State space representation

See also sides

time-domain technique

$$\begin{array}{rl}\dot{x}& =Ax+Bu\\ y& =Cx+Du\end{array}$$

• Linearly independent
• State vector: $x=[x_1,x_2,\dots ,x_n{]}^T$

transfer function to a state space representation

controller form

Given

$$G(s)=\frac{{\sum }_{i=1}^{n-1}{b}_i{s}^i+b_0}{s^n+{\sum }_{i=1}^{n-1}{a}_i{s}^i+a_0}=\frac{Y(s)}{U(s)}$$

We get controller canonical state space form:

$$\begin{array}{rl}\dot{x}(t)& =\left[\begin{array}{cccccc}0& 1& 0& \cdots & 0& 0\\ 0& 0& 1& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 1& 0\\ 0& 0& 0& \cdots & 0& 1\\ -a_0& -a_1& -a_2& \cdots & -a_{n-2}& -a_{n-1}\end{array}\right]x(t)+\left[\begin{array}{c}0\\ 0\\ ⋮\\ 0\\ 0\\ 1\end{array}\right]u(t)\\ y(t)& =\left[\begin{array}{ccccc}{b}_0& b_1& \cdots & b_{n-2}& b_{n-1}\end{array}\right]x(t).\end{array}$$

observer form

We get observer canonical state space form:

$$\begin{array}{rl}\dot{x}(t)& =\left[\begin{array}{cccccc}-a_{n-1}& 1& 0& \cdots & 0& 0\\ -a_{n-2}& 0& 1& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ -a_2& 0& 0& \cdots & 1& 0\\ -a_1& 0& 0& \cdots & 0& 1\\ -a_0& 0& 0& \cdots & 0& 0\end{array}\right]x(t)+\left[\begin{array}{c}{b}_{n-1}\\ b_{n-2}\\ ⋮\\ b_2\\ b_1\\ b_0\end{array}\right]u(t)\\ y(t)& =\left[\begin{array}{ccccc}1& 0& \cdots & 0& 0\end{array}\right]x(t).\end{array}$$