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Questions and Answers
What is the closed-loop transfer function for the given system?
What is the closed-loop transfer function for the given system?
What is the damping ratio ($\xi$) for the closed-loop system?
What is the damping ratio ($\xi$) for the closed-loop system?
What is the natural frequency ($\omega_n$) of the closed-loop system?
What is the natural frequency ($\omega_n$) of the closed-loop system?
What does the performance criterion P.O. represent?
What does the performance criterion P.O. represent?
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What does the ITAE performance criterion aim to minimize?
What does the ITAE performance criterion aim to minimize?
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Which parameter values will minimize the ITAE performance criterion for a step input?
Which parameter values will minimize the ITAE performance criterion for a step input?
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What is the closed-loop transfer function for the given system with a step input?
What is the closed-loop transfer function for the given system with a step input?
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What does the performance criterion P.O. represent?
What does the performance criterion P.O. represent?
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What is the damping ratio ($\xi$) for the closed-loop system?
What is the damping ratio ($\xi$) for the closed-loop system?
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What is the natural frequency ($\omega_n$) of the closed-loop system?
What is the natural frequency ($\omega_n$) of the closed-loop system?
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Study Notes
The Performance of Feedback Control Systems
- The steady-state error of a feedback control system is defined as E(s) = R(s) - Y(s) = [1 - T(s)]R(s), where R(s) is the input, Y(s) is the output, and T(s) is the transfer function of the system.
Steady-State Error
- The steady-state error is calculated by taking the limit of E(s) as s approaches 0: ess = lim(s→0) E(s) = 1 / (1 + K1G(0)), where K1 is the gain of the system and G(s) is the transfer function of the system.
Performance Indices
- A performance index is a quantitative measure of the performance of a system, chosen to emphasize important system specifications.
- A system is considered optimum when the system parameters are adjusted to minimize the performance index.
- Three common performance indices are:
- Integral of the Square of the Error (ISE): ∫e²(t)dt
- Integral of the Absolute Magnitude of the Error (IAE): ∫|e(t)|dt
- Integral of Time Multiplied by Absolute Error (ITAE): ∫t|e(t)|dt
Type of System Required for Different Inputs
- For a step input, a type 1 system (one integration) is required to achieve ess = 0.
- For a ramp input, a type 2 system is required.
Optimum Coefficients of T(s) Based on ITAE Criterion
- For a step input, the optimum coefficients of T(s) based on the ITAE criterion are: s + ωn s² + 1.4ωn s + ωn², where ωn is the natural frequency of the system.
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Description
Test your knowledge on the performance of feedback control systems and the calculation of steady-state error. Explore the concept of performance indices and their significance in evaluating system specifications.
