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Model Structure Determination for a Heat Pump System
Copyright © IFAC Identification and System Parameter Estimation, Budapest, Hungary 1991
MODEL STRUCTURE DETERMINATION FOR A HEA T PUMP SYSTEM K. M. Hangos* and L. Hallager** *Computer and Automation Institute, Hungarian Academy o/Sciences, p.a. Box 63 , H-1518 Budapest, Hungary **Department o/Chemical Engineering, Technical University 0/ Denmark, Building 229, 2800 Lyngby, Denmark
Abstract. In this paper principles of model structure determination by selection between candidate structures of underlying linearized ordinary differential equation models aredescribed. The principles are exemplified on a simple heat pump system. The identification procedure applying influence graph models, qualitative prediction tables taking results on structural stability analysis into account, is demonstrated for the example. The same methodology may be applied using other qualitative structural properties (e.g. qualitative correlation tables, structural observability and controllability analysis etc.) Structure identification methods based on qualitative models may complement conventional identification methods serving as structure determination tools in the early stages of modelling. Keywords System analysis, Chemical variables control, Linear differential equations, Influence graph models, Structure determination.
An influence graph is a weighted directed graph consisting of
INTRODUCTION
- nodes representing the variables (state, input and output) of the system to be modelled; - directed edges describing influence of the variable they start from to the growth rate (the accumulation or time derivative) of the variable they end at; - weight associated to the edges usually describing the sign (i.e. enhancing or damping) and, in some cases, the magnitude of the influence they represent.
The theory and methods for linear or linearized model structure determination form a difficult and sophisticated field of process identification. The methods usually involve parameter estimation of a set of mostly linear(ized) candidate models, followed by a decision step choosing between the candidate models (see e.g. Rasmussen and coworkers (1990)). An alternative approach is to use qualitative process knowledge and experimental data to determine the structure of linear models without parameter estimation. The purpose of this paper is to show how qualitative knowledge, qualitative models and structural properties may be used for structure identification.
An influence graph can be interpreted in different ways depending on: the mathematical form of the influence function (e.g. fvar at end (var_acstart), where the influence edge is direeteofrom var_acstart to var3cend) and
The field of qualitative modeling and its application is relatively new dating from the early 80s. The main approaches to qualitative physics include Forbus's Qualitative Process Theory (Forbus, 1984), Qualitative Physics by de Kleer and Brown (1984), Kuiper's Qualitative Simulation (Kuipers, 1986; 1989) and the graph analysis based methods of King (1983), Puccia and Levins (1985) and others.
- how the relation of influences influencing the same variable is understood. In general, a set of nonlinear ordinary differential equations (ODE) can be associated to each influence graph. In other words, only concentrated (lumped) parameter dynamic systems can be described by influence graphs . The set of associated non linear ODEs is called underlying ODE model and can be written in the following general form:
Methods suitable for structure determination of qualitative models are described in this paper for one of the most important classes of qualitative models namely by signed directed graphs (so called influence graphs). After introducing the basic notions about qualitative models and influence graphs, their structural properties, such as structural stability and qualitative prediction tables, are described. The structural properties and their differences in case of possible alternative models serve as a basis for structure determination. Finally the structure identification of a heat pump system in a pilot plant destillation column with suitable and intentionally modified model structures is presented.
dx = f
dt
x
(x)
(1)
where t time x vector of variables fx right hand side vector-vector function. The influence graph fixes the following structural properties of the underlying ODE model.
INFLUENCE GRAPHS, THEIR STRUCTURE AND STRUCTURAL PROPERTIES
1. The number and type of variables entering the right hand side function of the ODE for the ith variable is given by the starting variables of the influence edges ending at it. 2. The sign of the partial derivative of the nonlinear right hand side function with respect to its independent variables is given by the weights associated to the corresponding influence edges.
The notion of influence graphs emerged from the graph analysis based methods developed for modeling large ecological and biological systems (Puccia and Levins, 1985), for qualitative analysis of reaction kinetic behaviour (King, 1983) and for describing the structure of large control systems (Larsen and Evans, 1987).
S73
Linearized ODE model and the graph
The steady-state solution, of course, depends on c too:
If the underlying ODE model has at least one stationary (steady-state) solution, i.e. f x(x*) = 0
for some
x*
(5)
Qualitative prediction tables describe changes of the steady-state value of the variables when the parameters of the underlying ODE model change. The description is given in the following qualitative terms:
( 2)
then the original non linear underlying ODE can be linearized around x* using the lacobian matrix J of the function fx at the same point. This linearize4 ODE has the form dx = J(x - x*) dt
x*(c) .
fx(x*, c) =0,
+ : increase - : decrease no change ? : ambiguous (the direction of the change depends on the relation of the numerical weights in the influence graph which are unknown in the general case).
o:
( 3)
and there is a one-to-one correspondence between the structure of the influence graph and its linearized underlying ODE model.
Note that from a system engineering point of view one can regard parameters as input variables forming an input vector u=[cl ' ...., cm]T. The lacobian matrix of the right hand side function fx with respect to the element of this vector forms an input matrix B of a linear state space model corresponding to the underlying linearized model equations.
