GATE Electronics and Communications (EC) 2019 Solved Paper

Question : 53 of 65

Marks: +1, -0

Let the state-space representation of an LTI system be

˙{x} t A x(t)+B u(t), y( {t})

= {C} {x}( {t})+ {d} {u}( {t})

where A, B, C are matrices, d is a scalar, u(t) is the input to the system, and

y(t)

is its output. Let

{B}=

[\table 0 , 0 , 1]^{⊤}

and

{d}=0 .

Which one of the following options for

A

and

C

will ensure that the transfer function of this LTI system is

H(s)=\;{1}/{s^{3}+3 s^{2}+2 s+1} ?

Solution:

{H}( {s})=\;{1}/{ {s}^{2}+3 {s}^{2}+2 {s}+1}

∴ [\table ˙{ {x}}_{1} , 0 , 1 , 0 ; ˙{ {x}}_{2} , 0 , 0 , 1 ; ˙{ {x}}_{3} , -1 , -2 , -3][\table {x}_{1} ; {x}_{2} ; {x}_{3}]+[\table 0 ; 0 ; 1][ {u}]

\&[ {y}]=[\table 1 , 0 , 0][\table {x}_{1} ; {x}_{2} ; {x}_{3}]+[0][ {u}]

∴ A=[\table 0 , 1 , 0 ; 0 , 0 , 1 ; -1 , -2 , -3]

and

C=[\table 1 , 0 , 0]

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