Symbolic manipulation techniques for model simplification in object-oriented modelling of large scale continuous systems

Transcription

1 Mathematics and Computers in Simulation 48 (1998) 133±150 Symbolic manipulation techniques for model simplification in object-oriented modelling of large scale continuous systems Emanuele Carpanzano 1, Claudio Maffezzoni * Dip. di Elettronica de Informazione, Politecnico di Milano. P.za L. Da Vinci 32, Milan, Italy Received 24 March 1998; revised 14 July 1998; accepted 14 July 1998 Abstract In the present work, techniques for the symbolic manipulation of general nonlinear differential algebraic equation (DAE) systems are presented, and used for model simplification purposes, to support efficient simulation of large scale continuous systems in an object-oriented modelling environment. The specific problems addressed are efficient elimination of trivial equations by means of substitution, system block lower triangular (BLT) partitioning, and tearing, i.e. hiding of algebraic variables. Moreover, the weakening heuristic criterion for decoupling of large systems, via dynamic approximation, is studied. All these techniques have been successfully implemented and tested in MOSES (modular object-oriented software environment for simulation), in order to define a complete model simplification process. The results achieved by applying the discussed algorithms and criteria on serial multibody systems are illustrated. A brief overview on further known symbolic manipulation techniques is also given, comparing them with the proposed ones, throughout the paper. # 1998 IMACS/ Elsevier Science B.V. Keywords: Object-oriented modelling; Nonlinear DAE systems; Symbolic manipulation algorithms; Model simplification; Multibody systems 1. Introduction The object-oriented approach is becoming very popular in the field of modelling, especially with reference to complex physical system modelling. Following such an approach, a model is structured as closely as possible to the corresponding physical system; in particular, models are defined in acausal form, so that one software module is associated to one physical component, independently of the context in which it is used [12,15]. Complex models can be realised by aggregating more sub-models and their connections within a composite larger model, which may in turn be connected to other ÐÐÐÐ * Corresponding author. Tel.: ; fax: ; maffezzo@elet.polimi.it 1 carpanza@elet.polimi.it /98/$ ± see front matter # 1998 IMACS/Elsevier Science B.V. All rights reserved PII: S ( 9 8 )

2 134 E. Carpanzano, C. Maffezzoni / Mathematics and Computers in Simulation 48 (1998) 133±150 models. However, for continuous time modelling, by assembling the declarative equations of submodels, this form of model representation gives rise to larger scale nonlinear systems of differential algebraic equations (DAEs) of the form F t; y; _y; u; p ˆ0; (1) where F is a generic nonlinear n-vector function, y is the unknown variable n-vector, u is the input variable m-vector, p is the parameters vector and t is time. Once such a model is defined it is convenient to check the correctness 2 of the model before using it for analysis or design purposes, or before generating the simulation code. Moreover, especially for complex plants, the resulting DAE system is of very large dimension, so its numerical solution would require excessively long computation times. The simulation of a DAE system can be executed much more efficiently if a preliminary symbolic manipulation is performed in order to simplify the model [3,7,14]. The first problem, i.e. model verification, is not discussed in this work, the interested reader is therefore referred to the existing literature [2,14]. Consequently, it is here assumed that the considered DAE system (1) is mathematically correct. Instead, the problem of model simplification is dealt with, in particular, techniques for the symbolic manipulation of nonlinear DAE systems are presented to this aim. A new efficient substitution algorithm is introduced (Section 2), the block lower triangular (BLT) partitioning algorithm is briefly discussed (Section 3) and a flexible tearing algorithm is presented, which allows to easily implement both general and domain-specific heuristic rules, and which works in both the scalar and the vector cases (Section 4). In Section 5 the weakening heuristic criterion is outlined 3. Then, the whole model simplification process implemented in the modular object-oriented software environment for simulation (MOSES) is defined (Section 6), and, finally, the results achieved by applying the proposed techniques to serial multibody systems are discussed (Section 7). Remark 1. The minimum number of times that all or part of the equations of the DAE system (1) must be differentiated with respect to time in order to transform it into an explicit ODE form is defined the index of the system [1]. All known numerical DAE solvers have troubles with solving systems of index greater than 1. Unfortunately, high index problems are natural in object-oriented modelling [14], so symbolic manipulation techniques have been defined in order to reduce the index to 1 [3,15]. Since the object of the present paper is signally the study of symbolic manipulation techniques for model simplification purposes, it is here assumed that the considered DAE system (1) has index Eliminating trivial equations: the substitution algorithm The goal of the substitution algorithm is to reduce the dimension of the global DAE system, by identifying and eliminating all the trivial equations of the form X ˆW; X ˆW T (2) 2 In the general case a model is correct if it is syntactically and semantically correct, according to the rules of the modelling language, and if it represents a complete and consistent problem from a mathematical point of view [2]. 3 While the previous algorithms are harmless for the numerical integration, the weakening criterion, being a dynamic approximation, has an influence upon the numerical integration's accuracy, as pointed out in Remark 4.

