Laguerre Functions Research Articles

Design of active disturbance rejection controller for the system with unknown model using Laguerre functions

This paper concentrates on the design problem of linear active disturbance rejection control (LADRC) for uncertain systems, whose order and parameters are unknown. In general, LADRC has diversified structural forms due to different order and parameters selection. To clarify the prerequisites for constructing the LADRC strategy, a Laguerre-based LADRC (LLADRC) method is proposed, which provides controller design guidance for systems with unknown models by capturing the impulse response. Then, intensive attention is focused on establishing internal stability conditions of the feedback system and parameter tuning rules of the prescribed controller. To gain a better understanding, the internal correlation between LLADRC and standard LADRC of the same order is analyzed. It is shown that observer and controller gains of the standard LADRC can be tuned via a workable LLADRC to achieve better control performance. Finally, a numerical example and an application design of DC motor speed control are studied, both of which demonstrate the feasibility and validity of the proposed method.

Efficient spectral element method for the Euler equations on unbounded domains

Mitigating the impact of waves leaving a numerical domain has been a persistent challenge in numerical modeling. Reducing wave reflection at the domain boundary is crucial for accurate simulations. Absorbing layers, while common, often incur significant computational costs. This paper introduces an efficient application of a Legendre-Laguerre basis for absorbing layers for two-dimensional non-linear compressible Euler equations. The method couples a spectral-element bounded domain with a semi-infinite region, employing a tensor product of Lagrange and scaled Laguerre basis functions. Semi-infinite elements are used in the absorbing layer with Rayleigh damping. In comparison to existing methods with similar absorbing layer extensions, this approach, a pioneering application to the Euler equations of compressible and stratified flows, demonstrates substantial computational savings. The study marks the first application of semi-infinite elements to mitigate wave reflection in the solution of the Euler equations, particularly in nonhydrostatic atmospheric modeling. A comprehensive set of tests demonstrates the method's versatility for general systems of conservation laws, with a focus on its effectiveness in damping vertically propagating mountain gravity waves, a benchmark for atmospheric models. Across all tests, the model presented in this paper consistently exhibits notable performance improvements compared to a traditional Rayleigh damping approach.

Finite-frequency model order reduction of linear and bilinear systems via low-rank approximation

In this paper, we first investigate the finite-frequency model order reduction for linear systems based on low-rank Gramian approximations. An efficient algorithm for computing low-rank approximations of the finite-frequency and frequency-dependent Gramians based on Laguerre functions is proposed. The approach constructs the low-rank decomposition factors of the finite-frequency Gramians or frequency-dependent Gramians through a recursive formula of Laguerre functions expansion coefficient vectors and then combines the low-rank square root method and frequency-dependent balanced truncation method to obtain the reduced-order models. In this process, it avoids dealing with the matrix-valued functions and solving the related (generalized) Lyapunov matrix equations directly, making them computationally efficient. Furthermore, the above method is successfully extended to bilinear systems, and a corresponding efficient computation method for low-rank approximations of the finite-frequency Gramians of bilinear systems is derived. Finally, some numerical simulations are provided to illustrate the effectiveness of our proposed algorithms.

Frequency dependent polarizability of confined harmonic oscillators

The D-dimensional Schrödinger equation for a confined harmonic potential perturbed by a time-dependent dipole field is analysed. The model uses exact confined solutions for the harmonic potential whereas the time-dependent field is treated using first-order time-dependent perturbation theory. Exact analytic solutions are obtained in terms of generalised Laguerre functions. This system has applications to nanostructures where there are strong confinement potentials in some dimensions and example calculations are carried out to illustrate the effect of the dimension.

