Adaptive Tracking Control of a PMSM-Toggle System with a Clamping Effect

This paper discusses an adaptive control (AC) designed to track an energy-saving point-to-point (ESPTP) trajectory for a mechatronic system, which is a toggle mechanism driven by a permanent magnet synchronous motor (PMSM) with a clamping unit. To generate the PTP trajectory, we employed an adaptive real-coded genetic algorithm (ARGA) to search for the energy-saving trajectory for a PMSM-toggle system with a clamping effect. In this study, a high-degree polynomial was used, and the initial and final conditions were taken as the constraints for the trajectory. In the ARGA, the parameters of the polynomials were determined by satisfying the desired fitness function of the input energy. The proposed AC was established by the Lyapunov stability theory in the presence of a mechatronic system with uncertainties and the impact force not being exactly known. The trajectory was tracked by the AC in experimental results so as to be compared with results produced by trapezoidal and high-degree polynomials during motion.


Introduction
This paper discusses adaptive control (AC) designed to track an energy-saving point-to-point (ESPTP) trajectory for a PMSM-toggle system. In general, this example is referred to as point-to-point control, and it takes into account low acceleration and jerk-free motion [1]. Astrom and Wittenmark [2] presented a general methodology for the off-line tridimensional optimal trajectory planning of robot manipulators in the presence of moving obstacles. Planning robot trajectory by using energetic criteria provides several advantages. On one hand, it yields smooth trajectories and is easy to track, while reducing the stress in the actuators and manipulator structures. Moreover, the minimum amount of energy may be desirable in several applications, such as those with energy-saving control or a quantitatively limited energy source [3]. Examples of minimum-energy trajectory planning are provided in [4]. However, the selection of a suitable profile for a specific application is still a challenge since it affects overall servo performance. Thus, in this study, the authors designed the kinematics of the trajectory profiles for motion tracking control within a PTP trajectory.
The AC techniques proposed in this study are essential to providing stable, robust performance for a wide range of applications such as robot control [5][6][7][8][9] and process control [10]. Most such applications are inherently nonlinear. Moreover, a relatively small number of general theories exist for the AC of nonlinear systems [11]. Since the application of a mechatronic system has minimum-energy tracking control problems for elevator systems, the AC technique developed by Chen [12], who made use of conservation of energy formulation to design control laws for the fixed position control problem, was adopted to control the PMSM-toggle system in this study. In addition, an inertia-related Lyapunov function containing a quadratic form of a linear combination of position-and speed-error states was formulated.
The difference between previous studies [13][14][15][16][17][18] and this study is that this study takes the clamping unit into consideration. The main contribution of this study is that the proposed AC adapts not only to parametric uncertainties of mass variations, but also to external disturbances. The performance with external disturbances is validated through the results obtained both numerically and experimentally on the energy-saving point-to-point trajectory processes for a PMSM-toggle system with a clamping unit.

Electrical Model
The permanent magnet synchronous motor drive toggle system is shown in Fig. 1 where e τ is the electromagnetic torque, the variables r ω and r ω ɺ are the angular speed and acceleration of the rotor, respectively, m B is the damping coefficient, and m J is the moment of inertia. It is noted that , where p is the number of pole pairs and n is the ratio of the geared speed-reducer. Eqs. (2) and (3) represent the mathematical model of the PMSM. They give the voltage and motor torque variation with respect to time.

Impact Model
In this section, we consider the motion in a given stroke of the toggle mechanism undergoing impact when two objects collide over a very short period of time. The continuous force model approach [19] employs a logical spring-damper element to estimate the impact force between the two masses of the mechatronic system as, where l K is the elastic spring coefficient, z is the relative displacement or penetration between the surfaces of the two colliding bodies, z ɺ is the relative speed, and l D is the damping coefficient. For the time period of i i t t t − + ≤ ≤ , the differential-algebraic equation can also be rewritten in matrix form. The impact model of the toggle mechanism with a clamping unit is shown in Fig. 2.

Mathematical Model of the Toggle Mechanism
The toggle mechanism of the electrical injection molding machine was driven by a PMSM. The experimental setup and physical model are shown in Fig. 1(a) and Fig. 1(b), respectively. The differential-algebraic equations of the toggle mechanism are summarized in matrix form [20]. The matrix form of the equations can be written as:  [20]. The system Eq. (5) is an initial value problem and can be integrated by using the fourth order Runge-Kutta method.

