Bond graph formulation of an optimal control problem for linear time invariant systems

A recent communication has proposed a conjectural procedure for representing a category of optimal control problems in bond graph language [W. Marquis-Favre, B. Chereji, D. Thomasset, S. Scavarda, Bond graph representation of an optimal control problem: the dc motor example, in: ICBGM’05 International Conference of Bond Graph Modelling and Simulation, New Orleans, USA, January 23-27, 2005, pp. 239-244]. This paper aims at providing a fundamental theory for proving the effectiveness of this procedure. The class of problem that the procedure can deal with has been extended. Its application was formerly restricted to linear time invariant siso system. The systems considered now are linear time invariant mimo systems. The optimization objective is the minimization of dissipation and input. The developments concerning the optimal control problem are based on the Pontryagin maximum principle and the proof of the effectiveness of the procedure makes a broad use of the port-Hamiltonian concept. As a result, the bond graph representation of the given optimization problem enables the analytical system, which provides the optimal solution, to be derived. The work presented in this paper is the first step in research with perspectives towards formulating dynamic optimization problems in bond graph and, towards coupling this formulation with a sizing methodology using bond graph language and a state-space inverse model approach. This sizing methodology, however, is not the topic of this paper and thus is not presented here.     

A bond graph algorithm to determine the equilibrium state of a system

An algorithm is presented which enables one to determine the nature of the equilibrium state of a system with constant inputs by direct inspection of its bond graph representation. The algorithm determines whether or not there exists a unique equilibrium state (within a certain region). If not, the system may have no (or infinitely many) equilibrium states in the linear case.     

Bond graph fluid line models for inclusion with dynamic systems simulations

A procedure is presented whereby the control volume equations for one-dimensional, compressible gas dynamics are cast into first-order, state variable form. These equations are interpreted using causal bond graphs. The resulting bond graph is shown to reduce to the classic I-C chain under acoustic constraints and to a more recently developed model of low speed thermal energy transport subject to associated constraints.Through example it is demonstrated that the control volume bond graph is easily coupled to an overall system model and thus can be digitally simulated as part of the overall nonlinear state space representation. The result is that a very accurate gas dynamic model can be coupled with an overall dynamic system model without requiring a prohibitively large number of equations.     

Modelling of switching systems in bond graphs using the concept of switched power junctions

Modelling of switching systems using bond graph is a research issue. There have been many proposals in this context. However, there are many issues that need consideration. This paper proposes the concept of switched power junctions that is a generalisation of the conventional junction concept in bond graph modelling. This approach removes most of the difficulties associated with other methodologies. This method ensures that causality remains invariant during mode switching thus keeping the state vector and its dimension time invariant. Further, the switched power junction displays all feasible system modes at the same time on the same graph. This concept is further illustrated through simulation using examples of switching systems in the electrical domain.     

Examination of chaotic behaviours using bond graph model

In this paper, modelling and simulation of Chua's chaotic oscillator, which exhibits rich chaotic behaviours, are presented by using the bond graph model. Up to now modelling of Chua's chaotic oscillator using bond graph model is not yet developed. The non-linear resistor in the circuit is modelled in this contribution by linear time-invariant components and ideal switches using piecewise linearization approach. The bond graph model of all the circuit including switches is then generated. Simulations are provided via the computer program called as BOMAS using the obtained bond graph model. Finally, Chua's circuit is verified experimentally. It is shown that all experimental and simulation results well agree with the chaotic behaviours of Chua's circuit.     

Algebraic system theory: an analyst's point of view

A systematic development of the realization theory of finite dimensional constant linear systems is presented which synthesizes the various currently available approaches.Based on representation theorems for submodules and quotient modules of spaces of polynomial matrices and vectors, this paper combines the abstract algebraic ideas centering around module theory, the use of coprime factorizations of rational transfer functions and state space equations into a unified theory.     

Modelling and Passivity Based Control of switched systems from bond graph formalism: Application to multicellular converters

This paper addresses the modelling and control of switched systems with Boolean inputs. A generalization of Passivity Based Control (PBC) is proposed and fitted to bond graph formalism. The state equations of the equivalent average model are first deduced from the original bond graph using the notion of commutation cells and then interpreted according to Port Controlled Hamiltonian formalism. The whole approach is presented in a formal way. This method is then applied on a multicellular serial converter, which is widespread in power systems and of growing interest. The application of PBC associated to a modelling approach using commutation cells on a non-trivial example shows its efficiency to determine a generic controller, the number of elementary cells being considered as a parameter.     