Unbounded Domain Modeling in Axisymmetric Rotor Wake Analysis

Mitigating the effects of wind turbine wakes is a central part of wind farm design. This paper proposes a spectral solver for the axisymmetrical Ainslie wake model based on modified Laguerre basis functions over the semi-infinite radial domain and a marching scheme in the downstream direction. This orthogonal basis promises fast convergence and a low number of DOFs for discretizing the continuity and axial momentum equation, promising a computationally efficient method. The numerical implementation of the model could not be finished in time; preliminary results are presented but still show non-negligible conservation errors. The focus of this work lies, therefore, on a detailed derivation of the method and a discussion of the sources of numerical errors in the nonlinear terms, and due to the truncation of the spectral basis, a comparison of the solver outputs to other methods remains part of the ongoing work.

Orthonormal expansions for translation-invariant kernels

We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of ℒ2(R). This allows us to derive explicit expansions on the real line for (i) Matérn kernels of all half-integer orders in terms of associated Laguerre functions, (ii) the Cauchy kernel in terms of rational functions, and (iii) the Gaussian kernel in terms of Hermite functions.

The continuous wavelet transform for a Laguerre type operator on the half line

In this paper, we consider a Laguerre differential operator on half line by accomplishing harmonic analysis tools with respect to the operator. We study some definitions and properties of Laguerre continuous wavelet transform. We also explore generalized Laguerre Fourier transform and convolution product on half line associated with the Laguerre differential operator. Also a new continuous wavelet transform associated with Laguerre function is constructed and investigated.

Long-Horizon Constrained Model Predictive Direct Speed Control for PMSM Drives Based on Laguerre Functions

Long-Horizon Constrained Model Predictive Direct Speed Control for PMSM Drives Based on Laguerre Functions

Model order reduction for discrete time-delay systems based on Laguerre function expansion

We investigate model order reduction for discrete time-delay systems via orthogonal polynomial expansion in the frequency domain. The transfer function of systems is expanded in the framework of Laguerre function basis. We show that Laguerre coefficients of the states satisfy a linear system and reduced models generated by projection methods can preserve some Laguerre coefficients of the original systems. We prove that the subspace spanned by Laguerre coefficients is exactly a high order Krylov subspace, thereby leading to an efficient computation of projection matrices and a more accurate coefficient-matching property in the two-sided framework. Further the Laguerre expansion of systems is employed to enable an approximate but fast execution of balanced truncation for discrete time-delay systems. Specifically, Gramians are approximated based on the derived Laguerre coefficients, which circumvents the main bottleneck of balanced truncation methods. Finally, two numerical examples are simulated to demonstrate the feasibility and effectiveness of the proposed methods.

Improved Model Predictive Speed Control of a PMSM via Laguerre Functions

This paper proposes a model predictive speed control strategy for a surface-mounted permanent magnet synchronous motor by applying Laguerre functions. The model predictive controller (MPC) incorporates an integrator. A quadratic programming procedure is applied to solve the constrained optimization problem online. The paper also provides a solution for stability. The performance efficiency of the proposed scheme is validated by comparing the results with the performance of an optimal linear quadratic regulator, conventional state-space model predictive control, and a simple MPC algorithm with integral action. Extensive simulation results confirm the efficacy of the proposed scheme, showing that it achieves good steady-state performance while maintaining a fast dynamic response.

Low-rank balanced truncation of discrete time-delay systems based on Laguerre expansions

This paper introduces a novel model order reduction method based on low-rank Gramian approximations for discrete time-delay systems. Firstly, an efficient algorithm based on Laguerre functions to compute the low-rank decomposition factors of the controllability and observability Gramians for discrete time-delay systems is given, in which the low-rank factors satisfy the iterative recursive formulas of the expansion coefficients of the Laguerre functions. It effectively avoids the direct solutions of Gramians. Then, the reduced-order model of the discrete time-delay system is obtained by combining the low-rank square root method. Additionally, a modified algorithm that combines the dominant subspace projection method is introduced, which alleviates certain drawbacks of the above technique and enhances the stability in certain cases. Finally, two numerical examples are given to verify the accuracy and efficiency of our proposed algorithm.

