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The impact of decomposition on distributed model predictive control: A process network case study Davood Babaei Pourkargar, Ali Almansoori, and Prodromos Daoutidis Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b00644 • Publication Date (Web): 28 Jul 2017 Downloaded from http://pubs.acs.org on August 2, 2017

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Industrial & Engineering Chemistry Research

The impact of decomposition on distributed model predictive control: A process network case study Davood Babaei Pourkargar1 , Ali Almansoori2 , Prodromos Daoutidis1, * 1

Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA

2

Department of Chemical Engineering, The Petroleum Institute, Abu Dhabi, UAE

Abstract This paper addresses the impact of decomposition on the closed-loop performance and computational efficiency of model predictive control (MPC) of nonlinear process networks. Distributed MPC structures with different communication strategies are designed for regulation of an integrated reactorseparator process. Different system decompositions are also considered, including decompositions into local controllers with minimum interactions obtained via community detection methods. The closedloop performance and computational effort of the different MPC designs are analyzed. Through such a comprehensive comparison, trade-offs between performance and computation effort, and the importance of systematic choice of the system decomposition are documented.

Keywords: Model predictive control, distributed control, network decomposition, integrated process networks, process control

Introduction Model predictive control (MPC) is a well-accepted and widely used multivariable control strategy for chemical processes; it can directly accommodate different performance optimality criteria and can handle input and state constraints.1 The MPC design involves the repeated online solution of a constrained dynamic optimization problem. Therefore, its practical applicability depends on the solvability of the underlying optimization problem in real time. This issue becomes more crucial when considering a centralized control architecture for large-scale systems and nonlinear process dynamics, which are common characteristics of the systems in chemical process industries.2,3 * Corresponding

author, Tel: +1 (612) 625-8818, Fax: +1 (612) 626-7246, Email address: [email protected]

ACS Paragon Plus Environment

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One way to overcome the computational challenges of centralized model predictive control (CMPC) for large-scale nonlinear systems is to decompose the control problem into a set of smaller control problems with some level of communication and cooperation.4–6 The resulting distributed model predictive control (DMPC) structure is a middle ground between centralized and decentralized control structures in which the local controllers are designed independently of the other subsystems. Previous research in DMPC has mostly focused on feasibility, optimality and closed-loop stability for a given system decomposition.7–15 Yet, the choice of the decomposition is an essential factor for realizing the potential benefits of distributed control. Since this choice determines the number of subsystems and how the state variables and manipulated inputs are distributed among them, it directly affects the computational burden and performance of the closed-loop system. In process applications of DMPC, the system decomposition is usually obtained based on intuition, e.g. based on the layout of the physical units,14,15 or according to material/energy balance subsystems.7 A systematic framework for such a network decomposition, i.e. the identification of dynamic subsystems of physically distributed inputs, outputs, and states with favorable interaction characteristics in a distributed control setting, is a challenging open problem. Some initial attempts to address the optimal decomposition problem have focused on solving a multi-objective mixed integer nonlinear program to optimize performance and structure simplicity,16 and an open-loop performance metric.17 Optimal decompositions for state estimation have also been proposed based on observability, relative degree and sensitivity analysis between the system states and measured outputs.18 The available concepts and tools in network and graph theory can aid to address this large-scale system decomposition problem systematically. From a network theory point of view, the decomposition of a network corresponds to identifying weakly connected subsystems whereby the variables of each subsystem are strongly connected.19,20 Such an approach has been adopted to generate hierarchies of system decompositions with different degrees of decentralization based on input-output connectivity, following an agglomerative or divisive clustering procedure.21,22 An alternative decomposition method based on the maximization of the modularity of the system equation graph was recently proposed23 in which the network is partitioned based on modularity maximization of ”communities” that include inputs, states, and outputs. Such an approach inherently minimizes input-state or state-state interactions between communities and thus appears well-suited for the distributed control architecture design problem.


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In this paper, our goal is to provide a template for analyzing the impact of different system decompositions as well as different DMPC communication architectures on the closed-loop performance and computational effort of DMPC. Whereas several applications of DMPC have appeared, there does not exist a comprehensive assessment of the efficiency of different DMPC designs and the effect of the system decomposition. We seek to address this through a detailed case study on a benchmark process system that brings about the importance of both system decomposition and information sharing in achieving practically implementable control designs with satisfactory closed-loop performance. Specifically, sequential and iterative DMPC strategies are implemented to address the regulation problem for an integrated process system consisting of two continuous stirred tank reactors (CSTRs) and a liquid-vapor separator. Four different decompositions are considered obtained through modularity maximization or by intuition. Closedloop performance indices which penalize the magnitudes of manipulated inputs and regulation errors are evaluated for each decomposition under the sequential and iterative DMPC designs and additionally compared to those of a centralized MPC design. The average computation times are also calculated for the proposed MPC designs to evaluate which control architecture is suitable for real-time closed-loop process operation. The rest of the paper is organized as follows. Section ”Preliminaries” presents the mathematical description of the considered class of nonlinear integrated systems and its decomposition into distributed subsystems. Community-based network decomposition is briefly reviewed in Section ”Network decomposition”. In Section ”Model predictive control structure”, we present the MPC formulations for centralized, sequential and iterative distributed structures. Finally, we illustrate the system decomposition impact on closed-loop performance and the corresponding computational requirements for the reactor-separator integrated system.

