CONTINUOUS TIME TRANSFER FUNCTIONS

Calculate where I is the identity matrix, and A was the 4x4 matrix derived two pages ago under Derivation of State Space Representation.

s I = [\begin{matrix} s & 0 & 0 & 0 \\ 0 & s & 0 & 0 \\ 0 & 0 & s & 0 \\ 0 & 0 & 0 & s \end{matrix}]

A = [\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & - B & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & K_{p} B & ω_{n}^{2} & - 2 ζ ω_{n} \end{matrix}]

(s I - A) = [\begin{matrix} s & 0 & 0 & 0 \\ 0 & s & 0 & 0 \\ 0 & 0 & s & 0 \\ 0 & 0 & 0 & s \end{matrix}] - [\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & - B & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & K_{p} B & ω_{n}^{2} & - 2 ζ ω_{n} \end{matrix}]

(s I - A) = [\begin{matrix} s & - 1 & 0 & 0 \\ 0 & s + B & 0 & 0 \\ 0 & 0 & s & - 1 \\ 0 & - K_{p} B & - ω_{n}^{2} & s + 2 ζ ω_{n} \end{matrix}]

Calculate inverse

{(s I - A)}^{- 1}

using Row Reduction:

\begin{matrix} r_{1} ⇒ r_{1} \frac{1}{s} \end{matrix} [\begin{matrix} s & - 1 & 0 & 0 \\ 0 & s + B & 0 & 0 \\ 0 & 0 & s & - 1 \\ 0 & - K_{p} B & - ω_{n}^{2} & s + 2 ζ ω_{n} \end{matrix}] [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

Multiply first row by 1/s

\begin{matrix}  \end{matrix} [\begin{matrix} 1 & - \frac{1}{s} & 0 & 0 \\ 0 & s + B & 0 & 0 \\ 0 & 0 & s & - 1 \\ 0 & - K_{p} B & - ω_{n}^{2} & s + 2 ζ ω_{n} \end{matrix}] [\begin{matrix} \frac{1}{s} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

\begin{matrix} r_{2} ⇒ r_{2} \frac{1}{s + B} \end{matrix} [\begin{matrix} 1 & - \frac{1}{s} & 0 & 0 \\ 0 & s + B & 0 & 0 \\ 0 & 0 & s & - 1 \\ 0 & - K_{p} B & - ω_{n}^{2} & s + 2 ζ ω_{n} \end{matrix}] [\begin{matrix} \frac{1}{s} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

Multiply second row by 1/(s+B)

\begin{matrix}  \end{matrix} [\begin{matrix} 1 & - \frac{1}{s} & 0 & 0 \\ 0 & \frac{s + B}{s + B} & 0 & 0 \\ 0 & 0 & s & - 1 \\ 0 & - K_{p} B & - ω_{n}^{2} & s + 2 ζ ω_{n} \end{matrix}] [\begin{matrix} \frac{1}{s} & 0 & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

\begin{matrix} r_{1} ⇒ r_{1} + \frac{1}{s} r_{2} \\ r_{4} ⇒ r_{4} + (K_{p} B) r_{2} \end{matrix} [\begin{matrix} 1 & - \frac{1}{s} & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & s & - 1 \\ 0 & - K_{p} B & - ω_{n}^{2} & s + 2 ζ ω_{n} \end{matrix}] [\begin{matrix} \frac{1}{s} & 0 & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

\begin{matrix}  \end{matrix} [\begin{matrix} 1 & - \frac{1}{s} + \frac{1}{s} & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & s & - 1 \\ 0 & (- K_{p} B + K_{p} B) & - ω_{n}^{2} & s + 2 ζ ω_{n} \end{matrix}] [\begin{matrix} \frac{1}{s} & (0 + \frac{1}{s} \frac{1}{s + B}) & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & \frac{0}{+} & 0 & 1 \end{matrix}]

\begin{matrix}  \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & s & - 1 \\ 0 & 0 & - ω_{n}^{2} & s + 2 ζ ω_{n} \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & \frac{K_{p} B}{s + B} & 0 & 1 \end{matrix}]