The structure of a signed directed graph can be uniquely determined by its adjacency matrix extended by the weight associated to the edges. An element aij in the adjacency matrix is 0 if there is no directed edge from node i to node j and aij is equal to the weight of the edge if it exists. The occurrence matrix is defined as a transpose of the adjacency matrix (Roberts, 1976).
dx = A(x _ x*) + B(u - u*) dt
It is easy to see from the definition of the influence graph that the occurrence matrix of an influence graph supplemented with the sign weight of the edges gives 'I'=Sign(J), i.e. the signed boolean version of the lacobian matrix of the underlying ODE model.
(6)
Thus the occurrence matrix J of an influence graph supplemented with the weight of the edges gives the lacobian matrix of the underlying ODE model in the partitioned form for
Structural properties An influence graph gives uncomplete qualitative information about the structure of its underlying ODE model. However, it is possible to determine some properties of the underlying ODE model based only on this qualitative information. These properties will be calIed structural properties and they include stability (near a steady-state) and qualitative behaviour of the steady state values caused by parameter changes (described by prediction tables) . Other structural properties such as structural observability and controlIability can be defmed and analyzed (Larsen and Evans, 1987), but these are out of the scope of this paper.
The effect of the change in parameter ck can be found by differentiating Eq. (5) with respect to ck and solve it for the unknown variables
a x~
ax~ , ... , aC
aC k
i = 1, 2, ... ,n
(7 )
k
which gives
Structural stability A linear(ize4) system is called structurally stable if for every physically realistic (e.g. positive) value of the non zero elements in its state transition matrix J (in Eq. (3» a stable system is obtained, i.e. all the real parts of the eigenvalues of J are negative. Note that the eigenvalues of the state transition matrix J are the same as the eigenvalues of the associated influence graph (Roberts, 1976).
(8 )
where 'l'ik ' i,k=I, ... ,n are the elements of the lacobian matrix J .
In case of large systems or systems with unknown parameters it could be quite complicated or even impossible to compute the eigenvalues of the state transition matrix J to check the stability conditions. Using the graph theoretical properties of the influence graph it is, however, possible to get information about the stability of the underlying ODE model. For this purpose the notion of strong components of a directed graph can be used (Roberts, 1976). A strong component is a sub graph in a directed graph in which there exists at least one directed path from node i to node j for every pair (i,j) (and vice versa). Any directed graph can be uniquely decomposed to its pairwise disjunct strong components. Further details on structural stability analysis can be found elsewhere (Hangos, 1991)
The qualitative prediction table can be constructed from the signs of these values with a row for each parameter and a column for each variable. (For further details about qualitative prediction tables see (Puccia and Levins, 1985), (Hangos, 1990». IDENTIFICATION OF MODEL STRUCTURE FROM STEADY STATE CHANGES In principle any structural property may form the basis of structure identification. For the sake of simplicity, the qualitative prediction tables and stability analysis wilI be used here. The steps involved are:
Oualitative prediction tables In order to compute the qualitative prediction table, we have to denote explicitely the dependence of the underlying ODE model in Eq. (1) on the parameters c: dx = f x(x, c), Tt
n m fx' xe9t, ce9t .
I. Specification of a set of candidate model structures. 2. Determination of structural properties for each candidate model structure. 3. Selection of characteristic measurement conditions under the predictions are different for the candidates.
(4)
574
4. Measuring or performing experiments and measurement evaluation for the selected measurements as well as some other, non characteristic ones which are useful for checking purposes.
almost the same as that of the simplified flowsheet of the heat pump system (Fig. 1.) whereP8 is the freon pressure in the condenser, P9 is that before the compressors and PlO is that in the reboiler. A partly heuristic qualitative ODE model has been also constructed for the heat pump system (see Csaki et aI. , 1990) which supports the same candidate structures.
Note that in case of structural stability analysis and qualitative prediction tables prototype tools have been developed for automatic evaluation of these structural properties based on the signed boolean v~rsion of the. state transition and input matrices. The tools are Implemented In the programming language NIAL using a subroutine package for evaluation of structural properties of state space models (Jantzen, 1989). For the problem presented here the .aix?ve subroutines have been extended to handle quahtatlve arithmetics and compute qualitative prediction values.
The resulting influence graphs is shown on Fig. 2. The presence or absence of the direct influence from PlO to P8 could not be derived from our process knowledge thus two candidate open loop model structures have been selected with and without this direct influence. Both of these structures have been found to be ambiguously stable by structural stability analysis. Control loops
Different problems met during the above procedure may cause the need for intentional modification of the candidate model structures.