3 E. Carpanzano, C. Maffezzoni / Mathematics and Computers in Simulation 48 (1998) 133± Fig. 1. The proxyvar data structure. where X is a matrix (scalar) variable, while the matrix (scalar) W can be either a variable or a constant. Such equations are eliminated by substituting one of the involved variables with the other (or with its constant value), using the proper sign and transposed operators, in all the equations of the system. The DAE system structural consistency 4 is preserved, since for every eliminated unknown variable also an equation is eliminated. Here an algorithm is proposed that performs the considered operation both in the scalar and in the matrix case. To efficiently implement the algorithm the data structure shown in Fig. 1 is defined. To every variable (scalar or matrix), a structure called proxyvar is associated, which is composed by three elements: var contains a pointer to the variable that is equivalent to the considered variable, once the change of sign and the transpose operation described in sign ( or ) and transp (true if the variable has to be transposed, false otherwise), respectively, are performed. With this simple structure, it is possible to store directly in the variables the effect of the substitution algorithm, providing also an easy way to determine its actual value even if it is not part of the system being numerically solved. This is essential in order to have a complete inspectability of the system, i.e. to keep trace of the substitutions. In the following, we call tail a variable whose field var of the relative proxyvar structure contains the variable itself, and whose fields sign and transp contain and false, respectively. The simplified pseudo-code of the proposed substitution algorithm is shown below. The sign and the transposition operators are omitted for the sake of simplicity. Consequently, the proxyvar data structure is simply a pointer to the equivalent variable, as shown in Fig. 2. The notation proxyvar X!Y means that the pointer proxyvar associated to X is set to point to Y. The method returntailvarof: X returns the first tail variable or constant found, starting from the variable X. For example, if we had: X! Y! Z! W! W, the method would return the variable W. If the starting variable is encountered again during the path, e.g. X! V! N! X! V!, then there is a cycle of assignments, which is cut by pointing the proxyvar of X to the variable itself (X! X, i.e. X becomes a tail variable), and by returning the variable X itself. Fig. 2. Simplified proxyvar data structure. 4 A DAE system is structurally consistent (or non-singular) if it is possible to form a set of ordered pairs of variables and equations, such that each variable y i and each equation F i are only members of one pair, and for each pair (y i, F i ) the variable y i appears in F i. Such a set is called an output set [15].

4 136 E. Carpanzano, C. Maffezzoni / Mathematics and Computers in Simulation 48 (1998) 133±150 The method substitut: X with: (returntailvarof: X) substitutes all the occurrences of X in the system with the tail variable or the constant equivalent to X, and then removes X from the set of unknown variables (if a cycle is found by the method returntailvarof: X then no substitution is performed). Whenever a variable is substituted, its proxyvar pointer is set to point to the tail variable or constant that substitutes it, this is necessary to keep inspectability. The proposed algorithm is composed of the following phases: 1. initialisation: the proxyvar pointer of every variable is initialised; 2. main cycle: this cycle is executed as long as trivial equations are found in the system, and it consists of two minor cycles: proxyvar setting cycle: the equations of the system are analysed, when a trivial equation is found, i.e. an equation in one of the forms (2), the proxyvar pointers are properly updated; variables substitution cycle: again, all the equations are considered, and a variable is replaced by its equivalent variable whenever necessary.

5 E. Carpanzano, C. Maffezzoni / Mathematics and Computers in Simulation 48 (1998) 133± It is possible to verify that the proposed algorithm allows to resolve the following tricky cases. Multiple assignments. If the algorithm is applied to the equations: Y ˆ H; Q ˆ K; Y ˆ Q after the analysis of the first two equations the proxyvar pointers of the variables Y and Q point to H and K, respectively, i.e. Y! H and Q! K. Problems could arise when the third equation is considered. This problem is solved by the algorithm by setting the pointers as follows: Y! H! Q! K. The substitution algorithm partitions the set of variables in equivalence categories, i.e. groups of equivalent variables, which are represented by one of them in the system to be numerically solved (or by a constant). In the shown example all the variables belong to the same category, whose tail variable is K. Generation of new trivial equations. Consider the set of equations: X ˆ Y Z Q; Y ˆ 1; Z ˆ 1 By removing the two trivial equations and replacing variables Y and Z with the constants 1 and 1, respectively, the new trivial equation X ˆ Q is generated. It is easy to verify that the proposed algorithm manages to eliminate such trivial equations generated during the substitution's sequence. Cycles. If the cycle of assignments is found: X ˆ Y; Y ˆ Z; Z ˆ X then by applying the substitution algorithm, the proxyvar pointers are set as follows: X! Y; Y! Z; Z! X. In this case, there is not a tail variable in the equivalence category. This problem is solved by the returntailvarof: X method, which cuts the cycle by setting the proxyvar pointer of a variable of the cycle to point the variable itself, as previously discussed. The illustrated simplified version of the algorithm can easily be extended in order to properly deal with the sign and transposition operators too, by introducing the methods for their correct updating whenever required. 3. Partitioning the system: the BLT algorithm When dealing with systems of very large dimensions, an important symbolic manipulation step is the BLT-partitioning of the system [12], which allows to decompose the overall system into subsystems which can be solved in sequence. The incident matrix I of a given DAE system is defined as follows: I is a matrix whose rows and columns represent the equations and the variables of the system, respectively, and whose element i; j is not zero if the jth variable (or its derivative) is present in equation i and is 0 otherwise. The BLT-partitioning algorithm can be performed directly on the incidence matrix, by permuting its columns (unknowns) and rows (equations) so as to make it block lower triangular, as depicted in Fig. 3. This operation is important since it allows to deal with smaller subsystems when performing the subsequent symbolic and numerical operations. In particular, the tearing algorithm can be applied to each single block, as shown in Section 4.

