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Simplified tree-structured decomposition using bond graphs
Automatica 35 (1999) 755—760
Technical Communique
Simplified tree-structured decomposition using bond graphs Bruce H. Wilson*, Bora Eryilmaz Department of Mechanical, Industrial, and Manufacturing Engineering, Northeastern University, Boston, MA 02115, USA Received 13 October 1998; received in final form 3 November 1998
Abstract The value set of an uncertain transfer function facilitates a number of robust control problems. For the general case, value set synthesis is aided when the transfer function is decomposed into a tree structure with (mostly) disjoint uncertain parameters. We have developed a general and direct technique for obtaining such transfer functions. Bond graphs provide the machinery for the new technique. The method developed here simplifies tree-structured transfer function synthesis for physically derived symbolic plant models. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Bond graph; Tree-structured decomposition; Value set
1. Introduction Many robust control applications require the complexplane region that contains the frequency response of an uncertain transfer function G(s, q), where q represents a vector of uncertain parameters. A frequency-dependent region known as the value set defines the response envelope of this uncertain transfer function. The value set is defined by G( ju, Q)&+G( ju, q)"q3Q,LC,
(1)
where Q represents the known subspace that contains the uncertain parameter vector. The value set finds application in evaluating a given uncertain transfer function’s ability to satisfy the Nyquist criterion, in synthesizing QFT templates, and in determining a frequency response envelope. One means to determine a value set is to grid the uncertain parameters and evaluate the frequency response for all possible combinations. The grid method, however, not only introduces a combinatoric explosion into the analysis, but also underbounds the true value set. To address the combinatoric explosion imposed by gridding, researchers have worked to reduce the number of parameter sets that must be evaluated to synthesize ————— * Corresponding author. Tel.: (617)373 3808 fax: (617) 373-2921; e-mail: [email protected]. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Peter Dorato.
a value set. The objective is to identify testing sets, finite subsets of the original uncertainty space, which are used for synthesizing value sets of uncertain polynomials and uncertain transfer functions, provided that their uncertainty structures fit into specific forms. While improved analysis methods of completely general uncertain transfer functions remains an open topic, useful techniques exist for certain classes of transfer functions. A quite general transfer function form, for which a testing set can be found, is provided in Fu et al. (1995). This form is given by L h(s, c)"1#g (s) (p (s)#c2P (s))IG, G G G G
(2)
where g (s) and p (s) are real scalar functions and poly G nomials in s, c 3! LR, represent a partition of c, P (s) G G G are real vector polynomials with dimension N, and k are G nonzero integers (positive or negative). Fu et al. (1995) cite process control, where cascaded subplants with independent uncertainty comprise the overall plant, as an application where Eq. (2) is useful. Note that this form also captures independent, affine, and multilinear uncertainty structures, as well as independent real zero, pole, and gain variations and complex zero and pole variations. More background on this topic is available in Ackermann (1994) and Tempo and Blanchini (1996). Though the structure in Eq. (2) provides considerable generality, it will not be possible to express the characteristic polynomials and transfer functions associated with
0005-1098/99/$—see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 0 0 0 5 - 1 0 9 8 ( 9 8 ) 0 0 2 2 6 - X
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B. H. Wilson, B. Eryilmaz/Automatica 35 (1999) 755—760
many robust control engineering problems in this form or in other special forms mentioned above. While any polynomial or transfer function can be forced, through overbounding, to match a special form, this processes introduces conservativeness. The tree-structured decomposition (TSD) increases the quantity of uncertain polynomials and transfer functions whose value sets can be readily synthesized without introducing undesirable conservativeness (Ackermann, 1994; Barmish et al., 1990). TSD is a process of expressing an uncertain polynomial or uncertain transfer function as an algebraic function of simpler terms. Ideally, the coefficients of these terms will originate from disjoint subsets of the uncertain parameters, although this is not always possible. When the subsets are disjoint, value set synthesis is simplified. For uncertain transfer functions, our focus, the intent is to express G(s, q) as G(s, q)"f (G(s, q ), G(s, q ), 2 , G(s, q )) L with q 5q " for iOj, (3) G H where f ( ) ) includes simple algebraic operations (addition, multiplication, inversion) among its variables. If G(s, q) can be decomposed in this manner, the value set G( ju, Q) can be synthesized by (i) determining the boundary of each G ( ju, Q ), *G ( ju, Q ) and (ii) perG G G G forming the operations in f ( ) ), using the complex set boundaries *G ( ju, Q ). G G TSDs of transfer functions can be synthesized in several ways. For block diagrams whose blocks contain disjoint uncertain parameters, TSDs are synthesized by block diagram algebra. To illustrate, consider a block diagram with two uncertain transfer functions G (s, q ) and G (s, q ) cascaded in the forward loop and the uncer tain transfer function H(s, q ) in the feedback loop. With the input º(s) and the output ½(s), the associated transfer function G(s)"½(s)/º(s) can be expressed as G (s, q ) G (s, q ) G(s, q)" 1#G (s, q )G (s, q )H(s, q ) 1 " (4) G (s, q )G (s, q )#H(s, q ) , which has the desired tree-structured form. In addition to block diagrams whose blocks contain independent uncertain parameters, certain repetitive vibratory systems also lead to a TSD (Ackermann, 1991). Furthermore, an algorithm is available to extract a TSD from a characteristic polynomial or transfer function (Sienel, 1992). However, rather than employ an algorithm (which may not always succeed) to identify a TSD, it is preferable to obtain a tree-structured transfer function directly from a model, with as few algebraic manipulations as possible. To this end, this paper introduces bond graphs (Karnopp et al., 1990) as a means to synthesize
tree-structured transfer functions of lumped, passive, serially connected elements. This class includes physical models from a variety of domains, and goes well beyond simple block diagrams and repetitive vibratory systems. For example, this paper will provide techniques to synthesize a tree-structured transfer function of a combined hydraulic—mechanical—electrical system used in power generation.
