Optimal PID-Control on First Order Plus Time Delay Systems - PowerPoint PPT Presentation

Introduction Optimal Controller Tuning rule Conclusions Optimal PID-Control on First Order Plus Time Delay Systems Verification of the SIMC rules Chriss Grimholt and Sigurd Skogestad Norwegian University of Science and Technology February 19, 2014 C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 1 / 22

Introduction Optimal Controller Tuning rule Conclusions I NTRODUCTION The Questions ◮ How much can we gain by using PID instead of PI control? ◮ Is there a simple PID tuning rule that gives close to optimal performance? C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 2 / 22

Introduction Optimal Controller Tuning rule Conclusions T HE SYSTEM We investigating the optimal tuning for the process k ( τ 1 s + 1 ) e − θ s G ( s ) = Considering the cascade form PID controller � τ I s + 1 � K ( s ) = K C ( τ D s + 1 ) τ I s C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 3 / 22

Introduction Optimal Controller Tuning rule Conclusions Q UANTIFYING THE O PTIMAL C ONTROLLER Trade-off between ◮ Output performance } High controller gain (Tight control)  ◮ Robustness   ◮ Input usage Low controller gain (Smooth control)  ◮ Noise sensitivity  We focus on output performance and robustness. ◮ Output Performance: J – weighted average IAE ◮ Robustness: M S – peak sensitivity C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 4 / 22

Introduction Optimal Controller Tuning rule Conclusions R OBUSTNESS – M S 1 S = 1 + GK M S = max | S ( j ω ) | = � S � ∞ ω 10 1 1 Magnitude, | S | 0 . 5 M S Im L ( j ω ) 10 0 0 1 / M S − 0 . 5 − 1 10 − 1 10 0 10 1 10 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 Frequency, ω Re L ( j ω ) C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 5 / 22

Introduction Optimal Controller Tuning rule Conclusions P ERFORMANCE – J We consider unit step disturbances at two different locations: ◮ at plant output ◮ at plant input d i d o e u + y sp + y K ( s ) G ( s ) − C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 6 / 22

Introduction Optimal Controller Tuning rule Conclusions P ERFORMANCE – J � ∞ IAE = | e ( t ) | dt 0 � IAE do ( K ) + IAE di ( K ) � J ( K ) = 0 . 5 IAE ◦ IAE ◦ do di IAE ◦ do PID optimal controller for d o (at M S = 1 . 59) IAE ◦ di PID optimal controller for d i (at M S = 1 . 59) 1 . 5 1 . 5 d i d o 1 IAE di 1 IAE do Error, e ( t ) Error, e ( t ) 0 . 5 0 . 5 Time, t Time, t 0 0 2 4 6 8 10 2 4 6 8 10 − 0 . 5 − 0 . 5 C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 7 / 22

Introduction Optimal Controller Tuning rule Conclusions P ARETO OPTIMAL PLOTS – O PTIMAL J ( K ) VS . M S 4 3 Performance, J ( K ) 2 1 PO PID 0 1 1 . 5 2 2 . 5 3 Robustness, M S C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 8 / 22

Introduction Optimal Controller Tuning rule Conclusions P ARETO OPTIMAL PLOTS – O PTIMAL J ( K ) VS . M S 4 3 Performance, J ( K ) 2 1 PO PID Infeasible 0 1 1 . 5 2 2 . 5 3 Robustness, M S C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 9 / 22

Introduction Optimal Controller Tuning rule Conclusions P ARETO OPTIMAL PLOTS – O PTIMAL J ( K ) VS . M S 4 3 Performance, J ( K ) Sub-Optimal 2 1 PO PID 0 1 1 . 5 2 2 . 5 3 Robustness, M S C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 10 / 22

Introduction Optimal Controller Tuning rule Conclusions P ARETO OPTIMAL PLOTS – O PTIMAL J ( K ) VS . M S 4 3 Performance, J ( K ) Uninteresting 2 1 PO PID 0 1 1 . 5 2 2 . 5 3 Robustness, M S C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 11 / 22

Introduction Optimal Controller Tuning rule Conclusions T HE CASES G 1 ( s ) = e − s 1 ( s + 1 ) e − s G 2 ( s ) = 1 ( 8 s + 1 ) e − s G 3 ( s ) = G 4 ( s ) = 1 s e − s C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 12 / 22

Introduction Optimal Controller Tuning rule Conclusions O PTIMAL PI CONTROL 4 4 1 G ( s ) = e − s ( s + 1 ) e − s G ( s ) = 3 3 Performance, J ( K ) Performance, J ( K ) 2 2 PO PI 1 1 PO PI 0 0 1 1 . 5 2 2 . 5 3 1 1 . 5 2 2 . 5 3 Robustness, M S Robustness, M S 4 4 1 G ( s ) = 1 ( 8 s + 1 ) e − s G ( s ) = s e − s 3 3 Performance, J ( K ) Performance, J ( K ) 2 2 PO PI PO PI 1 1 0 0 1 . 5 2 2 . 5 3 1 . 5 2 2 . 5 3 1 1 Robustness, M S Robustness, M S C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 13 / 22

