Question

1. Consider the system described by: *(t) - 6 m (0) + veu(t): y(t) = 01 (1) 60 = {1, 1421 a) Find the state transition matrix and the impulse response matrix of the system. b) Determine whether the system is (i) completely state controllable, (ii) differentially control- lable, (iii) instantaneously controllable, (iv) stabilizable at time to = 0. c) Repeat part (b) for to = 1. d) Determine whether the system is (i) observable, (ii) differentially observable, (iii) instanta- neously observable, (iv) detectable at time to = 0. e) Find the (i) controllable subspace, (ii) observable subspace of the system at time to = 0. f) Given y(2) = y(2) = 1 and ult) = 2,0 St <2, is it possible to find (0)? If so, find it. If not, explain why. g) Given y(2) = y(2) = 1 and u(t) = 2,0 <t<2, find the orthogonal projection of x(0) on the observable subspace of the system at time to = 0. h) Is it possible to drive the system from c(0) = 0 to (ti) = for some finite tı > 0? If -1 so, find the minimum possible tı > 0 and the input uſt), 0 <t<ti, which drives the system from r(0) to (ti). If not, explain why. Hint: note that the input does not have any effect on the system for 0 <t<1 and recall your answer to part (iii) of part (c). j) Find the input u(t) which drives the system from x(0) = 0 to x(2) = , while minimizing t)u(t)dt. Hint: see Appendix C in your text book.

Answer 1

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