\begin{matrix} r_{3} ⇒ r_{3} \frac{1}{s} \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & s & - 1 \\ 0 & 0 & - ω_{n}^{2} & s + 2 ζ ω_{n} \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & \frac{K_{p} B}{s + B} & 0 & 1 \end{matrix}]

\begin{matrix}  \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{s}{s} & \frac{- 1}{s} \\ 0 & 0 & - ω_{n}^{2} & s + 2 ζ ω_{n} \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & 0 & \frac{1}{s} & 0 \\ 0 & \frac{K_{p} B}{s + B} & 0 & 1 \end{matrix}]

\begin{matrix} r_{4} ⇒ r_{4} + ω_{n}^{2} r_{3} \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{s}{s} & \frac{- 1}{s} \\ 0 & 0 & - ω_{n}^{2} & s + 2 ζ ω_{n} \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & 0 & \frac{1}{s} & 0 \\ 0 & \frac{K_{p} B}{s + B} & 0 & 1 \end{matrix}]

\begin{matrix}  \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{s}{s} & \frac{- 1}{s} \\ 0 & 0 & (- ω_{n}^{2} + ω_{n}^{2}) & (s + 2 ζ ω_{n}) + ω_{n}^{2} (\frac{- 1}{s}) \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & 0 & \frac{1}{s} & 0 \\ 0 & \frac{K_{p} B}{s + B} & 0 + ω_{n}^{2} (\frac{1}{s}) & 1 \end{matrix}]

\begin{matrix}  \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{- 1}{s} \\ 0 & 0 & 0 & (s + 2 ζ ω_{n} - \frac{ω_{n}^{2}}{s}) \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & 0 & \frac{1}{s} & 0 \\ 0 & \frac{K_{p} B}{s + B} & \frac{ω_{n}^{2}}{s} & 1 \end{matrix}]

\begin{matrix} r_{4} ⇒ \frac{r_{4}}{(s + 2 ζ ω_{n} - \frac{ω_{n}^{2}}{s})} \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{- 1}{s} \\ 0 & 0 & 0 & (s + 2 ζ ω_{n} - \frac{ω_{n}^{2}}{s}) \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & 0 & \frac{1}{s} & 0 \\ 0 & \frac{K_{p} B}{s + B} & \frac{ω_{n}^{2}}{s} & 1 \end{matrix}]

\begin{matrix}  \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{- 1}{s} \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & 0 & \frac{1}{s} & 0 \\ 0 & \frac{\frac{K_{p} B}{s + B}}{(s + 2 ζ ω_{n} - \frac{ω_{n}^{2}}{s})} & \frac{\frac{ω_{n}^{2}}{s}}{(s + 2 ζ ω_{n} - \frac{ω_{n}^{2}}{s})} & \frac{1}{(s + 2 ζ ω_{n} - \frac{ω_{n}^{2}}{s})} \end{matrix}]

\begin{matrix}  \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{- 1}{s} \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & 0 & \frac{1}{s} & 0 \\ 0 & \frac{s \frac{K_{p} B}{s + B}}{s (s + 2 ζ ω_{n} - \frac{ω_{n}^{2}}{s})} & \frac{s \frac{ω_{n}^{2}}{s}}{s (s + 2 ζ ω_{n} - \frac{ω_{n}^{2}}{s})} & \frac{s}{s (s + 2 ζ ω_{n} - \frac{ω_{n}^{2}}{s})} \end{matrix}]

\begin{matrix}  \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{- 1}{s} \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & 0 & \frac{1}{s} & 0 \\ 0 & \frac{s \frac{K_{p} B}{s + B}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} & \frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} & \frac{s}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} \end{matrix}]

\begin{matrix}  \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{- 1}{s} \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & 0 & \frac{1}{s} & 0 \\ 0 & \frac{K_{p} B s}{(s + B) (s^{2} + 2 ζ ω_{n} s - ω_{n}^{2})} & \frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} & \frac{s}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} \end{matrix}]

