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Simultaneous steady state and dynamic design of process systems using structural indicators
~
Computers and Chemical EngineeringSupplement(1999) S55-S58 CD \999 Elsevier Science Ltd. All rights reserved PII: S0098·\354/99/00\58·\
Pergamon
Simultaneous Steady State and Dynamic Design of Process Systems Using Structural Indicators Ashley M. Walsh and Ian T. Cameron Computer Aided Process Engineering Centre Department of Chemical Engineering The University of Queensland, Brisbane, Australia 4072. ABSTRACf
The simultaneous design of the steady-state and dynamic performance of a process has the ability to satisfy much more demanding dynamic performance criteria than the design of dynamics only by the connection of a control system. A method for designing process dynamics based on the use of a linearised systems' eigenvalues has been developed. The eigenvalues are associated with system states using the unit perturbation spectral resolution (UPSR), characterising the dynamics of each state. The design method uses a homotopy approach to determine a final design which satisfies both steady-state and dynamic performance criteria. A highly interacting single stage forced circulation evaporator system, including control loops, was designed by this method with the goal of reducing the time taken for the liquid composition to reach steady-state. Initially the system was successfully redesigned to speed up the eigenvalue associated with the liquid composition state, but this did not result in an improved startup performance. Further analysis showed that the integral action of the composition controller was the source of the limiting eigenvalue. Design changes made to speed up this eigenvalue did result in an improved startup performance. The proposed approach provides a structured way to address the design-control interface, giving significant insight into the dynamic behaviour of the system such that a systematic design or redesign of an existing system can be undertaken with confidence.
INTRODUCTION
Traditional process design methods first develop a design that satisfies a set of steady state performance requirements, and then shape the process dynamics through the connection of a control system. This is a less than satisfactory method. A design methodology in which a process' steady state and dynamic performance are designed simultaneously is required. The direct simulation of possible process designs is time consuming, requires a closed-loop system, and does not make much use of the known dynamic structure of a process. An alternative to direct simulation is the use of a method of dynamic design based on the characterisation of the dynamic structure of a process. The dynamic performance goals are quantified in terms of the required dynamic structure, and the relationship between the dynamic structure and the process design parameters is then used to determine appropriate design parameters to achieve the performance goals. DYNArvUC STRUCTURE
The key requirement for the proposed design methodology is a measure which quantifies a process' dynamic structure. A suitable measure of dynamic Structure must possess two qualities:
1. It must be possible to quantify the dynamic performance goals using the structural measures.
2. It must be possible to calculate the design parameters which result in a process with the desired dynamic structure. The measure of dynamic structure that has been used in this work is a process' eigenvalues and the Unit Perturbation Spectral Resolution (UPSR) matrix. The eigenvalues of a linear system are the inverse of the system's response times. It has been shown that it is possible to associate eigenvalues with states (Robertson, 1992, DeCarlo and Saeks, 1979, Wasynczuk and DeCarlo, 1981, Walsh and Cameron, 1996) to characterise the response rate of each state. This association process is also described as spectral association. For a model whose states represent physical quantities in a system, spectral association determines the sources of the different dynamic modes of a process. Consider the linear system in equation (1): dx(t) - - = Ax(t), x(O) = X o dt
(1)
The solution to this system can be expressed in the spectral resolution form with the matrix Z (Ross, 1974): x(t)
=Vexp(At)V·1x o = Zexp(;\t)
(2)
The eigenvalues ;\ and eigenvectors V of the coefficient matrix A satisfy equation (3):
556
Computers and Chemical Engineering Supplement (1999) S55-S58
f(r,X) = [
The UPSR matrix P is calculated from the eigenvectors
V: (4)
where ® represents the Hadamard product, Pjj = Vjj(V-I)ji
(5)
Walsh and Cameron (1996) has a more detailed discussion of the conceptual basis of the UPSR, but the important properties can be summarised as follows: o
The row and column sums are one. This reflects the nature of the UPSR in indicating the distribution of the system eigenvalues among the system states.
o
The columns of the UPSR matrix P can be rearranged into a block diagonal structure, with the same rearrangement made to the vector of eigenvalues A. These blocks indicate state-eigenvalue groupings, in which the eigenvalues of the group have their source in the states of the group.
o
The value Pij is a measure of the strength of the association between state XI and eigenvalue A.i. Where there is no association between a particular state and eigenvalue the UPSR matrix will have a zero element. A one-to-one association between a stateeigenvalue pair will be indicated by an element equal to one in the UPSR matrix.
