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(1)Singular perturbation analysis of a mode initialization algorithm for simulating mode switching systems, long version Krister Edstrom Department of Electrical Engineering Linkoping University, S-581 83 Linkoping, Sweden WWW: http://www.control.isy.liu.se Email: edstrom@isy.liu.se February 9, 1999. REGL. AU. ERTEKNIK. OL TOM ATIC CONTR. LINKÖPING. Report no.: LiTH-ISY-R-2082 Submitted to CDC '98 Technical reports from the Automatic Control group in Linkoping are available by anonymous ftp at the address ftp.control.isy.liu.se. This report is contained in the compressed postscript le 2082.ps.Z..

(2) Singular perturbation analysis of a mode initialization algorithm for simulating mode switching systems, long version1 Krister Edstrom Dept. of Electrical Engineering Linkopings Universitet SE-581 83 Linkoping, Sweden edstrom@isy.liu.se Abstract. i1. An initialization algorithm for the continuous states in mode switching systems is shown to give correct initial values. The mode switching systems are modeled with switched bond graphs, and the proof is based on singular perturbation theory.. i2. Figure 1: A simple electrical example.. 1 Introduction. using switched bond graph has also been introduced 3]. In this paper, the algorithm is presented, and it is shown that the algorithm yields a valid initial state for the new mode.. In this paper we deal with the problem of simulating a subclass of hybrid systems called mode switching systems. A mode switching system can be viewed as a collection of continuous models together with rules for switching between them. An intuitive picture is an automaton with continuous models in the vertices and switching rules as transitions.. It is also shown how this initialization procedure can be interpreted using singular perturbation theory. By replacing the instant changes in the model with fast dynamical behavior, the total dynamics of the model can be divided into one fast part, approximating the instant behavior of the switch, and one slow part. By decreasing, or possibly increasing, a parameter in the description of the fast dynamics, the di erence in time scales between the fast and slow dynamics can be increased. When this di erence becomes large enough, the slow dynamics can be considered constant while the fast dynamics reaches steady state. The initial values are achieved by assuming that the slow dynamics is constant, and that the fast dynamics instantly reaches steady state.. When simulating a physically modeled mode switching system, the state space may change size when changing modes. When the number of states is decreasing, the problem of initializing the new mode becomes nontrivial. From a physical point of view, the initialization rules come from a generalization of the principle of momentum conservation. This generalization includes, e.g., the charge conservation for electrical systems, and volume conservation for hydraulic systems. When the number of states does not change, or when the number of states increases, the initialization is trivial, since the states are continuous.. In this paper it is proven that the bond graph initialization algorithm presented in Section 3 yields exactly the same result as the initialization by dividing the dynamics into two time scales.. Example 1. Consider the two capacitors in Figure 1. When the switch is closed their behavior is determined by the principle of electrical charge conservation, stating that the total amount of charges over the two capacitors is kept constant. Hence this principle should be used to initialize the new mode.. In Section 2, the modeling language is presented. The language used is switched bond graphs, which is an extension of the (classical) bond graph language. In Section 3 the initialization of a new mode is discussed, and the discussion is divided into three parts. It is shown that the initialization is trivial when the number of states increases or is constant across the change of modes. The initialization algorithm for the third case, when the number of states decreases is presented in Section 3.3. It is also shown that the algorithm gives. The use of a generalization of momentum conservation has been discussed earlier in 7]. An initialization algorithm based on the physical behavior for simulation 1 A short version of this paper is published with the same title in 1998 IEEE Conference on Decision and Control.. 1.

(3) conditions, g and g , have to be added. g gives the condition for switching from state to , and g the condition for switching the other way.. initial values for the new mode, i.e., that the algorithm terminates and returns values for the states in the new mode. In Section 4 the algorithm in Section 3.3 is interpreted in a singular perturbation theory framework, and the correctness of the initialization values is conrmed. Finally, an electrical example is presented in Section 5.. EF. In this section the algorithm for initializing a new mode is presented. The algorithm presented here does not cover the general case. There are some assumptions made about the underlying bond graph to make the proofs in Section 4 easier. These assumptions and some denitions are presented in Section 3.1. In the initialization algorithm, the states are assumed to be continuous as far as possible. This assumption relies on the fact that physical systems are continuous even if the structure changes, assuming that no elements that store energy become connected. There is also a tradition when working with di erential equations, that their solutions are continuous if possible.. F. E. If the mode being left is conict free, there are three things that can happen when a switch changes state: There can be a change of causality at an R-element, a storage element or a source. If the change of causality is at a source, the mode entered will be a conict mode. This may indicate an error in the model or the system, but it might also be a valid model. This problem is discussed further in 3].. F. EF. g. Sf:0. Figure 2:. FE. 3 A Mode Initialization Algorithm. Switched bond graphs 8, 2, 4] are an extension of classical bond graphs in the sense that they also allow modeling of instant changes in the system. When using switched bond graphs, the bond graph language is extended with a new element, a switch 9]. With its two di erent states, and , the switch captures the mode switching behavior. Fixed in the -state, the switch is replaced with a zero e ort source, Se:0, and in the state, the switch is replaced with a zero ow source, Sf:0, see Figure 2. The causal assignment rules for the zero e ort and ow sources are the same as for the ordinary sources. This leads to di erent causality in the di erent modes, since the causality of a switch changes between its states. With di erent causality, the continuous models of the di erent modes may have a di erent number of states.. Sw. F. From a switched bond graph, simulation is algorithmic 2] if the initializations are trivial and no chattering occurs. A physically based initialization algorithm for switched bond graphs has been discussed in 3], and the initialization procedure with the mathematically equivalent hybrid bond graphs has been discussed in 7].. The bond graph language 5, 1] is a graphical language for making physical models of systems. It is based on energy conservation, and shows how the energy ows in the system. When using bond graphs, the system is modeled using a few standard elements, where each element describes a certain aspect of the energy behavior, e.g., if it is stored, dissipated, entered or transformed. The mathematical relations in the elements are described using two variables, e ort e and ow f , whose product is of the quantity power, P = e f .. Se:0. EF. E. 2 Switched Bond Graphs. E. FE. When the causality changes at an R-element, both modes will be conict free. This case is discussed in Section 3.2. The case when the change of causality is at a storage element is discussed in Section 3.3.. E. FE. g. If the mode entered is conict free we deal with two cases. The case when the mode being left has dependent storage elements is discussed in Section 3.4, and if the mode being left has no causal conict we are back in the conict free case, discussed in Section 3.2.. F. The two states of a switch element. Causal strokes are shown for the two states.. 3.1 Assumptions and denitions. Note that a switched bond graph can be viewed as a collection of classical bond graphs, each one describing a certain mode. To nd such a classical bond graph, all switches have to be xed in one of the two states, i.e., replaced with sources. There is one classical bond graph for each combination of switch states.. Since the general case is not considered, we will in this section specify the restrictions that have been made. Furthermore some terminology and notation will be introduced. Consider a mode switching system modeled with a switched bond graph. Assume that the bond graph has the following properties.. To complete the description of a switch, two transition 2.

