Multiobjective Control Structure Design: A Branch and Bound Approach

Article pubs.acs.org/IECR

Multiobjective Control Structure Design: A Branch and Bound Approach Vinay Kariwala†,§ and Yi Cao*,‡ †

School of Chemical and Biomedical Engineering, Nanyang Technological University, Singapore School of Engineering, Cranfield University, Cranfield, Bedford MK 43 0AL, U.K.

‡

ABSTRACT: Control structure design traditionally involves two steps of selections, namely the selection of controlled and manipulated variables and the selection of pairings interconnecting these variables. The available criteria for both selections require enumeration of every alternative. Hence, an exhaustive search can be computationally forbidding for large-scale processes. On the other hand, owing to the computational complexity, variables and pairings are often selected sequentially, which may result in suboptimal control structures. In this paper, an efficient branch and bound (BAB) method is proposed to select the variables and pairings together in a multiobjective optimization framework. As an illustration of the proposed multiobjective BAB framework, the minimum singular value rule and the μ-interaction measure are used as the criteria for selection of controlled variables and pairings, respectively. Numerical tests using randomly generated matrices and the large-scale case study of hydrodealkylation of toluene (HDA) process show that the BAB method is able to reduce the solution time by several orders of magnitude in comparison with exhaustive search.

1. INTRODUCTION Control structure design (CSD) deals with the selection of controlled and manipulated variables (CVs and MVs) and the pairings interconnecting these variables.1 Simply stated, the objective of CSD is to decide the following: Given the process, where shall the controllers be placed? This choice is often not obvious for processes encountered in practice. This problem is further complicated by the mass and energy integration among the different process units, which necessitates consideration of the whole plant together. CSD for complete chemical plants is also known as plantwide control. With its practical implications, CSD has been studied by several researchers.1−4 A review of many of the available techniques is provided by Larsson and Skogestad.5 While some of the available approaches involve extensive use of process knowledge and heuristics,4,6 some model-based systematic methods have also been recently proposed.1,7 In model-based systematic methods, the selection of variables and pairings is carried out sequentially, which often requires enumeration of every alternative. The rapid growth of alternatives with process dimensions makes CSD through an exhaustive search computationally forbidding for large-scale processes. Furthermore, the selection of CVs and MVs followed by pairing selection may result in suboptimal control structures. For example, the selected CVs and MVs may lead to highly interacting control loops for all possible pairings rendering decentralized control difficult. This necessitates the consideration of different tasks of CSD together in a multiobjective optimization framework such that a set of promising solutions (Pareto-optimal set) can be found. Then, the practicing engineer can select the control structure from the Pareto-optimal set by trading-off different selection criteria. The CSD problem is a combinatorial optimization problem, where binary decision variables are related to the selection of variables and pairings. Here, a variable or pairing can be © 2012 American Chemical Society

considered to be selected, if the decision variable is one and vice versa. Traditionally, multiobjective CSD problem has been solved by converting the multiobjective problem into an optimization problem with a single objective by weighing different objectives8 or by converting all but one of the objectives to constraints (ε-constraint method).9 In these approaches, the binary decision variables are relaxed as continuous variables. Subsequently, the Pareto-optimal set is obtained by repeatedly solving the mixed integer linear or nonlinear program (MILP or MINLP) with different weights or constraint limits. A drawback of these approaches is that the choice of weights and constraint limits is nontrivial. Furthermore, the Pareto-optimal set obtained using weighted objective function approach is not necessarily complete.10 Evolutionary algorithms10 can directly handle the multiobjective nature of CSD problem but do not guarantee global optimality of the solution. Recently, efficient branch and bound (BAB) methods have been developed for selection of CVs,11−13 MVs,14,15 and pairings 16,17 by posing them as subset selection and permutation problems, respectively. These BAB methods guarantee globally optimal solutions, while requiring several orders of magnitude lower computational times in comparison with exhaustive search. These methods, however, still need to be applied sequentially (CV and MV selection followed by pairing selection). Motivated by this drawback, we propose a novel BAB method to directly solve the multiobjective CSD problem in this paper. The proposed BAB framework is general and can handle most of the available criteria for the selection of CVs, MVs, and Received: Revised: Accepted: Published: 6064

July 13, 2011 April 8, 2012 April 10, 2012 April 10, 2012 dx.doi.org/10.1021/ie201510c | Ind. Eng. Chem. Res. 2012, 51, 6064−6070

Industrial & Engineering Chemistry Research

Article

Figure 1. Solution tree for selecting control structure for a system with four outputs and three inputs.