As a necessary structural element to the controlled heat pump system the influence graph model of a PI controller had to be constructed. This has been performed directly from the equations relating the controlled variable (Y) and the manipulated variable (U) in case of a general linearized process model and a linear PI controller with constant parameters, setpoint SET_Y and disturbance DIST:
1. In case of similar candidate model structures no characteristic measurement conditions can be identified for selecting one of them (all the qualitative predictions coincide). 2. In case of large and complicated model structur~s the majority of the qualitative predictions can be ambiguous (i.e. depending on the relative numerical weights of the influences, denoted by ? in the tables). 3. Some candidate model structures are unstable which makes it (practically) impossible to perform open loop experiments in or near the corresponding steady-states.
U(t) == kp(Y - SET _ Y) + kI
f (Y -
SET _ Y)dt
dY
dt == ay Y + b U U + aD DIST
(9) (10)
In order to obtain influences to the manipulated variable U Eq.(9) has to be differentiated assuming SET_ Y being constant:
In the above cases we have to modify the system itself and / or the candidate models in any of the following ways
dU
- lumping several parameters into a single quasi-parameter (in the case of complicated models) to obtain more definite predictions, - introducing new variables and influences to modify the stability behaviour of the system (in practical cases this means introducing controllers to stabilize the system).
dY
dt == k P dt + kI
Y - k I SET - Y
(11)
The resulting influences can be found if the term dY / dt is substituted to Eq. (11) from Eq. (10). If the value of the manipulated variable is of no interest for the modeling goal this structure can be simplified by lumping the variable U into the variable Y. If the qualitative behaviour of the steady state changes is of interest (as in the case of qualitative prediction tables) further simplification can be achieved by omitting the controlled variable Y with all the influences to it from the graph and composing new influences from its seq:oint SET_Y acting on those variables which were influenced by it. Later on this simplified approximating influence graph model will be used.
MODEL STRUCTURE ELEMENTS FOR TIffi HEAT PUMP SYSTEM The candidate influence graph models of the heat pump system of our pilot plant distillation column can be constructed from two principal elements (or subgraphs): one for the open loop structure of the heat pump and another for the PI controllers implemented on it. These elements will be investigated in detail in this section.
STRUcruRE DETERMINA nON AND VALIDATION FOR TIffi HEAT PUMP SYSTEM
The heat pump system The simplified flowsheet of the heat pump system for our distillation pilot plant can be seen in Fig. 1. with thevariables used in our models encircled. The circuit, which is operating with freon 114 as a heat transfer medium, has been designed for large heat transfer load variations. The heat pump works as follows. Most of the freon vapor is condensed in the reboiler (denoted by HERB on Fig. 1.). An extra condenser (HESCOND) removes an amount of heat corresponding to the heat introduced by the compressors (COMP). The condensed freon passes to a receiver and through a heat exchanger (HECI) to the freon evaporator (HECOND). The heat exchanger serves to superheat the freon gas before it enters the compressors. In this way the freon carries the heat caused by the condensation of the column top product from the freon evaporator to the reboiler where it is used to reboil the bottom product of the column. More details about the system can be found elsewhere (Hallager and coworkers, 1990).
The set of candidate model structure was determined by introducing two PI control loops into the candidate o~en loop structures to stabilize it. These loops control the hlgh pressure (PlO) and the low pressure (P8) variables respectively. Note that these loops describe the current control configuration only approximately since e.g. P8 is controlled by a cascade of two PID controllers (Hallager and coworkers, 1990). The experimental qualitative results (i.e. the sign of changes in the steady state values) of the setpoint changes of these controllers with infinite reflux have been used for model structure determination. The experimental data are summarized in Tables 2. and 4. Structure with two controllers The above mentioned simplifications are used to eliminate the controlled (state) variables P8 and PlO. The resulting influence graph is shown on Fig. 4. It can be seen that the variable P9 has become almost isolated and it does not affect any other variable in the graph thus it can also be removed. In this way only two state variables (PI and P3) remains wich describe the very simplified column dynamics. The dynamics of the heat pump is covered bv its controllers.
Open loop structures The influence graph of the heat pump system in open loop has been constructed from engineering experience and qualitative ODE model equations of the heat pump system. The effect of the distillation column is described via the lumped variables PI and P3. Otherwise the topology of the influence graph is
575
Janowski, R. (1987). An introduction to QSIM and qualitative simulation. Artificial Intelligence in Engineering,~, 65-71. Jantzen, J. (1989). Structural Tests Using NIAL. 4th IFAC Symposium on Computer Aided Design in Control Systems, CADCS'88, IFAC Proceeding Series, 2, 219-223. John, P. W. M. (1971). Statistical Design and Analysis of Experiments. Macmillan Co., New York. King, R. B. (1983). Chemical Applications of Topology and Group Theory. 14. Topological Aspects of Chaotic Chemical Reactions, Theoretica Chimica Acta (Berl.),~, 323-338. Kleer de J., Brown, J. S. (1984), A Qualitative Physics Based on Confluences. Artificial Intelligence, 24,7-84. Kuipers , B. (1986). Qualitative Simulation. Artificial Intelligence, 29, pp. 289-338. Kuipers, B. (1989). Qualitative Reasoning: Modeling and Simulation with Incomplete Knowledge. Automatica,~, 571-585. Larsen, P. M. , Evans, F. J. (1987). Structural Design of decentralized control systems. in NATO Advanced Study Institute (ed): The application of advanced computing concevts and techniques in control engill~ Tuscany (Italy). Puccia, C. J., Levins, R. (1985). Oualitative Modelling of Complex Systems: An Introduction to Loop Analysis and Time Averaging. Harvard Univer:sity Press, Cambridge (Massaxhusetts)-London (England). Rasmussen, K.H., Nielsen, C.S., Bay J\!lrgensen, S. (1990). Identification of Distillation Process Dynamics Comparing Process Knowledge and Black Box Based Approaches. Prep. ACC'90, San Diego, California. Roberts, S.F. (1976). Discrete Mathematical Models with Applications to Social. Biological and Environmental Problems. Premtice-Hall, Englewood Cliffs (New Yersey). Thomas, R. (1980). Logical description, analysis and synthesis of biological and other networks comprising feedback loops. In: Aspects of Chemical Evolution Advances in Chemical Physics, 247-282.