\begin{matrix} r_{3} ⇒ r_{3} + \frac{1}{s} r_{4} \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{- 1}{s} \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & 0 & \frac{1}{s} & 0 \\ 0 & \frac{K_{p} B s}{(s + B) (s^{2} + 2 ζ ω_{n} s - ω_{n}^{2})} & \frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} & \frac{s}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} \end{matrix}]

\begin{matrix}  \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{- 1}{s} + \frac{1}{s} \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & 0 + \frac{1}{s} \frac{K_{p} B s}{(s + B) (s^{2} + 2 ζ ω_{n} s - ω_{n}^{2})} & \frac{1}{s} + \frac{1}{s} \frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} & 0 + \frac{1}{s} \frac{s}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} \\ 0 & \frac{K_{p} B s}{(s + B) (s^{2} + 2 ζ ω_{n} s - ω_{n}^{2})} & \frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} & \frac{s}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} \end{matrix}]

\begin{matrix}  \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & \frac{K_{p} B}{(s + B) (s^{2} + 2 ζ ω_{n} s - ω_{n}^{2})} & \frac{1}{s} (1 + \frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}}) & \frac{1}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} \\ 0 & \frac{K_{p} B s}{(s + B) (s^{2} + 2 ζ ω_{n} s - ω_{n}^{2})} & \frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} & \frac{s}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} \end{matrix}]

\begin{matrix}  \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & \frac{K_{p} B}{(s + B) (s^{2} + 2 ζ ω_{n} s - ω_{n}^{2})} & \frac{1}{s} (\frac{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} + \frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}}) & \frac{1}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} \\ 0 & \frac{K_{p} B s}{(s + B) (s^{2} + 2 ζ ω_{n} s - ω_{n}^{2})} & \frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} & \frac{s}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} \end{matrix}]

\begin{matrix}  \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & \frac{K_{p} B}{(s + B) (s^{2} + 2 ζ ω_{n} s - ω_{n}^{2})} & \frac{1}{s} (\frac{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2} + ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}}) & \frac{1}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} \\ 0 & \frac{K_{p} B s}{(s + B) (s^{2} + 2 ζ ω_{n} s - ω_{n}^{2})} & \frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} & \frac{s}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} \end{matrix}]

\begin{matrix}  \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & \frac{K_{p} B}{(s + B) (s^{2} + 2 ζ ω_{n} s - ω_{n}^{2})} & \frac{1}{s} (\frac{s^{2} + 2 ζ ω_{n} s}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}}) & \frac{1}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} \\ 0 & \frac{K_{p} B s}{(s + B) (s^{2} + 2 ζ ω_{n} s - ω_{n}^{2})} & \frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} & \frac{s}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} \end{matrix}]

\begin{matrix}  \end{matrix} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & \frac{K_{p} B}{(s + B) (s^{2} + 2 ζ ω_{n} s - ω_{n}^{2})} & \frac{s + 2 ζ ω_{n}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} & \frac{1}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} \\ 0 & \frac{K_{p} B s}{(s + B) (s^{2} + 2 ζ ω_{n} s - ω_{n}^{2})} & \frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} & \frac{s}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} \end{matrix}]

{(s I - A)}^{- 1} = [\begin{matrix} \frac{1}{s} & \frac{1}{s (s + B)} & 0 & 0 \\ 0 & \frac{1}{s + B} & 0 & 0 \\ 0 & \frac{K_{p} B}{(s + B) (s^{2} + 2 ζ ω_{n} s - ω_{n}^{2})} & \frac{s + 2 ζ ω_{n}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} & \frac{1}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} \\ 0 & \frac{K_{p} B s}{(s + B) (s^{2} + 2 ζ ω_{n} s - ω_{n}^{2})} & \frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} & \frac{s}{s^{2} + 2 ζ ω_{n} s - ω_{n}^{2}} \end{matrix}]

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