Preliminaries We consider a general class of deterministic input-affine nonlinear processes described by the following state-space model x(t) ˙ = f (x(t)) + g(x(t)) u(t)


3

(1)


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where x(t) = [x1 (t) x2 (t) · · · xn (t)]T ∈ Rn presents the vector of the system state variables, t is time, the vector of manipulated inputs is denoted by u(t) = [u1 (t) u2 (t) · · · um (t)]T ∈ Rm , and f : Rn → Rn and g : Rn → Rn×m denote smooth locally Lipschitz functions. We assume that all of the state variables of the system can be measured at predetermined sampling times, x(tk ), where tk is the time of kth sampling. For convenience and without loss of generality, we consider that the origin is the equilibrium point for the unforced model which implies that f (0) = 0. Following a centralized control design for the nonlinear multivariable system of (1), a single controller is synthesized to compute the optimal value of all components of the manipulated input vector at each sampling time. The computational effort to implement such a centralized optimization problem may be dramatically elevated by increasing the number of states and manipulated variables; a possible alternative is to decompose the optimization problem and employ a distributed control strategy. In such a strategy, multiple distributed predictive controllers whose calculations are handled by separate processors in parallel, are designed to compute the manipulated inputs. Such a strategy necessitates the decomposition of the vector of system states and manipulated inputs of (1) into L interconnected subsystems whose dynamics can be described by x˙(i) (t) = f(i) (x(t)) + g(i) (x(t)) u(i) (t),

i = 1, . . . , L

(2)

u = [uT(1) uT(2) · · · uT(L) ]T

(3)

where T T T T x = [x(1) x(2) · · · x(L) ] ,

and f(i) and g(i) denote the components of the nonlinear functions of f and g in (1) associated with the input-state subsystems such that T T T T f = [ f(1) f(2) · · · f(L) ]

(4)

and L

g u = ∑ G(i) u(i) ,

G(i) = [0T · · · gT(i) · · · 0T ]T

(5)

i=1

where 0 indicates zero matrices of appropriate dimension. The state and manipulated variables of the system can then be accordingly represented in the following state-input pairs L (x, u) = (x(i) , u(i) ) i=1 ,

x(i) = [x(i),1 x(i),2 · · · x(i),ni ]T ,


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u(i) = [u(i),1 u(i),2 · · · u(i),mi ]T

(6)

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where L

L

n = ∑ ni ,

m = ∑ mi

i=1

(7)

i=1

In the remainder of this paper, we review the application of community-based methods to select the optimal decomposition for the general nonlinear process described by (1), and various DMPC implementations.

Network decomposition A large number of real-life networks in the computer, information, social, ecological and biological sciences are naturally structured in a way that allows the occurrence of groups which are more densely connected internally than with the rest of the network.20 Accordingly, such networks can be partitioned into subsets such that the links between vertices in a subset significantly exceed those between vertices in different subsets.24 The resulting subsets are called communities, and the procedure of identifying is called community detection. Modularity-based methods consider this problem as a maximization of a modularity measure.19 Modularity represents an excess fraction of edges which fall within a community compared to a random graph; thus decompositions with maximum modularity represent optimal ones in the sense that they capture the presence of communities with a statistically significant number of intra and inter community interactions. Modularity-based methods have recently been adopted to address the decomposition problem for nonlinear process networks. Community detection through modularity maximization on the unweighted system digraph, directly accounting for state interactions between the resulting subsystems, was proposed in.23 In both cases, recursive bisections are considered and appropriate versions of Newman’s spectral algorithm24 are employed to determine optimal partitions until further partitions do not increase modularity. The results of the latter approach will be used in this work.


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Model predictive control structure In this section, we present the optimization problem formulation for the MPC structures employed in this study. All of the predictive control structures were developed assuming full availability of the system states. The difference between the proposed DMPC formulations is due to the various communication strategies used to transfer the manipulated variable information over the network of local controllers (the readers may refer to6,8 for a detailed discussion).

CMPC The CMPC employs the solution of a single optimization problem to compute all of the manipulated inputs of the system. Figure 1 illustrates the schematic of CMPC design for control of a typical process. The CMPC for a class of systems described by (1), can be formulated by the following optimization problem subject to constraints and input bounds Z tk+N

minimize u

(xT P x + uT W u) dt

tk

subject to x˙ = f (x) + g(x) u umin ≤ u ≤ umax F (x, u,t) ≤ 0 G (x, u,t) = 0

(8)

where tk indicates the time of k th sampling and N is the number of required samplings in the prediction horizon. The positive definite weight matrices P ∈ Rn×n and W ∈ Rm×m , are adjusted to penalize the state regulation errors and manipulated variables in the objective function. The vectors umin and umax correspond to the lower and upper bounds considered for the manipulated inputs. The general nonlinear vector functions of F and G describe the inequality and equality constraints of the system, respectively. After computing the temporal profiles of all components of the manipulated input vector during the prediction horizon of [tk tk+N ], we apply the manipulated inputs at tk for t ∈ [tk tk+1 ), and then resolve the optimization problem for the next sampling period.


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DMPC The main idea of DMPC hinges on decomposing the centralized control problem into multiple local controller designs. As illustrated in Figures 2 and 3, some level of communication may also be considered between the local controllers to enhance the closed-loop performance. We could employ the same or different objective functions for the local controllers. By using the same objective function, each local controller considers the impact of the corresponding manipulated inputs on all states of the process. In such a control strategy called cooperative DMPC,7 each of the local controllers computes the corresponding components of the manipulated input vector by minimizing a global objective function. On the other hand, we could tailor a particular objective function to each local controller to consider only the states associated with the subsystem for which the local controller is designed. The resulting strategy which has a simpler objective function is called non-cooperative DMPC.8 The optimization problem for the i th local predictive controller associated with the i th subsystem is Z tk+N

minimize u(i)

tk

(xT P(i) x + uT(i)W(i) u(i) ) dt L

subject to x˙ = f (x) + ∑ G(i) u(i) i=1

umin (i)

(9)

≤ u(i) ≤ umax (i)

