Analysis of Interaction Effects on Plantwide Operability - American

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Analysis of Interaction Effects on Plantwide Operability Ridwan Setiawan and Jie Bao* School of Chemical Engineering, The University of New South Wales, UNSW, Sydney, NSW 2052, Australia ABSTRACT: Modern chemical plants are complex and consist of many process units interconnected in various configurations (e.g., recycle, bypass, and heat integration). Because of these interconnections, the unit interactions often impose limitations on the plantwide operability of chemical processes, especially on the performance of decentralized controller. In this article, a plantwide operability analysis approach that explicitly considers the interactions based on network perspective is presented. It provides a uniform framework for the assessment of plantwide stability, stabilizability, and achievable dynamic performance in the case of regulatory control using the concept of dissipative systems. To focus on the effects of process dynamics and the interaction between subsystems on the plantwide operability, this analysis was developed with the assumption of state-feedback control. The above analysis problem involves linear matrix inequalities (LMIs), which are convex and easy to solve.

1. INTRODUCTION In recent decades, increasing raw material and energy costs, combined with tighter operating conditions and growing environmental and safety requirements, have driven more developments of highly integrated modern chemical plants.1,2 Economic necessity has forced the extensive use of material recycles and heat integrations, which, in turn, creates strong interactions between different process units. Such interactions exacerbate the effects of disturbance propagation and often lead to the degradation of dynamic performance,13 particularly on processes using decentralized control. Process interaction becomes one of the main causes of plantwide operability problems in modern chemical plants, which was virtually nonexistent in the past while plants consisted only of units interconnected in a purely series configuration.4 Therefore, it is crucial to study the effects of unit interactions on the plantwide operability, preferably at the earlier stages of process design. There are some qualitative guidelines for improving plantwide operability (e.g., refs 2, 4, 5), most of which are largely based on industrial experiences and/or dynamic simulations. A significant body of literature has been developed for process operability/ controllability analysis, which determines whether a standalone process can be effectively controlled by a feedback control system (e.g., refs 1 and 611). Many of the existing operability analysis approaches can be used to estimate the level of dynamic performance that can be possibly achieved.7,8,12 These approaches, if used for plantwide operability analysis, treat the entire plantwide system as a single complex process. The plantwide operability, in terms of achievable performance, can be treated as a decentralized dynamic performance analysis. However, a tight bound of decentralized performance problem generally is very difficult to obtain, because it is inherently nonconvex.13,14 The effect of right half plane (RHP) zeros and time delays on decentralized control performance was investigated.15,16 A necessary and sufficient stabilizability condition for single complex multivariable processes with a decentralized controller was proposed.17 Decentralized performance, in terms of the achievable output variance, was also studied.1820 However, these simplified approaches are often limited to certain types of processes (e.g., open-loop stable systems15,16,18 or processes whose r 2011 American Chemical Society

undelayed dynamics are minimum phase18,19). In addition, most of the above approaches (e.g., refs 16 and 1820) are limited to fully decentralized (multiloop) control systems and thus cannot be directly used for plantwide operability analysis when each process unit is controlled by a multivariable controller (a block-decentralized structure). Much effort has also been exerted to develop interaction analysis methods, including the Dynamic Relative Gain Array (DRGA)21,23,24 and its extension for block-decentralized structure using Block Relative Gain (BRG).25,26 Based on the calculated relative gain, the best inputoutput pairings that minimize interactions can then be properly selected. The μ-interaction measure (μ-IM), developed based on structured singular values27,28 and a gap metric-based interaction measure for plantwide block-decentralized control,29,30 were also proposed. Both μ-IM and gap metric techniques treat the process interactions as uncertainties and analyze the effect of interactions based on robust control theory with a controller structural constraint. Because the known interactions are treated as uncertainties, these approaches can lead to conservative results. To explicitly address the interactions in plantwide processes, we recently developed a dynamic operability analysis framework from a network perspective for general nonlinear processes, which allows the assessment of plantwide stability, stabilizability, and control performance.31 The plantwide process is viewed as a network of process units and the overall plantwide operability is then obtained based on unit interactions and the dissipativity of each individual unit. Inspired by ref 32, the network perspective was developed to allow a clear distinction between the two layers of interconnections of which process units are interconnected via physical terminals while the controllers interconnect with the units through signal terminals.31 The structure of the physical interconnection layer, also known as the process network topology, can describe typical unit configurations in chemical plants such as units interconnected in series, recycle streams, and bypass streams. Thus, this approach can be used to study the effects of interactions on plantwide operability. In our past Received: December 17, 2010 Accepted: May 29, 2011 Revised: May 21, 2011 Published: May 30, 2011 8585

dx.doi.org/10.1021/ie1025198 | Ind. Eng. Chem. Res. 2011, 50, 8585–8602

Industrial & Engineering Chemistry Research work, the dissipativity of a process3337 was linked to its controllability to develop several process operability analysis tools.8,3840 However, the above framework requires the information on the dissipativity of nonlinear process units, which is often very difficult to obtain. This is perhaps the biggest hurdle in implementing such an approach by process design engineers who do not necessarily have nonlinear control expertise. In addition, because it is difficult to trade off dissipativity (i.e., the supply rates) between nonlinear process units, this approach can lead to conservative results. Furthermore, as an operability analysis framework for general nonlinear systems, it cannot be used to determine more-detailed dynamical performance achievable than the L 2-gain of the closed-loop system. The intention of this paper is to develop a more practical plantwide interaction analysis tool that addresses the above issues of current decentralized performance assessment. The network approach, similar to our previous work,31 is adopted in this work such that the effects of interactions between process units can be directly linked to the dynamic features of each process units (represented by their dissipativity) and how they are connected. To ease the difficulty in implementation of the plantwide operability analysis reported in ref 31, this approach is based on linearized models of process units. Although this means that this approach cannot be applied to highly nonlinear processes, it can often be sufficient in assessing operability in plantwide regulatory control. The use of linear models allows (i) optimization of dissipativity (supply rates) of process units leading to significantly less conservative results; (ii) obtaining of more-detailed dynamic performance achievable (e.g., frequency domain performance indicators); and (iii) the development of an easy-to-use numerical interaction analysis tool that can be used by process design engineers in the early stages of process design. Furthermore, this approach allows the parametrization of the supply rates, leading to a much less conservative analysis than that described in ref 31. This paper is organized as follows. Section 2 describes the new method to represent individual process units and process networks that allows the operability analysis based on network topology and dissipativity. The use of dissipativity for operability analysis and the main framework on interaction analysis are elaborated in Section 3. Discussions are then presented followed by an illustrative example to elaborate the procedure in performing the interaction and the plantwide operability analysis of large-scale chemical processes.