G(s) evaluated at the frequency ω is represented as G(jω) ∈  m×n and the steady-state gain matrix as G ∈ m×n. For Pn ∈ n(Nm), GPn denotes the permuted n × n submatrix of G with rows indexed by Pn. Then, the selected CVs correspond to the row indices of GPn, while the pairings are selected on the diagonal elements of GPn. With these preliminaries, a multiobjective CSD problem involves solving the following optimization problem

pairings. For illustration purposes, a biobjective CSD problem is considered in this work, where the minimum singular value (MSV) rule18 and μ-interaction measure (μ-IM)19 are used for selection of CVs and pairings, respectively. The computational efficiency of the proposed method is demonstrated using randomly generated matrices and the large-scale case study of the hydrodealkylation of toluene (HDA) process.20 These numerical tests show that the BAB method is able to reduce the solution time by several orders of magnitude in comparison with an exhaustive search. We point out that BAB methods have been used earlier for solving multiobjective combinatorial optimization problems. The application of BAB methods, however, has mostly been limited for solving MILPs.21,22 Only recently, BAB methods have been presented for solving multiobjective subset selection23 and multiobjective permutation problems.17 The multiobjective combinatorial problem considered in this paper is more general, as it involves solving the subset selection problem together with the permutation problem. It is the first time that a BAB approach has been proposed for such a problem. The rest of this paper is organized as follows: the notation used in this paper is standardized in section 2. The general multiobjective BAB framework for CV and pairing selection is proposed in section 3. In section 4, the BAB approach is used to solve the biobjective CSD problem with MSV rule and μ-IM as the criteria for selection of CVs and pairings, respectively. The efficiency of the BAB approaches is demonstrated in section 5 through some numerical tests and the work is concluded in section 6.

min {J1(Pn), J2 (Pn), ..., Jq (Pn)}

Pn ∈n(Nm)

s.t. Li(Pn) ≤ 0,

i = 1, 2, ..., r

(1) (2)

A solution Pn ∈ n(Nm) is considered feasible if it satisfies all the constraints in eq 2. A feasible solution Pjn is said to be dominated by another feasible solution Pin if Jk (Pni) ≤ Jk (Pnj)

∀ k ∈ Nq

(3)

with strict inequality occurring for at least one k ∈ Nq. All nondominated feasible solutions construct the Pareto-optimal set 7 , where every pair of Pin, Pjn ∈ 7 satisfy ∃ s,t ∈ Nq: Js (Pni) < Js (Pnj), Jt (Pni) > Jt (Pnj)

(4)

Remark 1 (MV Selection). CSD problems with more MVs than outputs require selection of a subset of available MVs and their subsequent pairings with outputs. Such problems can be handled in the same framework by interchanging the roles of MVs and CVs or in other words, using GT(s) as the transfer function matrix between inputs and outputs.

2. PRELIMINARIES In this paper, {a, b} denotes an unordered set, while (a, b) denotes an ordered set, both consisting of the elements a and b. Note that {a, b} = {b, a}, but (a, b) ≠ (b, a). We define Nn as the set of first n natural numbers, i.e. Nn = {1, 2, ..., n}, where the subscript n is explicitly used to denote the size of the set. For an m-element set, Xm with m > n, n(Xm) represents the ensemble of all possible permutations of n-element subsets of Xm. For example, for m = 3 and n = 2, the 2-element subsets of N3 = {1, 2, 3} are {1, 2}, {1, 3}, and {2, 3}. Thus, 2(N3) = {(1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)}. We assume that the set of MVs or inputs is given; see Remark 1. Then, for a process with m measurements or outputs y = {y1, y2, ..., ym} and n inputs u = {u1, u2, ..., un}, m > n, the CSD problem consists of selecting n CVs among the available m outputs and their subsequent pairings with inputs. For Pn = (p1, ..., pn) ∈ n(Nm), it is considered that {yp1, yp2, ..., ypn} are selected as CVs and {yp1 − u1, yp2 − u2, ..., ypn − un} as the pairings. For example, for m = 3 and n = 2, if P2 = (3, 2), the selected CVs are {y3, y2} and the selected pairings are {y3 − u1, y2 − u2}. G(s) denotes the stable transfer function matrix relating the inputs and outputs of the process. The transfer function matrix