(4) Assumption 1. changing the causality at n1 , i.e., changing causality at the bond, the causality will change at the other node as well. Since there is only one bond between the two nods, it immediately follows that there is a causal path between the nodes, and that it changes when changing causality at n1 .. A1 There are no multi-port elements in the bond graph. A2 There are no loops in the bond graph. A3 There are no transformers or gyrators.. Now consider the case when there is a junction j1 on the other side of b1 . When changing the causality at n1 , the causality at b1 will change. If a causal stroke is moved to the side of the bond closest to j1 , a causal stroke at another bond b2 connected to j1 has to moved away from j1 and vice versa. Note that the two bonds where the causality is changed are in the same causal path both before and after the change of causality.. Assumption A2 simplies the causal propagation algorithm, and we get a better overview of causal changes when changing modes. The third assumption is just to simplify notation. The algorithm can easily be extended to bond graphs with transformers and gyrators. We also need the following three denitions:. Denition 1 (Causal path). If b2 is connected to a node, that node is n2 , and the proof is clear since b1 and b2 are in the same causal path both before and after changing nodes, and since no other bond has changed causality. If there is a junction j2 on the other side of b2, we can repeat the arguments above, introducing a third bond b3 that is connected to j2 and changes causality when b2 changes causality.. Consider the bond graph as being a graph where the bonds are the edges and the bond graph nodes and junctions are the nodes. Let the graph be directed by the causal strokes in the bond graph. If there is a directed path in the graph between two nodes corresponding to bond graph nodes, there is a causal path 10] between these nodes in the bond graph.. The bonds b2 and b3 are in the same causal path since both changes causality. Since b1 and b2 are in the same causal path, we can conclude that b1 b2 and b3 are all in the same causal path both before and after the causality changes. We can also see that the causality has not changed at any other bond.. This denition has to be slightly modied if there are GY-elements in the bond graph 10].. Denition 2 (Causal dependence). Two bond graph elements are causally dependent if there is a causal path between them.. Repeat the procedure for new junctions until reaching a second node. Since there are no loops in the bond graph, the procedure will nally lead to a second node.. Denition 3 (Dependent set). In a bond graph with one storage element s with nonpreferred causality, the dependent set D of elements consists of s and all elements that are causally dependent on s.. For the special case when n1 is a switch the following corollary holds.. The following lemma shows how the causality changes, when changing causality at one element in the bond graph.. Corollary 1. When a switch changes states, there will be one other node that changes causality. There will also be a causal path between this node and the switch in both the mode being left and the mode being entered.. Lemma 1. Consider a bond graph for which Assumptions A1 and A2 hold.. 3.2 From a con ict free mode to a con ict free mode If the change of causality is at an R-element, as shown. When changing the causality at one node, n1 , the causality will change at this node and precisely one other node, n2 .. in Figure 3, the continuous states of the system are the same in both modes, meaning that they have the same physical interpretation. Therefore the two modes can be expressed in state space form with the same states: Mode 1: x_ = f1 (x u) (1) Mode 2: x_ = f2 (x u) (2). Furthermore, there is a causal path between n1 and n2 both before and after changing causality, and when changing modes it is only the causality along this causal path that changes.. Proof: The proof is constructive. First we will con-. sider the degenerate case when the bond b1 next to n1 is connected to a node n2 on the other side as well. When 3.

(5) the storage elements in {D2 . pk k = 2 : : : l2 are the sources in D2 , and pk k = l2 + 1 : : : n2 the sources in {D2 .. Since we have the same states in both modes, we can initialize the states variables in the new mode to get continuous state variables throughout the change of mode. R 1. Sw. Introduce a variable xk for each storage element sk . Since there is a storage element with non-preferred causality in mode M2 , there is one relation at a storage element that yields a di erentiation instead of an integration. This means that all xk k = 1 : : : n1 are not state variables in the overall model.. R Sw. C. 1. C. Figure 3: An example of switching between two conict free modes.. In the same way, introduce one input variable uk for each source pk . To show that all steps in the algorithm are possible to perform, the following lemma is needed.. The initialization rule for this type of transition is hence. x(t) = x(t ). Lemma 2. (3). 0. Consider a bond graph for which Assumptions A1, A2 and A3 hold.. where t is before the change of mode and t is after. Furthermore t t < " " > 0. 0. j. ;. 0. j. Any causal path from s1 , the storage element with nonpreferred causality, in M2 will lead to another storage element of the same type or to a source with causality that is the same as the preferred causality for the storage element.. 8. 3.3 From a con ict free mode to a mode with dependent states. When two states become dependent, in the generic case all state variables can not be continuous. The dependency between the states will lead to algebraic connections between the state variables. The algorithm presented in this section will nd these algebraic relations as well as all states that are continuous throughout the change of mode. The key to nding the relations is the causal propagation. R Sw. 0. Proof:. In the proof it is assumed that s1 is a Celement. For an I-element the proof is dual. We need to show that there are no causal paths from s1 to an Sf-, I- or R-element. Assume that there is a causal path from s1 to an Sfelement. Since there are no gyrators in the bond graph, and since the causality stroke on the bond next to s1 is on the same side of the bond as s1 , Sf-element will have e ort causality due to the assumed causal path between the elements. This can not appear with a proper causality propagation algorithm. By changing causality along the causal path between s1 and the Sf-element according to Lemma 1, a correct causal assignment is achieved, with preferred (e ort) causality at s1 and ow causality at the Sf-element.. R C. Sw. 0. C. Figure 4: An example of switching from a conict free mode to a mode with derivative causality.. We will rst introduce some notation. The mode being left is denoted M1 and the mode being entered M2. We assume that M1 is conict free and that M2 has one storage element with derivative causality. Storage elements in general are denoted with sk , where k is an index. The storage element in M2 with derivative causality will have index 1: s1 . Sources are in the same way denoted with pk .. If there is a causal path between two storage element, s1 and an I-element, where s1 has derivative causality, both elements will have derivative causality according to the assumed causal path between the elements. Both elements get preferred causality if the causality is changed along the causal path according to Lemma 1.. The dependent set in M2 is denoted D2 . s1 is then an element in D2 .. If there is a causal path between s1 and an R-element, the non-preferred causality at s1 can be relaxed by changing causality at the R-element, using the same arguments as above.. Let l1 be the number of storage elements and l2 the number of sources in D2 . s1 is the element in D2 with non-preferred causality, sk k = 2 : : : l1 the other storage elements in D2 , and sk k = l1 + 1 : : : n1 . As a consequence, D2 , the dependent set in M2 , will 4.