6 138 E. Carpanzano, C. Maffezzoni / Mathematics and Computers in Simulation 48 (1998) 133±150 Fig. 3. BLT-partitioned system. Techniques exist for permuting directly to BLT form, but there is not any known advantage over the usual two-stage approach: 1. permute entries onto the diagonal (usually called finding a transversal [8]); 2. use symmetric permutations to find the BLT form itself. Very economical procedures exist for constructing BLT-partitions with minimum-sized diagonal blocks, typically requiring O(n) O() operations for a matrix of order n with non-zero elements [8]. In MOSES Tarjan's algorithm has been implemented [17]. Remark 2. The first step of the BLT-partitioning algorithm, i.e. to permute the rows of the incidence matrix to make all diagonal elements non-zero, can also be viewed as a procedure which assigns to each variable y i a unique equation F j such that y i appears in F j, i.e. as a procedure to identify an output set. Consequently, if it is impossible to pair variables and equations in this way, then the system is structurally singular (see 4 ). Whenever a system turns out to be structurally singular, symbolic manipulation techniques can be used in order to give the user precise hints about what is wrong [2,14]. This point is not discussed in the paper, since it has been assumed that the considered DAE system (1) is mathematically correct, thus structurally non-singular. 4. Hiding of algebraic variables: the tearing algorithm It has been shown in the previous section that efficient algorithms exist to transform a DAE system to block lower triangular form, where some blocks of dimensions 1 are present along the diagonal. The algorithm guarantees that the dimensions of the diagonal blocks are kept as small as possible. Non-trivial blocks on the diagonal correspond to systems of equations that have to be solved simultaneously. To simplify the numerical solution of the overall DAE system, tearing is executed in a symbolic way, and consists of solving each block of the BLT-partitioned system for as many algebraic variables as possible; this way variables and equations of each block are split into two sets, so that the variables in the first set can be explicitly computed if the variables of the other set are known. In the following, the tearing problem will be studied with reference to a general DAE system of the form (1), which can represent either the whole DAE system or a subsystem corresponding to a block of the BLT-partitioned system. Consider the unknown variables set y of the system as two subsets x and z, x being the state variables (i.e. those variables which are present in the system together with their derivatives) and z being the algebraic variables. It is possible to rewrite the considered

7 E. Carpanzano, C. Maffezzoni / Mathematics and Computers in Simulation 48 (1998) 133± system in the form: F t; x; _x; z ˆ0 (3) omitting u and p for the sake of simplicity. The symbolic tearing of the DAE system is obtained by choosing the q-sub-vector ~z of z of minimum dimension such that Eq. (3) can be written as: z 1 ˆ g 1 t; x; _x;~z z 2 ˆ g 2 t; x; _x;~z; z 1.. z k ˆ g k t; x; _x;~z; z 1 ;...; z k 1 G t; x; _x;~z; z 1 ;...; z k ˆ0 (4) where z 1 ; z 2 ;...; z k are the remaining elements of z and once ~z has been extracted, g 1 ; g 2 ;...; g k are suitable scalar functions and G is a vector function of dimension n k. Moreover, each assignments z j ˆ g j t; x; _x;~z; z 1 ;...; z j 1 has to be obtained directly by solving one equation of Eq. (3) for z j, once j 1 equations of Eq. (3) have been solved for z 1 ; z 2 ;...; z j 1. After substitution of variables z 1 ; z 2 ;...; z k into G, the problem is turned to the solution of a reduced DAE system 5 G t; x; _x;~z ˆ0. In particular, we are interested in dividing the equations of the given DAE system into two subsets so that the first one (assignments) is as large as possible, and the second one (implicit equations) is as small as possible. It has been proven in [5] that the stated tearing problem is NP-complete. This means that the optimal solution for the tearing problem cannot be guaranteed in polynomial time, consequently approximation algorithms have to be considered. Numerous approximation tearing algorithms have been proposed in different contexts, but among them there is not a clear winner [8,13]. In this paper an efficient and flexible algorithm is summarised, which allows the application of both general and domain-specific heuristic rules, and which works in the vector case as well. Since the use of bipartite graphs allows dealing with the problem in a simple and efficient way, we define the associated bipartite graph (see Fig. 4) as follows. In the bipartite graph there are two sets of nodes: E-nodes (squares), one for each equation of (3), and V-nodes (circles), one for each algebraic variable of (3). There is an edge from an E-node e i to a V-node v j if and only if the equation associated to e i contains the variable associated to v j, and this edge is bold if and only if e i can be solved for v j. The bipartite graph can be used to work out a tearing algorithm, which turns out to be simple and efficient to implement directly on the incidence matrix of the DAE system 6. 5 Since DAE solvers produce a solution within the user-specified error bounds only for x and ~z (and not for _x), hidden variables' errors are not directly controlled [15]. If accuracy of z j is a real concern, the first j assignments must at least have the form: z i ˆ g i t; x; ~z; z 1 ;...;z i 1, for iˆ1, 2,..., j, i.e. they cannot contain derivatives. 6 A suitable data structure for the incidence matrix of a DAE system allows several symbolic manipulation algorithms to be efficiently performed; so it is reasonable to assume that this data structure is available in the symbolic manipulation environment [3], as discussed in Appendix A.

