Descriptor system techniques and software tools

The role of the descriptor system representation as basis for reliable numerical computations for system analysis and synthesis, and in particular, for the manipulation of rational matrices, is discussed and available robust numerical software tools are described.

💡 Research Summary

The paper presents a comprehensive treatment of descriptor‑system techniques as a unifying framework for the reliable numerical manipulation of rational matrices, together with a survey of the robust software tools that implement these techniques.

A descriptor system generalizes the standard state‑space model by allowing a possibly singular matrix E in the equations E ẋ = A x + B u, y = C x + D u (continuous time) or E x(k+1) = A x(k) + B u(k) (discrete time). The key requirement is that the matrix pencil A − λE is regular (determinant not identically zero). This representation naturally captures interconnected differential‑algebraic models, constrained mechanical systems, and certain economic processes, and it subsumes the ordinary state‑space case (E = I).

The central object of interest is the transfer‑function matrix (TFM) G(λ) = C(A − λE)⁻¹B + D, which is a matrix of rational functions. Direct manipulation of G(λ) in polynomial form is numerically hazardous because of coefficient scaling and cancellation. Instead, the paper advocates the realization problem: for any given rational matrix G(λ) there exists a quadruple (A − λE, B, C, D) such that (4) holds. Realizations are not unique; any invertible similarity transformation (U, V) yields an equivalent realization. This freedom is exploited to obtain minimal realizations, i.e., those with the smallest possible state dimension n.

Theorem 1 (Varga et al., 1981) characterizes minimality through five conditions: (i) finite controllability (rank