Sheridan College Smart Factory and Intelligent Controls Lecture Notes PDF

Sheridan College Smart Factory and Intelligent Controls  Lecture 7  Second-order Systems  Underdamped Transient-Response Specifications of Second Order Systems  Feedback Control Systems First-order Systems k First-order System τ  Step Response of...

Sheridan College Smart Factory and Intelligent Controls  Lecture 7  Second-order Systems  Underdamped Transient-Response Specifications of Second Order Systems  Feedback Control Systems First-order Systems k First-order System τ  Step Response of First-order System Example: If K=10 and τ =1.5s then Second-order Systems Second Order Systems 𝝎𝝎𝟐𝟐𝒏𝒏 𝒔𝒔𝟐𝟐 + 𝟐𝟐𝟐𝟐𝝎𝝎𝒏𝒏 𝒔𝒔 + 𝝎𝝎𝟐𝟐𝒏𝒏 Second Order Systems Second Order System Response to Unit-Step Input Figure below: response c(t) plotted against time for various values of the damping ratio ζ  Second Order Systems Damping characteristics of second order systems Second Order Systems Second Order Systems Second Order Systems Second Order Systems Second Order Systems Transient-Response Specifications  Time Response of Control Systems Time response of a dynamic system response to an input is expressed as a function of time. Input System Output The time response of any system has two components  Transient response  Steady-state response  System Transient-Response When the systems are subjected to inputs or disturbances, they cannot respond instantly, and they exhibit a transient response. The desired performance characteristics of control systems can be given in terms of transient response specifications. Consequently, the transient response characteristics make one of the most important factors in system design. Such performance characteristics are usually specified in terms of the transient response to unit-step input, u(t), since such an input is easy to generate.  Transient Response-Specifications Transient response requirements o Fast o Limited overshoot o Reach and remain close to the desired reference value in the minimum time possible Steady state requirements o Stability (the response stays bounded over time) o Small steady state error  Steady-State Error The difference between the set point and the measured value is called steady- state error or offset.  Steady-State Error The difference between the set point and the measured value is called steady- state error or offset. Underdamped Transient-Response Specifications of Second Order Systems  Underdamped Transient-Response Specifications of Second Order Systems 1. Delay time td 2. Rise time tr 3. Peak time tp 4. Settling time ts 5. Maximum overshoot Mp  Underdamped Transient-Response Specifications of Second Order Systems Delay Time. The delay time Td is the time needed for the response to reach half of its final value the very first time. Rise Time. The rise time Tr is the time required for the response to rise from 10% to 90% of its final value. Peak Time. The peak time Tp is the time required for the response to reach the first peak of the overshoot. Underdamped Transient-Response Specifications of Second Order Systems Settling Time. The settling time Ts is the time required for the response curve to reach and stay within 2% of the final value.  Underdamped Transient-Response Specifications of Second Order Systems Maximum (percent Overshoot). The maximum percent overshoot Mp is the maximum peak value of the response curve measured from the final value. cmax − cfinal M p% = × 100% cfinal Underdamped Transient-Response Specifications of Second Order Systems The lower the value of the damping ratio, the more oscillatory the response. Underdamped Transient-Response Specifications of Second Order Systems Comment: If we specify the values of Td , Tr , Tp , Ts and Mp , the shape of the response curve is virtually fixed as shown in the figure.  Underdamped Transient-Response Parameters Delay Time 1 + 0.7ζ Td = ωn Rise Time π −β Tr = ωd  1 − ζ 2  where β = tan −1    ζ    Settling Time 4 Ts = ( 2% criterion ) ζωn  Underdamped Transient-Response Parameters Peak Time π Tp = ωd Maximum Overshoot − The value of the response at the peak time Tp πζ − 1−ζ 2 M = p% e ×100% Notice that Mp% is function of only of the damping ratio Damping Ratio − The value of the response at the peak time Tp − ln ( M p % 100 ) ζ = π 2 + ln 2 ( M p % 100 )  Figure: relationship between the damping ratio ξ and the Maximum Overshoot (MP) Notice that no overshoot for ξ ≥1; overshoot becomes negligible for ξ > 0.7 Summary  Example: Given the transfer function: Find Tp, %OS, Ts, and Tr. SOLUTION: ωn and ζ are calculated as 10 and 0.75, respectively. Now substitute ωn and ζ into Eqs. and find, respectively, that Tp = 0.475 second, %OS = 2.838, and Ts = 0.533 second andTr = 0.23 second.  First vs Second Order Systems Comparison of first order and second order systems: - Varying a first-order system's parameter simply changes the speed of the response - Changes in the parameters of a second-order system can change the form of the response  Closed Loop Transfer Function C (s) G (s) =  Closed Loop Transfer Function R( s) 1 + G ( s) H ( s)  Due date for Lab Assignment: October 22nd, 11:59 pm. Access SLATE for Slides, Class notes, and Lab assignments Don't forget, we have our mid-term exam Tomorrow (in person). See you Tomorrow!