.Modelica_LinearSystems2.StateSpace.Internal.invariantZerosWithRealMatrix

Information

Syntax

InvariantZeros = StateSpace.Internal.invariantZerosWithRealMatrix(A,B,C,D)

Description

Computes the invariant zeros of a system in state space form:

der(x) = A*x + B*u
     y = C*x + D*u

The invariant zeros of this system are defined as the variables s that make the Rosenbrock matrix of the system

|	sI−A		−B	|
|				|
|	C		D	|

singular, whereby I is the identity matrix of the same size as A

This function applies the algorithm described in [1] where the system (A, B, C, D) is reduced to a new system (Ar, Br Cr, Dr) with the same zeros and with Dr of full rank.

The zeros are returned as a matrix InvariantZeros[:,2] where InvariantZeros[i,1] is the real and InvariantZeros[i,2] is the imaginary part of the complex zero i.

Example

  Modelica_LinearSystems2.StateSpace ss=Modelica_LinearSystems2.StateSpace(
    A=[1, 1, 1;0, 1, 1;0,0,1],
    B=[1;0;1],
    C=[0,1,1],
    D=[0]);

  Complex zeros[:];

algorithm
  zeros := Modelica_LinearSystems2.StateSpace.Analysis.invariantZeros(ss);
// zeros = {1, 0}

References

 [1] Emami-Naeini, A. and Van Dooren, P. (1982):

Computation of Zeros of Linear Multivariable Systems. Automatica, 18, pp. 415-430.
 

Interface

encapsulated function invariantZerosWithRealMatrix
  import Modelica;
  import MatricesMSL = Modelica.Math.Matrices;
  import Modelica_LinearSystems2.StateSpace;
  import Modelica_LinearSystems2;
  import Modelica_LinearSystems2.Math.Matrices;
  import Complex;
  input Real A[:, size(A, 1)] "State matrix of linear state space system";
  input Real B[size(A, 1), :] "Input matrix of linear state space system";
  input Real C[:, size(A, 1)] "Output matrix of linear state space system";
  input Real D[size(C, 1), size(B, 2)] "Feedforward matrix of linear state space system";
  output Real InvariantZeros[:, 2] "Finite, invariant zeros of linear state space system; size(Zeros,1) <= size(A,1); Zeros[:,1]: Real part, Zeros[:,2]: Imaginary part";
end invariantZerosWithRealMatrix;

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