Define the system type. C(s) = K(s+1) s +0.1 G(s) = s(s+ 3) a unit feedback Determine the position-velocity-acceleration error coefficients in the control system. steady state error Find for step, ramp and parabola input. controller and serial to it
Define the system type. C(s) = K(s+1) s +0.1 G(s) = s(s+ 3) a unit feedback Determine the position-velocity-acceleration error coefficients in the control system. steady state error Find for step, ramp and parabola input. controller and serial to it
Introductory Circuit Analysis (13th Edition)
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ISBN:9780133923605
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Publisher:Robert L. Boylestad
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![### Determining the System Type in Control Theory
To define the type of a given control system, we must examine its components and feedback loop. Let's consider the following system:
#### Controller and System:
\[ C(s) = \frac{K(s + 1)}{s + 0.1} \]
\[ G(s) = \frac{1}{s(s + 3)} \]
where:
- \( C(s) \) represents the transfer function of the controller.
- \( G(s) \) represents the transfer function of the plant (system to be controlled).
### Unit Feedback System:
We assume a unit feedback for the system, meaning the feedback loop has a transfer function of 1.
### Error Coefficients:
For a control system, the position, velocity, and acceleration error coefficients are fundamental in determining the steady-state errors for different types of inputs (step, ramp, and parabolic).
#### Position Error Coefficient (\( K_p \)):
\[ K_p = \lim_{{s \to 0}} G(s) \cdot C(s) \]
#### Velocity Error Coefficient (\( K_v \)):
\[ K_v = \lim_{{s \to 0}} s \cdot G(s) \cdot C(s) \]
#### Acceleration Error Coefficient (\( K_a \)):
\[ K_a = \lim_{{s \to 0}} s^2 \cdot G(s) \cdot C(s) \]
### Steady-State Errors:
The steady-state error for different types of inputs are calculated as follows:
#### For Step Input:
\[ e_{ss} = \frac{1}{1 + K_p} \]
#### For Ramp Input:
\[ e_{ss} = \frac{1}{K_v} \]
#### For Parabolic Input:
\[ e_{ss} = \frac{1}{K_a} \]
### Conclusion:
Determine the system type by calculating these error coefficients and analyze the steady-state error for the given step, ramp, and parabolic inputs. This systematic approach allows us to understand the behavior and accuracy of the control system.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d6fa171-8a96-4a91-bca7-267628414860%2Fb6768b7f-adda-4ba3-b625-248e5726f66e%2Fszc4xbj_processed.png&w=3840&q=75)
Transcribed Image Text:### Determining the System Type in Control Theory
To define the type of a given control system, we must examine its components and feedback loop. Let's consider the following system:
#### Controller and System:
\[ C(s) = \frac{K(s + 1)}{s + 0.1} \]
\[ G(s) = \frac{1}{s(s + 3)} \]
where:
- \( C(s) \) represents the transfer function of the controller.
- \( G(s) \) represents the transfer function of the plant (system to be controlled).
### Unit Feedback System:
We assume a unit feedback for the system, meaning the feedback loop has a transfer function of 1.
### Error Coefficients:
For a control system, the position, velocity, and acceleration error coefficients are fundamental in determining the steady-state errors for different types of inputs (step, ramp, and parabolic).
#### Position Error Coefficient (\( K_p \)):
\[ K_p = \lim_{{s \to 0}} G(s) \cdot C(s) \]
#### Velocity Error Coefficient (\( K_v \)):
\[ K_v = \lim_{{s \to 0}} s \cdot G(s) \cdot C(s) \]
#### Acceleration Error Coefficient (\( K_a \)):
\[ K_a = \lim_{{s \to 0}} s^2 \cdot G(s) \cdot C(s) \]
### Steady-State Errors:
The steady-state error for different types of inputs are calculated as follows:
#### For Step Input:
\[ e_{ss} = \frac{1}{1 + K_p} \]
#### For Ramp Input:
\[ e_{ss} = \frac{1}{K_v} \]
#### For Parabolic Input:
\[ e_{ss} = \frac{1}{K_a} \]
### Conclusion:
Determine the system type by calculating these error coefficients and analyze the steady-state error for the given step, ramp, and parabolic inputs. This systematic approach allows us to understand the behavior and accuracy of the control system.
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