Comparative Performance Analysis of Coordinated Model Predictive

Article pubs.acs.org/IECR

Comparative Performance Analysis of Coordinated Model Predictive Control Schemes in the Presence of Model−Plant Mismatch Abhay Anand,†,‡ Lakshminarayanan Samavedham,*,‡ and Sitanandam Sundaramoorthy§ †

Singapore-Delft Water Alliance and ‡Department of Chemical and Biomolecular Engineering, National University of Singapore, Singapore § Department of Chemical Engineering, Pondicherry Engineering College, Pondicherry, India ABSTRACT: Large-scale systems are formed by the interconnection of several subsystems, whose different spatial and temporal characteristics make them significantly heterogeneous. The optimal management of such systems must generally deal not only with issues related to large dimensionality and strong nonlinearity, but also with the presence of several interactions between the subsystems, which have a significant influence on the local control decisions and the overall system optimality. For such largescale systems, model predictive control (MPC) is an attractive control strategy and can be implemented in centralized or decentralized configurations. It has been shown that, to achieve a flexible and reliable control structure with optimum overall system performance, individual decentralized controllers have to be coordinated and driven toward the performance of a centralized controller. In this work, three coordination strategies that have been reported in the literature, viz., communication based coordination, cooperation based coordination, and price driven coordination, are evaluated for controlling multivariable processes. These three strategies have been evaluated on a benchmark chemical engineering system and on a quadruple tank system (via simulations), on the basis of their robustness, stability, and performance in comparison to that of a centralized MPC implementation. The ability to deal with a variety of model uncertainties and the coordination between the controllers within and across a hierarchy are some important aspects that have been investigated.

1. INTRODUCTION Control systems for complex, networked processes are often designed as a hierarchical structure.1 Usually, a multitiered hierarchy is considered. At the lowest regulatory layer, PID-type controllers are employed to regulate the individual process variables (control loops). In the next layer, local model predictive controllers (MPC) are designed using detailed models (which could be nonlinear) of different subsystems (units). A short time horizon is considered to predict and control the operating conditions of the system. This layer is succeeded by an MPC coordinator. At this level, information in the form of states, predicted input and output trajectories, etc. are transmitted between the various subsystems. The coordinator ensures that the goals of the higher level are attained and also manages information flow within the immediate lower layer (a layer containing local MPCs). At the top of the hierarchy, optimal plant performance is coupled with economic objectives and real time optimization (RTO) is performed over a long time horizon. A simple and abstract model of the system is used in this layer of the hierarchy to obtain targets for the lower levels. Since a simplified and steady state model of the process is being used at this level, the model needs to be periodically updated. At the topmost layer scheduling of the various processes within the system and plant-wide decision making is carried out over a long planning horizon generally on the order of weeks or months. This hierarchy is depicted in Figure 1. Studies in the literature have shown that there needs to be a certain level of integrity between the models used at the different levels of the control hierarchy. In the third level of the hierarchy, the coordinator works toward integrating the local model predictive controllers at the lower level. It incorporates the goals derived by the layers © 2012 American Chemical Society

Figure 1. Hierarchical control structure.

Received: Revised: Accepted: Published: 8273

August 17, 2011 April 27, 2012 May 24, 2012 May 24, 2012 dx.doi.org/10.1021/ie2018405 | Ind. Eng. Chem. Res. 2012, 51, 8273−8285

Industrial & Engineering Chemistry Research

Article

of the interactions between subsystems has a severe effect on the overall plant operation and forms the core of the coordinator design.13,14 The accuracy and validity of the models will change over time and result in deteriorating the performance of the controller. This necessitates the design of model predictive controllers and coordinators that are robust to parameter variations and disturbances. This work looks at the performance of different coordination strategies and also the effect of the accuracy of the interaction models on the performance of the coordinator. The robustness of a control strategy or a coordination strategy will be defined by its ability to deal with inaccuracies in the models.15 The ability to deal with a variety of model uncertainties and the coordination between the controllers within and across a hierarchy has been evaluated in this work. While there have been studies on some coordinated MPC strategies, there is a lack of critical comparisons between strategies evolving from different schools of thought. This paper presents a comparison of a few different MPC coordination techniques that have been proposed in the literature. These strategies have been evaluated on benchmark chemical engineering case studies with stability, computational effort, and robustness of the devised algorithms being the performance metrics. The merits and limitations of the various strategies and their fields of application have also been discussed.

above in its objective function and uses the information from the individual model predictive controllers at the lower level to drive the plant performance toward the overall optima. Information such as states, predicted trajectory information, and calculated control action at each time step are relayed between the local model predictive controllers to decide the best set of control actions. MPC is one of the most attractive control strategies2 for control of large-scale integrated systems. MPC, which originated in the 1970s, has evolved and developed into a successful strategy to control multivariable processes that are subject to operational constraints.3,4 MPC can be implemented in either a centralized or decentralized fashion. While centralized MPC leads to a plantwide optimum, it is computationally intensive, it is relatively difficult to implement, tune, and maintain, and it is characterized by poor fault tolerance. On the other hand, decentralized MPC is flexible, reliable, and easy to implement and maintain, but it leads to solutions that are not plant-wide optimum.5 Centralized and decentralized controllers define the limiting extremes of controller design. Networked systems are characterized by a large number of interactions between the individual subsystems, and these have an influence on the local control decisions and overall plant-wide optimality. In most systems/plants, neglecting these interactions or inadequately coordinating the subsystems will, in general, lead to suboptimal performance and even instability of the process. The performance deterioration of decentralized control6 compared to centralized control is due to ignoring or inadequately modeling subsystem interactions in the decentralized control algorithms. Over the past few years, with the necessity to control large-scale systems efficiently and optimally, distributed and coordinated control structures have been developed to address the shortcomings of both control paradigms.7,8 To achieve optimum plant operation, decentralized controllers have to be coordinated and driven toward achieving the performance of a centralized controller.9 This calls for the design of robust coordinators to provide a dynamic performance equivalent to that of a centralized control scheme while maintaining the existing decentralized structure. While each layer should be carefully designed, current research is oriented toward designing robust and computationally feasible coordination layers.10 Coordinated MPC works toward combining the advantages of both the centralized and decentralized control strategies while addressing their drawbacks. The decentralized structure of the system is maintained, but the performance is driven toward that of a centralized scheme. The coordinator (which is sandwiched between the RTO and the distributed MPC layer) coordinates the actions of the individual MPCs, relaying information among the various individual controllers to account for the interaction effects that exist between the different subsystems of the complex large-scale process. The coordinator uses information such as states, predicted output trajectory information, and calculated control action at each time step to decide the best set of control actions for each individual controller. Each individual controller has a well-defined model and cost function that govern their functioning. These are modified to a common structure to enable communication and cooperation between the individual controllers.11,12 Over the past decade, with the escalating importance of the economic efficiency of systems, plants are being designed in an increasingly complex manner with strongly interacting subsystems. The quantification of these complex interaction effects/ models between subsystems is a very challenging task. The effect

