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2.4  State space models



While normal mode models are appropriate for structures, state-space models allow the representation of more general linear dynamic systems and are commonly used in the Control Toolbox or Simulink. The standard form for state space-models is
 
{
.
x
 
} = [A] {x(t)} + [B] {u(t)}
{y} = [C] {x(t)} + [D] {u(t)}
    (2.8)

with inputs {u}, states {x} and outputs {y}. State-space models are represented in the SDT, as generally done in other Toolboxes for use with MATLAB, using four independent matrix variables a, b, c, and d (you should also take a look at the LTI state-space object of the Control Toolbox).


The natural state-space representation of normal mode models (2.4) is given by
 
{
.
p
 
..
p
 
} = [
0 I
-W2 -G
] {
p
.
p
 
} + [
0
T b
] {u(t)}
{y(t)} = [c 0] {
p
.
p
 
}
    (2.9)
Transformations to this form are provided by nor2ss and fe2ss. Another special form of state-space models is constructed by res2ss.

A state-space representation of the nominal structural model (2.1) is given by
 
{
.
q
 
..
q
 
} = [
0 I
-M-1K -M-1C
] {
q
.
q
 
} + [
0
M-1 b
] {u(t)}
{y(t)} = [c 0] {
q
.
q
 
}
    (2.10)
The interest of this representation is mostly academic because it does not preserve symmetry (an useful feature of models of structures associated to the assumption of reciprocity) and because M-1K is usually a full matrix (so that the associated memory requirements for a realistic finite element model would be prohibitive). The SDT thus always starts by transforming a model to the normal mode form and the associated state-space model (2.9).


The transfer functions from inputs to outputs are described in the frequency domain by
  {y(s)} = ( [C][s I-A]-1[B]+[D] ) {u(s)}     (2.11)
assuming that [A] is diagonalizable in the basis of complex modes, model (2.8) is equivalent to the diagonal model
 
( s [I] - [\ j \ ] ) {(s)} = [LT b] {u}
{y} = [c R] {(s)}
    (2.12)
where the left and right modeshapes (columns of [R] and [L]) are solution of
  {jL}T [A] = j{jL}T and [A] {jR} = j{jR}     (2.13)
and verify the orthogonality conditions
  [L]T [R] = [I] and [L]T [A] [R] = [\ j \ ]     (2.14)


The diagonal state space form corresponds to the partial fraction expansion
  {y(s)} =
2N
j=1
{cj}{jTb}
s-j
=
2N
j=1
[Rj]NS× NA
s-j
    (2.15)
where the contribution of each mode is characterized by the pole location j and the residue matrix Rj (which is equal to the product of the complex modal output {cj} by the modal input {jTb}).

The partial fraction expansion (2.15) is heavily used for the identification routines implemented in the SDT (see the section on the pole/residue representation ref .

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