The qualitative prediction table for this case is given in Table 1. Comparing the corresponding elements with the experimentally found qualitative prediction table (Table 2.) it can be seen that the predictions coincide which validates the simplified model structure describing the column dynamics. However the structure of the heat pump has been effectively removed by simplifications and the difference between the original candidate structures (i.e. the presence or absence of the direct influence from PlO to P8) cannot be investigated with these two active control loops. Structure with one controller In order to investigate the difference in the candidate model structures the heat pump with only one controller for the high pressure PlO was examined. The resulting influence graph is shown in Fig. 3. and it has been found ambiguously stable with the structural stability tests. The qualitative prediction table for this case is given in Table 3. describing the change in steady state values in state variables PI, P3, P8, P9 with respect to the change in SET_I0 for the two candidate structure (with or without direct influence from SET_I0 to P8). It can be seen that thepredictions for P8 and P9 are different and definit in this case. Comparing the predictions with the experimentally found data in Table 4. the structure with direct influence can be excluded. Furthermore the qualitative relationship between the weights li98 a89 < a99 a88
can be justified using the symbolic results of the values in the qualitative prediction table and the defir::te positive predictions for PI and P3 assuming the inequality above (see values in brackets in Table 3.). CONCLUSION The method of structure determination of linearized models based on qualitative models in the form of influence graphs and their structural properties such as structural stability analysis and qualitative prediction tables has been described in this paper. The method uses the qualitative results of the values of steady state changes from simple step response experiments. The method has been applied for the heat pump system of our pilot plant distillation column. Different influence graph structures of modeling controller:s and controlled variables as well as open loop structures have been investigated and valuated with the proposed method. Some guidelines to setup candidate model structures from process knowledge have also been indicated. It has been shown that fast (compared t::> the dynamics of the system) controllers can be used not only to stabilize the investigated system but also to simplify its dynamic structure and to focus attention of the structure determination to the 'interesting part of the system.
REFERENCES Csfri, Zs., Hangos, K.M., Hallager, L., J\!lrgensen, S. B. (1991). Qualitative Simulation for Start up a Distillation Column. Submitted to the IFAC Workshop on Computer Software Structures Integrated AIIKBS Systems in Process Control. Forbus, K. D. (1984). Qualitative Process Theory. Artificial Intelligence, 24, 85-168. Hallager, L., Jensen, N., Bay J\!lrgensen, S. (1990). An Industrial Scale Flexible Distillation Pilot Plant with an Indirect Heat Pump - Heat Pump Control Configuration. Spring NDF Workshop, Abo (Finland). Hangos, K.M. (1990). Experiment Design for Qualitative Model Structure Determination. Report DIR 90-03. Instituttet for Kemiteknik DTH, Lyngby. Hangos, K. M. (1991). Qualitative Process Modelling . accepted for the CPC-IV Conference, South Padre Island (USA).
Fig.l . Flowsheet of the heat pump with it variables encircled
576
TABLE I Qualitative prediction table for the Structure with two control loops CFi~. 4,)
PI
P3
+ +
+ +
TABLE 2 Experimental steady state chan~es (experimental Qualitative prediction tables) with two controlloops
Fig. 2.
Influence graph of a heat pump system open loop
~ posItive . . .mfluence ',.: parameter negativ influence
O •,
variable
--e
PI
P3
P8
P9
PlO
+ +
+ +
0 +
+ +
+ 0
(and CCYL)
TABLE 3 Qualitative prediction table for the structure with one control loop (Fig. 3.)
P8 SET_ 1O
I
without
I !
P9
with
+
+
PI
P3
1
1
1(+)
< if
3
98
1(+) 3
89 < ~ 3 88 >
\ \ PIe
\.
,I
TABLE 4 Experimental steady state changes (experimental Qualitative prediction tables) with one control loop
SET_le
\.....~
Fig.3.
Influence graph of a heat pump system with one controller
P8
P9
PI
P3
variable o , -,
+
+
+
+
.-'
parameter
~ positive influence
--e
negativ influence
'--
-......::;-'L.8u __- - <
",-, SET_la)
'-" Fig. 4.