3. BAB METHOD FOR MULTIOBJECTIVE CSD A BAB approach for subset selection was first proposed by Narendra and Fukunaga24 in the context of pattern recognition. This algorithm has been subsequently improved by several researchers.25−27 Cao and Saha15 introduced the concepts of fixed and candidate sets to describe the BAB principle. On the basis of these concepts, Cao and Kariwala11 proposed the bidirectional BAB method, which significantly improved the computation efficiency. In this section, these concepts are used to derive a general BAB method for multiobjective optimization problems involving subset selection (CV selection) together with permutation (pairing selection). Principle. Let S represent the multiobjective CSD problem in section 2. The BAB method branches the problem S into several nonoverlapping subproblems (partially selected CVs and assigned pairings). The subproblem Si is pruned, if a set of lower bounds on selection criteria over all complete solutions of Si is dominated by a member of the current Pareto-optimal set 7 ; otherwise, Si is branched further. Here, a complete solution refers to Pn ∈ n(Nm), i.e. pairings assigned for n CVs selected from m candidate outputs. Whenever a complete control structure is reached, which is also Pareto-optimal, 7 is 6065

dx.doi.org/10.1021/ie201510c | Ind. Eng. Chem. Res. 2012, 51, 6064−6070


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J̲ (:) ≥ Ji (Pnj) i

updated. Next, we provide further details of the three steps, namely branching, pruning, and updating. Branching. The implementation of BAB schemes requires a solution tree containing all possible alternatives. For combined CV and pairing selection problem, the solution tree for m = 4 and n = 3 is shown in Figure 1. The tree has (n + 1) levels, where level i corresponds to the ith input ui (except level 0, which corresponds to the empty root node). A node at level i represents a partially assigned pairing, where the label of the node denotes the output with which ui is paired. At level i, a node has (m − i) subnodes and the pairings assigned at a node are passed to all its subnodes. The tree has n!Cmn terminal nodes (marked by gray circles in Figure 1), which represent different pairing alternatives Pn ∈ n(Nm). Note that the solution tree described here is different from the solution tree for subset selection,11 which has only Cmn terminal nodes, and the solution tree for permutation,17 where the number of terminal nodes is n!. For ease of notation in the subsequent discussion, we introduce the concepts of fixed and candidate sets. Definition 1 (Fixed Set) . The fixed set Ff is the ordered set of f output indices representing the partially assigned pairing. Definition 2 (Candidate Set). The candidate set Cc is an unordered set, whose elements can be chosen to append Ff. On the basis of these definitions, a node in the solution tree is uniquely defined by its fixed and candidate sets, and is denoted as a 2-tuple, i.e. S = (Ff, Cc). Furthermore, the solution tree is branched as follows: Definition 3 (Branching Rule). A node S = (Ff, Cc) has c = (m − f) branches. The fixed and candidate sets of the ith subnode Si = (Fif+1, Cic−1), i = 1, 2, ..., c are defined as F if + 1 = (Ff , ci),

Cci− 1 = Cc \ci

with strict inequality occurring for at least one i ∈ Nq. Then the set : ⊆ n(Nm) cannot contain any Pareto-optimal solutions and hence can be pruned, i.e. none of the solutions contained in : need to be evaluated. Pruning can also be performed, if for any constraint Lk ≤ 0, k = 1, 2, ..., S , L̲ k (: ) ≥ 0, where L̲ k(: ) is a lower bound of Lk over the set : . Updating. Whenever the BAB method encounters a terminal node Pn, the current Pareto-optimal set 7 is updated. Pn is included in 7 , if for every Pjn ∈ 7 , ∃ i ∈ Nq: Ji (Pn) < Ji (Pnj)

On the other hand, Jk (Pn) ≤

∀ Pn ∈ :

(9)

is removed from 7 if (10)

