Generating state space equations from a bond graph with dependent storage elements using singular perturbation theory. Krister Edstrom Department of Electrical Engineering Linkoping University, S-58 83 Linkoping, Sweden WWW: http://www.control.isy.liu.se Email: edstrom@isy.liu.se August 25, 998 REGLERTEKNIK AUTOMATIC CONTROL LINKÖPING Report no.: LiTH-ISY-R-25 Submitted to ICBGM'99 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 25.ps.Z.
Generating state space equations from a bond graph with dependent storage elements using singular perturbation theory. Krister Edstrom (edstrom@isy.liu.se) Dept. of Electrical Engineering Linkopings Universitet SE-58 83 Linkoping, Sweden Introduction It is well known that when allowing derivative causality in a bond graph, the generated mathematical description will be a DAE [5] (Dierential Algebraic Equation) as a mathematical description. For simulation purposes this is in most cases a suitable description. There is however a well developed theory for ODE:s (Ordinary Dierential Equations) in state space form. The ODE:s can be used to, e.g., analyze, or construct a controller for, the system. When generating a state space description from a bond graph without any derivative causality [5], algebraic loops or causal loops, the states have clear physical interpretations. For an I-element in the mechanical domain, the state is the momentum of the mass, and for a C-element in the electrical domain, the state is the charge of the capacitor. When generating a state space description from a bond graph with derivative causality, there should still be a physical interpretation. When the derivative causality is generated by two masses that are stiy connected, the state variable in a generated state space description should be the total momentum of the two masses. In this paper we will consider the derivative causality as a consequence of simplifying the model by neglecting energy dissipation. If the energy dissipation is small enough, it is a natural model simplication to remove it from the model. To allow generation of a state space description from a bond graph with connected storage elements, we will rst add the energy dissipation in the bond graph using a linear R-element. The dissipation will then be removed from the mathematical description by letting the value of the parameter in the constitutive relation of the R-element tend to zero. There exist methods for generating a state space description from a bond graph [4]. The method presented here have two properties that make it interesting: First there is a physical interpretation of the generated states, and second it relies on an interpretation of the dependency of the storage elements as the connection between the storage elements containing energy dissipation that tends to zero.
2 Singular perturbations In a singular perturbation framework [3] it is possible to study a system where the dynamics have two dierent time scales. A linear systems that is singularly perturbed can be expressed in the following form: _(t) _x(t) A () A 2 () A 2 () A 22 () (t) x(t) () When the value of the parameter becomes very small, the states in will have a signicantly smaller time constant than the states in x. By letting go to zero, the time constant of will go to zero. When looking at the step response of the system, will reach its steady state before x changes value, assuming that is very small. The singular perturbed model is in standard form if det A () 6. If the system is in standard form it can for be rewritten as: A () A 2 () _x(t) A 2 () A 22 () x(t) ) (2) (t) (t) _x(t) x(t)?a? ()A 2() A 22 ()? A 2 ()A? ()A 2() By assigning, the state space is reduced, and we get a DAE description of the system. As seen in Equation (2) it is possible to rewrite the equations in state space form with as an output variable if det A () 6. 3 Generating the state space description The generation of a state space description is done in ve steps as shown below. The described procedure handles one C-element with derivative causality, denoted by s, but the method can be extended to hold for I-elements, and for multiple elements with derivative causality. The bond graph is assumed to be nice, meaning that it should include no other storage elements with derivative causality, no algebraic loops and no causal loops.. Start with the bond graph with dependent storage elements. Add an R-element to relax the derivative causality. If s is connected to a -junction it is necessary to add a -junction as well. When the derivative causality is relaxed, the number of states increases with one. Let the value of the parameter in the linear constitutive relation of the R-element be. 2. Derive a state space description using SCAP [2]. _x Ax + Bu This is possible due to our assumption about a nice bond graph, since this assumptions implies that there are no zero-order causal paths (paths between storage elements or between R-elements or causal loops) or elements with derivative causality in the bond graph [5]. 3. Generate a transformation matrix T from the bond graph with derivative causality: T x T is found using a standard bond graph equation generation algorithm along the causal paths between the storage element with derivative causality and other storage elements. The generation of the transformation matrix is described more in detail in Section 4.. 2
