12 Questions
What are the coefficients that will minimize the ITAE performance criterion for a step input in the general closed-loop transfer function?
b0, b1, ..., bn-1
What is the closed-loop transfer function for the given system?
b0 Y (s) T ( s) = R ( s) s + bn - 1 s n - 1 + ... + b1s + b0
What is the natural frequency of the closed-loop system?
110
What is the damping ratio of the closed-loop system?
7
How do you calculate the percent overshoot (P.O.) of a second-order system?
P.O. = 100 * e^(-πξ) / (1-ξ^2)
What is the closed-loop transfer function for the given system?
100/(s+2)(s+5) + 100
What is the percent overshoot (P.O.) of the second-order system?
29%
What are the coefficients that will minimize the ITAE performance criterion for a step input in the general closed-loop transfer function?
b0, b1, ..., bn-1
What is the natural frequency of the closed-loop system?
110
How do you calculate the damping ratio of the closed-loop system?
By finding the roots of the characteristic equation of the closed-loop transfer function
What is the percent overshoot (P.O.) of the second-order system?
0.909
What is the natural frequency of the closed-loop system in radians per second?
110
Study Notes
The Performance of Feedback Control Systems
- The steady-state error of a feedback control system is determined by the equation: E(s) = R(s) - Y(s) = [1 - T(s)]R(s)
- The system's transfer function, T(s), is given by: T(s) = K1G(s) / [1 + K1G(s)H(s)]
- The steady-state error, ess, is calculated as: ess = lim(s→0) E(s) = 1 / [1 + K1G(0)]
Performance Indices
- A performance index is a quantitative measure of a system's performance, chosen to emphasize important system specifications
- A system is considered optimal when its parameters are adjusted to reach an extremum value (usually a minimum) of the performance index
- The performance index must be a positive or zero value
Types of Performance Indices
- T ISE (Integral of the Square of the Error): ∫e²(t) dt from 0 to T
- T IAE (Integral of the Absolute Magnitude of the Error): ∫|e(t)| dt from 0 to T
- T ITAE (Integral of Time multiplied by Absolute Error): ∫t|e(t)| dt from 0 to T
Optimum Coefficients of T(s) Based on the ITAE Criterion
- For a step input, the optimum coefficients of T(s) are: s + ωn, s² + 1.4ωn s + ωn², s³ + 1.75ωn s² + 2
- To get ess = 0, a type 1 system is required for a step input, and a type 2 system is required for a ramp input
Test your knowledge of feedback control systems and performance indices with this quiz. Explore topics such as steady-state error, performance measures, and system specifications.
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