4. BIOBJECTIVE CSD PROBLEM To illustrate the solution of multiobjective CSD problem using BAB method, we consider a biobjective CSD problem using the minimum singular valve (MSV)18 rule as the CV selection criterion together with the μ-interaction measure (μ-IM)19 as the pairing criterion. In this section, these criteria and their lower bounds are discussed. 4.1. Minimum Singular Value. CV selection involves identifying a subset among the set of available measurements (subset selection). A survey of most of the methods available for CV selection is given by Van de Wal and de Jager.28 Recently, Skogestad29 introduced the concept of self-optimizing control, which involves selecting CVs such that in presence of disturbances, the loss incurred in implementing the operational policy by holding these CVs at constant set points is minimal, as compared to the use of an online optimizer. The choice of CVs based on the general nonlinear formulation of selfoptimizing control can be time-consuming. To quickly prescreen the alternatives, Skogestad and Postlethwaite18 presented the approximate minimum singular value (MSV) rule, which is a local method based on linearized steady-state model and second order accurate economic objective function. Some other local methods for selection of CVs in the selfoptimizing control framework have also been recently proposed.30,31 To present the MSV rule, let Ĝ = D1GD2 denote the scaled gain matrix, where D1 and D2 are the output and input scaling matrices, respectively.18,29 It is noted by Kariwala and Skogestad32 that D1 and D2 are independent of the selected CVs. Then, CVs can be selected using MSV rule by solving

(5)

(6)

where : ⊆ n(Nm) . For example, for the rightmost node on level 2 and its adjacent node with label 2 in the solution tree shown in Figure 1, : = {(4, 3, 1), (4, 3, 2)} and : = {(4, 2, 1), (4, 2, 3)}, respectively. Let J ̲ i(: ) be a lower bound of Ji, i = 1, 2, ..., q, calculated over all the elements of : , i.e. J̲ (:) ≤ Ji (Pn) i

Jk (Pnj)

Pjn

with strict inequality occurring for at least one k ∈ Nq. The BAB framework presented in this section is general and is suitable not only for multiobjective CSD problems but also for any other multiobjective problems which involves both subset selection and permutation. This is the first time, where a BAB approach has been used to solve multiobjective integrated subset selection and permutation problems.

where ci is the ith element of Cc. To illustrate the concepts of fixed and candidate sets, consider the rightmost node on level 2 in the solution tree shown in Figure 1. The fixed and candidate sets of this node are F2 = (4, 3) and C2 = {1, 2}, respectively. Similarly, the fixed and candidate sets of the adjacent node with label 2 are F2 = (4, 2) and C2 = {1, 3}, respectively. It can be easily established that the solution tree branched based on Definition 3 is nonredundant and every terminal node belongs to one and only one branch.17 Pruning. It is noticeable that the number of nodes in the solution tree shown in Figure 1 is much larger than the number of feasible control structures, i.e. n!Cmn . A BAB method gains its efficiency by pruning branches of the tree, which cannot lead to the optimal solution. To illustrate this, consider a node S = (Ff, Cc). Then, the ensemble of all n-element ordered sets that can be obtained by expanding Ff is given as : = {(Ff , Pn − f )|Pn − f ∈ n − f (Cc)}

(8)

max σ̲(Ĝ Xn)

X n ⊂ Nm

(11)

where σ̲ is the MSV and Ĝ Xn is the submatrix of Ĝ with rows indexed by the unordered set Xn. We point out that for processes with more MVs than outputs, the set of MVs can also be selected by maximizing MSV of the unscaled gain matrix.18 In either case, the goal is to square a matrix such that the MSV

(7)

Further, assume that the current Pareto-optimal set 7 contains a member Pjn such that 6066



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of the square matrix is maximized. Different BAB methods for solving this problem have been proposed.14,32 Cao and Kariwala11 have combined these approaches to propose a very efficient BAB method for CV selection based on the MSV rule. 4.2. μ-Interaction Measure. For appropriate input−output pairing selection, a number of criterion are available in the literature.18,33 In this paper, we use μ-IM19 as the pairing selection criteria. The μ-IM is useful for assessing the feasibility of stabilizing the closed-loop system through independent design of individual loops of the controller. To present this method, let the rows of the n × n dimensional transfer matrix GPn(s) be permuted such that the chosen pairings lie along the diagonal and G̃ Pn(s) represent the matrix consisting of the diagonal elements of GPn(s). Then, the diagonal controller K(s) stabilizing G̃ Pn(s) also stabilizes GPn(s), if19 σ̅(T̃(jω)) < μΔ−1(E(jω))

