Question

The transfer function of the system \(\frac{{Y\left( s \right)}}{{U\left( s \right)}}\) whose state-space equations are given below is:

\(\left[ {\begin{array}{*{20}{c}} {{{\dot x}_1}\left( t \right)}\\ {{{\dot x}_2}\left( t \right)} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&2\\ 2&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}\left( t \right)}\\ {{x_2}\left( t \right)} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 1\\ 2 \end{array}} \right]u\left( t \right)\)

\(y\left( t \right) = \left[ {\begin{array}{*{20}{c}} 1&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}\left( t \right)}\\ {{x_2}\left( t \right)} \end{array}} \right]\)
1. \(\frac{{\left( {s + 2} \right)}}{{\left( {{s^2} - 2s - 2} \right)}}\)
2. \(\frac{{\left( {s - 2} \right)}}{{\left( {{s^2} + s - 4} \right)}}\)
3. \(\frac{{\left( {s - 4} \right)}}{{\left( {{s^2} + s - 4} \right)}}\)
4. \(\frac{{\left( {s + 4} \right)}}{{\left( {{s^2} - s - 4} \right)}}\)

Answer 1

Correct Answer - Option 4 : \(\frac{{\left( {s + 4} \right)}}{{\left( {{s^2} - s - 4} \right)}}\)

From the given state space equations,

\(A = \left[ {\begin{array}{*{20}{c}} 1&2\\ 2&0 \end{array}} \right]\)

\(B = \left[ {\begin{array}{*{20}{c}} 1\\ 2 \end{array}} \right]\)

\(C = \left[ {\begin{array}{*{20}{c}} 1&0 \end{array}} \right]\)

Transfer function = C[sI - A]^-1 B + D

\(sI - A = \left[ {\begin{array}{*{20}{c}} s&0\\ 0&s \end{array}} \right] - \left[ {\begin{array}{*{20}{c}} 1&2\\ 2&0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {s - 1}&{ - 2}\\ { - 2}&s \end{array}} \right]\)

\({\left( {sI - A} \right)^{ - 1}} = \frac{1}{{{s^2} - s - 4}}\left[ {\begin{array}{*{20}{c}} s&2\\ 2&{s - 1} \end{array}} \right]\)

Transfer function \(= \left[ {\begin{array}{*{20}{c}} 1&0 \end{array}} \right]\frac{1}{{\left( {{s^2} - s - 4} \right)}}\left[ {\begin{array}{*{20}{c}} s&2\\ 2&{s - 1} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1\\ 2 \end{array}} \right]\)

\(= \frac{1}{{\left( {{s^2} - s - 4} \right)}}\left[ {\begin{array}{*{20}{c}} s&2 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1\\ 2 \end{array}} \right]\)

\(= \frac{{s + 4}}{{\left( {{s^2} - s - 4} \right)}}\)