h D D x D F x F V h B B x B LV DB RV RS RB LB DS DV frequency (1/min) magnitude of relative gain

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1 199 AIChE Annual Meeting, Noveber 1{1, Los Angeles, CA Paper 19h Partial Relative Gain: A New Tool for Control Structure Selection Kurt E Haggblo Process Control Laboratory Faculty of Cheical Engineering Abo Akadei University FIN{000 Abo (Turku) Finland E-ail: khaggblo@abo Fax: +8{{19 Abstract A new procedure for control structure selection based on the Relative Gain Array (RGA) is presented In addition to the RGA for the full openloop syste, the RGA for the syste under partial control is considered (PRG) This Partial Relative Gain provides necessary conditions for integral controllability and integrity solves the variable pairing proble in decentralized control ore reliably than the conventional RGA, which ay fail or be abiguous for systes larger than indicates when block-decentralized control should be considered 199 AIChE Annual Meeting, Paper 19h Characteristics of the RGA Interpretations of the RGA The Relative Gain Array (RGA) is a atrix of interaction easures for all possible single-input single-output variables The RGA (SISO) pairings between a set of indicates the preferable variable pairings in a decentralized (ultiloop SISO) control syste based on interaction considerations provides inforation about fundaental properties such as integral controllability, integrity, and robustness with respect to odelling errors and input uncertainty is not a true eaure of closed-loop interactions, which eans that the RGA ay fail for systes larger than The open-loop gain g ij will change by the factor 1, where ij ij is the relative gain for pairing output \ i" with input \ j ", when other control loops are closed This iplies: Variable pairings with positive ij as close to unity as possible should be preferred ij =1 indicates a \perfect" variable pairing Negative relative gains should be avoided < 0 results in a closed-loop syste which is only conditionally stable, at best Relative gains uch larger than unity should be avoided a syste with ij 1 ay be practically uncontrollable If g ij and ij =0,the relative gain does not indicate whether the variable pairing is feasible control depends entirely on other control loops 199 AIChE Annual Meeting, Paper 19h 199 AIChE Annual Meeting, Paper 19h

2 Liitations of Open-Loop RGA Two-Product Distillation Colun Exaple 1: Necessity to consider partially controlled syste Two-product distillation colun with total condenser (Haggblo and Waller, 1991): y(s) Gyu (s) 0 = z(s) G zu (s) Is 1 u(s) v(s) L h D D x D y =[x D x B ] T u =[L V ] T z =[h D h B ] T v =[D B ] T F x F RGA: T Gyu (s) 0 Gyu (s) 0 (s) = G zu (s) Is 1 G zu (s) Is 1 yu (s) 0 = 0 I V h B B x B where yu (s) =G yu (s) G yu (s) T (s) iplies that y should always be controlled by u and z by v (ie, the LV -structure) 199 AIChE Annual Meeting, Paper 19h 199 AIChE Annual Meeting, Paper 19h Distillation Control Structures Liitations of Open-Loop RGA agnitude of relative gain LV DB LS RV RS RB LB DS DV frequency (1/in) Frequency-dependent relative gains for dierent distillation control structures ( R = L=D, S = V=B) Exaple : Inadequacy of conventional variable pairing rule Exaple by Hovd and Skogestad (199): (1 s) G(s) = (1 + s) Variable pairing on 1 :19 :9 :19 1 : (G(s)) = ij = 1 (conguration C 1 ) results in a closed-loop tie constant CL 110, ij = (conguration C 1 ) results in a closed-loop tie constant CL AIChE Annual Meeting, Paper 19h 199 AIChE Annual Meeting, Paper 19h 8

