Question

Given the system below: dot{mathbf{x}}(t) = egin{bmatrix} -7 & -2 & 6 \ 2 & -3 & -2 \ -2 & -2 & 1 end{bmatrix} cdot mathbf{x}(t) + egin{bmatrix} 1 & 1 \ 1 & -1 \ 1 & 0 end{bmatrix} cdot mathbf{u}(t) mathbf{y}(t) = egin{bmatrix} -1 & -1 & 2 \ 1 & 1 & -1 end{bmatrix} cdot mathbf{x}(t) (A) (5 Points) Comment on the controllability and observability of the system (B) (10 Points) Perform controllable and observable canonical decompositions and identify the uncontrollable and unobservable modes as appropriate. (C) (10 Points) Diagonalize the system and identify the four subsystems associated with a Kalman decomposition as appropriate, i.e. controllable and observable, controllable and unobservable, uncontrollable and observable, and uncontrollable and unobservable.

          Given the system below:
dot{mathbf{x}}(t) = egin{bmatrix} -7 & -2 & 6 \ 2 & -3 & -2 \ -2 & -2 & 1 end{bmatrix} cdot mathbf{x}(t) + egin{bmatrix} 1 & 1 \ 1 & -1 \ 1 & 0 end{bmatrix} cdot mathbf{u}(t)
mathbf{y}(t) = egin{bmatrix} -1 & -1 & 2 \ 1 & 1 & -1 end{bmatrix} cdot mathbf{x}(t)
(A) (5 Points) Comment on the controllability and observability of the system
(B) (10 Points) Perform controllable and observable canonical decompositions and identify the uncontrollable and unobservable modes as appropriate.
(C) (10 Points) Diagonalize the system and identify the four subsystems associated with a Kalman decomposition as appropriate, i.e. controllable and observable, controllable and unobservable, uncontrollable and observable, and uncontrollable and unobservable.

Given the system below:
dotmathbfx(t) = eginbmatrix -7 -2 6 2 -3 -2 -2 -2 1 endbmatrix cdot mathbfx(t) + eginbmatrix 1 1 1 -1 1 0 endbmatrix cdot mathbfu(t)
mathbfy(t) = eginbmatrix -1 -1 2 1 1 -1 endbmatrix cdot mathbfx(t)
(A) (5 Points) Comment on the controllability and observability of the system
(B) (10 Points) Perform controllable and observable canonical decompositions and identify the uncontrollable and unobservable modes as appropriate.
(C) (10 Points) Diagonalize the system and identify the four subsystems associated with a Kalman decomposition as appropriate, i.e. controllable and observable, controllable and unobservable, uncontrollable and observable, and uncontrollable and unobservable.

Added by Emily J.

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