Hankel Approximation Research Articles

Optimal Rank‐1 Hankel Approximation in the Spectral Norm for Matrices with Multiple Largest Eigenvalue

AbstractWe extend a result from [2, 3] on the optimal rank‐1 Hankel approximation of real symmetric matrices with isolated largest eigenvalue to the case where the largest eigenvalue is not isolated. To illustrate our findings, we give an example where the optimal rank‐1 Hankel approximation is easily obtained and one where it does not exist.

Optimal rank-1 Hankel approximation of matrices: Frobenius norm and spectral norm and Cadzow's algorithm

We characterize optimal rank-1 matrix approximations with Hankel or Toeplitz structure with regard to two different norms, the Frobenius norm and the spectral norm, in a new way. More precisely, we show that these rank-1 matrix approximation problems can be solved by maximizing special rational functions. Our approach enables us to show that the optimal solutions with respect to these two norms have completely different structure and only coincide in the trivial case when the singular value decomposition already provides an optimal rank-1 approximation with the desired Hankel or Toeplitz structure. We also prove that the Cadzow algorithm for structured low-rank approximations always converges to a fixed point in the rank-1 case. However, it usually does not converge to the optimal solution, neither with regard to the Frobenius norm nor the spectral norm.

The Difference between Optimal Rank‐1 Hankel Approximations in the Frobenius Norm and the Spectral Norm

AbstractWe present new examples illustrating the fact that optimal Hankel structured rank‐1 approximation of matrices is usually different for the Frobenius and spectral norm. Further, we compare our results for optimal rank‐1 Hankel approximation to the results of the well‐known Cadzow algorithm [3].

Depth Map Reconstruction and Enhancement With Local and Patch Manifold Regularized Deep Depth Priors

The depth map captured by depth sensors (e.g., the time of flight (ToF) and Kinect) is often prone to low resolution, degradation, noise, and poor quality. This paper proposes a novel model for the robust depth estimation of RGB-D images through local and nonlocal manifold regularizations. The first stage called deep depth prior manifold (DDPM), is inspired partly by the deep depth prior (DDP) model, that is a deep convolutional neural network (CNN) integrated with a local manifold regularization term. The local neighboring relationships between depth pixels and color images are employed to promote smoothing in the results. The Laplacian Eigenmap technique used for local manifold modeling produces over-smooth depth map. To improve the quality of the reconstructed image, a nonlocal manifold modeling stage was suggested, where the similarity between the depth and the corresponding color image is determined by characterizing their matching aspects. These objectives are aggregated within an optimization problem. Moreover, to extract edges better considering visual nonlocal characteristics, the structured low-rank Hankel approximation was adopted to better eliminate depth degradations, and to extract highly promoted edges and sharp points. Three types of the degradations were handling in this work, containing undersampling, ToF-like, and Kinect-like degradations. Experimental results indicate that the proposed method outperformed the state-of-the-art restoration techniques on standard benchmark images, in terms of well-known criteria like PSNR.

Some Features of the construction of a family of atomic radial basis functions Plop r,a (x1,x2)

A lot of methods for solving boundary value problems using arbitrary grids, such as SDI (scattered data interpolation) and SPH (smoothed particle hydrodynamics), use families of atomic radial basis functions that depend on parameters to improve the accuracy of calculations. Functions of this kind are commonly called "shape functions". When polynomials or polynomial splines are used as such functions, they are called "basis functions". The term "radial" means that the carrier of the function is a disk or layer. The term "atomic" means that the support of the function is limited, ie the function is finite. In most cases, the term "finite" is used in English-language publications. The article presents an algorithm for constructing such a function, which is the solution of the functional-differential equation where - circle of radius r: , and . The function generated by this equation has two parameters: r and . Variation of these parameters allows to reduce the error in the calculations of the Poisson boundary value problem by several times. The theorem on the existence of such an unambiguous function is proved in the article. The proof of the theorem allows us to construct one-dimensional Fourier transform of this function in the form , where . Previously, function was calculated using its Taylor approximation (at ), and at – using the asymptotic Hankel approximation of the function . Thus in a circle of a point a fairly large error was found. Therefore, the calculation of the function in the range was carried out by Chebyshev approximation of this function in the range . Chebyshev coefficients (calculated in the Maple 18 system with an accuracy of 26 decimal digits) and the range were chosen by an experiment aimed at minimizing the overall error in calculating of the function . Thanks to the use of the Chebyshev approximation, the obtained function has more than twice less error than calculated by the previous algorithm. Arbitrary value of the function is calculated using a six-point Aitken scheme, which can be considered (to some extent) a smoothing filter. The use of Aitken's six-point scheme introduces an error equal to 6% of the total function calculation error , but helps to save a lot of time in the formation of ARBF in solving boundary value problems using the method of collocation.