2. Tree-structured decomposition using bond graphs Bond graphs are recognized as useful way to model systems, but have not been known to provide any particular advantage in a robust control context. However, previous work in (1) impedance relations in bond graphs (Karnopp and Rosenberg, 1968) and (2) the use of bond graphs in design synthesis (Redfield and Krishnan, 1993) provide a direct means to synthesize tree-structured transfer functions from bond graphs. While the use of bond graphs to synthesize these transfer functions will be developed, space limitations preclude more than a cursory explanation of the bond graph method itself, and readers are advised to consult the supplied references for more detail. The lumped model of the DC motor, torsional spring, and inertia pair in Fig. 1 has a corresponding bond graph representation, which is shown in Fig. 2.
Fig. 1. DC motor with compliant load.
Fig. 2. Bond graph for DC motor and flexible shaft.
B. H. Wilson, B. Eryilmaz/Automatica 35 (1999) 755—760
The lumped model and corresponding bond graph will be used to develop and illustrate the tree-structured transfer function synthesis technique. Note that the oneport impedance elements in the bond graph (e.g., J s) replace the usual elements (e.g., I : J ). The impedance J s for bond 6 in Fig. 2 implies that e /f "J s, where e and f represent effort and flow. The inertia impedance presented above comes from a set of equivalent element impedances, which are introduced in Karnopp and Rosenberg (1968). In mechanical translational systems, impedance is the force divided by velocity; in electrical systems impedance is the voltage divided by current. Other physical domains have their own definitions of impedance. In addition to the element impedances, junction-structure (two-port) impedance equivalents are also available (Redfield and Krishnan, 1993). For the bond graph in Fig. 2, and all bond graphs, several impedance identities apply, and these will now be derived. In a bond graph, a one-junction specifies that the sum of the efforts directed into the junction minus the sum of those leaving the junction equals zero. The one junction also implies that all bonds that are attached to the junction have the same flow. This is a direct generalized analogy of Kirchoff’s voltage law. In Fig. 2, the first one junction corresponds to the motor armature circuit. The remaining one junctions apply a torque summation to each inertia. As efforts sum to zero and each attached bond has the same flow, we obtain the following impedance (effort divided by flow) relation: L
e L e L G G" G& z "0 G f f G G G (one-junction impedance law),
(5)
where n corresponds to the quantity of bonds attached to the one junction to which this equivalence is applied. For zero junctions the roles of efforts and flows are reversed. Here, the sum of the flows entering and exiting a zero-junction equals zero, and all bonds attached to the junction has the same effort. A zero-junction is used to model parallel electrical circuits and mechanical elements with common forces or torques. In Fig. 2, a zero-junction is used to connect the torsional compliance to the two adjacent inertias; the sign convention of bonds 8 and 10 is such that the torque created by the compliance is applied equal and opposite to the two inertias. For the corresponding zero-junction impedance relation, we have L
g L f L G G" G & z\"0 G e e G G G (zero-junction impedance law),
(6)
where n refers to the quantity of bonds attached to the zero-junction.
