A robust solver using a continuation method for Nevanlinna-Pick interpolation with degree constraint

This note modifies a previous algorithm for solving a certain convex optimization problem, introduced by Byrnes, Georgiou, and Lindquist, to determine any Nevanlinna-Pick interpolant satisfying degree constraint. The modified algorithm is based on a continuation method with predictor-corrector steps and it turns out to be quite efficient and numerically robust.     

Matrix-valued Nevanlinna-Pick interpolation with complexity constraint: an optimization approach

Over the last several years, a new theory of Nevanlinna-Pick interpolation with complexity constraint has been developed for scalar interpolants. In this paper we generalize this theory to the matrix-valued case, also allowing for multiple interpolation points. We parameterize a class of interpolants consisting of "most interpolants" of no higher degree than the central solution in terms of spectral zeros. This is a complete parameterization, and for each choice of interpolant we provide a convex optimization problem for determining it. This is derived in the context of duality theory of mathematical programming. To solve the convex optimization problem, we employ a homotopy continuation technique previously developed for the scalar case. These results can be applied to many classes of engineering problems, and, to illustrate this, we provide some examples. In particular, we apply our method to a benchmark problem in multivariate robust control. By constructing a controller satisfying all design specifications but having only half the McMillan degree of conventional H/sup /spl infin// controllers, we demonstrate the advantage of the proposed method.     

A norm selection criterion for the generalized delta rule

The derivation of a supervised training algorithm for a neural network implies the selection of a norm criterion which gives a suitable global measure of the particular distribution of errors. The author addresses this problem and proposes a correspondence between error distribution at the output of a layered feedforward neural network and L(p) norms. The generalized delta rule is investigated in order to verify how its structure can be modified in order to perform a minimization in the generic L(p) norm. The particular case of the Chebyshev norm is developed and tested.     

一种基于变量熵求解约束满足问题的置信传播算法

在置信传播(belief propagation,BP)算法中,提出一种基于变量熵来挑选变量从而固定变量赋值的策略,用于求解一类具有增长定义域的随机约束满足问题.RB模型是一个具有增长定义域的随机约束满足问题的典型代表,已经严格证明它不仅存在精确的可满足性相变现象,而且可以生成难解实例.在RB模型上选取两组不同的参数进行数值实验.结果表明:在接近可满足性相变点时,BP引导的消去算法仍然可以非常有效地找到随机实例的解;不断增加问题的规模,算法的运行时间呈指数级增长;并且当控制参数(约束紧度)增加时,变量的平均自由度逐渐降低.     

Closed-loop shaping based on Nevanlinna-Pick interpolation with a degree bound

This note presents a novel method for shaping the frequency response of a single-input-single-output closed-loop system, based on the theory of Nevanlinna-Pick interpolation with degree constraint. The method imposes a degree bound on the closed-loop transfer function and searches for a function with a desired frequency response. Numerical examples illustrate the potential of the method in designing controllers with lower degrees than the ones obtained by conventional H/sup /spl infin// controller design methods with weighting functions.     

A collapse result for constraint queries over structures of small degree

Collapse results, which are central for understanding constraint database queries, show that in terms of the expressive power, a large class of queries collapses to a much smaller one, typically involving only a restricted form of quantification. Most collapse results have been proved over constraints involving a linear order, and proofs are typically rather nontrivial. In this short note we give an easy proof of a powerful form of collapse for a large class of constraints without a linear order, namely those in which all basic relations are of small degree.     

Gelfand-Yaglom-Perez theorem for generalized relative entropy functionals

The measure-theoretic definition of Kullback-Leibler relative-entropy (or simply KL-entropy) plays a basic role in defining various classical information measures on general spaces. Entropy, mutual information and conditional forms of entropy can be expressed in terms of KL-entropy and hence properties of their measure-theoretic analogs will follow from those of measure-theoretic KL-entropy. These measure-theoretic definitions are key to extending the ergodic theorems of information theory to non-discrete cases. A fundamental theorem in this respect is the Gelfand-Yaglom-Perez (GYP) Theorem [M.S. Pinsker, Information and Information Stability of Random Variables and Process, 1960, Holden-Day, San Francisco, CA (English ed., 1964, translated and edited by Amiel Feinstein), Theorem. 2.4.2] which states that measure-theoretic relative-entropy equals the supremum of relative-entropies over all measurable partitions. This paper states and proves the GYP-theorem for Rényi relative-entropy of order greater than one. Consequently, the result can be easily extended to Tsallis relative-entropy.     