F(i) (x(i) , u(i) ,t) ≤ 0 G(i) (x(i) , u(i) ,t) = 0 where the positive definite matrix P(i) can be adapted for each subsystem to synthesize cooperative or noncooperative DMPCs. Also, the positive definite matrix W(i) includes only the components corresponding to the manipulated inputs of the i th subsystem. In this paper, we implement two broadly used classes of DMPC with a non-cooperative structure, sequential and iterative DMPC, described in the following subsections. Sequential DMPC In this control structure, the states of the system are available to the local controllers at the sampling times and each local controller sends the computed future trajectories of its manipulated inputs and the input trajectories received from the preceding local controllers to the next local controller. Feedback of all of


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the system states is available to all of the local controllers at each sampling time. Specifically, the i th local controller receives the entire future trajectories of u(1) , u(2) , . . . , u(i−1) , evaluates the future input trajectory of u(i) based on the states x(tk ) from the measurement sensors at the sampling time tk and the latest received input trajectories, and sends the entire future trajectories of u(1) , u(2) , . . . , u(i) to the i + 1 th local controller and the first step input value of u(i) to its actuators.8 The main drawback of this structure is the subsequent computations of manipulated inputs by the local controllers which lead to an increase in computation time. Figure 2 shows how local controllers communicate over the network in this sequential scheme. Iterative DMPC Iterative DMPC is a fully interactive distributed control structure in the sense that the local controllers continuously share the latest values of their computed manipulated inputs over the network and revise them to converge to the optimal manipulated inputs. Figure 3 shows how the local controllers communicate in this scheme. Similar to the other distributed control structures presented previously, all of the local controllers receive the measurements of all of the system states, x(tk ), at the sampling time, tk , from the available sensors. Then each local controller computes its future input trajectory in an iterative manner. At each iteration, each local controller calculates its future manipulated input trajectory in parallel by employing the state measurements of the system and the latest manipulated inputs trajectories received from the other controllers. The local controllers exchange the future trajectories of the computed manipulated inputs over the network. The local controllers re-evaluate the manipulated inputs in parallel based on the state measurements and latest input variables until satisfying a termination condition.8 A variety of choices is available for the termination condition, e.g. the iterations may be terminated when the Euclidean norm of the difference between the vectors of computed manipulated inputs of two consecutive iterations is smaller than a desired threshold, and/or the iteration may be terminated when the number of iterations exceeds the maximum allowable iteration number to avoid an increase in computation time. Note that the parallel processing in computations of the manipulated inputs can dramatically decrease the required computation time compared to the centralized control structure and the sequential DMPC.


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Application to an integrated process network To implement the proposed MPC structures and illustrate the impact of employing different control architectures, we consider a typical reactor-separator process network. As shown in Figure 4 the process r

r

1 2 consists of two CSTRs where two series first-order exothermic reactions of the form of A − → B− → C take

place, and a liquid-vapor separator.7,15,25 The terms r1 and r2 denote the reaction rates for the conversion of feedstream component A to the desired product B, and the desired product B to the side product C, respectively. Fresh feedstreams which enter the first and second reactors contain only component A. The stream which leaves the first reactor and enters the second reactor contains all of the components. The outlet stream from the second reactor is then fed to the separator. There is a recycle stream to the first reactor, and a purge stream is considered to avoid the side product accumulation.


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Process model The following process model describes the hold up, temperature, and concentration dynamics of the reactors and separator dV1 = Ff 1 + FR − F1 dt dV2 = Ff 2 + F1 − F2 dt dV3 = F2 − FP − FR − F3 dt dT1 Ff 1 FR Q1 m = (T0 − T1 ) + (T3 − T1 ) + − (k11 xA1 ∆Hr1 + k21 xB1 ∆Hr2 ) dt V1 V1 ρC pV1 C p F1 Q2 m dT2 Ff 2 = (T0 − T2 ) + (T1 − T2 ) + − (k12 xA2 ∆Hr1 + k22 xB2 ∆Hr2 ) dt V2 V2 ρC pV2 C p dT3 F2 Q3 = (T2 − T3 ) + dt V3 ρC pV3 FR dxA1 Ff 1 = (xA0 − xA1 ) + (xAR − xA1 ) − k11 xA1 dt V1 V1 Ff 1 dxB1 FR = (xBR − xB1 ) − xB1 + k11 xA1 − k21 xB1 dt V1 V1 F1 dxA2 Ff 2 = (xA0 − xA2 ) + (xA1 − xA2 ) − k12 xA2 dt V2 V2 Ff 2 dxB2 F1 = (xB1 − xB2 ) − xB2 + k12 xA2 − k22 xB2 dt V2 V2 FP + FR dxA3 F2 = (xA2 − xA3 ) − (xAR − xA3 ) dt V3 V3 dxB3 F2 FP + FR = (xB2 − xB3 ) − (xBR − xB3 ) dt V3 V3

(10)

where t denotes the time, V is the volumetric hold up, T indicates the temperature, xA and xB are the mole fractions of the A and B components, and F denotes the volumetric flow rate. The subscripts 1, 2, and 3 correspond to the outlet stream properties of the two reactors and the separator, whereas 0, R and P denote the feed, recycle and purge streams, respectively. The physical properties of the flow streams such as density (ρ), heat capacity (C p ), and molality (m), are assumed to be constant. The terms Ff 1 and Ff 2 r

r

1 2 indicate the feed flow rates, ∆Hr1 and ∆Hr2 are the heats of reactions for A − → B and B − → C, respectively,

and Q denotes the heat flow rate provided by the jackets to the reactors and the separator. The mole


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fractions of the components in the recycle stream are computed assuming equilibrium αS xS3 , ∑S=A,B,C αS xS3 xC3 = 1 − xA3 − xB3

xSR =

S = A, B,C

(11)

where α is the volatility. The reaction rate coefficient in the reactor i and for the reaction r is obtained by the Arrhenius equation kri = kr0 exp(

−Er ), RTi

r = 1, 2,

i = 1, 2

(12)

where k0 is the pre-exponential factor, E the activation energy and R the universal gas constant. We also consider a small value of ε as the purge ratio FP = εFR

(13)

The parameters of the model are given in Table 1 and the stable steady state values are presented in Table 2.