757
A gyrator term relates efforts and flows on opposite sides of a gyrator, a DC motor. For the gyrator in Fig. 2, the following apply: e "K f ,
e f " . K
Dividing the first of these terms by the second, we obtain K z " (gyrator impedance law) z
(7)
which can be generalized to all gyrators. Finally, for transformers, the impedance relation z "mz (transformer impedance law)
(8)
where m is the transformer modulus; see Karnopp and Rosenberg (1968) for a derivation. This paper describes a procedure for synthesizing system impedance or admittance operators that have a TSD. The impedance relations provided in Eqs. (5)—(8) provide the machinery to obtain a tree-structured impedance operator from a bond graph. The next section demonstrates the use of these relations to synthesize a tree-structured transfer functions of two input—output pairs from the plant in Fig. 1.
3. Example of tree-structured transfer function synthesis Let us assume that we are interested in the admittance transfer function as seen by the voltage applied to the motor in Fig. 1 and that every parameter in the system has some uncertainty. If we wish to analyze the value set of this admittance at various driving frequencies, there exists a need to express the corresponding transfer function in such a manner that value set synthesis will be facilitated. Using the equivalences derived in the previous section, we can readily synthesize a tree-structured admittance function. Working from right to left in Fig. 2, the impedance equivalences provide the following relations: z "z #z "J s#B ,
(9a)
s z\"z\#z\" #z\ , K
(9b)
z "z #z #z "J s#B #z ,
(9c)
K z " R, z
(9d)
z "z #z #z "¸ s#R #z ,
(9e)
and the admittance y equals the reciprocal of z . By substituting z into z , z into z , and so on, the
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following admittance tree-structured transfer function results: 1 I (s) "½ (s)" . » (s) K R ¸ s#R # 1 J s#B # s 1 # K J s#B Q (10) When s"ju is substituted into Eq. (10) and parameter uncertainty is considered, a number of the sub-value sets in this equation will form simple Kharitonov rectangles. Thus for overall value set synthesis, only three complex set additions and four complex set inversions are required. The admittance transfer function (10) has the desired tree-structured form with disjoint uncertain parameters. For comparison, the same transfer function expressed as a conventional ratio of polynomials becomes ½ (s)" B B #K (J #J ) K (B #B ) (s#(( J>J ( ) s#( J JQ )s# Q J J ) * , s#a s#a s#a s#a where K (K#R (B #B )) a " ¸J J K (¸ (B #B )#R (J #J ))#B (R B #K) a " ¸J J ¸ (K (J #J )#B B )#R (B J #J B )#KJ a" ¸J J R J J #¸ (B J #J B ) a " (11) ¸J J The transfer function written in the conventional form has no obvious uncertainty structure; hence, no convenient testing set exists for Eq. (11). Therefore, the only available approaches to analyzing the value set are either gridding all eight uncertain parameters or overbounding the numerator and denominator so that they match a form for which results are available. The TSD in Eq. (10) obviates the need to apply these methods.
4. Discussion The use of a bond graph to synthesize a tree-structured transfer function is illustrated in the previous section. As bond graphs are a domain-independent modeling language for lumped physical elements, the procedures developed here can be applied to a wide variety of systems. The example demonstrates the recursive nature in which element impedances are used to synthesize system trans-
fer functions. In the case that the input—output variables are associated with a collocated sensor—actuator pair, each impedance is only used once in generating a treestructured transfer function, and the recursion is guaranteed to produce a transfer function with disjoint (nonrepeating) parameters. The synthesis and structure of Eq. (10) demonstrate this. The same guarantee cannot be provided for input—output variables associated with a noncollocated sensor—actuator pair. In such cases a tree-structured transfer function can still be synthesized, but disjoint parameters cannot be guaranteed. Two aspects of the procedure developed here merit clarification. The first concerns the potential confusion between the continued fractions encountered here and those found in network synthesis. The second concerns the need to use bond graphs to synthesize a tree-structured transfer function, given that block diagrams also constitute a modeling language. 4.1. Continued fractions: bond graphs and network synthesis Let us consider first the synthesis of a tree-structured transfer function such as Eq. (10). Bond-graph-based TSD begins with a symbolic physical model. A symbolic bond graph representation of this model is created, and energy-based impedance equivalences, Eqs. (5)—(8), are used to synthesize a symbolic transfer function with a continued fraction form. The process begins with an unparameterized physical model and concludes with a unique, exact symbolic tree-structured transfer function. Contrast this approach with the process of network synthesis (Baher, 1984). Network synthesis begins with a transfer function with numeric coefficients. Various mathematical expansion techniques — whose outcome depends on the coefficients — are employed to synthesize nonunique, approximate continued-fraction realizations of the transfer function, such as the Foster (1,2) and Cauer (1,2) realizations. The intent of these realizations is to synthesize a circuit whose behavior will approximate that of the original numeric transfer function. The expansions cannot be applied to a symbolic model. 4.2. Bond graphs and block diagrams Both bond graphs and block diagrams can be used to represent system dynamics and to synthesize treestructured transfer functions. This raises the question: What advantages, if any, do bond graphs have over block diagrams in synthesizing tree-structured transfer functions? The procedure to synthesize a tree-structured transfer function from a bond graph is now well known: input— output variables are identified, a system bond graph model is synthesized, and impedance equivalents are
B. H. Wilson, B. Eryilmaz/Automatica 35 (1999) 755—760
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Fig. 3. Block diagram for DC motor and flexible drive train.
