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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

(5.12) |

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

(5.13) |

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

(5.14) |

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 state-space model (5.13).

The transfer functions from inputs to outputs are described in the
frequency domain by

(5.15) |

assuming that [A] is diagonalizable in the basis of complex modes, model (5.12) is equivalent to the diagonal model

(5.16) |

where the left and right modeshapes (columns of [θ_{R}] and
[θ_{L}]) are solution of

(5.17) |

and verify the orthogonality conditions

(5.18) |

The diagonal state space form corresponds to the partial fraction
expansion

(5.19) |

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 (5.19) is heavily used for the identification routines implemented in the SDT (see the section on the pole/residue representation ref .

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