MPC architectures The integrated process described above has been widely used in previous control studies to illustrate a variety of DMPC designs; in these studies the system decomposition was performed based on intuition.7,15,25 After representing the process model by an equivalent equation graph and employing community detection through modularity maximization on the resulting graph, two system decompositions have been identified with very close modularity values.23 The first decomposition illustrated in Table 3 consists of three subsystems. This decomposition has the highest modularity value obtained by community detection.23 According to the structure of Subsystem 1, the flow rates of the feed to the first reactor and recycle stream, and inlet heat flow rate of the first reactor are assigned to control the hold up, temperature and concentrations of the first reactor. In Subsystem 2, the flow rates of the feed to the second reactor and outlet stream of the first reactor, and inlet heat flow rate of the second reactor are considered to control the hold up, temperature and concentrations of the second reactor. The rest of the control inputs which include flow rates of the outlet streams of the second reactor and separator, and the inlet heat flow rate of the separator are designated to address the control problem


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of the hold up, temperature and concentrations of components inside the separator by Subsystem 3. This decomposition therefore closely matches the physical topology of the process network. The second decomposition considered is presented in Table 3 and consists of four subsystems. This decomposition is a suboptimal one with a modularity value very close to the optimal one. In Subsystem 1, the feed flow rate and inlet heat flow to the first reactor are assigned to stabilize the hold up, temperature and concentrations of the first reactor. Subsystem 2 considers the flow rates of the feed to the second reactor and outlet stream of the first reactor, and inlet heat flow rate of the second reactor to control the hold up, temperature and concentrations of the second reactor. According to Subsystem 3, flow rates of the outlet streams of the second reactor and the separator, and the recycle flow rate are designated to regulate the hold up and concentrations of components inside the separator. Finally, the separator temperature is considered to be manipulated by its inlet heat flow rate in Subsystem 4. By considering the above decompositions, and the MPC formulations presented in Section ”Model predictive control structure”, we synthesized three and four layer DMPC structures alongside a CMPC design to address the regulation problem. The resulting DMPC structures involve a hierarchy of L local controllers, where L = 3 for the first control architecture and L = 4 for the second one. Each of the local model predictive controllers solves the optimization problem only for their manipulated inputs. The objective function of each local controller penalizes only its own manipulated inputs and the regulation errors of the corresponding states of the subsystem as presented in Table 3. Two intuitive decompositions of the system are also presented in Table 3; one based on the configuration of physical units (decomposition 3) and the other based on material/energy balances (decomposition 4). Both of these decompositions consist of three subsystems which leads to three layer DMPC designs. In the third decomposition based on the physical topology of the process network, the flow rates of feedstream and outlet stream, and the heat flow rate of the first CSTR are considered to control its properties. Also, the flow rates of feedstream and outlet stream, and the heat flow rate of the second CSTR are assigned to control its properties. The rest of the control inputs which include flow rates of the outlet stream of the separator, the recycle stream, and the inlet heat flow rate of the separator are designated to address the control problem of the hold up, temperature and concentrations of components inside the separator. In


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the fourth decomposition, the flow rates of the feedstreams and the recycle stream are assigned to control all of the concentrations in the process. The outlet flow rates of the units control the hold up of the units, and the heat flow rates are assigned to control the thermal dynamics.

Simulation results In this section, we initially employ the CMPC and the proposed DMPC designs to address the state regulation problem of the integrated process system described by (10)-(13) for the different decompositions described above. The number of samplings in the prediction horizon is set to N = 15, the sampling time of the MPC iterations is ∆ = 1.5 minutes, and thus the prediction horizon for solving the optimal control problem at each sampling is equal to N∆ = 22.5 minutes. We assume that all of the state measurements of the system (hold ups, temperatures, and concentrations) are synchronously available at the sampling times. The control objective is to stabilize the process at the stable steady state condition presented in Table 2 from the initial state values showed in Table 4. The manipulated variables associated with stream flow rates and heat flows in the optimal control problem are constrained to be positive and remain within ±%80 of their steady state values presented in Table 2. In addition, since the hold up dynamics are non-self regulating, i.e. they may tend towards a different steady state or be unstable if the corresponding flow rates are chosen freely, we add three equality constraints to the optimization problems to ensure that the time derivative of the hold up (right-hand side of the hold up dynamic equations) is always proportional to the negative values of the hold up variables. Such equality constraints make the closed-loop hold up dynamics very close for different decompositions. The following diagonal weighting matrices are applied to balance the importance of each state and input variable in the objective function of the optimal control problem of CMPC P = diag([104 104 104 1 1 1 105 105 105 105 105 105 ]) W = diag([103 103 103 103 103 103 10−2 10−2 10−2 ])

(14)

where diag(V ) denotes a square diagonal matrix with the elements of vector V on the main diagonal. In P, the first three diagonal elements correspond to the hold ups, the second three to temperatures and the rest to concentrations. The first six diagonal elements in W also indicate the importance of volumetric flow rates and the rest correspond to the heat flows. The elements of the diagonal matrices are chosen based


