Variance analysis of L2 model reduction when undermodeling—the output error case

In this contribution, variance properties of L₂ model reduction are studied. That is, given an estimated model of high order we study the resulting variance of an L₂ reduced approximation. The main result of the paper is showing that estimating a low-order output error (OE) model via L₂ model reduction of a high-order model gives a smaller variance compared to estimating a low-order model directly from data in case of undermodeling. This has previously been shown to hold for Finite Impulse Response models, but is in this paper extended to general linear OE models.     

On H2 model reduction of bilinear systems

The H₂ model reduction problem for continuous-time bilinear systems is studied in this paper. By defining the H₂ norm of bilinear systems in terms of the state-space matrices, the H₂ model reduction error is computed via the reachability or observability gramian. Necessary conditions for the reduced order bilinear models to be H₂ optimal are given. The gradient flow approach is used to obtain the solution of the H₂ model reduction problem. The formulation allows certain properties of the original models to be preserved in the reduced order models. The model reduction procedure developed can also be applied to finite-dimensional linear time-invariant systems. A numerical example is employed to illustrate the effectiveness of the proposed method.     

Robustified L2 boosting

Five robustifications of L₂ boosting for linear regression with various robustness properties are considered. The first two use the Huber loss as implementing loss function for boosting and the second two use robust simple linear regression for the fitting in L₂ boosting (i.e. robust base learners). Both concepts can be applied with or without down-weighting of leverage points. Our last method uses robust correlation estimates and appears to be most robust. Crucial advantages of all methods are that they do not compute covariance matrices of all covariates and that they do not have to identify multivariate leverage points. When there are no outliers, the robust methods are only slightly worse than L₂ boosting. In the contaminated case though, the robust methods outperform L₂ boosting by a large margin. Some of the robustifications are also computationally highly efficient and therefore well suited for truly high-dimensional problems.     

Practical L2 disturbance attenuation for nonlinear systems

In this work, a notion of generalized L₂-gain for nonlinear systems, where the gain is considered as a function of the state instead of a (global) constant, is presented. This new notion seems to be adequate to characterize the gain properties of several nonlinear systems which do not possess a uniform L₂-gain property (i.e. the L₂-gain depends on the operating point). Moreover, a notion of practical L₂-gain attenuation, which extends the standard definition and parallels (mutatae mutandis) the concepts of practical stability, is also proposed.     

Adaptive nonlinear excitation control with L2 disturbance attenuation for power systems

This paper presents a design approach to nonlinear feedback excitation control of power systems with unknown disturbance and unknown parameters. It is shown that the stabilizing control law with desired L₂ gain from the disturbance to a penalty signal can be designed by a recursive way without linearization. A state feedback law is presented for the case of the system with known parameters, and then the control law is extended to adaptive controller for the case when the parameters of the electrical dynamics of the power system are unknown. Simulation results demonstrate that the proposed controllers guarantee transient stability of the system regardless of the system parameters and faults.     

Robust multi-view L2 triangulation via optimal inlier selection and 3D structure refinement

This paper presents a new robust approach for multi-view L₂ triangulation based on optimal inlier selection and 3D structure refinement. The proposed method starts with estimating the scale of noise in image measurements, which affects both the quantity and the accuracy of reconstructed 3D points but is overlooked or ignored in existing triangulation pipelines. A new residual-consensus scheme within which the uncertainty of epipolar transfer is analytically characterized by deriving its closed-form covariance is developed to robustly estimate the noise scale. Different from existing robust triangulation pipelines, the issue of outliers is addressed by directly searching for the optimal 3D points that are within either the theoretical correct error bounds calculated by second-order cone programming (SOCP) or the efficiently calculated approximate ranges. In particular, both the inlier selection and 3D structure refinement are realized in an optimal fashion using Differential Evolution (DE) optimization which allows flexibility in the design of the objective function. To validate the performance of the proposed method, extensive experiments using both synthetic data and real image sequences were carried out. Comparing with state-of-the-art robust triangulation strategies, the proposed method can consistently identify more reliable inliers and hence, reconstruct more unambiguous 3D points with higher accuracy than existing methods.     

四旋翼无人机L1 自适应块控反步姿态控制器设计

针对四旋翼无人机的姿态控制问题,提出一种L1自适应块控反步控制方法.将四旋翼姿态运动模型转化为一类多输入多输出不确定非线性系统的形式;根据该系统严格反馈的结构特点,对外回路设计了块控反步控制器;针对内回路存在的外部干扰和内部参数摄动等不确定性,引入L1自适应控制思想补偿其影响.稳定性分析表明闭环系统内所有信号一致有界.仿真和姿态稳定实验验证了所提控制策略的有效性和鲁棒性.     