Influence graph of a heat pump system with two controllers
577
MODEL STRUCTURE DETERMINATION FOR A HEA T PUMP SYSTEM K. M. Hangos* and L. Hallager** *Computer and Automation Institute, Hungarian Academy o/Sciences, p.a. Box 63 , H-1518 Budapest, Hungary **Department o/Chemical Engineering, Technical University 0/ Denmark, Building 229, 2800 Lyngby, Denmark
Abstract. In this paper principles of model structure determination by selection between candidate structures of underlying linearized ordinary differential equation models aredescribed. The principles are exemplified on a simple heat pump system. The identification procedure applying influence graph models, qualitative prediction tables taking results on structural stability analysis into account, is demonstrated for the example. The same methodology may be applied using other qualitative structural properties (e.g. qualitative correlation tables, structural observability and controllability analysis etc.) Structure identification methods based on qualitative models may complement conventional identification methods serving as structure determination tools in the early stages of modelling. Keywords System analysis, Chemical variables control, Linear differential equations, Influence graph models, Structure determination.
An influence graph is a weighted directed graph consisting of
INTRODUCTION
- nodes representing the variables (state, input and output) of the system to be modelled; - directed edges describing influence of the variable they start from to the growth rate (the accumulation or time derivative) of the variable they end at; - weight associated to the edges usually describing the sign (i.e. enhancing or damping) and, in some cases, the magnitude of the influence they represent.
The theory and methods for linear or linearized model structure determination form a difficult and sophisticated field of process identification. The methods usually involve parameter estimation of a set of mostly linear(ized) candidate models, followed by a decision step choosing between the candidate models (see e.g. Rasmussen and coworkers (1990)). An alternative approach is to use qualitative process knowledge and experimental data to determine the structure of linear models without parameter estimation. The purpose of this paper is to show how qualitative knowledge, qualitative models and structural properties may be used for structure identification.
An influence graph can be interpreted in different ways depending on: the mathematical form of the influence function (e.g. fvar at end (var_acstart), where the influence edge is direeteofrom var_acstart to var3cend) and
The field of qualitative modeling and its application is relatively new dating from the early 80s. The main approaches to qualitative physics include Forbus's Qualitative Process Theory (Forbus, 1984), Qualitative Physics by de Kleer and Brown (1984), Kuiper's Qualitative Simulation (Kuipers, 1986; 1989) and the graph analysis based methods of King (1983), Puccia and Levins (1985) and others.
- how the relation of influences influencing the same variable is understood. In general, a set of nonlinear ordinary differential equations (ODE) can be associated to each influence graph. In other words, only concentrated (lumped) parameter dynamic systems can be described by influence graphs . The set of associated non linear ODEs is called underlying ODE model and can be written in the following general form:
Methods suitable for structure determination of qualitative models are described in this paper for one of the most important classes of qualitative models namely by signed directed graphs (so called influence graphs). After introducing the basic notions about qualitative models and influence graphs, their structural properties, such as structural stability and qualitative prediction tables, are described. The structural properties and their differences in case of possible alternative models serve as a basis for structure determination. Finally the structure identification of a heat pump system in a pilot plant destillation column with suitable and intentionally modified model structures is presented.
dx = f
dt
x
(x)
(1)
where t time x vector of variables fx right hand side vector-vector function. The influence graph fixes the following structural properties of the underlying ODE model.
INFLUENCE GRAPHS, THEIR STRUCTURE AND STRUCTURAL PROPERTIES
1. The number and type of variables entering the right hand side function of the ODE for the ith variable is given by the starting variables of the influence edges ending at it. 2. The sign of the partial derivative of the nonlinear right hand side function with respect to its independent variables is given by the weights associated to the corresponding influence edges.
The notion of influence graphs emerged from the graph analysis based methods developed for modeling large ecological and biological systems (Puccia and Levins, 1985), for qualitative analysis of reaction kinetic behaviour (King, 1983) and for describing the structure of large control systems (Larsen and Evans, 1987).
S73
Linearized ODE model and the graph
The steady-state solution, of course, depends on c too:
If the underlying ODE model has at least one stationary (steady-state) solution, i.e. f x(x*) = 0
for some
x*
(5)
Qualitative prediction tables describe changes of the steady-state value of the variables when the parameters of the underlying ODE model change. The description is given in the following qualitative terms:
( 2)
then the original non linear underlying ODE can be linearized around x* using the lacobian matrix J of the function fx at the same point. This linearize4 ODE has the form dx = J(x - x*) dt
x*(c) .
fx(x*, c) =0,
+ : increase - : decrease no change ? : ambiguous (the direction of the change depends on the relation of the numerical weights in the influence graph which are unknown in the general case).
o:
( 3)
and there is a one-to-one correspondence between the structure of the influence graph and its linearized underlying ODE model.
Note that from a system engineering point of view one can regard parameters as input variables forming an input vector u=[cl ' ...., cm]T. The lacobian matrix of the right hand side function fx with respect to the element of this vector forms an input matrix B of a linear state space model corresponding to the underlying linearized model equations.
