Discrete Control Sysfstem in LabVIEW.pdf
- 2. • Introduction • Mathematical Model – Discretization – We will make a Discrete version of the Model/Differential Equation • PID Controller – Discrete PI Controller – We will make a Discrete version of the standard continuous PI Controller • Control System – We make a basic Control System where the Discrete Model and Discrete PI Controller are used Contents
- 4. Introduction • We will simulate a 1. Order Process/Differential Equation –We will Implement a Discrete version of the Model and perform Simulations • We will create a basic Control System –We will make and Implement a Discrete PI Controller and perform Simulations
- 5. Control System Controller Process 𝑟 𝑢 𝑒 − Reference Value Control Signal 𝑦 𝑦 PID Controller The purpose with a Control System is to Control a Dynamic System, e.g., an industrial process, an airplane, a self-driven car, etc. (a Control System is “everywhere” today) Feedback Loop
- 7. 1. Order System 1. Order Process 𝑢 𝑦 ̇ 𝑥 = −𝑎𝑥 + 𝑏𝑢 𝑦 = 𝑥 In order to simulate this model in LabVIEW you can make a discrete version of the model, or you can implement it as a “Block Diagram” using the features in LabVIEW Control Design and Simulation Module Differential Equation of a 1. order System:
- 8. ̇ 𝑦 = 1 𝑇 (−𝑦 + 𝐾𝑢) Where 𝐾 is the Gain and 𝑇 is the Time constant ̇ 𝑦 = −𝑎𝑦 + 𝑏𝑢 Dynamic System 𝑢(𝑡) 𝑦(𝑡) Assume the following general Differential Equation: or: Where 𝑎 = ! " and 𝑏 = # " This differential equation represents a 1. order dynamic system Assume 𝑢(𝑡) is a step (𝑈), then we can find that the solution to the differential equation is: 𝑦 𝑡 = 𝐾𝑈(1 − 𝑒$ % ") Input Signal Output Signal (by using Laplace) 1. Order System
- 9. 100% 63% 𝐾𝑈 𝑡 𝑇 𝑦 𝑡 = 𝐾𝑈(1 − 𝑒/ 0 1) 𝐻 𝑠 = 𝑦(𝑠) 𝑢(𝑠) = 𝐾 𝑇𝑠 + 1 𝑦(𝑡) 1. Order Step Response Time constant 𝐾 is the Gain
- 10. Discretization We have the continuous differential equation: ̇ 𝑥 = −𝑎𝑥 + 𝑏𝑢 We apply Euler: ̇ 𝑥 ≈ & '(! $& ' "! Then we get: 𝑥 𝑘 + 1 − 𝑥 𝑘 𝑇) = −𝑎𝑥(𝑘) + 𝑏𝑢(𝑘) This gives the following discrete differential equation (difference equation): 𝑥 𝑘 + 1 = (1 − 𝑇!𝑎)𝑥(𝑘) + 𝑇!𝑏𝑢(𝑘) This equation can easily be implemented in any text-based programming language or the Formula Node in LabVIEW Where 𝑎 = ! " and 𝑏 = # "
- 11. Discrete Model in LabVIEW 𝑥 𝑘 + 1 = (1 − 𝑇!𝑎)𝑥(𝑘) + 𝑇!𝑏𝑢(𝑘)
- 13. Code
- 15. Control System Controller Process 𝑟 𝑢 𝑒 − Reference Value Control Signal 𝑦 𝑦 PID Controller The purpose with a Control System is to Control a Dynamic System, e.g., an industrial process, an airplane, a self-driven car, etc. (a Control System is “everywhere” today) Feedback Loop
- 16. PID Controller 𝑢 𝑡 = 𝐾*𝑒 + 𝐾* 𝑇+ ( , - 𝑒𝑑𝜏 + 𝐾*𝑇. ̇ 𝑒 P I D Proportional Gain Integral Time Derivative Time 𝐾. 𝑇/ 𝑇0 Tuning Parameters:
- 17. 𝑢! = 𝑢!"# + 𝐾$ 𝑒! − 𝑒!"# + 𝐾$ 𝑇% 𝑇&𝑒! 𝑒/ = 𝑟/ − 𝑦/ Discrete PI Controller that we can implement in different programming languages: 𝑢 𝑡 = 𝐾!𝑒 + 𝐾! 𝑇" 4 # $ 𝑒𝑑𝜏 PI Controller Very often we just need a PI Controller:
- 18. 𝑢 𝑡 = 𝐾"𝑒 + 𝐾" 𝑇# 4 $ % 𝑒𝑑𝜏 We start with the continuous PI Controller: ̇ 𝑥 ≈ 𝑥 𝑘 − 𝑥 𝑘 − 1 𝑇! We can use the Euler Backward Discretization method: Where 𝑇) is the Sampling Time Then we get: 𝑢& − 𝑢&'( 𝑇! = 𝐾" 𝑒& − 𝑒&'( 𝑇! + 𝐾" 𝑇# 𝑒& We derive both sides in order to remove the Integral: ̇ 𝑢 = 𝐾* ̇ 𝑒 + 𝐾* 𝑇+ 𝑒 Finally, we get: 𝑢7 = 𝑢7/8 + 𝐾9 𝑒7 − 𝑒7/8 + 𝐾9 𝑇: 𝑇;𝑒7 Where 𝑒' = 𝑟' − 𝑦' Discrete PI Controller
- 20. Control System Controller Process 𝑟 𝑢 𝑒 − Reference Value Control Signal 𝑦 𝑦 PID Controller The purpose with a Control System is to Control a Dynamic System, e.g., an industrial process, an airplane, a self-driven car, etc. (a Control System is “everywhere” today) Feedback Loop
- 21. Control System in LabVIEW
- 23. Discrete PI Controller 𝑒! = 𝑟! − 𝑦! 𝑢! = 𝑢!"# + 𝐾$ 𝑒! − 𝑒!"# + 𝐾$ 𝑇% 𝑇&𝑒! Discrete PI Algorithm:
- 24. Discrete PI Controller (Alternative Solution)
- 25. Control System in LabVIEW
- 27. Summary • A Basic Control System has been made using LabVIEW • Lots of Improvements can be made, e.g.,: – Improve GUI • More Features/Functionality, More Intuitive and more user-friendly – Improve Code Structure, e.g., use a State Machine principle – Make a more robust PI(D) Controller – Use and Test with a more complicated Process/Model – Find better PI(D) Parameters using different Tuning methods, e.g., Ziegler-Nichols, Skogestad, etc. – Connect and Control a Real Process using a DAQ Device – Etc.
- 28. Hans-Petter Halvorsen University of South-Eastern Norway www.usn.no E-mail: hans.p.halvorsen@usn.no Web: https://www.halvorsen.blog