Causal path input/output

The structural information contained in the ARRs, i.e. the information on which ARR depends on which component parameters can be obtained directly by inspection of causal paths in a diagnostic bond graph [1]. There is no need to derive equations and to eliminate unknowns in order to set up a mode-dependent FSM. To that end, causal paths from model inputs to inputs of sensor elements are considered. Elements that are traversed on these causal paths contribute to the ARR of a residual related to a sensor element. An output of a source or an element that is followed directly or indirectly by switches on the causal path to a sensor element provides an entry in the FSM equal to the product of the switch states. 

In contrast, the output of a controlled source is algebraically related to its input. If the latter is not an output of an independent source or an energy store with integral causality, then it can be expressed by means of such outputs by back propagation of causal paths in the junction structure and by eliminating intermediate variables. 

The outputs of resistors depend algebraically on their inputs. By back propagation along causal paths through the junction structure, their outputs can be expressed by outputs of sources either independent, or controlled ones and outputs of energy stores. The outputs of dependent sources do not need to be eUminated, since they have already been determined in the previous step. 

For storage ports, the derivative with respect to time of an output is a function of the input(s). By working back causal paths, the inputs can be expressed by outputs of other energy storage elements, of resistors, or sources. 

Note, that if there are causal paths between resistor ports then impUcit, algebraic equations will result. This means, that the output of the resistor port at one end of the causal path cannot be computed without knowing the output of the resistive port at the other end of the causal path. In this case, intermediate variables cannot be eliminated and expressed by system inputs and state variables by back propagation along causal paths. The mathematical model will be of the form of a DAE system that can be transformed into an ODE system if the system of coupled algebraic equations can be solved symbolically. 

The causally completed BG in Fig. B.8 indicates that the two capacitors C Ci, C C2 are in integral causality and that there are causal paths from R / 2 to R Ri, from R / 2 to R and from R R2 to R / <, via the controlled source MSe. These causal paths share bonds. The output V of the controlled source is algebraically dependent on its input Ud. The latter one depends algebraically on the capacitor voltages u, U2, and the resistor voltage m/jj-... 

If there were no causal paths between resistor ports then their outputs could be expressed by the two state variables mi, M2 and the input E by back propagation of causal paths. The result would be an ordered set of equations that could be computed in that order. Clearly, if state space matrices are needed, the outputs of the resistors could be inserted into the constitutive equations of the storage elements. 

Ngwompo and his co-authors [24] state that a LTI SISO system is structurally invertible if there is at least one causal path in the causal direct bond graph between the input variable and the output variable ([24, Proposition 1, p. 162]). Furthermore, they show how the state equations of the inverse system can be directly determined from a causal direct bond graph model or from a bicausal bond graph. (In order to support tasks such as bond graph-based system inversion, Gawthrop extended the concept of computational causality by introducing the notion of bicausality [19, 25].) Clearly, the state equations of the inverse model of a SISO system can be converted into a transfer function. 

Consider the simple linear electrical network depicted in Fig. 4.19. It can be viewed as an electrical analogue of the coupled hydraulic tank system considered in Section 4.4.1. A bond graph of the direct model with the two inputs I(t) and E(t) and the two outputs e and /2 appears in Fig. 4.20. There is one set of two disjoint input-output causal paths... 

The relative degree r of a linear system is defined in terms of the number of poles p and number of zeros z as r = p - z. As discussed by Fotsu Ngwombo et al. [22], r = scp where scp is defined as the shortest causal path from system input to system output in terms of Fig. 5.2c, this is the shortest causal path from the input flow /o to the output flow /2. In this case, the shortest causal path is... 

Definition 6.5 (Input/output (I/O) causal path) An input/output (FO) causal path starts from a modulated element and goes to a detector De or Df element). 

Procedure 1 (Output relative order (Fotsu-Ngwompo [15] and Wu and Youcef-Toumi [55])) In a bond graph representation in preferential integral causality of a system S, the relative order n[ of the output yi (and so the ith infinite zero order in row) is determined by the minimal order a causal path (Definition 6.7) can have between the output yt and any inputs. ... 

Example Inspection of Fig. 6.14 bicausal bond graph representation shows that among all the causal paths from the outputs z and 4> to the inputs F and Fj, the lower orders are —1 for both outputs. Thus their essential orders are equal to 1 and the specifications for heave and pitch velocities must be at least functions. Compared to the data given in Fig. 6.11 it can be concluded that the specifications verily the differentiability criterion and the methodology phases can be pursued. 

The variable influence path leading to x, is shown in Fig. 14. As a consequence, x, can only be influenced by variables that appear on the path causality is established by the input variables. This occurs because the output set assignment given to the set of relationships in the structural matrbc is unique, as it is made evident by rearranging the tabulation order of relationships and variables comprising the structural matrbc ... 

Causality The response of the system must be entirely determined by the applied perturbation that is, the output depends only on the present and past input values. A causal system cannot predict what its future input will be. Causal systems are also called physically realizable systems. If the system is at rest and a perturbation is applied at f = 0, the response must be 0 for f < 0. In the complex plane, the above criterion requires that for f < 0, oo = 0. Moreover, the integral on and inside a closed path C of an analytic function... 

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