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This paper investigates the application of the model predictive control (MPC) approach to control the speed of a permanent magnet synchronous motor (PMSM) drive system. The MPC is used to calculate the optimal control actions including system constraints. To alleviate computational effort and to reduce numerical problems, particularly in large prediction horizon, an exponentially weighted functional model predictive control (FMPC) is employed. In order to validate the effectiveness of the proposed FMPC scheme, the performance of the proposed controller is compared with a classical PI controller through simulation studies. Obtained results show that accurate tracking performance of the PMSM has been achieved.

Permanent magnet synchronous motors fed by PWM inverters are widely used for industrial applications, especially servo drive applications, in which constant torque operation is desired. In traction and spindle drives, on the other hand, constant power operation is desired [

The PMSM drive system has been controlled using a PI controller due to its simplicity. The PI controller, however, cannot provide good performance in both transient and load disturbance conditions. Several researchers have investigated the speed controller design of adjustable-speed PMSM systems to improve their transient responses, load disturbance rejection capability, tracking ability, and robustness [

The MPC controller generally requires a significant computational effort. As the performance of the available computing hardware has rapidly increased and new faster algorithms have been developed, it is now possible to implement MPC to command fast systems with shorter time steps, as electrical drives. Electric drives are of particular interest for the application of MPC for at least two reasons:

they fit in the class of systems for which a quite good linear model can be obtained both by analytical means and by identification techniques;

bounds on drive variables play a key role in the dynamics of the system; indeed, two main approaches are available to deal with system constraints: antiwindup techniques, widely used in the classical PI controllers, and MPC. The presence of the constraint is one of the main reasons why, for example, state space controllers have limited application in electrical drives.

In spite of these advantages, MPC applications to electrical drives are still largely unexplored and only few research laboratories are involved in them. For example, generalized Predictive Control (GPC)—a special case of MPC—has been applied to induction motors for only current regulation [

In this paper, a centralized MPC with large prediction horizon for PMSM speed control is presented. The proposed centralized scheme improves the control performance in a coordinated manner.

Another challenge of centralized MPC for PMSM is its large computational effort needed. To overcome this drawback, a functional MPC with orthonormal basis Laguerre function [

The dynamic model of the PMSM can be described in the

The basic principle in controlling the PMSM is based on field orientation. This is obtained by letting the permanent magnet flux linkage be aligned with the

Applying the field orientation concept (the electromagnetic torque is linearly proportional with

Model predictive control uses an explicit model of system to predict future trajectory of system states, and outputs. This prediction capability allows solving optimal control problem online, where prediction error (i.e., containing difference between the predicted output and reference output) and control input action are minimized over a future horizon, possibly subject to constraints on the manipulated inputs, states, and outputs. The optimization yields an optimal control sequence as input and only the first input from the sequence is used as the input to the system. At the next sampling interval, the horizon is shifted and the whole optimization procedure is repeated. The main reason for using this procedure, which is called receding horizon control (RHC), is that it allows compensating for future disturbance and modeling error.

The basic structure of model predictive control is depicted in Figure

Basic structure of model predictive control.

In this paper, the state space model of the system is used in the model predictive control. The general discrete form of the state space model used in model predictive control is of the form:

The final aim of model predictive control is to provide zero output error with minimal control effort.

Therefore, the cost function

The constraints of model predictive control include constraints of magnitude and change of input, state, and output variables that can be defined in the following form

In the classical model predictive control, the future control signal is modeled as a vector of forward shift operator with length of

A solution to this drawback is the use of functional MPC. In the functional MPC, future input is assumed to be a linear combination of a few simple base functions. In principle, these could be any appropriate functions. However in practice, a polynomial basis is usually used [

In this paper, orthonormal basis Laguerre function is used for modeling input trajectory. Laguerre polynomial is one of the most popular orthonormal base functions which has extensive applications in system identification [

