Generalized Sylvester Equation Research Articles

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.

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.

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.

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.

On the generalized Sylvester operator equation AX−Y B = C

In this paper, we study the necessary and sufficient conditions to have a solution pair ( X , Y ) to the generalized Sylvester equation AX−Y B = C where A , B , C ∈ L ( H ) are given. Moreover, we give some connections the generalized Fuglede-Putnam property and the solution pair ( X , Y ) of the operator equation AX−Y B = C. Finally, we consider some properties of the operator equation AX−Y B = C when A and B belong to the special classes of operators.

Cross-Gramian-Based Model Reduction for Descriptor Systems

In this paper, we explore model order reduction for large-scale square descriptor systems. A balancing-free square-root method is proposed. The balancing-free square-root method is based on two cross Gramians, one of which is known as the proper cross Gramian and the other as the improper cross Gramian. The proper cross Gramian is the unique solution of a projected generalized continuous-time Sylvester equation, and the improper cross Gramian solves a projected generalized discrete-time Sylvester equation. In order to compute the low-rank factors of these two cross Gramians, we extend the low-rank iteration of the alternating direction implicit method and the Smith method to the projected generalized Sylvester equations. We illustrate the effectiveness of the balance truncation method with one numerical example.

Functional interval observer design for linear time-varying systems with additive disturbances

This paper investigates the problem of the functional interval observer design for linear time-varying (LTV) systems with additive disturbances in both input and output channels. First, sufficient conditions for the existence of a functional interval observer for LTV systems are proposed. Based on the solution of a type of Sylvester matrix equations (generalized Sylvester equations, GSEs), completely parameterized expressions of functional interval observer coefficient matrices are established, yielding a simple and effective design approach for solving the interval estimation issue of the LTV system, while the free parameters in the expressions provide the design degrees of freedom that can be utilized to achieve additional system specifications. Furthermore, the developed observer may execute state estimation at the desired convergence rate, considerably improving estimate performance. Finally, a numerical example and a back-to-turn (BTT) aircraft control example are presented, with the results showing that the upper and lower bounds provided by the designed observer provide a better performance of interval estimation of the functional state variables, demonstrating the effectiveness of the proposed method.

A numerical study of moving boundary problem involving dual phase lag model of heat mass transfer during immersion frying

In this paper, we developed a dual phase lag model of heat mass transfer during immersion frying of foods. This model is a moving boundary problem of a couple system of second order hyperbolic partial differential equations. Modified finite element Legendre wavelet Galerkin method developed for its solution. The discretization in space and then application of Legendre wavelet Galerkin method converts our problem into a coupled system of generalized Sylvester equations. In a particular case, the solution of the present model is compared with exact solution and are approximately the same. L∞ error decreases as Legendre wavelet basis functions increase and strip size decreases. Whole analysis is done in dimensionless form. The effect of variability of moving interface on Fourier number, Luikov number, Posnov number, Kossovitch number, Biot number and relaxation times is discussed in detail.

Parametric design of functional observer for second‐order linear time‐varying systems

AbstractThis note constructs a functional observer for second‐order linear time‐varying (SOLTV) systems under framework of second‐order systems. Based on the assumption that the SOLTV system is observable, sufficient conditions for the existence of a second‐order functional observer are proposed. Combined with the solution of generalized Sylvester equation (GSE), the completely parameterized expressions of coefficient matrices of functional observer are proposed, which includes the free parameters representing degrees of freedom, and can be utilized to achieve some additional performance. Further, the design procedure is put forward, whose validity is verified through an example.

Reduced‐order functional observers for descriptor linear time‐invariant systems: a parametric method

SummaryThis article introduces the parametric design method of the reduced‐order functional observer for descriptor linear time‐invariant systems. First, existence conditions of reduced‐order functional observers are provided. Then, based on the solution of generalized Sylvester equations, completely parameterized expressions of the coefficients matrices of the reduced‐order functional observer are established, and the design algorithm of the reduced‐order functional observer is also proposed. The advantage of the proposed design method is that the observation error system can be transformed into a linear system with an expected eigenstructure. Finally, two examples are given to verify the effectiveness of the proposed method.

Output feedback predictive control for discrete quasilinear systems with application to spacecraft flying‐around

AbstractThis paper considers a spacecraft flying‐around problem modeled by discrete quasilinear system under sight coordinate, which can be regarded as an output tracking for discrete quasilinear system. An output feedback predictive control scheme is presented to deal with the flying‐around task. This scheme contains the design of output feedback and the optimization of tracking performance. On the one hand, the completely parametric form of output feedback based on generalized Sylvester equation is developed to stabilize the closed‐loop nominal system. On the other hand, a cost function, aimed to measure tracking performance, is optimized by the design of predictive control such that tracking performance is ensured. Further analysis proposes a necessary and sufficient condition based on Lyapunov method to maintain both bounded stability and tracking performance of the closed‐loop system caused by output feedback predictive control. Compared with the existing method, a numerical simulation is given to richly demonstrate the effectiveness and feasibility of the proposed control scheme.

Parametric design to reduced-order functional observer for linear time-varying systems

This article studies the parametric design of reduced-order functional observer (ROFO) for linear time-varying (LTV) systems. Firstly, existence conditions of the ROFO are deduced based on the differentiable nonsingular transformation. Then, depending on the solution of the generalized Sylvester equation (GSE), a series of fully parameterized expressions of observer coefficient matrices are established, and a parametric design flow is given. Using this method, the observer can be constructed under the expected convergence speed of the observation error. Finally, two numerical examples are given to verify the correctness and effectiveness of this method and also the aircraft control problem.

