Multi parametric model predictive control based on laguerre model for permanent magnet linear synchronous motors

Received May 26 2018 Revised Sep 13, 2018 Accepted Oct 10, 2018 The permanent magnet linear motors are widely used in various industrial applications due to its advantages in comparisons with rotary motors such as mechanical durability and directly creating linear motions without gears or belts. The main difficulties of its control design are that the control performances include the tracking of position and velocity as well as guarantee limitations of the voltage control and its variation. In this work, a cascade control strategy including an inner and an outer loop is applied to synchronous linear motor. Particularly, an offline MPC controller based on MPP method and Laguerre model was proposed for inner loop and the outer controller was designed with the aid of nonlinear damping method. The numerical simulation was implemented to validate performance of the proposed controller under voltage input constraints.


INTRODUCTION
The Permanent Magnet Linear Synchronous Motors (PMLSM) is extensively used in various industries due to the ability to directly create linear motion without gears or belts. Although the mediate mechanical actuators are eliminated, the system become weak robustness, in that external impact such as frictional force, endeffect, changed load and non-sine of flux cause damage to control performances. Generally, principle operation of synchronous linear motor is similar to permanent magnet synchronous motor; however their physical construction is different in [1]- [5], and various applications such as CNC Lathe [6], sliding door [7].
In recent years, there has been many researches for control problem of permanent magnet linear motor. An adaptive fuzzy neural network in [8] was proposed to control the permanent magnet linear synchronous motor. The authors in [9] presented a control design to regulate velocity based on PIself tuning combining with appropriate estimation technique at slow velocity zone, but if load is changed, PIself tuning controller will be not efficient. In order to overcome changed load, model reference control method based on Lyapunov stability theory employed in [10]. Additionally, the backstepping technique in [11], was applied to reduce influence of frictional force and controller is designed based on appropriate frictional estimated model. In [12], the advantage of that the sliding mode control applied in Linear Motor is that real position value tracks set point. However, the disadvantages of this method is that sliding surface is complicated and chattering problem occurred. It is clear that the previous researches do not mention position, velocity and current constraints. To solve this problem, the MPC approach in [13] was proposed as a single  [14] built a new mathematic model and use optimal control approach to result in linear quadratic regulation (LQR). However, the considered model did not include disturbance load as well as friction force. In addition, the implementation of this MPC controller on a microcontroller is very difficult because of calculation burden.
In this paper, we apply cascade control strategy to synchronous linear motor including an inner and an outer loop. The offline MPC controller based on MPP method in [15] was proposed for inner loop to make motor current to follow the reference signal from the outer controller. We modify optimization problem in the MPC controller by using a Laguerre Model approach in [13] to reduce the number of optimal variables. The major advantage of our MPC controller lies in the ability to solve constraints problem and reducing amount of calculation because the optimal problem is offline solved. The outer controller was designed based on nonlinear damping method in [16] to guarantee the error between real and reference velocity converge to arbitrary small value.

Laguerre orthogonal polynomials
As represented in [13] Laguerre polynomials are defined as follows: where a is a positive constant, 01 a  and 0,1,... i  . The application of Laguerre polynomials is mainly in the area of system identification, in which the discrete-time impulse response of a dynamic system is represented by a Laguerre model (see Wahlberg, 1991). In this work, based on Wang 2009, we obtain the main result: Suppose that the impulse response of a stable system is   Hk, then with a given number of terms N ,   Hk is written as: with 12 , ,..., N c c c are the coefficients to be determined from the system data. The discrete-time Laguerre functions are orthonormal functions, and with these orthonormal properties, the coefficients of the Laguerre network are defined by the following relation:

Model predictive control based on Laguerre function
In this section, we consider this discrete time linear system described by: In which, m x is vector of state variable and () uk is the input at the time k . Convert the Equation (1) as follows: Where: N is prediction horizon. From (7), the sake of designing the MPC controller is finding the sequence of input signal () uk minimizing this under cost function: Where , QR are positive definite matries. Denote that: Putting (7) into a form admitting variable (9) as; 0X Ax BU  (10) Where: is initial state of prediction horizon. Substituting (10) into the cost function (8): Where: In this work, we regard the sequence of signals The equation (12) is rewrited to: Substuting (13) model (4), we can obtain: 11 11 Substituting into the cost function we obtain: We set the optimization problem as:  Unconstrained MPC controller In this case, (7) was considered without any constraints on inputs and state variables: We consider prediction model (7) and constraints (15) Furthermore, by using ( ) (17): Thus, the optimization problem (14)

Multi parametric programming
In this section, we remind the results of multi parametric programming method (MPP) in [11]. The basic idea is that the space of parameter is separated into critical regions in that each critical regions, the solution of optimization problem is in the same form. Consider the quadratic cost function: And from (20),  in (24) must be satisfied: The results (25) and (26) are the basic of multi parametric programming method in this case. Based on the above results, the main steps of the off-line mp-QP solver are outlined in the following algorithm [11]: Step 1: Defining the current region be the whole space D of the vector of parameter  .
Step 2: Choose vector 0  in the current region.
Step 3: With 0   , find the optimal solution   00 , z  by QP method.
Step 4: Define the active and inactive constraints in case of   00 , z  , and then build those matrices: , , G WS.
Step 6: Characterize the 0 CR of x from (13) in which the optimal solution is in (12).
Step 7: Redefine the current region be the 0 D CR  and go to step 2.
Step 8: When all regions have been explored, exit.

MPC controller for current sub-system
As mentioned in [17] The current error model (31) is rewriten as: Obtaining discrete time model from (32) by using ZOH method: Lemma 1: As is presented in [7], by using controller (36) and disturbance observer (37), the state variables of system (35) converges to a ball centered at the origin.
Differentiating both side of (39) along solution of (38) with respect to t: By selecting control parameter constant 12 0 ,, L L K such that: And assume that d , since 2 k can be chosen arbitrarily large and from (59), Lemma 1 is proved.  Figure 1 and Figure 2 describe the responses of PMLSM in cases unconstrainted and constrainted MPC controller. In first case, Figure 1. display that actual trajectory's motor tracking designed trajectory very fast, but it requires the large started voltage. This is a reason to we must constraint the voltage input.

CONCLUSION
In this work, we apply cascade control strategy for polysolenoid linear motor to saparate the motor into outer and inner loop. The offline MPC controller based on MPP method was proposed for inner loop to make motor current to follow the reference signal from the outer controller. Optimization problem in the MPC controller was modified by using a Laguerre Model approach to reduce the number of optimal variables. The major advantage of our MPC controller lies in the ability to solve constraints problem and reducing amount of calculation because the optimal problem is offline solved. The outer controller was designed based on nonlinear damping method to guarantee the error between real and reference velocity converge to arbitrary small value.