Sylvester Equation Research Articles

A modified noise-tolerant ZNN model for solving time-varying Sylvester equation with its application to robot manipulator

Recently, Xiao et al. (2021) proposed an efficient noise-tolerant zeroing neural network (NTZNN) model with fixed-time convergence for solving the time-varying Sylvester equation. In this paper, we propose a modified version of their NTZNN model, named the modified noise-tolerant zeroing neural network (MNTZNN) model. It extends the NTZNN model to a more general form and then we prove that, with appropriate parameter selection, our new MNTZNN model can significantly accelerate the convergence of the NTZNN model. Numerical experiments confirm that the MNTZNN model not only maintains fixed-time convergence and noise-tolerance but also has a faster convergence rate than the NTZNN model under certain conditions. In addition, the design strategy of the MNTZNN is also successfully applied to the path tracking of a 6-link planar robot manipulator under noise disturbance, which demonstrates its applicability and practicality.

An Efficient Takagi–Sugeno Fuzzy Zeroing Neural Network for Solving Time-Varying Sylvester Equation

In this paper, we propose an efficient Takagi-Sugeno fuzzy zeroing neural network (TS-FZNN) activated by a new activation function for solving time-varying Sylvester equation. The self-adaptive convergence parameter is designed by the Takagi-Sugeno fuzzy logic system. Convergence and robustness of the proposed model are analyzed. Theoretical analysis shows that the proposed model not only has less fixed convergence time than the recently suggested ZNN models, but also is noise-tolerant. Numerical experiments are performed to illustrate its efficiency and effectiveness as well as the superior performance over the existing ZNN models for solving time-varying Sylvester equation, including the applicability of the proposed model to robot manipulator. The effects of model parameters on the dynamic response of the robotic manipulator system are also illustrated by the convergence rate of position errors of the robotic manipulator trajectory.

Partial Eigenstructure Assignment for Linear Time-Invariant Systems via Dynamic Compensator

This article studies the partial eigenstructure assignment (PEA) problem for a type of linear time-invariant (LTI) system. By introducing a dynamic output feedback controller, the closed-loop system is similar to a given arbitrary constant matrix, so the desired closed-loop eigenstructure can be obtained. Different from the normal eigenstructure assignment, only a part of the left and right generalized eigenvectors is assigned to the closed-loop system to remove complicated constraints, which reflects the partial eigenstructure assignment. Meanwhile, based on the solutions to the generalized Sylvester equations (GSEs), two arbitrary parameter matrices representing the degrees of freedom are presented to obtain the parametric form of the coefficient matrices of the dynamic compensator and the partial eigenvector matrices. Finally, an illustrative example and the simulation results prove the excellent effectiveness and feasibility of parametric method we proposed.

Solutions to the SU([formula omitted]) self-dual Yang–Mills equation

In this paper we aim to derive solutions for the SU(N) self-dual Yang–Mills (SDYM) equation with arbitrary N. A set of noncommutative relations are introduced to construct a matrix equation that can be reduced to the SDYM equation. It is shown that these relations can be generated from two different Sylvester equations, which correspond to the two Cauchy matrix schemes for the (matrix) Kadomtsev–Petviashvili hierarchy and the (matrix) Ablowitz–Kaup–Newell–Segur hierarchy, respectively. In each Cauchy matrix scheme we investigate the possible reductions that can lead to the SU(N) SDYM equation and also analyze the physical significance of some solutions, i.e. being Hermitian, positive-definite and of determinant being one.

Special Sylvester equations, PI and EP elements in rings with involution

In this paper, we study the general solution to certain special Sylvester equations, PI and EP elements in rings with involution. Especially, some practical necessary and sufficient solvability conditions for special Sylvester equations [Formula: see text] and [Formula: see text] are presented, and expressions of the general solution to these two equations when they are solvable are provided.

Cooperative Output Regulation of Large-Scale Wind Turbines for Power Reserve Control

This paper develops a distributed control framework to coordinate large-scale wind turbines for fairly sharing the power reserve request on a wind farm. A cooperative output regulation problem is formulated in a multi-agent framework, where the synchronization of turbine system outputs are established regarding the heterogeneity of turbine models. For this purpose, a homogeneous virtual system is created first and embedded into each turbine to generate the agreed trajectories across the wind farm in a distributed manner. The local power tracking of individual turbines is then enforced by the synthesized output regulator, whose control gains are determined by a variant of the Sylvester equation. The proposed approach issues the generator torque and blade pitch commands in a unified manner and is resistant to wind disturbances. The case studies are carried out through simulations and in a real laboratory-scale distributed communication network. The simulation results demonstrate the effectiveness of coordinated actions of wind turbines in response to the changing demands, under both constant and turbulent wind scenarios with the plug-and-play capability.