substituted recursively — in a single pass — to synthesize a transfer function. A simple, direct procedure suffices to synthesize the tree-structured transfer function, and during this process one can work with available symbols, z in the example of result, instead of the actual impe dance that corresponds to this term. This is especially important as terms become more complicated. A block diagram approach to TSD begins with a block diagram model of a system, which will generally be more complicated than a bond graph of the same system. This is due to the block diagram signals carrying only effort or flow information, whereas the bonds of a bond graph carry both effort and flow. A comparison between the bond graph model in Fig. 2 and its block diagram equivalent in Fig. 3 illustrates the more concise representation of system dynamics that a bond graph provides. This conciseness is even more evident as behavior representing compliant gear pairs, viscous friction in compliant elements, transformers, and gyrators is added to a model. Block diagrams can, of course, be made more compact using intermediate simplifications, but this requires additional effort on the part of the designer and may, in fact, make a block diagram difficult to interpret. Beyond the issue of conciseness, however, block diagram simplification must be performed to generate a tree-structured transfer function. This simplification requires time and skill, more so when the block diagram has multiple feedback loops. Let us consider the process of synthesizing I(s)/» (s) using Fig. 3. Four loops must GL be manipulated to synthesize this transfer function. During the simplification process, the designer must express each intermediate term as a tree-structured transfer function, which is not the usual form. Furthermore, to reduce the bookkeeping during this process, the designer must invent his or her own notation for the simplified blocks, unlike the symbols that are inherently available during bond-graph-based TSD.
5. Conclusion A body of literature on the TSD describes applications of tree-structured transfer functions, and methods are
available for analyzing these functions (Ackermann and Sienel; 1991; Rantzer and Gulman 1991). As the engineering community has found applications for tree-structured transfer functions, a procedure that simplifies their synthesis should be useful. A novel procedure for synthesizing such functions, applicable to a broad class of systems, is the principal contribution of this paper. The procedure developed here is built around a bond graph representation of system dynamics and impedance equivalences that are applied to a bond graph. The class of systems considered encompasses variable pairs of lumped, passive systems with no internal loops. When the input—output variables originate from a collocated sensor—actuator pair, the procedure guarantees a treestructured transfer function with disjoint parameters. The procedure will also generate a TSD for noncollocated pairs, but the disjoint property cannot be guaranteed. Regarding collocated pairs, as the impedance or admittance transfer functions synthesized in this paper result from a collection of passive generalized inductive, capacitive, resistive, transformer, and gyrator components, the resulting frequency response function (impedance or admittance) is known to be positive real. Further analysis of the transfer functions synthesized here and a demonstration of value set synthesis are beyond the scope of this note. However, the work presented here can be extended by identifying robust system properties (based on known impedance properties) and by applying tree-structured transfer functions to robust control analysis and design problems.
Acknowledgements The authors wish to thank Professor Robin Redfield, Professor Chris Hollot, Professor Bahram Shafai, Dr. Wolfgang Sienel and various anonymous reviewers for helpful comments regarding this paper. The authors also acknowledge the support of the U.S. NSF, Grant CMS-9410144.
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References Ackermann, J. (1991). Uncertainty structures and robust stability analysis. Proce. European Control Conf., Grenoble, France. Ackermann, J. (1994). Robust control: Systems with uncertain physical parameters. Berlin: Springer. Ackermann, J., & Sienel, W. (1991). On the computation of value sets for robust stability analysis. Proc. European Control Conf., Grenoble, France. Baher, H. (1984). Synthesis of electrical networks. New York: Wiley. Barmish, B. R., Ackermann, J. E., & Hu, H. (1990). The tree structured decomposition: A new approach to robust stability analysis. Proc. Conf. on Information Science and Systems, Princeton, NJ: Princeton University. Fu, M., Dasgupta, S., & Blondel, V. (1995). Robust stability under a class of nonlinear parametric perturbations. IEEE ¹rans. Automat. Control, 40(2), 213—223.
