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res2ss, ss2res

Purpose

Transformations between the residue res and state-space ss forms.

Syntax

SYS            = res2ss(RES)
SYS            = res2ss(RES,'AllIO')
[a,b,c,d]      = res2ss(res,po,idopt)
RES            = ss2res(SYS)
[res,po,idopt] = ss2res(a,b,c,d)

Description

The functions res2ss and ss2res provide transformations between the complex / normal mode residue forms res (see section 5.6) and the state space forms (see section 5.4). You can use either high level calls with data structures or low level calls providing each argument

ci=demosdt('demo gartid est')
SYS = res2ss(ci.Stack{'IdMain'});
RES = ss2res(SYS);
ID=ci.Stack{'IdMain'};
[a,b,c,d] = res2ss(ID.res,ID.po,ID.idopt);

Important properties and limitations of these transformations are

ss

res

Theory

For control design or simulation based on identification results, the minimal model resulting from id_rm is usually sufficient (there is no need to refer to the normal modes). The state-space form is then the reference model form.

As shown in section 2.8.1, the residue matrix can be decomposed into a dyad formed of a column vector (the modal output), and a row vector (the modal input). From these two matrices, one derives the [B] and [C] matrices of a real parameter state-space description of the system with a bloc diagonal [A] matrix

 {
1 
2 
} = [
[0][\ I \ ] 
− [\ ωj2 \ ]− [\ 2ζjωj \ ] 
] {
x1 
x2 
} + {
B1 
B2 
} {u(t)} 
{y(t)}  =  [C1   C2] {
x1 
x2 

where the blocks of matrices B1, B2, C1, C2 are given by

{
C1j 
C2j
}
=
 [Re
cψj
  Im
cψj
]  
1
ωj
1−ζj2
 [
ωj
1−ζj2
0
 ζjωj1
]
{
Bj1 
Bj2
}
=
2 [
1
−ζjωj
−ωj
1−ζj2
 
] [
Re
ψjTb
Im
ψjTb

Form the state space model thus obtained, FRFs in the xf format can be readily obtained using qbode. If the state space model is not needed, it is faster to use res2xf to generate these FRFs.

See also

demo_fe, res2xf, res2nor, qbode, id_rm, id_rc


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