在概述核思想与核函数、正则化技术的基础上,系统地介绍了经典线性算法的非线性核形式,同时分析它们的优缺点,井讨论了进一步发展的方向．     

求解含非线性参数的QoS路由的启发式算法

具有非线性参数的QoS路由分为含有非线性约束条件的QoS路由和含有非线性优化目标的QoS路由两类，它们都是NP问题．提出了两种启发式算法求解这两类QOS路由优化问题问题．对第一类问题，求解去掉非线性约束条件后的优化问题．如果找到的解满足非线性约束条件，则该解是最优解；否则在优化问题中添加一个新的线性约束，将已得到的解去掉，反复下去就可得到最终解．对第二类问题，将非线性优化目标换为约束条件中的线性参数，求解此优化模型，如果有解，则记录此时对应的非线性目标值．而后增加一个新的线性约束，去掉刚才得到的解，比较两次得到的非线性目标值，保留最小值．如果得到的解不满足该线性参数的约束条件，则算法结束；否则继续迭代．证明了两种算法的收敛性，并且时间复杂性为近似多项式时间．计算实例表明了算法的有效性．     

The large discretization step method for time-dependent partial differential equations

A new method for the acceleration of linear and nonlinear time-dependent calculations is presented. It is based on the large discretization step (LDS, in short) approximation, defined in this work, which employs an extended system of low accuracy schemes to approximate a high accuracy discrete approximation to a time-dependent differential operator.These approximations are efficiently implemented in the LDS methods for linear and nonlinear hyperbolic equations, presented here. In these algorithms the high and low accuracy schemes are interpreted as the same discretization of a time-dependent operator on fine and coarse grids, respectively. Thus, a system of correction terms and corresponding equations are derived and solved on the coarse grid to yield the fine grid accuracy. These terms are initialized by visiting the fine grid once in many coarse grid time steps. The resulting methods are very general, simple to implement and may be used to accelerate many existing time marching schemes.The efficiency of the LDS algorithms is defined as the cost of computing the fine grid solution relative to the cost of obtaining the same accuracy with the LDS methods. The LDS method’s typical efficiency is 16 for two-dimensional problems and 28 for three-dimensional problems for both linear and nonlinear equations. For a particularly good discretization of a linear equation, an efficiency of 25 in two-dimensional and 66 in three-dimensional was obtained.     

Efficient unsteady high Reynolds number flow computations on unstructured grids

Despite the advances in computer power and numerical algorithms over the last decades, solutions to unsteady flow problems remain computing time intensive. Especially for high Reynolds number flows, nonlinear multigrid, which is commonly used to solve the nonlinear systems of equations, converges slowly. The stiffness induced by the high-aspect ratio cells and turbulence is not tackled well by this solution method.In this paper, it is investigated if a Jacobian-free Newton-Krylov (jfnk) solution method can speed up unsteady flow computations at high Reynolds numbers. Preconditioning of the linear systems that arise after Newton linearization is commonly performed with matrix-free preconditioners or approximate factorizations based on crude approximations of the Jacobian. Approximate factorizations based on a Jacobian that matches the target residual operator are unpopular because these preconditioners consume a large amount of memory and can suffer from robustness issues. However, these preconditioners remain appealing because they closely resemble A^-1.In this paper, it is shown that a jfnk solution method with an approximate factorization preconditioner based on a Jacobian that approximately matches the target residual operator enables a speed up of a factor 2.5-12 over nonlinear multigrid for two-dimensional high Reynolds number flows. The solution method performs equally well as nonlinear multigrid for three-dimensional laminar problems. A modest memory consumption is achieved with partly lumping the Jacobian before constructing the approximate factorization preconditioner, whereas robustness is ensured with enhanced diagonal dominance.     

A Data-Reusing Nonlinear Gradient Descent Algorithm for a Class of Complex-Valued Neural Adaptive Filters

A complex-valued data-reusing nonlinear gradient descent (CDRNGD) learning algorithm for a class of complex-valued nonlinear neural adaptive filters is introduced and the affinity between the family of data-reusing algorithms and the class of normalised gradient descent algorithms is examined. Error bounds on the class of complex data-reusing algorithms are established and indicate the stability of such algorithms. Experiments on nonlinear inputs show the class of complex data-reusing algorithms outperforming the standard complex nonlinear gradient descent algorithms and converging to the normalised complex non-linear gradient descent algorithm without experiencing the stability problems commonly encountered with normalised gradient descent algorithms. This revised version was published online in June 2006 with corrections to the Cover Date.     

Energy functions for dissipativity-based balancing of discrete-time nonlinear systems

Most of the energy functions used in nonlinear balancing theory can be expressed as storage functions in the framework of dissipativity theory. By defining a framework of discrete-time dissipative systems, this paper presents existence conditions for their discrete-time energy functions along with algorithms to find them based on dynamic optimization problems. Furthermore, the important case of the nonlinear discrete-time versions of the controllability and observability functions, its properties and algorithms to find them are presented. These algorithms are illustrated with linear and nonlinear examples.     

A minimal residual class of algorithms for linear systems

A family of rapidly convergent algorithms to solve linear systems of equations are described. These methods are easy to implement. In an empirical comparison over a class of problems the presented algorithms were superior to several commonly used methods.     

An analysis of an implicit finite element algorithm for geometrically nonlinear problems of structural dynamics part 1. Stability

A series of implicit finite element algorithms for the geometrically nonlinear structural dynamics problem are proposed. The proposed algorithms require only the solution of a linear system at each time step. Thus, they are computationally efficient. In addition, the algorithms are discrete conservation laws. The conservative nature of the proposed schemes has a positive effect in providing a stable approximation. The stability of the algorithms is analyzed using energy methods. One of the proposed computational methods is shown to be unconditionally stable.     

分片线性规划及应用

提出通过分片线性逼近和分片线性规划,将非线性优化问题转化为一系列的线性规划进行求解的方法。讨论了分片线性规划的性质,证明了分片线性规划问题可以通过有限次线性规划得到求解,同时,给出了分片线性规划问题局部最优解的充要条件,并基于此构造了求解分片线性规划问题的下降算法。该算法与自适应链接超平面模型相结合,成功地对离心式冷水机组的工作点进行了优化。通过优化,机组的能耗比之当前工作点有了明显的下降,表明通过分片线性规划求解非线性优化问题的有效性。