Natural actor-critic with baseline adjustment for variance reduction

In this study, we discuss a baseline function for the estimation of a natural policy gradient with respect to variance, and demonstrate a condition in which an optimal baseline function that reduces the variance is equivalent to the state value function. However, outside of this condition, the state value could be considerably different from the optimal baseline. For such cases, an extended version of the NTD algorithm is proposed, where an auxiliary function is estimated to adjust the baseline, being state value estimates in the original NTD version, to the optimal baseline. The proposed algorithm is applied to simple MDPs and a challenging pendulum swing-up problem. This work was presented in part at the 13th International Symposium on Artificial Life and Robotics, Oita, Japan, January 31–February 2, 2008     

Convergent algorithms for frequency weighted L2 model reduction

This paper is concerned with computing an L₂-optimal reduced-order model for a given stable multivariable linear system in the presence of input and output frequency weightings. By parametrizing a class of reduced-order models in terms of an orthogonal projection and using manifold techniques as tools, both continuous and iterative algorithms are derived and their convergence properties are established. As an application, we show that an L₂ optimal reduced-order filter in the closed-loop sense can be computed using these algorithms.     

Frequency-weighted model reduction with ∞ error bounds

This paper considers frequency-weighted model reduction. The explicit lower and upper approximation error bounds are derived for certain classes of weighted model reduction problems. The approximation is based on some unweighted approximations and the error bounds are given in terms of the Hankel singular values of the weighted model.     

Image super-resolution by dictionary concatenation and sparse representation with approximate L0 norm minimization

This paper proposes a different image super-resolution (SR) reconstruction scheme, based on the newly advanced results of sparse representation and the recently presented SR methods via this model. Firstly, we online learn a subsidiary dictionary with the degradation estimation of the given low-resolution image, and concatenate it with main one offline learned from many natural images with high quality. This strategy can strengthen the expressive ability of dictionary atoms. Secondly, the conventional matching pursuit algorithms commonly use a fixed sparsity threshold for sparse decomposition of all image patches, which is not optimal and even introduces annoying artifacts. Alternatively, we employ the approximate L₀ norm minimization to decompose accurately the patch over its dictionary. Thus the coefficients of representation with variant number of nonzero items can exactly weight atoms for those complicated local structures of image. Experimental results show that the proposed method produces high-resolution images that are competitive or superior in quality to results generated by similar techniques.     

并联混合有源电力滤波器自适应L2增益鲁棒控制

为了抑制外界未知扰动和参数摄动对并联混合有源电力滤波器(SHAPF) 系统性能的影响, 提出一种新型的自适应L₂ 增益鲁棒控制策略. 首先建立含有扰动和参数摄动的SHAPF 欧拉-拉格朗日(EL) 数学模型, 得到了SHAPF 在dq 坐标系下的误差动态模型; 然后通过构造适当的Lyapunov 函数设计参数自适应控制率, 实现了对系统参数摄动的补偿, 进而利用阻尼注入方法设计系统的L₂增益鲁棒控制器, 以保证闭环系统的 gamma  耗散性. 仿真实验验证了所提出策略的正确性和有效性.

     

Low-rank matrix decomposition in L1-norm by dynamic systems

Low-rank matrix approximation is used in many applications of computer vision, and is frequently implemented by singular value decomposition under L₂-norm sense. To resist outliers and handle matrix with missing entries, a few methods have been proposed for low-rank matrix approximation in L₁ norm. However, the methods suffer from computational efficiency or optimization capability. Thus, in this paper we propose a solution using dynamic system to perform low-rank approximation under L₁-norm sense. From the state vector of the system, two low-rank matrices are distilled, and the product of the two low-rank matrices approximates to the given measurement matrix with missing entries, in L₁ norm. With the evolution of the system, the approximation accuracy improves step by step. The system involves a parameter, whose influences on the computational time and the final optimized two low-rank matrices are theoretically studied and experimentally valuated. The efficiency and approximation accuracy of the proposed algorithm are demonstrated by a large number of numerical tests on synthetic data and by two real datasets. Compared with state-of-the-art algorithms, the newly proposed one is competitive.     

