.Modelica_LinearSystems2.StateSpace.Internal.isDetectableSISO

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 neagtive 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) , ths system is transformed to 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.

Interface

encapsulated function isDetectableSISO
  import Modelica;
  import Modelica_LinearSystems2;
  import Modelica_LinearSystems2.StateSpace;
  import Modelica_LinearSystems2.Math.Complex;
  input StateSpace ss "State space system";
  output Boolean detectable;
end isDetectableSISO;