A process plant consists of several interlinked unit operations. The plant is multi-input and ... For a à MIMO process, PID controllers ... Sequential loop closing, e.g., (Shiu & Hwang, 1998). 3. ... where the two lower limits of reset time are. , = , , =.
International Conference on Process Engineering and Advanced Materials (ICPEAM 2018) 13 – 15 August 2018 Kuala Lumpur, Malaysia
RouthRouth-Hurwitz tuning method for stable/unstable timetime-delay MIMO processes Noraini Mohd1,2, Jobrun Nandong1,2,3 1Curtin
Malaysia Research Institute 2Department of Chemical Engineering 3Intelligent Systems, Design and Control (ISDCON) Curtin University, 98009 Miri, Sarawak MALAYSIA
Intelligent Systems, Design & Control Research Cluster
Outline • • • • • • •
Background Problem Statement SISO PID Control Stability Derivation of PID Tuning Relations Design Procedures for MIMO Examples Conclusions
Background A process plant consists of several interlinked unit operations. The plant is multi-input and multi-output (MIMO) in nature.
Example Sulfur-Iodine Thermochemical Cycle (SITC) Plant (Mohd, 2018)
Background • Majority (~90%) of industrial controllers are ProportionalIntegral-Derivative (PID) type (Astrom & Hagglund, 2001) • PID Controller, ideal form: 1 () = 1 + + , , are tuning parameters • A decentralized or multi-loop PID control system is widely used in process industry. • For a × MIMO process, PID controllers are used, thus, there are tuning parameters in total. total • Multi-loop PID tuning task is challenging due to process interactions and many tuning parameters.
Background • Five major approaches to multi-loop PID tuning (Huang et al., 2003) 1. Detuning, Detuning e.g., BLT (Luyben, 1986) 2. Sequential loop closing, closing e.g., (Shiu & Hwang, 1998) 3. Iterative or trialtrial-andand-error 4. Independent tuning , e.g., (Vu & Lee, 2010) 5. Simultaneous tuning, tuning e.g., (Nandong, 2015) • Simultaneous tuning is based on multi-scale control (MSC) scheme (Nandong & Zang, 2013) – Reduce tuning parameters to only 2 -3 MSC tuning parameters (Nandong, 2015) – MSC tuning parameters are dimensionless – Similar MSC tuning values can be applied to different unit operations
Problem Statement
Consider a × MIMO Process (PP) given as
() ⋯ () ⋮ ⋱ ⋮ = () ⋯ ()
where for a stable process, is given by
exp (−$ ) = , + 1
where , $ , > 0
and for an unstable process, is given by
exp (−$ ) = , − 1
where , $ , > 0
Find the values of )*+ , ,-+ , ,.+ for / = 1, 2, … which provide closed loop stability and desired performance criteria
SISO PID Control Stability • PID stability regions (of 2 , , ) can be established via several approaches: 1.
2. 3. 4.
• • •
Hermite-Biehler Theorem, e.g., see (Silva, Datta & Bhattacharyya, 2002) D-partition, e.g., see (Gryazina & Polyak, 2006) Nyquist plot, e.g., see (Fang, 2010) Routh-Hurwitz criteria, e.g., see (Seer & Nandong, 2017a,b)
For some processes, no stability region exists, exists e.g., some classes of unstable time-delay systems. Important to establish PID stability regions PID tuning values can be obtained within a stability region.
PID Stability Theorem • PID stability theorem based on Routh-Hurwitz criteria has been established (Seer & Nandong, 2017a,b) • From the PID theorem, a stability region based necessary criterion is ,5 < < ,578 Ω = 4 ,5 < < ,578 2 < 2 < 2 5
578
• Basic idea is to establish upper and lower limits on the PID parameters. Stability lies in between the maximum lower and minimum upper limits. • Proof is provided in (Seer & Nandong, 2017a)
Derivation PID stability region • Case A – Stable process =
9: ;?@) A: @B
, assume exp −$ ≅
>D@ BD@
where E =
• Closed-loop characteristic equation for PID controller is E G − 2 7H
+ E + G + 2 − E F + + 2 − E + 2 K =0 7I
7J
7L
? F
• where 2 = 2 2G is the loop gain determining the control performance. • Using PID stability theorem (Seer & Nandong, 2017a), the stability regions for the PID controller are established.
Stability Region for Stable Process • The stability region considering necessarynecessary- sufficient criteria > E Ω = 4 > MNO E, .5 Q R 0 < 2 < R: S
where
,5 Q =
9D B9
1+
A: >9AS
DBA: B9 AS >D
Note: Higher upper limit on loop gain implying a higher performance margin for the controller.