2. COORDINATED MODEL PREDICTIVE CONTROL Early formulations of coordinated MPC in the literature are based on the assumption that exchange (communication) of predicted trajectory information between subsystems is sufficient to account for interactions. It has been demonstrated that exchanging only interaction information among the subsystem controllers is not adequate to guarantee closed loop stability.16 This instability arises due to the contest between the local controllers as will be seen in the subsequent subsection. In addition to the communication of information, there needs to be cooperation between the controllers.17 A need to modify the objective functions as well as to incorporate interaction models into the local subsystem model arises. Such observations are the basis of coordination strategies. The main tasks of the coordinator are to provide information (such as states, predicted output trajectory, and calculated control action at each time step) to controllers, enabling them to derive interaction factors (the effect of one subsystem on the other) and also to modify the local optimization problem such that the coordinated performance of the local optimization problems is driven toward the performance of the centralized global optimization problem (goal coordination).18 Two common strategies used for coordinator design are19 (1) the interaction prediction principle and (2) the interaction balance principle. The interaction balance principle includes the interaction variables in addition to the input variables in the manipulated variable set of the local controllers, and then the coordinator works toward balancing the error between the desired (calculated) and real interaction variables. On the other hand, the interaction prediction principle considers only the input variables in the manipulated variable set, and then the coordinator works toward calculating the correct input variables after predicting and accounting for the effects of the interactions. These principles are fundamental to developing a coordinator for multiple MPCs, and these form the basis of all coordination techniques derived in the literature. In our work, 8274

dx.doi.org/10.1021/ie2018405 | Ind. Eng. Chem. Res. 2012, 51, 8273−8285


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the popular coordination techniques that have been evaluated are the communication based MPC coordination, cooperation based MPC coordination,16 and price driven MPC coordination.20 Consider a large-scale multiple input−multiple output (MIMO) system comprised of M individual subsystems. The discrete transfer function matrix formulation of the system, where ỹ denotes the output variable vector, ũ denotes the manipulated variable vector, and d̃ denotes the measured disturbance vector (the dimensions of which are (1 × M)) is as follows: y ̃ = Gp̃ u ̃ + Gd̃ d ̃

(1)

0

···

Gd 2(z) ··· ⋮

···

0

···

u ̃ = [ u1(z) u 2(z) ··· uM (z)]

(5)

d ̃ = [ d1(z) d 2(z) ··· dM(z)]

(6)

(15)

(16)

⎤ ⎥ ⎥ ··· 0 ⎥ ⎥ ⋮ ⋮ ⎥ ··· gPij− C + 1⎥⎦ ··· 0

0

0

g1ij

0

⋮

···

gPij− 1

gPij− 2

··· (gNij + 1 − gNij ) ⎤ ⎥ ij ij ij ij ⎥ (g4 − g2 ) ··· (gN + 2 − gN ) ⎥ ⎥ ⋮ ··· ⋮ ⎥ ij ij ij ij ⎥ (gP + 2 − g2 ) ··· (gN + P − gN )⎦

(17)

(g3ij − g2ij)

(18)

··· (g Ndĩ + 1 − g Ndĩ ) ⎤ ⎥ di di di di ⎥ (g4 − g2 ) ··· (g Ñ + 2 − g Ñ ) ⎥ ⎥ ⋮ ··· ⋮ ⎥ ⎥ (gPdi+ 2 − g2di) ··· (g Ndĩ + P − g Ndĩ )⎦ (g3di − g2di)

The prediction equation may be written as Yi(t ) = GiiΔui(t ) + hi(t ) + li(t )

(20)

where M

hi(t ) =

∑ GijΔuj(t ) j=1 j≠i

(7)

(21)

hi(t) denotes the effect of future inputs of other subsystems on the ith subsystem output.

(8) M

li(t ) =

∑ Fijxj(t ) + Fdixdi(t ) + YiM(t ) j=1

Yi = (Gi1Δu1 + Gi2Δu 2 + ... + GiM ΔuM )

(22)

In eq 22, li(t) is obtained by summing up the effect of past inputs of all M subsystems (first term), the effect of measured disturbances (second term), and the current measurements (third term). Equation 20, the prediction equation, forms the basis of all control calculations.

(9)

where Yi = [ yi (t + 1) yi (t + 2) ··· yi (t + P)]T

Nd = Ñ + P

(19)

Let P be the prediction horizon and C the control horizon. The prediction equation for the ith subsystem will then take the form

+ (Fi1x1 + Fi2x 2 + ... + FiMxM ) + (Fdixi) + YiM

(14)

⎡(g di − g di) 1 ⎢ 2 ⎢ di (g − g1di) Fdi = ⎢⎢ 3 ⎢⋮ ⎢ di di ⎣(gP + 1 − g1 )

In the process transfer function matrix G̃ p, the off-diagonal transfer functions represent the interaction models in the system. Each interaction model represents the effect of the input of one subsystem on the output of a different subsystem. For example, G̃ ij(z) is a transfer function model between the input to the ith subsystem and the output of the jth subsystem. Without loss of generality, it can be assumed that each subsystem is a single input−single output (SISO) system with one measured disturbance each. Each transfer function G̃ ij(z) is equivalent to Np number of step response coefficients, and each G̃ dj(z) is equivalent to Nd step response coefficients.

d

Np = N + P

⎡(g ij − g ij) 1 ⎢ 2 ⎢ ij (g3 − g1ij) ⎢ Fij = ⎢⋮ ⎢ ⎢(g ij − g ij) ⎣ P+1 1

(3)

(4)

d

(13)

⎡ g ij ⎢ 1 ⎢ ij g Gij = ⎢ 2 ⎢⋮ ⎢ ⎢ g ij ⎣ P

(2)

y ̃ = [ y1(z) y2 (z) ··· yM (z)]

di di di Gdi(z) ≡ [ g N g2 ··· g N ]

xdi = [Δdi(t − 1) Δdi(t − 2) ··· Δdi(t − Ñ ]T

The step response coefficient matrices are as follows:

Also

ij ij ij Gij(z) ≡ [ g1 g2 ··· g Np ]

(12)

YiM = [ yiM (t ) yiM (t ) ··· yiM (t )]T

G12(z) ··· G1M (z) ⎤ ⎥ G22(z) ··· G2M (z) ⎥ ⎥ ⋮ ⋮ ⋮ ⎥ ⎥ GM 2(z) ··· GMM (z)⎦ ⎤ 0 ⎥ ⎥ 0 ⎥ ⋮ ⎥ ⎥ GdM (z)⎦

xi = [Δui(t − 1) Δui(t − 2) ··· Δui(t − N )]T

The current measurements of the controlled variables are given as

and the disturbance transfer function matrix G̃ d is defined as ⎡Gd1(z) ⎢ ⎢0 Gd̃ = ⎢ ⎢⋮ ⎢ ⎣0

(11)

