functionisDetectableSISO

Information

This function checks whether a SISO state space system is detectable or not.

A system is detectable for the continuous-time case if all of the unobservable eigenvalues have negative real part or for the discrete-time case if all of the unobservable eigenvalues are in the complex unit circle respectively. Hence, a oberservable system is always detectable of course.

As observability is a dual concept of controllability, the concept of detectability is dual to stabilizability, that is, a system is detectable if the pair (A', C') is stabilizable. Therefore, the same algorithm to check stabilizability are applied to the dual pair (A', C') of the system:

To check stabilizability (see Modelica_LinearSystems2.StateSpace.Analysis.isStabilizable) , the system is transformed to upper controller Hessenberg form

| *

*

⋯

⋯

* |

|

*

|

| *

*

⋯

⋯

* |

|

0

|

Q*A*Q ' = H =

| 0

*

⋯

⋯

* |,

Q*b = q =

|

⋮

|,

c*Q = ( *, ⋯, * )

| ⋮

⋱

⋱

⋱

⋮ |

|

⋮

|

| 0

⋯

0

*

* |

|

0

|

The system can be partitioned to

H=[H11,H12; H21, H22], q=[q1;0],

where the pair (H11, q1) contains the controllable part of the system, that is, rank(H) = rank(H11). For stabilizability the H22 has to be stable.

Inputs

Outputs