L1PMA: A Fortran 77 package for best L1 piecewise monotonic data smoothing

Fortran 77 software is presented for the calculation of a best L₁ approximation to n measurements that include random errors by requiring k−1 sign changes in the first divided differences of the approximation or equivalently k monotonic sections, alternately increasing and decreasing. A dynamic programming algorithm separates the measurements into optimal disjoint sections of adjacent data and applies to each section a single L₁ monotonic calculation. The most distinctive feature of the algorithm is that it terminates at a global minimum in at most n³+O(kn²) computer operations, although this calculation can exhibit O(n^k) local minima, because the optimal positions of the turning points are also unknowns of the optimization process. The arithmetic operations involved in this calculation are comparisons mainly spent in finding the medians of subranges of data during the monotonic calculations. The package employs techniques for median and for best L₁ monotonic approximation, while full details of these techniques are specified. The package has been applied and tested on a variety of data that have substantial differences and showed quadratic behaviour in n. Some numerical results demonstrate the performance of the method. Further, there is a commentary on the division of the code into subroutines. Driver programs and numerical examples with output are provided to help new users of the method. Besides that piecewise monotonicity is a property of a wide range of functions, an important application of the method is in estimating turning points of a function from some noisy measurements of its values.     

H2 approximation of multiple input/output delay systems

Many physical multivariable processes can be sufficiently described as linear models with multiple input/output delays. To simplify the synthesis and analysis of control problem, a reduced-complexity model is often desired. In this paper, an H₂ model reduction scheme is introduced for stable linear systems with multiple input/output delays. The reduced model can be a finite dimensional linear model, or a linear model with a time delay. In the latter case the approximation can be improved drastically without increasing the order of the finite dimensional part. The stability is preserved in the approximating models by employing a parametrization of linear stable systems. The optimal parameters can be obtained by solving an optimization problem using a gradient-based method. Two chemical numerical examples are used to show the effectiveness of the proposed scheme.     

Improved robust H2 and H∞ filtering for uncertain discrete-time systems

This paper is concerned with the robust H₂ and H_∞ filtering problems for linear discrete-time systems with polytopic parameter uncertainty. We aim to derive a less-conservative design than existing linear matrix inequality (LMI) based sufficient conditions. It is shown that a more efficient evaluation of robust H₂ or H_∞ performance can be obtained by a matrix inequality condition which contains additional free parameters as compared to existing characterizations. When applying this new matrix inequality condition to the robust filter design, these parameters provide extra degrees of freedom in optimizing the guaranteed H₂ or H_∞ performance and lead to a less-conservative design.     

A parametric optimization approach to H∞ and H2 strong stabilization

This paper presents an approach for designing stable MIMO H_∞ and H₂ controllers by directly computing the norm-constrained stable transfer matrices Q in the H_∞ and H₂ suboptimal controller parameterizations. This is done by first converting the H₂ and H_∞ strong stabilization problems into some nonlinear unconstrained optimization problems through explicit parameterization of the norm-constrained Q's for any fixed order. Then, a two-stage numerical search is carried out by using a combination of a genetic algorithm and a quasi-Newton algorithm in order to reach an optimal solution. The effectiveness of the proposed algorithms is illustrated through some benchmark numerical examples.     

Identification of multi-input systems: variance analysis and input design issues

This paper examines the identification of multi-input systems. Motivated by an experiment design problem (should one excite the various inputs simultaneously or separately), we examine the effect of an additional input on the variance of the estimated coefficients of parametrized rational transfer function models, with special emphasis on the commonly used FIR, ARX, ARMAX, OE and BJ model structures. We first show that, for model structures that have common parameters in the input-output and noise models (e.g. ARMAX), any additional input contributes to a reduction of the covariance of all parameter estimates. We then show that the accuracy improvement extends beyond the case of common parameters in all transfer functions, and we show exactly which parameter estimates are improved when a new input is added. We also conclude that it is always better to excite all inputs simultaneously.     

基于稀疏流形聚类嵌入模型和L1范数正则化的标签错误检测

综合利用含错标签中的有用信息和数据结构中蕴含的鉴别信息,提出一种基于稀疏流形聚类嵌入模型和L1范数正则化的标签错误检测修正方法.首先,用稀疏流形聚类嵌入模型将数据投影到易分类的空间,利用标注正确的极少量样本和最近邻分类器获得新标签;然后,构造标签错误检测模型,获得仅含0、1元素的检测向量,正确、错误的标签分别对应着1、0的位置;最后,给出了相应的优化算法及收敛证明,并在相关实验上验证了算法的有效性.