Based on the prediction horizon and the model horizon, the numbers of step response coefficients are

where the process transfer function matrix G̃ p is defined as ⎡ G (z ) ⎢ 11 ⎢ G (z ) Gp̃ = ⎢ 21 ⎢⋮ ⎢ ⎣GM1(z)

Δui = [Δui(t ) Δui(t + 1) ··· Δui(t + C − 1)]T

(10) 8275



Article M

For each subsystem i, the cost function Fi to be minimized by the local MPC is written in the form:

hri(t ) =

j=1 j≠r j≠i

M

Fi =

∑ wrJr (Δui)

(23)

r=1

∑ wr = 1

Jr =

M

∑ wrJr (Δui) r=1 r≠i

(25)

Here the cost function Fi for the ith subsystem is the weighted sum of the cost functions Jj of all the M subsystems. For the ith subsystem the cost function Ji is of the form Ji =

(RSi

− Yi ) Q̃ i(RSi − Yi ) + ΔuiTR̃ iΔui T

(41)

Cr = GriTQ̃ r(RSr − Grr Δur − lr − hri)

(42)

M

Fi = wi Ji +

∑ wrJr r=1 r≠i

(43)

In the standard form the cost function is written as

(27)

Fi = (i = 1, 2, ..., M )

(40)

Q r = GriTQ̃ rGri

(26)

with Q̃ i = qiIP × P

1 T Δui Q r Δui − ΔuiTCr 2

Writing the cost function Fi

where RiS is a vector of an individual subsystem set point RSi defined as RSi = [1 1 ··· 1]T RSi(P terms)

(39)

This may be substituted in the cost function eq 33 and Jr(Δui) is rewritten in the standard form as

(24)

r=1

(38)

⇒hr (t ) = GriΔui(t ) + hri(t )

M

Fi = wJi i (Δui) +

∑ GrjΔuj(t )

(28)

1 T Δui Q̅ iΔui − ΔuiTCi̅ 2

(44)

Substituting eqs 30−32 and eqs 40−42 in eq 44, we get

and

M

R̃ i = rIi C × C

(i = 1, 2, ..., M )

Q̅ i = wi(GiiTQ̃ iGii + R̃ i) +

(29)

qi and ri are tunable weights and are selected depending on the system dynamics. For calculating hi, Δuj (j ≠ i) values, which are the control decisions calculated by the individual MPCs, are assumed to be known. Therefore, Ji can be written as

∑ wr(GriTQ̃ rGri) r=1 r≠1

(45) M

T ̃ i Ci̅ = wG i ii Q i(R S − li − hi) +

∑ wrGriTQ̃ r(RSr − Grr Δur r=1 r≠1

− lr − hri)

(46)

1 Ji = ΔuiTQ iΔui − ΔuiTCi 2

(30)

Q i = GiiTQ̃ iGii + R̃ i

(31)

Δuimin ≤ Δui ≤ Δuimax

(47a)

Ci = GiiTQ̃ i(RSi − li − hi)

(32)

uimin ≤ ui ≤ uimax

(47b)

yimin ≤ yi ≤ yimax

(47c)

The input rate, input, and output constraints for the ith subsystem are written in the form

The cost function Jr(Δui) (r ≠ i) can be calculated as follows: Jr = (RSr − Yr )T Q̃ r(RSr − Yr ) + ΔurTR̃ r Δur

(33)

Yr = Grr Δur + hr + lr

(34)

for i = 1, 2, ..., M

The input rate constraint eq 47a may be rewritten in the form

M

hr =

∑ GrjΔuj j=1 j≠r

⎡−Δu min ⎤ ⎡−IC × C ⎤ i ⎥ ⎥Δui ≤ ⎢ ⎢ ⎢⎣ Δuimax ⎥⎦ ⎣ IC × C ⎦

(35)

M

lr =

That is

∑ Frjxj + Fdrxdr + YrM

M1iΔui ≤ N1i ,

(36)

j=1

(48)

Rewriting hr as

i = 1, 2, ..., M

(49)

where M

hr (t ) = GriΔui(t ) +

∑ GrjΔuj(t ) j=1 j≠r j≠i

⎡−IC × C ⎤ ⎥; M1i = ⎢ ⎣ IC × C ⎦

(37) 8276

⎡−Δu min ⎤ i N1i = ⎢ max ⎥ ⎢⎣ Δui ⎥⎦

(50)



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implemented independently in an iterative manner. In the communication based coordination21 strategy, subsystem controllers exchange interaction information at every iteration. Since an MPC optimization scheme is being employed, trajectories for the input variables are available at each iteration and this information is exchanged between the subsystem controllers as illustrated in Figure 2.

Similarly, the input constraint eq 47b may be reformulated as described below: ⎡ u (t ) ⎤ ⎡1 0 0 ⎢ i ⎥ ⎡1⎤ ⎢ ⎢ ⎥ ⎢ ui(t + 1) ⎥ ⎢1 1 0 1 ⎢ ⎥ ⎢·⎥ ⎢· · 1 ⎢· ⎥ = ⎢ ⎥ui(t − 1) + ⎢ · ⎢ ⎥ ⎢· ⎥ ⎢· · · ⎢· ⎥ ⎢·⎥ ⎢· · · ⎢ u (t + c − 1)⎥ ⎢⎣1⎥⎦ ⎢⎣1 1 1 ⎣ i ⎦

· · · 1 · 1

· · · · 1 1

0⎤ ⎥ 0⎥ 0⎥ ⎥ ·⎥ ·⎥ 1 ⎥⎦

⎡ Δu (t ) ⎤ ⎢ i ⎥ ⎢ Δui(t + 1) ⎥ ⎢ ⎥ × ⎢· ⎥ ⎢· ⎥ ⎢· ⎥ ⎢ Δu (t + c − 1)⎥ ⎣ i ⎦

(51)

That is ui = c1ui(t − 1) + c 2Δui

Figure 2. Schematic of communication and cooperation based MPC coordination.

(52)

where ⎡1⎤ ⎢ ⎥ ⎢1⎥ · c1 = ⎢ ⎥ ⎢·⎥ ⎢·⎥ ⎢⎣1⎥⎦

and

⎡1 ⎢ ⎢1 ⎢· c2 = ⎢ ⎢· ⎢· ⎢⎣1

0 1 · · · 1

0 0 1 · · 1

· · · 1 · 1

· · · · 1 1

0⎤ ⎥ 0⎥ 0⎥ ⎥ ·⎥ ·⎥ 1 ⎥⎦

⇒M 2i Δui ≤ N2i ⎡− c 2 ⎤ M 2i = ⎢ ⎥; ⎣c2 ⎦

Each communication based MPC transmits the current state and input trajectory information to all interconnected subsystem MPCs through the coordinator. However, each individual controller has no knowledge of the cost functions of other controllers. The objectives of each subsystem MPC controller are frequently in conflict with the objectives of the controllers (MPCs) that control the other interacting subsystems. The equilibrium of such a strategy is driven to a noncooperative equilibrium or Nash equilibrium.22 Due to the noncooperative and competing effect, such a strategy is usually suboptimal and when interactions are strong, closed loop stability is not guaranteed. In the communication based coordination strategy, we substitute wi = 1 and wr = 0 ∀ r ≠ i in eqs 43, 45, and 46 as each local MPC optimizes its own independent objective. This leads to

