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.