Chapter 4, Problem 18SE

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

x_{k + 1} = Ax_k + Bu_k for k =0, 1,... (1)

where A is n × n, B is n × m, {x_k} is a sequence of “state vectors” in ℝⁿ that describe the state of the system at discrete times, and {u_k} is a control, or input, sequence. The pair (A, B) is said to be controllable if

$\text{rank}\left[\begin{array}{ccccc}B& AB& A^{2}B& \cdots & A^{n-1}B\end{array}\right]=n$ (2)

The matrix that appears in (2) is called the controllability matrix for the system. If (A, B) is controllable, then the system can be controlled, or driven from the state 0 to any specified state v (in ℝⁿ) in at most n steps, simply by choosing an appropriate control sequence in 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 web site (Case Study for Chapter 4).

18. Suppose A is a 4 × 4 matrix and B is a 4 × 2 matrix, and let u₀,...,u₃ represent a sequence of input vectors in ℝ².

1. a. Set x₀ = 0, compute x₁,..., x₄ from equation (1), and write a formula for x₄ 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 × 8.)
2. b. Suppose (A, B) is controllable and v is any vector in ℝ⁴. Explain why there exists a control sequence u₀,...,u₃ in ℝ² such that x₄ = v.