This work proposes a realistic solution to the control problem of sensorless induction motors. Due to some important aspects related to their construction and reliability, the induction motors are extensively used in many modern industrial applications. Considering that the system is facing the lack of hardware sensors, the proposed complex control strategies are based on the estimation of unavailable system variables and parameters. In order to control the rotor speed, two robust control strategies are proposed: a modified super-twisting adaptive technique and a model predictive technique. The tests performed under several practical assumptions show that the closed loop behaviour of the system is adequate, and the output variable follows the imposed time varying reference, despite the considered uncertainties and disturbances acting on the process.
Nowadays, induction motors are facing an interesting challenge from the perspective of modelling and sensorless control. This is mainly caused by some particular, inherited operating conditions. In the last decades, due to the environmental rules imposed by the international institutions, the induction motors have been proposed to be a reliable solution for the usual drive systems.
Regarding the control design of these systems, beside the classical scalar control and vector control strategies [
Two specific problems are found in practice: first, the models are uncertain [
The present work approaches a linked observer—estimator used to estimate the unmeasurable state and those parameters that are uncertain or unknown. The proposed reduced-order state observer is designed by using an appropriate linear transformation and provides the reconstruction of rotor fluxes. In what concern the estimation of unknown process parameters (e.g., the stator resistance) and of the load torque, acting as an external disturbance on the rotor, a parameter estimator and a disturbance observer were developed. The parameter estimator is derived from a typical one used in biotechnology applications [
Using the estimates provided by the proposed observers, two control strategies were proposed: a modified super-twisting algorithm (STA) and a robust model predictive control (RMPC), designed such that the output (i.e., rotor speed) follows a chosen time-varying reference.
The main objective of the super-twisting algorithm proposed by Levant in his work [
A super-twisting algorithm is typically used to impose zero values to the sliding variable and its time derivate in a finite time, to remove the chattering effect and to preserve the robustness by improving the disturbances rejection performance.
Our proposed approach uses an adaptive gain in the definition of the sliding surface in order to cope with time-varying disturbances acting on the system.
The second considered control algorithm is an optimal one, named model predictive control. This algorithm prevailed as an efficient method in widespread applications due to its optimal characteristics and some inherent features concerning the stability and robustness [
The main contributions of the paper consist in the design of a linked estimator-observer for unmeasurable/unknown variables of the process, and in the development of two novel modified robust control strategies that use the “software” information provided by the designed observers.
To emphasize the estimation, tracking, and robustness performances of the proposed algorithms, several realistic tests were performed, and some metrics defined in accordance with the tracking error were computed.
The combination of estimation and control algorithms leads to complex structures necessary for the general control objective. Because it is necessary to use combined information “software” from estimators and “hardware” from sensors, the general control structure must provide reliable solutions to problems related to convergence and to robustness (it is considered that the system is one subject to external disturbances). Also, the complexity of considered control strategies is an intrinsic one: the super-twisting algorithm requires a correct definition of the slip surface and the choices of the tuning parameters, due to the nonlinearities introduced by the control law, and moreover the predictive algorithm requires solving a minimization problem with constraints.
The fifth-order dynamical model of an induction motor [
The parameter
For systems (
For the process described above, the objective is to control the rotor speed (
Some practical assumptions are considered: The rotor fluxes and the controlled variable are unmeasurable The induction machine operates in a synchronous reference frame (
Thus, the controlled variable is the rotor speed; that is,
Therefore, we can formulate the following control problem: the considered output will asymptotically track some desired trajectories despite any external disturbances and uncertainties related to some time-varying process parameters and state variables (unknown or unmeasurable). To resolve this problem, we introduce state and disturbances observers as well as a parameter estimator, considering practical and technical hypotheses. Then, by means of these observers, we derive two control strategies. In order to capture the behaviour of the closed loop systems from practical operating conditions point of view, it was considered that a measurement noise acts on the input variables.
