Kalman decomposition
Encyclopedia
Kalman decomposition provides a mathematical means to convert a representation of any linear time-invariant
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.
where
Similarly, a discrete-time linear control system can be described as
with similar meanings for the variables. Thus, the system can be described using the tuple consisting of four matrices
.
Let the order of the system be
.
Then, the Kalman decomposition is defined as a transformation of the tuple
to 
as follows:
is an 
invertible matrix defined as
where
By construction, the matrix
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.
has matrices in the following form:
LTI system theory
Linear time-invariant system theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. It investigates the response of a linear and time-invariant...
control system to a form in which the system can be decomposed into a standard form which makes clear the observable
Observability
Observability, in control theory, is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. The observability and controllability of a system are mathematical duals. The concept of observability was introduced by American-Hungarian scientist Rudolf E...
and controllable
Controllability
Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control....
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.
Notation
The derivation is identical for both discrete-time as well as continuous time LTI systems. The description of a continuous time linear system is
where
-
is the "state vector", -
is the "output vector", -
is the "input (or control) vector", -
is the "state matrix", -
is the "input matrix", -
is the "output matrix", -
is the "feedthrough (or feedforward) matrix".
Similarly, a discrete-time linear control system can be described as
with similar meanings for the variables. Thus, the system can be described using the tuple consisting of four matrices
.Let the order of the system be
.Then, the Kalman decomposition is defined as a transformation of the tuple
to as follows:
is an invertible matrix defined as
where
-
is a matrix whose columns span the subspace of states which are both reachable and unobservable. -
is chosen so that the columns of 
are a basis for the reachable subspace. -
is chosen so that the columns of 
are a basis for the unobservable subspace. -
is chosen so that 
is invertible.
By construction, the matrix
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.Standard Form
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.
External links
- Lectures on Dynamic Systems and Control, Lecture 25 - Mohammed Dahleh, Munther Dahleh, George Verghese — MIT OpenCourseWare
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