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Robust Multiloop PID Controller Design: A Successive Semidefinite

Programming Approach

J . Bao, J . F. Forbes,*, and P. J . McLellan

School of Chemical Engineering and Industr ial Chemistry, The University of New South Wales, Sydney, New

South Wales, Australia 2052, Department of Chemical and Materials Engineering, University of Alberta,

Edmonton, Alberta, Canada T6G 2G6, and Department of Chemical Engineering, Queens University,

Ki n g st o n , O n t a r io , Ca n a d a , K7 L 3 N 6

The problem of robust multiloop proport iona l-integra l-derivat ive (PI D) controller tun ing formultivariable processes is addressed in this paper. The problem is formulated in the Hcontrolf r a m ew o r k, a n d t h e con t r ol ler p a r a m e t er s a r e d et e r m in ed b a s ed on b ot h u s er -s pe ci fi edp erform a n ce a n d rob u st s ta b i li ty . Th e P I D se tt in g s a re com p u ted b y solvin g a n on lin earop tim iz at ion p rob lem with m a trix in equ a lity con stra in ts , u sin g a su ccessive se m id e fin itep r og r a m m i n g a p pr oa c h . Th e p r op os ed m et h od i s i ll us t r a t e d b y a s im p le ca s e s t u d y t h a tinvestigates robust PID control of a distillation column.

1. Introduction

P roportiona l-integra l-derivat ive (PID ) contr ollers h a vebeen used extensively in the process industries sincethey are simple and often effect ive and represent thebasic building blocks available in many process controlsystems. Despite their wide spread use and considerablehistory, PID tuning is st ill an act ive area of research,both academic and industrial (e.g. , Wang and Cluett , 1

Astr om et a l.,2 and Hovd and Skogestad 3,4). During thepast f ive decades, a comprehensive PID tuning litera-ture ha s developed. The first significant tun ing meth odwa s proposed by Ziegler a nd N ichols.5 Analytical meth-ods to obtain PI D pa ra meters based on simple first- or

second-order transfer function models were developedby Rivera et al.6 and G aw throp and Nomikos.7 For morecomplicated transfer functions or transfer matrices formultiinput-multioutput (MIMO) systems models, nu-merical search procedures that minimize different per-formance objective functions were also proposed (Radkea n d I s e rma n n ,8 Zhuang and Atherton,9 Vega et al . ,10

Astr om et a l.2). A method for autotuning fully coupledmultivaria ble PID controllers from decentra lized r elayfeedback wa s presented by Wan g et a l .11

Since the process models used for controller designare often simplificat ions or a pproximat ions, i t is es-sential tha t the P ID tun ings obta ined by such methodsshould tolera te model-plant mismat ch. Un fortuna tely,the above controller design methods do not deal witht h e robu s t n es s is su e e x plici t ly a n d in ma n y de sig nschemes, only control performance is optimized. An H(robu s t ) P I D con t ro ller s y n t h es is me t h od wa s f irs tpresented by Gr imble.12 A genetic algorithm wa s thenproposed to determine P ID controller tuning t o achieveH o p t ima li t y by Ch e n e t a l . 13 I n b o t h o f t h e s e a p -proaches, t he H n orm of t h e we igh t e d s en s i t iv i t yfunction and complementary sensit ivity function are

minimized; however, these design a pproaches ar e lim-ited to s ingle-input-single-output (SI SO) m odels.

F o r con t ro l a p p lica t io n e n gin ee rs, mu lt iloop P I Dcontrollers can be preferred to the mult ivariable ap-proach. Many plants have older, or legacy, controlsystems that do not possess the capabilities to supportthe implementat ion of complex mult ivaria ble control-lers. For these plants, a mult iloop PID control schemedoes not require purchase of additional control systemhardware, as may be required to implement mult ivari-able controllers. Multiloop designs a lso ca n ha ve betterf a i lu r e t ol er a n ce t h a n t h e m u lt i va r i a b le a p pr oa c h ;however, mult iloop controllers ma y suffer from contr olperforma nce loss a nd even the insta bility of closed-loop

s y s t e m wh e n e a c h in div idu a l lo o p is t u n e d by u s in gSISO tuning methods. This results from the interactionsam ong different loops, w hich a decentralized controlstructure cannot deal with.14

The problem of multiloop robust P ID contr oller tu ningfor MIMO models is addressed in this paper. To mini-mize the performance loss due to the restrict ion of adecentralized controller structure, t he controller pa-ra meters a re computed ba sed on th e closed-loop syst emconsisting of the full process model and the multiloopcontroller. This leads to a less conservative designcompa red to th e decentra lized control approaches, wh ichuse the diagonal subsystem as a design model and treatt h e k n own in t era c t ion s a s u n ce rt a in t ies (M ora ri a n d

Zafiriou,

15

a n d S a m y u d i a e t a l .

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) . Hovd and Skoges-t a d 3,4 provide a sequentia l method for building a mul-tiloop design by closing one loop at a time, as well as agood int roduction to t he a rea of decentra lized contr ollerdesign. To achieve robustness, t he tu ning pr oblem posedin this paper is formulated in the Hcontr ol fra meworkwith constraints on the controller structure (i.e. , fixed-order, PID , an d decentr alized str ucture). This a pproachalso leads t o a syst ema tic and unified tool for mult iloopP ID t uning w ith user-specified performan ce an d robuststability, which needs much less heuristic user interac-t i o n t h a n m a n y o f t h e m a n u a l a n d t i m e - c o n s u m i n gapproaches mentioned a bove. A numerica l optimizat ionprocedure is proposed to solve th e str ucture-const ra inedH problem wit h in t h e f ra me wo rk of s e mide fin it e

programming (SDP), since the robust mult iloop PID

* Author to whom correspondence should be addr essed.P hone : (780) 492-0873. Fax: (780) 492-2881. E -m ai l :Fraser.Forbes@UAlberta.ca.

The U niversity of New S outh Wales. Un iversity of Alberta .

Queens U niversity.

3407In d. E ng. Chem. Res.1999, 38 , 3407-3419

10.1021/ie980746u CCC : $18.00 1999 American C hemical SocietyP ubl ish ed on Web 08/11/1999