Ind. Eng. Chem. Res. 2004, 43, 3685-3694
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Plantwide Control System Design: Extension to Multiple-Forcing and Multiple-Steady-State Operation Rong Chen, Thomas McAvoy,* and Evanghelos Zafiriou Department of Chemical Engineering, Institute for Systems Research, University of Maryland, College Park, Maryland 20742
A new approach to designing plantwide control systems that is based on a linear dynamic process model and output optimal control is extended in this paper. The design of a plantwide architecture is split into four stages, and the results from one stage are used as the input to the next. During the design process, transient responses are easily calculated, and they can be used to compare candidate architectures to one another to eliminate architectures with poor dynamic performance. The design methodology is facilitated through a user-friendly software package that makes use of the best currently available algorithms for solving output optimal control problems. In this paper, the design methodology is extended to cover cases where a set of step disturbances and/ or step set-point changes is important, as well as to cases where a common control system is required for multiple-steady-state operation. The approach is illustrated on the Tennessee Eastman process. 1. Introduction Increasingly, a plantwide perspective on designing control systems for an entire chemical plant is being considered. There are two key issues in the plantwide control design area: (1) how to qualitatively and/or quantitatively determine process interaction and (2) in the face of process interaction, how to systematically design an effective control structure. Information about process interactions can be extracted from a process model, which ranges from a simple qualitative model, e.g., a process flowsheet with steady-state data, to a complicated quantitative one, e.g., a first-principles nonlinear dynamic model. When a process model is available, the issue of how to extract process information and use the information in control structure design is very important. For example, the relative gain array1 (RGA) is a popular tool for representing interactions, and several RGA-based rules have been developed for selecting control loop pairings.2 The second key issue involves selecting controlled variables (measurements), manipulated variables, control configurations (decentralized or centralized control structures), and control laws.3 For this issue, a hierarchical design procedure is generally used to decompose the design problem into several stages. In designing a control system for an entire chemical plant, it is very difficult to find a global optimum solution to all of the control objectives to be achieved. A hierarchical design procedure can provide a systematic and practical way of finding satisfactory solutions in a reduced search space. Current approaches to plantwide control can be loosely categorized as optimization-based approaches, heuristic-based approaches, and a combination of these two approaches (hybrid approaches).4 Optimizationbased approaches try to find control structure candidates by using static or dynamic process models and quantitative methods from modern control and nonlinear optimization. Process-oriented approaches use heu* To whom correspondence should be addressed. Tel.: (301) 405-1939. Fax: (301) 314-9920. E-mail:
[email protected].
ristics, which are developed from engineering experience and process insight. Many authors believe that hybrid approaches, which combine heuristics and mathematically based methods, are more promising, as they can facilitate the design task in an efficient way by automatically generating and evaluating control structure candidates. In most of the current approaches, a hierarchical design procedure that decomposes the plantwide control problem into several stages and solves them in sequence is used by researchers because good scalability is retained. This paper extends an earlier hybrid design approach5 to cases where a set of step disturbances and/or step set-point changes is important. In addition, an extension to operation at multiple steady states is made. The Tennessee Eastman process6 is used for illustration. In the earlier hybrid approach,5 a methodology was presented that uses a linear dynamic process model in designing a plantwide control system. Nonlinear models can be linearized around an operating point to generate linear models. The methodology is a hierarchical design procedure that uses optimal static output feedback (OSOF) control.7,8 The methodology identifies feasible control structures for an entire chemical plant, which are determined by a combination of mathematical analysis and engineering judgment. There are three major characteristics of the methodology. First, the methodology extracts process information from a linear time-invariant (LTI) state space model that is developed at a local operating point. Because a dynamic model provides more process insight than a static model does, a dynamic model should be used in plantwide control design whenever it is available. Second, the optimal control based methodology is a hierarchical design procedure that consists of four stages that are extended from an original design procedure used by McAvoy and co-workers.4,9 In stage 1, the process model is scaled, and the controlled variables related principally to safe process operation are identified. In stage 2, decentralized control structures are determined for the stage 1 variables. In stage 3, either centralized or decentralized control structures are designed for the control of product
