Extensions of Skogestad’s SIMC tuning rules to oscillatory and unstable processes

1. Extensions of Skogestad’s SIMC tuning rules to oscillatory and unstable processes Henrik Manum, student, NTNU

2. Project goals • Extend the SIMC-rules to oscillatory and unstable processes • Reduce the model at hand to a first or second order plus delay model • can use existing SIMC-rules on the reduced model • Derive new rules based on the given model

3. Reminder of the SIMC PID tuning rules • Assume we have a model on one of the following forms: • SIMC-PID controller settings: Fast and robust

4. Processes covered • Stable process with pair of complex poles • Unstable process with single real RHP pole

5. Stable process with pair of complex poles • Divide the processes into three parts • Category A • Pure 2nd order underdamped system • Category B • Damped oscillations, but a clear peak in the frequency domain • Category C • Peak less than steady state gain Main focus today

6. Category B • Resonant peak, asymptotically: • Phase, empirical, from Bode-plot Conservative for all frequencies Most likely too complicated, but will use this as a starting-point.

7. Category B • Justification for gain-approximation Peak in gain for pure 2nd order under-damped process:

8. Category B

9. Category B • So, we use the maximum gain to stay safe in gain, and we use the empirical phase-approximation to get a model on the form with the approximations given in the previous slides

10. Category B • Direct synthesis of controller for the process (for setpoints) Pure I-controller

11. Category B • Performance and robustness evaluation of the resulting I-controller Want to solve this optimization problem for a PI-controller and compare the resulting controller to our I-controller

12. Category B • Naive solution to the optimization problem:

13. Category B Our I-controller

14. Category B Our I-controller

15. Category B • Pros and cons with this method of controller evaluation • Difficult to find a solution • SIMULINK model often diverges • The problem is most likely non-convex • Good graphical representation of trade-off between IAE and TV • Further work: • Look at method by Kristiansson and Lennartson. • Frequency based approach • Broader range of input-signals • Probably easier to use in practice

16. The remaining processes • Category C: Same procedure as category B, but with I derived a 1st order model with a resulting PI-controller • Categroy A: Pure oscillatory. Based on work done by prof. Skogestad, compared with method from literature • Unstable process: Reviewed work by prof. Skogestad and compared to a method found in literature PLEASE CONSULT THE REPORT IF YOU HAVE INTEREST

17. Summary • The goal of extending the SIMC rules to oscillatory and unstable processes has not been achieved, but we are closer to the goal than when we started • The author has learned a lot, including: • Frequency analysis • Robustness and performance measures in frequency domain • Properties of linear models in frequency domain in general • Optimization used in practice • Time domain analysis • Experience with Matlab on control problems

18. References • SIMC-rules • Optimal controller • For more references see the report

19. THE END