8 140 E. Carpanzano, C. Maffezzoni / Mathematics and Computers in Simulation 48 (1998) 133±150 Fig. 4. Reduced incident matrix (a) and corresponding bipartite graph (b). Since an assignment is easily obtained whenever there is an equation with only one unknown algebraic variable z k and the equation is solvable with respect to z k, an assignment is defined whenever, in the associated bipartite graph, an E-node e k is found having only one incident edge, and this edge is bold. Thus, to obtain as many assignments as possible, the graph has to be manipulated so as to get as many E-nodes with only one incident and bold edge as possible. To this aim the following algorithm is proposed.

9 E. Carpanzano, C. Maffezzoni / Mathematics and Computers in Simulation 48 (1998) 133± Every time the cycle is executed one of the following two operations is performed: 1. V-node v l, E-node e i and all their incident edges are removed from the bipartite graph; 2. V-node v r and all its incident edges are removed from the bipartite graph. In case 1 an assignment is identified, while in case 2 a choice is taken which gives rise to an implicit equation. In fact, in case 1 equation e i can be solved for the unknown algebraic variable v l, to obtain an assignment. In case 2, on the contrary, an implicit equation is automatically generated by removing V- node v r from the graph, since there will be one more E-node remaining at the end of the algorithm, so there will be one more implicit equation left at the end of the algorithm. The number of assignments obtained by performing the illustrated algorithm depends on the way V- node v r is chosen in the underlined statement. This choice can be done by making use of both general heuristic rules and domain-specific heuristic rules. A simple rule is, e.g., the following one: among the existing V-nodes, select a node v r, if any, that is connected by a non-bold edge to one of the E-nodes having minimum number of incident edges. Efficient general heuristic rules, and domain-specific rules for multibody systems are presented in [5]. Since the data structure for the incidence matrix of a DAE system does not depend on the equations and variables' type, the illustrated tearing algorithm can directly be executed on the incidence matrix of DAE system in vector form. Performing the tearing algorithm on vector equations can give better results than performing the same algorithm on the corresponding set of scalar equations, obtained by expanding every vector equation. In Section 7 it will be shown that, for serial multibody systems, the minimum number of implicit equations can be achieved by using only general heuristic rules, provided that the tearing algorithm is first performed on the equations written in vector form. Remark 3. The presented tearing algorithm can be applied to general nonlinear DAE systems, corresponding to models of whatever physical domain, since no hypothesis is done as regards the linearity of the system, and since general heuristic rules can be used, when no domain-specific rules are available. So, it does not require informations in the model library to direct the manipulation process. Tearing algorithms based on domain-specific heuristic rules can be defined, as for example in [9]. Such algorithms can give better results for a specific application domain, but cannot be applied to multidisciplinary systems, unless the necessary domain-specific heuristic rules are defined and implemented in the model library when new physical domains are considered. When dealing with DAE systems which are linear in the unknowns, it is possible to perform the computation in a fully symbolic way. This can be done either by symbolic Gaussian elimination or by applying Cramer's rule. When using standard Gaussian elimination, the sequence of the elimination process must take into account the numerical values of the matrix elements, in order to avoid divisions by zero. Alternatively, this sequence can be defined symbolically by the modeller through a technique called relaxing. Relaxing has the advantage that the sequence of computation can be determined before a simulation run starts. In particular, the computation can be done in a fully symbolic way, which would not otherwise be possible. More about relaxing, and its application in Dymola, can be found in [16]. When solving systems which are linear in the unknowns through the relaxing technique, it must be guaranteed that the predefined elimination sequence does not lead to divisions by zero, for all the possible values of the matrix elements. As is well known, systems which are linear in the unknowns can