10.1021/ie034015s CCC: $27.50 © 2004 American Chemical Society Published on Web 01/13/2004
3686 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
rate and quality. In stage 4, either centralized or decentralized control structures are designed for other controlled variables related to component inventory and unit operation control. The design methodology is illustrated below for the multiple-steady-state operation of the Tennessee Eastman process.6 Third, the optimal control based methodology is implemented using a highly automated computer-aided toolkit. The code requires a limited amount of interaction from a user who does not have to be an experienced control engineer. Some engineering judgment is required to determine design parameters and evaluate control structure candidates that are generated in each design stage. Because the plantwide control design problem is quite computationally intensive, significant attention has been paid to developing the numerical algorithms used to provide good scalability.5 The structure of this paper is as follows: First, a review of the optimal control problem solved in the earlier design approach is given. Then, extensions to a set of step disturbances and/or step set-point changes are discussed. Then, an extension to the case where a common control system operates at multiple steady states is given. Next, the multiple-steady-state formulation is illustrated on the Tennessee Eastman process. Finally, conclusions are given. 2. Basic Optimal Static Control Problem Solved Given a linear time-invariant state space model, an optimal static output feedback (OSOF) controller is designed to stabilize the system and bring the states from arbitrary initial values to zero, following a trajectory that minimizes a linear quadratic objective function (LQR). The basic formulation of the OSOF LQR design problem was initially studied by Levine and Athans,7 and detailed results on this problem are given by Lewis8 as follows
Process Model x˙ ) Ax + Bu
(1a)
y ) Cx
(1b)
x(0) ) x0
(1c)
u ) -Ky
(2)
Output Feedback
General Objective Function min J ) K
1
1
∞ (yTQy + uTRu) dt + ∑∑gijkij2 ∫ 0 2 2 i
(3)
The design equations for K and two auxiliary matrices, P and S, which result from the first-order necessary conditions for optimality, given by Lewis,8 are given by eqs 4
ACTP + PAC + CTKTRKC + CTQC ) 0
(4a)
ACS + SACT + X ) 0
(4b)
RKCSCT - BTPSCT + g*K ) 0
(4c)
T
X ) x0x0
(4d)
where AC ) A - BKC and g*K is a matrix with elements gij*kij. The solution given by eqs 4 depends on the initial states, x0. It is usual to assume that the initial states are uniformly distributed on the unit sphere, and when this assumption is made, the expected value of J in eq 3 is minimized and X is set equal to I, the identity matrix.7 When X ) I, the solution obtained applies to a nonspecific type of process forcing. In a number of cases, one might be interested in how a particular plantwide controller responds to a specific set of step disturbances and/or step set-point changes, e.g., a production rate change. Such alternative types of forcing are considered below. Because there is no explicit analytical solution for the OSOF controller K, numerical optimization routines are used to solve the three coupled nonlinear matrix eqs 4a-c simultaneously and to obtain K. The OSOF solution depends on how x, y, and u are scaled. The following scaling guidelines are recommended: Whether the states are scaled or not depends on whether they can be compared to one another. If states have physical meanings, e.g., they are obtained from a first-principles model, they should be scaled by either their steady-state values or the ranges of their desired movement. If states do not have physical meanings, e.g., the state space model is converted from a transfer function model, they are left unscaled. Manipulated variables (MVs) can be either valve opening percentages or set points of inner cascade controllers, and the MVs should always be scaled. If cascade controllers are used, they need to be proportional-only controllers so that they can be incorporated into eqs 1. One scaling method is to use the ranges of allowable valve movement. Another scaling method is to use physical valve ranges. The measurements can be scaled by either the physical ranges of their transmitters or the ranges of their desired movement. When the model is scaled, Q and R can be chosen as identity matrices. If it is desired to put more or less weight on measurements and/or manipulated variables, the diagonal elements of Q and R can be adjusted.
j
3. Alternative Optimal Static Control Problems where x represents the states, u represents the manipulated variables, y represents the measurements, Q and R are weight matrices, and gij is a weight on element kij in K. In most cases where eq 3 is solved to develop a plantwide control system, the gij’s are zero. However, in some cases, e.g., safety variables, a multiloop single-input-single-output (SISO) structure it is used. Then the gij’s can be used to force the off-diagonal elements of K to be zero so that the resulting controller has a diagonal structure. To make element kij small, a large value for the corresponding weight, gij, should be used.