Karnopp, D., Margolis, D., & Rosenberg, R. C. (1990). System dynamics: a unified approach. New York, NY: Wiley. Karnopp, D., & Rosenberg, R. (1968). Analysis and simulation of multiport systems. Cambridge, MA: MIT Press. Rantzer, A., & Gutman, P. (1991). Algorithm for addition and multiplication of value sets of uncertain transfer functions. Proc. 30th Conf. Decision Control, Brighton, England. Redfield R. C., & Krishnan, S. (1993). Dynamic system synthesis with a bond graph approach: Part I — Synthesis of one-port impedances. J. Dyn. Systems Measurement Control, 115, 357—363. Sienel, W. (1992). Algorithms for tree structured decomposition. Proc. 31st IEEE Conf. on Decision and Control, Tuscon, AZ, pp. 739—740. Tempo, R., & Blanchini, F. (1996). Robustness analysis with real parametric uncertainty. In W. S. Levine (Ed), ¹he Control Handbook, (Ch. 28, pp. 495—505). Boca Raton, FL; CRC Press.
Technical Communique
Simplified tree-structured decomposition using bond graphs Bruce H. Wilson*, Bora Eryilmaz Department of Mechanical, Industrial, and Manufacturing Engineering, Northeastern University, Boston, MA 02115, USA Received 13 October 1998; received in final form 3 November 1998
Abstract The value set of an uncertain transfer function facilitates a number of robust control problems. For the general case, value set synthesis is aided when the transfer function is decomposed into a tree structure with (mostly) disjoint uncertain parameters. We have developed a general and direct technique for obtaining such transfer functions. Bond graphs provide the machinery for the new technique. The method developed here simplifies tree-structured transfer function synthesis for physically derived symbolic plant models. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Bond graph; Tree-structured decomposition; Value set
1. Introduction Many robust control applications require the complexplane region that contains the frequency response of an uncertain transfer function G(s, q), where q represents a vector of uncertain parameters. A frequency-dependent region known as the value set defines the response envelope of this uncertain transfer function. The value set is defined by G( ju, Q)&+G( ju, q)"q3Q,LC,
(1)
where Q represents the known subspace that contains the uncertain parameter vector. The value set finds application in evaluating a given uncertain transfer function’s ability to satisfy the Nyquist criterion, in synthesizing QFT templates, and in determining a frequency response envelope. One means to determine a value set is to grid the uncertain parameters and evaluate the frequency response for all possible combinations. The grid method, however, not only introduces a combinatoric explosion into the analysis, but also underbounds the true value set. To address the combinatoric explosion imposed by gridding, researchers have worked to reduce the number of parameter sets that must be evaluated to synthesize ————— * Corresponding author. Tel.: (617)373 3808 fax: (617) 373-2921; e-mail: [email protected]. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Peter Dorato.
a value set. The objective is to identify testing sets, finite subsets of the original uncertainty space, which are used for synthesizing value sets of uncertain polynomials and uncertain transfer functions, provided that their uncertainty structures fit into specific forms. While improved analysis methods of completely general uncertain transfer functions remains an open topic, useful techniques exist for certain classes of transfer functions. A quite general transfer function form, for which a testing set can be found, is provided in Fu et al. (1995). This form is given by L h(s, c)"1#g (s) (p (s)#c2P (s))IG, G G G G
(2)
where g (s) and p (s) are real scalar functions and poly G nomials in s, c 3! LR, represent a partition of c, P (s) G G G are real vector polynomials with dimension N, and k are G nonzero integers (positive or negative). Fu et al. (1995) cite process control, where cascaded subplants with independent uncertainty comprise the overall plant, as an application where Eq. (2) is useful. Note that this form also captures independent, affine, and multilinear uncertainty structures, as well as independent real zero, pole, and gain variations and complex zero and pole variations. More background on this topic is available in Ackermann (1994) and Tempo and Blanchini (1996). Though the structure in Eq. (2) provides considerable generality, it will not be possible to express the characteristic polynomials and transfer functions associated with
0005-1098/99/$—see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 0 0 0 5 - 1 0 9 8 ( 9 8 ) 0 0 2 2 6 - X
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B. H. Wilson, B. Eryilmaz/Automatica 35 (1999) 755—760
many robust control engineering problems in this form or in other special forms mentioned above. While any polynomial or transfer function can be forced, through overbounding, to match a special form, this processes introduces conservativeness. The tree-structured decomposition (TSD) increases the quantity of uncertain polynomials and transfer functions whose value sets can be readily synthesized without introducing undesirable conservativeness (Ackermann, 1994; Barmish et al., 1990). TSD is a process of expressing an uncertain polynomial or uncertain transfer function as an algebraic function of simpler terms. Ideally, the coefficients of these terms will originate from disjoint subsets of the uncertain parameters, although this is not always possible. When the subsets are disjoint, value set synthesis is simplified. For uncertain transfer