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1

### GATE EE 2000

Subjective
Consider the state equation $$\mathop X\limits^ \bullet \left( t \right) = Ax\left( t \right)$$
Given : $${e^{AT}} = \left[ {\matrix{ {{e^{ - t}} + t{e^{ - t}}} & {t{e^{ - t}}} \cr { - t{e^{ - t}}} & {{e^{ - t}} - t{e^{ - t}}} \cr } } \right]$$

(a) Find a set of states $${x_1}\left( 1 \right)$$ and $${x_2}\left( 1 \right)$$ such that $${x_1}\left( 2 \right) = 2.$$
(b) Show that $$\,{\left( {s{\rm I} - A} \right)^{ - t}} = \Phi \left( s \right) = {1 \over \Delta }\left[ {\matrix{ {s + 2} & 1 \cr { - 1} & s \cr } } \right];$$ $$\Delta = {\left( {s + 1} \right)^2}$$
(c) From $$\Phi \left( s \right),$$ find the matrix $$A$$.

(a)
\eqalign{ & {x_1}\left( 1 \right) = 0.74\,{x_1}\left( 0 \right) + 0.37\,{x_2}\left( 0 \right) \cr & {x_2}\left( 1 \right) = - 0.37\,{x_1}\left( 0 \right) \cr}
(c) $$A = \left[ {\matrix{ 0 & 1 \cr { - 1} & { - 2} \cr } } \right]$$
2

### GATE EE 1998

Subjective
The state-space representation of a system is given by $$\left[ {\matrix{ {\mathop {{X_1}}\limits^ \bullet } \cr {\mathop {{X_2}}\limits^ \bullet } \cr } } \right] = \left[ {\matrix{ { - 5} & 1 \cr { - 6} & 0 \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right].$$
Find the Laplace transform of the state transistion matrix. Find also the value of $${x_1}$$ at $$t=1$$ if $${x_1}\left( 0 \right) = 1$$ and $${x_2}\left( 0 \right) = 0.$$

$$L\left[ {{e^{AT}}} \right] = \left[ {\matrix{ {{s \over {\left( {s + 2} \right)\left( {s + 3} \right)}}} & {{{ + 1} \over {\left( {s + 2} \right)\left( {s + 3} \right)}}} \cr {{{ - 6} \over {\left( {s + 2} \right)\left( {s + 3} \right)}}} & {{{s + 5} \over {\left( {s + 2} \right)\left( {s + 3} \right)}}} \cr } } \right]$$
$${x_1}\left( t \right)$$ at $$t=1$$ sec is $$0.121$$
3

### GATE EE 1997

Subjective
Determine the transfer function of the system having the following state variable representation:
\eqalign{ & X = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr { - 40} & { - 44} & { - 14} \cr } } \right]x + \left[ {\matrix{ 0 \cr 1 \cr 0 \cr } } \right]u \cr & y = \left[ {\matrix{ 0 & 1 & 0 \cr } } \right]x \cr}

Transfer function $$= {{{s^2} + 14s} \over {{s^3} + 14{s^2} + 44s + 40}}$$
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#### Questions Asked from State Variable Analysis

On those following papers in Marks 5
Number in Brackets after Paper Indicates No. of Questions
GATE EE 2002 (1)
GATE EE 2000 (1)
GATE EE 1998 (1)
GATE EE 1997 (1)

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