(6) 3. Derive an equation x1 = g1s (x2 : : : xl1 ) + g1p (u1 : : : ul2 ) expressing the state of s1 in the states of the other storage elements and the sources in D2 . Rewrite the equation as:. only include storage elements and sources with the same preferred causality. When changing modes, from mode M1 to mode M2 , the values of the state variables in M1 just before the mode change x(t ), are assumed to be known. The algorithm will then give the initial values of the state variables in mode M2, x(t).. x1 g1s (x2 : : : xl1 ) = g1p (u1 : : : ul2 ). 0. ;. The equation is derived using a standard bond graph equation generation algorithm for the given causality. That the equation will be of the form above follows from Lemma 2, by following causal paths from s1 , when using the equation generation algorithm presented in 2]. This equation should be ful

(7) lled immediately when entering the new mode, and can therefore be used to initialize the new mode. In Example 1 this equation states that the voltages across the two capacitors are equal when the switch is closed.. The idea behind the algorithm is that it is the variables in D2 that are discontinuous, and the state variables belonging to storage elements outside D2 are continuous throughout the change of modes. Assuming this, the initialization is easy for the states in {D2 , since they will keep their values throughout the change of mode. The values of the variables in D2 are found using equation generation along causal paths in D2 . To get better understanding of one of the steps in the algorithm we will discuss a certain type of equations that appear when generating equation inside D2 , equations of the following form:. x_ 1 (t) x_ k (t) = f (X (t)). 4. Derive l1 1 equations g2 : : : gl1 , expressing x_ k k 2 : : : l1 in the rate variable x_ 1 , using the causal path between sk and s1 . These equations correspond to Equation 4. The other causal paths leading to sk , can be neglected according to the discussion leading to Equation (6). ;. 2 f. (4). . x1 and xk are two variables corresponding to storage elements in D2 , and X consists of variables corresponding to storage elements in {D2 . The equation is hence a relation between variables in {D2 and derivatives of variables in D2 .. lim0. . !. t ;. f (X (s)) ds = 0. 5. Integrate both sides of Equation 8 across the mode change:. xk = k1 x1 + C k = 2 : : : l1 0. =. Zt. . 0. 0. Zt 0. f (X (s)) ds =. f (X (s)) ds = x1 (t ) xk (t ) 0. . 0. ;. ;. xk (t) k1 x1 (t) = xk (t ) k1 x1 (t ) k = 2 : : : l1 0. ;. !. x1 (t) xk (t) =. (9). From Equation (6) we know that the value of C becomes C = xk (t ) k1 x1 (t ), and l1 1 new initialization equations are achieved:. (5) ;. !. (8). The sign k1 before x_ 1 depends on the junction structure and the direction of the bonds along the causal path.. If using the notation g(t ) = lim 0 g(t ), and the slightly sloppy notation g(t) = lim 0 g(t + ) we see that: 0. g. x_ k = gk (x_ 1 ) = k1 x_ 1 k = 2 : : : l1. We have assumed that the variables in X (t) are continuous across the change of modes, and hence f (X (t)) is continuous across the change of mode. Since the mode change is instantaneous, the value of f (X (t)) will be the same immediately before and immediately after the change of mode. Hence. Z t+. (7). ;. 0. (10). Where x(t ) is the value of a state before the mode change, and x(t) the value after. In Example 1, where k = 2, the equation states that the total charge over the two capacitors is kept constant when the switch is closed. 0. (6). 0. We see that we can disregard the part of the bond graph in {D2 when deriving initial conditions for variables in D2 using relations between derivatives.. 6. Another n l equations are needed to get the initialization complete. Since all storage elements outside D2 have preferred causality, they will be continuous: ;. Algorithm 1 (Initialization algorithm) 1. Propagate causality in mode M2 .. xk (t) = xk (t ) k = l + 1 : : : n. 2. Find D2 , the dependent set in M2 .. 0. 5. (11).

(8) If both bonds are directed either towards or from a junction, there is a change of sign in one of the variables, independent of junction type.. 7. The three sets of equations, Equations (7), (10), and (11), form a system of equations that can be solved, see Lemma 4, to achieve initial values of the new mode.. Assume that one bond is directed towards the junction and the other from the junction. For both types of junction the relations between the variables are: e2 = e1 (16) f2 = f1 (17). We will in Lemma 4 prove that this initialization algorithm gives a solvable set of equations. To do this we need the following result.. Lemma 3. Consider two storage elements of the same kind with a causal path between them. Assume that the half arrows on the bonds next to the storage elements point towards its respective element.. With an odd number of junctions in the path where both bonds in the path are directed either from or towards the junction, the total number of sign changes is odd. Since 11 = ( 1)k , where k is the number of sign changes in one direction and 21 = ( 1)l , where l is the number of sign changes in the other direction, and k + l is odd, we get 11 21 = ( 1)k+l = 1 (18) Hence 11 and 21 are di erent. ;. When using the equation generation algorithm in 2] in one direction along the causal path, an equation relating the two states can be derived: x1 + 12 a2 x2 + f (x u) = 0 where 11 is either +1 or 1, and a2 > 0 is a constant. The term +12 a2 x2 comes from the causal path between the two storage elements considered, and f (x u) comes from all causal path leading to the element with derivative causality from other storage elements and sources. Compare with Equation (7). Another equation relating the state derivatives, the rates, can be achieved by generating equations along the causal path in the other direction and integrate the relation between the derivatives: x1 + 21 x2 = C where 21 is either +1 or 1, and C is a constant. Compare with Equation (10).. ;. ;. ;. ;. Example 2. Look at the two causally dependent elements in Figure 5. Assume that the relations at the C-elements are linear, with coecients c1 and c2 .. C2 C1 0 1 Figure 5: Two causally dependent C-elements.. If generating the equation from C1 the result will be:. ;. x1 + cc1 x2 = 0. The two signs are dierent, i.e., 12 = 21 .. (19). 2. ;. If generating the equation from C2 and integrating it the result will be: x1 x2 = C (20) We see that 11 = +1 and 21 = 1. Hence 11 = 21 .. Proof: Consider the energy bonds in the causal path. between the storage elements. Since the two bonds next to the storage elements point in di erent directions, there has to exist an odd number junctions in the path where both bonds in the path are directed either from or towards the junction.. ;. ;. ;. Lemma 4. The system of equations generated in Algorithm 1 is solvable when assuming that Equation (7) is linear: g1s (x2 : : : xl1 ) = 12 a2 x2 + : : : + 1l al1 xl1 (21) (22) g1p (u1 : : : ul2 ) = b1 u1 + : : : + bl2 ul2 where ak 0 k = 2 : : : l1 and each sign 1k k = 2 : : : l1 is either +1 or 1.. Now consider one of the junctions in the path. Assume that both bonds are directed either towards or from the junction. Let e1 and f1 be the variables corresponding to one of the bonds and e2 and f2 the variables corresponding to the other bond. If the junction is a 1-junction, the relation between the variables are: e2 = e1 (12) f2 = f1 (13) And for a 0-junction the relations are: e2 = e1 (14) f2 = f1 (15). . ;. ;. Proof:. If we look at Equations (7), (10) and (11) and rewrite them in matrix form, we get the following structure: T x(t) = xm (t ) (23) 0. ;. 6.

