Partial Relative Gain: A New Tool for Control Structure Selection

Paper 193h, 1997 AIChE Annual Meeting, November 16{21, Los Angeles, CA

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Partial Relative Gain: A New Tool for Control Structure Selection Kurt E. H
aggblom Process Control Laboratory, Faculty of Chemical Engineering,  Abo Akademi University, FIN{20500  Abo (Turku), Finland E-mail: khaggblo@abo., Fax: +358{2{2154479 Keywords: process control, control structure selection, decentralized control, integral controllability, integrity, relative gain array, partial relative gain, partial control

Abstract. A new procedure for analysis and selection

of control structures is presented. In addition to the relative gain array (RGA) for the entire openloop system, the RGA for the system under partial control is considered. This partial relative gain (PRG) provides necessary conditions for a control conguration to be integral controllable with integrity. The PRG also solves the variable pairing problem in cases where conventional use of the RGA fails or is ambiguous, which may occur for systems larger than 2 2. Furthermore, it may be inferred from the PRG when block-decentralized control should be considered.

1. Introduction

Relative gain analysis is a widely used technique in control system design for multivariable processes (Bristol, 1966 Shinskey, 1979, 1984 McAvoy, 1983). The analysis is based on a \Relative Gain Array" (RGA), which is a matrix of interaction measures for all possible single-input single-output (SISO) pairings between the variables considered. The RGA thus indicates the preferable variable pairings in a decentralized (multiloop SISO) control system based on interaction considerations. Further developments have shown that the RGA also provides information about fundamental properties such as integral controllability, integrity, and robustness with respect to modeling errors and input uncertainty (Grosdidier et al., 1985 Skogestad and Morari, 1987 Yu and Luyben, 1987 Campo and Morari, 1994). Despite its considerable popularity, conventional relative gain analysis has several limitations (Haggblom, 1995). To deal with the deciencies of the RGA, a number of complementing measures and procedures have been proposed. The Block Relative Gain (BRG) (Manousiouthakis et al., 1986) and the Niederlinski Index (NI) (Niederlinski, 1971) are two measures of similar simplicity as the RGA. The BRG is used to analyze the feasibility of a block-decentralized control structure and NI gives a necessary condition for closedloop stability of a (block) decentralized control system. c November 1, 1997 (UNPUBLISHED) Copyright


More comprehensive tests of integral controllability (unconditional stability of the closed-loop system) and integrity (failure tolerance) are considered in Grosdidier and Morari (1986), Mijares et al. (1986), Yu and Luyben (1986), Yu and Fan (1990), Skogestad and Morari (1992), Campo and Morari (1994). In general these tests, which tend to be rather complicated, provide conditions that are only necessary or only sucient. In practice, the various measures and procedures can be used to eliminate a number of inferior control structures, but not necessarily to nd the \best" control structure. In this paper a new, RGA based, procedure for control structure selection is presented. The Partial Relative Gain (PRG) for a subsystem is dened as the RGA for the subsystem with the rest of the system under integral feedback control, that is, a partially controlled system (Haggblom, 1996). In addition to the RGA for the entire open-loop system, the PRGs for relevant subsystems are considered. It is proved that integral controllability with integrity requires that all variable pairings in a decentralized control system correspond to positive relative gains in the RGA for the openloop system as well as positive PRG elements in all relevant subsystems. This means that infeasible control structures can easily be eliminated. The PRG also makes it possible explicitly to compare feasible control structures with each other. This is very useful since variable pairing based on the RGA for the open-loop system is unreliable for systems larger than 2 2. Furthermore, it can be inferred from the PRG when block-decentralized control should be considered (Haggblom, 1997a). The usefulness of the PRG is demonstrated by examples.

2. Basic Concepts A linear

system can be described by the model ( )= ( ) ( ) (1) where ( ) and ( ) are -dimensional vectors of inputs and outputs, respectively, and ( ) is a matrix of transfer functions. It is assumed that ( ) is stable and strictly proper, and that the steady-state gain matrix = (0) is nonsingular. n

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2.1. The relative gain array

The steady-state relative gain array (RGA) of the system ( ) is dened ( ) =  ;T (2) where  denotes element-by-element multiplication (often called the Hadamard or Schur product), and ;T is the transpose of ;1 . Let and  be partitioned as

  11 12 11 12 = =   (3a 3b) 21 22 21 22 where 22 is assumed to be nonsingular. Equation (2) and the inverse of a block matrix partitioned as in Eq. (3a) give 11( ) = 11  ;11T (4) where 11 = 11 ; 12 ;221 21 (5) is the Schur complement of 22 (Kailath, 1980). 11 is also the eective gain matrix of subsystem 11( ) when the rest of the system (i.e., 22( )) is closed under integral feedback control. From Eq. (4) it follows that the relative gain ij ( ) for the variable pairing i { j is given by (6) ij ( ) = ij ij where ij is the open-loop gain between i and j and ij is the apparent gain between the same variables when the rest of the system is under integral feedback control. Equations (3{5) adequately dene the relative gains in a square submatrix of the full RGA, but a dierent notation is useful in this paper. Let m denote a square submatrix of and m ( ) the corresponding submatrix of ( ). Then m ( ) = m  ;mT (7) where m is the eective gain matrix of subsystem m ( ) when the rest of the system is closed under integral feedback control. The submatrix m ( ) is dierent from the RGA for subsystem m ( ) when the rest of the system open. According to Eq. (2), the RGA for m is ( m ) = m  ;mT (8) G s

