Frequency Synthesis of Interconnected Homogeneous LTI Systems

The synthesis problem of the interconnection of homogeneous linear time-invariant systems such that the frequency response satisfies magnitude constraints is investigated. To this end, the usual synthesis approach of traditional filters, viewed as the interconnection of integrators $\frac{1}{s}$, is revisited based on the linear fractional transformation representation and the $\lbrace x,y,z\rbrace$-dissipative characterization. This approach, based on convex optimization, consists of two steps: the magnitude synthesis and the spectral factorization steps. When the systems are modeled by a lossless dissipative transfer function $T(s)$, it is demonstrated that each step can be extended. However, a factorization error appears when considering a general dissipative $T(s)$, preventing the direct extension of the two-step approach. It is then revealed how to overcome this issue by coupling both steps, generalizing thereby the usual synthesis approach. Finally, the interest of this work is illustrated through two applications: the design of $LC$-bandpass filters and the weighted $\mathcal {H}_{\infty }$-control of interconnected homogeneous systems.