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## Q. 7.EX.9

State Equations in Modal Canonical Form
A “quarter car model” [see Eq. (2.12)] with one resonant mode has a transfer function given by

$G(s) = \frac{2s + 4}{s^2(s^2 + 2s + 4)} = \frac{1}{s^2} − \frac{1}{s^2 + 2s + 4}.$               (7.19)

$\frac{Y(s)}{R(s)} = \frac{\frac{k_wb}{m_1 m_2}\left(s + \frac{k_s}{b}\right) }{s^4 + \left(\frac{b}{m_1} + \frac{b}{m_2} \right) s^3 + \left(\frac{k_s}{m_1}+\frac{k_s}{m_2}+\frac{k_w}{m_1} \right) s^2 +\left(\frac{k_w b }{m_1 m_2} \right) s+ \frac{k_w k_s }{m_1 m_2} } .$             (2.12)

Find state matrices in modal form describing this system .

## Verified Solution

The transfer function has been given in real partial-fraction form. To get state-description matrices, we draw a corresponding block diagram with integrators only, assign the state, and write down the corresponding matrices. This process is not unique, so there are several acceptable solutions to the problem as stated, but they will differ in only trivial ways. A block diagram with a satisfactory assignment of variables is given in Fig. 7.9.

Notice that the second-order term to represent the complex poles has been realized in control canonical form. There are a number of other possibilities that can be used as alternatives for this part. This particular form allows us to write down the system matrices by inspection:

$\pmb F = \begin{bmatrix} 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 \\ 0 & 0 & -2 & -4 \\ 0 & 0 & 1 & 0 \end{bmatrix}, \ \ \ \ \pmb G = \begin{bmatrix}1 \\0 \\ 1 \\ 0 \end{bmatrix} , \\ \pmb H = \begin{bmatrix} 0 & 1 & 0 & -1 \end{bmatrix} , \ \ \ \ J =0.$          (7.20)