In the previous section, the basic OSOF LQR design, in which the initial states are assumed to be uniformly distributed on the unit sphere, is presented. The process model used in the basic OSOF LQR design does not include step disturbances, and the OSOF controller is a regulator (drives states to zero) not a tracker (tracks nonzero set points). In a number of cases, it might be desirable to design a plantwide control system to reject specific step disturbances and/or track specific step setpoint changes. In this section, two alternative OSOF LQR design problems are formulated and solved that
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address these issues. Using results from these problems, one can calculate a common controller that is optimal for a specific set of step disturbances and/or step setpoint changes. In another case, where the process has multiple operating modes, engineers might also want to design a control structure works in all operating modes. The basic OSOF LQR design is also extended to handle this requirement. 1. Specific Step Disturbance Rejection/Step SetPoint Change Problem. In our approach, step disturbances and/or step set-point changes are considered. The size of these steps should correspond to the size of changes that can be expected to occur in an actual plant. For these forcings, the process model becomes
x˘ ) Ax + Bu + Wd0
(5a)
y ) Cx + Vd0
(5b)
x(0) ) 0
(5c)
where d0 is a scalar that gives the size of the step disturbance and W and V are vectors that give its effect on the states and measurements, respectively. The feedback controller used is
u ) -K(y - ySP)
(6)
Because an OSOF controller is used, steady-state offset is expected. For this case, the OSOF controller can be designed using minor extensions of the results given by Lewis.8 Assume that there exists a controller K that stabilizes the process. The steady-state values of the states, manipulated variables, and measurements are
new objective function shown in eq 11 subject to the constraints in eqs 9 and 10
min J ) K
1
1
∫∞(y˜ TQy˜ + u˜ TRu˜ ) dt + 2(ySP - yj)TZ(ySP 2 0 yj) +
C(A - BKC)-1[Wd0 + BKySP] (7b) u j ) -K(yj - ySP) ) -K[I + C(A - BKC)-1BK]Vd0 + KC(A - BKC)-1[Wd0] + K[I + C(A - BKC)-1BK]ySP (7c) If deviations from steady state are denoted by tildes, then the state, output, and manipulated variable deviations are
x˜ ) x(t) - xj
(8a)
y˜ ) y(t) - yj
(8b)
u˜ ) u(t) - u j
(8c)
Substituting eqs 8 into eqs 5 gives the dynamic process model as
dx˜ /dt ) Ax˜ + Bu˜
(9a)
y˜ ) Cx˜
(9b)
x˜ (0) ) -xj
(9c)
and the output feedback is changed from eq 6 to eq 10
u˜ ) -Ky˜
(10)
Now, an OSOF controller is designed that minimizes a
∑i ∑j gijkij2
2
(11)
where Z is a weight on the steady-state measurement offset. The new objective function is similar to eq 4.2.29 in Lewis.8 It should be pointed out that the objective function involves both transient response and steadystate offset. To make the output deviation, y˜ , small, Q can be increased. To make the steady-state offset small, Z can be increased. Therefore, a tradeoff exists in selecting weighting matrices. Often, as Z increases, the elements of the OSOF controller increase as well. Because an OSOF controller with large elements is not desirable (the appropriate absolute values of these elements should be below 10.0), Z cannot be selected too large. The new design equations, which can be derived on the basis of matrix algebra theory, are shown in eqs 12 (Chen10)
0 ) ACTP + PAC + CTKTRKC + CTQC
(12a)
0 ) ACS + SACT + xjxjT
(12b)
0 ) -BTAC -TPxjejT + RKCSCT - BTPSCT + BTAC -TCTZejejT + g*K (12c) ej ) (ySP - yj)
xj ) (A - BKC)-1[(BKV - W)d0 - BKySP] (7a) yj ) cxj + Vd0 ) [I + C(A - BKC)-1BK]Vd0 -
1
(12d)
The design equations consist of coupled matrix equations in which P, S, and K are independent variables. A MATLAB-based numerical routine, which is based on the Moerder and Calise’s method11 for the basic OSOF LQR design, was developed to solve these design equations (Chen10). 2. Several Step Disturbances/Step Set-Point Changes. In the Appendix, new results are presented for the case of n forcings that involve a common controller K. The following design equations are derived for this case
0 ) ACTPl + PlAC + CTKTRKC + CTQC (13a) 0 ) ACS + SACT + X
(13b)
n
X)
xjixjiT ∑ i)1
(13c)
n
xjiejiT + RKCSCT - BTPlSCT + ∑ i)1
0 ) -BTAC-TPl
n
ejiejiT + g*K ∑ i)1
BTAC-TCTZ
eji ) (yiSP - yji)
(13d) (13e)
The set of equations to be solved in this case is very similar to the set given by eqs 12 for the case of a single forcing. The key result derived in the Appendix is that
3688 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
the X term is calculated as the sum of the terms for the individual forcings. 3. Multiple-Steady-State Operation Design Problem. In some cases, a process has to operate in several different steady-state modes. Some control structures might be feasible for one operating mode but not acceptable for others. When the process is switched frequently between different modes, it is not desirable to change control structures at each switch. Therefore, a control structure that works in all of the operating modes is preferred. The basic OSOF controller design method, which involves generic forcing, is extended here to solve this problem. Assume the linearized process models for n operating modes are known as follows: For the ith model
x˘ ) Aix + Biu
(14a)
y ) Cix
(14b)
E[x(0) x(0)T] ) I