The structure of a signed directed graph can be uniquely determined by its adjacency matrix extended by the weight associated to the edges. An element aij in the adjacency matrix is 0 if there is no directed edge from node i to node j and aij is equal to the weight of the edge if it exists. The occurrence matrix is defined as a transpose of the adjacency matrix (Roberts, 1976).
dx = A(x _ x*) + B(u - u*) dt
It is easy to see from the definition of the influence graph that the occurrence matrix of an influence graph supplemented with the sign weight of the edges gives 'I'=Sign(J), i.e. the signed boolean version of the lacobian matrix of the underlying ODE model.
(6)
Thus the occurrence matrix J of an influence graph supplemented with the weight of the edges gives the lacobian matrix of the underlying ODE model in the partitioned form for
Structural properties An influence graph gives uncomplete qualitative information about the structure of its underlying ODE model. However, it is possible to determine some properties of the underlying ODE model based only on this qualitative information. These properties will be calIed structural properties and they include stability (near a steady-state) and qualitative behaviour of the steady state values caused by parameter changes (described by prediction tables) . Other structural properties such as structural observability and controlIability can be defmed and analyzed (Larsen and Evans, 1987), but these are out of the scope of this paper.
The effect of the change in parameter ck can be found by differentiating Eq. (5) with respect to ck and solve it for the unknown variables
a x~
ax~ , ... , aC
aC k
i = 1, 2, ... ,n
(7 )
k
which gives
Structural stability A linear(ize4) system is called structurally stable if for every physically realistic (e.g. positive) value of the non zero elements in its state transition matrix J (in Eq. (3» a stable system is obtained, i.e. all the real parts of the eigenvalues of J are negative. Note that the eigenvalues of the state transition matrix J are the same as the eigenvalues of the associated influence graph (Roberts, 1976).
(8 )
where 'l'ik ' i,k=I, ... ,n are the elements of the lacobian matrix J .
In case of large systems or systems with unknown parameters it could be quite complicated or even impossible to compute the eigenvalues of the state transition matrix J to check the stability conditions. Using the graph theoretical properties of the influence graph it is, however, possible to get information about the stability of the underlying ODE model. For this purpose the notion of strong components of a directed graph can be used (Roberts, 1976). A strong component is a sub graph in a directed graph in which there exists at least one directed path from node i to node j for every pair (i,j) (and vice versa). Any directed graph can be uniquely decomposed to its pairwise disjunct strong components. Further details on structural stability analysis can be found elsewhere (Hangos, 1991)
The qualitative prediction table can be constructed from the signs of these values with a row for each parameter and a column for each variable. (For further details about qualitative prediction tables see (Puccia and Levins, 1985), (Hangos, 1990». IDENTIFICATION OF MODEL STRUCTURE FROM STEADY STATE CHANGES In principle any structural property may form the basis of structure identification. For the sake of simplicity, the qualitative prediction tables and stability analysis wilI be used here. The steps involved are:
Oualitative prediction tables In order to compute the qualitative prediction table, we have to denote explicitely the dependence of the underlying ODE model in Eq. (1) on the parameters c: dx = f x(x, c), Tt
n m fx' xe9t, ce9t .
I. Specification of a set of candidate model structures. 2. Determination of structural properties for each candidate model structure. 3. Selection of characteristic measurement conditions under the predictions are different for the candidates.
(4)
574
4. Measuring or performing experiments and measurement evaluation for the selected measurements as well as some other, non characteristic ones which are useful for checking purposes.
almost the same as that of the simplified flowsheet of the heat pump system (Fig. 1.) whereP8 is the freon pressure in the condenser, P9 is that before the compressors and PlO is that in the reboiler. A partly heuristic qualitative ODE model has been also constructed for the heat pump system (see Csaki et aI. , 1990) which supports the same candidate structures.
Note that in case of structural stability analysis and qualitative prediction tables prototype tools have been developed for automatic evaluation of these structural properties based on the signed boolean v~rsion of the. state transition and input matrices. The tools are Implemented In the programming language NIAL using a subroutine package for evaluation of structural properties of state space models (Jantzen, 1989). For the problem presented here the .aix?ve subroutines have been extended to handle quahtatlve arithmetics and compute qualitative prediction values.
The resulting influence graphs is shown on Fig. 2. The presence or absence of the direct influence from PlO to P8 could not be derived from our process knowledge thus two candidate open loop model structures have been selected with and without this direct influence. Both of these structures have been found to be ambiguously stable by structural stability analysis. Control loops
Different problems met during the above procedure may cause the need for intentional modification of the candidate model structures.
As a necessary structural element to the controlled heat pump system the influence graph model of a PI controller had to be constructed. This has been performed directly from the equations relating the controlled variable (Y) and the manipulated variable (U) in case of a general linearized process model and a linear PI controller with constant parameters, setpoint SET_Y and disturbance DIST:
1. In case of similar candidate model structures no characteristic measurement conditions can be identified for selecting one of them (all the qualitative predictions coincide). 2. In case of large and complicated model structur~s the majority of the qualitative predictions can be ambiguous (i.e. depending on the relative numerical weights of the influences, denoted by ? in the tables). 3. Some candidate model structures are unstable which makes it (practically) impossible to perform open loop experiments in or near the corresponding steady-states.