∀ω∈

s.t. [Λ(G Pn)]ii > 0; −1 det(G PnG̃ Pn ) > 0

min {Jσ (Pn), Jμ (Pn)}

s.t. [Λ(G Pn)]ii > 0;

(20) (21)

where

(12)

Jσ (Pn) = σ̲−1(Ĝ Pn)

(22)

Jμ (Pn) = μΔ̅ (G Pn(I ◦G Pn)−1 − I )}

(23)

Note that the maximization of σ̲(G̃ Xn) in eq 11 is equivalent to minimization of σ̲−1(G̃ Xn). Furthermore, for any Xn ⊂ Nm, Xn ∈ n(Nm). Thus, the CVs can be selected through minimization of σ̲−1(G̃ Pn) over Pn ∈ n(Nm). We point out that for given Xn, Jσ(Pn) is the same for all Pn ∈ n(Xn), where n(Xn) ⊂ n(Nm). Thus, a BAB method based on Jσ(Pn) in eq 22 will involve redundant computations. To avoid such computational redundancy, we introduce the concept of semifixed set. Definition 4 (Semifixed Set). The semifixed set Fs is an unordered set, whose elements can be freely permuted to append Ff. Then, for CV selection, the union of Ff ∪ Fs is treated as the fixed set, while Cc\Fs is the corresponding candidate set. Remark 2 (Semifixed Set). Introducing the concept of semifixed set is the key to develop the BAB algorithm for the integrated subset and pairing selection problem. This is because for a particular subset of r variables, there are r! pairings, which have the same value of the subset selection criterion. The use of conventional BAB approach will involve a large amount of redundant computations. The concept of semifixed set is introduced here to replace the fixed set for subset selection related calculations to avoid such redundancy so that the efficiency of the algorithm is ensured. The application of a BAB method requires lower bounds on selection criteria for pruning purposes. To present these lower bounds, let : in eq 5 represent the ensemble of all possible solutions of the node S = (Ff, Fs, Cc). Then, a pair of lower bounds on Jσ(Pn) and Jμ(Pn) for all Pn ∈ : are given as11,16

(13)

(14)

J̲ (:) = max(σ̲−1(Ĝ Ff ∪ Fs), σ̲−1(Ĝ Ff ∪ Cc)) σ

(15)

J̲ (:) = ρ(G Ff G̃ Ff μ

The left-hand-side of the above condition is called the Niederlinski Index (NI). Then, pairing selection consists of solving the following optimization problem: min μΔ̅ (G Pn(I ◦G Pn)−1 − I )

i = 1, 2, ..., n

−1

where ◦ is the elementwise or Hadamard product. In particular, pairings are avoided on the negative elements of RGA, which is a necessary condition for integrity of the closed-loop system against loop failure.38,39 Furthermore, for decentralised control with integral control action, the following condition is sufficient to ensure the existence of a stabilizing controller:38

Pn ∈ n(Nn)

(19)

det(G PnG̃ Pn ) > 0

The condition in eq 12 is called μ-IM. This powerful result allows the designer to impose restrictions on the individual controllers, but still design the controller solely based on G̃ Pn(s) such that closed loop stability is ensured. Though originally proposed for stable systems, μ-IM has recently been extended to unstable systems as well.35 Note that if the pairings are chosen such that μΔ(E(jω)) is small at all frequencies, the restrictions on decentralized controller synthesis using independent design method is minimum. The term μΔ(E(jω)) can also be seen as a measure of generalized diagonal dominance.36 Thus, the use of BAB method is also useful to find the pairing for which GPn(jω) is most diagonally dominant. As the exact computation of μ is computationally intractable, we instead minimize the upper bound on μ (denoted as μ̅) obtained through D-scaling method.34 The μ-IM can be defined at any frequency. For simplicity, in the rest of the work, we use the steady-state μ-IM. Another useful criteria for pairing selection is based on relative gain array (RGA)37 defined as

−1 det(G PnG̃ Pn ) > 0

(18)

P ∈n(Nm)

−1

Λ(G Pn) = G Pn◦G Pn−T

(17)

