In control theory, a Kalman decomposition provides a mathematical means to convert a representation of any linear time-invariant (LTI) control system to a form in which the system can be decomposed into a standard form which makes clear the observable and controllable components of the system. This decomposition results in the system being presented with a more illuminating structure, making it easier to draw conclusions on the system's reachable and observable subspaces.
Definition Consider the continuous-time LTI control system
, ,
or the discrete-time LTI control system , .
The Kalman decomposition is defined as the realization of this system obtained by transforming the original matrices as follows:
, , , ,
where is the coordinate transformation matrix defined as
,
and whose submatrices are : a matrix whose columns span the subspace of states which are both reachable and unobservable. : chosen so that the columns of are a basis for the reachable subspace. : chosen so that the columns of are a basis for the unobservable subspace. : chosen so that is invertible. It can be observed that some of these matrices may have dimension zero. For example, if the system is both observable and controllable, then , making the other matrices zero dimension.
Consequences By using results from controllability and observability, it can be shown that the transformed system has matrices in the following form:
This leads to the conclusion that The subsystem is both reachable and observable. The subsystem is reachable. The subsystem is observable.
Variants A Kalman decomposition also exists for linear dynamical quantum systems. Unlike classical dynamical systems, the coordinate transformation used in this variant requires to be in a specific class of transformations due to the physical laws of quantum mechanics.
See also Realization (systems) Observability Controllability
References
External links Lectures on Dynamic Systems and Control, Lecture 25 - , Munther Dahleh, George Verghese — MIT OpenCourseWare