WOLFRAM SYSTEM MODELER

# integralExpT

Return the exponential, the integral of the exponential, and time-weighted integral of the exponential of a matrix # Wolfram Language

In:= `SystemModel["Modelica.Math.Matrices.integralExpT"]`
Out:= # Information

This information is part of the Modelica Standard Library maintained by the Modelica Association.

```(phi,gamma,gamma1) = Matrices.integralExp(A,B);
(phi,gamma,gamma1) = Matrices.integralExp(A,B,T=1);
```

#### Description

This function computes the exponential phi = e^(AT) of matrix A and the integral gamma = integral(phi*dt)*B and the integral integral((T-t)*exp(A*t)*dt)*B, where A is a square (n,n) matrix and B, gamma, and gamma1 are (n,m) matrices.

The function calculates the matrices phi,gamma,gamma1 through the equation:

```                                 [ A B 0 ]
[phi gamma gamma1] = [I 0 0]*exp([ 0 0 I ]*T)
[ 0 0 0 ]
```

The matrices define the discretized first-order-hold equivalent of a state-space system:

```      x(k+1) = phi*x(k) + gamma*u(k) + gamma1/T*(u(k+1) - u(k))
```

The first-order-hold sampling, also known as ramp-invariant method, gives more smooth control signals as the ZOH equivalent. First-order-hold sampling is, e.g., described in

K. J. Åström, B. Wittenmark:
Computer Controlled Systems - Theory and Design
Third Edition, p. 256

# Syntax

(phi, gamma, gamma1) = integralExpT(A, B, T)

# Inputs (3)

A Type: Real[:,size(A, 1)] Type: Real[size(A, 1),:] Default Value: 1 Type: Real

# Outputs (3)

phi Type: Real[size(A, 1),size(A, 1)] Description: = exp(A*T) Type: Real[size(A, 1),size(B, 2)] Description: = integral(phi)*B Type: Real[size(A, 1),size(B, 2)] Description: = integral((T-t)*exp(A*t))*B

# Revisions

Release Notes:

• July 31, 2002 by Nico Walther
Realized.