Let a causal LTI system be characterized by the following differential equation- with initial rest condition-d2ydt2-7dydt-10yt-4xt-5dxtdtWhere- xt and yt are the input and output respectively- The impulse response of the system is -ut is the unit step function-

D^2ydt^2+7dydt+10y=4x+5dxdt (s^2+7s+10) Y(s)=(4+5s)X(s) Y(s)X(s)=5s+4s^2+7s+10 Impulse response = L^ 1(Transfer function) =L^ 1 [ 5s+4(s+2)(s+5) ] =L^ 1 [ ...

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Control Systems

Impulse Response-2

Let a causal ...

Question

Let a causal LTI system be characterized by the following differential equation, with initial rest condition.
$\frac{d^{2} y}{d t^{2}} + 7 \frac{d y}{d t} + 10 y (t) = 4 x (t) + 5 \frac{d x (t)}{d t}$

Where, x(t) and y(t) are the input and output respectively. The impulse response of the system is [u(t) is the unit step function]

2 e^{- 2 t} u (t) - 7 e^{- 5 t} u (t)

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- 2 e^{- 2 t} u (t) + 7 e^{- 5 t} u (t)

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7 e^{- 2 t} u (t) - 2 e^{- 5 t} u (t)

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- 7 e^{- 2 t} u (t) + 2 e^{- 5 t} u (t)

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Solution

The correct option is B $- 2 e^{- 2 t} u (t) + 7 e^{- 5 t} u (t)$
$\frac{d^{2} y}{d t^{2}} + 7 \frac{d y}{d t} + 10 y = 4 x + 5 \frac{d x}{d t}$

$(s^{2} + 7 s + 10) Y (s) = (4 + 5 s) X (s)$

$\frac{Y (s)}{X (s)} = \frac{5 s + 4}{s^{2} + 7 s + 10}$

Impulse response = $L^{- 1}$ (Transfer function)

$= L^{- 1} [\frac{5 s + 4}{(s + 2) (s + 5)}]$

$= L^{- 1} [- \frac{2}{s + 2} + \frac{7}{s + 5}]$

$= - 2 e^{- 2 t} u (t) + 7 e^{5 t} u (t)$

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Let a causal LTI system be characterized by the following differential equation, with initial rest condition. d2ydt2+7dydt+10y(t)=4x(t)+5dx(t)dt Where, x(t) and y(t) are the input and output respectively. The impulse response of the system is [u(t) is the unit step function]

The correct option is B −2e−2tu(t)+7e−5tu(t)d2ydt2+7dydt+10y=4x+5dxdt (s2+7s+10)Y(s)=(4+5s)X(s) Y(s)X(s)=5s+4s2+7s+10 Impulse response = L−1(Transfer function) =L−1[5s+4(s+2)(s+5)] =L−1[−2s+2+7s+5] =−2e−2tu(t)+7e5tu(t)