Basic State-Space Model Examples

Example 1

Find the state-space model for:

$$\text{dddot}x+a_1\ddot{x}+a_2\dot{x}+a_{3}x=u$$

Since the highest order is $\text{dddot}x$, we use $x,\dot{x},\ddot{x}$ as state variables in a vector:

$$\bar{x}=\left[\begin{array}{c}x\\ \dot{x}\\ \ddot{x}\end{array}\right]$$

We can then write the derivative of $\bar{x}$ in terms of the state variables, which gives us a three equation system including the original equation we wanted to model:

$$\dot{\bar{x}}=\left[\begin{array}{c}\dot{x}\\ \ddot{x}\\ \text{dddot}x\end{array}\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -a_3& -a_2& -a_1\end{array}\right]\left[\begin{array}{c}x\\ \dot{x}\\ \ddot{x}\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]u$$

The first row gives:

$$\dot{x}=\left[\begin{array}{ccc}0& 1& 0\end{array}\right]\left[\begin{array}{c}x\\ \dot{x}\\ \ddot{x}\end{array}\right]+0u=\dot{x}$$

The second row gives:

$$\ddot{x}=\left[\begin{array}{ccc}0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ \dot{x}\\ \ddot{x}\end{array}\right]+0u=\ddot{x}$$

The third row gives:

$$\text{dddot}x=\left[\begin{array}{ccc}-a_3& -a_2& -a_1\end{array}\right]\left[\begin{array}{c}x\\ \dot{x}\\ \ddot{x}\end{array}\right]+1u=-a_{3}x-a_2\dot{x}-a_1\ddot{x}+u$$

Example 2