3)dx 2 5.S(8 x? +3

x? 2. dx x? +9

d 2r = 2 . 3x² . In 2 3. dx

This paragraph from the text defines the term "controllable". 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 xk+1 = Axk+ Buk for k = = 0, 1,... (1) where A is n xn, B is n xm, {x} is a sequence of "state vectors" in R" that describe the state of the system at discrete times, and {u} is a control, or input, sequence. The pair (A, B) is said to be controllable if rank B AB A² B ... A"-¹B] = 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 R") in at most n steps, simply by choosing an appropriate control sequence in R" Determine of the following matrix pair is controllable: [0.9 1.0 0 A = 0 -0.9 0 B = , 0 0 0.5]