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Chapter 4: Q30E (page 191)

Write the difference equations in Exercises 29 and 30 as first order systems, \({x_{k + {\bf{1}}}} = A{x_k}\), for all \(k\).

\({y_{k + {\bf{3}}}} - \frac{{\bf{3}}}{{\bf{4}}}{y_{k + {\bf{2}}}} + \frac{{\bf{1}}}{{{\bf{16}}}}{y_k} = {\bf{0}}\)

Short Answer

Expert verified

\(A = \left[ {\begin{array}{*{20}{c}}0&1&0\\0&0&1\\{ - \frac{1}{{16}}}&0&{\frac{3}{4}}\end{array}} \right]\)

Step by step solution

01

Write vectors \({x_k}\)and \({x_{k + {\bf{1}}}}\)

Vectors \({x_k}\) and \({x_{k + 1}}\) can be expressed as

\({x_k} = \left[ {\begin{array}{*{20}{c}}{{y_k}}\\{{y_{k + 1}}}\\{{y_{k + 2}}}\end{array}} \right]\) and \({x_{k + 1}} = \left[ {\begin{array}{*{20}{c}}{{y_{k + 1}}}\\{{y_{k + 2}}}\\{{y_{k + 3}}}\end{array}} \right]\).

02

Write \({x_{k + {\bf{1}}}}\) in the matrix form

The matrix formfor \({x_{k + 1}}\) is

\(\begin{aligned} {x_{k + 1}} &= \left[ {\begin{array}{*{20}{c}}0&1&0\\0&0&1\\{ - \frac{1}{{16}}}&0&{\frac{3}{4}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{y_k}}\\{{y_{k + 1}}}\\{{y_{k + 2}}}\end{array}} \right]\\ &= A{x_k}\end{aligned}\).

So, matrix \(A = \left[ {\begin{array}{*{20}{c}}0&1&0\\0&0&1\\{ - \frac{1}{{16}}}&0&{\frac{3}{4}}\end{array}} \right]\).

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Most popular questions from this chapter

Let \({\mathop{\rm u}\nolimits} = \left[ {\begin{array}{*{20}{c}}1\\2\end{array}} \right]\). Find \({\mathop{\rm v}\nolimits} \) in \({\mathbb{R}^3}\) such that \(\left[ {\begin{array}{*{20}{c}}1&{ - 3}&4\\2&{ - 6}&8\end{array}} \right] = {{\mathop{\rm uv}\nolimits} ^T}\) .

Question 18: Suppose A is a \(4 \times 4\) matrix and B is a \(4 \times 2\) matrix, and let \({{\mathop{\rm u}\nolimits} _0},...,{{\mathop{\rm u}\nolimits} _3}\) represent a sequence of input vectors in \({\mathbb{R}^2}\).

  1. Set \({{\mathop{\rm x}\nolimits} _0} = 0\), compute \({{\mathop{\rm x}\nolimits} _1},...,{{\mathop{\rm x}\nolimits} _4}\) from equation (1), and write a formula for \({{\mathop{\rm x}\nolimits} _4}\) involving the controllability matrix \(M\) appearing in equation (2). (Note: The matrix \(M\) is constructed as a partitioned matrix. Its overall size here is \(4 \times 8\).)
  2. Suppose \(\left( {A,B} \right)\) is controllable and v is any vector in \({\mathbb{R}^4}\). Explain why there exists a control sequence \({{\mathop{\rm u}\nolimits} _0},...,{{\mathop{\rm u}\nolimits} _3}\) in \({\mathbb{R}^2}\) such that \({{\mathop{\rm x}\nolimits} _4} = {\mathop{\rm v}\nolimits} \).

[M] Let \(A = \left[ {\begin{array}{*{20}{c}}7&{ - 9}&{ - 4}&5&3&{ - 3}&{ - 7}\\{ - 4}&6&7&{ - 2}&{ - 6}&{ - 5}&5\\5&{ - 7}&{ - 6}&5&{ - 6}&2&8\\{ - 3}&5&8&{ - 1}&{ - 7}&{ - 4}&8\\6&{ - 8}&{ - 5}&4&4&9&3\end{array}} \right]\).

  1. Construct matrices \(C\) and \(N\) whose columns are bases for \({\mathop{\rm Col}\nolimits} A\) and \({\mathop{\rm Nul}\nolimits} A\), respectively, and construct a matrix \(R\) whose rows form a basis for Row\(A\).
  2. Construct a matrix \(M\) whose columns form a basis for \({\mathop{\rm Nul}\nolimits} {A^T}\), form the matrices \(S = \left[ {\begin{array}{*{20}{c}}{{R^T}}&N\end{array}} \right]\) and \(T = \left[ {\begin{array}{*{20}{c}}C&M\end{array}} \right]\), and explain why \(S\) and \(T\) should be square. Verify that both \(S\) and \(T\) are invertible.

If A is a \({\bf{6}} \times {\bf{8}}\) matrix, what is the smallest possible dimension of Null A?

In Exercise 18, Ais an \(m \times n\) matrix. Mark each statement True or False. Justify each answer.

18. a. If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A.

b. Row operations preserve the linear dependence relations among the rows of A.

c. The dimension of the null space of A is the number of columns of A that are not pivot columns.

d. The row space of \({A^T}\) is the same as the column space of A.

e. If A and B are row equivalent, then their row spaces are the same.
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