National Instruments 370753C-01 stair( )

Chapter 6 State-Space Design

ans (a scalar) = 1

zeros(SysM)

ans is null

stair( )

[SysT,T,nc] = stair(Sys,tol)

The stair( ) converts a dynamic system to staircase form. In the

staircase form, the A and B system matrices are linearly transformed so that

they are partitioned into controllable and uncontrollable parts. By duality,

converting the transpose of a system into staircase form results in its being

separated into observable and unobservable parts. The matrices are

partitioned as shown in the following example:

where Ac is controllable, Auc is uncontrollable, and Acuc represents the

coupling from the uncontrollable states to the controllable part of the

representation. There is no coupling from the controllable states to the

uncontrollable ones.

T is a transformation matrix between the original system, Sys, and the

system in staircase form, SysT.

The transformations are as follows:

The optional tolerance, tol, indicates the threshold value beneath which

numbers in the transformed matrices should be rounded to zero.

The staircase algorithm used to partition the system derives from the

VanDooren algorithms. For further details, refer to [Van79], [Van81],

and [BeV88].

Astair

Auc 0

Acuc Ac

Bstair

Astair T1–AT=

Bstair T1–B=

Cstair CT=