## 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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