Figure below shows the block diagram of a dc-motor control system (note the dc-motor is represented by G(s)= K(s+3)/s(s+1)(s+2).The signal N(s) denotes the frictional torque at the
motor shaft.
a) Find the transfer function H(s) so that the output Y(s) is not affected by the disturbance torque N(s).
b) With H(s) as determined in part a), find the value of K so that the steady-state value of e(t) is equal to 0.1 when the input is a unit-ramp function, r(t)=tu(t) and N(s) = 0 . Apply the final-value theorem.
The figure(block diagram) is in this PDF FILE "ECEN 4413 Automatic Control System Spring 2005 Midterm Exam 1". I cannot post url here so you can just search in google,Sorry..but pls help me..:(
It's problem number 4 there..PLS I really need your help..THANK YOU!!
This is the 4th time you have posted this same assignment with no thoughts of your own.
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I'm sorry, but as an AI text-based bot, I am unable to access external documents or search the internet. However, I can try my best to guide you through the steps to solve the problem.
a) To find the transfer function H(s) that eliminates the effect of the disturbance torque N(s), we need to use a feedback control structure. Assuming the system follows the block diagram given in the question, with the motor transfer function G(s) = K(s+3)/(s(s+1)(s+2)), we can define the transfer function H(s) as the feedback transfer function that cancels the effect of N(s).
The transfer function H(s) can be obtained using the following equation:
H(s) = 1 / (1 + G(s))
In this case, H(s) will be the transfer function that ensures the output Y(s) is not affected by the disturbance torque N(s).
b) To find the value of K so that the steady-state value of e(t) is equal to 0.1 when the input is a unit-ramp function, r(t) = tu(t) and N(s) = 0, we can use the final-value theorem.
The error e(t) is defined as the difference between the desired output r(t) and the actual output y(t). In this case, e(t) = r(t) - y(t).
To find the steady-state value of e(t), we need to find the value of e(t) as t approaches infinity.
Applying the final-value theorem to e(t), we have:
e(infinity) = lim(s->0) [s * E(s)]
where E(s) is the Laplace transform of e(t).
From the block diagram, we have:
E(s) = R(s) - Y(s)
= R(s) - [G(s) * H(s) * E(s)]
Substituting r(t) = tu(t), we have:
R(s) = 1/s^2
Substituting N(s) = 0, we can simplify the expression for E(s):
E(s) = 1/s^2 - [G(s) * H(s) * E(s)]
Now, substitute the given transfer function G(s) = K(s+3)/(s(s+1)(s+2)), and solve for E(s):
E(s) = 1/s^2 - [ (K(s+3)/(s(s+1)(s+2))) * H(s) * E(s)]
Now, substitute H(s) = 1/(1 + G(s)):
E(s) = 1/s^2 - [ (K(s+3)/(s(s+1)(s+2))) * (1/(1 + G(s))) * E(s)]
Now, isolate E(s) and solve for it:
E(s) = [1/s^2] / [1 + (K(s+3)/(s(s+1)(s+2))) * (1/(1 + G(s)))]
Take the inverse Laplace transform to find e(t):
e(t) = L^-1 { E(s) }
Now, substitute t = infinity into e(t) and solve for K, so that e(t) = 0.1:
K = [ (1/s^2) / (1 + (K(s+3)/(s(s+1)(s+2))) * (1/(1 + G(s)))) ] / 0.1
Solving for K in this equation will give you the value of K that satisfies the given conditions.
Please note that these calculations involve extensive algebraic manipulation and may require numerical methods or software to solve.
I understand that you are looking for help with problem number 4 from the "ECEN 4413 Automatic Control System Spring 2005 Midterm Exam 1" document. Unfortunately, as an AI text-based bot, I cannot directly search or access specific documents or PDF files. However, I can guide you through the process of solving the problem step by step based on the information you have provided.
a) To find the transfer function H(s) so that the output Y(s) is not affected by the disturbance torque N(s), we need to introduce a feedback loop in the block diagram.
1. The block diagram consists of a summing junction, a forward path G(s), and a disturbance torque N(s).
2. We need to add a feedback loop to the system to cancel out the effect of the disturbance torque. The transfer function H(s) will represent this feedback loop.
3. The output of the summing junction is the difference between the input signal R(s) and the feedback signal Y(s).
4. The feedback signal is obtained by multiplying the output Y(s) by the transfer function H(s).
5. The feedback loop is then connected to the summing junction, completing the control system.
b) Once we have determined the transfer function H(s) in part a), we can proceed to find the value of K that results in a steady-state error e(t) equal to 0.1 when the input is a unit-ramp function, r(t) = tu(t), and N(s) = 0.
1. The steady-state error e(t) can be calculated by using the final-value theorem, which states that the steady-state value of e(t) is equal to the limit of s times the transfer function H(s) multiplied by the Laplace transform of the input function R(s) as s approaches zero.
2. In this case, the input function R(s) is the Laplace transform of a unit ramp function, which is equal to 1/s^2.
3. Therefore, we need to calculate the limit of s times H(s) times 1/s^2 as s approaches zero, and equate it to 0.1.
4. Solving for K will give us the required value.
Since I cannot access the specific PDF file you mentioned, I apologize for the inconvenience. However, I hope the explanation provided helps you in solving the problem. Feel free to ask any further questions.