functionintegralExpT
Return the exponential, the integral of the exponential, and time-weighted integral of the exponential of a matrix
Extends from Modelica.Icons.Function (Icon for functions).
Information
(phi,gamma,gamma1) = Matrices.integralExpT(A,B); (phi,gamma,gamma1) = Matrices.integralExpT(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
Inputs
| Type | Name | Default | Description |
|---|---|---|---|
| Real[:,size(A, 1)] | A | ||
| Real[size(A, 1),:] | B | ||
| Real | T | 1 |
Outputs
| Type | Name | Default | Description |
|---|---|---|---|
| Real[size(A, 1),size(A, 1)] | phi | = exp(A*T) | |
| Real[size(A, 1),size(B, 2)] | gamma | = integral(phi)*B | |
| Real[size(A, 1),size(B, 2)] | gamma1 | = integral((T-t)*exp(A*t))*B |
Revisions
Release Notes:
- July 31, 2002
by Nico Walther
Realized.