Stability Region for Unstable Process • Case B - Consider unstable process =
, assume exp −$ ≅
9: ;?@) A: @ >
>D@ BD@
where E =
? F
• Applying the same PID theorem as in the Case A, stability region is E < < G
Ω = 4 > MNO ,5 T , ,5 FT R: RU < 2 < R VW R U
S
• where the two lower limits of reset time are ,5 T =
DA:
A: >AS
, ,5 FT =
XW XVJ
1+R
R: VXRS : VWYX RS VW
Some Remarks • For the unstable process I.
Stability lies in between minimum and maximum allowable performance limits. II. PID controller cannot stabilize the process if i.
G < E (upper limit is less than lower limit of )
i. ii.
> E
ii.
R: RS
U (upper limit is less than lower limit of 2) < R RVW U
III. For the stability region to exist < G
Case A – Tuning Relations • Tuning relations are given as
= [\ E , ]^_[_ [\ > 1 [G G , ]^_[_ 0 < [G < 1 2 = 2G G = [ MNO E, ,5 Q , ]^_[_ [ > 1 $
• The dimensionless tuning parameters are `a , `b and `+
• Similar values of [G , [\ and [ can be used for different × MIMO processes • Detailed tuning procedure can be referred to the full paper.
Case B – Tuning Relations • Tuning relations are given as = [\ G − E + E,
[G G 1 − 1− 2 = [G 2G − E
= [
G $
]^_[_ 0 < [\ < 1 ,
MNO ,5 T , ,5 FT
]^_[_ 0 < [G < 1 ,
]^_[_ [ > 1
Example 1 – Tyreus’ Distillation Column (Tyreus, 1979) −1.986exp (−0.71) 5.24exp (−60) 5.98exp (−2.24) 66.67 + 1 400 + 1 14.29 + 1 0.02exp (−0.59) −0.33exp (−0.68) 2.38exp (−0.42) = 7.14 + 1 F 2.38 + 1 F 1.43 + 1 F 0.37exp (−7.75) −11.3exp (−3.79) −9.81exp (−1.59) 22.22 + 1 21.74 + 1 F 11.36 + 1
Model
Dimensionless Settings [G = 0.6 [ = 5 [\ = 4 PID parameters 2 = −14.18, −10.86, −0.2185 = 200, 24.2, 29.4 = 1.42, 1.36, 3.18
Closed-loop setpoint tracking responses for Example 1
Conclusion • New PID tuning algorithms based on RouthRouth-Hurwitz criteria have been developed for a class of stable/unstable MIMO processes with time-delays. • The tuning algorithms reduce the problem of finding PID tuning values to only 3 dimensionless parameter values, values regardless of the size of MIMO system involved.
References • • • • •
•
•
Åström, K. J., & Hägglund, T. (2001). The future of PID control. Control engineering practice, 9(11), 1163-1175. Fang, B. (2010). Computation of stabilizing PID gain regions based on the inverse Nyquist plot. Journal of Process Control, 20(10), 1183-1187. Gryazina, E. N., & Polyak, B. T. (2006). Stability regions in the parameter space: Ddecomposition revisited. Automatica, 42(1), 13-26. Huang, H. P., Jeng, J. C., Chiang, C. H., & Pan, W. (2003). A direct method for multi-loop PI/PID controller design. Journal of Process Control, 13(8), 769-786. Mohd, N. (2018). Plantwide Control and Simulation of Sulfur-Iodine Thermochemical Cycle Process for Hydrogen Production (Doctoral dissertation, Curtin University). Nandong, J., & Zang, Z. (2013). High-performance multi-scale control scheme for stable, integrating and unstable time-delay processes. Journal of process control, 23(10), 1333-1343. Nandong, J. (2015). Heuristic-based multi-scale control procedure of simultaneous multi-loop PID tuning for multivariable processes. Journal of Process Control, 35, 101-112.
References • •
•
• •
•
Seer, Q. H., & Nandong, J. (2017a). Stabilization and PID tuning algorithms for second-order unstable processes with time-delays. ISA transactions, 67, 233-245. Seer, Q. H., & Nandong, J. (2017b). Stabilising PID tuning for a class of fourthorder integrating nonminimum-phase systems. International. Journal of Control, 1-17. Shiu, S. J., & Hwang, S. H. (1998). Sequential design method for multivariable decoupling and multiloop PID controllers. Industrial & engineering chemistry research, 37(1), 107-119. Silva, G. J., Datta, A., & Bhattacharyya, S. P. (2002). New results on the synthesis of PID controllers. IEEE transactions on automatic control, 47(2), 241-252. Tyréus, B. D. (1979). Multivariable control system design for an industrial distillation column. Industrial & Engineering Chemistry Process Design and Development, 18(1), 177-182. Vu, T. N. L., & Lee, M. (2010). Independent design of multi-loop PI/PID controllers for interacting multivariable processes. Journal of Process control, 20(8), 922933.
Acknowledgements • This work is supported by Curtin Malaysia Research Institute (CMRI). • Financial support by Curtin University Malaysia for attending this conference is deeply appreciated.
Thank You
