(Solved): 3.1. Find the state-variable model for the system shown in Figure 2.16(a) and considered in Exampl ...

3.1. Find the state-variable model for the system shown in Figure 2.16(a) and considered in Example 2.5: a. When \( x \) and \( v=\dot{x} \) are state variables b. When \( z \) and \( w=\dot{z} \) are state variables The inputs are \( f_{a}(t) \) and \( g \). Take the energy stored in the spring as the output. Figure 2.16 (a) Translational system with vertical motion. (b) Free-body diagram. Recall from Example \( 2.5 \) in the Chapter 2 notes that: \[ \begin{array}{c} K x_{0}=M g \\ x=x_{0}+z \end{array} \] Note: It is not always possible to write the desired outputs in standard state-space form \( \boldsymbol{y}=\boldsymbol{C q} \boldsymbol{+ D u} \), e.g., if the outputs are not linear functions of the states. If this is the case, just state that it cannot be done and why.