(14c)
Assume that a common controller, K, exists to stabilize all of the process models. The LQR design problem is formulated as follows: Find an OSOF controller that minimizes the objective function
min J ) K
[
1
1
∞ ∑l 2∫0 (yTQly + uTRlu) dt + 2∑i ∑j gijkij2
]
(15)
subject to eqs 2 and 14. The design equations, which are new, are given by Chen10 as
AC,lTPl + PlAC,l + ClTKTRlKCl + ClTQlCl ) 0 AC,lSl + SlAC,lT + X ) 0 RK
∑l (ClSlClT) - ∑l (BlTPlSlClT) + ∑l (g*K) ) 0 AC,l ) Al - BlKCl X ) I
(16)
The design equations consist of coupled matrix equations in which Pl, Sl, and K are independent variables. It can be noted that to extend eqs 16 to the case of a specific step forcing, the forcing would have the same effect on each state for each mode of operation. It is likely that step disturbances would have different effects in different operating modes, and therefore, the approach used in section 3.1 above cannot be used in this case. Also, it is possible that K can stabilize a plant at several steady states, but that instability could result during a switch between different steady states in the real plant. This possibility is not considered in the plantwide architecture methodology discussed here. A MATLAB-based numerical routine, which is based on the Moerder and Calise’s method11 was developed to solve eq 16. The use of eq 16 on the Tennessee Eastman process is illustrated next. 4. Application of Plantwide Control Design Methodology to Multiple-Mode Operation of the Tennessee Eastman Process The hierarchical plantwide control design procedure, which contains four design stages, is considered here. The methodology was discussed earlier in detail by Chen and McAvoy,5 and in this paper, a new application of
the methodology is presented. In 1993, a plantwide control test problem, the Tennessee Eastman Challenge (TEC) process, was published by Downs and Vogel.6 The TEC problem is based on an actual industrial process, and a nonlinear dynamic model is given. A flowsheet for the TEC process is given in Figure 1. The TEC process contains four unit operations: a two-phase CSTR with exothermic, irreversible, gas-phase reactions; a vapor-liquid separator (modeled as a partial condenser); a product stripper; and a gas header. There are eight chemical components (labeled A-H) in the process, and three of them (A-C) are effectively noncondensable. A first-principles dynamic model that includes 50 states, 12 manipulated variables, and 41 measurements, is available in Fortran code. Downs and Vogel6 discussed the different operating modes for the TEC process and gave a base-case design that was not optimized for the operating costs. Three optimal operating modes (1, with a G/H mass ratio of 50/50; 2, with a G/H mass ratio of 10/90; and 3, with a G/H mass ratio of 90/10) were published by Ricker.12 These modes result from minimizing the operating costs using the information provided by Downs and Vogel6 about what product mixes are desired. In all optimal operations, the reactor agitator rate is fixed at its maximum speed, the recycle flow around the compressor is fixed at it maximum value, and the steam flow is fixed at its minimum value. Therefore, for the optimal operations, these three manipulated variables are not available for control, and as a result, only nine manipulated variables are used for the plantwide control design. In this section, the goal is to design a single plantwide controller for all three optimal operating modes of the TEC process simultaneously. In the plantwide control design methodology, an OSOF controller is designed on the basis of a set of preselected measurements and a set of the manipulated variables that are available in a particular stage. Then, control structure candidates are determined using both mathematical analysis and engineering judgment. For each control structure candidate, a corresponding centralized controller or decentralized controller is automatically tuned, and process transients are generated on the basis of the linearized model so that a user can compare the control performances of different plantwide architectures. The flowsheet of the hierarchical design procedure is shown in Figure 2. The procedure vertically decomposes a plantwide control design problem into three subproblems on the basis of the priorities of the control objectives. The three subproblems are (1) controlling the variables related principally to safety issues, (2) controlling the variables related to production rate and product quality, and (3) controlling the component balances and unit operations. The output of the current design stage, including controllers, is the input to the next stage. The steps in the design procedure are as follows: Stage 1. Input and Preparation. Stage 1 considers all variables associated with safe operation, e.g., liquid levels, variables with limits, and variables that have a very slow response so that they respond in a manner that is similar to pure integrating variables. The major tasks of the stage 1 design involve scaling the process model and identifying measurements for safety and other slow-responding variables that have to be controlled in the stage 2 design. Scaling is required because the elements in the OSOF controller should be dimen-
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3689
Figure 1. Tennessee Eastman process.
Figure 2. Outline of the hierarchical design procedure.
