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The differential equation of a control system having input x(t)
and output y(t)
is given as
d2y(t)dt2+9dy(t)dt+20y(t)=20x(t)
The output response for the system for unit step input is given by (y(0)=0)
−1+5e−4t−4e−5t.
1−e−4t−e−5t
1−5e−4t+4e−5t.
None of these?
and output y(t)
is given as
d2y(t)dt2+9dy(t)dt+20y(t)=20x(t)
The output response for the system for unit step input is given by (y(0)=0)
−1+5e−4t−4e−5t.
1−e−4t−e−5t
1−5e−4t+4e−5t.
None of these?
Most Upvoted Answer
The differential equation of a control system having input x(t) and ou...
Analysis:
To find the output response of the system for a unit step input, we need to solve the given differential equation. Given the output response is y(0) = 0, we can proceed to calculate the output response using the given differential equation and initial conditions.
Solution:
1. Characteristic Equation:
The characteristic equation of the given differential equation is:
s^2 + 9s + 20 = 0
(s + 4)(s + 5) = 0
s = -4, -5
2. Homogeneous Solution:
The homogeneous solution is given by:
y_h(t) = c1e^(-4t) + c2e^(-5t)
3. Particular Solution:
For a unit step input x(t) = 1, the particular solution is a constant.
y_p(t) = k
4. Overall Solution:
The overall solution is the sum of homogeneous and particular solutions:
y(t) = y_h(t) + y_p(t)
y(t) = c1e^(-4t) + c2e^(-5t) + k
5. Applying Initial Condition:
Given y(0) = 0, we have:
0 = c1 + c2 + k
From the given output response, we can find c1, c2, and k.
6. Final Output Response:
After calculating the constants c1, c2, and k from the given output response, we can determine the final output response for a unit step input.
Therefore, none of the provided options matches the output response for the system with the given differential equation and initial condition.
To find the output response of the system for a unit step input, we need to solve the given differential equation. Given the output response is y(0) = 0, we can proceed to calculate the output response using the given differential equation and initial conditions.
Solution:
1. Characteristic Equation:
The characteristic equation of the given differential equation is:
s^2 + 9s + 20 = 0
(s + 4)(s + 5) = 0
s = -4, -5
2. Homogeneous Solution:
The homogeneous solution is given by:
y_h(t) = c1e^(-4t) + c2e^(-5t)
3. Particular Solution:
For a unit step input x(t) = 1, the particular solution is a constant.
y_p(t) = k
4. Overall Solution:
The overall solution is the sum of homogeneous and particular solutions:
y(t) = y_h(t) + y_p(t)
y(t) = c1e^(-4t) + c2e^(-5t) + k
5. Applying Initial Condition:
Given y(0) = 0, we have:
0 = c1 + c2 + k
From the given output response, we can find c1, c2, and k.
6. Final Output Response:
After calculating the constants c1, c2, and k from the given output response, we can determine the final output response for a unit step input.
Therefore, none of the provided options matches the output response for the system with the given differential equation and initial condition.
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The differential equation of a control system having input x(t) and output y(t) is given asd2y(t)dt2+9dy(t)dt+20y(t)=20x(t)The output response for the system for unit step input is given by (y(0)=0)−1+5e−4t−4e−5t.1−e−4t−e−5t1−5e−4t+4e−5t. None of these?
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