10 142 E. Carpanzano, C. Maffezzoni / Mathematics and Computers in Simulation 48 (1998) 133±150 always be solved symbolically through Cramer's rule, i.e. using determinants. However, this method may be extremely inefficient for medium to large system sizes. An efficient linear system solver, using Cramer's rule, has been realised and implemented in Omola [4]. The techniques briefly discussed in this remark require restrictions on the linearity and on the dimensions of the considered problem (symbolic solution through Cramer's rule), or practical experience of the modeller (use of domain-specific heuristic rules), or both of them (use of the relax operator in the model library). On the contrary, the algorithm discussed in this section is general, in the sense that it requires no restrictions on the considered DAE system and no specific skill of the modeller or of the user. 5. The weakening heuristic criterion When dealing with physical systems, it may frequently happen that a certain variable has a weak dynamical influence in a certain equation or group of equations, typically for two reasons: (a) its absolute relevance is small; (b) the dynamics associated to that variable is relatively slow with respect to the dynamics associated to the variables dominating the said equation(s). As an example of the first type, the case of the momentum equation for a liquid flowing in a pipe can be considered: the fluid temperature has a negligible effect in the equation, which consists of causing ``small'' variations of the liquid density and viscosity. An example of the second type is the electrical part of a direct-current motor, in whose equations the back-electromotive force depends on the rotor angular speed: here the angular speed, being a mechanical variable, may change only slowly with respect to the dynamics of the motor's electrical part. It should be noticed that a weakness declaration can have various causes; e.g. it can be due to different dynamics in the system, but it can also be caused by specific structure and sizing of certain components in a plant. So, the weakening criterion can be defined as a problem-dependent heuristic criterion based on dynamic decoupling. A variable can be declared weak by the model developer 7 in the considered equation(s) if either one (or both) of the reasons (a) and (b) apply. When a variable y j is declared weak in an equation F i, the numerical solution of the DAE system will be performed by using in F i, in any integration interval, the value of y j evaluated at the interval initial time. This means that if a variable is declared weak in an equation, then at a given integration step, its value can be considered as known; it follows that the 1- element i; j of the incidence matrix can be substituted by a 0-element, when performing the previously discussed symbolic manipulation algorithms. In particular, the weakness attribute may strongly improve the results of the BLT-partition (by allowing to decouple the global DAE system into more subsystems), and/or of the tearing algorithm (by allowing a more efficient splitting in assignments and implicit equations for single BLT-blocks). The use of the weakening criterion in object-oriented modelling, and its applications in MOSES, are also discussed in [6,11]. Remark 4. The manipulation techniques discussed in the previous sections have no influence upon the numerical integration process; on the contrary, the weakening heuristic criterion introduces extra-errors 7 A variable may be declared weak in an equation both while building the model library and while assembling a plant model.

11 E. Carpanzano, C. Maffezzoni / Mathematics and Computers in Simulation 48 (1998) 133± into the numerical integration. In particular, such errors are due to handling the weakened variables as constants while other variables of the system are being computed. As a consequence, the choice of weak variables has to be performed by an expert modeller, in order to guarantee that weakening does not significantly affect the accuracy of the numerical integration process. Remark 5. It has to be checked by the symbolic manipulation environment that by substituting the 1- elements of the incidence matrix corresponding to weak relations with 0-elements, the DAE systems' structural consistency is preserved, otherwise a suitable message has to be given to the modeller, pointing out the improper weak declarations [3]. Remark 6. In this paper only pure symbolic approaches have been considered, but, for the sake of completeness, it has to be pointed out that recently a new method for solving DAE systems efficiently, called inline integration, using a mixed symbolic and numeric approach, has been proposed. Following this method, discretisation formulae representing the numerical integration algorithm are symbolically inserted into the DAE model, and the symbolic manipulation algorithms treat these additional equations in the same way as the physical equations of the model itself. It has been shown that this uniform treatment of physical equations and discretisation formulae often leads to a significant model simplification. More about inline integration, and its implementation in Dymola, can be found in [10]. 6. The model simplification process Now that the used symbolic manipulation algorithms have been illustrated, the model simplification process, whose Petri net representation is shown in Fig. 5, can be defined. This consists of the following steps: 1. all the vector equations are collected in a set and the substitution algorithm is applied on it; 2. the tearing algorithm is applied on the reduced set of vector equations; 3. every vector equation is expanded into scalar ones; 4. the substitution algorithm is applied on the set of scalar equations; 5. the BLT-partition of the scalar equations is performed by applying Tarjan's algorithm; 6. first the assignments of each block obtained through the vector tearing are put in the final set of assignments of the block 8, then the tearing algorithm is applied to the remaining scalar equations of the considered block. When performing the BLT-partition and the tearing algorithm, the 1-elements of the incidence matrix corresponding to relations declared weak by the modeller are replaced by 0-elements, as discussed in Section 5. Throughout the model simplification process, zeros are exploited whenever an equation is modified, or a vector equation is expanded, in order to reduce the complexity of the equations. 8 An equation is an assignment for a block only if it is solved for one of the variables of the considered block, as a consequence not necessarily all the assignments obtained through the vector tearing are put in the final sets of assignments.