Dynamic behavior modelling of the silicone-ethanol actuator based on principal dynamic modes of the Laguerre model

The silicone-ethanol actuator is an emerging artificial muscle gaining attention because of its simple construction/stimulation and high-force generation. Most related researches are focused on structural modification methods, while behavior modeling has received less attention. Although the previous models are highly accurate, they suffer from the lack of ethanol content as an input, highly feedback dependency, and fixed output range. In this research, to address the mentioned concerns, the actuator was made according to the conventional procedure and subjected to stimulation, and displacement/blocked-force recording by special custom hardware. The displacement/blocked-force was not limited to a fix amount in order to evaluate ethanol leakage. To predict displacement/blocked-force, a non-autoregressive model was developed with the applied energy, temperature, and ethanol content as inputs. The Laguerre functions was applied to ensure dynamic matching between input-output. Finally, Principal Dynamic Mode analysis was used to optimize the number of filter banks in the model. The results showed that the proposed model provide a reasonable balance between accuracy, interpretability, and simplicity. Also, we found that the prediction of displacement/blocked-force is mainly influenced by the core-temperature/applied-energy, respectively. The proposed model focuses on the effect of stimulation signal and disregards the structural parameters due to their variety in different applications. By considering the both structural parameters and excitation signal, a more comprehensive model can be developed to predict actuator dynamic behavior.

Predictive controllers for synchronous reluctance motor drive systems

This paper proposes the design and implementation of predictive controllers for synchronous reluctance motor drive systems to enhance their dynamic responses. The predictive speed and current controllers in this paper are designed in systematic procedures. The predictive speed controller is implemented by using Laguerre function procedure. The Laguerre function is used to simplify the algorithm and to minimize the execution time of the digital signal processor. For predictive current controller, a finite control set method improves the current tracking ability. The measured currents are used to predict the future phase-current based on the motor model. The optimal control inputs of both predictive controllers are determined by using a cost function minimization method. Experimental results show the proposed drive system provides a wide adjustable speed range, from 2 r/min to 1800 r/min. It has better performance than a proportional-integral (PI) controller including fast rise time, which is 0.9 second, small steady state error, which is 0.32 r/min, and small current ripples. A 32-bit floating-point digital signal processor, TMS-320-F-28335 DSP, is employed to implement the control algorithms.

Autonomous predictive maintenance of quadrotor UAV with multi-actuator degradation

Abstract With the wide application of quadrotor unmanned aerial vehicles (UAVs), the requirements for their safety and reliability are becoming increasingly stringent. In this paper, based on the feedback of airframe performance health perception information and the predictive function control strategy, the autonomous maintenance of a quadrotor UAV with multi-actuator degradation is realised. Autonomous maintenance architecture is constructed by the predictive maintenance (PdM) idea and the Laguerre function model predictive pontrol (LF-MPC) strategy. Using the two-stage Kalman filter (TSKF) method, based on the established UAV degradation model, the aircraft state and actuator degradation state are predicted simultaneously. For the predictive perception of system health, on the one hand, the system health degree (HD) based on Mahalanobis distance is defined by the degree of airframe state deviation from the expected state, and then the failure threshold of the UAV is obtained. On the other hand, according to the degradation state of each actuator, a comprehensive degradation variable fused with different weight coefficients of multiple actuators degradation is used to obtain the probability density function (PDF) of remaining useful life (RUL) prediction. For the autonomous maintenance of system health, the LF-MPC weight matrixes are adjusted adaptively in real-time based on the HD evaluation, to achieve a compromise balance between UAV performance and control effect, and greatly extend the working time of UAV. Simulation results verified the effectiveness of the proposed method.

