Feedback Control Systems and Performance Indices Quiz

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