Consider the following linear time-invariant system: 1 b x = ( )x + (b) = (...

   

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Consider the following linear time-invariant system: 1 b₁ x = ( )x + (b) = ( 6 ) x x= x+ y= 4.1.1 What conditions on b₁ and b₂ make the system controllable? 4.1.2 What conditions on c₁ and c2 make the system observable? 4.1.3 Verify your calculations in MATLAB using the "ctrb" and "obsv" commands. 4.2 Consider the following linear time invariant system: x=1 1 1 1 [1 1 10x+0-1u y = [1_0_0]x 201 0 2 4.2.1 Is the system controllable? Show it by finding the controllability index of the controllability matrix. Compare to MATLAB using the "ctrb" command. 4.2.2 If not fully controllable derive the controllability canonical decomposition form and compare to what can be obtained using the MATLAB "ctbrf" command. Is the system stable? Is the system stabilizable? 4.2.3 Is the system observable? Find the observability matrix and its rank and compare it to MATLAB calculation using the "obsv" command. 4.2.4 If the system is not fully observable find the observability canonical decomposition form and compare it to the model and transformations obtained using MATLAB "obsvf" command. Consider the following linear time-invariant system: 1 b₁ x = ( )x + (b) = ( 6 ) x x= x+ y= 4.1.1 What conditions on b₁ and b₂ make the system controllable? 4.1.2 What conditions on c₁ and c2 make the system observable? 4.1.3 Verify your calculations in MATLAB using the "ctrb" and "obsv" commands. 4.2 Consider the following linear time invariant system: x=1 1 1 1 [1 1 10x+0-1u y = [1_0_0]x 201 0 2 4.2.1 Is the system controllable? Show it by finding the controllability index of the controllability matrix. Compare to MATLAB using the "ctrb" command. 4.2.2 If not fully controllable derive the controllability canonical decomposition form and compare to what can be obtained using the MATLAB "ctbrf" command. Is the system stable? Is the system stabilizable? 4.2.3 Is the system observable? Find the observability matrix and its rank and compare it to MATLAB calculation using the "obsv" command. 4.2.4 If the system is not fully observable find the observability canonical decomposition form and compare it to the model and transformations obtained using MATLAB "obsvf" command.