This paper presents and validates a new proposal for effective speed vector control of induction motors based on linear Generalized Predictive Control (GPC) law. The presented GPCPI cascade configuration simplifies the design with regard to GPCGPC cascade configuration, maintaining the advantages of the predictive control algorithm. The robust stability of the closed loop system is demonstrated by the poles placement method for several typical cases of uncertainties in induction motors. The controller has been tested using several simulations and experiments and has been compared with Proportional Integral Derivative (PID) and Sliding Mode (SM) control schemes, obtaining outstanding results in speed tracking even in the presence of parameter uncertainties, unknown load disturbance, and measurement noise in the loop signals, suggesting its use in industrial applications.
The Model Predictive Control (MPC) groups a set of controllers which are based on the model of the system and the known future reference for optimal control signal calculation. The operational principle of predictive control is to calculate in advance the control signal required by the system, when the future input reference that will be applied is known beforehand [
Predictive algorithms are often implemented using two or more GPC blocks to control several loops of the electrical machine, and usually they are connected in cascade form [
All predictive control schemes are based on the minimization of a cost function. In the GPC this implies solving a quadratic programming problem in the case where physical constraints are introduced in the optimization. If no constraints are considered, an analytical solution can be obtained. In this sense, it is known that all real systems have constraints, such as saturation values, frequencies, and time limits of the actuators [
In addition, even if the delay time of the electric motor systems is usually small, sometimes it can be long enough so that its compensation improves significantly the system behaviour [
Finally, the robustness of the GPC regulator is another aspect included by some authors in the controller design [
After all these considerations and taking into account the complexity in the design of predictive controllers and their important computational costs, this paper presents an induction motor speed indirect vector control that combines the GPC algorithm with PI regulators, proposing a simple, robust, and effective design which provides better dynamical behaviour than other speed regulators such as other GPC, PID, and Sliding Mode (SM). The rest of the paper is organized as follows. In Section
The objective of this paper is to demonstrate experimentally that the GPC algorithm can be used in speed regulation of induction motors in an effective way with a simple, robust, and stable design, offering faster speed tracking than other algorithms such as PID or SM, allowing being implemented in industrial applications.
The proposed speed regulator combines a GPC scheme with two PI current regulators. The dynamics of the induction motor is regulated using a distributed control in cascade form: the stator, (
The dynamics of the motor can be described by the stator voltage equations and the rotor flux equation, expressed all in the
The employed symbols are described as follows:
viscous friction coefficient,
moment of inertia,
magnetizing inductance,
stator inductance,
rotor inductance,
rotor resistance,
stator resistance,
number of poles,
leakage factor,
electromagnetic torque,
load or disturbance torque,
synchronous rotating speed,
mechanical rotor speed,
electrical rotor speed,
rotor flux vector,
direct and quadrature components of the rotor flux vector,
stator current vector,
direct and quadrature components of the stator current vector,
stator voltage vector,
direct and quadrature components of the stator voltage vector.
Figure
Diagram of GPC speed control of induction motor with the PI current control and SVPWM.
Next, the dynamics of the induction motor system will be calculated in order to design the GPC.
As it is known that the synchronous speed can be expressed as follows:
External representation of the induction motor with the PI current control and SVPWM.
Usually the rotor flux is held constant, fixing the rotor flux current command (
The electromagnetic torque of the induction motor in steady state, taking into account that torque and rotor flux current components are decoupled in` the
Considering that the SVPWM and the VSI modules have neither dynamics nor gain in diagram of Figure
Now, if the dynamics associated to the two current loops are neglected, because they are much faster than the rest, the following second order transfer function of the induction motor is obtained as follows:
Taking into account the consideration to obtain (
The design of the GPC controller is carried out using the first order transfer function of the motor. As the GPC controller is defined in discrete time, the transfer function must be transformed into a discrete time transfer function. Then, using the ZOH (ZeroOrder Hold) discretization method, it is obtained that
Taking into account GPC theory and employing the CARIMA model [
The GPC algorithm involves applying a control sequence that minimizes a multistage cost function of the form:
The prediction of the future output is as follows:
Taking into account that
Thus, from (
Assuming that the
The minimum of the cost function (
As the receding horizon strategy is used, the control signal applied to the process is obtained from the first element of
Now replacing the variables
Diagram of the induction motor speed control scheme using GPC linear regulator.
The analytical solution of the cost function minimization is possible only if the control signal (
The two current PI regulators use the same tuning values. The higher the bandwidth chosen for these controllers, the faster the current loops dynamics are. However, in practice, any real system’s bandwidth is limited physically. In the employed experimental platform (Section
Induction motor parameters (manufacturer).
Parameter  Symbol  Value 

