0 1. Consider an autonomous system (t) = [ O (a) Determine the eigenvalues and eigenvectors...

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0 1. Consider an autonomous system à(t) = [ O (a) Determine the eigenvalues and eigenvectors of A. (b) Show that the state transition matrix of the system is eºt [ 2. Consider the *₁ (t) *₂ (t) *3 (t) = 1 0 › system i(t) = [ ₁ ] z(1) + [ à ] u(t) x(t) (a) Compute the controllability gramian as a function of a where t is the last digit of your student number. If the last digit of your student number is 0, you will take t as 0.5. (b) For which values of a, is the system controllable? (Use controllability gramian) 3. The dynamical equations of a system are given as = لیا = r(t) where a and w are two real numbers. cos(wt) sin(wt) x₂(t)- x₁(t) + u(t) x₁ (t) + x3 (t) 2 x2(t) x₂ (t)- x3(t) (a) Calculate the state transition matrix of the system. (b) Analyze the stability and controllability of the system. 0 1. Consider an autonomous system à(t) = [ O (a) Determine the eigenvalues and eigenvectors of A. (b) Show that the state transition matrix of the system is eºt [ 2. Consider the *₁ (t) *₂ (t) *3 (t) = 1 0 › system i(t) = [ ₁ ] z(1) + [ à ] u(t) x(t) (a) Compute the controllability gramian as a function of a where t is the last digit of your student number. If the last digit of your student number is 0, you will take t as 0.5. (b) For which values of a, is the system controllable? (Use controllability gramian) 3. The dynamical equations of a system are given as = لیا = r(t) where a and w are two real numbers. cos(wt) sin(wt) x₂(t)- x₁(t) + u(t) x₁ (t) + x3 (t) 2 x2(t) x₂ (t)- x3(t) (a) Calculate the state transition matrix of the system. (b) Analyze the stability and controllability of the system.