12 144 E. Carpanzano, C. Maffezzoni / Mathematics and Computers in Simulation 48 (1998) 133±150 Fig. 5. Petri net representing the model simplification process. The proposed model simplification process has been implemented in MOSES [11], and the results achieved by applying it to serial multibody systems are presented in the following section. 7. Application to serial multibody systems in MOSES In this section, the whole model simplification process is applied to two serial multibody systems (Section 7.1), different tearing rules are compared (Section 7.2) and the weakening criterion is explored (Section 7.3).

13 E. Carpanzano, C. Maffezzoni / Mathematics and Computers in Simulation 48 (1998) 133± Applications of the whole model simplification process The first example illustrates how the entire manipulation process operates, by considering simple 3- link and 6-link serial multibody systems, without motors and controllers and with no flexibility of the transmission gears. The starting numbers of equations of the considered models are: 3-link robot: vector equations and 36 scalar equations, 6-link robot: vector equations and 72 scalar equations. The results obtained for each manipulation step are the following ones: Step 1 ± vector substitution 3-link robot: 48 trivial vector equations are eliminated (63 vector equations remain), 6-link robot: 96 trivial vector equations are eliminated (129 vector equations remain). Step 2 ± vector tearing 3-link robot: all the remaining 63 vector equations are transformed into assignments, 6-link robot: all the remaining 129 vector equations are transformed into assignments. Step 3 ± expansion of the vector equations 3-link robot: total number of scalar equations obtained is 255 (189 are assignments), 6-link robot: total number of scalar equations are eliminated is 519 (387 are assignments). Step 4 ± scalar substitution 3-link robot: 153 trivial scalar equations are eliminated (102 scalar equations remain), 6-link robot: 279 trivial scalar equations are eliminated (240 scalar equations remain). Step 5 ± BLT-partition 3-link robot: 35 blocks are obtained (1 of size 68 (the first) and 34 of size 1), 6-link robot: 74 blocks are obtained (1 of size 167 (the first) and 73 of size 1). Step 6 ± scalar tearing 3-link robot: the first block is split into five implicit equations and 63 assignments; as regards the other 34 blocks, one contains one implicit equation, while the remaining contain one assignment each, 6-link robot: the first block is split into 11 implicit equations and 156 assignments; as regards the other 73 blocks, one contains one implicit equation, while the remaining contain one assignment each. Notice that in the outlined model simplification process the tearing algorithm has been executed, both in the scalar case and in the vector case, by using only general heuristic rules, as is discussed in the sequel Comparison among different heuristic tearing rules In order to compare the results obtained by applying the proposed tearing algorithm, with different heuristic rules, to the systems considered in the previous section, Table 1 can be analysed, where the

14 146 E. Carpanzano, C. Maffezzoni / Mathematics and Computers in Simulation 48 (1998) 133±150 Table 1 Results achieved by performing the proposed tearing algorithm Considered cases Rules used Robots Vector Scalar General Specific 3-Link 6-Link 1 No Yes Yes No No Yes Yes Yes Yes Yes Yes No 6 12 number of implicit equations obtained after the application of the algorithm is shown. First, the algorithm is applied directly on the expanded system, i.e. treating only scalar equations, by using only general heuristic rules (row 1) and by using also domain-specific rules (row 2). Then, the algorithm is applied also on the vector equations, before expanding them (row 3), by applying only general heuristic rules. From the results shown in Table 1, it can be noticed that, if we perform the algorithm only on the expanded system (scalar case, rows 1 and 2), the use of domain-specific rules is very important, since by using these rules the optimal solution, i.e. the minimum number of implicit equations, is achieved. In fact, the number of resulting equations is equal to the number of state variables of the corresponding model (row 2). On the other hand, when the algorithm is first applied to the DAE system in vector form and then to the expanded scalar form, we achieve the optimal solution even without domain-specific heuristic rules (row 3). A more in-depth discussion on this topic can be found in [5] Testing the weakening criterion In this section, the possible benefit of using the weakening criterion is illustrated through a simple example, and the results obtained by applying it to the model of a six degree of freedom robot are reported. Consider a simple brushless motor directly applied to a load inertia. A control loop using an analogue PI controller is used to control the motor torque and to improve its dynamical response. In this system, the electrical dynamics is much faster than the mechanical one. The motor model is constituted by: (a) the equations of the mechanical part: J _! m ˆ m l f (5) f ˆ 0 k f! m (6) m ˆ k m i (b) the equation of the electrical part: (7) v ˆ L Ri k v f! m g (c) the PI controller equation: v ˆ k P i 0 i k I Z i 0 i dt (8) (9)