Speed Control of an Uncertain Heavy-Duty Vehicle Using Improved IMC Technique

An improved internal model control (IMC) for speed control and disturbance rejection of the heavy-duty vehicle (HDV) is proposed. The limitation of conventional IMC is the steady-state error in accelerating and decelerating mode of the vehicle. In this work, the conventional low-pass filter is modified and replaced by a filter named as robust filter and its parameters are determined using simple designed formula as proposed in this work. The clustering and Hankel approximation approaches are used for order reduction, and $${H}_\infty $$ control technique is employed for the design of improved IMC structure. The proposed control techniques are developed to achieve the better speed regulation and disturbance rejection performances of HDV system in the influence of road grade disturbance.

Intelligent modified internal model control for speed control of nonlinear uncertain heavy duty vehicles

The objective of this paper is to control the speed of heavy duty vehicle (HDV) through angular position of throttle valve. Modified internal model control (IMC) schemes with fuzzy supervisor as an adaptive tuning are proposed to control the speed of HDV. Internal model (IM) plays a key role in design of various IMC structures with robust and adaptive features. The motivation to design an IM is to produce nearly stable performance as of the system itself. Clustering algorithm and Hankel approximation based model order reduction techniques are used for the design of suitable IM. The time domain performance specifications such as overshoot, settling time, rise time and integral error performance indices such as the integral of the absolute error and the integral of the square of error are taken into consideration for performance analysis of HDV for various uncertainties.

Direction-finding methods for cyclostationary signals in the presence of coherent sources

By exploiting the signal cyclostationarity, conjugate cyclic MUSIC presented by Gardner (1994), has been shown to be effective in performing signal-selective direction finding. However, a direct application of conjugate cyclic MUSIC in the presence of coherent signals of interest (SOIs) will result in ambiguity. To tackle the drawback of conjugate cyclic MUSIC, we present a scheme called the Hankel approximation method (HAM) in conjunction with conjugate cyclic MUSIC to cope with the performance deterioration due to coherent SOIs for a uniform linear array. Theoretical analysis shows that at least 2K sensor elements are required to resolve K coherent SOIs. Making use of the forward/backward (F/B) technique presented by Pillai and Kwon (1989), F/B HAM is further developed, which requires [3K/2] sensor elements to decorrelate K coherent SOIs, where [x] represents the integer part of x. Several simulation examples are also presented for illustrating the effectiveness of the proposed methods.

A note on Hankel approximation of all-pass functions

The paper extends the results obtained by Fuhrmann and Ober on Hankel approximation. With an arbitrary transfer function G we can associate—using left-and right-coprime factorization—two all-pass functions U ̄ and U and also a function R∗ (called the characteristic function of G) which is connected with the LQG controller of the function G. We analyse the connections between the Hankel operators H U ̄ ∗ , H R∗ and H ̂ U , H ̂ R , and also establish a one-to-one correspondence between the optimal Hankel approximation of H R∗ and that of H U ̄ ∗ with some additional property. Finally, the duality results on Nehari complements and Hankel approximations are extended.

Continuity of Best Hankel Approximation and Convergence of Near-Best Approximants

Consider a bounded Hankel operator $\Gamma$ with $s$-numbers $s_0\ge s_1\ge \cdots$ and a sequence of bounded Hankel operators $\Gamma_n$ converging to $\Gamma$ in the operator norm. In this paper, it is shown that for each $k$ with $s_{k-1} > s_k \ge s_{k+1} \ge \cdots$, the rational symbols of the best rank-$k$ Hankel approximants of $\Gamma_n$ converge uniformly to the corresponding rational symbol of the best rank-$k$ Hankel approximant of $\Gamma$. Based on this continuity result, the convergence of the near-best Hankel approximants corresponding to a fairly general class of truncated Hankel operators is discussed.

An extension of A-A-K Hankel approximation theory using state-space formulation

This paper presents a state-space extension of A-A-K Hankel approximation theory. Conventional A-A-K theory can only be applied to solve minimum-degree approximation problems. Here, we extend the A-A-K results and show that it can also be generalized to minimum-norm problems for multivariable systems. State-space formula for generating enough independent Hankel singular vectors corresponding to a prescribed singular value is derived and is used to construct all the optimal Hankel approximants for a given rational function matrix.

