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Consider the state – space model

\(\left[ {\begin{array}{*{20}{c}} {{{\dot x}_1}}\\ {{{\dot x}_2}}\\ {{{\dot x}_3}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 1}&1&0\\ 0&{ - 1}&0\\ 0&0&{ - 2} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}}\\ {{x_2}}\\ {{x_3}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0\\ 4\\ 0 \end{array}} \right]u\)

\(y = \left[ {\begin{array}{*{20}{c}} 1&1&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}}\\ {{x_2}}\\ {{x_3}} \end{array}} \right]\)

System is


1. Controllable and observable
2. Uncontrollable and observable
3. Uncontrollable and Unobservable
4. Controllable and Unobservable

1 Answer

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Best answer
Correct Answer - Option 2 : Uncontrollable and observable

Concept:

Controllability:

It is the internal states of the system are changed from one value to another value in a finite time by a finite input, then we can say the system is controllable otherwise it is not controllable.

To check Controllability, consider the controllability matrix (ϕc)

\(\left[ {{\phi _c}} \right] = {\left[ {{A^0}B\;\;\;{A^1}B} \right]_{2 \times 2}}\;\;\;;\left[ {{\phi _c}} \right] = {\left[ {{A^0}B\;\;\;\;\;{A^1}B\;\;\;\;{A^2}B} \right]_{3 \times 3}}\)

If |ϕc| = 0; then the system is uncontrollable

If |ϕc| ≠ 0, then the system is controllable.

Observability:

If the internal states of the system can be evaluated from the output of the system of any time, then we can say, the system is observable otherwise it is not observable.

To check observability, consider the observability matrix

\(\left[ {{\phi _0}} \right] = {\left[ {{C^T}\;\;\;\;A{C^T}} \right]_{2 \times 2}}\; \to {2^{nd}}\;order\:\:\:;\:\:\:\left[ {{\phi _0}} \right] = {\left[ {{C^T}\;\;\;\;\;A{C^T}\;\;\;\;\;{A^2}{C^T}} \right]_{3 \times 3}}\)

If |ϕo| = 0; Then system is not-observable

If |ϕo| ≠ 0; then the system is observable.

Calculation:

The controllability matrix is given by:

\({Q_c} = \left[ {\begin{array}{*{20}{c}} B&{AB}&{{A^2}B} \end{array}} \right]\)

\(= \left[ {\begin{array}{*{20}{c}} 0&4&{ - 8}\\ 4&{ - 4}&4\\ 0&0&0 \end{array}} \right] \Rightarrow \left| {{Q_c}} \right| = 0\)  → Uncontrollable

The observability  matrix is given by:

\({Q_o} = \left[ {\begin{array}{*{20}{c}} C\\ {CA}\\ {C{A^2}} \end{array}} \right]\)

\({Q_o} = \left[ {\begin{array}{*{20}{c}} 1&1&1\\ { - 1}&0&{ - 2}\\ 1&1&4 \end{array}} \right] \Rightarrow \left| {{Q_o}} \right| \ne 0\) → Observable

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