functions, our focus, the intent is to express G(s, q) as G(s, q)"f (G(s, q ), G(s, q ), 2 , G(s, q )) L with q 5q " for iOj, (3) G H where f ( ) ) includes simple algebraic operations (addition, multiplication, inversion) among its variables. If G(s, q) can be decomposed in this manner, the value set G( ju, Q) can be synthesized by (i) determining the boundary of each G ( ju, Q ), *G ( ju, Q ) and (ii) perG G G G forming the operations in f ( ) ), using the complex set boundaries *G ( ju, Q ). G G TSDs of transfer functions can be synthesized in several ways. For block diagrams whose blocks contain disjoint uncertain parameters, TSDs are synthesized by block diagram algebra. To illustrate, consider a block diagram with two uncertain transfer functions G (s, q ) and G (s, q ) cascaded in the forward loop and the uncer tain transfer function H(s, q ) in the feedback loop. With the input º(s) and the output ½(s), the associated transfer function G(s)"½(s)/º(s) can be expressed as G (s, q ) G (s, q ) G(s, q)" 1#G (s, q )G (s, q )H(s, q ) 1 " (4) G (s, q )G (s, q )#H(s, q ) , which has the desired tree-structured form. In addition to block diagrams whose blocks contain independent uncertain parameters, certain repetitive vibratory systems also lead to a TSD (Ackermann, 1991). Furthermore, an algorithm is available to extract a TSD from a characteristic polynomial or transfer function (Sienel, 1992). However, rather than employ an algorithm (which may not always succeed) to identify a TSD, it is preferable to obtain a tree-structured transfer function directly from a model, with as few algebraic manipulations as possible. To this end, this paper introduces bond graphs (Karnopp et al., 1990) as a means to synthesize
tree-structured transfer functions of lumped, passive, serially connected elements. This class includes physical models from a variety of domains, and goes well beyond simple block diagrams and repetitive vibratory systems. For example, this paper will provide techniques to synthesize a tree-structured transfer function of a combined hydraulic—mechanical—electrical system used in power generation.
2. Tree-structured decomposition using bond graphs Bond graphs are recognized as useful way to model systems, but have not been known to provide any particular advantage in a robust control context. However, previous work in (1) impedance relations in bond graphs (Karnopp and Rosenberg, 1968) and (2) the use of bond graphs in design synthesis (Redfield and Krishnan, 1993) provide a direct means to synthesize tree-structured transfer functions from bond graphs. While the use of bond graphs to synthesize these transfer functions will be developed, space limitations preclude more than a cursory explanation of the bond graph method itself, and readers are advised to consult the supplied references for more detail. The lumped model of the DC motor, torsional spring, and inertia pair in Fig. 1 has a corresponding bond graph representation, which is shown in Fig. 2.
Fig. 1. DC motor with compliant load.
Fig. 2. Bond graph for DC motor and flexible shaft.
B. H. Wilson, B. Eryilmaz/Automatica 35 (1999) 755—760
The lumped model and corresponding bond graph will be used to develop and illustrate the tree-structured transfer function synthesis technique. Note that the oneport impedance elements in the bond graph (e.g., J s) replace the usual elements (e.g., I : J ). The impedance J s for bond 6 in Fig. 2 implies that e /f "J s, where e and f represent effort and flow. The inertia impedance presented above comes from a set of equivalent element impedances, which are introduced in Karnopp and Rosenberg (1968). In mechanical translational systems, impedance is the force divided by velocity; in electrical systems impedance is the voltage divided by current. Other physical domains have their own definitions of impedance. In addition to the element impedances, junction-structure (two-port) impedance equivalents are also available (Redfield and Krishnan, 1993). For the bond graph in Fig. 2, and all bond graphs, several impedance identities apply, and these will now be derived. In a bond graph, a one-junction specifies that the sum of the efforts directed into the junction minus the sum of those leaving the junction equals zero. The one junction also implies that all bonds that are attached to the junction have the same flow. This is a direct generalized analogy of Kirchoff’s voltage law. In Fig. 2, the first one junction corresponds to the motor armature circuit. The remaining one junctions apply a torque summation to each inertia. As efforts sum to zero and each attached bond has the same flow, we obtain the following impedance (effort divided by flow) relation: L
e L e L G G" G& z "0 G f f G G G (one-junction impedance law),
(5)
where n corresponds to the quantity of bonds attached to the one junction to which this equivalence is applied. For zero junctions the roles of efforts and flows are reversed. Here, the sum of the flows entering and exiting a zero-junction equals zero, and all bonds attached to the junction has the same effort. A zero-junction is used to model parallel electrical circuits and mechanical elements with common forces or torques. In Fig. 2, a zero-junction is used to connect the torsional compliance to the two adjacent inertias; the sign convention of bonds 8 and 10 is such that the torque created by the compliance is applied equal and opposite to the two inertias. For the corresponding zero-junction impedance relation, we have L
g L f L G G" G & z\"0 G e e G G G (zero-junction impedance law),
(6)
where n refers to the quantity of bonds attached to the zero-junction.
