Question

Question: Determine if the matrix pairs in Exercises 19-22 are controllable.

22. (M) \(A = \left( {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\{ - 1}&{ - 13}&{ - 12.2}&{ - 1.5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\0\\0\\{ - 1}\end{array}} \right)\).

Short answer

The matrix pair \(\left( {A,B} \right)\) is controllable.

Step-by-step solution

Define the rank of a matrix

Therank of matrix\(A\), denoted by rank\(A\), is the dimension of the column spaceof \(A\).

Write the augmented matrix

Calculate the rank of the matrix \(\left( {\begin{array}{*{20}{c}}B&{AB}&{{A^2}B}&{{A^3}B}\end{array}} \right)\) to determine whether the matrix pair \(\left( {A,B} \right)\) is controllable.

Write the augmented matrix as shown below:

\(\left( {\begin{array}{*{20}{c}}B&{AB}&{{A^2}B}&{{A^3}B}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}1&0&0&{ - 1}\\0&0&{ - 1}&{.5}\\0&{ - 1}&{.5}&{11.45}\\{ - 1}&{.5}&{11.45}&{ - 10.275}\end{array}} \right)\)

Convert the matrix into row-reduced echelon form

Consider the matrix \(A = \left( {\begin{array}{*{20}{c}}1&0&0&{ - 1}\\0&0&{ - 1}&{.5}\\0&{ - 1}&{.5}&{11.45}\\{ - 1}&{.5}&{11.45}&{ - 10.275}\end{array}} \right)\).

Use the code in MATLAB to obtain the row-reduced echelon form of the matrix.

\(\begin{array}{l} > > {\mathop{\rm A}\nolimits} = \left( {1\,\,\,\,0\,\,\,\,0\,\,\, - 1;\,0\,\,\,0\,\,\, - 1\,\,\,.5;\,0\,\,\, - 1\,\,\,.5\,\,\,11.45;\,\, - 1\,\,\,.5\,\,\,11.45\,\,\, - 10.275} \right)\\ > > {\mathop{\rm U}\nolimits} = {\mathop{\rm rref}\nolimits} \left( {\mathop{\rm A}\nolimits} \right)\end{array}\)

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

The matrix has four pivot columns, so the rank of the matrix is 4.

Determine whether the matrix pairs are controllable

The pair \(\left( {A,B} \right)\) is said to becontrollable if rank\(\left( {\begin{array}{*{20}{c}}B&{AB}&{{A^2}B}& \cdots &{{A^{n - 1}}B}\end{array}} \right) = n\).

The rank of the matrix is 4. Therefore, the matrix pair \(\left( {A,B} \right)\) is controllable.