Model order reduction based on low-rank decomposition of the cross Gramian

This paper considers model order reduction based on low-rank decomposition of the cross Gramian for linear systems. The proposed approach uses the low-rank factors of the cross Gramian to generate approximate balanced model for the large-scale system. Then, the reduced-order model is obtained by truncating the states corresponding to the small approximate Hankel singular values. The low-rank factors are directly constructed from the expansion coefficients of impulse responses in the space spanned by Legendre polynomials by solving block tridiagonal linear systems. In contrast to balanced truncation related approaches which require to solve Sylvester equation to compute the full cross Gramian, the proposed method needs to solve sparse linear equations, and only one singular value decomposition is applied to a low-dimensional matrix. The stability preservation of the reduced model is briefly discussed. And in combination with the dominant subspace projection method, the reduction procedure is modified to alleviate the shortcoming, which may unexpectedly lead to unstable systems even though the original one is stable. Finally, numerical experiments are given to demonstrate the effectiveness of the proposed methods.

Iterative optimal solutions of linear matrix equations for hyperspectral and multispectral image fusing

For a linear matrix function f in X in {mathbb {R}}^{mtimes n} we consider inhomogeneous linear matrix equations f(X) = E for E ne 0 that have or do not have solutions. For such systems we compute optimal norm constrained solutions iteratively using the Conjugate Gradient and Lanczos’ methods in combination with the More–Sorensen optimizer. We build codes for ten linear matrix equations, of Sylvester, Lyapunov, Stein and structured types and their T-versions, that differ only in two five times repeated equation specific code lines. Numerical experiments with linear matrix equations are performed that illustrate universality and efficiency of our method for dense and small data matrices, as well as for sparse and certain structured input matrices. Specifically we show how to adapt our universal method for sparse inputs and for structured data such as encountered when fusing image data sets via a Sylvester equation algorithm to obtain an image of higher resolution.

Parametric design method for partial eigenstructure assignment of second-order linear systems via observer-based state feedback

In this article, a parametric method of observer-based partial eigenstrunt in second-order linear time-invariant (LTI) systems is explored. First, an observer is constructed to approximate the full sate vector when the entire state vector is not available for measurement. Meanwhile, the separation principle of the first-order linear systems is extended to second-order linear systems. Second, by partitioning the system matrix of the open-loop system into two parts, the satisfactory and unsatisfactory eigenstructures are selected. On the premise of retaining the satisfactory part in the open-loop system, the unsatisfactory eigenstructure assignment problem can be converted into the solutions of a type of generalized Sylvester equations (GSE). Third, by introducing a group of arbitrary parameters, complete parametric expressions of the proportional plus derivative (PD) state feedback controller can be obtained. The main advantage of the proposed method is that it provides all the design degrees of freedom and reduces the design complexity of the controller. Finally, the correctness of the proposed method is verified by two examples and simulation results. Using the degrees of freedom of the parameters, the gain matrices are optimized through the optimization index provided by the optimization function, and better performance is obtained.

An Accelerated Double-Integral ZNN with Resisting Linear Noise for Dynamic Sylvester Equation Solving and Its Application to the Control of the SFM Chaotic System

The dynamic Sylvester equation (DSE) is frequently encountered in engineering and mathematics fields. The original zeroing neural network (OZNN) can work well to handle DSE under a noise-free environment, but may not work in noise. Though an integral-enhanced zeroing neural network (IEZNN) can be employed to solve the DSE under multiple-noise, it may fall flat under linear noise, and its convergence speed is unsatisfactory. Therefore, an accelerated double-integral zeroing neural network (ADIZNN) is proposed based on an innovative design formula to resist linear noise and accelerate convergence. Besides, theoretical proofs verify the convergence and robustness of the ADIZNN model. Moreover, simulation experiments indicate that the convergence rate and anti-noise ability of the ADIZNN are far superior to the OZNN and IEZNN under linear noise. Finally, chaos control of the sine function memristor (SFM) chaotic system is provided to suggest that the controller based on the ADIZNN has a smaller amount of error and higher accuracy than other ZNNs.

Improved Recurrent Neural Networks for Text Classification and Dynamic Sylvester Equation Solving

Improved Recurrent Neural Networks for Text Classification and Dynamic Sylvester Equation Solving

Applications of the Sylvester equation for the lattice BKP system

Applications of the Sylvester equation for the lattice BKP system

Applications of the Sylvester equation for the lattice BKP system

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A computationally effective time-restricted stability preserving [formula omitted]-optimal model order reduction approach

Several approaches for reducing model order on the definite time segments have become the topic of investigation in a series of papers that bring challenges during application in a large-scale setting. The subject of discussion of this paper is the computationally efficient time-restricted H2-optimal model order reduction method of higher dimensional sparse systems that requires the solutions of time-restricted Lyapunov and Sylvester equations. Our discussion is on developing the algorithms to solve these matrix equations that face difficulty when calculating the matrix exponential of the large-scale matrices. As a result, an efficient remedy is also proposed to compute the matrix exponential. Our ideas are also evaluated for index-1 descriptor systems apart from the generalized structure. Numerical analyses are conducted on several benchmark examples to illustrate how accurate and efficient our suggested approaches are by comparing them with the existing methods.

