Matrix full rank, Controllability and Observability: Revisit Basic Concepts

In control theory using the state-space method, it is often assumed that a matrix is of full rank to further mathematical development. Few people stop to ask two questions: (1) How far is the ‘distance’ between the given full-rank matrix and a non-full-rank matrix?, and (2) How does this ‘distance’ affect control? In robust system analysis and design, in addition to verifying that a system is stable, one also measures how far it is from an unstable system. The simplest stability measurement for a single-input-single-output (SISO) linear system is the gain margin and phase margin. Similarly, beyond a ‘Yes’ or ‘No’ answer, for a full-rank matrix in control, we should ask the above two questions. Pursuing such questions leads to a lot of interesting and useful results. Some of them are well-documented in the literature; some are yet to be fully explored. This paper illustrates some existing research and proposes some future studies. In general, we can review all control lemmas and theorems based on matrix full-rank conditions to explore further studies. The study can also extend to non-linear systems. The basic nonlinear system controllability (observability) tests are based on if the relevant matrices, constructed using Lie Bracket (Lie Derivative), are of full rank. Furthermore, similar studies can be on another important matrix property: positiveness, and on more challenging research into the degrees of controllability and observability for networked control systems.