2. SYSTEM REPRESENTATION The representation of large-scale systems in this framework is performed using the network view of the constituent subsystems. 2.1. Network Perspective. The network perspective for operability analysis was first presented in ref 31, where a large-scale system is represented as the network of individual process units and its control system. The plantwide operability is then analyzed based on the interconnection topology and the dissipativity of each closed-loop subsystem. In this paper, this representation is modified to allow the operability analysis of plantwide processes using block-decentralized controllers. As shown in Figure 1, the decomposition of the plantwide systems is based on the boundary of each individual process unit, which results in a block-decentralized case. The control structure is then restricted to a block diagonal, implying that each unit is locally controlled. This approach has the following distinctive features, compared to ref 31: (1) The dissipativity of each closed-loop subsystem (including the controller and process unit) is used to produce much-less-conservative plantwide operability analysis results.

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Figure 1. Network of process units in closed-loop mode.

Figure 2. The ith process unit (physical and information flows).

(2) A set of performance variables and a frequency-dependent weighting function matrix are used to facilitate the determination of the detailed achievable dynamic plantwide performance. Based on the above network perspective, each individual subsystem as shown in Figure 2 is represented as a two-port system, with one port describing the physical and the other describing signal inputs and outputs: (1) The physical interconnection layer where different process units are interconnected physically through the interconnecting inputs (~u) and outputs (~y). In addition, the collective of process units interact with the environment through external plant input d (which is assumed to be the plant disturbance) and external output ye (which is a subset of ~y). (2) The information interconnection layer where physical process units interconnect with the controllers through the variables ^y (representing the process measurements) and û (representing the controller output signals to the actuators). In this work, the network representation shown in Figure 1 enables the analysis of detailed plantwide dynamic performance, which is achievable as follows: (1) A vector z (which contains some process variables) is introduced to represent the control performance. In process control, it is often important to attenuate the effects of disturbance (d) on some process variables of interest (e.g., the product purity of a distillation column 8586

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Industrial & Engineering Chemistry Research or the temperature of a reactor). By choosing z to be these process variables (in this case, the product purity or temperature), the achievable plantwide dynamic performance of the closed-loop process can be inferred from the system from d to z (denoted as Tzd(s)). (2) A weighting function matrix W(s) is used to facilitate the assessment of detailed dynamic performance. The plantwide dynamic performance can be conveniently represented by W(s), with the L 2-gain constraint on the system from d to ~z (system T~zd). Here, in this paper, the truncated L 2 norm of the input and output is used, leading to the following mathematical representation of the constraint: ||(WTzd)dT||2 < γ||dT||2 with γ e 1. Note that d and z include all disturbance and process variables in all process units in the entire plant. With the network perspective, this approach estimates the actual plantwide performance achieved by decentralized controller with the presence of interactions between different process units. This gives a more-realistic performance assessment than the local performance that is addressed in many existing analysis methods based on “independent design”.28,30,41 The topology of a large-scale process describes the interconnection configuration of different process units. In network view, it is represented as an interaction matrix H as shown in Figure 1, which is written as follows: " # " # ~ H ~u ¼ ~y ð1Þ He ye ~ describes the interconnection between The upper part (H) process units, while the interconnections of all process outputs to the external environment are captured in the lower part (He). 2.2. Models of Process Units. Because of the existence of the two-layer interconnection, each process unit is also modeled accordingly to reflect different types of inputs and outputs. The resulting model of the unit is called a two-port system (illustrated as the separating dashed line in Figure 2), with one physical port and one information port. In this article, process unit i (i = 1, ..., N) is represented as follows: 2 3 2 32 3 Ai xi Bi, 1 Bi, 2 Bi, 3 x_ i 6 6 6 ~yi 7 7 6 Ci, 1 Di, 11 Di, 12 Di, 13 7 76 6 ~ui 7 7 7 6 7 6 7 ð2Þ Σi : 6 6z 7 ¼ 6C 7 6 7 4 i5 4 i, 2 Di, 21 Di, 22 Di, 23 5 4 di 5 ^yi Ci, 3 Di, 31 Di, 32 Di, 33 ûi 2 Σi, cl :

3 2 Ai þ Bi, 3 Ki, 1 x_ i 6 6 6 x_ i, w 7 7 6 Bi, w ðCi, 2 þ Di, 23 Ki, 1 Þ 6 7 6 6 ~y 7 ¼ 6 C þ D K i, 13 i, 1 4 i 5 4 i, 1 ~zi Di, w ðCi, 2 þ Di, 23 Ki, 1 Þ

where Ki ¼

h

i Ki, 1

Ki, 2

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With its local (decentralized) feedback controller Ki (i = 1, ..., N), "

#

"

xk, i ¼ Ki ^yi

# ð3Þ

the closed-loop subsystem Σi has the following representation: 2

Σi, cl

3 ^x_ i 6 7 7 :6 4 ~yi 5 ¼ zi

2

32 3 î, 2 ^xi B 7 6 7 7 6 ^ Di, 12 54 ~ui 7 5 ^ i, 22 di D

î, 1 B ^ Di, 11 ^ i, 21 D

î A 6 6C ^ 4 i, 1 ^ i, 2 C

ð4Þ

where ^x is the combined states of both the individual process unit and its local controller. When block-decentralized statefeedback controllers ~u = Kixi (i = 1, ..., N) are used, eq 4 becomes 2 3 x_ i 6 6 ~y 7 7 4 i5 ¼ zi

Σi, cl :

