.Modelica_LinearSystems2.StateSpace.Internal.trianUpperHess

Information

This function computes an triangular matrix from an upper Hessenberg matrix by stepwise annihilation of the subdiagonal elements.

A -> QA = T

It is assumed that the original matrix has upper hessenberg form. Additionally the vector b is transformed in the same way

b -> Qb = q

The function is primarily used to calculate the transfer function gain from a SISO state space system in observer Hessenberg form

    ( *   *   ...   ...    * )          ( * )
    ( *   *   ...   ...    * )          ( . )
A = ( 0   *   ...   ...    * ),    b =  ( . ),   c = ( 0, ..., 0, * )
    ( .   .    .     .     . )          ( * )
    ( 0  ...   0     *     * )          ( * )

If A is upper Hessenberg and T = Q*A is triangular then obviously H(s) = Q*(s*I -A) = s*I - T.

Further on, if T is triangular then also H = sI - T is and the element l_nn of L = inv(H) is given by 1/h_nn. The frequency response G(s0)for a given s0 that is neither zero nor pole of the system can be calculated by

G(s0)  = c*(s0*I -A)-1*b = c*(s0*I -A)-1 *Q*Q-1*b = c*(Q-1*(s0*I -A))-1*Q-1*b = c*H-1(s0)*q

and because only the n'th element of c is different to zero the gain k is given by

    q_nn*c_nn     product(s0 - poles_i)
k = ---------- * ----------------------
       h_nn       product(s0 - zeros_i)

Interface

encapsulated function trianUpperHess
  import Modelica;
  import Modelica_LinearSystems2.Math.Matrices;
  import Modelica_LinearSystems2.Math.Vectors;
  input Real H[:, :] "Upper Hessenberg matrix";
  input Real b[size(H, 1)];
  output Real Ht[size(H, 1), size(H, 2)];
  output Real bt[size(b, 1)];
end trianUpperHess;