sionless and have values in a relatively small range to be compared with one another. The other task in stage 1 is to select a set of manipulated variables and a set of measurements to be used in the stage 2 design. The input of the hierarchical design procedure includes the following information: (a) a state space linear timeinvariant process model as described by eq 1; (b) process flowsheet and steady-state process data for state variables, manipulated variables, and measurements; (c) operating ranges of the measurements and manipulated variables, which are typically used in scaling the model; (d) control objectives, which are used to define controlled variables; (e) process constraints, which are also used to define controlled variables, including both hard and soft constraints in the process operation (Hard constraints, e.g., safety-related issues, cannot be violated at any time. Soft constraints can be violated over a short period of time. However, if the violation is not corrected, operating performance suffers. For example, a valve might have a constraint on how frequently it can move.) and (f) process insight and engineering judgment. Using this information, the dynamic process model is scaled and its eigenvalues are calculated. The eigenvalues are used to determine unstable, integrating, and
slow-responding modes. Determining which measurements to use to control these modes is comparatively more difficult, and in our approach, this determination consists of three operations. First, a set of controlled variables is identified on the basis of control objectives and process constraints. For example, if the pressure in a gas loop has a high limit, this pressure is a controlled variable. Second, measurements are assigned to these controlled variables. In some cases, a controlled variable can be measured at different locations, e.g., the gas-loop pressure can be measured at any unit in the gas loop. Therefore, it is necessary to select the best location to measure the process variable using engineering judgment or some numerical tools. In other cases, there are high correlations among some of the measurements, and these correlations indicate that it is better not control these variables simultaneously. Third, a numerical method based on an eigenvalue analysis of the state space model is used to identify unstable and slow-responding measurements. The reason for using the eigenvalue analysis is that the OSOF controller must stabilize the plant, and consequently, it requires that the positive and zero eigenvalues in the state space model should be detected in the measurements. If the process is open-loop-unstable, it is necessary to determine which measurements are best to detect the instability. Details of the eigenvalue approach are given by Chen and McAvoy.5 For optimal operations 1 and 2 (50/50 and 10/90), there are eigenvalues with positive real parts, which can be eliminated by controlling the reactor cooling water temperature with its control valve. For the third optimal operation (90/10), no positive eigenvalue is present in the state space model. Because the process should operate at any of the three optimal operations and a common control structure is required, the same controlled variables are required for all three optimal operations in the stage 2 design. The results of design stage 1 generate two choices for controlled variables to
3690 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
Finally, Chen and McAvoy5 described a sensitivity approach that involves solving several optimal control problems in addition to a base-case problem. In each problem, one manipulated variable is given preferential weighting while the remaining manipulated variables are weighted so as to restrict their movement. A sensitivity table is generated from the solutions to these problems, and from this table potential SISO pairings are identified. The outputs of design stage 2 include a set of feasible control structure candidates that are implemented using proportional-only controllers. These controllers are incorporated into the model for use in later stages. Transients can be calculated easily for design stages 2-4 to evaluate control structures. The calculations carried out in stages 2-4 are essentially the same. These calculations are discussed in detail in Chen and McAvoy.5 For the TEC process, there are potentially nine manipulated variables available in design stage 2. However, in the statement of the Tennessee Eastman problem, the A,C, and D feed streams have frequency constraints placed on their movements. As a result, it is not desirable to use these manipulated variables to control the measurements found in design stage 1, as a rapid response is required for the reactor pressure, temperature, and level identified in stage 1. The OSOF controller is calculated by solving eq 16 for the remaining six manipulated variables, and it is shown in Table 2 with large elements (absolute value greater than 0.1) in boldface. The sensitivity matrix is shown in Table 3 with acceptable values (between 0.2 and 5) in boldface. If the elements in the same position of Tables 2 and 3 are both in boldface, the corresponding loop pairings are accepted as a viable control architecture. Five control structures, which are listed in Table 4, are accepted as candidates. At this point, other tools such as the relative gain array (RGA) can be used to screen candidates. Two of the five candidates in Table 4 fail an RGA screening because they involve pairing on negative RGA values. As a result, only three control structure candidates are considered to be passed on to design stage 3. Next, proportional-only controllers for these three candidate structures are calculated. For the TEC process, there is no need to tightly control the separator and stripper levels, and controllers for these variables are tuned as averaging controllers. The gains for the averaging level controls are simply chosen as +1 or -1 (%/%) (depending on the sign of the process gain) for the unscaled model. The controller gains for the remaining three loops are calculated by solving eq 16, and the results are given
Table 1. Controlled Variables for Stage 2 structure 1a
structure 2b
reactor pressure reactor level separator level stripper level
reactor pressure reactor level reactor temperature separator level stripper level
a Where the reactor temperature is controlled by the reactor cooling water flow. b Where the reactor cooling water temperature is controlled by the reactor cooling water flow.