Estimating Eigenvalues for a class of Dynamic systems

In a previous work an approach was presented for extracting information about the eigenvalues of a linear, time-invariant dynamic system directly from a graphical model. In this paper a generalization is given of the results previously obtained. For some particular cases we can give the entire spectrum. For these cases the spectrum is shown as a function of the structure of an abstract form of the bond graph model. For more general cases, we give upper limits on the imaginery part, and upper and lower limits on the real parts, of the eigenvalues. In contrast to most existing methods, the information about the eigenvalues is generated prior to deriving the state equations. When suitably automated, the results obtained here can provide a considerable reduction in the computational effort required to get information about eigenvalues. This feature is particularly useful in an interactive design context.     

Stabilization irrespective of bounds of uncertain variations for linear uncertain systems with delays

In this study, the stabilizability irrespective of the bounds of uncertain parameters and time delays is investigated for linear uncertain delay systems. For uncertain systems without delays, a linear time-varying or time-invariant uncertain system has been shown to be stabilizable independent of the bounds of uncertain variations if and only if the system has a particular geometric configuration called an antisymmetric stepwise configuration (ASC) or a generalized antisymmetric stepwise configuration (GASC), respectively. In this study, fundamental approaches to investigating the stabilizability of delay systems with specific uncertainty structures such as ASCs or GASCs are presented. For a class of 3-dimensional systems, it is shown here that if a linear time-varying or time-invariant uncertain delay system has an ASC or a GASC, respectively, then the system can be stabilized, however large the given bounds of delays and uncertain parameters might be.     

Consensus of linear multi-agent systems based on full-order observer

This paper considers distributed consensus problem of multi-agent systems consisting of general linear dynamics with a time-invariant communication topology. A distributed full-order observer type consensus protocol based on relative output measurements of neighbor agents is proposed. It is found that the consensus problem of linear multi-agent systems with a directed communication topology having a spanning tree can be solved if and only if all subsystems are asymptotically stable. Some necessary and sufficient conditions are obtained for ensuring consensus in multi-agent systems. The design technique is based on algebraic graph theory, Riccati inequality and linear control theory. Finally, simulation example is given to illustrate the effectiveness of the theoretical results.     

Order Reduction of Linear System Models with a Time-Frequency Domain Method

A mixed method of model-order reduction is introduced, which is based on the linear state-space representation of the original system. The derivation of its transfer function matrix expression uses the Leverrier algorithm and the application of well known partial-fraction expansion techniques together with the concept of dominant eigenvalues on the transfer function of the system. The method is general, relatively simple to apply and yields fairly accurate reduced-order models. As an illustration the method has been applied successfully to a tenth-order two-input two-output time-invariant linear model of a practical power system.     

Pseudo bond graphs for generalized compartmental models in engineering and physiology

True bond graphs, which use effort and flow variables whose product is power, can in principle be used to describe all types of physical systems. However, many system models do not use power variables and yet can be represented usefully as pseudo bond graphs. Pseudo bond graphs have been used particularly for open systems in which it is convenient to consider control volumes or compartments with boundaries across which mass can flow. In this paper, we show how the bond graph methods used for conductive and convective heat transfer can be generalized to account for diffusion, convection, and accumulation of a variety of physical quantities and how pseudo bond graphs can aid in constructing and representing such models. These models are known in mathematical biology as “compartmental models” and it is a main contribution of this paper to show that the same pseudo bond graphs apply to thermofluid and physiological dynamic models. The bond graphs build in some conservation principles automatically and yet have the flexibility to incorporate general multiport laws when necessary. Thus the pseudo bond graphs can exhibit system structure as do other network graphs and are very general in nature.     

Stabilization and passification of distributed-order fractional linear systems using methods of preservation

This paper studies the stabilization and passification of a class of distributed-order linear time-invariant systems, by using methods of preservation in the frequency domain. Results about preservation of stability and passivity of classical linear time-invariant systems are extended to one more general family of matrix functions. Based on these results, a new approach to the problems of stabilization and passification of distributed-order linear time-invariant systems is presented. Also a result that extends the known techniques for pole placement of classical linear time-invariant systems to the new class of distributed-order linear time-invariant systems is given. Examples are given to show the validity of theoretical results.     