(9) where T. 2 1 ; ; 21 6 . 6 6 . = 66 ; .l1 1 6 4 0.. ... 0. and. 12 a2 ;. ::: 0 3 ::: 0 7. ::: :::. 1 0. 00 10. ::: 0 7 ::: 0 7 5. :::. 0. 00. 1. 0 0. 0 0. 0. 0. .. .. .. .. 2 x1 (t) 3 x2 (t) 6 x3.(t) 7 6 .. 777 6 6 xl1 (t) 7 x(t) = 6 6 xl1 +1 (t) 7 6 7 6 (t ) 7 4 xl1 +2 .. 5. . xn1 (t). 0. 00 00. 13 a3 ::: ; 0 :::. .. .. .... .. .. .... 1l1 al1. .. .. .. .. . . .. 77 . . . .7. .. .. Mode 2:. x_ = f2 (x u) where x = (xTa xb )T .. (24). .. .. . . .. . . :::. 1.. Since the derivative of x is calculated in a new way in mode 2, it may be discontinuous, but a variable with discontinuous and bounded derivative is continuous. Hence the state variables are continuous.. 2 g1p (u1 (t )::: xl (t )) 3 2 ; 21 x1 (t )+x2 (t ) 7 6 6 ; 31 x1 (t. )+x3 (t ) 7 77 6 .. 6 77 (25) 6 0 ; x ( t )+ x ( t ) l1 1 1 l1 xm (t ) = 6 77 6 x (t ) l +1 1 6 75 6 xl1 +2 (t ) 4 .. 0. 0. 0. 0. 0. 0. 0. One problem is that the state variable xb in mode 2 is not a state variable in mode 1, and it has to be known to perform the initialization. In a bond graph frame this is not a problem since xb is a state variable in the constitutive relation for a storage element. If the values at all such variables are calculated during simulation, the initialization procedure will work.. 0. 0 0. .. xn1 (t ) 0. The right hand side of Equation (23) only contain old values of x1 : : : xn1 and is therefore known. The system of equations is solvable if there are n1 linearly independent equations on the left hand side.. R Sw. To prove full rank of T , we will try to achieve an upper triangular matrix with positive diagonal elements by eliminating the elements in positions (2 1) (3 1) : : : (l1 1). We do this by adding multiples of columns 2 through l1 to column 1. When all these elements are eliminated, the element in position (1 1) has to be positive.. 0. R C. Sw. 0. C. Figure 6: Switching from a mode with derivative causality to a conict free mode.. 4 Singular Perturbation Analysis. To eliminate the element in position (2 1), add 21 times column 2 to column 1. The element in position (1 1) will then be T11+ = 1 21 12 a2 . Since, from Lemma 3, 21 12 = 1 T11+ = 1 + a2 . We know that a2 0, therefore the value of T11 has increased (or is the same). By eliminating the other elements, with the same arguments, we see that the nal value of T11 1.. The initialization procedures described in Section 3.2 and 3.4 has been motivated by arguing that the states are continuous because the state derivatives are piecewise continuous and bounded. Here the initialization algorithm in Section 3.3 will be analyzed.. ;. ;. . . We have in Section 3.3 presented an initialization algorithm for modes with derivative causality at one storage element and shown that the algorithm yields initial values for the new mode. The next step is to argue that the achieved values are correct by using the theory of singular perturbations 6]. We will start by considering the instant change of the state variables as a very fast continuous change. Then there will be two time scales in the model one fast describing the dynamics earlier modeled as instant changes, and one slow, describing the rest of the dynamics. If we separate the two time scales enough, the slow time scale can be considered constant while the fast dynamics reaches steady state. We will show that the steady state value of the fast dynamics and the constant value of the slow dynamics are the values achieved by Algorithm 1 in Section 3.3.. To summarize: By column operations we can remove the elements below the diagonal. For each element we remove we know that we will not decrease the value of the element in position (1 1). Hence we will get an upper diagonal matrix with non-zero elements in the diagonal. The matrix T will therefore have full rank, and be invertible.. 3.4 From mode with dependent states to a con ict mode.. When the derivative causality is relaxed at a storage element, the number of state variables increases with one. In the continuous mathematical description this means that an algebraic relation between variables has been replaced with a dynamic relation. Mode 1:. x_ a = f1a (xa x_ b u) xb = f1b (xa xb u). (28). To come to this conclusion, we will rst replace the switch that changes position with a linear R-element, where e = R f , and assume that the constant R changes value when the mode change appear. This will. (26) (27) 7.

(10) 4.2 The main theorem. lead to a model where the causality does not change when changing modes. We have transferred the mode switching model to a corresponding time varying continuous model. Hence the states will be continuous throughout the mode change.. The change of basis, introduced to get the system in a form where it is clear that the system is singularly perturbed, uses a transformation matrix that is derived from Equations (7), (10) and (11). It is the same matrix that in Lemma 4 is proved to have full rank. With the new states y = (y1 : : : yn )T , and the old states x = (x1 : : : xn )T , we have the following transformation:. In order to simplify the discussion in this section we will only consider C-elements as storage elements in the dependent set D. Therefore we will give the parameter R in the R-element in the corresponding classical bond graph a small value " in the new mode. The proofs for I-elements will be dual, and the parameter value of the added R-element has to be R = 1=", where " is small.. y = Tx. (29). where T is found in Equation (24). With this change of basis the following theorem holds.. When generating equations from the bond graph describing the continuous model, we will get a state space description where the right hand side depends on ". The state space model can by a change of basis be transformed to a model where it is clear that the system is singularly perturbed. Furthermore, after the change of basis, the partition into fast and slow dynamics will be clear from the state space description. The interpretation of the fast and the slow dynamics will be the expected. The slow dynamics corresponds to the generalized momentum, which is kept constant during a change of mode, and the fast dynamics correspond to Equation (10), quantities in the system that instantly changes.. Theorem 1. Consider a bond graph with linear elements for which Assumptions A1 and A2 and A3 hold. In the linear switch bond graph model, replace the switch whose state is changing when changing modes with a linear R-element with parameter value ". Derive the state space equations for the bond graph. Make a change of basis of the state space equation according to Equation (29) with T de