Stator resistance 

0.81 
Rotor resistance 

0.57 
Magnetizing inductance 

0.117774 H 
Stator inductance 

0.120416 H 
Rotor inductance 

0.121498 H 
Nominal rotor flux 

1.01 Wb 
Number of poles 

4 
Nominal torque 

49.3 N m 
Moment of inertia 

0.057 kg m^{2} 
Viscous friction coefficient 

0.015 N m/(rad/s) 
Temperature coeff. Al/Cu 

0.0039 K^{−1} 
As to the speed regulation, in the first case, a PID speed controller has been used to measure the delay time of the system. This PID speed regulator has also been designed in the frequency domain, using a bandwidth which is 10 times smaller than the PI current loop regulators [
A slidingmode speed regulator for induction motor is also implemented in order to be compared with the proposed GPC regulator. This advanced speed regulator was presented by the authors in [
The tuning of the GPC regulator requires choosing the values of two horizons and two weighting factors. The control horizon,
Several stable cases of GPC speed regulator designs.
GPC speed regulator design  

D1 (efficient 1) 
D2 (efficient 2) 
U1 
U1 
U2 
U2 
 
PI currents regulators design  
 
Stator nominal electr. parameters 

Us1: 

Us2: 
The control weighting factor,
The closed loop stability of the motor
In this sense, it is necessary to translate the GPC controlled system parameters to its equivalent RST controlled system parameters [
The robust stability is analysed taking into account the parametric uncertainties of the induction motor in two limit real cases of each control loop detailed in Table
First, the following two uncertainty cases for the limits of robustness, related with the mechanical parameters of the induction motor, are considered in the proposed GPC. The U1 takes
Table
Nichols diagram of the GPC and PID speed loops.
Nichols diagram of the PI current loops.
The employed platform is composed by a PC with
Using D1 design for the GPC controller, simulation and experimental tests are carried out with a trapezoidal speed reference of 1445 rpm and 0.33 Hz, adding an square form load torque of 30 Nm (starting from the second period of the speed reference).
Results are shown in Figure
Simulation and real tests of the GPCPI model with D1 design with unknown load torque.
Figure
Comparative experimental tests: speed responses (a) and torque reference currents (b) of GPC D1 controller versus PI regulator.
In Figure
Comparative experimental tests: speed responses of GPC1 D1 controller versus GPC2 D1 regulator.
Figure
Comparative experimental tests: speed responses of GPC1 D1, SM, PI, and PID speed regulators.
Figures
Comparative experimental tests: speed responses of GPC1 D1, SM, PI, and PID speed regulators for constant acceleration case.
Comparative experimental tests: speed responses of GPC1 D1, SM, PI, and PID speed regulators for variable acceleration case.
Therefore, the use of the first order transfer function model is justified because the first order model simplifies the computational cost and the controlled performance contains less oscillations than the second order case.
One of the issues that usually exists in real applications is load disturbance, and, in the previous tests, the proposed GPC speed controller has demonstrated its performance even in presence of this effect. Moreover, parametric uncertainties that is, the change of values in the induction motor parameters can arise. These have been considered as U1 and U2 in Table
In this sense, the graphs of Figure
Simulation and real tests of the proposed GPC (D2 design) speed regulator with highlevel noise in the two feedback loops and unknown load torque.
The contribution of this work consists of the combination of the GPC algorithm in the speed loop with a PI based control in the current loops, using an easy and effective design, where the robust stability is demonstrated for typical induction motors’ parametric uncertainties. The GPC speed controller design is based on the first order model of the induction motor with a delay time, which is compensated. This regulator design is simpler to implement than other predictive proposed schemes, as neither constraints nor robustness terms have been taken into account.
The proposed controller has been tested using various simulation and experimental tests in the presence of the parametric uncertainties, unknown load disturbance, and measurement noise in the loop signals: the rotor speed and the stator current. The experiment demonstrates the effectiveness of the approach. Moreover, the presented results also show that the GPC speed regulator is considerably faster than the classic PID and slightly faster than the advanced SM speed controller, using the same computational cost. This work demonstrates that the GPCPI controller is an effective speed control algorithm, in both adverse and acceptable conditions, its robustness is clearly shown, the proposed control scheme is also easy to tune and to implement in a real system, and therefore it can be used in industrial applications.
The authors are very grateful to the UPV/EHU for its support throughout the project GUI07/08 and the Basque Government for the support to this work throughout the project SPE09UN12.