## Modelica_LinearSystems2.Utilities.Import

Functions to import data in a Modelica environment

### Information

Extends from Modelica.Icons.Package (Icon for standard packages).

### Package Content

Name | Description
--- | ---
Examples | Demonstrate the usage of the Import functions
linearize | Linearize a model after simulation up to a given time

## Modelica_LinearSystems2.Utilities.Import.linearize

Linearize a model after simulation up to a given time

### Information

This function initializes a Modelica model and then simulates the model with its default experiment options until time instant "t_linearize". If t_linearize=0, no simulation takes place (only initialization). At the simulation stop time, the model is linearized in such a form that

- all top-level signals with prefix "input" are treated as inputs u(t) of the model ,
- all top-level signals with prefix "output" are treated as outputs y(t) of the model,
- all variables that appear differentiated and that are selected as states at this time instant are treated as states x of the model.

Formally, the non-linear hybrid differential-algebraic equation system is therefore treated as the following ordinary equation system at time instant t_linearize:

        der(x) = f(x,u)

             y = g(x,u)

Taylor series expansion (linearization) of this model around the simulation stop time t_linearize:

       u0 = u(t_linearize)

       y0 = y(t_linearize)

       x0 = x(t_linearize)

and neglecting higher order terms results in the following system:

       der(x0+dx) = f(x0,u0) + der(f,x)*dx + der(f,u)*du

          y0 + dy = g(x0,u0) + der(g,x)*dx + der(g,u)*du

where der(f,x) is the partial derivative of f with respect to x, and the partial derivatives are computed at the linearization point t_linearize. Re-ordering of terms gives (note der(x0) = 0):

       der(dx) = der(f,x)*dx + der(f,u)*du + f(x0,u0)

            dy = der(g,x)*dx + der(g,u)*du + (g(x0,u0) - y0)

or

       der(dx) = A*dx + B*du + f0

            dy = C*dx + D*du

This function returns the matrices A, B, C, D and assumes that the linearization point is a steady-state point of the simulation (i.e., f(x0,u0) = 0). Additionally, the full Modelica names of all inputs, outputs and states shall be returned if possible (default is to return empty name strings).

### Inputs

Name | Description
--- | ---
modelName | Name of the Modelica model
t_linearize | Simulate until T_linearize and then linearize [s]

### Outputs

Name | Description
--- | ---
A[nx, nx] | A-matrix
B[nx, nu] | B-matrix
C[ny, nx] | C-matrix
D[ny, nu] | D-matrix
inputNames[nu] | Modelica names of inputs
outputNames[ny] | Modelica names of outputs
stateNames[nx] | Modelica names of states