The use of the UPSR and eigenvalues to characterise process dynamics satisfies the requirements set out above for ~ structural measure in the following ways: l. The eigenvalues have a direct physical interpretation: they measure the speed of response of the different dynamic modes of the system.
2': The relationship between the eigenvalues A and the design parameters p can be inverted in order to determine the system design that has the eigenvalues that should satisfy the dynamic performance goals.
[p]
A(X ) - AIUg" (r )] , X= Y '¥(X)
(6)
in which ,¥(X) are the steady-state criteria. and y is a vector of the operating-point variables. In the evaporator case-study that follows, the steady-state equations are the set of differential and algebraic equations that model the system dynamics. The homotopy parameter r is varied from zero to one. The design parameters p and operating point y solved at the different values of r such that 1=0. In this way the design parameters become a function of the homotopy parameter, as given by X(r): f'(r.Xtr)
=0
(7)
A function called HOMSOLV has been written for the MATLABTM environment to solve the redesign homotopy problem. The most important details of this function can be summarised as: o Weights can be applied to alI elements of the optimisation variable vector X and objective function vector f. At the very least weighting must be used to . normalise the optimisation variables and objective function elements. A variable step size method is used. There are two constraints on the step size. Firstly, it must be possible to find a solution for the next value of r along the homotopy. Secondly, constraints placed on the maximum change in the optimisation variables must be satisfied. • The values of the optimisation variables X at each point along the homotopy are solved using Newton's method with a line search.
o
• A combination of the UPSR and eigenvalue continuity is used to determine eigenvalue-to-state associations from one step to the next. The UPSR is always arranged in a block-diagonal structure, with eigenvalue continuity within these blocks maintained as the homotopy parameter r is varied.
PROBLEM FORMULA nON AND SOLUTION
THE EVAPORATOR
After specific dynamic performance goals have been set, they must be translated into an objective function containing the structural measures. Steady-state performance criteria can also be included in the objective function. In this way a design that satisfies both dynamic and steady-state performance criteria can be found. In this work the system eigenvalues are being used as the measure of dynamic structure, with the UPSR determining eigenvalue-to-state associations. The objective function is formulated in terms of changing the eigenvalues associated with particular states. A homotopy solution method is used for solving what is actually a redesign problem . The method is described as redesign because using a homotopy involves moving from an initial design to a final design . Dynamic and steady state criteria are combined in one function in the form:
The evaporator system consists of a forced circulation evaporator and condenser, as shown in Figure 1. Weak liquor is introduced to the evaporator, where the concentration of non-volatile material is increased by boiling. Water vapour from the overhead of the evaporator then passes through a condenser, and is removed as condensate. Concentrated liquor is removed from the evaporator bottom. Three control loops are used to regulate the system. The level of liquid in the separator is controller by varying the flow of product liquor. The product composition is controlled by varying the steam flowrate to the shell side of the evaporator component. The pressure of the process-side vapour in the evaporator is controlled by varying the cooling water flowrate to the condenser. All the control loops use conventional proportional-integral (PI) controllers.
Computers and Chemical Engineering Supplement (/999) S55-S58 Cooling Wat.r F200. noD
PIOO,TIOO
557
Redesign of the Composition Eigenvalue
The first redesign, designated SSDES, attempted to improve the startup performance by increasing the magnitude of the eigenvalue associated with the state X2.
P2
Evaporator
The problem was set up to include both the dynamic and steady-state design objectives. The objective function had the form given in equation (6). The target eigenvalue is given by:
FJ
Condensate
=(1- r) j,,°X2 + r (4 j,,°X2) j,,~2 =-0.137 min"!
j"target (r)
(8)
where j,,~2 is the eigenvalue associated with X2 in the original design. The redesign was successfully solved. Figure 1 The evaporator: process flow diagram. However, Figure 2 shows that the startup profile of the composition X2 for the new design does not improve significantly on the original design. The system consists of 11 componerits and 15 states, as shown in Table 1. 26,----,------.-----, Table 1 Evaporator components and states. Component
States
Description
Evaporator shell
MASS
process-side liquid mass
and separator.