Introduction Optimal Controller Tuning rule Conclusions O PTIMAL PID C OMPARED WITH O PTIMAL PI 4 4 1 G ( s ) = e − s ( s + 1 ) e − s G ( s ) = 3 3 Performance, J ( K ) Performance, J ( K ) 2 2 No benefit from using PID PO PI 1 1 PO PI P O P I D PO PID 0 0 1 1 . 5 2 2 . 5 3 1 1 . 5 2 2 . 5 3 Robustness, M S Robustness, M S 4 4 1 G ( s ) = 1 ( 8 s + 1 ) e − s G ( s ) = s e − s 3 3 Performance, J ( K ) Performance, J ( K ) 2 2 PO PI PO PI 1 1 P O P I D PO PID 42% Increase in performance 0 0 1 . 5 2 2 . 5 3 1 . 5 2 2 . 5 3 1 1 Robustness, M S Robustness, M S C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 14 / 22

Introduction Optimal Controller Tuning rule Conclusions O PTIMAL PID TUNING ◮ τ D is independent of M S and the time constant τ 1 for lag dominated processes ( τ 1 /θ � 2 . 5). ◮ Optimal controller for delay dominated processes ( τ 1 /θ � 2 . 5) is on the boarder between cascade and ideal controller realization. Normalized controller gain, K C k θ/τ 1 1 M S = 1 . 20 Integral and derivative time 0 M = 1 . 5 S 3 M S = 1 . 59 0 M = 1 . 7 S 0 . 8 M S = 2 . 00 M S = 2 . 00 τ I /θ 0 . 6 M = 1 . 7 0 S 2 M S = 1 . 59 M S = 1 . 5 0 M S = 1 . 20 0 . 4 τ D almost independent of M S 1 0 . 2 τ D /θ τ D ≈ 0 . 48 Two coinciding zeros 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Process time constant τ 1 /θ Process time constant τ 1 /θ C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 15 / 22

Introduction Optimal Controller Tuning rule Conclusions O PTIMAL PID – C ASCADE VS . I DEAL Delay dominated processes τ 1 /θ < 3: Performance can be improved by using Ideal controller. � 1 + 1 � K Ideal = K ′ I s + τ ′ D s C τ ′ 3 e − s G ( s ) = Performance, J ( K ) ( s + 1 ) Cascade 2 Ideal 1 0 1 1 . 5 2 2 . 5 3 Robustness, M S C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 16 / 22

Introduction Optimal Controller Tuning rule Conclusions T HE SIMC PI RULES FOR FOPTD SIMC: Probably the best simple PID tuning in the world K C = 1 τ 1 (1) k τ c + θ τ I = min { τ 1 , 4 ( τ c + θ ) } (2) where τ c is the tuning constant. τ c = θ is recommended for tight control. However, only PI tuning for FOPTD C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 17 / 22

Introduction Optimal Controller Tuning rule Conclusions O PTIMAL PID CONTROL C OMPARED WITH SIMC 4 4 1 G ( s ) = e − s ( s + 1 ) e − s G ( s ) = 3 3 Performance, J ( K ) Performance, J ( K ) θ 5 θ . 1 2 5 2 . = 1 θ θ = θ θ 5 τ c 1 5 1 . c 0 = 0 . τ = SIMC (P)I = = τ c τ c c P I τ τ c M C S I PO PI 1 1 PO PI P O P I D PO PID 0 0 1 1 . 5 2 2 . 5 3 1 1 . 5 2 2 . 5 3 Robustness, M S Robustness, M S 4 4 θ 5 . 1 = 1 c G ( s ) = 1 θ τ 5 ( 8 s + 1 ) e − s G ( s ) = s e − s 1 . = τ c 3 3 Performance, J ( K ) Performance, J ( K ) θ 1 θ 1 = = τ c τ c θ θ 5 5 . . 0 0 = 2 = 2 τ c τ c SIMC PI PO PI SIMC PI PO PI 1 1 P O P I D PO PID 0 0 1 . 5 2 2 . 5 3 1 . 5 2 2 . 5 3 1 1 Robustness, M S Robustness, M S C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 18 / 22

Introduction Optimal Controller Tuning rule Conclusions P ROPOSED : SIMC PID RULES FOR FOPTD K C = 1 τ 1 (3) k τ c + θ τ I = min { τ 1 , 4 ( τ c + θ ) } (4) τ D = θ/ 3 (5) C. Grimholt and S. Skogestad (NTNU) Optimal PID-Control on FOPTD Systems February 19, 2014 19 / 22