15 E. Carpanzano, C. Maffezzoni / Mathematics and Computers in Simulation 48 (1998) 133± Fig. 6. Control scheme of the motor model ((- - -) weak relation). where! m is the rotor angular velocity, m, l, and f are the active torque, the load torque and the friction torque, respectively, i and v are the terminal current and voltage, J, 0, k f, k m, L, R and k v are suitable constants, while k P and k I are the PI parameters, and i 0 is the current setpoint. In Eq. (8) the ``mechanical variable''! m is put within curly parentheses to indicate that! m is a weak variable in that equation; in fact! m, being a mechanical variable, is subject to slow variations with respect to the electrical dynamics. This way, the symbolic manipulation algorithms split the system of equations into two almost independent subsystems with the causal structure of Fig. 6. As a consequence, if the external perturbation is such that the fast electrical dynamics is excited, then the solver will integrate the electrical part equation with the last available value of! m, generally using a smaller integration step than the one for the mechanical part. Once the value of the current i is available, the mechanical subsystem will be solved afterwards with its own integration step. Table 2 summarises the results of a system simulation 9, in response to a square wave excitation of i 0, in two cases: in the first one the DAE system is solved globally (row 1), in the second one! m is declared weak in (8) and, consequently, two cascaded DAE subsystems are solved (row 2). It is apparent that the step-size of the mechanical part is larger in the second case (while having the same tolerances on the system variables). Here, the application of the weakening criterion does not give a definite advantage in terms of simulation time: this is due to the little overload necessary to realise this technique in computation, which is more visible in small size systems. The benefit of weakening may be very relevant to industrial robot simulation, where the electrical part is made by a few simple equations (to be solved with small integration steps), while the mechanical part is made by complex 3D mechanical equations (to be solved almost independently with larger integration steps). In MOSES, the weakening criterion has been applied to the six link robot SMART 6.12.R, which is a serial multibody system equipped with brushless motors and PI controllers. The global system has been split by the model simplification process in two subsystems, one related to the electrical and control part of the six joints, and one containing all the equations of the mechanical part. Table 2 Weakening: simulation results with the simple motor model Number of steps of the mechanical part Number of steps of the electrical part Global system Decoupled system Computation time (s) 9 Performed with the numerical DAE-solver DASSL [1].

16 148 E. Carpanzano, C. Maffezzoni / Mathematics and Computers in Simulation 48 (1998) 133±150 Table 3 Weakening: simulation results with the SMART 6.12.R model Number of steps of the mechanical part Number of steps of the electrical part Global system Decoupled system Computation time (s) The electrical and control subsystem consists of 12 state equations and 12 state variables (the motor current and the controller state of each link), while the mechanical one consists of 24 equations and 24 state variables (12 describing the joints' variables and 12 representing the flexibility of the transmission gears). The simulation results are presented in Table 3, where it can be noticed how the use of the weakening criterion allows a significant reduction of the computation time. 8. Concluding remarks In the paper symbolic manipulation techniques have been presented for decoupling and order reduction of general nonlinear DAE systems, which have been assumed to be mathematically correct and of index 1. These techniques have been applied to simplify object-oriented models of large scale continuous systems, in order to perform efficient simulations of their behaviour. In particular, a new efficient substitution algorithm, for eliminating (scalar and matrix) trivial equations, has been introduced; the decomposition of the overall system into subsystems, which can be solved in sequence, has been discussed; and a flexible tearing algorithm has been proposed, which allows the use of both general and domain-specific heuristic rules and which works in the vector case too. Moreover, the weakening criterion has been studied. The complete model simplification process implemented in MOSES has been outlined, and applications to robotic systems have been shown, where the proposed manipulation techniques have been used to reduce the computational burden of the simulation process. Since there is a growing consent on using the object-oriented approach to model large scale and heterogeneous systems, and mathematical simulation is the means to practically use such models, model simplification techniques, like the ones considered in this paper, are becoming of great importance. In the next future it will be necessary to extend such techniques in order to deal with hybrid systems, which incorporate both continuous time and discrete-event dynamics [2]. In particular, the use of heuristic criteria, as for example weakening, will become a crucial point to simplify large complex models. Appendix A A. Efficient data structures for equations and incidence matrices In order to execute the proposed manipulation algorithms, symbolic operations have to be performed on the equations of the considered DAE system and on its incidence matrix. For example, it is necessary to establish which variables can be made explicit in a certain equation, to identify the bold

