Kalman decomposition

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.[1]

See also

References

  1. Zhang, Guofeng; Grivopoulos, Symeon; Petersen, Ian R.; Gough, John E. (February 2018). "The Kalman Decomposition for Linear Quantum Systems". IEEE Transactions on Automatic Control. 63 (2): 331–346. doi:10.1109/TAC.2017.2713343. hdl:10397/77565. ISSN 1558-2523. S2CID 10544143.
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