Internal stability of interconnected systems

In this paper, internal stability of interconnected systems is considered. It is shown that a system consisting only of single-input/single-output (SISO) plants is internally stable if and only if ΔΠ_ip_i(s) has all its roots in the open left half of the complex plane, where p_i(s) are the denominators of the plant transfer functions and Δ is the system determinant same as in the Mason's formula. This theorem is also extended to the case where the system may have multi-input and/or multi-output plants. Several typical control schemes are employed as illustrative examples to demonstrate the simplicity and usefulness of these results in internal stability analysis and stabilization synthesis.