3 Partial Control Exaple (cont'd) Variable pairing on ij =1 (cong C 1 ) Closing control loop y {u results in the transfer atrix (1 s) :9 1: G 10 (s) = (1 + s) :1 :9 for the reaining subsyste The RGA for the partially controlled syste is ( G10 (s)) = :98 :98 :98 :98 ) Control perforance will be very sluggish Variable pairing on ij = (cong C 1 ) Closing control loop y {u 1 results in a partially controlled syste with the RGA ( G0 (s)) = :19 :19 :19 :19 ) No particular control probles Controllability Measures Linear n n syste: y(s) =G(s)u(s) The Relative Gain Array (G) =G G T Let G and be partitioned as G11 G G = 1 11 = 1 G 1 G 1 where G is assued to be nonsingular Then where 11 (G) =G 11 G T 11 G 11 = G 11 G 1 G 1 G 1 is the Schur copleent of G Note: G11 is also the eective gain atrix of subsyste G 11 (s) when the rest of the syste (ie, G (s)) is closed under integral feedback control 199 AIChE Annual Meeting, Paper 19h AIChE Annual Meeting, Paper 19h 10 Controllability Measures Integral Controllability and Integrity The Relative Gain Array (cont'd) Let G denote a square subatrix of G and (G) the corresponding subatrix of (G) Then (G) =G G T is the eective gain atrix of subsyste G (s) when the rest of the syste is closed under integral feedback control where G The RGA for subsyste G (s) with the rest of the syste open is (G )=G G T The Block Relative Gain The BRG for MIMO control of subsyste G (s) is dened B (G) =G G 1 A syste G(s) is integral controllable with integrity (ICI) if there exists a controller such that the closed-loop syste is unconditionally stable and reains stable when individual controllers are arbitrarily brought in and out of service The closed-loop syste has this property if it reains stable when the gains of all individual controllers are siultaneously detuned by a factor in the range 0 < 1 as well as when the gains of any cobination of individual controllers are set to 0 Theore ICI For variable pairing along the diagonal, G(s) is ICI only if N(G) > 0andN(G k ) > 0 for all principal subatrices G k of size k k, k = n 1 An equivalent condition is ii (G) > 0, i = 1 n,and ii (G k ) > 0, i =1 k 199 AIChE Annual Meeting, Paper 19h AIChE Annual Meeting, Paper 19h 1

4 The Partial Relative Gain Let G (s) denote the transfer atrix of subsyste G (s) when the rest of the syste G(s) is under integral feedback control The partial relative gain (PRG) for subsyste G (s) is then P (G) =( G )= G G T The PRG provides inforation that other easures do not: Theore PRG P (G) = (G) =G G T = (G )=G G T = B (G) =G G 1 For variable pairing along the diagonal, G(s) is ICI only if ii (G) > 0, i =1 n, and ii ( G k ) > 0, i = 1 k, for all principal subsystes G k (s) of size k k, k = n 1 If, in addition to ii (G) > 0, the Niederlinski index N(G) > 0, the condition for k = is redundant Proof of Theore PRG (Partial) Consider a syste G(s) and a principal subsyste G (s) containing the ith input and output of G(s) Denote by G ii (s) the subsyste obtained by exluding the variables of G (s), except the ith ones RGA: PRG: ) ) G = G gii G ii (G) =G G T ( G )= G G T G ( G )= (G) G gii ii ( G )= ii (G)( G ) ii ( G ) ii is the eective gainofg ii excluding G is closed ) ii (Gii )=gii =( G ) ii when the syste Cobination of the last two equations gives ii ( G )= ii (G)= ii (Gii ) ) RGA conditions of Theore ICI and Theore PRG are equivalent 199 AIChE Annual Meeting, Paper 19h AIChE Annual Meeting, Paper 19h 1 Application of the PRG Case study of Petlyuk distillation colun Model by Wol and Skogestad (199): 1: 19: 0: 0:0 G 1: 18: 0:10 1: = : 8:9 0: 0:10 :80 :09 0:1 :1 :0 :8 0:11 0:001 8:998 9:08 0:000 0:8990 (G) = 8:91 8: 1:0 0:0000 1:08 1:19 0:0 0:0998 Potential control congurations according to RGA: Cong Pairings Relative Gains C 1 f1{1,{,{,{g f: 0:90 1:0 1:g C 1 f1{,{,{1,{g f0:11 0:90 : 1:g C 1 f1{1,{,{,{g f: 9:1 1:0 0:10g C 1 f1{,{,{1,{g f0:11 9:1 8: 0:10g C 1 f1{1,{,{,{g f: 0:0 0:00 1:g C 1 f1{,{,{1,{g f0:00 0:0 8: 1:g Closing loop 1{1 gives: G 0 = ( G 0 )= 0:80 0:1 1:08 0:18 0:9 0:108 0:081 0:1 :091 0:10 0:0 0:900 0:0 1:0 0:0000 1:1190 0:18 0:099 Theore PRG: C 1 is not ICI C 1 appears very good Closing loop {1 gives: 0:11 0:81 0:00 ( G 0:18 0:1 0:8 0 )= 1:0 0:18 0:100 ) C 1 is not ICI, C 1 appears very good 199 AIChE Annual Meeting, Paper 19h AIChE Annual Meeting, Paper 19h 1