2

Ai þ Bi, 3 K 6 6C þD K i, 13 4 i, 1 Ci, 2 þ Di, 23 K

32 3 xi Bi, 2 7 6 7 6 7 Di, 12 5 4 ~ui 7 5 ð5Þ Di, 22 di

Bi, 1 Di, 11 Di, 21

Assume that the state-space representation of the weighting function Wi is " Wi :

x_ i, w ~zi

#

" ¼

Ai, w Ci, w

Bi, w Di, w

#"

xi, w zi

# ð6Þ

then the closed-loop subsystem in eq 4 is given as follows: 3 2 î A ^x_ i 6 6 ^ i, 2 7 6 x_ i, w 7 6 Bi, w C 7 6 6 6 ~yi 7 ¼ 6 C ^ 5 4 4 i, 1 ^ i, 2 ~zi Di, w C 2 Σi, cl :

0 Ai, w 0 Ci, w

î, 1 B ^ i, 21 Bi, w D ^ Di, 11 ^ i, 21 Di, w D

32 3 î, 2 ^xi B 7 76 7 6 ^ i, 22 7 Bi, w D 7 6 xi, w 7 7 7 6 ^ Di, 12 5 4 ~ui 5 ^ Di, w Di, 22 di

ð7Þ The corresponding closed-loop model, using a decentralized state-feedback controller, is given as follows:

Bi, 3 Ki, 2 Ai, w þ Bi, w Di, 23 Ki, 2 Di, 13 Ki, 2 Ci, w þ Di, w Di, 23 Ki, 2

Bi, 1 Bi, w Di, 21 Di, 11 Di, w Di, 21

32 3 xi Bi, 2 76 7 6 7 Bi, w Di, 22 7 7 6 xi, w 7 7 6 7 Di, 12 5 4 ~ui 5 Di, w Di, 22 di

ð8Þ

Definition 1 (Dissipative Systems42,43). Consider a linear time-invariant system

3. DISSIPATIVE SYSTEMS AND PLANTWIDE OPERABILITY 3.1. Dissipativity of Interconnected Systems. Dissipativity is an inputoutput property. The concept of linear dissipative systems is defined as follows.

x_ k, i ûi

Σ:

" # " x_ A ¼ y C

B D

#" # x u

ð9Þ

where x ∈ Rn1, u ∈ Rm1, and y ∈ Rp1. It is dissipative if a non-negative function of its states ϕ(x): X f Rþ, called the 8587


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storage function (with ϕ(x(0)) = 0), can be found such that, for any τ g 0: Z τ wðuðtÞ, yðtÞÞ dt ð10Þ ϕðxðτÞÞ ϕðxð0ÞÞ e

then it is inputoutput stable. Furthermore, if Q = I, S = 0, and R = γ2I, then the system L 2-gain has an upper bound finite gain of γ; i.e., jjyT jj2 e γjjuT jj2

0

"u∈L

2e

where subscript T denotes truncation. Without truncation, the aforementioned L 2-gain is equivalent to the H ¥ norm for a linear system with input u and output y. The dissipativity of an interconnected dynamical system shown in Figure 3 is given as follows:

where w(u,y) is a real valued function of the system’s inputs and outputs, which is called the supply rate. With different supply rates, the dissipativity of a process is related to different inputoutput properties such as phase, gain, and their combination,31,44 and was found to be related to the process operability.8 In this work, a quadratic supply rate is considered as follows:

ϕ_ Σ e ~y Q1~y þ 2~y ½ S1 T

wðuðtÞ, yðtÞÞ ¼ yðtÞT QyðtÞ þ 2yðtÞT SuðtÞ þ uðtÞT RuðtÞ

T

" # " #T " ~u ~u R1 þ S2 R3T d d

R3 R2

#" # ~u ð13Þ d

~ is a static system and ~u = H~ ~ y, the dissipativity of the Since H overall interconnected system becomes

ð11Þ where Q ∈ Rpp, S ∈ Rpm, R ∈Rmm are constant matrices with Q = QT and R = RT. This type of system is called (Q,S,R)dissipative to emphasize the specific structure of the associated supply rate in eq 11. Stability analysis based on (Q,S,R)-dissipative systems was developed by Hill and Moylan.33,45,46 The dissipativity of an LTI system can be determined using the KalmanYakubovichPopov (KYP) lemma.47 The system described in eq 9 is (Q,S,R)-dissipative if there exists a P ∈ Rnn > 0 such that 32 2 3T 2 3 I 0 0 0 P 0 I 0 6 7 T7 6 76 60 I 7 7 6 6 7 6 7 6 0 R 0 S 76 0 I 7 > 0 ð12Þ 7 6 A B 7 6 P 0 0 6 0 54 A B 7 4 5 4 5 0 S 0 Q C D C D

~ T R1 HÞ~ ~ þH ~ T ST1 þ H ~ y ϕ_ Σoverall e ~yT ðQ1 þ S1 H ~ T R3 Þd þ dT R2 d þ 2~yT ðS2 þ H

ð14Þ

3.2. Interaction Effects on Plantwide Stability. In this frame~ can be used to describe different types of unit interconnecwork, H tions commonly encountered in chemical plants, including: (1) Downstream connection is the simplest type of interconnection, which includes series, parallel, and bypass configurations. The interaction effects caused by this type of interconnection are not significant that the overall plantwide operability is dependent solely on the operability of each individual unit. 4 It is ~ reflected by the lower triangular section of the H structure. (2) Local recycle describes every stream of which the products of a process unit are partially/totally fed back into that same process unit. It is captured by the block diagonal ~ section of the H-matrix. In general, the plantwide operability is not affected by local recycle if each unit is properly controlled on a local level. (3) Global recycle is used to describe any recycle (up flowing) streams (for energy integration and material recycle) between process units within a plantwide process. The global recycle streams often lead to operability problems, including the plantwide instability and the deteriorated control performance. The upper trian~ gular section of the H-matrix describes the topology of global recycle. The following example illustrates the use of interaction matrix ~ for interaction analysis, highlighting its significance in providH ing the information on the consequences of different interconnection types in plantwide operability analysis. Example 1. Figure 4 shows a simple system consisting of four units in series and bypass arrangement. Eighty percent (80%) of the outlet from Unit 1 flows to the second unit while

In this case, the storage function is defined as ϕ(x) = xTPx. The inputoutput stability and gain of a (Q,S,R)-dissipative system can be inferred from matrix Q as shown below. Theorem 1 (L 2-gain of (Q,S,R)-Dissipative System34). If the dynamical system Σ in eq 9 is (Q,S,R)-dissipative with Q < 0,

Figure 3. Interconnected systems.