X2
liquid composition
M2
process-side vapour mass
EMASS
energy holdup in shell
MIOO
steam-side vapour mass
Condenser
1'201
cooling water exit temp.
L-sensor
L2M
measurement
P-sensor
P2M
measurement
X-sensor
X2M
measurement
Product valve
V2
valve opening
CWvalve
V200
valve opening
Steam valve
V 100
valve opening
L-controller
LSUM
integral action
P-controller
PSUM
integral action
X-controller
XSUM
integral action
Redesign Goals
Robertson (1992) attempted to redesign the evaporator system with the goal of reducing the startup time. The method used was a single Newton step. In this work the evaporator will also be redesigned in order to reduce the startup time. More specifically, the time taken for the liquid composition X2 to reach steady-state will be reduced. In the initial design, the composition reaches steady-state after approximately 250 minutes. The redesign goal is to reduce the time taken for the liquid composition to reach steady-state after the final startup sequence disturbance at 120 minutes. (See Figure 2)
25
••••••••••••••••
,"
_24
• 0.
eo
••
••
••
.. '0
~ !=\I
~ 23 - - Initial Design
- - -SSDES ....... XCTAU
21 L 100
"--
150
" - -_ _----'
200
250
time (minutes)
Figure 2 Startup performance of different designs. Despite the failure of the SSDES design to achieve the startup performance goals, the new design was successful in achieving the desired change in the liquid composition eigenvalue. The most significant changes made by the SSDES redesign are: • The process-side volume, which consists of the tube and separator volumes, is greatly reduced. This reduces the process-side liquid inventory, increasing the sensitivity of the mass fraction X2. • The composition controller gain is increased. This increases the gain on both the proportional and integral action of the controller. The change in the X2 eigenvalue is the direct result of the increased proportional control gain. The change in the integral action is shown below to cause the small improvement seen in the startup performance.
S58
Computers and Chemical Engineering Supplement (/999) 555-558
Using the Spectral Resolution to Determine Eigenvalue Contributions
The small improvement in the SSDES startup performance suggests that the composition X2 dynamics are not strongly dependent on the associated eigenvalue. The spectral resolution can be used to test this proposition. The spectral resolution matrix Z in equation (2) can be calculated at any instant in time through startup. This allows the contributions of the different eigenvalues to the dynamic responses of the system states to be determined. In particular, the dynamic response of the liquid composition can be expressed in the form: 15
X2(t) = 25+ 'LZ2je
A'l
(9)
J
je l
Figure 3 shows the contributions of the different eigenvalues to the dynamic response of the liquid composition during the interval 100 minutes to 250 minutes. The eigenvalue associated with the composition controller XSUM is the main source of the dynamic response of X2. This shows that the design objective should be reformulated to target the XSUM eigenvalue, rather than the X2 eigenvalue.
1 .-----~---~.----...,
.......... ,.
••
0
•••
0
••
--
/
c,
-3
I
I/~AXSUM
I I , II 1(1 I II
-4 100
_ _--'-
L-~
'--
150 200 time (minutes)
(10)
't
Decreasing the reset time of the controller will increase the magnitude of the associated eigenvalue. The redesign XCTAU speeds up the integral action of the composition controller by decreasing the reset time from 30 minutes to 10 minutes. The eigenvalue associated with XSUM changes from -0.0393 minol to -0.0879-0.0993j min", With the new design the eigenvalues associated with the states X2 and XSUM form a complex conjugate pair, indicating a greater level of interaction between the proportional and integral actions of the composition controller, including some capacity for oscillatory behaviour. Figure 2 shows the startup performance of the XCTAU redesign. The time taken for the composition to reach steady-state is significantly reduced. CONCLUSIONS AND RECOMENDATIONS
The HOMSOLV function has been successfully used to redesign the evaporator system. This redesign included both steady-state criteria, and a dynamic criterion to change the eigenvalue associated with the liquid composition state. This demonstrates the ability to redesign a system to have a dynamic structure defined by its UPSR and eigenvalues. The liquid composition eigenvalue redesign failed to achieve the required startup performance goals because of an incorrect translation of the dynamic performance goals into a dynamic structure quantified by the UPSR and eigenvalues. The spectral resolution and composition controller retuning both showed that the slow startup dynamics of the liquid composition are a result of the dynamics of the integral action of the composition controller, which was not the target of the SSDES redesign.
