Solution Bounds of Sylvester Equations with applications to Problems of Observer Design

Control systems theory is a largely important and interesting area in the world of technology and engineering, and is also an important area of research, in which the research aims to design and analyze control systems, with particular reference to the stability and optimal performance of the system. In linear control theory, there are two equations which are widely used to determine these two factors mentioned above, namely the Riccati equation which determines how a system should be designed to achieve its optimal performance, and the Lyapunov equation which determines the stability of the system, both equations of which are matrix equations. Although the solutions of these equations are important to the design and analysis of control systems, it is not always necessary to obtain the exact solutions to these equations, but instead bounds on their solutions. Because of this, much effort has been made by researchers in deriving various types of bounds for these equations for both continuous-time and discrete-time versions, including bounds when the equations are subject to perturbations in the coefficients. These bounds include matrix bounds, eigenvalue bounds, norm bounds, trace bounds and determinant bounds [5]. In particular, Shi, P and C. E. de Souza derived some bounds on symmetric solutions of the continuous algebraic Riccati equation under perturbations on its coefficients.

There is another matrix equation that appears in the analysis and design of linear control systems, namely the Sylvester equation [3]. This equation is very similar to the Lyapunov equation; in fact, the Lyapunov is merely a special case of the Sylvester equation. It has applications in observer design in which it is used to solve pole placement problems, with an extension to reduced order observers. Although many numerical algorithms have been developed for the solution of this equation, these algorithms only give us approximate solutions, and not bounds on the actual solution. It would appear from journals, lists of references and website findings, that little of the work on solution bounds for the Riccati and Lyapunov equations has been applied to the Sylvester equation. Furthermore, there would appear to be no (or very little) analysis of this equation when its coefficients are subject to perturbations.

It follows from what has been done for the Lyapunov and Riccati equations to employ the methods used for these equations to the continuous-time and discrete-time Sylvester equations. Also, it would be possible to employ the perturbation analyses used for these equations to the Sylvester equation. The main aims of the research are as follows:


  1. To derive various types of solution bounds of the Sylvester equations which arise in the solution of pole placement problems in observer design with extensions to reduced order observer problems. The methods employed will be parallel to those used for the Lyapunov and Riccati equations.
  2. To undertake a perturbation analysis of the Sylvester equations similar to the perturbation analyses of the Lyapunov and Riccati equations.

For more information please contact Peng Shi.