.Modelica_LinearSystems2.DiscreteStateSpace.'constructor'.fromStateSpace

Information

Syntax

dss = 'constructor'.fromStateSpace(ss, Ts)
dss = 'constructor'.fromStateSpace(ss, Ts, method)

Description

This function derives a linear time invariant difference equation system in state space form

x(Ts*(k+1))        = A * x(Ts*k) + B * u(Ts*k)
y(Ts*k)            = C * x(Ts*k) + D * u(Ts*k)
x_continuous(Ts*k) =     x(Ts*k) + B2 * u(Ts*k)

with

• Ts - the sample time,
• k - the index of the actual sample instance (k=0,1,2,3,...),
• t - the time,
• u(t) - the input vector,
• y(t) - the output vector,
• x(t) - the discrete state vector (x(t=Ts*0) is the initial state),
• x_continuous(t) - the state vector of the continuous system from which the discrete block has been derived (details see below),
• A, B, C, D, B2 - matrices of appropriate dimensions.

from continuous state space form

der(xc(t)) = ss.A * xc(t) + ss.B * us(t)
    yc(t)  = ss.C * xc(t) + ss.D * uc(t)

The applied discretization method is selected by the user from

• ExplicitEuler - Discretization with explicit Euler integration,
• ImplicitEuler - Discretization with implicit Euler integration,
• Trapezoidal - Discretization with trapezoidal integration (Tustins method, recommended),
• ImpulseExact - Exact discretization for impulse inputs,
• StepExact - Exact discretization for step inputs (zero-order hold equivalent),
• RampExact - Exact discretization for ramp inputs (first-order hold equivalent).

Example

  import dss=Modelica_LinearSystems2.DiscreteStateSpace;
  import Modelica_LinearSystems2.StateSpace;

  StateSpace ss=StateSpace(A = [1], B = [1], C = [1], D = [0]);
  Modelica.Units.SI.Time Ts=0.1;
  Modelica_LinearSystems2.Types.Method method=Modelica_LinearSystems2.Types.Method.Trapezoidal;

public
  DiscreteStateSpace dss;

algorithm
  dss := dss.'constructor'.fromStateSpace(ss, Ts);

  //or just:
  //dss := dss(ss=ss, Ts=Ts, method=method);

  //  dss.A = [1.1053],
  //  dss.B = [0.11080],
  //  dss.C = [1],
  //  dss.D = [0.0526],
  //  dss.Ts = 0.1,
  //  dss.B2 = [0.0526],
  //  dss.method = Modelica_LinearSystems2.Types.Method.Trapezoidal

See also

fromMatrices2

Interface

function fromStateSpace
  import Modelica;
  import MatricesMSL = Modelica.Math.Matrices;
  import Modelica_LinearSystems2;
  import Modelica_LinearSystems2.Utilities.Types.Method;
  input Modelica_LinearSystems2.StateSpace ss "Continuous linear state space system";
  input Modelica.Units.SI.Time Ts "Sample time";
  input Modelica_LinearSystems2.Utilities.Types.Method method = Modelica_LinearSystems2.Utilities.Types.Method.Trapezoidal "Discretization method";
  output Modelica_LinearSystems2.DiscreteStateSpace dss(redeclare Real A[size(ss.A, 1), size(ss.A, 2)], redeclare Real B[size(ss.B, 1), size(ss.B, 2)], redeclare Real C[size(ss.C, 1), size(ss.C, 2)], redeclare Real D[size(ss.D, 1), size(ss.D, 2)], redeclare Real B2[size(ss.B, 1), size(ss.B, 2)]) "Discrete state space system";
end fromStateSpace;