Virtual Prototyping of Bond Graphs Models for Controller Synthesis through Energy and Power Shaping Sergio J. Junco Departamento de Electrónica Facultad de Ingeniería – Universidad Nacional de Rosario S2000EKE Rosario, Argentina Phone: +54-341-480.8543, Fax: +54-341-480.2654, E-mail: sjunco@fceia.unr.edu.ar Abstract – The foundations of a physically based method for control system synthesis in the Bond Graph (BG) domain are presented. Its rationale is stating the control problem as a stabilization one, which is solved via shaping the closed-loop energy and power flow functions. Procedurally, the first step is proposing a Target BG whose energy and power flow properties are in accordance with the control system objectives. Next, via virtual prototyping/manipulation of the Controlled BG (the open-loop plant BG-model), a Virtual BG equivalent to the Target BG is constructed, which allows calculating the control law as the final step, for which three alternatives techniques are given. Physical intuition –assisted by causal analysis and component exchange- helps in the phases of virtual prototyping, while the BG-version of Lyapunov’s Direct Method provides theoretical support to the methodology. The method works well for linear and nonlinear problems of equilibrium stabilization, regulation with internal stability and variable-reference tracking, as it is shown on the two simple examples handled throughout the paper. Keywords: Bond Graphs, Energy- and Power-shaping Based Control, Lyapunov Methods. I. INTRODUCTION The choice of Bond Graphs is most convenient for control system analysis and design in the physical domain. Indeed, BGs directly show the energy storage and power flow processes in the system, thus allowing for the physical interpretation of the control problem. The ease of mathematical manipulation on causally augmented BGs complements the previous feature with an essential requirement for controller computation. Developing a BG-supported methodology for controller design aims at providing simpler control systems respecting physical constraints, and also at facilitating the design tasks when multidisciplinary teams are responsible of them. General BG-methods to solve standard linear control problems [Bertrand et al., 1997; 2001], as well as analysis and control synthesis techniques for some classes of nonlinear systems [Grujic and Dauphin-Tanguy, 2000], [Junco, 1993; 2001a,b], [Wu and Youcef-Toumi, 1995], [Yeh, 2001] are available. This paper revisits the author’s own results on the direct application of Lyapunov Second Method on BGs and their formal use to controller design [Junco, 1993; 2001a,b]. This re-examination is made with the objective of providing a methodology based on physical intuition on the one hand, and supported by theoretical methods rigorously assuring the correctness of the results, on the other. Constructing a Control Lyapunov Function [Sontag, 1998] insuring stability of an equilibrium point is reinterpreted in this context as shaping the closed-loop energy function and injecting some damping to force its negative definiteness. Two concepts are introduced in order to express these undelying principles in the BGdomain: the Target BG, which captures these energy and power-flow properties, and the Virtual BG, which being equivalent to the Target BG allows for the computation of the control law. Section II reviews the available results on stability and stabilization using Lyapunov Second Methods directly on BGs, and suggests the energetic reinterpretation. Section III develops the main results with the help of two simple example systems. Section IV summarizes the methodology in the form of a design procedure. Finally, the conclusions are presented in Section V. II. EQUILIBRIUM STABILIZATION WITH LYAPUNOV DIRECT METHOD ON BGS A. Background on Lyapunov Second Method on BG. To fix ideas consider the controlled dynamical system x(t)=f(x(t),u(t)) & , where x and u denote state and input vectors, respectively. Without loss of generality suppose that for u º 0 the state space origin is an equilibrium point (EP), i.e., f(0,0)=0 . Analyzing the stability of the origin using Lyapunov’s second method basically implies choosing a scalar positive-definite function (pdf) V(x) (written V(x) >0) and studying the sign of f (x) L V ( f (x) L V B (x(t))/ dV dt , or (x( )) V t & or V& , for short, is the orbital or Lie-derivative of V, i.e., its time-derivative along the orbits, or trajectories, or solution paths of the state equation system). This is usually done evaluating the scalar product between the gradient of V(x) and the vector .....