Hilbert Method Research Articles

Fault Diagnosis of Piezoelectric Sensor Patches for Vibration Control Based on Multifeature Fusion and Improved SVM

The fault diagnosis of piezoelectric sensor patches is very important for the stability of the vibration control system and fault‐tolerant control technology. In order to improve the accuracy of fault self‐diagnosis of piezoelectric sensor patches, singular value decomposition (SVD) and Hilbert marginal spectrum method are proposed to extract multiple features of each IMF component and conduct feature fusion, and a support vector machine (SVM) based on particle swarm optimization (PSO) is designed for fault identification of different eigenvalues. In the experiment, the broken and degumming piezoelectric patches are simulated. Firstly, under the excitation of the square wave signal with no noise signal, when the SVD value and the maximum amplitude of Hilbert marginal spectrum are used as the fusion eigenvalue together, the diagnostic results show that the recognition accuracy can reach 95%, compared with the recognition accuracy of 70% and 80%, respectively, when the two are used as eigenvalues alone; the recognition result under fusion eigenvalue is obviously better than that of the latter. Secondly, in order to highlight the effectiveness of this method, the aforementioned experiment is conducted under the excitation of the square wave signal interfered by 0.5 dBW–1 dBW noise signal. The experimental results show that the fault recognition of the fused eigenvalue under different noise intensity signals is still superior to that of the single eigenvalue.

N-soliton solutions for the nonlocal two-wave interaction system via the Riemann–Hilbert method** Project supported by the National Natural Science Foundation of China (Grant Nos. 11331008 and 11522112).

In this paper, a nonlocal two-wave interaction system from the Manakov hierarchy is investigated via the Riemann–Hilbert approach. Based on the spectral analysis of the Lax pair, a Riemann–Hilbert problem for the nonlocal two-wave interaction system is constructed. By discussing the solutions of this Riemann–Hilbert problem in both the regular and non-regular cases, we explicitly present the N-soliton solution formula of the nonlocal two-wave interaction system. Moreover, the dynamical behaviour of the single-soliton solution is shown graphically.

Optimal Rotor Fault Detection in Induction Motor Using Particle-Swarm Optimization Optimized Neural Network

This study examined and presents an effective method for detection of failure of conductor bars in the winding of rotor of induction motor in low load conditions using neural networks of radial-base functions. The proposed method used Hilbert method to obtain the stator current signal push. The frequency and signal amplitude of the push stator were used as the input of the neural network and the network outputs were rotor fault state, and the number of conductive bars with broken fault. Moreover, particle-swarm optimization algorithm was used to determine the optimal network weights and neuron penetration radius in the neural network. The results obtained from the proposed method showed the optimal and efficient performance of the method in detecting conductive bars broken fault in induction motor in low load conditions.

Intermediate-frequency error identification of precision optical surface based on the Hilbert spectrum and instantaneous frequency filtering method

To evaluate the intermediate-frequency error of a precision optical surface, a method for more accurate separation and identification of the intermediate-frequency error is proposed. This method is based on the Hilbert spectrum and the instantaneous frequency separation method. Experimental results show that the intermediate-frequency error spectrum can be obtained using the method of instantaneous frequency separation based on the Hilbert spectrum. This method separates only the intermediate-frequency error component (0.03 mm − 1 ≤ f ≤ 8.33 mm − 1). Also, the method accurately identifies the nonqualified component distribution (at sampling points 200, 840, and 1000). Therefore, this algorithm is a useful guide for deterministic processing of optical components.

Gap Probability at the Hard Edge for Random Matrix Ensembles with Pole Singularities in the Potential

We study the Fredholm determinant of an integrable operator acting on the interval $(0,s)$ whose kernel is constructed out of the $\Psi$-function associated with a hierarchy of higher order analogues to the Painlevé III equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probability at the hard edge of unitary invariant random matrix ensembles perturbed by poles of order $k$ in a certain scaling regime. Using the Riemann--Hilbert method, we obtain the large $s$ asymptotics of the Fredholm determinant. Moreover, we derive a Painlevé type formula of the Fredholm determinant, which is expressed in terms of an explicit integral involving a solution to a coupled Painlevé III system.

Universality limits for generalized Jacobi measures

In this paper universality limits are studied in connection with measures which exhibit power-type singular behavior somewhere in their support. We extend the results of Lubinsky for Jacobi measures supported on [−1,1] to generalized Jacobi measures supported on a compact subset of the real line, where the singularity can be located in the interior or at an endpoint of the support. The analysis is based upon the Riemann–Hilbert method, Christoffel functions, the polynomial inverse image method of Totik and the normal family approach of Lubinsky.