(53) (54)

⎡−u min + c u (t − 1)⎤ 1 i i ⎥ N2i = ⎢ ⎢⎣ u max − c u (t − 1) ⎥⎦ 1 i i

(55)

Finally, the output constraint eq 47c may be expressed as M3i Δui ≤ N3i

(56)

where M3i

⎡−Gii ⎤ ⎥; =⎢ ⎢⎣Gii ⎥⎦

N3i

⎡−y min + l + h ⎤ i i i ⎥ = ⎢ max ⎢y ⎥ ⎣ i − li − hi ⎦

⎡ Mi ⎤ ⎢ 1⎥ Mi = ⎢ M 2i ⎥ ; ⎢ ⎥ ⎢⎣ M i ⎥⎦ 3

Ci̅ = GiiTQ̃ i(RSi − li − hi)

(61)

where M

hi =

∑ GijΔuj j=1 j≠i

(58)

⎡Ni ⎤ ⎢ 1⎥ Ni = ⎢ N2i ⎥ ⎢ ⎥ ⎢⎣ N i ⎥⎦ 3

(60)

(57)

Combining all three constraints eqs 47a, 47b, and 47c for the ith subsystem, we may write MiΔui ≤ Ni

Q̅ i = GiiTQ̃ iGii + R̃ i

(62)

and M

li =

∑ Fijxj + Fdixdi + YiM j=1

(63)

The controllers now solve the optimization problem defined in eqs 43 and 58 using the values derived for Q̅ i and C̅ i in eqs 60 and 61. The local controllers are exchanging interaction information through the modified prediction equation (eqs 62 and 63), but they optimize different individual objective functions and these independent objective functions may often be conflicting in nature. As a result, despite having knowledge of the local control decisions, the individual controllers try to achieve their own

(59)

The method of quadratic programming (QP) is used by each local MPC to minimize the cost function defined by eq 44 subject to the constraints defined by equation eq 58. 2.1. Communication Based Coordination. Every networked system is comprised of a number of individual subsystems subject to individual objectives and constraints. Generally, the control algorithms for these subsystems are 8277



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individual optima. In other words, the controllers are working with the same resources (control variables) but toward satisfying different objectives leading to a contest between the individual controllers. As a result, when the interactions are strong, the individual controllers fail to converge to a single optimal control decision. This drives the equilibrium of such a strategy toward a noncooperative equilibrium.21,23,24 2.2. Cooperation Based Coordination. The cooperation based coordination strategy works toward improving the performance of an existing decentralized control structure by allowing the subsystem controllers to exchange interaction information and also support each other in driving the performance toward that of a centralized controller.25 To overcome the drawbacks associated with communication based coordination strategies (noncooperative and competing controllers), the following modification has been incorporated in the cooperation based coordination strategy: The local objective functions of each subsystem MPC controller are converted to a common global objective function. This is achieved by using a weighted convex sum of the individual objective functions as the new objective function as indicated in eqs 64, 65, and 66 and also illustrated in Figure 2.

Figure 3. Schematic of price driven MPC coordination.

equating the total demand from all subsystems to overall resource availability.30,31 The price vector is iteratively adjusted until the constraints are satisfied. Newton’s algorithm is used to adjust the price vector iteratively. By introducing an auxiliary variable ei, the prediction equation, eq 20, is rewritten as Yi(t ) = GiiΔui(t ) + ei(t ) + li(t )

The “linking constraint” or the “resource constraint” is given by eq 68.

M

Fi =

∑ wrJr (Δui)

M

(64)

r=1

ei(t ) −

∑ GijΔuj = 0 j=1 j≠i

M

∑ wr = 1

vi = [ΔuiT eiT ]T

M

∑ wrJr (Δui) r=1 r≠i

(68)

Next we define an auxiliary vector vi as

(65)

r=1

Fi = wi Ji (Δui) +

(67)

= [Δui(t ) ... Δui(t + C − 1)ei(t + 1) ... ei(t + P)]T (66)

(69)

Now

Weights wr and wi are assigned to the various objectives heuristically based on the physical or economic significance of the variables being optimized/controlled at each subsystem. Subsystems or output variables that have a more significant effect on the overall plant operations as designated by the process engineer would be weighed more significantly than the others. In our studies, we have weighed all objectives equally in the cooperation based coordination strategy. The control problem derived in eqs 23−59 is solved to determine the optimal control profile. Since all the local MPC controllers are solving an optimization problem with the same objective function, the optimal control profile generated at all iterates of the cooperative based coordination is plant-wide feasible and closed loop stable (pareto optimal).26 2.3. Price Driven Coordination. One other technique used for coordinating multiple MPC controllers is based on separating the centralized optimization problem into a number of disjoint, independent subproblems.20,27 This is brought about by the addition of auxiliary variables to the decision variable set. The objective functions are rewritten in terms of the Lagrange multipliers, and the Lagrange multipliers are iteratively adjusted to satisfy the constraints. In the price driven coordination strategy,28,29 the large-scale system is decomposed into a number of subsystems based on the principle of separability. Figure 3 illustrates the price driven coordination scheme. Auxiliary variables (ei) in the form of price vectors (vi) and resource constraints (eq 68) are introduced to the existing control problem. The optimum plant performance is obtained by

Yi(t ) = Giĩ vi + li

(70)

where Giĩ = [Gii IP × P ]

(71)

Now the global objective function is written as a sum of M disjoint objective functions corresponding to the individual subsystems. M

J=

∑ Ji (vi) i=1

Ji (vi) = (RSi − Yi )T Q̃ i(RSi − Yi ) + viTR̅ ivi

(72) (73)

where ⎡ R̃ 0 ⎤ i ⎥ R̅ i = ⎢ ⎢⎣ 0 R iIP × P ⎥⎦

(74)

Now Ji (vi) =

1 T vi Q ivi − vC i i 2

(75)

where

8278

T Q i = Giĩ Q̃ iGiĩ + R̅ i

(76)

T Ci = Giĩ Q̃ i(RSi − li)

(77)



Article

Rewriting the resource constraint M

ei −

⎡−u min + c u (t − 1)⎤ i 1 i ⎥ N1i = ⎢ ⎢⎣ u max − c u (t − 1) ⎥⎦ i 1 i

⎡− c 2 ⎤ M 2i = ⎢ ⎥; ⎣c2 ⎦

∑ GijΔuj = O̅ j=1 j≠i

(88)

In summary, the optimization problem is posed as

(78)

M

where

J (v ) =

min

O̅ = [0 0 ··· 0]TP times

(79)