Figure 4. System interconnected in series, bypass, local, and global recycle. 8588



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the balance flows to the fourth unit. This flow-split is denoted as R2,1 and R4,1, respectively, with the subscripts of R explaining the destination and origin of each flow. Part of the outlet from Unit 3 (50%) is locally recycled back to that unit. The outlet of Unit 4 is fully recycled back to the system into Unit 1. The topology of the process interconnection is ~ as follows: then represented by interaction matrix H 2 3 0 ~u1 6 6 6 R2, 1 I 6 ~u2 7 7 6 7¼6 6 0 6 ~u 7 4 4 35 ~u4 R4, 1 I 2

2

0 6 6 0:8I ¼6 6 0 4 0:2I

0 0

0 0

R3, 2 I 0

R3, 3 I R4, 3 I

0 0 I 0

0 0 0:5I 0:5I

One can obviously see that the above condition is equivalent to "

Q1 ST1

S1 R1

#"

# I ~ >0 H

ð17Þ

~ is (R1,ST1 , which, in turn, is equivalent to the condition that H Q1)-dissipative. 0 Since the (Q,S,R)-dissipativity of a system is not unique, the stability condition in eq 16 often provides conservative results. This can be overcome by parametrizing the supply rates of all subsystems as shown below. Proposition 2. For a plantwide process that consists of N (Qi,1, Si,1,Ri,1)-dissipative process units, the plantwide process is input output stable if there exist

32 3 ~y1 R1, 4 I 76 6 ~y2 7 7 0 7 76 7 7 6 0 5 4 ~y3 7 5 0 ~y4

32 3 I ~y1 6 7 7 6 7 0 7 6 ~y2 7 7 6 7 07 5 4 ~y3 5 ~y4 0

#T "

I ~ H

ð15Þ

" i ¼ 1; :::; N

λi > 0 such that

~ T R1 H ~ þH ~ T ST1 þ H ~ 0, "i = 1, ..., N) with λ1.λ2. 3 3 3 .λN, such that Qconnection in constraint (21) is negative definite. A more systematic way to parametrize the supply rates is to parametrize the storage function ϕ = xTPx. This can be done by treating the P matrix in constraint (12) as a decision variable, resulting in the most compatible (Q,S,R)-dissipativity from all N units (constrained by the interconnection) in the process network. This approach is adopted in rest of the development for plantwide analysis in this paper. 3.3. Plantwide Stability Analysis. The plantwide stability can be determined from the overall dissipativity of the entire process system, which, in turn, can be derived by combining the dissipation inequalities of N subsystems given in constraint (13), ~ In together with the dissipativity of the interaction matrix H. this case, the storage function of the plantwide system is the summation of the storage functions of all subsystems. Proposition 3. Consider a plantwide process with N units locally controlled by block-decentralized controllers. The dissipativity of the ith closed-loop subsystem of process unit and its controller Σi,cl in eq 4 is given as ϕ_ i e ~yTi Qi, 1~yi þ 2~yTi ½ Si, 1

N

0 I 0 Bi, 1 Di, 11

0 Ri, 1 Ri,T3 0 Si, 1

0 Ri, 3 Ri, 2 0 Si, 2

3 0 7 0 7 7 I 7 7>0 Bi, 2 7 5 Di, 12

Pi 0 0 0 0

3 0 7 STi, 1 7 7 STi, 2 7 7 0 7 5 Qi, 1

ð25Þ

Matrix inequality 25 is nonlinear, because K i, Q i ,S i,R i , and P i are decision variables. However, it can be reduced to a set of LMIs using an extension to the Elimination Lemma (Lemma 2 in Appendix C). In order to facilitate the development of stabilizability analysis, the following lemma is required. Lemma 1. Let the matrices P = PT > 0, Q = QT < 0, R = RT, and S. The (Q,S,R)-dissipativity condition, based on the KYP lemma given in constraint (12), is equivalent to 2

I 6 6 0 6 6 AT 4 BT

ð23Þ

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3T 2 0 0 7 6 6 0 I 7 7 6 6 ~ CT 7 5 4 P DT 0

0 ~ R 0 ~S

32 P~ 0 I 76 6 0 0 ~ST 7 76 6 T 0 0 7 54 A ~ BT 0 Q

3 0 7 I 7 7 T 7 > 0 ð26Þ C 5 DT



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with 2

0 6 6 0 6 6 P 4 0

P 0 0 0

0 R 0 S

32

0 0 76 6 0 ST 7 76 6 0 7 5 4 P~ Q 0

0 ~ Q 0 ~ST

3

~ P 0 0 0

0 7 ~S 7 7¼I 07 5 ~ R

ð27Þ

Proof is given in Appendix D. Proposition 3 can be extended for plantwide stabilizability analysis as follows. Proposition 4. A plantwide process with N process units (with the model of the ith unit described in eq 2 and an interaction ~ is stabilizable (in terms of inputoutput stability from matrix H) d to ~y) with block-decentralized controllers Ki, "i = 1, ..., N if ~ i,1 = Q ~ Ti,1 < 0, Q ~ i,2 = Q ~ Ti,2 < 0, ~Si,1, there exist matrices P~i = P~Ti > 0, Q T ~ i,1 > 0 ("i = 1, ..., N), such that ~ i,1 = R R 32 2 3T 2 3 ~ i, 1 ~STi, 1 R 0 I I 76 6 6 DT 7 7 6 6 T 7 7 6 ~ i, 1 ð28Þ 0 7 4 i, 11 5 4 ~Si, 1 Q 5 4 Di, 11 5 > 0 ~ i, 2 DTi, 12 DTi, 12 0 0 Q 2

