MULTIVARIABLE CONTROLLER TUNING

Transcription

1 MULTIVARIABLE CONTROLLER TUNING Karl H. Johansson, Bn Jams, Gryham F. Bryant, and Karl J. Åström Abstract Th problm of tuning individual loops in a multivariabl controllr is invstigatd. It is shown how th prformanc of a spcific loop rlats to a row in th controllr matrix. Svral intrprtations of this rlation ar givn. An algorithm is also prsntd that stimats th modl rquird for th tuning via a rlay fdback xprimnt. Th algorithm dos not nd any prior information about th systm or th controllr. Th rsults ar illustratd by an xampl.. Introduction Poorly tund control loops rprsnt a larg conomic cost for industry [5, 4]. Control paramtrs ar oftn st to dfault valus or ar manually tund in an ad hoc way. Th rason for this is that thr is a grat lack of tools for tuning industrial controllrs systmatically. Nowadays thr xist mthods for automatic tuning of SISO control loops, which hav bn widly accptd and implmntd in svral commrcial controllrs []. Many control loops ar, howvr, coupld and th intraction has to b considrd in th control dsign to gain improvd prformanc [8]. Most modrn multivariabl control dsign mthods rquir a full modl of th procss []. In many cass such a modl is not availabl and physical modling or systm idntification may rquir a prohibitiv nginring ffort. Furthrmor, it is hard, or impossibl, to impos a crtain control structur on standard multivariabl dsign mthods. Thrfor, thr is a nd for simpl mthods of tuning multivariabl controllrs; particularly mthods that compromis optimality for nginring fficincy. This papr focus on th problm of rtuning an xisting multivariabl control systm. A framwork is dvlopd whr it is possibl to driv th influnc of rtuning on loop on th ovrall closd-loop prformanc. A badly tund loop can in this way b improvd by changing crtain lmnts of th controllr matrix. Tuning a loop corrsponds to changing a row in th Dpartmnt of Automatic Control, Lund Institut of Tchnology, Box 8, S- Lund, Swdn, fkall,kjag@control.lth.s. Exposur Managmnt, Bank of Amrica, Elmfild Road, Bromly, London, Unitd Kingdom, @compusrv.com. Industrial Systms Group, Cntr for Procss Systms Enginring, Imprial Collg of Scinc, Tchnology and Mdicin, Exhibition Road, London, SW7 BT, Unitd Kingdom, g.bryant@ic.ac.uk. controllr matrix; hnc, to solv a SIMO control dsign problm. Svral quantitis usful for stimating th influnc of a controllr row on th closd-loop systm ar drivd. Th information rquird for this typ of dsign is also discussd togthr with how this information can b obtaind. It is shown that no prior knowldg of th procss dynamics or of th controllr dynamics is ndd, if a modling xprimnt basd on rlay fdback is usd. In xisting work on xtnding th auto-tuning mthod for SISO control systms dvlopd in [] to MIMO systms, ithr on rlay is usd for ach xprimnt by closing on loop at a tim [7, 6, 9, 7] or all loops ar st undr rlay fdback simultanously [3, 9,, ]. A major drawback with th lattr approach is that instad of giving stationary limit cycls th rlays can induc vry complicatd oscillations [9, 9]. Thr xist no rsults in trms of plant data for whn this may or may not happn. Basd on a succssful rlay xprimnt a controllr is dsignd. Most authors limit th control structur to a dcntralizd configuration of SISO PID controllrs [, 9, 3, 7, ]. Dcoupling dsign is drivd in [6, ]. Tuning cascad controllrs (MISO controllrs) is considrd in [7, ]. For a survy on rlay fdback mthods s [3, 9]. Th outlin of th papr is as follows. Sction prsnts som rsults that ar usful for loop tuning. Rtuning a row in th controllr matrix is formalizd. In Sction 3 it is shown that th rquird information about th systm can b obtaind from an xprimnt with SISO rlay fdback. Sction 4 dscribs an application to a modl of a nw laboratory procss. Som concluding rmarks ar givn in Sction 5. An xtndd vrsion of this papr is givn in [9].. Loop Tuning Suppos that a multivariabl control systm with unsatisfactory closd-loop prformanc is givn. Th basic ida is to adjust crtain lmnts of th controllr matrix in ordr to improv th closd-loop bhavior. In gnral, such an adjustmnt will affct all loops in th systm. Th challng is to obtain this ffct on th dsird loop without dgrading th prformanc of th othr loops. This sction givs rsults which nabls th dsignr to comput th ffct of an adjustmnt of a singl loop on th ovrall closd-loop bhavior.