/
~ -2
I
1..=--
....J
250
Figure 3 Spectral Resolution of Liquid Composition. Redesign ofthe Composition Controller Eigenvalue
The composition controller state XSUM models the integral action of this PI controller. There are two ways in which the evaporator system design can be changed in order to speed up the eigenvalue associated with the state XSUM. The first is to set up a design problem similar to that used for the redesign of the X2 eigenvalue, which can then be solved with HOMSOLV. The second is to make direct changes t~. the composition controller tuning. The speed of the integral action of a PI controller is determined by its reset time 'to If no significant interaction occurs between the integral action and other dynamics of a system, then the eigenvalue associated with the integral action will be approximated by:
Further work needs to be done on determining how to translate dynamic performance goals into quantitative requirements of the process dynamic structure. REFERENCES
Robertson, G.A., 1992, Mathematical Modelling of Startup and Shutdown Operations of Process Plants, PhD Thesis, The University of Queensland, Chapt. 5-6. DeCarlo, R.A. and Saeks, R., 1979, "A Root Locus Technique for Interconnected Systems", IEEE Trans. Sys. Man. And Cyb., Vol. SMC-9, No. I, pp. 53-55. Wasynczuk and DeCarlo, 1981, "The Component Connection Model and Structure Preserving Model Order Reduction", Automatica, Vol. 17, No.4, pp. 619626. Walsh and Cameron, 1996, "Dynamic System Characterisation for Design Purposes", Chemeca 96, Vol. 2, pp. 85-90. Ross, S.L., 1974, Differential Equations, 2nd Ed., Wiley International, New York.
Computers and Chemical EngineeringSupplement(1999) S55-S58 CD \999 Elsevier Science Ltd. All rights reserved PII: S0098·\354/99/00\58·\
Pergamon
Simultaneous Steady State and Dynamic Design of Process Systems Using Structural Indicators Ashley M. Walsh and Ian T. Cameron Computer Aided Process Engineering Centre Department of Chemical Engineering The University of Queensland, Brisbane, Australia 4072. ABSTRACf
The simultaneous design of the steady-state and dynamic performance of a process has the ability to satisfy much more demanding dynamic performance criteria than the design of dynamics only by the connection of a control system. A method for designing process dynamics based on the use of a linearised systems' eigenvalues has been developed. The eigenvalues are associated with system states using the unit perturbation spectral resolution (UPSR), characterising the dynamics of each state. The design method uses a homotopy approach to determine a final design which satisfies both steady-state and dynamic performance criteria. A highly interacting single stage forced circulation evaporator system, including control loops, was designed by this method with the goal of reducing the time taken for the liquid composition to reach steady-state. Initially the system was successfully redesigned to speed up the eigenvalue associated with the liquid composition state, but this did not result in an improved startup performance. Further analysis showed that the integral action of the composition controller was the source of the limiting eigenvalue. Design changes made to speed up this eigenvalue did result in an improved startup performance. The proposed approach provides a structured way to address the design-control interface, giving significant insight into the dynamic behaviour of the system such that a systematic design or redesign of an existing system can be undertaken with confidence.