I 6 6 0 6 T Ui^ T6 6 Ai 6 BT 4 i, 1 BTi, 2

3T 2

0 7 I 7 7 T 7 Ci, 1 7 DTi, 11 7 5 DTi, 12

0 6 6 0 6 6 P~i 6 6 0 4 0

P~i 0 0 0 0

0 ~ i, 1 R 0 ~Si, 1 0

0 ~STi, 1 0 ~ i, 1 Q 0

32 0 I 7 0 76 6 0 6 7 6 T 0 7 7 6 Ai T 7 0 56 4 Bi, 1 T ~ B Q i, 2 i, 2

~ 1 þ ~S1 H ~T þ H ~T < 0 ~ ~ST1 þ H ~R ~1H Q

inequality described by constraint (25) if both LMIs (constraints (28) and (29)) are feasible. Based on the result in Lemma 1, the condition given in eq 53 can be reduced to constraint (28) and constraint (24) is equivalent to constraint (30). 0 The feasibility of the LMIs given in Proposition 4 guarantees the existence of a block-decentralized controller K = diag{K1 , 3 3 3 ,KN } that can stabilize the respective plantwide process. Furthermore, the state-feedback controller gains K i ("i = 1, ..., N) can be calculated by substituting the solutions obtained from Proposition 4 into constraint (25). 3.5. Control Performance Achievable. The control performance achievable can be represented by the minimum L 2-gain of the closed-loop system from disturbance d to the weighted performance variable ~z, which, in turn, can be determined by the dissipativity of the following closed-loop system: " #T " ~ T R1 H ~ þH ~ T ST1 þ H ~ Q1 þ S1 H ~y ϕ_ e T ~ Q3 þ S3 H ~z # " #T " ~ T R3 S2 þ H ~y d þ d T R2 d þ2 S4 ~z

3

0 7 I 7 7 T 7 ^ Ci, 1 7Ui > 0 DTi, 11 7 5 DTi, 12

0 6 6 0 6 6 0 6 6 P 4 i 0

0 Ri, 1 0 0 Si, 1

0 0 Ri, 2 0 0

Pi 0 0 0 0

32

0 0 76 STi, 1 7 6 0 76 6 0 7 76 0 6 0 7 5 4 P~i 0 Qi, 1

0 ~ i, 1 Q 0 0 ~STi, 1

0 0 ~ i, 2 Q 0 0

P~i 0 0 0 0

ð29Þ

Qj ¼ diagfQ1, j , 3 3 3 , QN, j g,

ð30Þ

Sj ¼ diagfS1, j , 3 3 3 , SN, j g

ð31Þ ~1 = U U = 0, [UU ] is full rank, Q where Ui = ~ ~ ~ ~ ~ ~ ~ diag{Q 1,1, ..., Q N,1}, S1 = diag{S1,1, ..., SN,1}, R = diag{R 1,1, ..., ~ 2 = diag{Q ~ 1,2, ..., Q ~ N,2}. ~ N,1}, and Q R Proof. The stabilizability is proven based on Proposition 3. The plantwide system is stabilizable if there exist controllers Ki ("i = 1, ..., N) such that constraint (25), with " # Ri, 1 Ri, 3 >0 Pi > 0 Qi, 1 < 0 Ri,T3 Ri, 2 [BTi,3

DTi,13]T,

T

^

^

and constraint (24) are satisfied. To simplify this analysis, the phase condition between the disturbance d and the interconnecting output ~y and the interactions between the disturbance d and interconnecting input ~u are ignored, leading to Ri,3 = 0 and Si,2 = 0. Since the disturbance does not play a direct role in the interactions between process units, this simplification does not cause conservativeness in the plantwide dissipativity analysis. Lemma 2 (given in Appendix C) guarantees the existence of K = diag{K1, 3 3 3 ,KN} to the nonlinear matrix

ð32Þ

j ¼ 1, 2, 3 j ¼ 1, 2, 3, 4

Rj ¼ diagfR1, j , 3 3 3 , RN, 1 g

3

0 ~Si, 1 7 7 7 0 7 7¼I 0 7 5 ~ i, 1 R

#" # ~y ~z

where

and 2

~ T ST3 Q3 þ H Q2

j ¼ 1, 2, 3

The overall closed-loop storage function given in eq 32 is obtained by summarizing the storage functions of all N units with the dissipativity of the ith unit (i = 1, ..., N) is given below: " #T " #" # " #T " #" # ~yi ~ui Qi, 1 Qi, 3 Si, 1 Si, 2 ~yi ~yi þ2 ϕ_ i e ~zi Qi,T3 Qi, 2 ~zi ~zi Si, 3 Si, 4 di "

~ui þ di

#T "

Ri, 1 Ri,T3

Ri, 3 Ri, 2

#"

~ui di

# ð33Þ

~ y. together with the unit interconnection, given as ~u = H~ Assume that the plantwide process is stabilizable (i.e., Q1 þ ~ þH ~ T S T1 þ H ~ < 0), by only taking into account the ~ T R1H S1H phase condition between ~u and ~y (i.e., Si,2 = Si,3 = Si,4 = 0) and ignoring the gain conditions between d and ~y, and between ~u and ~z (i.e., Qi,3 = Ri,3 = 0), constraint (32) can be reduced to ϕ_ e ~zT Q2~z þ dT R2 d

ð34Þ

The above inequality gives the upper bound of L 2-gain of the plantwide system from d to the weighted process variable ~z. This L 2-gain is similar to the H ¥ system norm on the truncated input and output space. The formulation given in constraint (34) is more desirable, because it allows the signal of 8591



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2

d ∈ L 2e (e.g., nonvanishing disturbances). Based on Theorem 1, if Q2 = I and R2 = γ2I, then jj~zT jj2 ¼ jjðWTzd ÞdT jj2 e γjjdT jj2

I 6 6 0 6 6 0 6 6A ^ 6 i, w 6C ^ 4 i, w1 ^ i, w2 C

ð35Þ

Applying the gain condition in constraint (34), Propositions 3 and 4 can be extended to determine the achievable control performance quantitatively, in terms of disturbance attenuation by following the procedures described below. Procedure 1 (Achievable Plantwide Disturbance Attenuation with Existing Controllers). Consider a plantwide process with existing block-decentralized controllers with the augmented plant model given in eq 7. The required control performance, in terms of disturbance attenuation specified by the weighting function W, is achievable if γ e 1, obtained from the following minimization problem: min