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on the magnitude of the state variables and manipulated inputs under open-loop process operation in such a way that the multiplication of each element and the corresponding system state or manipulated input remain in the same order of magnitude. In all of the proposed non-cooperative DMPC designs, the local model predictive controllers use the same weighting components associated with the states and inputs of their own subsystems, i.e. for the first control architecture (whose system decomposition presented in Table 3) we have P(1) = diag([104 0 0 1 0 0 105 105 0 0 0 0]) P(2) = diag([0 104 0 0 1 0 0 0 105 105 0 0]) P(3) = diag([0 0 104 0 0 1 0 0 0 0 105 105 ])

(15)

W(1) = W(2) = W(3) = diag([103 103 10−2 ]) and for the second control architecture P(1) = diag([104 0 0 1 0 0 105 105 0 0 0 0]) P(2) = diag([0 104 0 0 1 0 0 0 105 105 0 0]) P(3) = diag([0 0 104 0 0 0 0 0 0 0 105 105 ]) P(4) = diag([0 0 0 0 0 1 0 0 0 0 0 0])

(16)

W(1) = diag([103 10−2 ]), W(2) = diag([103 103 10−2 ]) W(3) = diag([103 103 103 ]), W(4) = 10−2 . Accordingly, we obtain the local weight matrices for the third decomposition P(1) = diag([104 0 0 1 0 0 105 105 0 0 0 0]) P(2) = diag([0 104 0 0 1 0 0 0 105 105 0 0]) P(3) = diag([0 0 104 0 0 1 0 0 0 0 105 105 ])

(17)

W(1) = W(2) = W(3) = diag([103 103 10−2 ]) and fourth decomposition P(1) = diag([0 0 0 0 0 0 105 105 105 105 105 105 ]) P(2) = diag([104 104 104 0 1 0 0 0 0 0 0 0]) P(3) = diag([0 0 0 1 1 1 0 0 0 0 0 0])

(18)

W(1) = W(2) = diag([103 103 103 ]), W(3) = diag([10−2 10−2 10−2 ]). We utilize sequential quadratic programming (SQP) to solve the nonlinear constrained optimal control problem at each sampling time.26 Note that due to the non-convex nature of the above centralized and


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distributed optimization problems, reaching a global minimization point may not be guaranteed using standard nonlinear optimization solvers. However, we specify sufficiently high values of maximum allowed number of iterations and function evaluations for the optimization problems to ensure that the optimizing procedure is not terminated before converging to the optimal point. In addition, note that for general nonlinear constrained systems, there is no guarantee that the optimal manipulated input solution and the performance indices of the DMPCs will converge to the solution and performance index of a CMPC due to the non-convex nature of the optimization problems. Two indices are defined to evaluate the closed-loop performance of the proposed control architectures during the process operation

Z tf

ISE =

eT (t) P e(t) dt

0

Z tf

ISC =

(19) T

u¯ (t)W u(t) ¯ dt 0

where ISE and ISC indicate the integrals of squared regulation errors, e = x − xss , and the deviation of control actions from their steady state values, u¯ = u − uss . The steady state values of the system states and manipulated inputs vectors are denoted by xss and uss , respectively, whose components are presented in Table 2. These two performance indices correspond to the regulation and control effort costs in the objective function of the centralized MPC for the entire process operation interval [0 t f ]. Thus a generalized index defined by ISE + ISC is a meaningful indicator to compare the closed-loop performance of the different control architectures. Note that the values of the diagonal weighting matrices affect the values of the performance indices, however the relative magnitudes of the performance indices remained unchanged in simulations we performed with different sets of weighting matrices (results are omitted for brevity). In addition, we observed that the weighting matrix values do not affect the computation time significantly. Figure 5 shows the temporal profiles of the regulation errors for the closed-loop process operation under the CMPC design. We observe that the regulation errors converge to zero with smooth transients and without any chattering which confirms the stabilization of the state variables of the system at their steady state values. The temporal profiles of the required manipulated inputs computed by the centralized control architecture are shown in Figure 6. The values of the control actions are presented in deviation form. Note that the value of each control action remains constant between sampling times for ∆ = 1.5


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minutes until receiving the next value from the controller. We observe that the control actions gradually converge to their steady state values. For brevity, we only present the results for the iterative DMPC and summarize the results of the sequential architecture only by its performance indices. For the convergence criteria for the iterations at each sampling time, we set 0.01 as the desired threshold for the summation of the Euclidean norm of the difference between the vectors of the computed manipulated inputs of two consecutive iterations. We also consider a sufficiently high value of 10 for the maximum allowed number of iterations. The temporal profiles of the regulation errors and manipulated inputs for the proposed decompositions under iterative DMPC are presented in Figures 7 and 8. The system states and manipulated inputs again gradually converge to their steady state values without any chattering. A faster stabilization is obtained using the first (optimal community-based) and third (based on the physical topology of units) decompositions of the system; recall that these decompositions are quite similar in that they are based on the physical layout of the units. Table 5 presents the performance indices for the proposed control architectures during a 2 hr operation. We note that decomposition 1 and 3 perform the best, with decomposition 1 being more favorable in terms of the ISC index. The values of the generalized index (i.e. ISE + ISC) increase by increasing the number of subsystems to which the original system is decomposed (compare decompositions 1 and 2 in both the sequential and iterative architectures) which reflects an expected performance degradation over the fully centralized controller. The noticeable increase in the value of the performance index of the distributed control architecture under the fourth decomposition is due to the significantly larger values of the regulation errors compared to the centralized control architecture and the other distributed control architectures. We also observe that the type of DMPC design (the way that local predictive controllers communicate over the network) does not have significant effect on the performance indices. Figure 9 presents the average computation times required by the CMPC and DMPC architectures during the 2 hr operation where the simulations are performed in MATLABr using a 3.4 GHz Intelr CoreTM i7−6700 processor. The solution time statistics of the CMPC and the iterative DMPC for different decompositions are described in ”Supplementary information” section. We addressed the computation time and number of iterations at each sampling time during closed-loop process operation. We observe