2 r _ ē ȳ H r m m K G y m K _ G r Figur Opning of control loop m. Notation Assum that thr xists a stabl closd-loop systm comprising a procss G and a nominal controllr K, both with m inputs and m outputs. Dnot th manipulatd variabl or procss input u (u,..., ) T,th controlld variabl or procss output y (y,...,y m ) T, and th rfrnc or st-point r (r,...,r m ) T.Th controllr matrix K acts on th rror signal (,..., m ) T r y.hnc,y Gu and u K.Th aim of th tuning procdur is to improv th prformanc of on loop by adjusting appropriat lmnts of th controllr matrix. Without loss of gnrality, considr loop m and dfin th following partitions: G m m G G, K m m K k. () Partition th vctors u ( T, ) T, y (ȳ T, y m ) T, r ( r T,r m ) T,and (ē T, m ) T corrspondingly, so that (u,..., ) T tc. Thn ε T m K k k + +k m m, whr ε T m (,...,,) and k i, i,...,m,arth lmnts of k. Row m of th controllr matrix K thus contains th coupling from th rror to th control signal. Figur shows th closd-loop systm with th signal path brokn. Any snsibl choic of th controllr row k that improvs th prformanc of loop m, rquirs at last knowldg of th SIMO transfr matrix from to in this partially opn systm. W dnot this transfr matrix H (I+G K ) G and assum that it is stabl. Th block diagram of Figur shows xplicitly th contribution of controllr row m to th fdback control of th systm. Th transfr matrics of th full multivariabl closd-loop systm can asily b dscribd in trms of thos for th systm with H acting as a procss and k as a controllr. In othr words, th multivariabl control dsign problm for G is rducd to a SIMO control problm for H with MISO controllr k. Paramtrization and stability It is simpl to calculat th ffct of nw or rdsignd controllr row lmnts of th singl-loop opning ap- k Figur Contribution of controllr row k. proach. Th input snsitivity function is givn by S i : (I + KG) and th output snsitivity function by S o : (I+ GK). Th diagonal lmnt m of th snsitivity matrix S i capturs much of th prformanc in loop m. By th dfinition of H and k, w hav ε T ms i ε m /( kh). Knowldg of H alon is thus sufficint to comput th transfr function for loop m that rsults from a particular choic of k. Th closd-loop transfr matrics ar affin functions in th Youla paramtr Q : (I+KG) K if G is stabl []. For xampl, th snsitivity and complmntary snsitivity matrics with rfrnc to procss inputs ar S i I QG and T i QG, rspctivly, and th corrsponding matrics with rfrnc to procss outputs ar S o I GQ and T o GQ. Th closdloop transfr matrics ar also affin functions in q : k/( kh). This mvctor of transfr functions is th Youla paramtr for th partially opn systm. Som calculations giv th rlation btwn q and Q as Q K (I + G K ) + K H q(i + G K ). Paramtrization of stabilizing controllr rows and columns ar studid in [8]. Naturally, any adjustmnts of controllr row m must b mad in such a way that th closd-loop systm rmains stabl. Th following rsult follows from th Nyquist thorm. Assum th closd-loop systm is stabl with controllr row k. Ltkb rplacd by k, whr k is such that no unstabl mods ar canclld and that th numbr of opn-loop RHP pols dos not chang. Thn th adjustd closd-loop systm rmains stabl if and only if N ( kh,) N( kh,), whrn (f(s),z)is th numbr of clockwis ncirclmnts of th point z by th imag of th usual Nyquist contour D N undr th map f as it is travrsd in a clockwis dirction.