∑ Ji (vi) i=1

subject to

Expanding eq 78, we may write

M ivi ≤ N i

e1 − G12Δu 2 − G13Δu3 − ... − G1M ΔuM = 0 e 2 − G22Δu 2 − G23Δu3 − ... − G2M ΔuM = 0 e3 − G32Δu 2 − G33Δu3 − ... − G3M ΔuM = 0

(89)

and M

(80)

∑ Aivi = O̅

⋮

i=1

eM − GM 2Δu 2 − GM 3Δu3 − ... − GMM ΔuM = 0

Using Lagrange multipliers (price vectors), the optimization problem is solved for the ith subsystem,

Equation 80 may be regrouped as ⎡OP × C ⎢ ⎢−G21 ⎢ ⎢−G31 ⎢⋮ ⎢ ⎢⎣−GM1

⎡−G12 IP × P ⎤ ⎥ ⎢ OP × P ⎥ ⎢OP × C ⎥⎡ Δu1⎤ ⎢ ⎥ + −G32 OP × P ⎥⎢ ⎣ e1 ⎦ ⎢ ⎥ ⎢⋮ ⋮ ⎥ ⎢ ⎢⎣−GM 2 OP × P ⎥⎦

⎡−G1M ⎢ ⎢−G2M ⎢ + ⎢−G3M ⎢⋮ ⎢ ⎢⎣OP × C

OP × P ⎤ ⎥ IP × P ⎥ ⎥⎡ Δu 2 ⎤ ⎥ + ... OP × P ⎥⎢ ⎣e2 ⎦ ⎥ ⋮ ⎥ OP × P ⎥⎦

⎡O̅ ⎤ OP × P ⎤ ⎢ P × 1⎥ ⎥ ⎢O̅P × 1⎥ OP × P ⎥ ⎥ ⎥⎡ ΔuM ⎤ ⎢ ⎥ = ⎢O̅P × 1⎥ OP × P ⎥⎢ ⎣e ⎦ ⎢ ⎥ ⎥ M ⋮ ⎢⋮ ⎥ ⎥ ⎢O ⎥ IP × P ⎦⎥ ⎣ ̅P × 1 ⎦

gi(λ) = min subject to

The constraints are Δuimin ≤ Δui ≤ Δuimax

(83a)

uimin ≤ ui ≤ uimax

(83b)

(84)

In terms of vi

M ivi ≤ N i

(85)

where

⎡− I ⎤ M1i = ⎢ ⎥ ; ⎣I ⎦

Cĩ = Ci − AiTλ

(92)

T Ci = [Giĩ Q̃ i(RSi − li)]

(93)

0 2C × P ⎤ ⎥ 0 2C × P ⎥ ⎥; −IP × P ⎥ ⎥ IP × P ⎦

⎡Ni ⎤ ⎢ 1 ⎥ ⎢ i ⎥ N i = ⎢ N2 ⎥ ⎢ αP × 1⎥ ⎢α ⎥ ⎣ P × 1⎦

⎡−Δu min ⎤ i N1i = ⎢ max ⎥ ⎥⎦ ⎢⎣ Δui

(94)

∑M i=1

3. CASE STUDIES Several case studies were analyzed to evaluate the performance of the coordination strategies. However, only the results from two systems (a generic benchmark system and a quadruple tank system) which exemplify the nature of the coordination strategies and bring out their uniqueness are provided here. A tuning strategy derived by Shridhar et al.32 was used to tune the individual multivariable model predictive controllers. The performances of the various control algorithms were investigated, and the results are provided and discussed. The performance metric is measured in terms of deviations from the set point (sum of squared errors (SSE)). Model based controllers based on transfer function models (convolution models) were derived and implemented in MATLAB version 7.8.0.347. The developed codes can be made available to researchers on request. 3.1. Shell Benchmark Problem. The Shell benchmark problem is on the control of a heavy oil fractionator. The output variables are the compositions at the top and side draws and the reflux temperature, while the manipulated variables are the top and side draw rates and also the reflux heat duty. The generic benchmark developed by Li et al.33 is a standard test problem

for, i = i, 2, . . ., M

⎡ Mi ⎢ 1 ⎢Mi Mi = ⎢ 2 ⎢ 0P × C ⎢ ⎣ 0P × C

(91)

where g is the gradient of Aivi and k is the iteration number. Note that IA×A is used to denote an identity matrix of dimensions A × A and OA×B is used to denote a matrix of zeros of dimensions A × B.

(82)

i = 1, 2, ..., P

(90)

T Q i = [Giĩ Q̃ iGiĩ + R̅ i]

λ k + 1 = λ k − αkg k

(81)

M

−A ≤ ei ≤ A ;

i = 1, 2, ..., M

λ (in eq 90) is adjusted using Newton’s algorithm until convergence (gk = 0):

which may be written as

i=1

M ivi ≤ N i ;

where

k

∑ Aivi = O̅

1 T vi Q ivi − viTCĩ 2

(86)

(87) 8279



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Figure 4. The cooperation based coordination algorithm was computationally very intensive but was observed to asymptotically converge to the centralized controller performance as seen in Figure 5. On the other hand, the price driven coordination was able to converge much faster to an improved solution. In Table 1, the average number of optimization calls per controller has been provided. The number of optimization calls is used as a measure of computational intensity. While the centralized MPC invokes the optimization routine only 500 times, the size of the optimization problem is much larger. All other controller strategies solve optimization problems of the same dimension, and the average computational times per control cycle for the different coordination algorithms are similar. This ensures that the number of optimization calls is a reliable metric for gauging the computational requirements of the different coordination algorithms. As seen in Table 1, the price driven coordinator requires considerably less computational effort compared to the cooperation based coordinator and outperforms the other coordination algorithms on a performance per computational effort basis. On the other hand, the cooperation based coordination algorithm produces a feasible solution at every iteration, and hence the number of iterations can be limited as desired. The performance was quantified by the “total SSE”, which is the sum of the SSEs of all response variables. The main task of the coordinator is to derive the effects of interactions between subsystems. The quantification of these interactions is not straightforward, and the exact interaction models may not be estimated accurately in the real world. It is of prime importance to gauge the effect of the mismatches in the interaction models on the performance of the coordinators. This would aid the selection and design of the most robust coordinator for real-world applications. This study is more important in a distributed setting as the existing distributed decentralized control configuration would be designed by explicitly ignoring these interaction effects. Therefore, the task of estimating the interaction effects to be utilized by the coordinator becomes even more challenging. The robustness of the control algorithms was studied by introducing model−plant mismatches.34 The communication based strategy does not guarantee stability as discussed in section 2.1. To overcome this drawback, the communication based algorithm was modified into the cooperation based coordination algorithm. All further comparisons are only between the cooperation based coordination and price driven coordination strategies. The models used by the MPC controller were modified by varying the gain, dead time, and time constants of the off-diagonal terms in the transfer function matrix G(s). Three off-diagonal models were randomly chosen and the transfer function properties were altered. In the simulation results provided below, mismatches were introduced in the transfer functions G12, G23, and G31 in G(s). In the presence of mismatches, the performances of the coordination algorithms were quantified based on the “percent deviation”, which is an indication of how much the uncertainty in the process models affects the performance of any coordination algorithm compared to its performance when there is no mismatch. Higher percent deviation values indicate that the particular coordination algorithm is more sensitive to model−plant mismatches.