U(t) == kp(Y - SET _ Y) + kI
f (Y -
SET _ Y)dt
dY
dt == ay Y + b U U + aD DIST
(9) (10)
In order to obtain influences to the manipulated variable U Eq.(9) has to be differentiated assuming SET_ Y being constant:
In the above cases we have to modify the system itself and / or the candidate models in any of the following ways
dU
- lumping several parameters into a single quasi-parameter (in the case of complicated models) to obtain more definite predictions, - introducing new variables and influences to modify the stability behaviour of the system (in practical cases this means introducing controllers to stabilize the system).
dY
dt == k P dt + kI
Y - k I SET - Y
(11)
The resulting influences can be found if the term dY / dt is substituted to Eq. (11) from Eq. (10). If the value of the manipulated variable is of no interest for the modeling goal this structure can be simplified by lumping the variable U into the variable Y. If the qualitative behaviour of the steady state changes is of interest (as in the case of qualitative prediction tables) further simplification can be achieved by omitting the controlled variable Y with all the influences to it from the graph and composing new influences from its seq:oint SET_Y acting on those variables which were influenced by it. Later on this simplified approximating influence graph model will be used.
MODEL STRUCTURE ELEMENTS FOR TIffi HEAT PUMP SYSTEM The candidate influence graph models of the heat pump system of our pilot plant distillation column can be constructed from two principal elements (or subgraphs): one for the open loop structure of the heat pump and another for the PI controllers implemented on it. These elements will be investigated in detail in this section.
STRUcruRE DETERMINA nON AND VALIDATION FOR TIffi HEAT PUMP SYSTEM
The heat pump system The simplified flowsheet of the heat pump system for our distillation pilot plant can be seen in Fig. 1. with thevariables used in our models encircled. The circuit, which is operating with freon 114 as a heat transfer medium, has been designed for large heat transfer load variations. The heat pump works as follows. Most of the freon vapor is condensed in the reboiler (denoted by HERB on Fig. 1.). An extra condenser (HESCOND) removes an amount of heat corresponding to the heat introduced by the compressors (COMP). The condensed freon passes to a receiver and through a heat exchanger (HECI) to the freon evaporator (HECOND). The heat exchanger serves to superheat the freon gas before it enters the compressors. In this way the freon carries the heat caused by the condensation of the column top product from the freon evaporator to the reboiler where it is used to reboil the bottom product of the column. More details about the system can be found elsewhere (Hallager and coworkers, 1990).
The set of candidate model structure was determined by introducing two PI control loops into the candidate o~en loop structures to stabilize it. These loops control the hlgh pressure (PlO) and the low pressure (P8) variables respectively. Note that these loops describe the current control configuration only approximately since e.g. P8 is controlled by a cascade of two PID controllers (Hallager and coworkers, 1990). The experimental qualitative results (i.e. the sign of changes in the steady state values) of the setpoint changes of these controllers with infinite reflux have been used for model structure determination. The experimental data are summarized in Tables 2. and 4. Structure with two controllers The above mentioned simplifications are used to eliminate the controlled (state) variables P8 and PlO. The resulting influence graph is shown on Fig. 4. It can be seen that the variable P9 has become almost isolated and it does not affect any other variable in the graph thus it can also be removed. In this way only two state variables (PI and P3) remains wich describe the very simplified column dynamics. The dynamics of the heat pump is covered bv its controllers.
Open loop structures The influence graph of the heat pump system in open loop has been constructed from engineering experience and qualitative ODE model equations of the heat pump system. The effect of the distillation column is described via the lumped variables PI and P3. Otherwise the topology of the influence graph is
575
Janowski, R. (1987). An introduction to QSIM and qualitative simulation. Artificial Intelligence in Engineering,~, 65-71. Jantzen, J. (1989). Structural Tests Using NIAL. 4th IFAC Symposium on Computer Aided Design in Control Systems, CADCS'88, IFAC Proceeding Series, 2, 219-223. John, P. W. M. (1971). Statistical Design and Analysis of Experiments. Macmillan Co., New York. King, R. B. (1983). Chemical Applications of Topology and Group Theory. 14. Topological Aspects of Chaotic Chemical Reactions, Theoretica Chimica Acta (Berl.),~, 323-338. Kleer de J., Brown, J. S. (1984), A Qualitative Physics Based on Confluences. Artificial Intelligence, 24,7-84. Kuipers , B. (1986). Qualitative Simulation. Artificial Intelligence, 29, pp. 289-338. Kuipers, B. (1989). Qualitative Reasoning: Modeling and Simulation with Incomplete Knowledge. Automatica,~, 571-585. Larsen, P. M. , Evans, F. J. (1987). Structural Design of decentralized control systems. in NATO Advanced Study Institute (ed): The application of advanced computing concevts and techniques in control engill~ Tuscany (Italy). Puccia, C. J., Levins, R. (1985). Oualitative Modelling of Complex Systems: An Introduction to Loop Analysis and Time Averaging. Harvard Univer:sity Press, Cambridge (Massaxhusetts)-London (England). Rasmussen, K.H., Nielsen, C.S., Bay J\!lrgensen, S. (1990). Identification of Distillation Process Dynamics Comparing Process Knowledge and Black Box Based Approaches. Prep. ACC'90, San Diego, California. Roberts, S.F. (1976). Discrete Mathematical Models with Applications to Social. Biological and Environmental Problems. Premtice-Hall, Englewood Cliffs (New Yersey). Thomas, R. (1980). Logical description, analysis and synthesis of biological and other networks comprising feedback loops. In: Aspects of Chemical Evolution Advances in Chemical Physics, 247-282.