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that the computational burden of solving the CMPC problem is noticeably greater than in the distributed control scenarios, which limits the applicability of such a centralized control design. For an improvement in computation time, the proposed DMPC structures take advantage of reducing the number of decision variables in each of the distributed optimization problems. By employing the sequential DMPC, the local predictive controllers operate in series, i.e. each local controller requires the values of the manipulated variables from the previous controller. On the other hand, the local optimization problems for each subsystem using the iterative DMPC structure are executed in parallel followed by an exchange of information between local predictive controllers. Due to such parallel processing, we observe a reduction in the computation burden of the iterative DMPC compared to the sequential DMPC, for all system decompositions (Figure 9). Further, the control architecture based on the first decomposition (which has optimal modularity) shows a slightly faster convergence of the iterative DMPC at each sampling time than all other decompositions. We also investigate the performance of the iterative DMPC for the presented decompositions in the presence of measurement noise and for the output regulation problem. Specifically, we consider the state regulation problem in the presence of a white-noise with a signal-to-noise ratio of 40 dB per sample in all the state measurements. For the output regulation problem, we consider the volumetric hold up, temperature, and product concentration in the CSTRs and separator as the system’s outputs which can be measured at each sampling time, along with the same manipulated inputs. The optimization objective function is modified to penalize only these outputs. Similar to the state regulation problem, for the weighting matrices we choose 104 for the diagonal elements corresponding to the hold up regulation errors, 1 for the temperature regulation errors, 105 for the product concentration regulation errors, 103 for the stream flow rates, and 10−2 for the heat flow rates. We omit for brevity the detailed presentation of the state and output profiles, and focus just on the closed-loop performance and computation cost. Figure 10 compares the performance indices for the centralized and iterative DMPC architectures considering the different system decompositions during a 2 hr operation. We observe that the performance indices vary only slightly, and decomposition 1 and 3 still perform the best, with decomposition 1 having a closer performance index to the centralized architecture. Figure 11 shows the corresponding average


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computation times during the 2 hr operation. By considering the required computation time for decompositions 1 and 3 which have the best closed-loop performance, we observe that the control architecture based on the first decomposition (with optimal modularity) shows a slightly faster convergence. In summary, we observe that the iterative DMPC architecture is the most efficient computationally compared to the centralized MPC and sequential DMPC design, while no significant difference in closedloop performance can be observed between sequential and iterative DMPC. We also document that the optimal decomposition obtained by the community detection method, which minimizes the interactions between the subsystems, enables closed-loop performance close to that of the CMPC, while reducing significantly the computation effort. Suboptimal decompositions exhibit also relatively short computation times but not as good closed-loop performance.

Concluding remarks The effect of system decomposition on the closed-loop performance and computation time of various DMPC designs was studied, for a benchmark integrated process system. Sequential and iterative DMPC schemes were developed aiming to reduce the computation time compared to a CMPC scheme while maintaining similar performance. Four different system decompositions were considered and evaluated. It was shown that iterative DMPC is generally more computationally efficient. The system decompositions obtained by community detection result in very good performance (close to that of CMPC), while reducing computational times significantly. These conclusions were based on a single case study, under a specific set of model parameters, using a specific optimization solver. Yet, they confirm an expected computational advantage of iterative versus sequential DMPC, and illustrate the importance of proper system decomposition to realize the benefits of DMPC, as well as the value of optimal decompositions based on community detection. As such, the current study provides valuable insights and a template for additional comparative studies towards developing practically implementable DMPC designs for large-scale plants. Note that in this study we chose to focus on the state feedback case in order to isolate and investigate the effects of decomposition on the controller design problem itself. Extending this study to the case of estimation and the combined estimation and control problem (output feedback control) will be the subject of the future


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research.

Supporting information Solution time statistics of the iterative DMPC for different decompositions

Acknowledgment Financial support from the Petroleum Institute, Abu Dhabi, UAE is gratefully acknowledged.


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References 1. Rawlings, J.; Mayne, D. Model predictive control: Theory and design. Nob Hill Publishing, Madison, 2009. 2. Huang, R.; Harinath, E.; Biegler, L. Lyapunov stability of economically oriented NMPC for cyclic processes. J. Process Contr. 2011, 21(4), 501–509. 3. Lopez-Negrete, R.; D’Amato, F.; Biegler, L.; Kumar, A. Fast nonlinear model predictive control: Formulation and industrial process applications. Comput. Chem. Eng. 2013, 51, 55–64. 4. Camponogara, E.; Jia, D.; Krogh, B.; Talukdar, S. Distributed model predictive control. IEEE Contr. Syst. Mag. 2002, 22, 44–52. 5. Mayne, D. Model predictive control: Recent developments and future promise. Automatica 2014, 50(12), 2967–2986. 6. Scattolini, R. Architectures for distributed and hierarchical model predictive control: A review. J. Process Contr. 2009, 19, 723–731. 7. Christofides, P.; Liu, J.; Munoz de la Pena, D. Networked and Distributed Predictive Control. Springer-Verlag, London, 2011. 8. Christofides, P.; Scattolini, R.; Munoz de la Pena, D.; Liu, J. Distributed model predictive control: A tutorial review and future research directions. Comput. Chem. Eng. 2013, 51, 21–41. 9. Doan, M.; Keviczky, T.; De Schutter, B. An iterative scheme for distributed model predictive control using fenchel’s duality. J. Process Contr. 2011, 21, 746–755. 10. Dunbar, W. Distributed receding horizon control of dynamically coupled nonlinear systems. IEEE Trans. Automat. Contr. 2007, 52, 1249–1263. 11. Dunbar, W.; Murray, R. Distributed receding horizon control with application to multi-vehicle formation stabilization. Automatica 2006, 42, 549–558.


