Editing State-space representation (section)

=== Transfer function ===
The "[[transfer function]]" of a continuous time-invariant linear state-space model can be derived in the following way:

First, taking the [[Laplace transform]] of 
<math display="block">\dot{\mathbf{x}}(t) = \mathbf{A} \mathbf{x}(t) + \mathbf{B} \mathbf{u}(t)</math>

yields
<math display="block">s\mathbf{X}(s)-\mathbf{x}(0) = \mathbf{A} \mathbf{X}(s) + \mathbf{B} \mathbf{U}(s). </math>
Next, we simplify for <math>\mathbf{X}(s)</math>, giving
<math display="block">(s\mathbf{I} - \mathbf{A})\mathbf{X}(s) =\mathbf{x}(0)+ \mathbf{B}\mathbf{U}(s) </math>
and thus
<math display="block">\mathbf{X}(s) =(s\mathbf{I} - \mathbf{A})^{-1}\mathbf{x}(0)+ (s\mathbf{I} - \mathbf{A})^{-1}\mathbf{B}\mathbf{U}(s). </math>

Substituting for <math>\mathbf{X}(s)</math> in the output equation

<math display="block">\mathbf{Y}(s) = \mathbf{C}\mathbf{X}(s) + \mathbf{D}\mathbf{U}(s),</math>
giving
<math display="block">\mathbf{Y}(s) = \mathbf{C}((s\mathbf{I} - \mathbf{A})^{-1}\mathbf{x}(0)+ (s\mathbf{I} - \mathbf{A})^{-1}\mathbf{B}\mathbf{U}(s)) + \mathbf{D}\mathbf{U}(s). </math>

Assuming zero initial conditions <math>\mathbf{x}(0) =\mathbf{0} </math> and a [[Single-input single-output system|single-input single-output (SISO) system]], the [[transfer function]]  is defined as the ratio of output and input <math>G(s)=Y(s)/U(s)</math>. For a [[MIMO|multiple-input multiple-output (MIMO) system]], however, this ratio is not defined. Therefore, assuming zero initial conditions, the [[transfer function matrix]] is derived from
<math display="block">\mathbf{Y}(s) = \mathbf{G}(s) \mathbf{U}(s) </math>

using the method of equating the coefficients which yields

<math display="block">\mathbf{G}(s) = \mathbf{C}(s\mathbf{I} - \mathbf{A})^{-1}\mathbf{B} + \mathbf{D} . </math>

Consequently, <math>\mathbf{G}(s)</math> is a matrix with the dimension <math>q \times p</math> which contains transfer functions for each input output combination. Due to the simplicity of this matrix notation, the state-space representation is commonly used for multiple-input, multiple-output systems. The [[Rosenbrock system matrix]] provides a bridge between the state-space representation and its [[transfer function]].