Closed-Form Expressions of Coprime Factors for Second-Order Systems

Coprime factors express control systems in a fractional representation. Their computation requires solutions to continuous-time algebraic Riccati equations. We provide a closed-form solution to the generalized filtering algebraic Riccati equation that leads to closed-form expressions for the coprime factors for second-order systems. Furthermore, for second-order linear time-invariant systems, these expressions can be written directly in terms of the second-order system matrices, which have structural properties. Their utility, in terms of computation time, is illustrated by designing a coprime factor-based Youla–Kucera controller for large-size systems with changing parameters. Specifically, we showed, through the examples of reconfigurable satellite networks and spring–mass–damper systems, that the closed-form expressions for coprime factors can be computed much faster than the standard way, which results in faster implementation of the Youla–Kucera controller. Hence, they are very useful for computing coprime factors for large-size systems, especially for real-time computation with limited resources.