Representaci贸n en espacio de estado
Sea un sistema de la forma:
\[y^{(n)} + a_1y^{(n-1)} + a_{n-1}\dot{y} + a_ny = b_0u^{(n)} + b_1u^{(n-1)}+ \cdots + b_{n-1}\dot{u} + b_nu\]Se definen los cambios de variables
\[\begin{aligned} x_1 &= y - \beta_0u\\ x_2 &= \dot{y} - \beta_0\dot{u} - \beta_1u = \dot{x}_1 - \beta_1u\\ x_3 &= \ddot{y} - \beta_0\ddot{u} - \beta_1\dot{u} - \beta_2u = \dot{x}_2 - \beta_2u\\ &\vdots\\ x_n &= y^{(n-1)} - \beta_0u^{(n-1)} - \beta_1u^{(n-2)} - \cdots - \beta_{n-2}\dot{u} - \beta_{n-1}u = \dot{x}_{n-1} - \beta_{n-1}u\\ \end{aligned}\]Donde
\[\begin{aligned} \beta_0 &= b_0\\ \beta_1 &= b_1 - a_1\beta_0\\ \beta_2 &= b_2 - a_1\beta_1 - a_2\beta_0\\ \beta_3 &= b_3 - a_1\beta_2 - a_2\beta_1 - a_3\beta_0\\ \vdots\\ \beta_{n-1} &= b_{n-1} - a_1\beta_{n-2} - \cdots - a_{n-2}\beta_1 - a_{n-1}\beta_0\\ \end{aligned}\]De esta forma, el cambio de variable queda de la siguiente forma:
\[\begin{aligned} \dot{x_1} &= x_2 + \beta_1u\\ \dot{x_2} &= x_3 + \beta_2u\\ \vdots\\ \dot{x_{n-1}} &= x_n + \beta_{n-1}u\\ \dot{x_n} &= -a_nx_1 - a_{n-1}x_2 - \cdots - a_1x_n + \beta_nu\\ \end{aligned}\]Donde:
\[\beta_n = b_n -a_1\beta_{n-1} - \cdots - a_{n-1}\beta_1 - a_{n-1}\beta_0\]Por lo tanto la forma matricial queda de la siguiente forma:
\[\dot{x} = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & & \vdots\\ 0 & 0 & 0 & \cdots & 1\\ -a_n & -a_{n-1} & -a_{n-2} & \cdots & -a_1\\ \end{bmatrix}\begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_{n-1}\\ x_n\\ \end{bmatrix} + \begin{bmatrix} \beta_1\\ \beta_2\\ \vdots\\ \beta_{n-1}\\ \beta_n\\ \end{bmatrix}u\] \[y = \begin{bmatrix}1&0&\cdots&0\end{bmatrix}x + \beta_0u\]Caso 2x2
\[\dot{x} = \begin{bmatrix} 0&1\\ -a_2&-a_1\\ \end{bmatrix}x + \begin{bmatrix} \beta_1\\ \beta_2\\ \end{bmatrix}u\ , \qquad y=[1\ \ 0] x + \beta_0u\]Caso 3x3
\[\dot{x} = \begin{bmatrix} 0&1&0\\ 0&0&1\\ -a_3&-a_2&-a_1\\ \end{bmatrix}x + \begin{bmatrix} \beta_1\\ \beta_2\\ \beta_3\\ \end{bmatrix}u\ , \qquad y=\begin{bmatrix}1&0&0\end{bmatrix} x + \beta_0u\]Funci贸n de transferencia
Transformando al dominio de Laplace:
\[\ddot{y} + a_1\dot{y} + a_2y = b_0\ddot{u} + b_1\dot{u} + b_2u\] \[\downarrow\] \[s^2Y(s) + a_1sY(s) + a_2Y(s) = b_0s^2U(s) + b_1sU(s) + b_2U(s)\] \[\dfrac{Y(s)}{U(s)} = \dfrac{b_0s^2 + b_1s + b_2}{s^2+a_1s+a_2}\]