Pi , Qi, 1 , Si, 1 , Ri, 1

The augmented plant given in eq 7 includes the weighting function Wi on the process variables of interest zi of each unit i, "i = 1, ..., N. The L 2-gain of the plantwide system from the process disturbance d to the weighted variables of interest ~z has an upper bound of γ. A value of γ e 1 is required to guarantee that the dynamic control performance specified by the weighting function is achievable. In operability analysis, it is desirable to include input constraint in the assessment of the achievable control performance. Variable z can be augmented with the controller output û as follows: " # " # Wz ~z ð37Þ ¼ ^z ¼ Duû ~z2 where Du is the weighting function on the controller output. In this case, constraint (34) becomes #" # " #T " I 0 ~z ~z ð38Þ þ dT γ2 d ϕ_ e 0 I ~z2 ~z2

Σi, cl

3 2 x_ i 6 7 Ai þ Bi, 3 Ki, 1 6 x_ i, w 7 6 Bi, w ðCi, 2 þ Di, 23 Ki, 1 Þ 6 7 6 7 6 6Ci, 1 þ Di, 13 Ki, 1 :6 6 ~yi 7 ¼ 6 6 6 ~zi 7 4Di, w ðCi, 2 þ Di, 23 Ki, 1 Þ 4 5 Du Ki, 1 ~zi, 2

2

Bi, 3 Ki, 2 Ai, w þ Bi, w Di, 23 Ki, 2 Di, 13Ki, 2 Ci, w þ Di, w Di, 23 Ki, 2 Du Ki, 2

Bi, 1 Bi, w Di, 21 Di, 11 Di, w Di, 21 0

7 7 7 7 7 î, w2 7 B 7 ^ i, w12 7 D 5 ^ i, w22 D

0 6 6 0 6 6 0 6 6 Pi 6 6 0 4 0

0 Ri, 1 0 0 Si, 1 0

0 I 0

î, w1 B ^ i, w11 D ^ i, w21 D

0 0 γ2 I 0 0 0

0 0 I

Pi 0 0 0 0 0

0 STi, 1 0 0 Qi, 1 0

3 0 7 0 7 7 0 7 7 0 7 7 0 7 5 I

3

7 7 7 7 7>0 ^ Bi, w2 7 7 ^ i, w12 7 D 5 ^ i, w22 D

ð36Þ

where

where Ki = [Ki,1Ki,2]. Based on the augmented plant model above, the L 2-gain of the system from d to ^z is denoted as ||(T^zd)dT||2 e γ||dT||2. Procedure 2 (Achievable Plantwide Disturbance Attenuation). Consider a plantwide process stabilizable using block-

decentralized controllers Ki ("i = 1, ..., N) with the augmented plant model is given in eq 39. The required control performance, in terms of disturbance attenuation specified by the weighting function W and controller output constraints specified by Du, is achievable if γ e 1, calculated from the following minimization problem: min

~ i, 1 , ~S i, 1 , R ~ i, 1 ~i, Q P

γ2

~ i,1 = Q ~ Ti,1 < 0, R ~ i,1 = R ~ Ti,1 > 0 ("i = subject to P~i = P~Ti > 0, Q 1, ..., N), matrix inequalities 30, 31 2

I 6 6 0 6 6 6 0 6 T 6D 4 i, 11 DTi, 12

3 Bi, 2 7 Bi, w Di, 22 7 7 7 Di, 12 7 Di, w Di, 22 7 5 0

0 I 0 DTi, 21 DTi, w DTi, 22 DTi, w 2

3T 2 ~ i, 1 R 0 7 6 7 6 07 6 0 7 6 I7 6 0 7 6 6~ 07 5 4 Si, 1 0 0

I 6 6 0 6 6 6 0 6 DT 4 i, 11 DTi, 12

3

xi 6 6 xi, w 7 7 7 6 6 ~u 7 4 i 5 di

3T 2

I 6 6 0 6 6 0 6 6A ^ 6 i, w 6C 4 ^ i, w1 ^ i, w2 C

which can be used to include the closed-loop L 2-gain from d to û in the plantwide control performance analysis. The new augmented plant model becomes 2

î, w1 B ^ i, w11 D ^ i, w21 D

0 0 I

2

γ2

subject to Pi = PTi > 0, Qi,1 = QTi,1 < 0 ("i = 1, ..., N), the LMI in constraint (24), and

0 I 0

ð39Þ

8592

0 γ2 I 0 0 0

0 I 0 DTi, 21 DTi, w DTi,22 DTi, w

0 0 γ2 I 0 0 3

0 7 07 7 I7 7>0 07 5 0

~STi, 1 0 0 ~ i, 1 Q 0

3 0 7 0 7 7 7 0 7 7 0 7 5 I

ð40Þ



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and 2

0 6 6 0 6 6 6 0 " #T 6 6 0 6 I ^T 6 0 Ui Mi, w 6 6 P~ 6 i, 1 6 6 P~i, 3 6 6 0 4 0 where