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12. Franco, E.; Magni, L.; Parisini, T.; Polycarpou, M.; Raimondo, D. Cooperative constrained control of distributed agents with nonlinear dynamics and delayed information exchange: A stabilizing recedinghorizon approach. IEEE Trans. Automat. Contr. 2008, 53, 324–338. 13. Liu, J.; Chen, X.; Munoz de la Pena, D.; Christofides, P. Sequential and iterative architectures for distributed model predictive control of nonlinear process systems. AIChE J. 2010, 56, 2137–2149. 14. Stewart, B.; Wright, S.; Rawlings, J. Cooperative distributed model predictive control for nonlinear systems. J. Process Contr. 2011, 21, 698–704. 15. Tippett, M.; Bao, J. Distributed model predictive control based on dissipativity. AIChE J. 2013, 59(3), 787–804. 16. Al-Gherwi, W.; Budman, H.; Elkamel, A. Selection of control structure for distributed model predictive control in the presence of model errors. J. Process Contr. 2010, 20, 270–284. 17. Motee, N.; Sayyar-Rodsari, B. Optimal partitioning in distributed model predictive control. In Proceedings of the American Control Conference, Denver, CO, 2003, 5300–5305. 18. Yin, X.; Arulmaran, K.; Liu, J.; Zeng, J. Subsystem decomposition and configuration for distributed state estimation. AIChE J. 2016, 62(6), 1995–2003. 19. Fortunato, S. Community detection in graphs. Phys. Rep. 2010, 486(3), 75–174. 20. Girvan, M.; Newman, M. Community structure in social and biological networks. Proc. Natl. Acad. Sci. 2002, 99(12), 7821–7826. 21. Heo, S.; Daoutidis, P. Control-relevant decomposition of process networks via optimization-based hierarchical clustering. AIChE J. 2016, 62(9), 3177–3188. 22. Heo, S.; Marvin, W.; Daoutidis, P. Automated synthesis of control configurations for process networks based on structural coupling. Chem. Eng. Sci. 2015, 136, 76–87. 23. Jogwar, S.; Daoutidis, P. Community-based synthesis of distributed control architectures for integrated networks. Chem. Eng. Sci. 2017, 172, 434–443.


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24. Leicht, E.; Newman, M. Community structure in directed networks. Phys. Rev. Lett. 2008, 100(11), 118703. 25. Stewart, B.; Venkat, A.; Rawlings, J.; Wright, S.; Pannocchia, G. Cooperative distributed model predictive control. Syst. Contr. Lett. 2010, 59(8), 460–469. 26. Fletcher, R. Practical Methods of Optimization. Wiley, New York, 1987.


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List of Tables 1

Process parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2

Steady state values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3

System decompositions (1: optimal community-based decomposition, 2: suboptimal communitybased decomposition, 3: intuitive decomposition based on the configuration of physical units, and 4: intuitive decomposition based on the material/energy balance). . . . . . . . . 28

4

Initial state values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5

Performance indices for MPC architectures during 2 hrs closed-loop process operation. . . 30


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List of Figures 1

Centralized MPC structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2

Sequential DMPC structure for system decomposition into M subsystems. . . . . . . . . . 32

3

Iterative DMPC structure for system decomposition into M subsystems. . . . . . . . . . . 33

4

Reactor-separator integrated process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5

Temporal profiles of the state variables of the system under CMPC. . . . . . . . . . . . . . 35

6

Temporal profiles of the manipulated variables computed by CMPC. The value of each control action remains constant between sampling times for ∆ = 1.5 minutes until receiving the next value from the controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7

Temporal profiles of the state variables of the system under iterative DMPC for the optimal community-based decomposition (blue; decomposition 1), the suboptimal communitybased decomposition (red; decomposition 2), decomposition based on the physical configuration of units (yellow; decomposition 3), and decomposition based on the material/energy balance (green; decomposition 4). . . . . . . . . . . . . . . . . . . . . . . . . 37

8

Temporal profiles of the manipulated variables computed by iterative DMPC for the optimal community-based decomposition (blue; decomposition 1), the suboptimal communitybased decomposition (red; decomposition 2), decomposition based on the physical configuration of units (yellow; decomposition 3), and decomposition based on the material/energy balance (green; decomposition 4). The value of each control action remains constant between sampling times for ∆ = 1.5 minutes until receiving the next value from the controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38


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9

Computation times required by MPC structures during 2 hrs closed-loop process operation for the optimal community-based decomposition (decomposition 1), the suboptimal community-based decomposition (decomposition 2), decomposition based on the physical configuration of units (decomposition 3), and decomposition based on the material/energy balance (decomposition 4). The blue bar corresponds to CMPC design, the red bars to sequential DMPC, and the green bars to iterative DMPC. . . . . . . . . . . . . . . . . . . 39

10

Performance indices of applying MPC structures during 2 hrs closed-loop process operation for the problems of state regulation (blue bars), measurement noise (red bars), and output regulation (yellow bars) where 1: optimal community-based decomposition, 2: suboptimal community-based decomposition, 3: decomposition based on the physical configuration of units, and 4: decomposition based on the material/energy balance. . . . . 40

11

Computation times required by MPC structures during 2 hrs closed-loop process operation for the problems of state regulation (blue bars), measurement noise (red bars), and output regulation (yellow bars) where 1: optimal community-based decomposition, 2: suboptimal community-based decomposition, 3: decomposition based on the physical configuration of units, and 4: decomposition based on the material/energy balance. . . . . . . . . . . . . 41


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Table 1: Process parameters.