INTRODUCTION
Traditional process design methods first develop a design that satisfies a set of steady state performance requirements, and then shape the process dynamics through the connection of a control system. This is a less than satisfactory method. A design methodology in which a process' steady state and dynamic performance are designed simultaneously is required. The direct simulation of possible process designs is time consuming, requires a closed-loop system, and does not make much use of the known dynamic structure of a process. An alternative to direct simulation is the use of a method of dynamic design based on the characterisation of the dynamic structure of a process. The dynamic performance goals are quantified in terms of the required dynamic structure, and the relationship between the dynamic structure and the process design parameters is then used to determine appropriate design parameters to achieve the performance goals. DYNArvUC STRUCTURE
The key requirement for the proposed design methodology is a measure which quantifies a process' dynamic structure. A suitable measure of dynamic Structure must possess two qualities:
1. It must be possible to quantify the dynamic performance goals using the structural measures.
2. It must be possible to calculate the design parameters which result in a process with the desired dynamic structure. The measure of dynamic structure that has been used in this work is a process' eigenvalues and the Unit Perturbation Spectral Resolution (UPSR) matrix. The eigenvalues of a linear system are the inverse of the system's response times. It has been shown that it is possible to associate eigenvalues with states (Robertson, 1992, DeCarlo and Saeks, 1979, Wasynczuk and DeCarlo, 1981, Walsh and Cameron, 1996) to characterise the response rate of each state. This association process is also described as spectral association. For a model whose states represent physical quantities in a system, spectral association determines the sources of the different dynamic modes of a process. Consider the linear system in equation (1): dx(t) - - = Ax(t), x(O) = X o dt
(1)
The solution to this system can be expressed in the spectral resolution form with the matrix Z (Ross, 1974): x(t)
=Vexp(At)V·1x o = Zexp(;\t)
(2)
The eigenvalues ;\ and eigenvectors V of the coefficient matrix A satisfy equation (3):
556
Computers and Chemical Engineering Supplement (1999) S55-S58
f(r,X) = [
The UPSR matrix P is calculated from the eigenvectors
V: (4)
where ® represents the Hadamard product, Pjj = Vjj(V-I)ji
(5)
Walsh and Cameron (1996) has a more detailed discussion of the conceptual basis of the UPSR, but the important properties can be summarised as follows: o
The row and column sums are one. This reflects the nature of the UPSR in indicating the distribution of the system eigenvalues among the system states.
o
The columns of the UPSR matrix P can be rearranged into a block diagonal structure, with the same rearrangement made to the vector of eigenvalues A. These blocks indicate state-eigenvalue groupings, in which the eigenvalues of the group have their source in the states of the group.
o
The value Pij is a measure of the strength of the association between state XI and eigenvalue A.i. Where there is no association between a particular state and eigenvalue the UPSR matrix will have a zero element. A one-to-one association between a stateeigenvalue pair will be indicated by an element equal to one in the UPSR matrix.
The use of the UPSR and eigenvalues to characterise process dynamics satisfies the requirements set out above for ~ structural measure in the following ways: l. The eigenvalues have a direct physical interpretation: they measure the speed of response of the different dynamic modes of the system.
2': The relationship between the eigenvalues A and the design parameters p can be inverted in order to determine the system design that has the eigenvalues that should satisfy the dynamic performance goals.
[p]
A(X ) - AIUg" (r )] , X= Y '¥(X)
(6)
in which ,¥(X) are the steady-state criteria. and y is a vector of the operating-point variables. In the evaporator case-study that follows, the steady-state equations are the set of differential and algebraic equations that model the system dynamics. The homotopy parameter r is varied from zero to one. The design parameters p and operating point y solved at the different values of r such that 1=0. In this way the design parameters become a function of the homotopy parameter, as given by X(r): f'(r.Xtr)
=0
(7)
A function called HOMSOLV has been written for the MATLABTM environment to solve the redesign homotopy problem. The most important details of this function can be summarised as: o Weights can be applied to alI elements of the optimisation variable vector X and objective function vector f. At the very least weighting must be used to . normalise the optimisation variables and objective function elements. A variable step size method is used. There are two constraints on the step size. Firstly, it must be possible to find a solution for the next value of r along the homotopy. Secondly, constraints placed on the maximum change in the optimisation variables must be satisfied. • The values of the optimisation variables X at each point along the homotopy are solved using Newton's method with a line search.
o
• A combination of the UPSR and eigenvalue continuity is used to determine eigenvalue-to-state associations from one step to the next. The UPSR is always arranged in a block-diagonal structure, with eigenvalue continuity within these blocks maintained as the homotopy parameter r is varied.
