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Question

# In the signal flow diagram given in the figure, u1 and u2 are possible inputs whereas y1 and y2 are possible outputs. When would the SISO system derived from this diagram, be controllable and observable?

A
When u1 is the only input and y1 is the only output.
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B
When u2 is the only input and y1 is the only output.
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C
When u1 is the only input and y2 is the only output.
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D
When u2 is the only input and y2 is the only output.
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Solution

## The correct option is B When u2 is the only input and y1 is the only output.Equations from flow diagram, ˙x1=5x1−2x2+u1;y1=x1 ˙x2=2x1+x2+u1+u2;y2=x1−x2 ˙x=[5−221]Ax+[1011]Bu;y=[101−1]Cx Considering the SISO cases: 1. u1 (input) and y1 (output): A del=[5−221]; B del=[11] and C = [1 0] O del[CCA]=[105−2] ⇒Rank(O)≠2⇒non−controllable 2. u2 (input) and y1 (output): A del[5−221]; B del=[01]and C del=[1−1] O del=[105−2] ⇒ Rank (O) = 2 ⇒ Observable C del=[0−211] ⇒ Rank(C) = 2 ⇒ controllable 3. u2(input) and y2 (output): Rank(O)≠2⇒non−observable Rank(C)=2|⇒controllable 4. u1 (input) and y2 (output): Neither observable and non-controllable

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