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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

Exercises 23-26 concern a vector space V, a basis \(B = \left\{ {{{\bf{b}}_{\bf{1}}},....,{{\bf{b}}_n}\,} \right\}\) and the coordinate mapping \({\bf{x}} \mapsto {\left( {\bf{x}} \right)_B}\).

Show that the coordinate mapping is onto \({\mathbb{R}^n}\). That is, given any y in \({\mathbb{R}^n}\), with entries \({y_{\bf{1}}}\),….,\({y_n}\), produce u in V such that \({\left( {\bf{u}} \right)_B} = y\).

If the null space of A \({\bf{7}} \times {\bf{6}}\) matrix A is 4-dimensional, what is the dimension of the column space of A?

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

17. A submatrix of a matrix A is any matrix that results from deleting some (or no) rows and/or columns of A. It can be shown that A has rank \(r\) if and only if A contains an invertible \(r \times r\) submatrix and no longer square submatrix is invertible. Demonstrate part of this statement by explaining (a) why an \(m \times n\) matrix A of rank \(r\) has an \(m \times r\) submatrix \({A_1}\) of rank \(r\), and (b) why \({A_1}\) has an invertible \(r \times r\) submatrix \({A_2}\).

The concept of rank plays an important role in the design of engineering control systems, such as the space shuttle system mentioned in this chapter’s introductory example. A state-space model of a control system includes a difference equation of the form

\({{\mathop{\rm x}\nolimits} _{k + 1}} = A{{\mathop{\rm x}\nolimits} _k} + B{{\mathop{\rm u}\nolimits} _k}\)for \(k = 0,1,....\) (1)

Where \(A\) is \(n \times n\), \(B\) is \(n \times m\), \(\left\{ {{{\mathop{\rm x}\nolimits} _k}} \right\}\) is a sequence of “state vectors” in \({\mathbb{R}^n}\) that describe the state of the system at discrete times, and \(\left\{ {{{\mathop{\rm u}\nolimits} _k}} \right\}\) is a control, or input, sequence. The pair \(\left( {A,B} \right)\) is said to be controllable if

\({\mathop{\rm rank}\nolimits} \left( {\begin{array}{*{20}{c}}B&{AB}&{{A^2}B}& \cdots &{{A^{n - 1}}B}\end{array}} \right) = n\) (2)

The matrix that appears in (2) is called the controllability matrix for the system. If \(\left( {A,B} \right)\) is controllable, then the system can be controlled, or driven from the state 0 to any specified state \({\mathop{\rm v}\nolimits} \) (in \({\mathbb{R}^n}\)) in at most \(n\) steps, simply by choosing an appropriate control sequence in \({\mathbb{R}^m}\). This fact is illustrated in Exercise 18 for \(n = 4\) and \(m = 2\). For a further discussion of controllability, see this text’s website (Case study for Chapter 4).

A homogeneous system of twelve linear equations in eight unknowns has two fixed solutions that are not multiples of each other, and all other solutions are linear combinations of these two solutions. Can the set of all solutions be described with fewer than twelve homogeneous linear equations? If so, how many? Discuss.

In Exercise 5, find the coordinate vector \({\left( x \right)_{\rm B}}\) of x relative to the given basis \({\rm B} = \left\{ {{b_{\bf{1}}},...,{b_n}} \right\}\).

5. \({b_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{3}}}\end{array}} \right),{b_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{ - {\bf{5}}}\end{array}} \right),x = \left( {\begin{array}{*{20}{c}}{ - {\bf{2}}}\\{\bf{1}}\end{array}} \right)\)
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