Improved eigenvalue sensitivity for finite-precision digital controller realisations via orthogonal Hermitian transform

An improved eigenvalue sensitivity is presented for digital controller realisations via the orthogonal Hermitian transform, subject to finite word length (FWL) effects. This approach can preserve the stability of the closed-loop system when the designed stabilising digital controllers are actually implemented with FWL by using the mode of fixed-point arithmetic. A performance index defined by eigenvalue sensitivity of the closed-loop system is evaluated by the mixed matrix-2/ Frobenius norms, so that the eigenvalues of the closed-loop system, from infinite precision after using an FWL implemented digital controller, become limited precision, the influence suffered is minimal. Then, the optimal similarity transformation for the controller is obtained via the orthogonal Hermitian transform. Thus, a minimum bit number used for implementing the stabilising digital controllers can be obtained from the performance index and the optimal similarity transformation under certain stability criteria. The main contributions are that this approach provides an analytical closed-form solution for the optimal transformation and leads to the implementation of the stabilising controllers with a lower bit number when using this optimal one. Finally, a numerical example is used to illustrate the effectiveness of the proposed scheme.