A BAB method to efficiently solve the optimization problem in eqs 16 and 17 is presented by Kariwala and Cao.16 4.3. Combined CV and Pairing Selection. When both CV selection using MSV rule and pairing selection using μ-IM are considered together, the multiobjective optimization problem is given as

where μ denotes the structured singular value34 computed with a diagonally structured Δ, T̃ (s) = G̃ PnK(s)(I + G̃ PnK(s))−1 and E(s) = (G Pn(s) − G̃ Pn(s))G̃ Pn (s)

i = 1, 2, ..., n

−1

− I)

(24) (25)

where ρ is the spectral radius. The lower bounds in eqs 24 and 25 can be used together to check whether the node is dominated by a member of the current Pareto optimal set and hence can be pruned. The constraints in eqs 20 and 21 are checked at the terminal nodes before updating the Paretooptimal set 7 .

(16) 6067



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The lower bounds in eqs 24 and 25 require the calculations of the singular values and eigenvalues, respectively, which are time-consuming, and hence are the performance bottleneck for the BAB algorithm. To improve the computational efficiency, the following determinant based pruning conditions have been proposed for single objective BAB algorithms:11,17 T det(Ĝ Ff ∪ FsĜ Ff ∪ Fs − Bσ I ) < 0 ⇒ σ̲−1(Ĝ Ff ∪ Fs) > Bσ

(26)

T

det(Ĝ Ff ∪ CcĜ Ff ∪ Cc − Bσ I ) < 0 ⇒ σ̲−1(Ĝ Ff ∪ Cc) > Bσ det(BμI − (G Ff G̃ Ff

−1

− I )) < 0 ⇒ ρ(G Ff G̃ Ff

−1

(27)

− I ) > Bμ (28)

det(BμI + (G Ff G̃ Ff

−1

− I )) < 0 ⇒ ρ(G Ff G̃ Ff

−1

− I ) > Bμ (29)

Note that in the bidirectional BAB algorithm, the conditions eqs 26 and 27 are used for upward and downward pruning, respectively.11 In this work, condition eq 26 has been modified such that it can be used with the semifixed set. These pruning conditions are particularly efficient when they are applied to prune subnodes through the Schur determinant formula because the main computation load, the matrix inverse, only needs to be calculated once for all subnodes.11,17 Here, the upward pruning condition is checked for all the elements in Cc\Fs to determine the elements, which would cause the pruning condition to be satisified, once these elements are appended to Ff ∪ Fs. Subsequently, any elements satisyfing the condition eq 26 are removed from Cc. In this way, multiple candidates are discarded based on calculation of one matrix inverse. Similarly, to carry out downward pruning at the subnode level, all the elements in Cc\Fs are checked to find the elements, whose elemination will lead to the satisfaction of the pruning condition eq 27. All the elements satisfying condition eq 27 are included in the semifixed set, i.e. these elements cannot be eliminated, hence have to be fixed for subset selection, but not fixed for pairing, so that multiple candidates are moved to the semifixed set based on calculation of one matrix inverse. To use the determinant conditions in eqs 26−29, the best upper bounds, Bσ and Bμ, have to be estimated through the following result. Theorem 1. Let the vectors ) σ and ) μ contain the MSV and μ-IM values of the current Pareto-set 7 , respectively. For the node S = (Ff, Fs, Cc), let bσ be the lower bound of the MSV calculated using eq 24. For the subset of 7 defined as P̃ = {Pn ∈ 7 |) σ(Pn) ≤ bσ}, let Bμ = maxPn∈P̃ ) μ(Pn). Then, if ρ(GFf G̃ Ff−1 − I) ≥ Bμ, all n-element solutions of S are dominated by at least one member of 7 and hence can be pruned. Similarly, let bμ be the lower bound of the μ-IM calculated using eq 25. For the subset of 7 defined as P̃ = {Pn ∈ 7 |) μ(Pn) ≤ bμ}, let Bσ = maxPn∈P̃ ) σ(Pn). Then, if17 max(σ̲−1(Ĝ Ff ∪ Fs), σ̲−1(Ĝ Ff ∪ Cc)) ≥ Bσ

Figure 2. Flowchart of branch and bound algorithm.