2

Mi, w

ATi 6 6 0 ¼6 6 BT 4 i, 1 BTi, 2

CTi, 2 BTi, w ATi, w DTi, 21 BTi, w DTi, 22 BTi, w 2

CTi, 1 0 DTi, 11 DTi, 12

~i, 1 P P~i ¼ 4 ~ Pi, 3 Ui ¼

h

BTi, 3

DTi, 23 BTi, w

0 0 0 0 0 P~Ti, 3 P~i, 2 0 0

CTi, 2 DTi, w CTi, w DTi, 21 DTi, w DTi, 22 DTi, w

0 0 ~ i, 1 R 0 0 0 0 ~Si, 1 0 3

0 7 07 7 07 5 0

3 P~Ti, 3 5 P~i, 2

DTi, 13

DTi, 23 DTi, w

0 0 0 γ2 I 0 0 0 0 0

ð42Þ

ð43Þ

DTu

iT

4. DISCUSSIONS This paper presents a uniform framework for stabilizability and plantwide performance of block-decentralized controllers, using a network perspective. The interactions between process units are captured based on the topology of the process network. Compared with many existing decentralized performance methods based on the full model of plantwide process, this network approach has the following advantages: (1) It formulates the plantwide operability analysis in a convex problem with linear matrix inequalities that can be easily solved. (2) Insights into the interaction effects on plantwide operability can be obtained to help pinpoint the sources of operability problems. (3) The “true” plantwide achievable performance with the presence of interactions between process units is obtained, as opposed to those “independent” analysis and design methods for decentralized control, where the interaction effects are only considered for the assessment of overall stability, not performance. Although the use of the theory of dissipative systems (especially the quadratic (Q,S,R)-dissipativity) and network view for plantwide operability analysis is not uncommon,31,34 existing results are often conservative, mainly caused by the “sufficient only” stability condition and the nonunique (Q,S,R)-dissipativity. Plantwide operability analysis relies on the compatibility between the (Q,S,R)-dissipativity of each subsystem (instead of individual dissipativity). As such, the conservativeness of the results can be significantly reduced by parametrizing the supply rates. In this

0 0 0 0 γ2 I 0 0 0 0

~i, 1 P ~i, 3 P 0 0 0 0 0 0 0

~Ti, 3 P ~i, 2 P 0 0 0 0 0 0 0

0 0 ~STi, 1 0 0 0 0 ~ i, 1 Q 0

3 0 7 0 7 7 7 0 7 7" # 0 7 7 I 7 0 7 U^ > 0 Mi, w i 7 0 7 7 0 7 7 0 7 5 I

ð41Þ

paper, the supply rates are parametrized in two ways: (1) application of multipliers (λi > 0); (2) optimizing storage functions of all subsystems simultaneously, leading to compatible supply rates. In general, the dissipativity based analysis of interconnected systems can be conservative because the detailed process and controller models are replaced by the less-informative (more “coarse”) inputoutput properties represented by the dissipativity conditions. Therefore, the later such a replacement is implemented, the less conservative results will be derived. In this paper, the plantwide stability, stabilizability and performance conditions are developed based on the dissipativity of each closed-loop subsystem, instead of the open-loop process units and controllers, so that the detailed models of individual process units and controllers are not thrown away “too early”. In this work, the plantwide stabilizability and performance is obtained with state-feedback controllers. While the plantwide achievable performance by output feedback controllers is often different (worse) than that of state-feedback controllers, because of the effects of the dynamics of the state observer, the interaction effects on the plantwide dynamic operability is not directly related to observer dynamics. The results obtained from the proposed analysis reflect the degradation of the performance caused only by the process dynamics, the interaction between its subsystems and the limitation imposed by block-decentralized control structure, not the controller itself. It is possible to extend the results to the output feedback case. However, the resulting problem is nonconvex, because it involves bilinear matrix inequalities (which can be solved using BMI techniques). Based on the network view, the operability analysis allows the performance requirements to be specified from all disturbance variables to any variables of interest (WTzd) accounting the unit interactions, given in terms of the L 2-gain, as shown in constraint (34). The weighting function matrix W(s) specifies detailed frequency-dependent performance requirement, such as the required disturbance attenuation in a frequency range (the typical desirable performance is shown later in this paper in Figure 6). Separate weighting functions (elements of W) can be used for each elements of z to specify different performance requirements for individual process variables. The proposed stabilizability and achievable control performance analysis should be carried out, based on the constraints on the controller output (i.e., the manipulated variables) and the permissible process output, which are related to the operating window analysis that determines the minimum and maximum limits on operating variables (as detailed in refs 4 and 50). By normalizing each variable using the steady-state and maximum 8593


Industrial & Engineering Chemistry Research permissible values, the plantwide operability analysis is assessed subject to the constraints of the above allowed operating region. In this work, the constraint on the controller output is considered in terms of its truncated 2-norm, which is represented by the L 2gain of the closed-loop system from disturbance d to controller output û, if the disturbance is also bounded by its truncated 2-norm. Another types of controller output constraints can be based on the maximum values of û represented by their ¥-norms. Such constraints can be translated to the generalized H 2-norm or L 1-norm of the closed-loop system from d to û when the disturbance is bounded by its 2-norm or ¥-norm, respectively. However, such formulations require solving nonconvex problems with nonlinear matrix inequalities and often lead to very conservative results (e.g., signals with sustained oscillations and those with a single overshoot followed by small variations can have the same ¥-norm).

Figure 5. Example of a chemical process network.

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The procedure for the plantwide operability analysis proposed in this paper can be summarized as follows: (1) Develop the linearized model for each subsystem around the nominal operating point. (2) Normalize the variables based on their steady-state and maximum permissible values, based on the process operating window of each process unit in the context of plantwide operation. (3) Obtain the structure of H representing the topology of the physical interconnection, based on the flowsheet of the large-scale process. (4) Conduct the interaction analysis based on the topology and identify the possible operability problems that may arise from unit interactions. (5) Perform the analysis of interaction effects on plantwide stability, stabilizability, and achievable performance as required.

5. ILLUSTRATIVE EXAMPLE In this section, the operability analysis based on the work presented in this paper was performed on a process, as shown in Figure 5. Example 3. A continuous stirred tank reactor (CSTR) is used to produce C by reacting A and B in water W. Since the reaction is exothermic, a heat removal system is installed as a water cooling jacket around the reactor. Material B is expensive and, therefore, is recycled back into the CSTR. A liquidliquid extraction (LLE) unit is used to recover B prior to recycling using a solvent S. The raffinate (rich in B) is heated to the required temperature T, using heat exchanger 1 (denoted as HE 1). The extract of the LLE (rich in C) is the product of the process that is fed into downstream units for further purification. A is fed directly to the CSTR from a storage tank, while reactant B is heated using heat exchangers 2 and 3 (denoted as HE 2 and HE 3, respectively) prior to entering the reactor. The plantwide process is regulated using a block-decentralized control strategy. 5.1. System Modeling. In this case study, each unit is modeled based on the first principle of mass and energy balance with the linearized model described using the state-space representation, as given in eq 2: (1) The states of each subsystem reflects the physical inventory of the unit, such as the total mass or energy (in terms of temperature).

