5.4 State space models
While normal mode models are appropriate for structures, statespace
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
spacemodels 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}.
Statespace 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 statespace object of the Control Toolbox).
The natural statespace representation of normal mode models
(5.4) is given by
Transformations to this form are provided by nor2ss and fe2ss.
Another special form of statespace models is constructed by res2ss.
A statespace representation of the nominal structural model (5.1) is given by
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^{−1}K 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 statespace 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] − [^{\} λ_{j} _{\} ]  ⎞
⎠  {η(s)} =
[θ_{L}^{T} b] {u} 

{y} = [c θ_{R}] {η(s)}


(5.12) 
where the left and right modeshapes (columns of [θ_{R}] and
[θ_{L}]) are solution of
{θ_{jL}}^{T} [A] = λ_{j}{θ_{jL}}^{T} and
[A] {θ_{jR}} = λ_{j}{θ_{jR}}
(5.13) 
and verify the orthogonality conditions
[θ_{L}]^{T} [θ_{R}] = [I] and
[θ_{L}]^{T} [A] [θ_{R}] = [^{\} λ_{j} _{\} ]
(5.14) 
The diagonal state space form corresponds to the partial fraction
expansion
{y(s)} =   {cψ_{j}}{ψ_{j}^{T}b} 

s−λ_{j} 
 =
  
(5.15) 
where the contribution of each mode is characterized by the
pole location λ_{j} and the residue matrix R_{j} (which is equal to the product of the complex modal
output {cθ_{j}} by the modal input
{θ_{j}^{T}b}).
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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