The qualitative prediction table for this case is given in Table 1. Comparing the corresponding elements with the experimentally found qualitative prediction table (Table 2.) it can be seen that the predictions coincide which validates the simplified model structure describing the column dynamics. However the structure of the heat pump has been effectively removed by simplifications and the difference between the original candidate structures (i.e. the presence or absence of the direct influence from PlO to P8) cannot be investigated with these two active control loops. Structure with one controller In order to investigate the difference in the candidate model structures the heat pump with only one controller for the high pressure PlO was examined. The resulting influence graph is shown in Fig. 3. and it has been found ambiguously stable with the structural stability tests. The qualitative prediction table for this case is given in Table 3. describing the change in steady state values in state variables PI, P3, P8, P9 with respect to the change in SET_I0 for the two candidate structure (with or without direct influence from SET_I0 to P8). It can be seen that thepredictions for P8 and P9 are different and definit in this case. Comparing the predictions with the experimentally found data in Table 4. the structure with direct influence can be excluded. Furthermore the qualitative relationship between the weights li98 a89 < a99 a88
can be justified using the symbolic results of the values in the qualitative prediction table and the defir::te positive predictions for PI and P3 assuming the inequality above (see values in brackets in Table 3.). CONCLUSION The method of structure determination of linearized models based on qualitative models in the form of influence graphs and their structural properties such as structural stability analysis and qualitative prediction tables has been described in this paper. The method uses the qualitative results of the values of steady state changes from simple step response experiments. The method has been applied for the heat pump system of our pilot plant distillation column. Different influence graph structures of modeling controller:s and controlled variables as well as open loop structures have been investigated and valuated with the proposed method. Some guidelines to setup candidate model structures from process knowledge have also been indicated. It has been shown that fast (compared t::> the dynamics of the system) controllers can be used not only to stabilize the investigated system but also to simplify its dynamic structure and to focus attention of the structure determination to the 'interesting part of the system.
REFERENCES Csfri, Zs., Hangos, K.M., Hallager, L., J\!lrgensen, S. B. (1991). Qualitative Simulation for Start up a Distillation Column. Submitted to the IFAC Workshop on Computer Software Structures Integrated AIIKBS Systems in Process Control. Forbus, K. D. (1984). Qualitative Process Theory. Artificial Intelligence, 24, 85-168. Hallager, L., Jensen, N., Bay J\!lrgensen, S. (1990). An Industrial Scale Flexible Distillation Pilot Plant with an Indirect Heat Pump - Heat Pump Control Configuration. Spring NDF Workshop, Abo (Finland). Hangos, K.M. (1990). Experiment Design for Qualitative Model Structure Determination. Report DIR 90-03. Instituttet for Kemiteknik DTH, Lyngby. Hangos, K. M. (1991). Qualitative Process Modelling . accepted for the CPC-IV Conference, South Padre Island (USA).
Fig.l . Flowsheet of the heat pump with it variables encircled
576
TABLE I Qualitative prediction table for the Structure with two control loops CFi~. 4,)
PI
P3
+ +
+ +
TABLE 2 Experimental steady state chan~es (experimental Qualitative prediction tables) with two controlloops
Fig. 2.
Influence graph of a heat pump system open loop
~ posItive . . .mfluence ',.: parameter negativ influence
O •,
variable
--e
PI
P3
P8
P9
PlO
+ +
+ +
0 +
+ +
+ 0
(and CCYL)
TABLE 3 Qualitative prediction table for the structure with one control loop (Fig. 3.)
P8 SET_ 1O
I
without
I !
P9
with
+
+
PI
P3
1
1
1(+)
< if
3
98
1(+) 3
89 < ~ 3 88 >
\ \ PIe
\.
,I
TABLE 4 Experimental steady state changes (experimental Qualitative prediction tables) with one control loop
SET_le
\.....~
Fig.3.
Influence graph of a heat pump system with one controller
P8
P9
PI
P3
variable o , -,
+
+
+
+
.-'
parameter
~ positive influence
--e
negativ influence
'--
-......::;-'L.8u __- - <
",-, SET_la)
'-" Fig. 4.
Influence graph of a heat pump system with two controllers
577