3 .5 H.5 K G r Figur Thr important points on th Nyquist curv. W ε T m 3. Extndd Rlay Exprimnt A rlay fdback xprimnt is a simpl and robust way of doing closd-loop idntification. Th stup for th original SISO xprimnt is simply to rplac th SISO controllr by a rlay []. Th main advantags of an idntification xprimnt basd on rlay fdback ar () that th frquncy of th xcitation signal is nar th cross-ovr frquncy of th opn-loop systm, () that th xprimnt is don in closd loop, and (3) that no prior knowldg about th procss dynamics is ndd. Th frquncy of th rlay output is clos to optimum in th sns that it is in th band whr th stimatd modl has to b accurat to support a satisfying control dsign. Evn if no controllr is prsnt in th loop during th xprimnt, th rlay itslf givs a high-gain fdback. This mans, for instanc, that th procss is automatically kpt clos to its oprating point during th xprimnt. A drawback with th original rlay fdback xprimnt is in som cass its lack of xcitation. Thrfor, w introduc a modification of th standard rlay xprimnt, by simply stimating two points on th Nyquist curv instad of on. It is wll-known that with a filtr in sris with th rlay, any point on th Nyquist curv can b stimatd []. This ida has bn xplord for SISO systms in [5, 6]. Prsson [3] invstigatd th amount of procss information ndd for control dsign in numbr of points and thir location on th Nyquist curv. Thr crucial points ar markd with crosss in Figur 3. Point is dtrmind by a standard rlay xprimnt, whras Point is dtrmind from an xprimnt with a rlay and an intgrator in sris. Th mthod can b intrprtd as putting a filtr W in sris with th rlay. Th filtr is initially st to W and thn to W /s. Togthr with stady-stat data, th gaind information is sufficint to driv a modl of th form b s+b G(s). () s 3 +a s +a s+a 3 Th controllr tuning dscribd in Sction is basd on knowldg of th column vctor H. Th st-up for Figur 4 Rlay xprimnt for idntifying H. an xtndd rlay xprimnt to idntify H is shown in Figur 4, compar with Figur. Th block with ε T m picks out rror signal m. Th rlay is thus connctd btwn W m and. This givs an oscillation with frquncis dtrmind by H m, which is typically th most important transfr function for controllr tuning in loop m. From masuring ē and m, w can stimat all lmnts of H. Wsummarizthmthodinth following algorithm. ALGORITHM SIMO RELAY EXPERIMENT. St W and wait for a stationary oscillation. Masur th frquncy ω and driv th rspons for ach lmnt H i.. St W /s and wait for a stationary oscillation. Masur th frquncy ω and driv th rspons for ach lmnt H i. 3. Frz th rlay output and wait for stadystat and driv th stady-stat gains for ach lmnt H i. 4. Estimat H i as in () basd on th rsponss and th corrsponding frquncis ω and ω. Th amounts of tim rquird for a stationary oscillation in Stp and Stp ar small. Exprimnts show that stationarity is oftn rachd aftr thr four rlay switchs. Not that Algorithm automatically givs highst priority to th last lmnt of H in th sns that th xcitation frquncis ar adjustd to suit H m.this mans also that if H,...,H m giv small rsponss around th cross-ovr frquncy of H m, thn th stimats of H,...,H m ar probably poor. Howvr, bcaus th lmnts ar small, th lack of accuracy has only a small influnc on th control prformanc.

