In the signal flow diagram given in the figure, \({u_1}\;and\;{u_2}\) are possible inputs whereas \({y_1}\;and\;{y_2}\) are possible output. When would the SISO system derived from this diagram be controllable and observable?

This question was previously asked in

GATE EE 2015 Official Paper: Shift 1

Option 2 : When \({u_2}\) is the only input and \({y_1}\) is the only output

CT 1: Ratio and Proportion

3742

10 Questions
16 Marks
30 Mins

From the signal flow graph, we have

\({\dot x_1} = {u_1} + 5{x_1} - 2{x_2}\)

\({\dot x_2} = {u_2} + {u_1} + 2{x_1} + {x_2}\)

\({y_1} = {x_1}\)

\({y_2} = - {x_2} + {x_1}\)

\(\left[ {\begin{array}{*{20}{c}} {{{\dot x}_1}} \\ {{{\dot x}_2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 5&{ - 2} \\ 2&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 1&0 \\ 1&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{u_1}} \\ {{u_2}} \end{array}} \right]\)

\(\left[ {\begin{array}{*{20}{c}} {{y_1}} \\ {{y_2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&0 \\ 1&{ - 1} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \end{array}} \right]\)

If u_{1} is zero, u_{2} is only input, and y_{1} is only output. Then, the above expression becomes

\(\left[ {\begin{array}{*{20}{c}} {{{\dot x}_1}} \\ {{{\dot x}_2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 5&{ - 2} \\ 2&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0 \\ 1 \end{array}} \right]\left[ {{u_2}} \right]\)

\({y_1} = \left[ {\begin{array}{*{20}{c}} 1&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \end{array}} \right]\)

To check the controllability, we form the matrix as

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

Since, determinant of the above matrix is 2(≠0), so its rank will be 2, ie. Equal to the number of variables. Hence, the system is controllable.

Again, to check the observability of the system, we form the matrix

\(\left[ {C:CA} \right] = \left[ {\begin{array}{*{20}{c}} 1&0 \\ {54}&{ - 2} \end{array}} \right]\)

Determinant of the above matrix is -2, so its rank will be 2, ie. Equal to the number of variables. Hence, the system is observable. Thus, the given system is controllable and observable when u_{2} is only input and y_{1} is only output.