(11) ned in Equation (24). The structure of the state space form will then be the following:. "y_ a + "a "a y "11 12 1 1 11 A21 A22 y2:n y_2:n = u 1:l + B11 "B12. In Section 4.1 some notation is introduced, and in the following section, Section 4.2, a theorem stating that Gr is singularly perturbed is found and it is also shown that the initial conditions given by Algorithm 1 are achieved using singular perturbation theory as well. The rest of Section 4 is used to prove the theorem in Section 4.2.. B21 B22. 2. (30). ul2 +1:n. where a11 = 0. Furthermore 6. B11 u1:l2 = gp 1 a11. (31). where g1p is de

(12) ned in Equation 7.. 4.1 Notation. The rest of Section 4 contains the proof of the theorem, but before the proof is presented, the initial values of Algorithm 1 will be conrmed.. There are two bond graphs considered, one including the switch that changes state, and one where it is replaced with an R-element. Denote the bond graph with the switch by Gsw and the bond graph with the R-element Gr . Denote the switch by sw and the Relement with r. The bond graphs are very similar, the only thing that di ers, except the replaced element, is a change of causality. The number of storage elements n1 and the number of sources n2 are therefore the same in both graphs.. Lemma 5. For the system described by Equations (30) and (31), the initial values of Algorithm 1 are achieved when " 0 if the states y2:n are continuous while y1 reaches steady state !. Proof: Of Lemma 5] By letting " tend to zero we get. The dependent set in Gsw when sw changes states is denoted by D. The number of storage elements in D is l1 and the number of sources in D is l2 . The storage element with derivative causality is denoted s1 . We will use the notation D also when discussing Gr , meaning exactly the same storage elements and sources that belongs to D in Gsw .. the following equations.. 0 a 0 y 11 1 y_2:n = A21 A22 y2:n B 0 u 1:l + 11 B21 B22. 8. 2. ul2 +1:n. (32).

(13) Assuming that the slow dynamics, i.e., y2:n , is constant and that y1 is determined by the rst row in Equation (32), we get the following initialization equations. y (t) = B11 u1:l2 (t) = gp (t) (33). 4.3 Lemmas describing structure in the bond graph Under the assumption that s1 is a C-element, the fol-. (34). Consider a bond graph for which Assumptions A1 and A2 and A3 hold.. a11 y2:n (t) = y2:n (t ) 1. lowing lemma holds.. Lemma 6. 1. 0. Rewrite this in terms of the old state variables x using that. y1 = x1 g1s yk = xk k1 x1 k = 1 : : : l1 yk = xk k = l1 + 1 : : : n1 ;. ;. Let s1 be a C-element. Consider the causal path between s1 and sj , where sj D. This path meets the path between s1 and sk , sk D, k = j , at a 1-junction. 2. (35) (36) (37) (38). 2. Proof:. Both sj and sk have e ort causality. The junction where the paths meet has to allow more that one e ort causality propagated to it. According to the causality rules for junctions, the junction has to be a 1-junction.. according to how y is dened in Equation 29. Then we get. x1 (t) = g1s (t) + g1p (t) (39) xk (t) = k1 x1 (t) + xk (t ) k1 x1 (t ) k = 1 : : : l1 (40) xk (t) = xk (t ) k = l1 + 1 : : : n1 (41) 0. ;. 6. The following result is then immediate from the duality of e ort and ow in bond graphs.. 0. Corollary 2 If s1 is an I-element the junction where the paths meet is a 0-junction.. 0. These are exactly the same initialization rules as achieved by Algorithm 1.. From Lemma 2 and the denition of dependent set, we know that in Gsw all causal paths in D lead to s1 . We have a similar result for Gr .. Outline of the proof of Theorem 1: In the. remainder of Section 4 a proof of Theorem 1 can be found. Here the ideas behind the proof are briey discussed.. Lemma 7. If replacing the Sw-element, sw, with an R-element, r, there will be a causal path from each element in D to r. These are also the only causal paths leading to r.. One of the key ideas of the proof is that " is only introduced in Gr at r. There are no other places in the bond graph were this variable occurs. To use this fact, expressions for the e ort and ow variables at r are derived.. Proof: We know that there is a causal path between. sw and s1 in Gsw , otherwise s1 would not have derivative causality. When replacing the switch with r, the. One can also note that old state variables that corresponds to elements outside of D can not be dependent of the variables at r due to the causal paths in Gr .. causality is changed at both elements to get the proper causality. We know from Lemma 1 that the causality also changes along the path between the two elements.. Using these two ideas, expressions for the derivatives of the old state variables x_ in the new state variables y are found. Then the derivatives of the new state variables y_ are expressed in x_ .. Look at one causal path from an element sk in D leading to s1 in Gsw . Such a path exists according to the denition of D. There is a 1-junction where this path, and the path from sw meet according to Lemma 6.. In Section 4.3 some lemmas describing structure of the bond graphs are presented. The structure is used in the proofs of forthcoming lemmas.. In Gr there is a path from sk to the junction since there was a path from sk to s1 in Gsw and the causality from the junction to sk has not changed when replacing sw according to Lemma 1. That there is a causal path from the junction to r is also clear since there is a causal path between s1 and r, and the junction lies in between the elements. That the path will not be broken at the junction follows from the causality rules for junctions, see Figure 7. If the causality changes at the two bonds next to the junction along the path between s1 and. The values of the variables at the bond next to r are derived in Section 4.4. Using this information, the derivatives of the old state variables are derived in the new state variables in Section 4.5. Finally, in Section 4.6, the derivatives of the new state variables are expressed. 9.