基于模糊性能指标的广义预测控制器参数调整

针对广义预测控制算法在控制时域中求得的M个控制量,利用模糊模拟技术对系统的约束进行检验,不断修正目标函数中控制量的加权系数,充分利用系统预测控制量的信息,增强系统的鲁棒性,并满足系统的约束。     

A new generalized computational framework for finding objectorientation using perspective trihedral angle constraint

This paper investigates a fundamental problem of determining the position and orientation of a three-dimensional (3-D) object using a single perspective image view. The technique is focused on the interpretation of trihedral angle constraint information. A new closed form solution based on Kanatani's formulation is proposed. The main distinguishing feature of the authors' method over the original Kanatani formulation is that their approach gives an effective closed form solution for a general trihedral angle constraint. The method also provides a general analytic technique for dealing with a class of problem of shape from inverse perspective projection by using “angle to angle correspondence information.” A detailed implementation of the authors' technique is presented. Different trihedral angle configurations were generated using synthetic data for testing the authors' approach of finding object orientation by angle to angle constraint. The authors performed simulation experiments by adding some noise to the synthetic data for evaluating the effectiveness of their method in a real situation. It has been found that the authors' method worked effectively in a noisy environment which confirms that the method is robust in practical application     

Design of optimal quantizers with an entropy constraint

Source encoding of transform domain coefficients can be used to obtain pictures of very good quality at low bit rates. Such a scheme requires that the entropy of the quantizer output be fixed at a prespecified value. The design of optimal quantizers subject to an entropy constraint is discussed in this paper and their specifications for three different input distributions associated with the transform coefficients is given.     

A criterion for structural stability of quadruples of matrices defining generalized dynamical systems

A characterization in terms of the rank of a matrix is proved for structurally stable quadruples related to generalized dynamical systems

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under a local equivalence relation by realizing a geometric approach.     

Conditional Akaike information criterion for generalized linear mixed models

In this study, a model identification instrument to determine the variance component structure for generalized linear mixed models (glmms) is developed based on the conditional Akaike information (cai). In particular, an asymptotically unbiased estimator of the cai (denoted as caicc) is derived as the model selection criterion which takes the estimation uncertainty in the variance component parameters into consideration. The relationship between bias correction and generalized degree of freedom for glmms is also explored. Simulation results show that the estimator performs well. The proposed criterion demonstrates a high proportion of correct model identification for glmms. Two sets of real data (epilepsy seizure count data and polio incidence data) are used to illustrate the proposed model identification method.     

New criterion for generalized diagonally dominant matrices and M-matrices

We provide some new sufficient and necessary conditions for identifying generalized (strictly) diagonally dominant matrices and obtain some criteria for judging (nonsingular) M-matrices.     

Linear quadratic control with stability degree constraint

The problem of static state feedback Linear Quadratic (LQ) optimal control subject to a prescribed degree of stability for the closed-loop system is considered in this paper. A necessary optimality condition is given via the Lagrange multiplier method. A globally convergent algorithm is provided to solve the optimization problem. It is shown that the algorithm recovers the standard LQ feedback gain provided the desired stability degree is small enough to be within the range by the standard LQ design. As for other cases the optimum occurs on the boundary of the α-region. A numerical example shows that the proposed algorithm provides a better design compared to the existing methods.     

A shaping limitation of rational sensitivity functions with a degree constraint

This note concerns a certain shaping limitation of sensitivity functions. The focus is on a frequency-wise infimum of gains of rational sensitivity functions with a degree constraint. An explicit infimum is derived for a special case. The result is useful for determining the inability of sensitivity functions of low degrees to achieve a specification in the frequency domain, and thus for motivating the use of higher degree sensitivity functions to fulfill the specification.     

Remarks on control design with degree constraint

The purpose of this note is to highlight similarities and differences between two alternative methodologies for feedback control design under constraints on the McMillan degree of the feedback system. Both sets of techniques focus on uniformly optimal designs. The first is based on the work of Gahinet and Apkarian and that of Skelton , while the other is based on earlier joint work of the authors with C. I. Byrnes.     

Some results on generalized residual entropy

Shannon’s entropy plays an important role in the context of information theory. Since this entropy is not applicable to a system which has survived for some units of time, the concept of residual entropy has been developed in the literature. Here we generalize the residual entropy by choosing a convex function ? with ?(1) = 0. In this paper, some orderings and aging properties have been defined in terms of the generalized residual entropy function and their properties have been studied. Quite a few results available in the literature have been generalized and some distributions (viz. uniform, exponential, Pareto, power series, finite range) have been characterized through the generalized residual entropy.     

On the generalized nyquist stability criterion

This paper starts with a straightforward self-contained treatment of the generalized Nyquist stability criterion based on the eigenloci of the open-loop transfer function matrix. The derivation is straightforward and does not invoke algebraic function theory, branch cuts, and Riemann surfaces. A similar criterion is derived for the distributed case where the open-loop transfer function matrices belong to the recently proposed algebraB(sigma_{0})^{m times m}[9], [14]. This leads to a new graphical sability test based on eigenloci which applies to distributed, possibly unstable, open-loop transfer functions.