Hilbert method toward a multiscale analysis from kinetic to macroscopic models for active particles

This paper develops a Hilbert type method to derive models at the macroscopic scale for large systems of several interacting living entities whose statistical dynamics at the microscopic scale is delivered by kinetic theory methods. The presentation is in three steps, where the first one presents the structures of the kinetic theory approach used toward the aforementioned analysis; the second step presents the mathematical method; while the third step provides a number of specific applications. The approach is focused on a simple system and with a binary mixture, where different time-space scalings are used. Namely, parabolic, hyperbolic, and mixed in the case of a mixture.

Riemann—Hilbert method and N—soliton for two—component Gerdjikov-Ivanov equation

We consider the Riemann—Hilbert method for initial problem of the vector Gerdjikov—Ivanov equation, and obtain the formula for its N-soliton solution, which is expressed as a ratio of (N +1)×(N +1) determinant and N × N determinant. Furthermore, by applying asymptotic analysis, the simple elastic interactions of N-soliton are confirmed, and the shifts of phase and position are also explicitly displayed.

The asymptotic behavior of the discrete holomorphic map Za via the Riemann–Hilbert method

We study the asymptotic behavior of the discrete analogue of the holomorphic map $z^a$. The analysis is based on the use of the Riemann-Hilbert approach. Specifically, using the Deift-Zhou nonlinear steepest descent method we prove the asymptotic formulae which was conjectured in 2000 by the first co-author and S.I.~Agafonov.

Uniform asymptotics for discrete orthogonal polynomials on infinite nodes with an accumulation point

In this paper, we develop the Riemann–Hilbert method to study the asymptotics of discrete orthogonal polynomials on infinite nodes with an accumulation point. To illustrate our method, we consider the Tricomi–Carlitz polynomials [Formula: see text] where [Formula: see text] is a positive parameter. Uniform Plancherel–Rotach type asymptotic formulas are obtained in the entire complex plane including a neighborhood of the origin, and our results agree with the ones obtained earlier in [W. M. Y. Goh and J. Wimp, On the asymptotics of the Tricomi–Carlitz polynomials and their zero distribution. I, SIAM J. Math. Anal. 25 (1994) 420–428] and in [K. F. Lee and R. Wong, Uniform asymptotic expansions of the Tricomi–Carlitz polynomials, Proc. Amer. Math. Soc. 138 (2010) 2513–2519].

Plancherel–Rotach type asymptotics of the sieved Pollaczek polynomials via the Riemann–Hilbert approach

We study the uniform asymptotics for the orthogonal polynomials with respect to weights composed of both absolutely continuous measure and discrete measure, by taking a special class of the sieved Pollaczek polynomials as an example. The Plancherel–Rotach type asymptotics of the sieved Pollaczek polynomials are obtained in the whole complex plane. The Riemann–Hilbert method is applied to derive the results. A main feature of the treatment is the appearance of a new band consisting of two adjacent intervals, one of which is a portion of the support of the absolutely continuous measure, the other is the discrete band.

Strong asymptotics for the Pollaczek multiple orthogonal polynomials

The asymptotic properties of multiple orthogonal polynomials with respect to two Pollaczek weights with different parameters are considered. This set of weights is a Nikishin system generated by two measures with unbounded supports; moreover, the second measure is discrete. During the last years, multiple orthogonal polynomials with respect to Nikishin systems of this type have found wide applications in the theory of random matrices. Strong asymptotic formulas for the polynomials under consideration are obtained by means of the matrix Riemann–Hilbert method.

Applying empirical mode decomposition (EMD) and entropy to diagnose circuit breaker faults

In this paper, a new method to extract characteristic parameters from vibration signal of HV (high voltage) CB (circuit breakers) is proposed based on the empirical mode decomposition (EMD)-entropy. First, the normal signal and several fault signals are decomposed with empirical mode, and then extraction and selection the IMF function. Second, calculating the signal envelope after selection in IMF using Hilbert method. Finally, on each envelope signals using time integral of entropy, acquiring the EMD-characteristic entropy vector. The fault signal characteristic entropy vector and normal signal's comparison determine fault types. In practice, this method is very convenient, and can be applied directly to diagnose the fault signals and its species.

Global asymptotics of the Szegő–Askey polynomials

The Szegő–Askey polynomials are orthogonal polynomials on the unit circle. In this paper, we study their asymptotic behavior by knowing only their weight function. Using the Riemann–Hilbert method, we obtain global asymptotic formulas in terms of Bessel functions and elementary functions for z in two overlapping regions, which together cover the whole complex plane. Our results agree with those obtained earlier by Temme [Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle, Constr. Approx. 2 (1986) 369–376]. Temme's approach started from an explicit expression of the Szegő–Askey polynomials in terms of an2F1-function, and followed by integral methods.