4 u u 5 5 tim Figur 5 Extndd rlay xprimnt for minimum phas systm. Th rror signal (dashd) is ngligibl compard to (solid) tim Figur 7 Extndd rlay xprimnt for nonminimum phas systm. Th rror signals (dashd) and (solid) ar of th sam magnitud. Im R Figur 6 Nyquist curvs of H for minimum phas systm. Th crosss ar stimatd frquncy points from rlay fdback xprimnts. Th small crosss corrspond to H and th larg to H. A third-ordr stimat of H is also shown (solid lin). Th frquncy rspons of H is ngligibl compard to th rspons of H. 4. Exampl In this sction th rtuning procdur is applid to a multivariabl lvl control problm. Th considrd systm is a normalizd modl of th quadrupl-tank laboratory procss dscribd in [9, ]. Th systm including modls for actuators and snsors is givn by G 5 (s + ) γ γ s + (s + ) γ γ (s + ) s +. Th paramtrs γ,γ [,]ar dtrmind by how two valvs ar st prior to an xprimnt. Th systm G has a RHP zro if and only if γ + γ (,].Nxt w study th systm for on minimum-phas stting and on nonminimum-phas stting. Minimum phas systm Lt γ γ 4/5. Thn G has zros in 5/4and 3/4, so th systm is minimum Th rtuning procdur in this papr has bn applid to th ral laboratory procss in [4]. Im R Figur 8 Nyquist curvs of H for nonminimum phas systm, compar Figur 6. Th frquncy rsponss of H and H ar of th sam magnitud. phas. Lt K diag{, } b th initial controllr. Th rspons of th xtndd rlay xprimnt dscribd in Algorithm is shown in Figur 5. Th rspons of is small compard to. This is furthr illustratd in Figur 6, whr th small crosss ar th stimatd frquncy points for H and th larg crosss th points for H. Th dashd curvs ar th Nyquist curvs for th tru systms, whras th solid curv is a third-ordr stimat of H. Th rsult from th rlay xprimnt indicats that w can nglct th influnc of H and simply rtun th last lmnt of k. Th PI controllr k (, (s + 3)/s ) givs th pols 4.9 and.±4.6ifor th scond diagonal lmnt of S i. Not that th tuning hr corrsponds to applying SISO mthods. For this xampl th MIMO charactristics of th systm ar insignificant. Nonminimum phas systm Lt us now chang th valvs so that γ γ /5. Thn G has zros in 5/ and/, so th systm is nonminimum phas. Lt K diag{.,.} b th initial controllr. Figur 7 shows th rsult of th rlay xprimnt. Th stimatd Nyquist curvs (solid) ar shown in Figur 8, togthr with th tru ons (dashd). W

5 s that th intraction is svr, so it is probably not sufficint to only rtun th scond loop. If a rlay xprimnt is also don in th first loop, it is straightforward to driv a multivariabl controllr, for xampl basd on dcoupling. 5. Conclusions It was shown how a poorly tund multivariabl controllr can b rtund through a simpl closd-loop xprimnt basd on rlay fdback and controllr row dsign. In particular, th cas with on bad loop was discussd. Th standard SISO rlay fdback xprimnt in [] was xtndd to giv bttr xcitation and a mor accurat modl, which sms to b ncssary for many MIMO control dsigns. Svral rsults on how a row in th controllr matrix affcts th closd-loop prformanc wr drivd. No fully automatic procdur was dscribd in th sns of automatic tuning for SISO systms. It is blivd that this can only b don if th considrd class of systms is mor limitd than in this papr. It was pointd out through an xampl that for simpl multivariabl control systms th proposd mthod agrs with automatic SISO tuning. For difficult MIMO control problms th mthod still provids a solid ground for controllr dsign. 