(14) sw = r when replacing sw, all causal paths leading to s1 in Gsw will lead to r in Gr .. b2. Hence all elements in D will have causal paths leading to r.. b1. That no other element leads to r follows from symmetry. If changing back from r to the switch, the causality would also change back. But with the same arguments, all paths leading to r would be directed to s1 , and the paths leading to s1 all come from elements in D. s1. Sw. k. 1. 1. R. 1. b3. 13. Figure 8: A 1-junction. Proof: From a bond graph equation generation al-. s1. s. 23. 12. gorithm it follows that sign ij is negative if bond bi and bj both are directed from the junction or towards the junction. Since ji depends on the directions of the same bonds, ji = ij. k. s. Figure 7: Causality changes at the junctions.. Equations (42) can now be rewritten as. e1 = 12 e2 + 13 e3 e2 = 12 e1 + 23 e3 e3 = 13 e1 + 23 e2 Solve Equation (48) for e1 e1 = 12 e2 12 23 e3 Compare the signs in front of e3 in Equations. The following lemma gives useful information about the dependency of di erent variables.. Lemma 8. The eort and ow variables of the bonds connected to elements outside D r are not dependent on the variables at r.. ;. Proof: All causal paths to r comes from elements in D, i.e., sources or storage elements with preferred causality. If a causal path goes from a bond outside D r into D it will hence reach a storage element or. and (50). Then we see that. 13 = 12 23 ;. ). 12 = 13 23 ;. (47) (48) (49) (50) (47) (51). a source. Hence an equation generation algorithm will stop there, and not continue to r.. With help of this lemma, the variables at the bond next to r will be expressed in the new basis y. To do this we will start by examining the rst row in Equation 23: x1 12 2x2 : : : 1l1 l1 xl1 (52) = g1p (u1 (t) : : : ul2 (t) We see from Equation 29 that the left hand side equals y1 . If rewriting Equation 52 we get:. 4.4 Variable values at r, the R-element. The value of the variables at the bond next to r is important in the proof of Theorem 1. In this section expressions for the two variables are derived. We will start with a lemma that gives a relation between the signs that corresponds to a 1-junction.. ;. Proposition 1 Consider a 1-junction with three connecting bonds as. g. ;. y1 g1p (u1 (t) : : : ul2 (t) = 0 ;. in Figure 8. The three dierent causalities that can be propagated through the junction, correspond to the following three equations: e1 = 12 e2 + 13 e3 (42) e2 = 21 e1 + 23 e3 (43) (44) e3 = 31 e1 + 32 e2 The signs depend on the directions of the energy halfarrows and have the following properties: ij = ji i j 1 2 3 i = j (45) 12 = 13 23 (46) 2 f. ;. (53). We will now show that the left hand side of Equation (53) is proportional to the e ort variable at r.. Lemma 9. The equation describing the eort variable er at the bond next to r in states and inputs is a constant r = 0 times the left hand side in Equation (53), that is 6. er = r (x1 g1s g1p ) = r (y1 g1p ) and hence the ow variable fr is fr = "r (y1 g1p ) ;. 6. ;. ;. ;. ;. 10. (54) (55).

(15) Proof: First some notation is introduced. The e ort. Sw. variables at the storage elements in Gsw are denoted esw 1 : : : l1 . The corresponding e ort k , where k variables in Gsw are denoted erk . The distinction between the storage variables will clarify the proof. In the same way the e ort variables are denoted esw k in Gsw and erk in Gr , where k l1 + 1 : : : l1 + l2 . 2 f. ksw. g. 2 f. sk. The constitutive relations for the elements in D give the following relationship between e orts, states and inputs:. xk esw k = ck k 1 : : : l1 esw k = uk l1 k l1 + 1 : : : l1 + l2 2 f. ;. 2 f. 1. In the proof we will rst consider Gsw . In this bond graph we will use signs to describe di erent parts of the junction structure. Then we can derive an expression sw for esw 2 : : : l1 + l2 and 1 in the signs and ek , k x k hence in the signs, ck , k 2 : : : l1 , and uk , k 1 : : : l2 . 2 f. 2 f. f. ks1sk. Figure 10: The junction where the paths meet.. g. g. 2. g. and hence. From step 3 in Algorithm 1, we know that the value of x1 is derived using standard bond graph equation generation. With the given causality x1 = c1 esw 1 . From Equation (7) it follows that 1 s p esw 1 = c (g1 + g1 ): 1. esw 1 = =. (58). 1. state or input, superposition can be used to nd an expression for esw 1 in state variables and inputs. lX 1 +l2 k=2. esw 1k. k=2 l1 X. ks1 ks1sk ksk esw k. ks1 ks1sk ksk xc k. k=2 lX 1 +l2. k=l1 +1. (61). k. ks1 ks1sk ksk uk. l1. ;. Replace the switch with r. Use superposition to determine the value of esw r :. (59). er =. sw where esw 1k is the part of e1 coming from element k in D.. lX 1 +l2 k=1. erk. (62). Consider the path between r and sk shown in Figure 11. The junction where the path meets the path from s1 is also shown. Note that for each k the signs are the same as in Gsw , since only the switch changes, not the bond graph structure.. To determine esw 1k look at the bond graph in Figure 9, where the path from s1 to sk is shown. The other paths can be neglected due to superposition. The 1-junction where this path meets the path from the switch is depicted in the gure. Three signs, ks1 ksk and ksw are introduced to describe the junction structure in the paths.. R. ksw. Three signs are also introduced at the junction according to Figure 10 and Lemma 1.. sk. From this we can conclude that s1 s1sk sk esw esw 1k = k k k k. lX 1 +l2. +. Since esw is a sum of e orts, each coming from a certain. esw 1 =. s1. ksksw ks1sw. (57). g. ks1. Figure 9: The path between element sk and s1.. (56). g. 1. ksk. g. ksk. 1. ks1. s1. Figure 11: The path between element sk and R.. (60) 11.

(16) The last equality comes from Equation (58).. In the same way as above we get. erk = ksw ksksw ksk erk and hence. We can conclude that er = r (x1 g1s gps) where r = s1sw = 0, since s1sw is a sign and c1 is a coecient in c1 the constitutive relation for s1 . From the constitutive relation of r, e = "f it is clear that fr = "r (x1 g1s gps ). (63). ;. er = 1sw 1s1sw 1s1 er1 +. lX 1 +l2. = 1sw 1s1sw 1s1 xc 1 + +. k=2 l1 X. 1. lX 1 +l2. k=l1 +1. k=2. ;. ksw ksksw ksk erk ksw ksksw ksk xc k k. In this section, the derivatives of the old state variables will be expressed in the old state variables, the inputs, and y1 . This is done using a standard equation generation algorithm following the causal paths to storage elements and sources. To include y1 in the description, equation generation along the paths leading to r will end at r. It is clear from Lemma 9 that the variables at r depend on y1 .. ksw ksksw ksk uk l1 ;. ksw ks1sw ks1 = s1sw. (65). Lemma 10. If deriving equations in Gr , expressing the state derivative at sk , a storage element in D, in y1 , x, and u, by following causal paths either to r or to storage elements and sources, the equation will have the following form:. Use this to rewrite Equation (64):. er = s1sw xc 1 l X 1. 1. s1sw ks1sw ks1 ksksw ksk xc k. +. k=l1 +1. x_ k = "r (y1 g1p ) + fk (x u) ;. (66). k. k=2 lX 1 +l2. s1sw ks1sw ks1 ksksw ksk uk. 6. ;. l1. ks1sk = ks1sw ksksw. Proof:. We know from Lemma 7 that there is a direct causal path from sk to r. We also know from Lemma 9 that the ow variable at the R-element is fr = "r (y1 g1p), where r = 0.. (67). ;. ;. Use this to rewrite Equation (66): ;. k=l1 +1. = s1sw. . s1sw ks1sk ks1 ksk uk. l1. ;. ;. ;. k=l1 +1. Lemma 11. (68). l1 x1 X s1sk s1 sk xk k k k ck c1 k=2 lX 1 +l2. ks1sk ks1 ksk uk l1. Expressed in the new state variables x_ 1 becomes. x_ k = "r (y1 g1p ) +