757
A gyrator term relates efforts and flows on opposite sides of a gyrator, a DC motor. For the gyrator in Fig. 2, the following apply: e "K f ,
e f " . K
Dividing the first of these terms by the second, we obtain K z " (gyrator impedance law) z
(7)
which can be generalized to all gyrators. Finally, for transformers, the impedance relation z "mz (transformer impedance law)
(8)
where m is the transformer modulus; see Karnopp and Rosenberg (1968) for a derivation. This paper describes a procedure for synthesizing system impedance or admittance operators that have a TSD. The impedance relations provided in Eqs. (5)—(8) provide the machinery to obtain a tree-structured impedance operator from a bond graph. The next section demonstrates the use of these relations to synthesize a tree-structured transfer functions of two input—output pairs from the plant in Fig. 1.
3. Example of tree-structured transfer function synthesis Let us assume that we are interested in the admittance transfer function as seen by the voltage applied to the motor in Fig. 1 and that every parameter in the system has some uncertainty. If we wish to analyze the value set of this admittance at various driving frequencies, there exists a need to express the corresponding transfer function in such a manner that value set synthesis will be facilitated. Using the equivalences derived in the previous section, we can readily synthesize a tree-structured admittance function. Working from right to left in Fig. 2, the impedance equivalences provide the following relations: z "z #z "J s#B ,
(9a)
s z\"z\#z\" #z\ , K
(9b)
z "z #z #z "J s#B #z ,
(9c)
K z " R, z
(9d)
z "z #z #z "¸ s#R #z ,
(9e)
and the admittance y equals the reciprocal of z . By substituting z into z , z into z , and so on, the
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following admittance tree-structured transfer function results: 1 I (s) "½ (s)" . » (s) K R ¸ s#R # 1 J s#B # s 1 # K J s#B Q (10) When s"ju is substituted into Eq. (10) and parameter uncertainty is considered, a number of the sub-value sets in this equation will form simple Kharitonov rectangles. Thus for overall value set synthesis, only three complex set additions and four complex set inversions are required. The admittance transfer function (10) has the desired tree-structured form with disjoint uncertain parameters. For comparison, the same transfer function expressed as a conventional ratio of polynomials becomes ½ (s)" B B #K (J #J ) K (B #B ) (s#(( J>J ( ) s#( J JQ )s# Q J J ) * , s#a s#a s#a s#a where K (K#R (B #B )) a " ¸J J K (¸ (B #B )#R (J #J ))#B (R B #K) a " ¸J J ¸ (K (J #J )#B B )#R (B J #J B )#KJ a" ¸J J R J J #¸ (B J #J B ) a " (11) ¸J J The transfer function written in the conventional form has no obvious uncertainty structure; hence, no convenient testing set exists for Eq. (11). Therefore, the only available approaches to analyzing the value set are either gridding all eight uncertain parameters or overbounding the numerator and denominator so that they match a form for which results are available. The TSD in Eq. (10) obviates the need to apply these methods.