A novel zeroing neural network for dynamic sylvester equation solving and robot trajectory tracking

To solve the theoretical solution of dynamic Sylvester equation (DSE), we use a fast convergence zeroing neural network (ZNN) system to solve the time-varying problem. In this paper, a new activation function (AF) is proposed to ensure fast convergence in predefined times, as well as its robustness in the presence of external noise perturbations. The effectiveness and robustness of this zeroing neural network system is analyzed theoretically and verified by simulation results. It was further verified by the application of robotic trajectory tracking.

The integrability, equivalence and solutions of two kinds of integrable deformed fourth-order matrix NLS equations

Based on the higher-order restricted flows, the first type of integrable deformed fourth-order matrix NLS equations, that is, the fourth-order matrix NLS equations with self-consistent sources (FMNLSSCS), is derived. By virtue of the $${\bar{\partial }}$$ -dressing method, the second type of integrable deformed fourth-order matrix NLS equations called the fourth-order matrix NLS–Maxwell–Bloch system (FMNLS-MB) is presented. We prove the equivalence of the FMNLSSCS and the FMNLS-MB successfully. Furthermore, N-soliton solutions are explicitly obtained by means of the Cauchy matrix method starting from corresponding Sylvester equation.

Partial eigenstructure assignment for descriptor high-order linear systems via proportional plus derivative state feedback: A parametric approach

In this paper, we present a parametric method to design proportional plus derivative (PD) state feedback for the problem of partial eigenstructure assignment (PESA) in a type of descriptor high-order linear time-invariant (LTI) systems. By dividing the original eigenstructure into replaced and remaining parts and using the solutions to the high-order generalized Sylvester equations (HGSEs), complete parameterized expressions of PD feedback controller and the replaced part of the right eigenvector matrix are established. With the proposed approach, the unsatisfactory eigenstructure in the open-loop system is changed into the expected eigenstructure in the closed-loop system, while the satisfactory part in the open-loop system is retained. Meanwhile, the regularity of the closed-loop system can also be well-guaranteed by the selection of arbitrary parameters. Finally, a numerical example and a flexible-joint robot system are presented to verify the feasibility and effectiveness of the proposed method, respectively.

Randomization and the valuation of guaranteed minimum death benefits

In this article, we focus on death-linked contingent claims (GMDBs) paying a random financial return at a random time of death in the general case where financial returns follow a regime switching model with two-sided phase-type jumps. We approximate the distribution of the remaining lifetime by either a series of Erlang distributions or a Laguerre series expansion, whose capability to fit the tail of the observed mortality data turns out to be much better than the commonly used series of exponential distributions.More precisely, we develop a Laurent series expansion of the discounted Laplace transform of the subordinated process at an Erlang distributed time, which leads to explicit formulae for European-type GMDB as well as related risk measures such as the Value-at-Risk (VaR) and the Conditional-Tail-Expectation (CTE). We further concentrate upon path-dependent GMDBs with lookback features like dynamic fund protection or dynamic withdrawal benefits, by relying on a Sylvester equation approach. The advantage of our approaches is that our results are of semi-closed form, avoiding numerical Fourier inversion or Monte-Carlo simulation, leading to fast evaluation. This is necessary in risk-management, in particularly for nested simulation in the framework of Solvency II. Several numerical experiments are included.Our results have implications beyond life-insurance and GMDBs, namely in all situations where randomization or Erlangization replaces known quantities, like, for example, model parameters, by random variables. In Finance, it is for example well-known that a random maturity time leads to much more convenient valuation formulas that well approximate its non-random counterpart.

Functional Detectability and Asymptotic Functional Observer Design

The objective of this article is to present a new solution to the functional observer design problem. First, a new definition for functional detectability is given; then, some algebraic conditions for linear multivariable systems to be functional detectable are presented. They generalize and coincide with the existing detectability condition required in the design of reduced and full-order Luenberger observers. Then, necessary and sufficient conditions for existence of an asymptotic functional observer are given and complete those existing results in the literature. The connection with the Sylvester equation and its solution is also given. The functional observer parameters are obtained from the Sylvester equation and the functional detectability condition. Necessary and sufficient conditions for stability of functional observers are given in the form of matrix inequalities based on two approaches: the first approach is based on stability analysis of the solution of the Sylvester equation, and the second approach is based on the Frobenius canonical form and its spectrum, which leads to a static output feedback formulation. Necessary and sufficient conditions and some simple sufficient conditions in the form of linear matrix inequality are given for the functional observer design. Moreover, the detectability of the considered system is not required. Two numerical examples are given to illustrate the presented results.

Observer-Based Model Reference Tracking Control of the Markov Jump System with Partly Unknown Transition Rates

This paper deals with the model reference tracking control problem of linear systems based on the observer for Markov jump systems with unknown transition rates. The main contributions are as follows: Firstly, we designed a descriptor observer for a given model by the matrix transformation. Then, a tracking control law composed of a feedforward compensator and feedback control law was designed by calculating variations based on the designed observer. The feedback part can stabilize the system. The feedforward part is the complete parametric feedforward tracking compensator. The two parts can be solved separately, and a controller that can make the system stable is proposed under the condition that transition rates are partially unknown through the Lyapunov stability theory. The feedforward parametric solution is given by the generalized Sylvester equation. The algorithm and criteria are proved by several examples and compared with the existing conclusions.