5. NUMERICAL TESTS In this section, we demonstrate the efficiency of the developed biobjective BAB method through numerical examples. First, we use randomly generated matrices to check the average performance of the BAB algorithm. Subsequently, the BAB method is applied for combined CV and pairing selection for the HDA process.20 All these tests are carried out on a notebook with Intel Core Duo Processor T2400 (1.83 GHz, 2MB RAM) using MATLAB 2007b. 5.1. Random Tests. First, the efficiency of the biobjective BAB algorithm is examined through some randomly generated matrices. The first test consists of selecting n CVs out of 2n measurements and their pairings with inputs. The BAB algorithm is applied to 1000 randomly generated 2n × n matrices for every n ranging from 2 to 9. The average computation time and number of node evaluations required by the BAB algorithm are shown in Figure 3. In the second test, the BAB algorithm is applied to 1000 4n × n random matrices and the average computation time and number of node evaluations are shown in Figure 4. For comparison purposes, the estimated computation time 4n and the number of node evaluations (n!* 2n n and n!* n ) required by a brute-force approach are also shown in Figures 3 and 4. For each n, the computation time required for a brute-force approach is estimated by multiplying the number of alternatives with the time required for evaluating the MSV and μ̅(E) of an n × n matrix (averaged over 1000 instances). Both tests show that the developed BAB algorithm is able to reduce the computation time by several orders of magnitude in comparison with brute force search. For 2n × n and 4n × n processes, the BAB algorithm is able to handle cases until n = 8 and n = 7 in 1 min, respectively. The brute force approach is

(30)

all n-element solutions of S are dominated by at least one member of 7 and hence can be pruned. A simplified flowchart for recursive implementation of the proposed BAB algorithm is shown in Figure 2. 6068



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Figure 3. Random tests for CSD for 2n × n systems: (a) computation time and (b) number of nodes evaluated. Figure 5. Selection criteria values of the Pareto-optimal set for the HDA process.

6. CONCLUSIONS The sequential selection of controlled or manipulated variables followed by pairing selection may lead to nonoptimal control structures. A multiobjective branch and bound (BAB) method is developed to find the Pareto-optimal set of control structures, which provides a clear picture of the trade-off between different selection criteria for the practicing engineer. The application of the proposed method is illustrated using a biobjective optimization problem with minimum singular value (MSV) rule and μ-interaction measure (μ-IM) as the selection criteria for controlled variables and pairings, respectively. Numerical tests are used to demonstrate the computational efficiency of the BAB method over brute force search. In the future, the computational efficiency of the BAB framework will be tested using more than two selection criteria and the method will also be extended to selection criteria other than MSV rule and μ-IM.

Figure 4. Random tests for CSD for 4n × n systems: (a) computation time and (b) number of nodes evaluated.

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only able to handle processes until n = 5 and n = 4, respectively, in the same time limit. 5.2. HDA Case Study. The developed algorithm is also applied to select CVs and pairings for the HDA process, which was used as a case study earlier by Cao and Kariwala.11 Details of self-optimizing control of this process are provided by Araujo et al.20,40 This process has 8 MVs; hence, 8 CVs are to be selected from 129 available measurements and to be paired with MVs. It is interesting to note that this problem has 8!* 8129 ≈ 6.15 × 1016 alternatives, whereas MATLAB requires about 2 ms to calculate the μ-IM of an 8 × 8 matrix. Therefore, if this problem were to be solved using the brute force approach, it would require 4 million years to find the globally optimal solution. In comparison, the developed BAB algorithm only requires about 3 min to solve the CSD problem. The Pareto-optimal set has 88 members, whose objective function values are shown in Figure 5. The Pareto-optimal set gives a clear overall picture for a designer to select the CVs and pairings which provide a good trade-off between the MSV rule and the μ-IM criteria. The results indicate that such a solution corresponds to a point in the low-left corner of the plot in Figure 5, which has nearly the largest MSV and the nearly the smallest μ̅(E). However, it would be difficult to identify this control structure by selecting the CVs and pairings sequentially.

AUTHOR INFORMATION

Corresponding Author

*E-mail: y.cao@cranfield.ac.uk. Phone: +44-1234-750111. Fax: +44-1234-754685. Present Address §

ABB Global Industries and Services Limited, Mahadevpua, Bangalore 560048, India. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The first author gratefully acknowledges the financial support from Office of Finance, Nanyang Technological University, Singapore, through grant no. M52120046.

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Multiobjective Control Structure Design: A Branch and Bound Approach

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