for the evaluation of new control strategies. It is a multivariable and highly constrained process with a high level of interactions between the subsystems. This problem is also characterized by input constraints, input rate constraints, and output constraints. The model of the process is y = G(s)u + Gd(s)d with the transfer function matrices: ⎡ 4.05e−27s 1.77e−28s 5.88e−27s ⎤ ⎢ ⎥ ⎢ 50s + 1 60s + 1 50s + 1 ⎥ ⎢ ⎥ 5.39e−18s 5.72e−14s 6.90e−15s ⎥ G (s ) = ⎢ ⎢ 50s + 1 60s + 1 40s + 1 ⎥ ⎢ ⎥ −20s 4.42e−22s 7.20 ⎥ ⎢ 4.38e ⎢⎣ 33s + 1 44s + 1 19s + 1 ⎥⎦ ⎡ 1.44e−27s ⎤ ⎢ ⎥ ⎢ 40s + 1 ⎥ ⎢ −15s ⎥ ⎥ Gd(s) = ⎢ 1.83e ⎢ 20s + 1 ⎥ ⎢ ⎥ ⎢ 1.26 ⎥ ⎣ 32s + 1 ⎦

and

The constraints include |yi | ≤ 0.5,

|ui| ≤ 0.5,

|Δui| ≤ 0.5

for i = 1, 2, 3

The interactions are quantified using the relative gain array (RGA). The RGA for this system is ⎡ 2.0757 − 0.7289 − 0.3468 ⎤ ⎥ ⎢ ⎢ 3.4242 0.9343 − 3.3585 ⎥ ⎢⎣−0.7946 0.7946 4.7053 ⎥⎦

The presence of significant off-diagonal terms indicates the severe interaction that exists in the system. Simulations indicated that the effects of these interactions were strong, and although the decentralized control strategy yielded a closed loop stable solution, its performance was significantly degraded compared to the centralized performance. A set point change of magnitude 0.1 was introduced to the three subsystems at sampling instants 10, 20, and 30, respectively. Also, step disturbances of magnitude 0.1 were introduced in the three subsystems at sampling instants 200, 250, and 300. In this system, all the coordination strategies were able to provide stable closed loop responses but with significantly differing performances as shown in Table 1 and also seen in Table 1. Performance Indices for Shell Benchmark Case Study control algorithm centralized MPC decentralized MPC commun based MPC coop. based MPC (200 iterations) coop. based MPC (1000 iterations) price driven MPC

total SSE

no. optimization calls per controller

av computational time per control cycle (ms)

0.097 0.604 0.524

500 500 2644

16.07 2.49 9.06

0.391

9565

32.92

0.1038

10373

36.86

0.183

3290

31.15

⎛ SSEno mismatch − SSE with mismatch ⎞ % deviation = ⎜ ⎟ ·100 SSEno mismatch ⎝ ⎠ (95) 8280



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Figure 4. Comparison of the Shell oil system output variables’ response under different control configurations.

Table 3. Performance Indices in the Presence of Model−Plant Mismatch (Dead Time Mismatch)

Table 2. Performance Indices in the Presence of Model−Plant Mismatch (Gain Mismatch) SSE for coop. based MPC (200 iterations)

% dev

SSE for price driven MPC

% dev

+50 +10 0 (no mismatch) −10 −50

unstable 0.731 0.391 0.429 0.443

unstable 86.95 − 9.72 13.30

0.782 0.201 0.183 0.188 0.203

327.32 9.83 − 2.73 10.93

SSE for coop. based MPC (200 iterations)

% dev


% dev

+50 +10 0 (no mismatch) −10 −50

0.467 0.413 0.391 0.397 0.416

19.43 5.63 − 1.53 6.39

0.207 0.187 0.183 0.185 0.187

13.11 2.19 − 1.09 2.19

seen in Tables 2 and 3. The direction of mismatch, especially in the gain, was found to have a significant effect on the controller performance, and cooperation based coordination was unable to handle large positive mismatches (overestimated gain values) as seen in Table 2. Though the performance of price driven coordination deteriorated in the presence of overestimated gain values, it was still able to converge to a feasible solution and resulted in a stable response. Underestimation of the gains in the process models did not have very significant effects on the coordination algorithm performances, and price driven coordination was found to deviate less than cooperation based coordination. The trends were seen to be similar for mismatches in gain, dead time, and time constants, and similar studies were carried out for mismatches in different transfer function models resulting in similar trends (not presented here for the sake of brevity). The applicabilty of the cooordination algorithm for MIMO systems was verified by reconfiguring the Shell benchmark case study with MIMO controllers. The same system which was previously decomposed into three SISO subsystems was now decomposed into a MIMO subsystem and a SISO subsystem as depicted by matrix 96. The upper block matrix represents the first two input−two output subsystem, and the lower block

Figure 5. Convergence of cooperation based coordinated MPC to centralized performance.

gain mismatch (%)

dead time mismatch (%)

Price driven coordination was found to be more robust and deviated less from the base value (no mismatch performance) as 8281



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matrix represents the second one input−one output subsystem. ⎡⎡ −27s ⎢ ⎢ 4.05e ⎢ ⎢ 50s + 1 ⎢⎢ −18s ⎢ ⎢ 5.39e ⎢ ⎣ 50s + 1 ⎢ ⎢ 4.38e−20s ⎢ ⎣ 33s + 1

⎤ 1.77e−28s ⎤ 5.88e−27s ⎥ ⎥ 60s + 1 ⎥ 50s + 1 ⎥ ⎥ ⎥ 5.72e−14s ⎥ 6.90e−15s ⎥ 60s + 1 ⎦ 40s + 1 ⎥ ⎥ ⎡ 7.20 ⎤ ⎥ 4.42e−22s ⎥⎥ ⎢⎣ 19s + 1 ⎦ ⎦ 44s + 1

(96)

Two MPC controllers were designed for the system, and the simulation results are provided in Table 4. It was seen that while Table 4. Performance Indices for Shell Benchmark Case Study (MIMO Configuration) control algorithm

total SSE

centralized MPC decentralized MPC comm. based MPC coop. based MPC (200 iterations) coop. based MPC (1000 iterations) price driven MPC

0.097 0.416 0.377 0.231 0.1034 0.163

Figure 6. Quadruple tank system.