(17) k (y u). !. ;. 1. ;. (71). where r = 0 and

(18) k ( ) does not depend on 1" .. ;. 6. Proof: Since the transformation matrix T is invertible, we know that x = T 1y. Then dene

(19) k (y u) = fk (T 1 y u).. Compare Equation (61) with Equation (68). Then we see that s1sw 1 (x1 c1 esw ) er = s1sw ( xc 1 esw 1 )= 1 c1 1 (69) s 1 sw s s = c (x1 g1 gp ) ;. 6. From Lemma 7 we also know that no other paths lead directly to r. Then all other paths will end at a storage element or a source without passing r. Since the Relement is the only place where " is introduced, we can conclude that f ( ) does not depend on ".. l1 X er = s1sw xc 1 s1sw ks1sk ks1 ksk xc k 1 k k=2 lX 1 +l2. (70). where r = 0 and f ( ) does not depend on ". x is the set of all old state variables and u is the set of all inputs.. By using Lemma 1 we can express the sign ks1sk in ks1sw and ksksw .. ;. ;. 4.5 Structure of the equation in the old state variables. (64). Since the path between the R-element and s1 does not change for di erent values of k, we know that the sign describing the path is independent of k.. +. ;. 6. ;. ;. ;. Lemma 12. If deriving equations in Gr , expressing the derivative at a storage sk element outside D, the equation will. ;. 12.

(20) Then Equation 77 can be rewritten. look like:. x_ k = fk (x u). y_1 = k"y1 (y1 g1p ) +

(21) (y u) (80) From Lemma 11 we know that

(22) k does not depend k is independent of 1" , we can on 1" , and since also P l 1 conclude that

(23) = k=1 k

(24) k is independent of 1" . The proof is complete if we can conclude that ky1 = 0. To do this we will examine the way y1 was derived in. (72). ;. where fk ( ) does not depend on ". x is the set of all state variables and u is the set of all inputs.. Proof: It follow from the proof of Lemma 10.. 6. This lemma tells us that l 1 linear combinations of x_ 1 : : : x_ l that are independent of 1" can be found.. the proof a little bit closer. If the sources in D are removed from the bond graph we can from Lemma 9 conclude that y = 1e (81). ;. 4.6 Structure of the equation in the new state variables In this Section the derivatives of the new state variables y_ will be expressed in the new state variables. Lemma 13. r r fr = "r y1 1. We can now express er in the old state variables, x by following the causal paths to storage elements, and we know from Lemma 7 that the causal paths from r leads to storage elements in D. To conclude, we can express y1 as. If y_1 is expressed in y1 : : : yn the equation will have the following structure:. y_1 = k"y1 (y1 + g1p ) +

(25) (y u). (73). l1 X 1 y1 = k c1 xk (83) r k=1 k where r = 0, k = 1 and ck = 0. ck is the constant in the constitutive relation of storage element sk and k is the sign that describes the bondgraph structure between r and sk .. where ky1 = 0, and

(26) is independent of 1" . 6. Proof:. To proof the statements, an expression for. y_1 will be derived.. 6. From Equations (24) and (29) we know that. y1 = x1 12 a2 x2 13 a3 x3 : : : 1l1 al1 xl1 (74) If denoting the coecient in front of xk by k the expression for y1 can be rewritten. ;. ;. ;. l X 1. y1 =. k=1. Di erentiate Equation 75. l X 1. y_1 =. k=1. (75). k x_ k. (76). l X k ( "r (y1 g1p ) +

(27) k (y u)) 1. ;. k=1. Now introduce. ky1 = and.

(28) =. l X 1. k=1. l X 1. k=1. j. 6. y_1 = 1. l X 1. r k=1. k c1 x_ k k. (84). Now we will express x_ k in y1 . All dependencies of other state variables are not interesting since we are looking at the coecient in front of y1 . Since the only causal path inside D that leads to sk comes from r according to Lemma 7, and since all causal paths that leads out from D does not depend on y1 according to Lemma 8. Deriving equations along the path from sk to r to express x_ k in y1 we get. Use Lemma 11 to rewrite the right hand side:. y_1 =. j. Di erentiate Equation 83. ;. k xk. (82). x_ k = fk = k fr = k "r y1 (85) That the sign is k follows from Lemma 3, and the. (77). ;. ;. ;. third equality follows from Equation (82).. k r. k

(29) k. When combining Equations (83) and (84) we get. (78). l1 2 X y_1 = 1" c k y1 k=1 k. (86). ;. Since each ck is positive, ky1 = ckk is negative, and hence not zero. 2. (79). ;. 13.

