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5.4  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

   
  {ẋ} = [A] {x(t)} + [B] {u(t)} 
  {y} = [C] {x(t)} + [D] {u(t)} 
     (5.8)

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 (5.4) is given by

   
 {
p′ 
p″ 
} = [
0I 
−Ω2−Γ
] {
p 
p′ 
} + [
0 
φT b 
] {u(t)} 
  {y(t)} = [cφ   0]  {
p 
p′ 
}
    (5.9)

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 (5.1) is given by

  
 {
q′ 
q″ 
} = [
0I 
−M−1K−M−1C
] {
q 
q′ 
} + [
0 
M−1 b 
] {u(t)} 
  {y(t)} = [c   0]  {
q 
q′ 
}
    (5.10)

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 (5.9).


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)}     (5.11)

assuming that [A] is diagonalizable in the basis of complex modes, model (5.8) is equivalent to the diagonal model

   
  ⎛
⎝		s [I] − [\ λj \ ]⎞
⎠		{η(s)} =  [θLT b] {u} 
{y} = [c θR] {η(s)} 
    (5.12)

where the left and right modeshapes (columns of [θR] and [θL]) are solution of

   {θjL}T [A] = λj{θjL}T   and    [A] {θjR} = λj{θjR}     (5.13)

and verify the orthogonality conditions

   [θL]T [θR] = [I]   and    [θL]T [A] [θR] = [\ λj \ ]     (5.14)


The diagonal state space form corresponds to the partial fraction expansion

    {y(s)} = 
2N
∑
j=1
{cψj}{ψjTb}
s−λj
 = 
2N
∑
j=1
[Rj]NS× NA
s−λj
     (5.15)

where the contribution of each mode is characterized by the pole location λj and the residue matrix Rj (which is equal to the product of the complex modal output {cθj} by the modal input {θjTb}).

The partial fraction expansion (5.15) is heavily used for the identification routines implemented in the SDT (see the section on the pole/residue representation ref .


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