The constraints are |yi| ≤ 5, |ui| ≤ 10, and |Δui| ≤ 1 for i = 1, 2. The RGA for the quadruple tank system is ⎡1.0667 − 0.0667 ⎤ ⎥ ⎢ ⎣−0.0667 1.0667 ⎦

Table 5. Performance Indices for the Quadruple Tank Case Study (Minimum Phase Configuration)

⎡ 3.7γ ⎤ 3.7(1 − γ2) 1 ⎢ ⎥ (62s + 1)(23s + 1) ⎥ ⎢ 62s + 1 G (s ) = ⎢ ⎥ 4.7(1 − γ1) 4.7γ2 ⎢ ⎥ ⎢⎣ (30s + 1)(90s + 1) 90s + 1 ⎥⎦

and

⎡−0.2250 1.2250 ⎤ ⎥ ⎢⎣ 1.2250 − 0.2250 ⎦

depending on the value of γ1 and γ2 (the former for γ1 = γ2 = 0.8 and the latter for γ1 = γ2 = 0.3). This indicates the presence of interacting subsystems and, depending on the valve positions (value of γ1 and γ2), the interaction effects (off-diagonal terms) become significant and ignoring them leads to severe instabilities. With a valve opening of 0.8 (γ1 = γ2 = 0.8), the system exhibits minimum phase characteristics, and with a valve opening of 0.3 (γ1 = γ2 = 0.3) the system exhibits nonminimum phase characteristics. The system was simulated under both minimum and nonminimum phase characteristics by manipulating the position of the external valve, and the different controller coordination algorithms were compared under both the minimum and nonminimum phase scenarios. The desired water level set points of the two lower tanks were increased by 1 unit at sampling instants 4 and 100. Also, disturbances (Gd) in the form of inflows to the upper level tanks were introduced at sampling instants 150 and 200. Under minimum phase behavior, all coordination algorithms were able to provide a closed loop stable solution with performances better than a decentralized controller and also close to the centralized controller performance as seen in Table 5

the performances of all the algorithms are better than the previously reported SISO control configuration, the relative performance trends of the different coordination algorithms remain the same. The performance improvement over the SISO control configuration is due to the fact that the MIMO subsytem includes two of the previously mentioned interaction models in the system model. As two of the interaction effects are implicitly modeled into the subsystem, the overall performance improves. 3.2. Quadruple Tank System. The quadruple tank system described in the literature35,36 was employed as a simulated test bed for evaluating different coordination strategies. It is desired to control the level of water in the lower two tanks (tank 1 and tank 2) using two pumps. The flows from pumps 1 and 2 (v1 and v2) are manipulated to control the water levels in the lower tanks 1 and 2, respectively (y1 and y2). The outflows from tanks 3 and 4 are the disturbances. γ1 and γ2 represent the valve positions that distribute the flow of water between the lower and upper tanks and can be tuned to change the dynamics of the system as seen in Figure 6. In this work, we employ the model of the quadruple tank system37 in the form y = G(s)u + Gd(s)d with the transfer function matrices:th

⎡ 1 ⎢ 20s + Gd(s) = ⎢ ⎢ 0.5 ⎢⎣ 30s +

or

control algorithm centralized MPC decentralized MPC comm. based MPC coop. based MPC (2 iterations) coop. based MPC (10 iterations) price driven MPC

⎤ ⎥ 1⎥ ⎥ ⎥ 1⎦

total SSE

no. optimization calls per controller

av computational time per control cycle (ms)

1.346 1.460 1.437 1.457

500 500 560 687

9.1 3.68 4.26 4.64

1.348

938

5.34

1.387

668

4.84

and also in Figure 7. Again, on a performance per computational effort basis, the price driven coordinator was seen 8282



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Figure 7. Comparison of the four tank system output variables’ response under different control configurations.

Table 6. Performance Indices for the Quadruple Tank Case Study (Nonminimum Phase Configuration) control algorithm

total SSE

centralized MPC decentralized MPC comm. based MPC coop. based MPC (2 iterations) coop. based MPC (10 iterations) price driven MPC

8.328 unstable unstable 9.088 8.461 unstable

Table 7. Performance Indices for the Quadruple Tank Case Study (Minimum Phase Configuration) in the Presence of Model−Plant Mismatch

to outperform the other coordination algorithms (see Table 5). This was not the case with a nonminimum phase behavior, which was seen to deteriorate the performance of the controllers, making the system harder to control. A nonminimum phase behavior resulted in severe interactions between subsystems, and only the cooperation based coordination resulted in a closed loop stable solution while all other coordination strategies failed. The details are summarized in Table 6. Similar to the previous case study, model−plant mismatches were introduced to the interaction models to analyze the robustness of the coordination algorithms. The mismatches were introduced to the system when a minimum phase behavior was exhibited. Though all the coordination algorithms were able to result in stable closed loop responses, it was observed that the coordination algorithms were most sensitive to gain overestimates compared to the other mismatches. Once again, price driven coordination was found to be more robust than cooperation based coordination and deviated less from the base values as seen in Table 7. In the case of severe overestimates of the process gains, the price driven coordination was seen to significantly outperform cooperation based coordination with a 3.57% lesser deviation.

gain mismatch (%)

SSE for coop. based MPC (2 iterations)

% dev


% dev

+50 +10 0 (no mismatch) −10 −50

1.736 1.491 1.457 1.472 1.501

8.18 2.33 − 1.03 3.02

1.451 1.415 1.387 1.401 1.426

4.61 2.02 − 1.01 2.81

4. CONCLUSIONS Coordinating multiple model predictive controllers has been shown to significantly improve the performance of decentralized control strategies, driving them toward the control performance of the centralized controller. Communication of information between controllers was seen to be insufficient to guarantee closed loop stability in the case of the quadruple tank system and even in the Shell benchmark problem where it yielded a closed loop stable response; its performance was significantly poorer compared to the other coordination strategies. To overcome this drawback, objective functions of the local controllers had to be modified to enable the subsystems to cooperate toward a pareto optimal solution. The price driven coordination was also seen to effectively coordinate the local controllers with a significantly lower computational demand. To quanitfy the computational demands of the various control algorithms, the number of optimization calls and the average computational time per control cycle were measured and compared. Cooperation based coordination is the only strategy that asymptotically converges to the centralized controller performance, albeit at a higher computational cost. On the other hand, it was also observed that price driven coordination 8283



