Standard control theory assumes causality: the current state determines future behavior. You observe the state, compute a control input, apply it, and observe the result. The state at time t depends only on past states and inputs. This assumption is so fundamental it rarely gets stated.
Descriptor systems break it. In descriptor systems, the relationship between states involves implicit algebraic constraints — the system's behavior at time t can depend on future inputs, not just past ones. The differential-algebraic structure means the system isn't just evolving forward; it's also satisfying constraints that look backward. These arise naturally in circuit models, chemical processes, and constrained mechanical systems where physical laws impose relationships that don't respect the arrow of time.
The standard data-driven control toolkit — behavioral theory, Willems' fundamental lemma — assumes causality. When the system is non-causal, the collected input-output trajectories don't contain enough information to predict behavior, because the data was generated under causal operation but the system requires non-causal reasoning.
A new innovation-based approach handles this by decomposing the descriptor system into its causal and non-causal parts, applying data-driven predictive control to each with appropriate handling. The non-causal part is resolved through the algebraic constraints, while the causal part uses standard trajectory-based prediction. The result: data-driven control that works even when the system's mathematics points both forward and backward in time.