State Space Formulation of the
Piano String FDS

• We can now write the entire system, ignoring the nonlinear hammer for the moment, as
  $$\begin{eqnarray*} {\bf y}_{n} &=& {\bf A_{1}y_{n-1}}+{\bf A_{2}y_{n-2}}+{\bf A_{3}y_{n-3}} \\ y_{out} &=& {\bf C_1 y_{n}} \end{eqnarray*}$$
  

• Collecting the displacements into a state vector, we can rewrite this as
  $$\begin{eqnarray*} \left [\begin{matrix}{\bf y}_{n} \cr {\bf y}_{n-1} \cr {\bf y}_{n-2}\end{matrix}\right ] &=& \left [\begin{matrix} {\bf A_{1}} & {\bf A_{2}} & {\bf A_{3}} \cr {\bf I_{N}} & - & - \cr - & {\bf I_{N}} & - \end{matrix}\right] \left [\begin{matrix}{\bf y}_{n-1} \cr {\bf y}_{n-2} \cr {\bf y}_{n-3}\end{matrix}\right ] \\ y_{out} &=& \left [\begin{matrix}{\bf C_1} & {\bf0} & {\bf0}\end{matrix}\right ] \left [\begin{matrix}{\bf y}_{n-1} \cr {\bf y}_{n-2} \cr {\bf y}_{n-3}\end{matrix}\right ] \end{eqnarray*}$$
  

  or

  $$\begin{eqnarray*} {\bf {\bf x}_{n}} &=& {\bf A}{\bf {\bf x}_{n-1}} \\ y_{out} &=& {\bf C{\bf x}_{n}} \end{eqnarray*}$$