be considered for design stage 2. These structures are listed in Table 1. In design stage 2, decentralized control structures are designed for both structure choices. For the purpose of illustration, only structure 2 is considered in this paper. If n control structure candidates are generated in design stage 1, the remaining stages have to be executed n times, with one of the design stage 1 control structures implemented each time. Stage 2. Controller Design for Structures Identified in Design Stage 1. The goal of design stage 2 is to generate decentralized control structure candidates for the variables that are identified in stage 1. Typically, a SISO loop is used to control a critical variable in a process. As a result, in design stage 2, a multiloop SISO control architecture is used to increase the reliability of the plantwide control system. The OSOF controller, K, contains information about process dynamics and interactions that can be used in designing a SISO architecture. After K is calculated, the next question is how to extract the information from K and use it for SISO control structure design. Because the process model has been scaled, K is dimensionless, and the magnitudes of the elements in K can be compared to one another. To extract information about process dynamics, the simplest metric is the absolute value of each element in the OSOF controller. Generally, an element with absolute value close to zero indicates a weak relation between the manipulated variable and the measurement. The following general rules can be used for control structure design: (1) If a row of the OSOF controller contains only small elements (e.g., absolute values less than 0.1), the corresponding manipulated variable should not be included in the control structure design. (2) If a column of the OSOF controller contains only small elements (e.g., absolute values less than 0.1), the corresponding measurement should not be included in the control structure design. (3) For decentralized control structures, if an element of the OSOF controller is small (e.g., absolute values less than 0.1), the corresponding pairing should not be used. Table 2. OSOF Controller in Design Stage 2, Structure 2 manipulated variable
reactor P
reactor L
reactor T
separator L
stripper L
E purge separator bottom stripper bottom reactor cooling water temperature condenser cooling water
-0.0487 -0.2615 0.0959 -0.0461 -0.2227 -0.3390
0.1540 0.1116 -0.1249 -0.2359 0.0207 -0.3206
0.0664 0.2333 -0.0185 0.0643 1.0976 0.1937
-0.2114 -0.8032 -1.4073 -1.1688 -0.9395 0.2917
0.1200 0.7026 0.7750 -0.4900 0.5087 0.2108
Table 3. Sensitivity Matrix for Design Stage 2, Structure 2 manipulated variable
reactor P
reactor L
reactor T
separator L
stripper L
E purge separator bottom stripper bottom reactor CWT condenser CW
-0.4784 0.4497 1.9773 0.6671 0.4754 2.6672
0.6332 -1.1093 7.5697 3.9317 -0.0415 2.6828
-0.5447 1.3556 -0.0858 -0.4159 0.9467 -12.1063
3.5529 -24.1201 3.5350 14.5012 -1.8422 1.3151
-0.4368 1.9983 1.1805 1.9514 0.6802 1.6366
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3691 Table 4. RGA Test on Possible Control Candidates in Design Stage 2, Structure 2 candidates
reactor P
reactor L
reactor T
separator L
stripper L
RGA
1 2 3 4 5
purge purge purge purge condenser CW
E E E condenser CW E
reactor CWT reactor CWT reactor CWT reactor CWT reactor CWT
separator bottom condenser CW condenser CW separator bottom separator bottom
stripper bottom stripper bottom stripper bottom stripper bottom stripper bottom
pass fail fail pass pass
Table 5. Controller Tuning Results from Design Stage 2, Structure 2
candidate
candidate no. in Table 5
K for reactor P
K for reactor L
K for reactor T
1 2 3
1 4 5
-0.8330 -0.7776 0.5192
0.4742 -0.2691 -0.4320
1.8684 1.2082 1.8326
Table 6. Identified Control Structures for Design Stage 3, Structure 2 candidate
stage 2 candidate
production rate
G (mol %)
1 2 3 4
1 1 2 3
C condenser CW C C
D D D D
in Table 5. The controllers determined in design stage 2 are incorporated into the dynamic process model. In design stage 3, the set points of loops closed in design stage 2 can be used as manipulated variables. Stage 3. Control Structure for Production/Quality Variables. The goal of the stage 3 design is to generate control structure candidates, which can be either centralized or decentralized, for the product rate and quality variables. For the TEC process, the production rate is measured, and the product quality is chosen as the percentage of G in product. These two measurements are used in solving eq 16 in stage 3. One important question in the stage 3 design is how to determine whether a centralized or a decentralized control system should be implemented from the point of view of the process dynamics and interaction. This problem is simply attacked by using process simulation based on the linearized model. The following heuristics are proposed: (1) The transients produced by implementing proportional-only diagonal control (when the OSOF controller contains only diagonal terms) indicate the performance of a totally decentralized control structure. (2) The transients produced by implementing a multivariable controller (when the OSOF controller is a full matrix) indicate the performance of a multivariable control structure. These two transients can be compared to estimate the benefit of using a multivariable