6. Rfrncs [] K. J. ÅSTRÖM and T. HÄGGLUND. Automatic tuning of simpl rgulators with spcifications on phas and amplitud margins. Automatica,, pp , 984. [] K. J. ÅSTRÖM and T. HÄGGLUND. PID Controllrs: Thory, Dsign, and Tuning. Instrumnt Socity of Amrica, Rsarch Triangl Park, NC, scond dition, 995. [3] K. J. ÅSTRÖM, T.H.LEE, K.K.TAN, and K. H. JOHANS- SON. Rcnt advancs in rlay fdback mthods a survy. In IEEE SMC Confrnc, pp. 66 6, Vancouvr, WA, 995. Invitd papr. [4] W. L. BIALKOWSKI. Drams vs rality: A viw from both sids of th gap. In Control Systms 9, Whistlr, B.C., Canada, 99. [5] D. B. ENDER. Procss control prformanc: Not as good as you think. Control Enginring, 4:, pp. 8 9, 993. [6] M. FRIMAN and K. V. WALLER. Autotuning of multiloop control systms. Ind. Eng. Chm. Rs., 33, pp , 994. [7] C. C. HANG, A.P.LOH, and V. U. VASNANI. Rlay fdback auto-tuning of cascad controllrs. IEEE Trans. on Control Systms Tchnology, :, pp. 4 45, 994. In 34th IEEE Confrnc on Dcision and Control, Nw Orlans, LA, 995. [9] K. H. JOHANSSON. Rlay fdback and multivariabl control. PhD thsis ISRN LUTFD/TFRT--48--SE, Dpartmnt of Automatic Control, Lund Institut of Tchnology, Lund, Swdn, Novmbr 997. [] K. H. JOHANSSON and J. L. R. NUNES. A multivariabl laboratory procss with an adjustabl zro. In 7th Amrican Control Confrnc, Philadlphia, PA, 998. [] J. M. MACIEJOWSKI. Multivariabl Fdback Dsign. Addison-Wsly, Rading, MA, 989. [] Z. J. PALMOR, Y. HALEVI, and N. KRASNEY. Automatic tuning of dcntralizd PID controllrs for TITO procsss. Automatica, 3:7, pp., 995. [3] P. PERSSON. Towards Autonomous PID Control. PhD thsis ISRN LUTFD/TFRT--37--SE, Dpartmnt of Automatic Control, Lund Institut of Tchnology, Lund, Swdn, April 99. [4] V. RECICA. Automatic tuning of multivariabl controllrs. Mastr thsis ISRN LUTFD/TFRT SE, Dpartmnt of Automatic Control, Lund Institut of Tchnology, Lund, Swdn, 998. [5] T. S. SCHEI. A mthod for closd loop automatic tuning of PID controllrs. Automatica, 8:3, pp , 99. [6] T. S. SCHEI. Automatic tuning of PID controllrs basd on transfr function stimation. Automatica, 3:, pp , 994. [7] S.-H. SHEN and C.-C. YU. Us of rlay-fdback tst for automatic tuning of multivariabl systms. AIChE Journal, 4:4, pp , 994. [8] F. G. SHINSKEY. Controlling Multivariabl Procsss. Instrumnt Socity of Amrica, Rsarch Triangl Park, NC, 98. [9] V. U. VASNANI. Towards Rlay Fdback Auto-Tuning of Multi-Loop Systms. PhD thsis, National Univrsity of Singapor, 994. [] Q. G. WANG, B. ZOU, T. H. LEE, and Q. BI. Auto-tuning of multivariabl PID controllrs from dcntralizd rlay fdback. Automatica, 33:3, pp , 997. [] P. ZGORZELSKI, H. UNBEHAUEN, and A. NIEDERLINSKI. A nw simpl dcntralizd adaptiv multivariabl rgulator and its application to multivariabl plants. In IFAC th Trinnial World Congrss, pp , Tallinn, Estonia, 99. [] M. ZHUANG and D. P. ATHERTON. Optimum cascad PID controllr dsign for SISO systms. In IEE Control 94, pp. 66 6, Warwick, 994. [3] M. ZHUANG and D. P. ATHERTON. PID controllr dsign for a TITO systm. IEE Proc. Control Thory Appl., 4:, pp., 994. [8] B. JAMES and G. F. BRYANT. A paramtrization for automatic loop-by-loop multivariabl controllr dsign.