Bicausality-based Procedures for Transfer and Transmission Matrix Determination of Single Source Single Load Linear Systems

International audienceIn linear SISO systems the energy flows from the source to the load through each actuating power line. Bond graph representation is well suited for studying the energy transfer in these systems because, in most of the systems, we can consider that energy flows through a power line embedded in two-port subsystems. Moreover, with bond graph language, we benefit from the ease of analytical exploitation introduced by the calculus causality and bicausality concepts.In the bond graph representation of a system the power in some port is associated to a power bond to which two power variables are attached. The determination of the transmission matrix linking two power variables is an efficient tool for studying the energy transfer. Furthermore it is often useful to study the bi-directional energy transfer i.e. from the source to the load and vice versa. The latter case (namely from the load to the source) is quite useful for the energy sizing of asystem from its dynamic specifications on an output variable.This paper proposes different procedures for determining direct and inverse transmission matrices and the transfer matrix between an internal bond and an environmental one for Single Source Single Load Linear systems represented by bond graphs. The procedures are based on the bicausality concept and the loop rule

Accounting for sizing parameters and boundary geometry in the bond graph representation of energetic elements

International audienceIn this paper we present how to take into account parameters and variables that were not initially present in the bond graph representation of some components. These bond graph considerations are justified mainly by design arguments but from the automatics point of view this is closely related to the state representation concept. We present this idea through the examples of a mechanical spring and a simple thermodynamic system. In each case one solution is to substitute a multiport energy storage element to the initial one port element. It emphasizes the close relationship between the order of this component and the output variables that we want to observe in this component

Sizing of an Electric Power Steering system on dynamic and energetic criteria

International audienceThis paper presents the methodology to size a mechatronic system on dynamic and energetic criteria. The methodology is based on the establishment of the inverse model from the bond graph representation of the system by using the bicausality concept. By means of an automotive example, we illustrate the methodology and we present a solution to extend it to more complex problem

A bond graph representation of an optimal control problem: the example of the DC motor

International audienceOptimization has become of great importance in present day Systems Engineering. Also, bond graph language proves to be a very efficient tool for modelling, analysing and designing mechatronic systems from an energy and dynamic point of view. The idea presented in this paper is to combine an optimization formulation in the optimal control context with bond graph language. The objective is to transpose the optimization problem into bond graph formalism so that its exploitation will solve this optimization problem. This paper, being a primary investigation, restricts itself to an optimal control problem formulation applied to the example of the classical DC motor. Developments are based on the use of Pontryagin's classical Maximum Principle where the cost function is expressed as the integral of a quadratic form of the state space vector and the control input to be determined. This type of expression may correspond to a certain energy loss-minimization. It is shown that the formulation of this optimal control problem in bond graph makes logical use of the adjoint system concept. Later variable mapping enables an augmented bond graph representation of the whole problem to be set up. Finally the bicausality assignment to this augmented bond graph representation furnishes the solution to the optimal control problem under consideration

Tolerance synthesis using bond graph inversion and fuzzy logic

International audienceIn the context of mechatronic systems design, this paper addresses a parameter tolerance synthesis with respect to specifications including output epistemic uncertainties. The methodology proposed here concerns uncertainties modelled with fuzzy logic. The procedure relies on output uncertainties propagation through an inverse model. Design parameter tolerance is then synthesized. The results are validated injecting designed parameters in the direct model. The methodology is illustrated on a linear model with specifications including combined uncertainties

Methodology of tolerance synthesis using bond graph

International audienceThis paper presents a methodology of parametric tolerance synthesis with respect to output aleatory uncertainty specifications. It relies on density function propagation through the inverse model. The resulting parameter density function is then used to synthesize a confidence interval suitable for sizing purpose. As an illustration, parametric tolerance synthesis on a DC motor rotating a load is processed

Bond Graph Representation of Standard Interconnection Model

International audienceThe study of the robustness of a system's parametric uncertainties is based on state representations which separate the nominal part of the system from the uncertain part. The most used form is the standard interconnection model. Recent works have been formulated so as to find this representation graphically by the bond graph approach. A new procedure is proposed in this paper to determine an uncertain model adapted to the study of robustness and for robust control. The advantage of this procedure is in simplifying the resulting graphical model

Applying causality and bicausality to multi-port elements in Bond Graphs

International audienceThe introduction of the bicausality concept in the bond graph language has allowed new analytical methodologies of a system, for instance in the context of model inversion, mechatronic system sizing and control. The causality assignment generally imposes the way these constitutive relations have to be used. In the case of linear multi-port elements, derivative causality or of bicausality is not necessarily possible. The conditions for the existence of a causal configuration are related to the form of the constitutive relation of the multi-port element. In this paper, we propose to inspect this condition and then to focus on the use of the causality applied to the linear multi-port elements. We show that the constitutive relations of any linear multi-port element may be used to determine quickly what kind of causality assignment does exist and what could be determined using different schemes of calculus. It clearly appears that this approach may be applied in other contexts and may have interesting applications on system sizing, identification and control

Structural analysis by bond graph approach: Duality between causal and bicausal procedure

Postprint version.International audienceThe infinite structure of linear time-invariant systems has been principally used to solve control problems. Nevertheless, this system characterization appears interesting in the design and sizing of mechatronic systems as well. Indeed, based on the bond graph language and inverse modelling, a methodology has already been developed for sizing mechatronic systems according to energy and dynamic criteria. One of the novelties of this methodology is its structural analysis step. This step enables structural properties to be deduced and helps in the formulation of the specifications. The aim of this paper is to add new graphical procedures to the structural analysis step to determine some structural properties (infinite pole orders and relative orders) from the inverse model (bicausal bond graph model). The structural analysis of the inverse model remains interesting since the essential orders are immediately obtained on the bicausal model. A discussion is carried out regarding the duality between the causal and bicausal procedures

Discrete IDA-PBC control law for Newtonian mechanical port-Hamiltonian systems

This paper deals with the stability of discrete closed-loop dynamics arising from digital IDA-PBC controller design. This work concerns the class of Newtonian mechanical port-Hamiltonian systems (PHSs), that is those having separable energy being quadrating in momentum (with constant mass matrix). We first introduce a discretization scheme which ensures a passivity equation relatively to the same storage and dissipation functions as the continuous-time PHS. A discrete controller is then obtained following the IDA-PBC design procedure applied to the discrete PHS system. This method guarantees that, from an energetic viewpoint, the discrete closed-loop behavior is similar to the continuous one. Under zero-state observability assumption, closed-loop stability then follows from LaSalle principle. The method is illustrated on an inertia wheel pendulum model