We assume that we have hardware sensors for the stator currents along the
For the process described by the dynamical model (
Under
We will use an appropriate linear transformation:
From (
The performance of the observer (
Recall the hypothesis
Let us define the positive definite Lyapunov candidate function:
Thus,
Therefore,
The OBE is defined, without loss of generality, under the next hypotheses:
Then, from (
Thus, for the initial nonlinear system, the following linked observer-estimator is proposed:
Moreover, based on (
To estimate the possible unknown external disturbances (the load torque), the following observer is proposed for the initial nonlinear model [
Let
If the following hypotheses hold,
then,
Based on observer (
Under the previous assumptions (see Section
We define an appropriate sliding surface [
The control law is a defined as
The time derivative of (
Using relations (
Thus,
The existence of the abovementioned control component (
In the literature, usually, the super-twisting controller has the following form [
The convergence of system (
However, in accordance with our physical restrictions imposed for the control input, we propose the following modified form of the relation:
Moreover, to improve the robustness of considered controller, we will suppose that the gain
Therefore, the vector of control inputs can be expressed as
The discrete model which will be used is a linear approximation of system (
So, by imposing
We define the discrete-time model:
The matrices
Let us consider the constrained minimization problem:
where
Problem (
Then, the solution is
Moreover, the receding horizon strategy [
The sufficient conditions (proper terminal cost, adequate prediction horizon, and so on) to ensure the convergence of the closed loop system were defined in [
The general scheme of proposed estimation and control is depicted in Figure
The estimation and control scheme proposed for the induction machine.
Estimator (
It is worth noting that for the abovementioned values, matrix
The tuning matrix, used for the estimation of
The estimations of the unmeasured state variables, of the unknown variables
Evolution of
Evolution of
The time profile of
The presented graphics show an asymptotic convergence of the proposed observer despite the variation of the stator resistance and the considered external disturbance of the process, represented by the load torque exerted on the rotor.
The behaviour in closed loop with the proposed control strategies (super-twisting and predictive) is presented in Figures
Evolution of output by using the proposed super-twisting control algorithm.
Evolution of output by using the proposed robust predictive control algorithm.
The control strategies were carried out using the following tuning parameters and weighting matrices (obtained by trial and error method) and the control input constraints: Super-twisting controller: Predictive controller:
The equilibrium points used in linearization process were determined by relation (
The first considered scenario is the “ideal” case, when the measured variables (the stator currents) are not perturbed.
Figures
Evolution of control inputs.
Evolution of optimal control inputs.
The components of control input on
From the presented graphics, it can be noticed that the control laws have the ability to maintain the output close to its reference. The control aim was attained even if for the design we used less a priori information about the process and despite the considered variation of the process parameter (the stator resistance, see Figure
Time profile of the stator resistance.
Time-varying profile of the load torque.
Moreover, the second scenario aims to verify the robustness of control algorithms, from another perspective: the presence of noise in the acquisition of the measurable variables (the stator currents). It is considered that the measurements are corrupted with white noise (variation of 10% from their nominal values). The tuning parameters and weighting matrices were described above.
The evolutions of the output variable and control effort are presented in Figures
Evolution of output—super-twisting control (noisy measurements).
Time profiles of control inputs (noisy measurements).
Evolution of output—predictive controller (noisy measurements).
Time profile of control input (noisy measurements).
Figure
Time profiles of stator currents.
The tests were accomplished in the MATLAB environment [
For a better comparison, the controlled system performance was also analysed by using some metrics, defined in accordance with the tracking error as Integral time absolute error, Mean absolute magnitude of the error,
The obtained values of the abovementioned metrics are highlighted in Table
Performance criteria results.
Performance indices | Modified STA | Modified STA (noisy measurements) | Predictive control | Predictive control (noisy measurements) |
---|---|---|---|---|
ITAE | 29.19 | 29.27 | 15.14 | 15.29 |
MAE | 2.34 | 2.35 | 2.16 | 2.17 |
The proposed estimation and control strategies can be applied to many practical situations involving the use of an induction motor, mostly in a sensorless layout. The rotor speed is controlled with partial data provided by the hardware sensors and by using the software sensors to provide the unknown or unmeasurable variables of the process.
The present work approached two realistic control strategies dedicated to sensorless induction motors. Two complex robust controllers were proposed: a modified super-twisting adaptive (STA) technique and a model predictive (MPC) technique. The STA approach used an adaptive gain for the sliding surface to handle the time-varying disturbances acting on the system. The MPC used an objective function that casts the disturbances variable, obtaining in this way an improved robustness of the controller.
Due to the lack of useful measurements, some specific observers were designed in order to successfully implement the robust controllers. More precisely, an innovative linked observer—estimator—was designed and used to reconstruct the rotor fluxes. Also, a parameter estimator and a disturbance observer were developed to cope with parameter uncertainties and load torque estimation.
The overall estimation and control schemes were tested under several practical assumptions concerning the induction motor. The behaviour of the closed loop system for both robust control schemes is satisfactory, taking into account the realistic and harsh simulation scenarios.
The simulation results and the computed performances indices showed that the best control results were obtained in the case of MPC, which provided better results from the robustness point of view.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This work was supported by the European Regional Development Fund, through the Competitiveness Operational Program (TISIPRO project, ID: P_40_416/105736, 2016–2021).