Closed-loop performance of MPC depends on the magnitude of prediction horizon

In this section, the Laguerre-based model predictive control and exponentially weighted model predictive control are combined in order to alleviate computational effort and reduce numerical problems. At first, a discrete model predictive control with exponential data weighting is designed. The input, state, and output vectors are changed in the following way:

The optimal control trajectory with transformed variables can be achieved by solving the new objective function and constraints:

After solving new objective function with new variables, the calculated input trajectory should be transformed into standard variable with the following equation:

choosing of the proper tuning parameter

transforming the system parameters (

the objective function with its constraints is created based on (

optimizing objective function based on Laguerre polynomial and then calculating unknown Laguerre coefficients;

calculating input chain from (

the calculated weighted input chain is transformed into unweighted input chain using (

The block diagram of the field-oriented PMSM with the proposed FMPC is shown in Figure

input weight matrix:

Block diagram of the proposed PMSM speed control system.

The constraints are chosen such that the

The constraints imposed on the control signal are hard, whereas the constraints on the states are soft, that is, small violations can be accepted. The constraints on the states are chosen so as to guarantee signals stay at physically reasonable values as follows:

The speed error is fed to the speed controller (FMPC) in order to generate the torque current command

The entire system has been simulated on the digital computer using the Matlab/Simulink/Powerlib software package. The motor used in the simulation procedure has the following specifications [

PMSM: 1.5 kw, 240 V, 2-pole, 4250 rpm,

Computer simulations have been carried out in order to validate the effectiveness of the proposed scheme. The simulation tests are carried out using Matlab/Simulink software package. Wherever, the state space model of the permanent magnet synchronous motor is programmed with the functional model predictive algorithms in MATLAB work space.

The MPC control algorithm depends on the solution of a constrained optimization problem. Most designers choose

choose the control interval such that the plant’s open-loop settling time is approximately 20–30 sampling periods (i.e., the sampling period is approximately one-fifth of the dominant time constant),

choose prediction horizon to be the number of sampling periods used in step 1,

use a relatively small control horizon, for example, 3–5.

Selection of suitable values of a and

In the proposed system under study, the parameters of the FMPC are adjusted to be

It is assumed that the machine follows a certain speed trajectory starting from 400 rad/sec., stepped to 300 rad/sec. at time

Speed response of the PMSM system based on FMPC and PI controllers.

Using FMPC controller

Using PI controller

Torque response of the PMSM system based on FMPC and PI controllers.

Using FMPC controller

Using PI controller

Rotor position response of the PMSM system based on FMPC and PI controllers.

Figures

Stator current response of the PMSM system based on the FMPC controller.

Stator current response of the PMSM system based on PI controller.

The performance of the PMSM scheme with the proposed FMPC controller is investigated at low speed (10 rad/sec). The load torque is assumed to be stepped from 2 N·m. to 5 N·m. at time

Low speed response at variable load based on proposed FMPC and PI controllers.

Torque response based on FMPC and PI controllers at low speed.

Also in the FMPC, the unknown variables are 16 times less than the classical MPC. In each time interval, the calculation time needed for classical MPC is 4.6 ms, whereas this time is reduced to 0.48 ms in the FMPC. This is a great computational advantage of using functional MPC.

In this paper, a centralized functional model predictive controller is proposed to control the speed and torque of the permanent magnet synchronous motor drive system. The proposed predictive controller uses orthonormal Laguerre functions to describe control input trajectory which reduces real-time computation largely. Also, exponential data weighing is used to decrease numerical issue, particularly with large prediction horizon. Constraints are imposed on both the

Computer simulations have been carried out in order to evaluate the effectiveness of the proposed controller. The results proved that the proposed system has accurate tracking performance at low speeds as well as high speeds. Also, small ripple contents are noticed in the torque and stator current waveforms. Moreover, the proposed controller has significantly better performance relative to PI controller especially at starting and load change conditions. The main reasons of this superiority are centralized structure of the proposed controller which reduces negative interaction between local control actions, proper constraints that improve optimal calculation of control trajectory, and, finally, using large prediction horizon which gives a performance close to global.