Parameters Values Units Parameters Values Units ρ 1000 Kg/m3 Cp 4.2 KJ/KgK xA0 1 − T0 359.1 K k1 2.77 × 103 1/s k2 2.5 × 103 1/s E1 5 × 104 KJ/Kmol E2 6 × 104 KJ/Kmol ∆H1 −6 × 104 KJ/Kmol ∆H2 −7 × 104 KJ/Kmol αA 5 − αB 1 − αC 0.5 − R 8.314 KJ/KmolK ε 0.02 − m 0.00279 Kmol/Kg


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Table 2: Steady state values.

Parameters Values V1ss 1 V3ss 1 T2ss 427.1 ss xA1 0.536 ss xA2 0.545 ss xA3 0.298 Ffss1 5.04 FRss 17 F2ss 27.08 Qss 715.3 × 103 1 Qss 568.7 × 103 3

Units Parameters Values m3 V2ss 0.5 m3 T1ss 432.4 K T3ss 432.1 ss − xB1 0.448 ss − xB2 0.438 ss − xB3 0.670 m3/hr Ffss2 5.04 m3/hr F1ss 22.04 m3/hr F3ss 9.74 KJ/hr Qss 579.8 × 103 2 KJ/hr


27

Units m3 K K − − − m3/hr m3/hr m3/hr KJ/hr


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Table 3: System decompositions (1: optimal community-based decomposition, 2: suboptimal communitybased decomposition, 3: intuitive decomposition based on the configuration of physical units, and 4: intuitive decomposition based on the material/energy balance).

Decomposition Subsystem 1 1 2 3 1 2 2 3 4 1 3 2 3 1 4 2 3

Input State Ff 1, FR, Q1 V1, T1, xA1, xB1 Ff 2, F1, Q2 V2, T2, xA2, xB2 F2, F3, Q3 V3, T3, xA3, xB3 F f 1 , Q1 V1, T1, xA1, xB1 Ff 2, F1, Q2 V2, T2, xA2, xB2 F2, F3, FR V3, xA3, xB3 Q3 T3 Ff 1, F1, Q1 V1, T1, xA1, xB1 Ff 2, F2, Q2 V2, T2, xA2, xB2 F3, FR, Q3 V3, T3, xA3, xB3 Ff 1, Ff 2, FR xA1, xA2, xA3, xB1, xB2, xB3 F1, F2, F3 V1, V2, V3 Q1, Q2, Q3 T1, T2, T3


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Table 4: Initial state values.

Parameters V10 V30 T20 0 xA1 0 xA2 0 xA3

Values Units Parameters Values Units 0.7 m3 V20 0.7 m3 1.5 m3 T10 388.7 K 386.3 K T30 390.6 K 0 0.890 − xB1 0.110 − 0 0.886 − xB2 0.113 − 0 0.748 − xB3 0.251 −


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Table 5: Performance indices for MPC architectures during 2 hrs closed-loop process operation.

Control Architecture CMPC Decomp. Decomp. Sequential Decomp. Decomp. DMPC Decomp. Decomp. Iterative Decomp. Decomp.

1 2 3 4 1 2 3 4

Performance Index ISE ISC ISE+ISC 4252.39 1394.73 5647.12 4063.55 2097.59 6161.14 10831.71 127.02 10958.73 4024.27 2447.93 6472.2 9364.43 231.96 9596.39 4063.21 2098.03 6161.24 10840.35 126.04 10966.39 4024.14 2447.40 6471.54 9364.43 231.96 9596.39


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Figure 1: Centralized MPC structure.


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Figure 2: Sequential DMPC structure for system decomposition into M subsystems.


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Figure 3: Iterative DMPC structure for system decomposition into M subsystems.


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Figure 4: Reactor-separator integrated process.


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Figure 5: Temporal profiles of the state variables of the system under CMPC.


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Figure 6: Temporal profiles of the manipulated variables computed by CMPC. The value of each control action remains constant between sampling times for ∆ = 1.5 minutes until receiving the next value from the controller.


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Figure 7: Temporal profiles of the state variables of the system under iterative DMPC for the optimal community-based decomposition (blue; decomposition 1), the suboptimal community-based decomposition (red; decomposition 2), decomposition based on the physical configuration of units (yellow; decomposition 3), and decomposition based on the material/energy balance (green; decomposition 4). ACS Paragon Plus Environment

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Figure 8: Temporal profiles of the manipulated variables computed by iterative DMPC for the optimal community-based decomposition (blue; decomposition 1), the suboptimal community-based decomposition (red; decomposition 2), decomposition based on the physical configuration of units (yellow; decomposition 3), and decomposition based on the material/energy balance (green; decomposition 4). The value of each control action remains constant between sampling times for ∆ = 1.5 minutes until receiving the next value from the controller.


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Figure 9: Computation times required by MPC structures during 2 hrs closed-loop process operation for the optimal community-based decomposition (decomposition 1), the suboptimal community-based decomposition (decomposition 2), decomposition based on the physical configuration of units (decomposition 3), and decomposition based on the material/energy balance (decomposition 4). The blue bar corresponds to CMPC design, the red bars to sequential DMPC, and the green bars to iterative DMPC.


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Figure 10: Performance indices of applying MPC structures during 2 hrs closed-loop process operation for the problems of state regulation (blue bars), measurement noise (red bars), and output regulation (yellow bars) where 1: optimal community-based decomposition, 2: suboptimal community-based decomposition, 3: decomposition based on the physical configuration of units, and 4: decomposition based on the material/energy balance.


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Figure 11: Computation times required by MPC structures during 2 hrs closed-loop process operation for the problems of state regulation (blue bars), measurement noise (red bars), and output regulation (yellow bars) where 1: optimal community-based decomposition, 2: suboptimal community-based decomposition, 3: decomposition based on the physical configuration of units, and 4: decomposition based on the material/energy balance.


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Table of Contents/Abstract Graphics


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