4. Discussion The use of a bond graph to synthesize a tree-structured transfer function is illustrated in the previous section. As bond graphs are a domain-independent modeling language for lumped physical elements, the procedures developed here can be applied to a wide variety of systems. The example demonstrates the recursive nature in which element impedances are used to synthesize system trans-
fer functions. In the case that the input—output variables are associated with a collocated sensor—actuator pair, each impedance is only used once in generating a treestructured transfer function, and the recursion is guaranteed to produce a transfer function with disjoint (nonrepeating) parameters. The synthesis and structure of Eq. (10) demonstrate this. The same guarantee cannot be provided for input—output variables associated with a noncollocated sensor—actuator pair. In such cases a tree-structured transfer function can still be synthesized, but disjoint parameters cannot be guaranteed. Two aspects of the procedure developed here merit clarification. The first concerns the potential confusion between the continued fractions encountered here and those found in network synthesis. The second concerns the need to use bond graphs to synthesize a tree-structured transfer function, given that block diagrams also constitute a modeling language. 4.1. Continued fractions: bond graphs and network synthesis Let us consider first the synthesis of a tree-structured transfer function such as Eq. (10). Bond-graph-based TSD begins with a symbolic physical model. A symbolic bond graph representation of this model is created, and energy-based impedance equivalences, Eqs. (5)—(8), are used to synthesize a symbolic transfer function with a continued fraction form. The process begins with an unparameterized physical model and concludes with a unique, exact symbolic tree-structured transfer function. Contrast this approach with the process of network synthesis (Baher, 1984). Network synthesis begins with a transfer function with numeric coefficients. Various mathematical expansion techniques — whose outcome depends on the coefficients — are employed to synthesize nonunique, approximate continued-fraction realizations of the transfer function, such as the Foster (1,2) and Cauer (1,2) realizations. The intent of these realizations is to synthesize a circuit whose behavior will approximate that of the original numeric transfer function. The expansions cannot be applied to a symbolic model. 4.2. Bond graphs and block diagrams Both bond graphs and block diagrams can be used to represent system dynamics and to synthesize treestructured transfer functions. This raises the question: What advantages, if any, do bond graphs have over block diagrams in synthesizing tree-structured transfer functions? The procedure to synthesize a tree-structured transfer function from a bond graph is now well known: input— output variables are identified, a system bond graph model is synthesized, and impedance equivalents are
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Fig. 3. Block diagram for DC motor and flexible drive train.
substituted recursively — in a single pass — to synthesize a transfer function. A simple, direct procedure suffices to synthesize the tree-structured transfer function, and during this process one can work with available symbols, z in the example of result, instead of the actual impe dance that corresponds to this term. This is especially important as terms become more complicated. A block diagram approach to TSD begins with a block diagram model of a system, which will generally be more complicated than a bond graph of the same system. This is due to the block diagram signals carrying only effort or flow information, whereas the bonds of a bond graph carry both effort and flow. A comparison between the bond graph model in Fig. 2 and its block diagram equivalent in Fig. 3 illustrates the more concise representation of system dynamics that a bond graph provides. This conciseness is even more evident as behavior representing compliant gear pairs, viscous friction in compliant elements, transformers, and gyrators is added to a model. Block diagrams can, of course, be made more compact using intermediate simplifications, but this requires additional effort on the part of the designer and may, in fact, make a block diagram difficult to interpret. Beyond the issue of conciseness, however, block diagram simplification must be performed to generate a tree-structured transfer function. This simplification requires time and skill, more so when the block diagram has multiple feedback loops. Let us consider the process of synthesizing I(s)/» (s) using Fig. 3. Four loops must GL be manipulated to synthesize this transfer function. During the simplification process, the designer must express each intermediate term as a tree-structured transfer function, which is not the usual form. Furthermore, to reduce the bookkeeping during this process, the designer must invent his or her own notation for the simplified blocks, unlike the symbols that are inherently available during bond-graph-based TSD.
5. Conclusion A body of literature on the TSD describes applications of tree-structured transfer functions, and methods are
available for analyzing these functions (Ackermann and Sienel; 1991; Rantzer and Gulman 1991). As the engineering community has found applications for tree-structured transfer functions, a procedure that simplifies their synthesis should be useful. A novel procedure for synthesizing such functions, applicable to a broad class of systems, is the principal contribution of this paper. The procedure developed here is built around a bond graph representation of system dynamics and impedance equivalences that are applied to a bond graph. The class of systems considered encompasses variable pairs of lumped, passive systems with no internal loops. When the input—output variables originate from a collocated sensor—actuator pair, the procedure guarantees a treestructured transfer function with disjoint parameters. The procedure will also generate a TSD for noncollocated pairs, but the disjoint property cannot be guaranteed. Regarding collocated pairs, as the impedance or admittance transfer functions synthesized in this paper result from a collection of passive generalized inductive, capacitive, resistive, transformer, and gyrator components, the resulting frequency response function (impedance or admittance) is known to be positive real. Further analysis of the transfer functions synthesized here and a demonstration of value set synthesis are beyond the scope of this note. However, the work presented here can be extended by identifying robust system properties (based on known impedance properties) and by applying tree-structured transfer functions to robust control analysis and design problems.
Acknowledgements The authors wish to thank Professor Robin Redfield, Professor Chris Hollot, Professor Bahram Shafai, Dr. Wolfgang Sienel and various anonymous reviewers for helpful comments regarding this paper. The authors also acknowledge the support of the U.S. NSF, Grant CMS-9410144.
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