Second-order Sylvester Matrix Equations Research Articles

Lanczos version of BCR algorithm for solving the generalised second‐order Sylvester matrix equation

It is well known that the biconjugate residual (BCR) algorithm and its variants are powerful procedures to find the solution of large sparse non-symmetric systems equation A x = b . In this study, the authors develop the Lanczos version of BCR algorithm for computing the solution pair ( V , W ) of the generalised second-order Sylvester matrix equation E V F + G V H + B V C = D W E + M , which includes the second-order Sylvester, Lyapunov and Stein matrix equations as special cases. The convergence results show that the algorithm with any initial matrices converges to the solutions within a finite number of iterations in the absence of round-off errors. Finally, two numerical examples are provided to support the theoretical findings and to testify the effectiveness and usefulness of the algorithm.

Hinf Tracking for a Class of Second-Order Dynamical Systems

The problem of tracking for second-order dynamical systems subject to disturbance input is considered by using a complete parametric design approach. The goal of this problem is to design a controller such that the output of the controlled system robustly asymptotically tracks the output of the reference model, and the transfer function from the exogenous disturbance to the tracking error meets a prescribed norm upper bound constraint. Based on the complete parametric solution to a class of generalized second-order Sylvester matrix equations, complete parameterizations for all the controller gain matrices are established in terms of two set of freedom parameters. Also, based on these parametric gain matrices, the prescribed norm upper bound constraint is transformed into an equivalent constraint condition which restricts the choice of the freedom parameters. An example is utilized to show the effect of the proposed approach.

Efficient iterative method for solving the second-order Sylvester matrix equation EVF2−AVF−CV=BW

The second-order Sylvester matrix equation EVF2−AVF−CV=BW (including the generalised Sylvester matrix equation, normal Sylvester matrix equation and Lyapunov matrix equation as special cases) over unknown matrix pair [V, W], has wide applications in many fields. In the present study, the authors propose an iterative method to solve the second-order Sylvester matrix equation. The proposed iterative method does not depend on the Jordan form of the matrix F. By this iterative method, the solvability of the matrix equation can be determined automatically over unknown matrix pair [V, W]≠0. When the matrix equation is solvable, its solution pair can be obtained within finite iterative steps, and its least Frobenius norm solution pair can be obtained by choosing suitable initial matrix pair. Furthermore, its optimal approximation solution pair to a given matrix pair can be derived by finding the least norm solution pair of a new matrix equation. A numerical example is given to show the efficiency of the proposed method.

Parameterisation of PID eigenstructure assignment in second-order linear systems

This paper considers the design of Eigen Structure Assignment (ESA) in second-order linear systems via Proportional-Integral-Derivative (PID) feedback. Based on two groups of parametric solutions to a type of so-called Second-order Sylvester Matrix Equations (SSMEs), two complete parametric methods for the proposed ESA problem are presented. Both methods give simple complete parametric expressions for the PID feedback gain matrices and the closed-loop eigenvector matrices. In the case of prescribed closed-loop eigenvalues the first method depends on a series of singular values decompositions and thus is numerically simple and reliable. In the case of undetermined closed-loop eigenvalues the second method utilises the right factorisation and allows the closed-loop eigenvalues to be set undetermined and sought via certain procedures in actual control system designs. The two methods offer all the design degrees of freedom, which can be utilised to satisfy certain performance specifications in real applications. An illustrative example shows the effect of the proposed approaches.

On the Parametric Solution to the Second-Order Sylvester Matrix EquationEVF2−AVF−CV=BW

This paper considers the solution to a class of the second-order Sylvester matrix equationEVF2−AVF−CV=BW. Under the controllability of the matrix triple(E,A,B), a complete, general, and explicit parametric solution to the second-order Sylvester matrix equation, with the matrixFin a diagonal form, is proposed. The results provide great convenience to the analysis of the solution to the second-order Sylvester matrix equation, and can perform important functions in many analysis and design problems in control systems theory. As a demonstration, an illustrative example is given to show the effectiveness of the proposed solution.

Solution to the second-order Sylvester matrix equation MVF/sup 2/+DVF+KV=BW

This note provides a complete general parametric solution (V,W) to the generalized second-order Sylvester matrix equation MVF/sup 2/+DVF+KV=BW, with F being an arbitrary square matrix. The primary feature of this solution is that the matrix F does not need to be in any canonical form, or may be even unknown a priori. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems involving second-order dynamical systems.

Unified parametric approaches for high-order integral observer design for matrix second-order linear systems

A type of high-order integral observers for matrix second-order linear systems is proposed on the basis of generalized eigenstructure assignment via unified parametric approaches. Through establishing two general parameteric solutions to this type of generalized matrix second-order Sylvester matrix equations, two unified complete parametric methods for the proposed observer design problem are presented. Both methods give simple complete parametric expressions for the observer gain matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the right factorization of the system, and allows eigenvalues of the error system to be set undetermined and sought via certain optimization procedures. A spring-mass-dashpot system is utilized to illustrate the design procedure and show the effect of the proposed approach.

Parametric Eigenstructure Assignment in Second-Order Descriptor Linear Systems

This note considers eigenstructure assignment in second-order descriptor linear systems via proportional plus derivative feedback. It is shown that the problem is closely related with a type of so-called second-order Sylvester matrix equations. Through establishing two general parametric solutions to this type of matrix equations, two complete parametric methods for the proposed eigenstructure assignment problem are presented. Both methods give simple complete parametric expressions for the feedback gains and the closed-loop eigenvector matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable. The second one utilizes the right factorization of the system, and allows the closed-loop eigenvalues to be set undetermined and sought via certain optimization procedures. An example shows the effect of the proposed approaches.

Two parametric approaches for eigenstructure assignment in second-order linear systems

This paper considers eigenstructure assignment in second-order linear systems via proportional plus derivative feedback. It is shown that the problem is closely related to a type of so-called second-order Sylvester matrix equations. Through establishing two general parametric solutions to this type of matrix equations, two complete parametric methods for the proposed eigenstructure assignment problem are presented. Both methods give simple complete parametric expressions for the feedback gains and the closed-loop eigenvector matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the right factorization of the system, and allows the closed-loop eigenvalues to be set undetermined and sought via certain optimization procedures. An example shows the effectiveness of the proposed approaches.