control architecture. In the results presented below, only SISO structures are presented. For design stage 3, the four control structure candidates that result when eq 16 is solved are given in Table 6. These candidates are determined by pairing only manipulated variables that have gain and sensitivity elements in the desired range. Also, because the production rate and quality variables respond more slowly than the stage 2 controlled variables, the A, C, and D feed flows are used in the optimal control calculation along with the other manipulated variables. Once an architecture is identified as a candidate, controllers for it are tuned and incorporated into the dynamic model to be used in design stage 4. Stage 4. Control Structure for Remaining Variables. The goal of the stage 4 design is to generate
control structure candidates, which can be either centralized or decentralized, for maintaining component balances and controlling unit operations. One question that is particularly important for the stage 4 design is how to determine which variables to use to control component inventories. Control objectives and process constraints can provide hints for unit operation control, but normally, they give no help for identifying uncontrolled chemical components. Generally, the inventory of a component in a chemical process has to be controlled unless the inventory is selfregulating or made self-regulating by closing other loops. A Downs drill table6 is used for checking component balances after a control structure has been designed. In the stage 4 design, a Downs drill table is made for each control structure candidate given by the stage 3 design to identify any uncontrolled chemical species. The measurements for these uncontrolled chemical components need to be determined. After the number of control loops for the remaining uncontrolled chemical inventories is identified, the same number of manipulated variables is consumed. Therefore, after the component balances loops are closed, the remaining degrees of freedom can be used for unit operation control or process optimization. For design stage 4, the four plantwide control structures that result are listed in Table 7. If a SISO control architecture is used in design stage 4 engineering judgment is used to determine loop pairings, and then, eq 16 is solved to tune the resulting loops. At the end of the stage 4 design, a set of plantwide control structures is generated. They can be multiloop SISO control (with loop pairings) or multivariable control. Transients based on the linearized model and the OSOF controllers are very simple to calculate and can be used to evaluate control performance. A typical transient is shown in Figures 3-5 for candidate 4 in Table 8. For the purpose of illustration, the first step upset (IDV1)6 is applied, and process transients at the three operating modes are generated to evaluate the control performance. These transients are useful in screening alternative control architectures. The best architectures are selected to be tested via nonlinear dynamic simulation. 5. Conclusions and Summary In this paper, extensions to a recently published approach to designing plantwide control systems have been presented. The plantwide design methodology is based on a linear dynamic process model, and output optimal control is used to develop plantwide control architectures. Plantwide architecture design is divided into stages, and the results from one stage are used as the input to the next. During the design of a plantwide control system, transient responses are easily calculated and can be used to compare candidate architectures to one another. In this paper, the design methodology is extended to cover cases where it is desired to have an architecture that can handle several step disturbances and/or step set-point changes. Also, an extension to
3692 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
Figure 3. Transients in the presence of IDV1 for candidate 4 in 50/50 operation.
Figure 4. Transients in the presence of IDV1 for candidate 4 in 10/90 operation. Table 7. Plantwide Control Structure Candidates for the TEC Process, Structure 2 candidate
reactor P
reactor L
reactor T
1
purge
E
reactor CWT
2
purge
E
reactor CWT
3
purge
condenser CW reactor CWT
4
condenser CW E
reactor CWT
production rate
separator L stripper L separator bottom separator bottom separator bottom separator bottom
cases where a common control system is required for multiple-steady-state operation is presented. The approach for multiple-steady-state operation is illustrated on the Tennessee Eastman process. The extensions presented here should enhance the utility of the optimal control based plantwide design methodology. Appendix Initially, two separate forcings are considered, and then the approach is generalized to n forcings. A
stripper bottom stripper bottom stripper bottom stripper bottom
C
G A (%) (%)
B (%)
D
A
condenser CW D
A
C
D
A
C
D
A purge
C E (%) (%)
condenser temp condenser CW
C E
common controller and a common model are used for both forcings. For each forcing, eqs 9 can be written as
dx˜ i/dt ) Ax˜ i + Bu˜ i y˜ i ) Cx˜ i
(A-1)
x˜ i(0) ) -x˜ i where i ) 1, 2. Equation 10 describes the common
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3693
Figure 5. Transients in the presence of IDV1 for candidate 4 in 90/10 operation.