(30) Lemma 14. R. y_2 : : : y_l1 are l1 1 linearly independent linear combinations of x_ 1 : : : x_ l1 that are independent of 1" . ;. ksw. Proof: In this proof we will consider the equations. sk. dening y2 : : : yl1 . By di erentiating these equations, expressing them in fr we will conclude that the terms including fr cancel each other.. ;. ;. s1. fkr = ksk ksw fr f1r = ks1 ksw fr. By di erentiating the expression we get. y_k = x_ k k1 x_ 1 = fkr k1 f1r. ks1. From the gure we see that. (87). ;. 1. Figure 13: The path between element sk and R.. From Equation (29), we know that for each k k = 2 : : : l1 ,. yk = xk k1 x1. ksk. (88). (90) (91). Now combine Equations (88), (89), (90), and (91).. where fkr is the ow at the k:th element in D.. y_k = fkr k1 f1r = fkr ks1 ksk f1r = ksk ksw fr ks1 ksk ks1 ksw fr = ksk ksw fr ks1 ksw fr ;. The sign k1 arises from Gsw by following the causal path from sk to s1 . We introducing two signs, ks1 and ksk along the path by splitting at the junction where the path meets the path from the switch, according to Figure 12. Since all ows are equal at a 1-junction, we. ;. ;. =0. ;. (92). Sw. Lemma 15. ksw sk. ksk. 1. ks1. y_l1 +1 : : : y_n1 are independent of 1" . s1. Proof:. From step 3 in Algorithm 1 we know that l1 + 1 : : : n1 . Since the corresponding storage elements does not belong to D, the equation does not depend on the variables at r according to Lemma 8, and hence not on 1" .. yk = xk x. Figure 12: The path between element sk and s1 . can conclude that. k1 = ks1 ksk. 2 f. g. Proof: Of Theorem 1] Combine the results of Lemmas 13, 14 and 15. (89). Our next step is to derive expressions for fkr and f1r in fr . We will do this by following causal path that leads from elements sk and s1 . However we will only consider the two paths leading either from sk or s1 to r. The reason for this is that the contribution from the other parts does not depend on ". Since all paths leading to the R-element come from elements in D, paths not leading directly from an element in D to r will never reach r. Then it would have to pass an element in D, but elements in D corresponds to either an input or a state variable.. 5 An Electrical Example To illustrate the algorithm and the proof, Example 1 is slightly extended, see Figure 14. The switched bond graph model of the circuit is shown in Figure 15. If starting the simulation with the switch open, i.e., in mode M1 , and closing the switch after some time, entering mode M2 , a reduction of the state space will occur, and the initialization of the states in the mode M2 will be non-trivial. Mode M2 with propagated causality is shown in Figure 16. The dependent set D2 consists of the two storage elements C1 and C2 . Three state variables are introduced, one for each C-element, even though the dimension of the state space is two.. Consider the same junction in Gr , the junction where the paths from s1 , sk and r meet, see Figure 13. The signs are the same as in Figure 12, since the bond graph structure is the same. 14.

(31) A relation between the states in D2 is derived: x1 = c1 x2 , and rewritten: x1 c1 x2 = 0. Then a relation c2 c2 between the rates is derived: x_ 2 = x_ 1 . Integrate both sides and rewrite the equation: x2 + x1 = 0. The left hand side is constant throughout the mode change: x2 (t+ ) + x1 (t+ ) = x2 (t) + x1 (t). ;. ;. C3. R1. C2. U. An initialization rule for the state corresponding to C3 is also needed. Since the element is not in the dependent set, the variable is continuous: x3 (t+ ) = x3 (t).. C1. These three equations together form a solvable set of equations, and its solution is the initial state value of the new mode. 01 c1 01 0x (t+)1 0 0 1 @1 1c2 0A @x12 (t+)A = @x1(t) + x2 (t)A (93) x3 (t) 0 0 1 x3 (t+ ). Figure 14: An electrical example.. ;. To show the connection to a continuous model the switch is replaced with an R:"-element as in Figure 17. The state space description derived from this model is. Se. C3. C2. C1. 1. 0. 1. C3. C2. C1. 1. 0. 1. Se. Sw. e. R:". R1. R1. Figure 17: The corresponding classical bond graph.. Figure 15: A switched bond graph model of the circuit.. 0x_ 1 0 1 @x_ 12 A = @ "c1"c x_ 3 0 001 + @ r1 A u ;. 1. 1. 1 0x 1 1 1 A@ A x 2 + rc 0. 1. "c2 1 + 1 ) ;( "c2 1 r1 c2 r1 c2. ; ;. 1 3. 1. r1 c3. x3. 1. 1. r1. Se. C3. C2. C1. 1. 0. 1. (94) The transformation matrix for the change of basis is the one found in Equation (93): 01 c1 01 c2 T = @1 1 0A (95) 0 0 1 Hence the new state space description is y_ = TAT 1y + TBu (96) where ;. Se:0. R1. ;. Figure 16: Mode M2 with propagated causality.. 0 TAT 1 = B @ ;. 15. ;. r1 (c1 +c2 )2 +"c21 "c1 (c2 +c1 )c2 r1 1 r1 (c2 +c1 ) 1 ; r1 (c2 +c1 ). c1 c2 r1 (c2 +c1 ) 1 ; r1 (c2 +c1 ) 1 r1 (c2 +c1 ). c1 1 c2 r1 c3 C 1 ; r1 c3 A 1 ; r1 c3. (97).

(32) and. 0 TB = @. c1 1 r11c2 ; r11 A ; r1. ;. (98). By multiplying the rst row with " the system is on the desired form. By letting " 0 and the rst line in the equation will have the solution y1 (t+ ) = 0. The second and third lines describe the slow dynamics, and are therefore assumed to be constant during the mode change: y2 (t+ ) = y2 (t) y3 (t+ ) = y3 (t). These are exactly the same initial conditions as derived from Equation (93). !. It is noticeable that the e ort variable shown in Figure 17 is proportional to y1 , e = c11 y1.. References. 1] F.E. Cellier. Continuous system modeling. Springer-Verlag, rst edition, 1991. 2] K. Edstrom. Simulation of mode switching systems using switched bond graphs. Linkopings Universitet, December 1996. Lic. thesis No. 586. 3] K. Edstrom. Mode initialization when simulating switched bond graphs. In Proc. of 2nd IMACS International Multiconference: CESA'98, Computational Engineering in Systems Applications. IMACS, April 1998. 4] K. Edstrom, J.-E. Stromberg, and J. Top. Aspects on simulation of switched bond graphs. In Proceedings IEEE 35th Conference on Decision and Control, 1996. 5] D.C. Karnopp, D.L. Margolis, and R.C. Rosenberg. System dynamics, A uni ed approach. Wiley Interscience, 1990. 6] P. Kokotovic, H.K. Khalil, and J. O'Reilly. Singular perturbation methods in control: Analysis and design. Academic Press, 1986. 7] P.J. Mosterman. Hybrid system dynamics: A hybrid bond graph modeling paradigm and its application in diagnosis. PhD thesis, Graduate School of Vanderbilt University, 1997. 8] J.-E. Stromberg. A mode switching modelling philosophy. PhD thesis, Linkopings Universitet, 1994. 9] J.-E. Stromberg, J. Top, and U. Soderman. Variable causality in bond graphs caused by discrete e ects. In Proc. First Int. Conf. on Bond Graph Modeling and Simulation (ICBGM'93), volume 25 of SCS Simulation Series, pages 115{119, 1993. 10] J. van Dijk. On the role of bond graph causality in modelling mechatronic systems. PhD thesis, University of Twente, 1994. 16.

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