A note on the solvability for generalized Sylvester equations

A note on the solvability for generalized Sylvester equations

Superpixelwise Collaborative-Representation Graph Embedding for Unsupervised Dimension Reduction in Hyperspectral Imagery

Recently, graph-embedding framework has been developed for dimensionality reduction (DR) and classification of hyperspectral images (HSI). However, it suffers from intraclass difference and interclass similarity in complex scenarios. In this article, an unsupervised DR method called superpixelwise collaborative-representation graph embedding (SPCRGE) is proposed for the HSI classification. In SPCRGE, homogeneous regions called superpixels are generated by grouping spectral-similar and spatially adjacent pixels. Pixels in one homogeneous region come from one class with high probability. Then, Laplacian regularized superpixelwise collaborative representation (SPCR) of a query pixel, i.e., using all pixels in its superpixel to represent the pixel, is obtained by solving a generalized Sylvester equation to extract commonality and maintain individuality of the pixel to some extent. Finally, a global projection matrix to a low-dimensional space is calculated by reducing the discrepancy between SPCRs and the original spectral features, and reducing the differences between pixels from one superpixel and increasing the differences between pixels from different superpixels simultaneously. Superior classification performances on several HSI datasets demonstrate the effectiveness of the proposed SPCRGE.

A parametric approach of partial eigenstructure assignment for high-order linear systems via proportional plus derivative state feedback

<abstract><p>In this paper, a partial eigenstructure assignment problem for the high-order linear time-invariant (LTI) systems via proportional plus derivative (PD) state feedback is considered. By partitioning the open-loop system into two parts (the altered part and the unchanged part) and utilizing the solutions to the high-order generalized Sylvester equation (HGSE), complete parametric expressions of the feedback gain matrices of the closed-loop system are established. Meanwhile, a group of arbitrary parameters representing the degrees of freedom of the proposed method is provided and optimized to satisfy the stability of the system and robustness criteria. Finally, a numerical example and a three-axis dynamic flight motion simulator system example with the simulation results are offered to illustrate the effectiveness and superiority of the proposed method.</p></abstract>

Topology identification of heterogeneous networks: Identifiability and reconstruction

This paper addresses the problem of identifying the graph structure of a dynamical network using measured input/output data. This problem is known as topology identification and has received considerable attention in recent literature. Most existing literature focuses on topology identification for networks with node dynamics modeled by single integrators or single-input single-output (SISO) systems. The goal of the current paper is to identify the topology of a more general class of heterogeneous networks, in which the dynamics of the nodes are modeled by general (possibly distinct) linear systems. Our two main contributions are the following. First, we establish conditions for topological identifiability, i.e., conditions under which the network topology can be uniquely reconstructed from measured data. We also specialize our results to homogeneous networks of SISO systems and we will see that such networks have quite particular identifiability properties. Secondly, we develop a topology identification method that reconstructs the network topology from input/output data. The solution of a generalized Sylvester equation will play an important role in our identification scheme.

Parametric control to a type of descriptor quasi-linear high-order systems via output feedback

This study presents a parametric approach to design output feedback for descriptor quasi-linear high-order systems. Utilizing the solution to the high-order generalized Sylvester equations (HGSEs), the more unified parametric form of output feedback is developed. With the proposed approach, the closed-loop is transformed into a linear constant system with an expected eigenstructure. Meanwhile, the regularity of the closed-loop system is maintained easily and effectively. Finally, a synchronization problem of Genesio-Tesi and Coullet systems is proposed to illustrate the effectiveness of the parametric approach.

Parametric control to permanent magnet synchronous motor via proportional plus integral feedback

This study investigates a parametric approach to design the proportional-integral (PI) controller for a permanent magnet synchronous motor (PMSM). Based on the solutions of the generalized Sylvester equation, the generally parametric expressions of PI controller and right eigenvector matrix are obtained. By using the parametric approach, the closed-loop system is converted into a linear time-invariant system with an expected eigenstructure. Further, a numerical example is put forward to illustrate the effectiveness and feasibility of the proposed parametric approach.

Parametric control to quasi-linear systems based on dynamic compensator and multi-objective optimization

This paper considers a parametric approach for quasi-linear systems by using dynamic compensator and multi-objective optimization. Based on the solutions of generalized Sylvester equations, we establish the more general parametric forms of dynamic compensator and the left and right closed-loop eigenvector matrices, and give two groups of arbitrary parameters. By using the parametric approach, the closed-loop system is converted into a linear constant one with a desired eigenstructure. Meanwhile, it also proposes a novel method to realize multi-objective design and optimization. Multiple performance objectives, containing overall eigenvalue sensitivity, $H_2$ norm, $H_\infty$ norm and low compensation gain, are formulated by arbitrary parameters, then robustness and low compensation gain criteria are expressed by a comprehensive objective function which contains each performance index weighted. By utilizing degrees of freedom (DOFs) in arbitrary parameters, we can optimize the comprehensive objective function such that an optimized dynamic compensator is found to satisfy the robustness and low compensation gain criteria. Finally, an example of attitude control of combined spacecrafts is presented which proves the effectiveness and feasibility of the parametric approach.

A parametric method to design dynamic compensator for high-order quasi-linear systems

This research proposes a parametric method to design dynamic compensator for high-order quasi-linear systems. According to the solutions of high-order generalized Sylvester equations, the completely parametric expressions of both the left and right eigenvector matrices and dynamic compensator are established. Based on the proposed method, we transform the closed-loop system into a linear time-invariant system with the desired eigenstructure. Further, the synchronization problem on Genesio–Tesi system and Coullet system is put forward to prove that the parametric method is effective and feasible.