Stability analysis of multiloop, multirate sampled systems.

Two analytical procedures are presented for determining the stability of multirate systems. These procedures are oriented toward digital computer implementation and offer substantial advantages over existing techniques from the computational standpoint. Frequency domain identities are developed which allow the expression of a system's sampled variables in terms of one basic sampling rate. The first method is restricted to systems whose sampling rates are such that the ratio of any two of them is an integer (or its reciprocal). A closed form of the system characteristic equation is obtained by applying the identities to expressions relating the various sampled quantities. Application of the second method requires only that the ratio of any two sampling rates be a rational number. The sampling identities are applied to the system closed-loop relation. Successive shifts in the Laplace variable are introduced to obtain a set of independent algebraic equations that are written in matrix form. The determinant of the matrix is the characteristic equation of the system. Nyquist stability techniques are employed to determine the stability of the system through the numerical evaluation of the characteristic determinant. Both procedures are illustrated by applying them to an example of a multiloop, multirate feedback system.