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The problem of active disturbance rejection control of induction motors is tackled by means of a generalized PI observer based discrete-time control, using the delta operator approach as the methodology of analyzing the sampled time process. In this scheme, model uncertainties and external disturbances are included in a general additive disturbance input which is to be online estimated and subsequently rejected via the controller actions. The observer carries out the disturbance estimation, thus reducing the complexity of the controller design. The controller efficiency is tested via some experimental results, performing a trajectory tracking task under load variations.

To obtain high performance control of electric machines there has been a growing interest in the design of controllers based on the discrete-time model of the system. In the case of induction motors, the system is continuous in nature, being necessary to obtain a sampled-time model. Preliminary studies on the sampling of continuous time nonlinear systems can be found in [

There exists a variety of control strategies for the induction motor that depend on the difficulty to measure parameters while their closed loop behavior is found to be sensitive to their variations. Generally speaking, the designed feedback control strategies have to exhibit a certain robustness level with respect to unknown bounded additive disturbance, in order to guarantee an acceptable performance. It is possible to (online or offline) obtain estimates of the motor parameters [

One variant of this scheme resorts to a local internal model characterization of the lumped disturbance using a representative element of a family of discrete-time polynomial signals of fixed degree. This results in a local self-updating polynomial model of the uncertainty which can be estimated, in an arbitrarily close manner, via a suitable extended linear observer of generalized proportional integral (GPI) nature [

For the case of the induction motor control, we consider a robust controller design based upon a simplified discrete model of the system including additive, completely unknown, disturbance inputs lumping nonlinearities and external disturbances whose effect is to be determined in an online fashion by means of a discrete-time linear observer of the GPI. The gathered knowledge will be used in the appropriate canceling of the assumed disturbances themselves while reducing the underlying control problem to a simple linear feedback control task. The control scheme thus requires knowledge of a reduced set of the motor parameters to be implemented.

The controller design for the induction motor is carried out within the philosophy of the classical field oriented controller scheme and implemented through a flux simulator or reconstructor (see Chiasson [

The control configuration for the first stage inherently includes a

In this case, the sampled time system is not defined purely in terms of the time-shift operator but in terms of the unified operator approach proposed by Goodwin et al. [

Here, the discrete-time GPI control has been proposed using the delta operator approach taking advantages of the high sampling rates and advantages of working directly in the sampled time system with respect to the continuous scheme, such as the faster implementation in a digital controller.

The remainder of the paper is organized as follows. Section

In this section, some preliminary concepts regarding the

The domain of possible nonnegative “times”

A time function

The

We will consider

The unified integration operation

The integration operator corresponds to the antiderivative operator.

In the case of the unified transform theory, the generalized exponential

The solution of (

If all the roots

Stability region for delta operator.

There is a close connection between the forward shift operator

We will just point out to the transform properties to be used throughout the work. A more extensive list of the delta transform properties is found in [

Consider the two-phase equivalent mathematical model

A simple way to obtain a discretization of the flux observer is using

An observer for this discretized system is given by

In order to analyze the stability of the discrete-time flux estimator, the estimation error is defined as:

The flux simulator variable,

It is assumed that only the shaft’s angular position,

The motor parameters are assumed to be known.

The load torque

Let us assume that the sampling period

Under the above assumptions, consider the induction motor dynamic model (

The proposed control strategy is based on a simplified vision of the system model (

From (

The control law

In this subsection, a methodology of disturbance estimation by means of a delta operator discrete-time observer, which can be associated to an extended Luenberger like linear observer, is developed.

The ideal performance of control systems and its dual estimation is to achieve zero steady-state errors in an asymptotic fashion. Given the uncertainty of unified disturbance signals (regarding external disturbances and the dynamics of the system) involved in the dynamics of the inner and outer loops of the proposed control scheme, it is necessary to make an approach to a generic model for signals. The approximation used and which is simpler to determine the internal model is given by the approximation of the truncated Taylor series. These families of functions with respect to disturbance signals are in agreement with the model

The disturbance inputs

The construction of the delta generalized proportional integral disturbance observer for

Define the observation error as

The disturbance estimation procedure is the dual counterpart of disturbance rejection mechanism which resides in the application of

Let us use (

ADRC-GPI observer-based controllers use an internal model approximation of the perturbation functions to reconstruct and reject the perturbations. Under this disturbance model approximation setting, several authors have applied it to different areas. Parker and Johnson used a first-order perturbation approximation to model wind speed perturbations in a wind turbine operating in region 3 [

The parameter

The ultimate bounded of the estimation errors, produced by the GPI observer, is strongly dependent on the product of poles magnitudes of the dominant characteristic polynomial for the estimation error. Given a desired ultimate value the use of a lager

GPI observers are bandwidth limited by the roots location of the estimation error characteristic polynomial. Generally, the larger the observer bandwidth is, the more accurate the estimation will be. However, a large observer bandwidth will increase noise sensitivity. Then, the selection of the roots of the estimation error characteristic polynomial affects the bandwidth of the GPI observer and also the influence of measurement noises on the estimations. Therefore, GPI observers are usually tuned in a compromise between disturbance estimation performance (set by the internal model approximation degree) and noise sensitivity.

The trajectory tracking problem is formulated in terms of the angular velocity. The disturbance observer, however, is treated in terms of the angular position second-order dynamics. This allows an alternative estimation of the angular velocity,

For the estimation of

Consider the observation error

The proof is similar to that of the previous proposition.

A two-stage feedback control law is considered for this system. In the first stage (outer loop), the angular position of the motor shaft is forced to track a reference signal

Consider again the linear system (

From (

Assuming a proper observer behavior related to system (

The characteristic polynomial of the tracking error,

Let

To assess the control approach, some experiments were carried out in a test bed including a controlled load, by means of a controlled coupled DC motor. The experimental induction motor has the following parameters: ^{2}],

The controller was devised in a MATLAB-xPC Target environment using a sampling period of

The reference trajectory of the velocity consisted in a series of rest to rest transitions with values

Figure

Velocity tracking results.

Regulation of the flux magnitude and control input.

Trajectory tracking of the stator currents and associated disturbance estimations.

Mechanical lumped disturbance estimation.

The main advantage of the control algorithm, using the xPC target environment, in a single tasking execution mode was the minimization of the execution time; in the case of the discrete-time control scheme, this time was

In this work, a discrete-time disturbance observer based control was proposed to solve the problem of controlling an induction motor. The discrete-time process based on the delta operator allows a faster digital control implementation scheme as well as some easy tuning strategies for both control and observer processes in relation to the pole placement for the closed loop tracking (and injection) errors. The presence of the observer in the control loop makes the proposed scheme quite simple and easy to implement. Besides, it is accurate in presence of different nature disturbance inputs.

The degree of polynomial approximation of the disturbance input, denoted by

Even though the control loops were proposed for first-order plants, the proposed observer based control can be extended without loss of generality to higher order systems.