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6. From Antsaklis & Michel. A Linear Systems Primer. Exercise: 5.4. Consider the state equation, x[k + 1] = egin{bmatrix} 1 & 1 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 end{bmatrix} x[k] + egin{bmatrix} 0 \ 1 \ 1 end{bmatrix} u[k], quad y[k] = egin{bmatrix} 1 & 1 & 0 \ 0 & 1 & 0 end{bmatrix} x[k]. (a) Is x^1 = egin{bmatrix} 3 \ 2 \ 2 end{bmatrix} reachable? If yes, what is the minimum number of steps required to transfer the state from the zero state to x^1? What inputs do you need? (b) Determine all states that are reachable. (c) Determine all states that are unobservable. (d) Optional - Extra Credit, 2 extra points!! If dot{x} = Ax + Bu is given with (A, B) as in (a), what is the minimum time required to transfer the state from the zero state to x^1? What is an appropriate u(t)?

          6. From Antsaklis & Michel. A Linear Systems Primer. Exercise: 5.4. Consider the state equation,
x[k + 1] = egin{bmatrix} 1 & 1 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 end{bmatrix} x[k] + egin{bmatrix} 0 \ 1 \ 1 end{bmatrix} u[k], quad y[k] = egin{bmatrix} 1 & 1 & 0 \ 0 & 1 & 0 end{bmatrix} x[k].
(a) Is x^1 = egin{bmatrix} 3 \ 2 \ 2 end{bmatrix} reachable? If yes, what is the minimum number of steps required to transfer the state from the zero state to x^1? What inputs do you need?
(b) Determine all states that are reachable.
(c) Determine all states that are unobservable.
(d) Optional - Extra Credit, 2 extra points!! If dot{x} = Ax + Bu is given with (A, B) as in (a), what is the minimum time required to transfer the state from the zero state to x^1? What is an appropriate u(t)?

6. From Antsaklis Michel. A Linear Systems Primer. Exercise: 5.4. Consider the state equation,
x[k + 1] = eginbmatrix 1 1 0 0 1 0 0 0 1 endbmatrix x[k] + eginbmatrix 0 1 1 endbmatrix u[k], quad y[k] = eginbmatrix 1 1 0 0 1 0 endbmatrix x[k].
(a) Is x^1 = eginbmatrix 3 2 2 endbmatrix reachable? If yes, what is the minimum number of steps required to transfer the state from the zero state to x^1? What inputs do you need?
(b) Determine all states that are reachable.
(c) Determine all states that are unobservable.
(d) Optional - Extra Credit, 2 extra points!! If dotx = Ax + Bu is given with (A, B) as in (a), what is the minimum time required to transfer the state from the zero state to x^1? What is an appropriate u(t)?

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