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

(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 [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^Tb}).

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 .