17 E. Carpanzano, C. Maffezzoni / Mathematics and Computers in Simulation 48 (1998) 133± Fig. 7. Equation and corresponding binary tree. elements in the reduced incidence matrix, and to solve equations for one of their variables, to obtain the assignments. These operations are performed both in the scalar and in the vector case. The efficiency of the discussed algorithms, as regards the model simplification, the storage requirements and the computing time, strongly depends on the efficiency of the data structures implemented to represent equations and matrices. The symbolic manipulation of equations is quite easy, both in the scalar case and in the vector case, if an equation is represented as a binary tree, like in the example shown in Fig. 7. To tree nodes that are not leaves there are associated operators, e.g. ˆ,,,, /, der, sin, cos, etc. To the leaves, there are associated variables, parameters, or numerical constants [12]. As the incidence matrix is concerned, since this is usually sparse for large systems, different data structures can be used for storing, accessing and manipulating without undue overheads [8]. In MOSES Fig. 8. Data structure for the incidence matrix.

18 150 E. Carpanzano, C. Maffezzoni / Mathematics and Computers in Simulation 48 (1998) 133±150 a sparse matrix is stored through two arrays [3], one, say R, for the rows (equations) and one, say C, for the columns (variables). Each element of R, e.g. R m, represents a row of the matrix and contains two pointers: one points to the equation associated to the row (Eq m ), while the other one points to an array (AR m ), whose elements point to the variables appearing in Eq m. A similar description can be given for each element of C, as outlined in Fig. 8. References [1] K.E. Brenan, S.L. Campbell, L.R. Petzold, The Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, SIAM Series, Classics in Applied Mathematics, [2] E. Carpanzano, Model Verification in Object-Oriented Modelling of Large Scale Nonlinear Hybrid Systems, CESA'98, IMACS Multiconference on Computational Engineering in Systems Applications, Nabeul-Hammamet, Tunisia, 1±4 April [3] E. Carpanzano, F. Formenti, Symbolic manipulation of DAE systems, Thesis (in Italian), Politecnico di Milano, April [4] E. Carpanzano, F. Formenti, Solution of symbolic linear systems in Omsim using Cramer's rule, Internal report, number ISRN LUTFD2/TFRT-7524-SE, Department of Automatic Control, Lund Institute of Technology, October [5] E. Carpanzano, R. Girelli, The tearing problem: definition, algorithm and application to generate efficient computational code from DAE systems, Second MATHMOD, IMACS Symposium on Mathematical Modelling, 5±7 February 1997, Technical University Vienna, Austria, pp. 1039±1046. [6] F. Casella, C. Maffezzoni, Exploiting Weak Interactions in Object Oriented Modelling, EUROSIM Simulation News Europe, February [7] F.E. Cellier, H. Elmqvist, Automated Formula Manipulation Supports Object Oriented Continuous System Modelling, IEEE Control System, April 1993, pp. 28±38. [8] I.S. Duff, A.M. Erisman, J. Reid, Direct Methods for Sparse Matrices, Clarendon Press, Oxford, 1986, pp. 252±260. [9] H. Elmqvist, M. Otter, Methods for tearing systems of equations in object oriented modelling, in: Guasch, Huber (Eds.), Proceedings of the Conference on Modelling and Simulation, 1994, pp. 326±332. [10] H. Elmqvist, M. Otter, F.E. Cellier, Inline integration: a new mixed symbolic/numeric approach for solving differentialalgebraic equation systems, Keynote Address, Proceedings of ESM'95, European Simulation Multiconference, Prague, Czech Republic, 5±8 June 1995, pp. 23±34. [11] C. Maffezzoni, R. Girelli, MOSES: modular modelling in an object oriented database, Mathematical and Computer Modelling of Dynamical Systems, Vol. 4, No. 2, 1998, pp. 121±147. [12] C. Maffezzoni, R. Girelli, P. Lluka, Generating Efficient Computational Procedures from Declarative Models, Simulation Practice and Theory, vol. 4, 1996, pp. 303±317. [13] R.S. Mah, Chemical Process Structures and Information Flow, Butterworths Series in Chemical Engineering, 1993, pp. 151±180. [14] S.E. Mattsson, Simulation of object-oriented continuous time models, Math. Comput. Simulation 39(5)(6) (1995) 513± 518. [15] S.E. Mattsson, M. Andersson, K.J. AÊ stroèm, Object oriented modelling and simulation, in: D.A. Linkens (Ed.), CAD for Control Systems, Marcel Dekker, New York, 1993, pp. 56±66. [16] M. Otter, H. Elmqvist, F.E. Cellier, ``Relaxing'' ± a symbolic sparse matrix method exploiting the model structure in generating efficient simulation code, in: Proceedings of the Symposium on Modelling, Analysis, and Simulation, CESA'96, IMACs MultiConference on Computational Engineering in Systems Applications, Lille, France, 1996, vol. 1, pp. 1±12. [17] R.E. Tarjan, Depth-first search and linear graph algorithms, SIAM J. Comput. 1(2) (1972) 146±160.