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(5) Kariwala, V. Fundamental Limitations on Achievable Decentralized Performance. Automatica 2007, 43, 1849−1854. (6) Cui, H.; Jacobsen, E. Performance limitations in decentralized control. J. Process Control 2002, 12, 485−494. (7) Camponogara, E.; Jia, D.; Krogh, B.; Talukdar, S. Distributed model predictive control. IEEE Control Syst. Mag. 2002, 22, 44−52. (8) Siljak, D.; Zecevic, A. Control of large scale systems: beyond decentralized feedback. Annu. Rev. Control 2005, 29, 169−179. (9) Jia, D.; Krogh, B. Distributed model predictive control. Proceedings of the IEEE American Control Conference; 2001. (10) Aske, E.; Strand, S.; Skogestad, S. Coordinator MPC for maximizing plant throughput. Comput. Chem. Eng. 2008, 32, 195−204. (11) Venkat, A.; Rawlings, J.; Wright, S. Implementable distributed model predictive control with guaranteed performance properties. Proceedings of the IEEE American Control Conference; 2006. (12) Venkat, A.; Rawlings, J.; Wright, S.Distributed model predictive control of large-scale systems. In Assessment and Future Directions of Nonlinear Model Predictive Control, 1st ed.; Springer: Berlin, Germany, 2007. (13) Negenborn, R.; van Overloop, P.; Keviczky, T.; Schutter, B. D. Distributed model predictive control for irrigation canals. Networks Heterog. Media 2009, 4, 359−380. (14) Negenborn, R.; van Overloop, P.; Schutter, B. D. Coordination of local controllers in large-scale water systems. In Proceedings of the 9th International Conference on Hydroinformatics; 2010. (15) Badwe, A.; Patwardhan, R.; Shah, S.; Patwardhan, S. C.; Gudi, R. Quantifying the effect of model-plant mismatch on controller performance. J. Process Control 2010, 20, 408−425. (16) Rawlings, J.; Stewart, B. Coordinating optimization based controllers: New opportunities and challenges. J. Process Control 2008, 18, 839−845. (17) Liu, J.; de la Pena, D. M.; Christofides, P. Distributed model predictive control of nonlinear process systems. AIChE J. 2009, 55, 1171−1184. (18) Scheu, H.; Marquardt, W. Literature Survey on Hierarchical and Distributed Nonlinear MPC, Including Analysis and Comparison, and Description of the Resulting Methodological Framework; Report D3.1.1; 2009. (19) Mesarovic, M.; Macko, D.; Takahara, Y. Two coordination principles and their application in large scale systems control. Automatica 1970, 6, 261−270. (20) Cheng, R.; Forbes, J.; Yip, W. Price-driven coordination method for solving plant-wide MPC problems. J. Process Control 2007, 17, 429− 438. (21) Venkat, A. Distributed Model Predictive Control: Theory and Applications. Ph.D. Thesis, University of Wisconsin, Madison, WI, USA, 2006. (22) Liu, J.; Chen, X.; de la Pena, D. M.; Christofides, P. Sequential and iterative architectures for distributed model predictive control of nonlinear process systems. AIChE J. 2010, 56, 2137−2149. (23) Anand, A.; Samavedham, L.; Sundaramoorthy, S. Towards Coordinating Local Model Predictive Controllers for Multi-Reservoir Management. In Proceedings of the 5th International Symposium on Design, Operation and Control of Chemical Processes; 2010. (24) Anand, A.; Joshua, G.; Sundaramoorthy, S.; Samavedham, L. Coordinating Multiple Model-Predictive Controllers for Multi-Reservoir Management. In Proceedings of the 8th IEEE International Conference on Networking, Sensing and Control; 2011. (25) Stewart, B.; Wright, S.; Rawlings, J. Cooperative distributed model predictive control for nonlinear systems. J. Process Control 2011, 21, 698−704. (26) Maestre, J. M.; de la Pena, D. M.; Camacho, E. F. Distributed Model Predictive Control Based on a Cooperative Game. Optim. Control Appl. Methods 2010, 32, 153−176. (27) Cheng, R.; Forbes, J.; Yip, W. Dantzig-Wolfe decomposition and plant-wide MPC coordination. Comput. Chem. Eng. 2008, 32, 1507− 1522. (28) Marcos, N.; Forbes, J.; Guay, M. Coordination of distributed model predictive controllers for constrained dynamic processes. In

was able to produce the same level of performance as the cooperation based controller at a lower computational effort. The cooperation based coordinator had an added advantage of producing a closed loop stable and feasible solution at every iteration. Based on the available computational resources and desired level of performance enhancement, the cooperation based coordination can be stopped at any arbitrary iteration. This indicated that, if the desired performance enhancement could be attained by the price driven coordination strategy, it would be computationally advantageous to implement it over the cooperation based coordination strategy. While studying the robustness of the coordination algorithms, price driven coordination was found to be a more robust control strategy as it deviated less from its base performance in the presence of mismatches. As the different algorithms are derived from fundamentally different methodologies, the interaction factors are utilized differently in the various algorithms. This resulted in some algorithms being more sensitive to the accuracy of the interaction models. In the presence of large mismatches where the cooperation based coordination failed, the price driven coordination was still able to come out with a feasible and stable solution. The system dynamics and level of interaction were also found to have a significant effect on the performance of the coordination algorithms. For the quadruple tank system with nonminimum phase system behavior, only the cooperation based coordination strategy was found to be closed loop stable. This necessitates a good understanding of the system dynamics before choosing an appropriate coordination algorithm, especially for systems with multivariable process zeros, like the quadruple tank system. This will help control practitioners to select the best coordination algorithm based on a priori knowledge of the system behavior and the extent of parametric uncertainties. Though coordinating multiple controllers improves the closed loop performances significantly, they come at the cost of increased communications between the controllers and a higher computational effort. Further research is needed to reduce the required communication between the local subsystem controllers by analyzing the subsystem interactions. Also, coordinators that are robust to errors and losses in communication need to be designed.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The research presented in this work was carried out as part of the SDWA Multi-Objective Multiple-Reservoir Management research program (R-264-001-005-272).

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REFERENCES

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Proceedings of the International Symposium on Advanced Control of Chemical Processes; 2009. (29) Marcos, N.; Forbes, J.; Guay, M. Distributed linear quadratic control of large-scale processes via prediction-driven coordination. In Proceedings of the 8th World Congress of Chemical Engineering; 2009. (30) Lasdon, L. Duality and decomposition in mathematical programming. IEEE Trans. Syst. Sci. Cybern. 1968, 4, 86−100. (31) Rantzer, A. Dynamic dual decomposition for distributed control. In Proceedings of the American Control Conference; 2009. (32) Shridhar, R.; Cooper, D. A Tuning Strategy for Unconstrained SISO Model Predictive Control. Ind. Eng. Chem. Res. 1997, 36, 729− 746. (33) Li, S.; Zhang, Y.; Zhu, Q. Nash-optimization enhanced distributed model predictive control applied to the Shell benchmark problem. Inf. Sci. 2005, 170, 329−349. (34) Badwe, A.; Gudi, R.; Patwardhan, R.; Shah, S.; Patwardhan, S. C. Detection of model-plant mismatch in MPC applications. J. Process Control 2009, 19, 1305−1313. (35) Alvarado, I.; Limon, D.; de la Pena, D. M.; et al. A comparative analysis of distributed MPC techniques applied to the HD-MPC four tank benchmark. J. Process Control 2011, 21, 800−815. (36) Rosinova, D.; Markech, M. Robust control of a quadruple tank process. ICIC Express Lett. 2008, 2, 231−238. (37) Johansson, K. The Quadruple-Tank Process: A Multivariable Laboratory Process with an Adjustable Zero. IEEE Trans. Control Syst. Technol. 2000, 8, 456−465.

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Comparative Performance Analysis of Coordinated Model Predictive

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