5 Closing loop { gives: 1:9 :9 ( G 10 )= :81 :008 :98 0:808 :9 1:811 0:000 ) C 1 and C 1 (!) are not ICI Congurations C 1 and C 1 pass all PRG tests for ICI Closing loops f1{1, {g in C 1 and f1{, {1g in C 1 gives: ( G 00 )=(G 00 )= 0:100 0:8980 0:8980 0:100 Conclusions A new procedure for screening of decentralized control structures has been presented In addition to the RGA for the full open-loop syste, the RGA for the syste under partial control is considered This partial relative gain (PRG) provides necessary conditions for integral controllability with integrity The PRG can solve the variable pairing proble when conventional use of the RGA fails or is abiguous The PRG ay indicate when block-decentralized control should be considered Partial relative gains: 0:8980 for C 1, 0:100 for C 1 ) C 1 should be preferred over C AIChE Annual Meeting, Paper 19h AIChE Annual Meeting, Paper 19h 18 Applications of the PRG Exaple : Eliination of infeasible congurations Exaple by Capo and Morari (199): G 1 1 0: = 0: 0: 0: 0: 0:8000 :0000 :1 0:0 :0000 :0000 1: 0: (G) = 0:0 0: 0:18 1: :1 1: 0:0 0:18 Congurations that ay be ICI according to the RGA: C 1, C 1, C 1, C 1, C 1 and C 1 Exaple (cont'd) Closing control loop y {u : G 10 = ( G10 )= 0 0 0: 1 0 0: 0: 0 1:0 0:0 0 1:0 0:0 1:0 0:0 0 Only conguration C 1 ay be ICI Conguration C 1 passes all tests for ICI based on Theore PRG Congurations C 1, C 1 and C 1 can be eliinated because they have negative Niederlinski indices 199 AIChE Annual Meeting, Paper 19h AIChE Annual Meeting, Paper 19h 0

6 Ranking of feasible congura- Exaple : tions Applications of the PRG Exaple by Niederlinski (191), Mijares et al (198): 1:0 0:1 1:0 G = 0: 0: 0:1 0: 0:8 0: 0:90 0:019 0:80 (G) = 0:00 0:911 0:109 0:191 0:8 0:11 Congurations with variable pairings on positive relative gain values: C 1, C 1, C 1, C 1 Exaple (cont'd) Coparison of C 1 and C 1 by closing loop y 1 {u 1 : 0: 0:0 G 0 = 0:8 0:0 ( G0 0:8 0:1 )= 0:1 0:8 The PRG iplies C 1 >C 1 By closing other loops it can be shown that C 1 > fc 1 C 1 g >C 1 All congurations satisfy the ICI requireents of Theore PRG 199 AIChE Annual Meeting, Paper 19h AIChE Annual Meeting, Paper 19h Applications of the PRG Exaple : Block-decentralized control Exaple studied by Alatiqi and Luyben (198), Grosdidier and Morari (198): :09 : 0: 0:9 G :1 :9 0:0 1: = 1: :11 :1 :8 11:8 1:0 0:10 :9 :108 0:900 0:9 0:0 :008 : 0:09 1:91 (G) = 0:088 0:0 1:9 0:19 :0088 :88 0:08 0:88 Only conguration with all variable pairings on positive relative gain values is C 1 C 1 satises all necessary ICI conditions of Theore PRG Exaple (cont'd) Closing loop y {u : ( G10 )= 0:199 0:0 1: 0:01 1:1 0: 0: 0:000 0: The PRG suggests that the variable pairings fy 1 {u 1 y {u g are inferior to fy 1 {u y {u 1 g Closing loop y {u in addition to y {u : ( G100 )= 0: 0: 0: 0: This PRG iplies that conguration C 1 would be better than C 1 However, C 1 is not ICI (NI < 0) The contradiction suggests a block-decentralized control structure with variable pairings fy {u y {u (y 1 y ){(u 1 u )g 199 AIChE Annual Meeting, Paper 19h 199 AIChE Annual Meeting, Paper 19h