PROBLEM FORMULA nON AND SOLUTION
THE EVAPORATOR
After specific dynamic performance goals have been set, they must be translated into an objective function containing the structural measures. Steady-state performance criteria can also be included in the objective function. In this way a design that satisfies both dynamic and steady-state performance criteria can be found. In this work the system eigenvalues are being used as the measure of dynamic structure, with the UPSR determining eigenvalue-to-state associations. The objective function is formulated in terms of changing the eigenvalues associated with particular states. A homotopy solution method is used for solving what is actually a redesign problem . The method is described as redesign because using a homotopy involves moving from an initial design to a final design . Dynamic and steady state criteria are combined in one function in the form:
The evaporator system consists of a forced circulation evaporator and condenser, as shown in Figure 1. Weak liquor is introduced to the evaporator, where the concentration of non-volatile material is increased by boiling. Water vapour from the overhead of the evaporator then passes through a condenser, and is removed as condensate. Concentrated liquor is removed from the evaporator bottom. Three control loops are used to regulate the system. The level of liquid in the separator is controller by varying the flow of product liquor. The product composition is controlled by varying the steam flowrate to the shell side of the evaporator component. The pressure of the process-side vapour in the evaporator is controlled by varying the cooling water flowrate to the condenser. All the control loops use conventional proportional-integral (PI) controllers.
Computers and Chemical Engineering Supplement (/999) S55-S58 Cooling Wat.r F200. noD
PIOO,TIOO
557
Redesign of the Composition Eigenvalue
The first redesign, designated SSDES, attempted to improve the startup performance by increasing the magnitude of the eigenvalue associated with the state X2.
P2
Evaporator
The problem was set up to include both the dynamic and steady-state design objectives. The objective function had the form given in equation (6). The target eigenvalue is given by:
FJ
Condensate
=(1- r) j,,°X2 + r (4 j,,°X2) j,,~2 =-0.137 min"!
j"target (r)
(8)
where j,,~2 is the eigenvalue associated with X2 in the original design. The redesign was successfully solved. Figure 1 The evaporator: process flow diagram. However, Figure 2 shows that the startup profile of the composition X2 for the new design does not improve significantly on the original design. The system consists of 11 componerits and 15 states, as shown in Table 1. 26,----,------.-----, Table 1 Evaporator components and states. Component
States
Description
Evaporator shell
MASS
process-side liquid mass
and separator.
X2
liquid composition
M2
process-side vapour mass
EMASS
energy holdup in shell
MIOO
steam-side vapour mass
Condenser
1'201
cooling water exit temp.
L-sensor
L2M
measurement
P-sensor
P2M
measurement
X-sensor
X2M
measurement
Product valve
V2
valve opening
CWvalve
V200
valve opening
Steam valve
V 100
valve opening
L-controller
LSUM
integral action
P-controller
PSUM
integral action
X-controller
XSUM
integral action
Redesign Goals
Robertson (1992) attempted to redesign the evaporator system with the goal of reducing the startup time. The method used was a single Newton step. In this work the evaporator will also be redesigned in order to reduce the startup time. More specifically, the time taken for the liquid composition X2 to reach steady-state will be reduced. In the initial design, the composition reaches steady-state after approximately 250 minutes. The redesign goal is to reduce the time taken for the liquid composition to reach steady-state after the final startup sequence disturbance at 120 minutes. (See Figure 2)
25
••••••••••••••••
,"
_24
• 0.
eo
••
••
••
.. '0
~ !=\I
~ 23 - - Initial Design
- - -SSDES ....... XCTAU
21 L 100
"--
150
" - -_ _----'
200
250
time (minutes)
Figure 2 Startup performance of different designs. Despite the failure of the SSDES design to achieve the startup performance goals, the new design was successful in achieving the desired change in the liquid composition eigenvalue. The most significant changes made by the SSDES redesign are: • The process-side volume, which consists of the tube and separator volumes, is greatly reduced. This reduces the process-side liquid inventory, increasing the sensitivity of the mass fraction X2. • The composition controller gain is increased. This increases the gain on both the proportional and integral action of the controller. The change in the X2 eigenvalue is the direct result of the increased proportional control gain. The change in the integral action is shown below to cause the small improvement seen in the startup performance.