Nonlinear Model Predictive Control for a Dynamic Positioning Ship Based on the Laguerre Function

In this paper, we present a novel nonlinear model predictive control (NMPC) algorithm based on the Laguerre function for dynamic positioning ships to solve the problems of input saturation, unknown time-varying disturbances, and heavy computation. The nonlinear model of a dynamic positioning ship is presented as a linear model, transformed from a standard affine nonlinear state-space model by precise feedback linearization. The environmental disturbance is overcome using an integrator. The time cost of the proposed nonlinear control algorithm is decreased by inducing the Laguerre function to describe the feedback-linearization system input increments. The Laguerre function reduces the matrix dimensions of the nonlinear optimization problem. The simulation results for a DP supply vessel showed that the novel algorithm maintained the effective control performance of the original nonlinear model predictive control algorithm and had a reduced computation load to satisfy the requirements of real-time operation.

High-accuracy solution of the Dirac equation for three-dimensional hydrogen atom using a variationally optimized diagonalization method

High-accuracy solution of the Dirac equation for three-dimensional (3D) hydrogen atom is theoretically presented using a variationally optimized diagonalization method. By imposing the asymptotical conditions, a complete set of orthogonal Laguerre basis functions is constructed explicitly for the solution of the Dirac equation, which eliminates the singularity difficulty at the point nucleus and gives a convergent integral. A fully explicit expression of the matrix elements of the Dirac Hamiltonian has been derived from the method of the generating function of Laguerre polynomials. The relative errors of the relativistic energies are ranged from 11 to 14 digits, showing that the calculated relativistic energies are in excellent agreement with the exact energies. These results show that my proposed orthogonal Laguerre basis set is extremely efficient and stable in calculating high-accuracy solution of the Dirac equation for 3D hydrogen atom.

A Simplified Constrained Predictive Position Control for PMSM Drives With Laguerre Functions

A Simplified Constrained Predictive Position Control for PMSM Drives With Laguerre Functions

MPC-Based Exponential Weight Laguerre Function With Non-Singular Terminal SMC for Four-Wheel Independent Drive Electric Vehicles

MPC-Based Exponential Weight Laguerre Function With Non-Singular Terminal SMC for Four-Wheel Independent Drive Electric Vehicles

Research on the Optimal Output Power of Proton‐Exchange Membrane Fuel Cell Based on Improved Model Predictive Control Strategy

To achieve optimal power output of proton‐exchange membrane fuel cell (PEMFC) under load variation and noise disturbance, this article proposes a recursive subspace model predictive control based on Laguerre function (RSMPCL) strategy. First, a fractional PEMFC identification model is established, and an optimal power setter is introduced to provide an optimal power value corresponding to the current load. Second, during the predictive process, a subspace predictor is updated in real time where a Householder transformer is applied to decrease the computation time of the frequent QR decomposition, and a variable forgetting factor is introduced to enhance the sensitivity and accuracy of predictive control. In the control process, a Laguerre function is employed to transform the problem of computing the input increment at the next step into calculating the coefficients of the input variables. This transformation reduces the computation time of the control quantities. Finally, load and disturbance tests conducted on a dSPACE semiphysical simulation platform indicate that compared with constrained subspace model predictive control, the proposed RSMPCL strategy can help the fuel cells achieve the optimal power point with control accuracy, robustness, and calculation speed.

Quantum mechanical approach to plasma waves with helical wavefront

Plasma waves with helical wavefront are studied theoretically from the quantum mechanical viewpoint and are shown to produce a spinning motion of a charged macroparticle in a complex plasma. The electrostatic helical perturbations are described by the wave function for a Laguerre–Gaussian beam mode with the radial/angular mode numbers n/l. The interaction and the transfer of angular momentum from the wave to a particle are analyzed by the method of second quantization with the help of the Feynman diagram. Laguerre function, instead of the Born approximation, is introduced to describe plasma waves with helical wavefront. A pair of dust particles in a complex plasma exchange a quasiparticle (virtual plasmon) resulting in the acquisition of angular momentum, which makes a dust particle spin in motion with rotational frequency Ωϕ. The resonance condition ω−kvz−lΩϕ=0 and the conservation of angular momentum IdΩϕ=lℏ determine the rotational frequency, where ω and k are frequency and axial wave number of the helical wave, and vz and Id are axial velocity and the moment of inertia of a dust particle.