A Bond-Graph Method for Flatness-Based Dynamic Feedback Linearization Controller Synthesis: Application to a Current-Fed Induction Motor

This paper presents a method for controller synthesis in the bond graph domain. A dynamic feedback linearization and decoupling controller of rotor speed and rotor-flux amplitude of a current-fed induction motor is derived on the basis of a flatness analysis entirely conducted on a two-input nonlinear bond graph model of the motor. Once the rotor speed has been found as the first flat output, a technique which uses a variational bond graph and its associated quotient bond graph (modulo the differential of the first flat output) allows identifying the second flat output as being the angle of the rotor flux-linkage space vector. The flat output parameterization of the control outputs is later used to derive the control law. Simulation results are given to demonstrate the control system performance. INTRODUCTION The main concern of this paper is a methodological approach on bond graphs (BG) to the development of control laws for nonlinear physical systems, thereby linking theoretical methods of nonlinear control with physically based modeling formalisms. Some available results for multivariable models, based on differential geometry theory, exploit the notion of relative degree for the input-output decoupling, disturbance rejection and exact linearization problems. Nevertheless, these techniques which have a nice graphical (BG) interpretation in the linear case (Bertrand et al. 1997), cannot easily extended to nonlinear BGs. Moreover, they are restricted to systems without unstable zero dynamics. A relatively recent and elegant technique that solves complex control problems is based on the property of flatness introduced in (Fliess et al. 1995). This technique, that enables to overcome the problem of unstable zero dynamics and to solve trajectory planning and tracking problems, has shown itself well suited to be performed on graphical models of dynamic systems. The most effective method to analyze the flatness property is based on the freeness of the tangent (variational) module associated with the nonlinear model. Indeed, in most cases the flatness property itself and the differential parameterization of the system variables by the flat outputs are not easily pointed out directly on the nonlinear system model. A variational approach introduced in a previous work for nonlinear BG models (Achir and Sueur 2005) enables to identify the flat outputs of non linear systems. Although the proposed algorithm is easily applicable in case of single input BGs, it comes up against the problem of computing the Kronecker indices in the multi input case. Nevertheless, it was shown that many BG tools developed for linear systems are no longer valid due to the fact that the model coefficients are time dependent and therefore new rules dealing with such problems have been proposed. In this paper, the synthesis on the BG domain of a dynamic feedback linearization and decoupling controller of rotor speed and rotor-flux amplitude of a current-fed induction motor (IM) is proposed. As this is a multivariable (two-input, two-output) application of flatness, the method of the quotient BG is introduced in order to help obtaining the second flat output once the first flat output variable has been guessed on the base of physical grounds. This technique is an adaptation to the BG domain of a previous result (Fossas et al. 1998). The article is organised as follows: Second Section presents a BG model of the current-fed IM which is well adapted to perform the flatness analysis that follows in the same section. Third Section addresses the controller synthesis of the IM. Next, Fourth Section presents some simulation results demonstrating the closed-loop performance of the control system. The conclusions are presented in Fifth Section. Some mathematical background on flatness, the concept of ring BG are recalled in the Appendix. FLATNESS ANALYSIS ON A BG MODEL OF THE IM BG Model of Current-Fed Induction Motor A BG model of a current-fed induction motor convenient for the purposes of this paper is given in Fig. 1. It is set forth in a stationary (or stator fixed) reference frame with orthogonal axes d and q. The controlled flow sources impress the stator currents isd,q. The energy variables associated to both generalized inertia components I:Lr are Φrd,q, the rotor flux linkage components. The inertia I:J is related to the angular momentum Jω; linear friction with coefficient b is assumed. The coefficients M and Lr denote the mutual and rotor self-inductance, respectively; Rr is the rotor resistance, and p the number of pole-pairs. Applying standard BG equation-reading procedures (Karnopp et al. 1990) on each of the three 1-junctions associated to the mentioned energy storages, this BG yields the set of voltage and mechanical balance equations (1) and (2), respectively. Note that the driving torque (3) is exclusively supplied by the diagonalplaced MGYs (moduli pMΦrx/Lr) on the left of the BG, while both MGYs on the right (moduli pΦrx) provide a zero net effort on the mechanical 1-junction. Equations (4) and (5) relate the pair of BG-currents (id , iq) to (isd , isq) and (ird , irq), the stator and rotor currents, respectively.