S58
Computers and Chemical Engineering Supplement (/999) 555-558
Using the Spectral Resolution to Determine Eigenvalue Contributions
The small improvement in the SSDES startup performance suggests that the composition X2 dynamics are not strongly dependent on the associated eigenvalue. The spectral resolution can be used to test this proposition. The spectral resolution matrix Z in equation (2) can be calculated at any instant in time through startup. This allows the contributions of the different eigenvalues to the dynamic responses of the system states to be determined. In particular, the dynamic response of the liquid composition can be expressed in the form: 15
X2(t) = 25+ 'LZ2je
A'l
(9)
J
je l
Figure 3 shows the contributions of the different eigenvalues to the dynamic response of the liquid composition during the interval 100 minutes to 250 minutes. The eigenvalue associated with the composition controller XSUM is the main source of the dynamic response of X2. This shows that the design objective should be reformulated to target the XSUM eigenvalue, rather than the X2 eigenvalue.
1 .-----~---~.----...,
.......... ,.
••
0
•••
0
••
--
/
c,
-3
I
I/~AXSUM
I I , II 1(1 I II
-4 100
_ _--'-
L-~
'--
150 200 time (minutes)
(10)
't
Decreasing the reset time of the controller will increase the magnitude of the associated eigenvalue. The redesign XCTAU speeds up the integral action of the composition controller by decreasing the reset time from 30 minutes to 10 minutes. The eigenvalue associated with XSUM changes from -0.0393 minol to -0.0879-0.0993j min", With the new design the eigenvalues associated with the states X2 and XSUM form a complex conjugate pair, indicating a greater level of interaction between the proportional and integral actions of the composition controller, including some capacity for oscillatory behaviour. Figure 2 shows the startup performance of the XCTAU redesign. The time taken for the composition to reach steady-state is significantly reduced. CONCLUSIONS AND RECOMENDATIONS
The HOMSOLV function has been successfully used to redesign the evaporator system. This redesign included both steady-state criteria, and a dynamic criterion to change the eigenvalue associated with the liquid composition state. This demonstrates the ability to redesign a system to have a dynamic structure defined by its UPSR and eigenvalues. The liquid composition eigenvalue redesign failed to achieve the required startup performance goals because of an incorrect translation of the dynamic performance goals into a dynamic structure quantified by the UPSR and eigenvalues. The spectral resolution and composition controller retuning both showed that the slow startup dynamics of the liquid composition are a result of the dynamics of the integral action of the composition controller, which was not the target of the SSDES redesign.
/
~ -2
I
1..=--
....J
250
Figure 3 Spectral Resolution of Liquid Composition. Redesign ofthe Composition Controller Eigenvalue
The composition controller state XSUM models the integral action of this PI controller. There are two ways in which the evaporator system design can be changed in order to speed up the eigenvalue associated with the state XSUM. The first is to set up a design problem similar to that used for the redesign of the X2 eigenvalue, which can then be solved with HOMSOLV. The second is to make direct changes t~. the composition controller tuning. The speed of the integral action of a PI controller is determined by its reset time 'to If no significant interaction occurs between the integral action and other dynamics of a system, then the eigenvalue associated with the integral action will be approximated by:
Further work needs to be done on determining how to translate dynamic performance goals into quantitative requirements of the process dynamic structure. REFERENCES
Robertson, G.A., 1992, Mathematical Modelling of Startup and Shutdown Operations of Process Plants, PhD Thesis, The University of Queensland, Chapt. 5-6. DeCarlo, R.A. and Saeks, R., 1979, "A Root Locus Technique for Interconnected Systems", IEEE Trans. Sys. Man. And Cyb., Vol. SMC-9, No. I, pp. 53-55. Wasynczuk and DeCarlo, 1981, "The Component Connection Model and Structure Preserving Model Order Reduction", Automatica, Vol. 17, No.4, pp. 619626. Walsh and Cameron, 1996, "Dynamic System Characterisation for Design Purposes", Chemeca 96, Vol. 2, pp. 85-90. Ross, S.L., 1974, Differential Equations, 2nd Ed., Wiley International, New York.