controller that will be used. If x, y, and u are defined as
()
()
( )
x y u x ≡ x1 , y ≡ y1 , u ≡ u1 2 2 2
(A-2)
then eqs A-1 can be written as
[
dx˜ /dt ) y˜ )
]
[ ]
∫0∞(y˜ T Q0
(A-3)
K 0 y˜ 0 K
K
[ ] [ ]
ACTP1 + P1AC + CTQC + CTKTRKC ) 0 ACTP2 + P2AC + CTQC + CTKTRKC ) 0
[ ])
∞ Q y˜ T ∫ 0 ( 0 2
0 R 0 y˜ + u˜ T u˜ dt + Q 0 R 1 SP 1 Z 0 SP (y - yj) + gijkij2 (A-4) (y - yj)T 0 Z 2 2 i j
∑∑
[
]
(A-5)
1 2
[(
CTQC 0 d T (x˜ Px˜ ) ) -x˜ T dt 0 CTQC
((
))
[ ]
∫0∞(y˜ T Q0
[ ])
0 R 0 1 y˜ + u˜ T u˜ dt ) tr(PX0) (A-10) Q 0 R 2
where
( )
-xj X0 ) -xj 1 (-xj1T - xj2T) 2
(A-11)
As a result of the structure of P and the properties of the trace, tr(PX0) can be written as
tr(PX0) ) tr(P1X1 + P1X2) ) tr[P1(X1 + X2)]
such that
(A-9)
Because P1 and P2 satisfy the same equation, it can be concluded that P1 ) P2. Using the trace, then, the integral in eq A-4 can be written as (Lewis8)
Consider the integral term in eq A-4 and assume that a symmetric P matrix with the following structure exists (Lewis8)
P 0 P) 1 P 0 2
)] )]
or
where AC ) A - BKC. The objective function to be optimized is
1
)( [( ) (
CTQC 0 CTKTRKC 0 + x˜ ) x˜ T 0 0 CTQC CTKTRKC AC 0 T A 0 P + P C A x˜ (A-8) 0 AC 0 C
xj x˜ (0) ) - xj 1 2
min J )
[ ])
0 R 0 1 y˜ + u˜ T u˜ dt ) x˜ (0)TPx˜ (0) (A-7) Q 0 R 2
[(
-x˜ T
C 0 x˜ 0 C
u˜ ) -
1 2
where P is determined by solving
AC 0 x˜ 0 AC
[ ] [ ] ()
vanishes as t goes to infinity, and hence, using eq A-6, the integral in eq A-4 is given by
(A-12)
where
Xi ) -xji(-xjiT)
+
)]
CTKTRKC 0 x˜ (A-6) 0 CTKTRKC
For an asymptotically stable closed-loop system, x˜ (t)
(A-13)
Equations A-10-A-13 imply that, for the case of two forcings, the X term in the OSOF design equations is simply the sum of the Xi’s for the individual forcings
X ) X1 + X2
(A-14)
3694 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
If a Hamiltonian is defined as8
H ) tr(P1X) + tr(gS) + (y
SP
Literature Cited
[ ]
Z 0 SP - yj) (y - yj) + 0 Z gijkij2 (A-15) T
∑i ∑j
where
g ≡ ACTP1 + P1AC + CTQC + CTKTRKC
(A-16)
and S is the matrix of Lagrange multipliers, then one can follow the same steps for deriving eq 12 in the case of a single forcing (Chen10) to obtain the design equations for a K matrix that is optimal for the case of two forcings. The resulting equations that need to be solved are
0 ) ACTP1 + P1AC + CTKTRKC + CTQC 0 ) ACS + SACT + X 2
xjiejiT + RKCSCT ∑ i)1
0 ) -BTAC-TP1
(A-17) 2
ejiejiT + g*K ∑ i)1
BTP1SCT + BTAC-TCTZ eji ) (yiSP - yji)
It is straightforward to show that, for n forcings, all that is required in the above derivation is to have the two summations in eq A-17 go from 1 to n and to have X be the sum of the individual xjixjiT terms.
(1) Bristol, E. On a New Measure of Interaction for Multivariable Process Control. IEEE Trans. Autom. Control 1966, AC-11, 133-134. (2) McAvoy, T. Interaction Analysis; Monograph Series, Instrument Society of America: Research Triangle Park, NC, 1983. (3) Larsson, T.; Skogestad, S. Plantwide Control: Review and a New Design Procedure. Model., Identif. Control 2000, 21, 209240. (4) McAvoy, T. Synthesis of Plantwide Control Systems Using Optimization. Ind. Eng. Chem. Res. 1999, 38, 2984-2994. (5) Chen, R.; McAvoy, T. Plantwide Control System Design: Methodology and Application to a Vinyl Acetate Process. Ind. Eng. Chem. Res. 2003, 42, 4753-4771. (6) Downs, J.; Vogel, E. A Plant-Wide Industrial Process Control Problem. Comput. Chem. Eng. 1993, 17, 245-255. (7) Levine, W.; Athans, M. On the Determination of the Optimal Constant Output Feedback Gain for Linear Multivariable Systems. IEEE Trans. Autom. Control 1970, AC-15, 44-48. (8) Lewis, F. Applied Optimal Control & Estimation; PrenticeHall: Englewood Cliffs, NJ, 1992. (9) McAvoy, T.; Ye, N. Base Control for the Tennessee Eastman Problem. Comput. Chem. Eng. 1994, 18, 383-413. (10) Chen, R. An Optimal Control Based Plantwide Control Design Methodology and Its Applications. Ph.D. Dissertation, University of Maryland, College Park, MD, 2002. (11) Moerder, D.; Calise, A. Convergence of a Numerical Algorithm for Calculating Optimal Output Feedback Gains. IEEE Trans. Autom. Control 1985, 30, 900-903. (12) Ricker, N. Optimal Steady-State Operation of the Tennessee Eastman Challenge Process. Comput. Chem. Eng. 1995, 19, 949-959. (13) Luyben, W.; Tyreus, B.; Luyben, M. Plantwide Process Control; McGraw-Hill: New York, 1999.
Received for review July 23, 2003 Revised manuscript received October 9, 2003 Accepted November 6, 2003 IE034015S
