Contents I. Introduction
Direct current (dc) motors have been extensively used in variable speed applications since their flux and torque can be independently controlled by the field and the armature current. However, they have disadvantages due to mechanical commutator and brushes, which limit the performance in high-speed and high-voltage operating conditions. Induction motors, on the other hand, are much more difficult to control but have definite advantages since they: 1) have no commutator, no brushes, and no rotor windings in squirrel cage motors; 2) have a simple rugged structure; 3) can tolerate heavy overloading; and 4) can produce higher torques by a lower weight, a smaller size, and a lower rotating mass. The developments in vector and direct torque control techniques have made induction motors suitable for high performance servo applications such as automated production, transportation or rail traction systems, provided that accurate torque or flux estimation and critical parameter identification (winding resistances and load torque) are achieved to guarantee high performance and high power efficiency [1]. In particular, the knowledge of the rotor resistance is crucial for achieving high efficiency of induction motors controlled by indirect field oriented controls, whereas the estimation of the rotor flux angle is needed by direct field-oriented controls. Finally, the estimation of the stator fluxes (and electromagnetic torque) is necessary for direct torque controls. Nonlinear rotor flux observers for induction motors were first designed in the Seventies under the name of bilinear observers [2], [3], then a reduced order nonlinear observer was presented in [4], finally a complete theory for rotor flux observers, including observers with arbitrary rate of convergence, was successively developed in [5] and [6]. Since flux observers have been found to be very sensitive to winding resistance variations (typically due to the machine heating), adaptive flux observers have been designed for sensored and sensorless induction motor drives (see [7]–[10]). A pioneering work in this field is [10], which presented a method for the on-line adaptation of the rotor time constant. The flux observers can be made simultaneously adaptive with respect to both rotor and stator resistances [11], [12]. Additional work on state estimation and parameter identification in induction motors can be found in [13]–[36]. In particular, [13] is focused on the effect of parameter uncertainties on the observer performance. Kalman-like and adaptive observers are presented in [14] and [15], least square techniques are used in [16]–[18] while results based on linearizations around equilibrium points are proposed in [19] and [20] (this latter paper concerns the sensorless regenerative control of induction motors). Gradient techniques are adopted in [21] for parameter estimation in induction motors at standstill; sliding mode observers that are adaptive with respect to motor uncertain parameters are presented in [22]–[25]; a discrete time observer is proposed in [26]. Rotor resistance and mutual inductance are estimated in [27] while rotor resistance identification in steady-state conditions is discussed in [28] and [29]. The rotor time constant is estimated in [30] and [31] by simple on-line identification schemes while a current perturbation signal is injected in [32] for the estimation of rotor and stator resistances. Off-line parameter estimations are treated, for example, in [33]–[35]. A review on induction motor parameter estimation techniques can be found in [36]. Robust solutions are proposed in [37]–[39]. As can be observed, a number of different approaches have been used. However, no solution among the aforementioned ones is simultaneously characterized by: 1) overall structural simplicity with no use of sign functions, high gains or output time derivatives that lead to well-known implementation difficulties and high noise sensitivity; 2) rigorous derivation of persistency of excitation (PE) conditions that are naturally related to motor observability and parameter identifiability and only depend on exogenous signals; and 3) exponential convergence to zero of all the estimation errors (including the ones corresponding to the three critical parameters) so that the uncertain rotor flux modulus reference that minimizes the power loss at steady state can be actually estimated and imposed (see Section V). In this brief, (see [40] for its preliminary version) a novel nonlinear adaptive observer for induction motors is proposed. It is constituted by: 1) an adaptive stator flux observer which, under PE conditions with a clear physical interpretation, is able to estimate the motor fluxes and to identify the rotor resistance; 2) a stator resistance identifier whose design is performed, under physical requirements of stator resistance identifiability at steady state (verified to hold in experiments and simulations and analogous to the ones presented in [41] but with a rigorous two-time-scales analysis being here carried out), on a different time scale (see [42] and [43] and its preliminary versions in [44] and [45] for a similar approach for parameter estimation in induction motors) to isolate its estimation from the estimation of stator fluxes and rotor resistance; and 3) a load torque identifier to be used in conjunction with the adaptive stator flux observer and the stator resistance identifier. New insights on the behavior of an intuitively inspired observer (of structure similar to the one presented in [41]) are thus given through a detailed stability proof which, in contrast to [41], does not rely on linearization arguments around constant operating conditions. The benefits of the proposed estimation strategy compared with the existing solutions are as follows: 1) a stability analysis is performed and exponential convergence is guaranteed; 2) the use of redundant estimates as well as the assumption of bounded stator currents integrals (which is required in [11] and [12] and may be unrealistic in the case of biased current measurements) are definitely removed; 3) no sign functions are used either in the adaptation as in [25] or in the observer as in [22]; 4) the dynamic order of the adaptive flux observer is reduced to six (it is higher in [11] and [12]); and 5) the obtained PE inequality only depends on the machine operating conditions and is guaranteed to be satisfied in the typical case of constant motor speed and flux modulus and nonzero electromagnetic torque. Although the convergence analysis of the error dynamics holds only locally (whereas the algorithms with redundant estimates proposed in [11], [12], [14], and [15] are globally convergent), satisfactory simulation and experimental results confirm the effectiveness of the proposed approach even in practical applications, i.e., in the presence of measurement noise, modeling errors, and discretization effects.
In contrast to several other observers in the literature, which estimate the parameters on the same time-scale, sufficiently high rate of convergence is obtained in simulation and experiments.