Bond graphs in control: Physical state variables and observers

Bond graph modeling techniques yield state equations intimately related to energy storage in physical systems. The easy physical interpretation of the state variables from a bond graph model aids in the realization of modern automatic control schemes involving state variable feedback. When it is inconvenient or impossible or measure certain state variables, the useful device of an observer may be used to estimate the missing state variables. It is shown that the same bond graph used to model the system can be used to derive a complete or reduced order observer. The partial observers use derivative causality in a new way and in some cases, the effects of completely unknown disturbances may be accounted for.     

Reduction of models of large scale lumped structures using normal modes and bond graphs

Bond graphs are an extremely useful modeling procedure for representing the actual energy exchange mechanisms of interacting dynamic systems. Governing state equations are straightforwardly obtained from the bond graph; however, for large structures, a restrictively large number of equations can result. A procedure is developed whereby the original equations are reduced to a form suitable for modal decomposition. The resulting modes are reinterpreted in bond graph form with the resulting model being an extremely accurate system representation while requiring only a fraction of the original number of equations. The procedure is demonstrated through example.     

Bond graph models of structured parameter uncertainties

Taking into account uncertainties in model parameter values is a crucial point for studying robustness in modeling and in control. This paper proposes to construct in a systematic and graphical procedure two forms of the linear state equation usually found in the literature for dealing with the robustness problem in the case of structured uncertainties on parameters. It is shown how to model uncertainties in a bond graph approach, in the case of additive or multiplicative parametric variations.     

Non-reduced optimal observers for time-varying linear systems

A method is presented for constructing a nonreduced observer for a finite dimensional linear system. The systems considered are nth order, time-invariant or time-varying forced linear systems which are assumed to be bounded and uniformly completely state reconstructible. The observer dynamics are derived from an optimal estimation formulation which is related to the concept of observability. The cost functional used in this formulation includes a parameter which determines the relative weighting of the system output and the estimate of the initial system state. The result is a practical method of estimating the state of a linear system from noiseless measurements of the input and output. The results are also of theoretical significance in that a direct connection between reconstructibility and the existence of observers is established.     

Hilbert—Schmidt Systems with Finite-Dimensional State Spaces

The problem of characterizing the input-output behaviour of those linear systems which admit internal representations with finite-dimensional state spaces is studied in this paper. The systems are assumed to be Hilbert-Schmidt systems, and the main result is derived using well-known properties of the kernels of the associated Hankel operators. A necessary and sufficient condition is derived for a system to admit a completely observable and completely accessible internal representation with a finite-dimensional state space.     

A new deep graph attention approach with influence and preference relationship reconstruction for rate prediction recommendation

Graph neural networks have been frequently applied in recommender systems due to their powerful representation abilities for irregular data. However, these methods still suffer from the difficulties such as the inflexible graph structure, sparse and highly imbalanced data, and relatively shallow networks, limiting rate prediction ability for recommendations. This paper presents a novel deep dynamic graph attention framework based on influence and preference relationship reconstruction (DGA-IPR) for recommender systems to learn optimal latent representations of users and items. The entire framework involves a user branch and an item branch. An influence-based dynamic graph attention (IDGA) module, a preference-based dynamic graph attention (PDGA) module, and an adaptive fine feature extraction (AFFE) module are respectively constructed for each branch. Concretely, the first two attention modules concentrate on reconstructing influence and preference relationship graphs, breaking imbalanced and fixed constraints of graph structures. Then a deep feature aggregation block and an adaptive feature fusion operation are built, improving the network depth and capturing potential high-order information expressions. Besides, AFFE is designed to acquire finer latent features for users and items. The DGA-IPR architecture is formed by integrating IDGA, PDGA, and AFFE for users and items, respectively. Experiments reveal the superiority of DGA-IPR over existing recommendation models.     

Dynamic forces in multiport mechanics: Direct and Lagrangian formulations

Holonomic systems can be represented by a bond graph in which inertial variables are related to generalized variables using a multiport displacement-modulated transformer structure. It is shown that the part of the generalized forces due to inertial elements can be found either by direct calculation or by the operations of Lagrange's equations. Although no gyrators are needed for the representation, gyrational coupling may exist. Alternative representations may involve gyrators explicitly and it is possible to construct kinetic co-energy functions for these gyrators for use in Lagrange's equations.