PopovBelevitchHautus

PopovBelevitchHautus, commonly referred to as the Popov–Belevitch–Hautus (PBH) criterion, is a set of rank tests used to assess controllability and observability of linear time-invariant systems. Named after Viktor Popov, J. Belevitch, and H. Hautus, it provides necessary and sufficient conditions that are often easier to verify than the traditional controllability and observability matrices.

For a continuous-time system x' = Ax + Bu, y = Cx + Du, with state x in R^n, input u in R^m, and output y in R^p, and for λ ranging over the complex numbers with det(λI − A) = 0 (i.e., the eigenvalues of A), the PBH tests are:

- Controllability: (A,B) is controllable if and only if rank [λI − A, B] = n for every eigenvalue λ of A.

- Observability: (A,C) is observable if and only if rank [λI − A; C] = n for every eigenvalue λ of A.

Analogous statements hold for discrete-time systems with x_{k+1} = Ax_k + Bu_k, y_k = Cx_k + Du_k, using the same eigenvalue-based conditions.

The PBH criterion is valued for directly linking eigenstructure to controllability and observability, avoiding the explicit construction of controllability and observability matrices. It is applicable to both continuous- and discrete-time models and is widely used in analysis, model reduction, and controller or observer design. In some literature, it is also referred to as the Hautus test.