Descriptor and Singular Systems
Descriptor systems mix differential and algebraic equations, capturing constraints and interconnections that ordinary state-space models cannot.
When E is singular
A descriptor system is written as E x-dot equals A x plus B u, where the matrix E may be singular. When E is invertible the system reduces to ordinary state space, but a singular E means some equations are algebraic constraints rather than dynamics. Such models arise naturally in constrained mechanics, electrical networks with algebraic Kirchhoff laws, and interconnected subsystems where conservation relations link variables instantaneously.
Regularity and index
A descriptor system is regular if the pencil sE minus A is not identically singular, which is required for solutions to exist and be unique. The nilpotency index of the pencil measures how many times inputs must be differentiated to solve for the algebraic variables; index one is benign, while higher indices signal impulsive modes that demand careful numerical treatment. The solution decomposes into a slow, dynamic part and a fast, algebraic-impulsive part.
Control considerations
Analysis and design extend state-space tools with new subtleties. Stability requires not only that finite eigenvalues lie in the left half plane but also that there are no impulsive modes, admissibility. Controllability splits into reachability of the dynamic modes and controllability at infinity of the impulsive modes. Many results are cast as generalized LMIs involving E.
- Mixes differential dynamics with algebraic constraints
- Requires a regular matrix pencil for well-posedness
- Index measures impulsive-mode severity
- Stability needs admissibility, not just left-half-plane poles
Descriptor form is the natural language for modeling by interconnection: assemble subsystems and their coupling constraints without eliminating variables, which is often impossible or destroys sparsity. Modern simulation and DAE solvers work directly in this form.
For a design-stage plant built from coupled physical subsystems with conservation constraints, a descriptor model preserves structure that manual elimination would obscure, aiding both simulation and controller synthesis. It generalizes ordinary state-space theory.