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.
The derivation is identical for both discrete-time as well as continuous time LTI systems. The description of a continuous time linear system is
x ˙ ( t ) = A x ( t ) + B u ( t ) y ( t ) = C x ( t ) + D u ( t ) where
x is the "state vector",
y is the "output vector",
u is the "input (or control) vector",
A is the "state matrix",
B is the "input matrix",
C is the "output matrix",
D is the "feedthrough (or feedforward) matrix".
Similarly, a discrete-time linear control system can be described as
x ( k + 1 ) = A x ( k ) + B u ( k ) y ( k ) = C x ( k ) + D u ( k ) with similar meanings for the variables. Thus, the system can be described using the tuple consisting of four matrices ( A , B , C , D ) . Let the order of the system be n .
Then, the Kalman decomposition is defined as a transformation of the tuple ( A , B , C , D ) to ( A ^ , B ^ , C ^ , D ^ ) as follows:
A ^ = T − 1 A T B ^ = T − 1 B C ^ = C T D ^ = D T is an n × n invertible matrix defined as
T = [ T r o ¯ T r o T r o ¯ T r ¯ o ] where
T r o ¯ is a matrix whose columns span the subspace of states which are both reachable and unobservable. T r o is chosen so that the columns of [ T r o ¯ T r o ] are a basis for the reachable subspace. T r o ¯ is chosen so that the columns of [ T r o ¯ T r o ¯ ] are a basis for the unobservable subspace. T r ¯ o is chosen so that [ T r o ¯ T r o T r o ¯ T r ¯ o ] is invertible.By construction, the matrix T 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 T = T r o , making the other matrices zero dimension.
By using results from controllability and observability, it can be shown that the transformed system ( A ^ , B ^ , C ^ , D ^ ) has matrices in the following form:
A ^ = [ A r o ¯ A 12 A 13 A 14 0 A r o 0 A 24 0 0 A r o ¯ A 34 0 0 0 A r ¯ o ] B ^ = [ B r o ¯ B r o 0 0 ] C ^ = [ 0 C r o 0 C r ¯ o ] D ^ = D This leads to the conclusion that
The subsystem ( A r o , B r o , C r o , D ) is both reachable and observable.The subsystem ( [ A r o ¯ A 12 0 A r o ] , [ B r o ¯ B r o ] , [ 0 C r o ] , D ) is reachable.The subsystem ( [ A r o A 24 0 A r ¯ o ] , [ B r o 0 ] , [ C r o C r ¯ o ] , D ) is observable.