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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−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 (5.13).
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 [θR] and [θL]) 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 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 (5.19) is heavily used for the identification routines implemented in the SDT (see the section on the pole/residue representation ref .