On Maximally Inflected Hyperbolic Curves

In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert's method we show that for any integers $$d$$ d and $$r$$ r such that $$4\le r \le 2d^2-2d$$ 4 ≤ r ≤ 2 d 2 - 2 d , there is a non-singular hyperbolic curve of degree $$2d$$ 2 d in $${\mathbb R}^2$$ R 2 with exactly $$r$$ r line segments in the boundary of its convex hull. We also give a complete classification of possible distributions of inflection points among the ovals of a maximally inflected non-singular hyperbolic curve of degree $$6$$ 6 .

Causal Discovery via Reproducing Kernel Hilbert Space Embeddings

Causal discovery via the asymmetry between the cause and the effect has proved to be a promising way to infer the causal direction from observations. The basic idea is to assume that the mechanism generating the cause distribution p(x) and that generating the conditional distribution p(y|x) correspond to two independent natural processes and thus p(x) and p(y|x) fulfill some sort of independence condition. However, in many situations, the independence condition does not hold for the anticausal direction; if we consider p(x, y) as generated via p(y)p(x|y), then there are usually some contrived mutual adjustments between p(y) and p(x|y). This kind of asymmetry can be exploited to identify the causal direction. Based on this postulate, in this letter, we define an uncorrelatedness criterion between p(x) and p(y|x) and, based on this uncorrelatedness, show asymmetry between the cause and the effect in terms that a certain complexity metric on p(x) and p(y|x) is less than the complexity metric on p(y) and p(x|y). We propose a Hilbert space embedding-based method EMD (an abbreviation for EMbeDding) to calculate the complexity metric and show that this method preserves the relative magnitude of the complexity metric. Based on the complexity metric, we propose an efficient kernel-based algorithm for causal discovery. The contribution of this letter is threefold. It allows a general transformation from the cause to the effect involving the noise effect and is applicable to both one-dimensional and high-dimensional data. Furthermore it can be used to infer the causal ordering for multiple variables. Extensive experiments on simulated and real-world data are conducted to show the effectiveness of the proposed method.

Asymptotics of a Fredholm Determinant Corresponding to the First Bulk Critical Universality Class in Random Matrix Models

We study the determinant det(I−KPII) of an integrable Fredholm operator KPII acting on the interval (−s, s) whose kernel is constructed out of the Ψ-function associated with the Hastings–McLeod solution of the second Painleve equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the unitary ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann–Hilbert method, we evaluate the large s-asymptotics of det(I−KPII) .

A novel autofocusing method using the angle of hilbert space for microscopy

Autofocusing technology is indispensable for routine use of microscopes on a large scale in biological field. The autofocusing method using the angle of Hilbert space is brought forward to measure whether the image is focused or not. The angle of Hillbert space can be used to evaluate accurately the similarity degree of two images. The experiment results show that the autofocusing method can decrease the computational cost and get accuracy for real-time biological and biomedical images with noise robustness. The focus curves are smooth and possess the unimodality, the monotonicity and the symmetry. Compared with other classic and optimum focus method, the Hilbert method demonstrates its robustness to noise and can improve the focus speed. The experiments showed that the proposed method can increase the overall performance of an autofocus system and has strong applicability in various autofocusing algorithms.

Detection of inter-turn short-circuit in induction motors using Park–Hilbert method

In this paper, a new method, called Park–Hilbert (P–H), is presented and used to detect inter-turn short circuit in three-phase induction motors. This technique is based on spectral analysis, via the FFT, of a new generated signal called PVMP–H which represents the Park’s vector modulus computed starting from the analytical signals obtained from the three-phase current by the Hilbert transform. A theoretical analysis is performed in order to show how the proposed method works. Various experimental tests carried out on a 3 kW three-phase induction motor, illustrate the efficiency of this new method and show that some spectral components of the PVMP–H are extremely sensitive to the occurrence of shorted turns. However, it is shown that the information about the motor state is distributed on many spectral components and in order to gather these useful information, the present paper suggests the monitoring of a new fault indicators computed starting from the most sensitive harmonics (basing on the multiple signature processing). This technique is applied on the proposed signal PVMP–H; and the experimental tests confirm that a small number of shorted turns can be effectively detected using these new fault indicators even under low load. A comparative study with the conventional MCSA and EPVA methods is also performed. The obtained results show the superiority of the P–H method and prove the effectiveness of the proposed fault indicators for extracting useful information from any motor signal.

Improvement of the Hilbert Method via ESPRIT for Detecting Rotor Fault in Induction Motors at Low Slip

The traditional Hilbert method to detect broken rotor bar fault in induction motors is reviewed and its major drawbacks are clearly revealed, namely, deteriorating or even completely failing when a motor operating at low slip due to the fixed constraints of fast Fourier transform (FFT) is used in this method. To overcome this, the estimation of signal parameters via rotational invariance technique (ESPRIT) is then introduced to replace FFT, and an improved Hilbert method is thus presented by conjugating the Hilbert transform and ESPRIT together. Experimental results of a small motor in a laboratory and a large motor operating on an industrial site are reported to demonstrate the effectiveness of the improved Hilbert method.