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2.2. The block relative gain

The block relative gain (BRG) is a useful feasibility measure of block-decentralized control, where multiple-input multiple-output (MIMO) control is considered for part of the system (Manousiouthakis et al., 1986 Grosdidier and Morari, 1987 Chiu and Arkun, 1990). The steady-state BRG for MIMO control of subsystem m ( ) is dened (9) Bm ( ) = m ;m1 According to Eqs. (7) and (9), the BRG for a SISO subsystem is equivalent with the corresponding relaG

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2 tive gain. The BRG is thus a generalization of the RGA.

2.3. The Niederlinski index Let ~ denote the matrix obtained by setting to zero G

all elements of that do not correspond to an inputoutput pairing in a given (block) decentralized control structure. The Niederlinski index (NI) for the control structure dened by ~ is then (Chiu and Arkun, 1990) (10) G~ ( ) = det( ) det( ~) For a fully decentralized control structure, det( ~) is equal to the product of the gains corresponding to the input-output pairings. Thus, if the variable pairings Q are along the diagonal of , det( ~) = i ii . In that case, the Niederlinski index is denoted by ( ) (i.e., the subscript ~ is dropped for convenience). G

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2.4. Integral controllability and integrity

In this section some denitions and theorems are given that are relevant for control of a system ( ) by a decentralized feedback controller with integral action. It is assumed that the inputs and the outputs are ordered in such a way that the controller is diagonal. The section is adapted from Grosdidier et al. (1985), Chiu and Arkun (1990) and Campo and Morari (1994). Denition ICI. A system ( ) is integral controllable with integrity (ICI) if there exists a controller such that the closed-loop system is unconditionally stable and remains stable when individual controllers are arbitrarily brought in and out of service. The closed-loop system has this property if it remains stable when the gains of all individual controllers are simultaneously detuned by a factor in the range 0  1 as well as when the gains of any combination of individual controllers are set to 0. Denition DIC. A system ( ) is decentralized integral controllable (DIC) if there exists a controller such that the closed-loop system is decentralized unconditionally stable. The closed-loop system has this property if it remains stable when the gains of any combination of controllers are detuned by individual factors i in the range 0  i  1. Theorem ICI. ( ) is ICI only if ( ) 0 and ( m ) 0 for all principal submatrices m of size , = 2 . . . ; 1. An equivalent condition is 0, = 1 . . . , and ii ( m ) 0, = ii( ) 1 ... . Usually DIC is a desired property since it allows the individual controllers to be arbitrarily detuned. The known tests for DIC tend to be rather complicated, however. Furthermore, they are, in general, only necessary but not sucient, or sucient but conservative. Since DIC implies ICI according to the denitions, Theorem ICI gives necessary conditions for DIC. G s

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3. Limitations of Open-Loop RGA

The conventional variable pairing rule for decentralized control based on the RGA is introduced and some important properties of the RGA are highlighted. A fundamental limitation of the open-loop RGA and the inadequacy of the conventional variable pairing rule are illustrated by two examples.

3.1. Variable pairing based on RGA

The RGA was originally proposed as a tool for pairing controlled and manipulated variables in decentralized (multiloop SISO) control systems (Bristol, 1966). From Eq. (6) it follows that the open-loop gain ;1 ij between i and j changes by the factor ij when the rest of the system is closed by integral feedback control. This implies that variable pairings corresponding to positive relative gains as close to unity as possible should be preferred. Negative relative gains or relative gains much larger than unity should be avoided and large negative relative gains are particularly undesirable (Bristol, 1966). It has been shown, under fairly general assumptions regarding the system and the controller, that a decentralized control system with a variable pairing i { j having ij 0 has at least one of the following properties (Grosdidier et al., 1985): (a) the closed-loop system is unstable (b) the loop i { j is unstable by itself (when all other loops are open) (c) the closedloop system is unstable if the loop i { j is open. Consequently, a variable pairing on a negative relative gain gives a closed-loop system which is conditionally stable, at best. This explains the relative gain conditions in Theorem ICI. According to the conventional pairing rule, a variable pairing with ij = 0 should not be considered. However, it is known that such a variable pairing may result in a stable and well-performing closed-loop system. The feasibility of this kind of variable pairing is entirely dependent on other control loops (Shinskey, 1984 Haggblom, 1996, 1997b). Hence, such a control system lacks integrity. This is not always a serious drawback, however. In distillation, for example, all so-called material balance control structures contain variable pairings having a relative gain equal to zero. g

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