Figure 6. Maximum singular value of the sensitivity function. 8594


Industrial & Engineering Chemistry Research (2) The inputs and outputs of each subsystem are either physical mass/energy flows or external inlets assumed to be disturbance variables. (3) the inputs are subdivided into three categories: the interconnecting inputs, external disturbances and manipulated variables (MVs), as previously described in Section 2. The detailed model of each unit is available in Appendix A. All state/input/output variables (x, d, û, ^y, ~u, ~y, and z) are normalized as follows: ~unormalized ¼

~u ~uss ~umax ~uss

with ~uss denotes the steady-state value and ~umax denotes the design maximum value, which are given in Tables B1B5 in Appendix B. This normalization is necessary to ensure that all the variables have comparable magnitudes. In this work, the permissible maximum and minimum values are assumed to be symmetrical about the steady-state values. 5.2. Topology and Interaction Analysis. The physical inputs and outputs of all units in this example, as shown in Figure 5, are combined, based on their corresponding orders. The topology of the physical interconnection H can then be represented as follows:

~ describes the interconnection The upper part of H (i.e., H) within the plantwide process as having no local recycles (as shown by the zeros on the entries of the bolded block diagonal section), two global recycles (Streams 2 and 3) and multiple downstream connections (as shown by the lower triangular section), while the lower part (i.e., He) shows the streams flowing out from the process into the environment (Streams 5, 11, 14, and 16). 5.3. Stability and Stabilizability Analysis. The exothermic reactor in this example has rendered the overall open-loop plantwide process unstable. Following Proposition 4, it was found that there exists a block-decentralized control system that can stabilize the plantwide process. The operability analysis then proceeds to the assessment of the achievable plantwide control performance. 5.4. Control Performance Analysis. The control performance analysis of two different configurations of the plantwide process were studied. The first configuration is the original plantwide process as shown in Figure 5, while configuration 2 does not include HE2, leading to a simpler process network. It is assumed that the design of HE3 and its operating window are

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unchanged in configuration 2. In order to ensure the same operating windows in both cases, the feed to HE3 (Stream 13) must be preheated (by an extra utility heat exchanger), which has the adverse effect of increasing utility cost. However, the use of such utility heat exchanger does not affect the plantwide interaction analysis. In terms of steady-state economy, the original configuration has better energy integration, thus potentially leading to a more profitable plant design. However, as pointed by Morari,7 processes with more heat integration often have stronger dynamic interactions, which, in turn, can degrade the control performance or even risk the plant stability. Although the second configuration may have a higher steady-state running cost, because of the increase of utility required, it results in better dynamic controllability (smaller control error, etc.), which may lead to more profitable plant operation. This dynamic operability analysis result can be used in conjunction with the steady-state analysis to find a process design optimized for overall economy of process operation. Based in Figure 5 and the models given in Appendix A, the disturbance variables include d, which can be represented as h i with d ∈ L 2e d ¼ T1 T7 F12 T12 (in the second configuration, F12 is replaced by F13 and T12 is replaced by T13). The following variables of interest are considered for control performance analysis: h i z ¼ Tr The1, h The3, c In this illustrative example, the required plantwide control performance specifications are given as follows: (1) All disturbance effects are attenuated by a factor of 100 at steady state; and (2) Disturbances are attenuated within ∼330 s (with a bandwidth of 0.015 rad/s).The singular value plot of the required plantwide sensitivity function (from disturbance d to process variables z) is shown in Figure 6 (the solid curve). The following weighting function (whose inverse should have a singular value plot of the required sensitivity function) is chosen to reflect the above performance requirement: 2 3 0 0 Wr, 1 6 7 6 7 6 Whe1, 1 0 7 W ¼6 0 7 4 5 0 0 Whe3, 1 2

I

6 25s þ 1 6 60 ¼ 100 5000s þ 1 6 4 0

0 I 0

0

3

7 7 07 7 5 I

ð45Þ

To limit the closed-loop controller gain (from the disturbance to the controller output), its weighting is chosen as Du = 0.2I. Both W and Du are used to form the final weighted closed-loop system T~zd. Applying Procedure 2 for the second configuration leads to the L 2 -gain of T~zd with an upper bound of γ = 0.97 ( 1). Therefore, the control performance specifications are relaxed for the first configuration to (1) the disturbance effect on The3,c is attenuated by a factor of 2 at steady state; and (2) disturbance effects are attenuated within ∼1.5 h (with a bandwidth of 8.9 10 4 rad/s) on T he3,c and 600 s (with a bandwidth of 8.4 10 3 rad/s) on Tr and The1,h . The maximum singular value plot of the required plantwide sensitivity function is shown in Figure 6 (the dashed curve), and the following weighting function is chosen: 2 3 0 0 Wr, 1 6 7 6 7 6 7 0 W 0 7 W ¼6 he1, 1 6 7 4 5 0 0 Whe3, 1 45s þ 1 100 6 9000s þ 1 6 6 6 0 ¼6 6 6 4 0

magnitude values. However, the controller for the second case exhibits a faster response. The operability analysis presented in this case study provides the decision-making foundation of selecting alternative process designs in addition to the economic considerations. Based on the steady-state plant economy alone, the first configuration is more appealing because the use of HE2 reduces the energy duty required by the process. However, this is at the expense of a poorer plantwide dynamic performance (e.g., larger variations of process variables, which attract economic penalties). There are always tradeoffs among alternative process designs, in terms of different aspects of plant economy, which should be considered by design engineers to make judicious choices. The operability analysis approach developed in this paper can assist in this decisionmaking process.

3

2

0 45s þ 1 100 9000s þ 1 0

0

7 7 7 7 0 7 7 7 300s þ 1 5 2 1200s þ 1

ð46Þ The above relaxed performance was found to be achievable by the first configuration. It is interesting to see that, although the open-loop time constant of each process unit is 0 0 I results in constraint (26). This concludes the proof.

0

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

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Analysis of Interaction Effects on Plantwide Operability - American

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