System Characteristic Polynomial Research Articles

The feedback interconnection of lumped linear time-invariant systems

This paper considers the feedback interconnection of two multi-input multi-output subsystems characterized by rational transfer functions Ĝ 1 and Ĝ functions are not assumed to be proper nor exponentially stableThe effect of output disturbances on stability is taken into account. Ten examples are given to show that instabilities may appear anywhere around the loop. Next, under a sequence of successively more restrictive assumptions, we prove four sets of necessary and sufficient conditions for the exponential stability of the system. Using coprime factorizations, we obtain four equivalent expressions for the system characteristic polynomial. Two stability tests are derived, the first one is based exclusively on transfer functions, the second is based on the characteristic polynomial. The paper ends by providing translation rules for reformulating all definitions and theorems for the discrete-time case (i.e. instead of Laplace transforms use Z-transforms, etc.).

Multirate sampled-data systems analysis via vector operators

The primary difficulties of both the time-domain switch decomposition method and the frequency-domain decomposition method are overcome by introducing certain matrix operators and performing spectral factorization of resulting matrices of polynomials in the z -transform variable. Topological operations of the switch-decomposition method are simplified. This new approach eliminates the need to solve a system of equations with rational polynomial coefficients such as arises in the frequency-decomposition method. The determination of a multirate sampled-data system's characteristic polynomial no longer requires the evaluation of a determinant of rational polynomial elements. New results on obtaining modified z -transforms from standard z -transforms at a faster rate and vice versa are presented.