Hankel operators and best Hankel approximation on the half-plane

The objective of this paper is to give an interrelation between Hankel oeprators on the unit disc and Hankel operators on the half-plane. As an application, the AAK result on the half-plane is established and the rate of best Hankel approximation on the halfplane is derived.

Characterization and use of the degrees of freedom in theH∞problem

This paper characterizes the degrees of freedom in the H ∞ 1-block problem and subsequently uses these degrees of freedom to obtain desirable properties in the Hankel approximation. In particular, for a special class of transfer function matrices, the degrees of freedom can be deployed to obtain station-arity of the approximation system gramians which, in turn, imposes flatness on all of the error-system singular values as well as providing completely stable projections in the context of a super-optimal algorithm. Subsequently, the class constraints are relaxed in order to generalize the results.

One-step extension approach to optimal Hankel-norm approximation and H/sup infinity /-optimization problems

A methodology is presented for Hankel approximation and H/sup infinity /-optimization problems that is based on a new formulation of a one-step extension problem which is solved by the Sarason interpolation theorem. The parameterization of all optimal Hankel approximants for multivariable systems is given in terms of the eigenvalue decomposition of an Hermitian matrix composed directly from the coefficients of a given transfer function matrix phi . Rather than starting with the state-space realization of phi , the authors use polynomial coefficients of phi as input data. In terms of these data, a natural basis is given for the finite dimensional Sarason model space and all computations involve only manipulations with finite matrices.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Model reduction of 2-D systems with frequency error bounds

Some explicit frequency domain infinity -norm error bounds for 2-D model reduction of denominator separable systems are derived. They are based on 1-D balanced model reduction and Hankel approximation methods. The evaluation of these error bounds only involves the computation of Hankel singular values of two 1-D transfer matrices. >

An efficient algorithm on optimal Hankel-norm approximation for multivariable systems

This note provides a novel methodology for Hankel approximation and H/sup infinity /-optimization problems, based on a new formulation of the one-step extension problem, which is solved by the Sarason interpolation theorem. The present method does not require an initial balanced realization, and the parameterization of all optimal solutions is given in terms of the eigenvalue decomposition of a Hermitian matrix composed directly from the coefficients of the given transfer function matrix. >

Rate of convergence of schmidt pairs and rational functions corresponding to best approximants of truncated hankel operators

The problem of approximating Hankel operators of infinite rank by finite-rank Hankel operators is considered. For efficiency, truncated infinite Hankel matrices Γn of Γ are utilized. In this paper for any compact Hankel operator Γ of the Wiener class, we derive the rate of l2-convergence of the Schmidt pairs of Γn to the corresponding Schmidt pairs of Γ. For a certain subclass of Hankel operators of the Wiener class, we also obtain the rate of l1-convergence. In addition, an upper bound for the rate of uniform convergence of the rational symbols of best rank-k Hankel approximants of Γn to the corresponding rational symbol of the best rank-k Hankel approximant to Γ asn → ∞ is derived.

A novel approach to the synthesis of recursive digital filters with linear phase

The synthesis of recursive digital filters with linear phase is approached by means of existing approximation techniques. The design is based on a finite impulse response (FIR) filter that represents the ideal filter in terms of both magnitude and phase characteristics. The recursive digital filter with linear phase is then synthesized as an approximation of the FIR model where optimal Hankel approximation theory as well as the model reduction techniques are used. The causality and stability of the designed filter are guaranteed. This technique is used for designing both one-dimensional and two-dimensional recursive digital filters with linear phase, and is illustrated by a few numerical examples. >

System reduction via truncated Hankel matrices

The problem of approximating Hankel operators of finite or infinite rank by lower-rank Hankel operators is considered. For efficiency, truncated Hankel matrices are used as the intermediate step before other existing algorithms such as theCF algorithms are applied to yield the desirable approximants. If the Hankel operator to be approximated is of finite rank, the order of approximation by truncated Hankel operators is obtained. It is also shown that when themths-number is simple, then rational symbols of the best rank-m Hankel approximants of thenth truncated Hankel matrices converge uniformly to the corresponding rational symbol of the best rank-m Hankel approximant of the original Hankel operator asn tends to infinity.

A DESIGN METHODOLOGY FOR TWO-DIMENSIONAL ANALOG FILTERS WITH LINEAR PHASE

A new methodology is proposed for the synthesis of two-dimensional analog filters with linear phase. The Fourier series theory is employed to design a noncausal Laguerre type filter and Hankel approximation machinery is then used to synthesize the causal, stable two-dimensional analog filter with linear phase. The proposed design technique is illustrated by design of a lowpass filter.