4. Perform the change of basis of the state space description: where _ A + Bu A T AT? ; B T B Multiply the rst row by. The resulting state space description will have the following form where A; is invertible, see Section 4: A; + A ; A; A ; A; + B B u 5. Let!. _ A; A ; A; + B u Since A; is invertible we can according to Equation (2) replace with in the last n? equations: _ A; + B u (3) The resulting state space equations describe the dynamics of the bond graph with derivative causality. 4 Details and proofs In this section some details in the algorithm are explored. A theorem and a lemma stating some the results are formulated, but the proofs are left out and can be found in []. 4. Generation of a transformation matrix Here we will take a closer look at step 3 in the algorithm in Section 3. To generate the T matrix, we will use the bond graph in which the dependent storage elements are, i.e., the bond graph without the additional R-element. We will assume that the number of storage elements in the bond graph is n. An algorithm for generating T will look like this:. Propagate causality in the bond graph. 2. Find all storage elements that are connected to the storage element with derivative causality, s, by causal paths. Assume that there are l such storage elements, s included. Consider only the part of the bond graph connecting these elements. 3. Derive an equation x a 2 x 2 + : : : + a l x l, where x is the state variable in the constitutive relation for s and x 2 ; : : : ; x l are state variables belonging to constitutive relations for the other storage elements. Use a standard bond graph equation generation algorithm. Rewrite the equation as: x? a 2 x 2? : : :? a l x l (4) 3
4. Derive l? equations relating the rates of the state variables: _x ; : : : ; _x l : _x j j _x ; j 2; : : : ; l (5) This is also done with a standard bond graph equation generation algorithm. Rewrite the equations as _x j? j _x ; j 2; : : : ; l (6) and integrate: x j? j x ; j 2; : : : ; l (7) 5. Let the coecients in left hand sides of Equations (4) and (8) form a matrix T : T B @ a 2 a 3 : : : a l? : : :? 2 : : :.......? l : : : C A (8) 6. Extend T with an identity matrix to get a matrix T of dimension n, the number of storage elements in the bond graph: T T (9) I To show that T is of full rank we have the following lemma: Lemma If the bond graph has no algebraic loops, no causal loops and one storage element with derivative causality that is dependent of one or several other storage elements, then the following statement holds: T is invertible if we assume that at each C-element the energy half-arrow points towards the C-element. Proof: See []. 4.2 A state space form after the change of basis After the change of basis with the transformation matrix it will be clear that the system is a singular perturbed system in standard form. Theorem Let the bond graph have no algebraic loops, no causal loops and one storage element with derivative causality that is dependent of one or several other storage elements. Derive the state space description from the bond graph with the added R-element. _x Ax + Bu () Derive the matrix T from the bond graph without the R-element. Perform the change of basis to get a new state space description: _ A + Bu; where T x () 4
Hence A T AT? ; and B T B (2) If multiplying the rst row in Equation (), the structure of the matrices A; + A A ; A; A ; A; and B A and B will be the following: B B u (3) where A; 6. Proof: See []. Since A; 6 it is clear that the system is singular perturbed and in standard form. 5 Example We will derive a state space description from the bond graph in Figure. R C Se C 2 Figure : A bond graph example. The rst step is to introduce an R-element according to Figure 2. Its constitutive relation is assumed to be linear with parameter. R C Se C 2 R 2 Figure 2: The bond graph with causality relaxed by an R-element. Derive a state space description from the bond graph in Figure 2 using SCAP: _x _x 2? +r rc? c x x 2 + r x and x 2 are the states in the constitutive relations for the two C-elements and u is the eort given by the Se -element. u 5
Derive equations in both directions along the causal path between the two C-elements in Figure. x 2 c 2 c x _x? _x 2 Introduce two new state variables using the derived equations. c 2 c x? x 2 2 x + x 2 The transformation matrix is then: T c? Change the basis, and multiply the rst row with : A? c 2 2 + 2 r+2cr+ r c(c+)r c(c+)r?? (c+)r (c+)r! ; B cr r Let! : _ 2? c+ c?? (c+)r (c+)r + u 2 r The state space description is thus _ 2? 2 + u (c + c 2 )r r There is a physical interpretation of the new state variable. If the bond graph models an electrical circuit, the causal path between the two C-elements shows that there are two capacitors connected in parallel. The two capacitors will behave as one with capacitance c + c 2 and with a state variable that describes the sum of the charges. This is exactly the structure of the state space equation above. References [] K. Edstrom. Singular perturbation analysis of a mode initialization algorithm for simulating mode switching system, long version. Technical report, Linkopings Universitet, 998. [2] D.C. Karnopp, D.L. Margolis, and R.C. Rosenberg. System dynamics, A unied approach. Wiley Interscience, 99. [3] P. Kokotovic, H.K. Khalil, and J. O'Reilly. Singular perturbation methods in control: Analysis and design. Academic Press, 986. [4] A. Rahmani and G. Dauphin-Tangui. Symbolic determination of state matrices from bond graph model with derivative causality. In Proc. of Computational Engineering in System Applications. IMACS, 998. [5] J. van Dijk. On the role of bond graph causality in modelling mechatronic systems. PhD thesis, University of Twente, 994. 6