Energy-Saving Trajectory Planning
This section discusses how the AC was designed to track an ESPTP trajectory for a PMSM-toggle system. The degrees of the polynomial depended on the number of end-point conditions, which are desired for smoothness in the resulting motion. In the simplest case, the motion is defined during the initial time t 0 and final time T, and it satisfies the end-point conditions of position, velocity and acceleration at any time. From the mathematical point of view, the problem is then to find a function such that This problem can be easily solved by considering the polynomial function where each coefficient , is a real number, and m a is a non-negative real number. The were determined such that the end-point constraints were satisfied. A high-degree polynomial was used to describe the trajectory, and it satisfied the desired constraints of position, velocity and acceleration at the end points.
For the PTP trajectory, we considered the profile with zero initial displacement, 1 h θ displacement at the final time T, a displacement and speed of 0 at t0, and t T = . Thus, we obtained the following end-point conditions: By using these four conditions and substituting them into equation (9)

T a T a T a T a T a T a T a T a T T
It is seen from Eqs. (10) and (11)  The PMSM is considered thermodynamically as an energy converter. It takes electrical energy from a controlled input and then outputs it as mechanical work to drive the toggle mechanism system with a clamping unit. The input absolute electrical energy (IAEE) to the system is defined as where q i is the electric current and q v is the voltage command.

Adaptive Real-Coded Genetic Algorithm
It is important that crossover probability and mutation probability are set correctly for the genetic algorithms; improper settings will cause algorithms to only find local optimums and will also cause premature convergence. Therefore, an efficient method that allows for fast settings is essential. To resolve this, a mechanism to adjust the crossover probability and mutation probability according to the algorithmic performance was considered. In this study, the adaptive real-coded genetic algorithm for polynomial coefficient identification of the ESPTP trajectory of the PMSM-toggle system was employed.
In equations (10) and (11) , , , , , , , , a a a a a a a a a = z .
The procedure carried out for the ARGA is shown in Fig. 3. In this study, the procedure was reproduced through roulette wheel selection, while the crossover and uniform mutation were carried out through the methods described in [21].

Fitness Function
How the fitness function is defined is the key to the genetic algorithm since the fitness function is a figure of merit and could be computed by using any domain knowledge. In the proposed ESPTP trajectory planning problem, the researchers defined the input energy function as the fitness function ( ) f z as (14) where m is the total number of samples and ( ) i E z is the input energy of the th i sampling time.

Adaptive Probability Law
To reduce premature convergence and improve the convergence rate of the traditional real-coded genetic algorithm (TRGA), the adaptive probabilities of crossover and mutation were used in the ARGA. "Crossover" is the breeding of two parents to produce a single offspring which possesses features of both parents and thus may turn out better or worse than either parent according to the objective function. The primary purpose of mutation is to introduce variation, help bring back certain essential genetic traits, and avoid premature convergence of the entire feasible space caused by certain super chromosomes.
To reduce premature convergence and improve the convergence rate of the TRGA, the adaptive probabilities rule [21] of crossover and mutation were used in the ARGA. The where max ,  15) and (16), it can be seen that the adaptive c Γ and m Γ vary with fitness functions. c Γ and m Γ increase when the population tends to get stuck at a local optimum (when attraction basins are found around locally optimal points) and decrease when the population is scattered in the solution space.

Increasing Function
For the sake of tracking the motion profile of the mechatronic system, the trajectory displacement needs to be designed as a monotonically increasing function from the start point to the end point. In this study, ( ) v , t 0 t T ≤ ≤ was assumed as the monotonically increasing function: where the subscript i represents the th i sampling time. This constraint of the monotonically increasing function had to be included in the procedure of the ARGA as shown in Fig. 3.

Adaptive Control Design
In this study, the researchers used the law of AC to describe what happens when two objects collide. "To adapt" means to change a behavior to conform to new circumstances. The AC law can control the two objects and balance the speed. The AC system is shown in Fig. 4 is the experimentally measured state as found by use of a linear encoder system. In order to design an AC, the researchers rewrote Eq. (5) as a second-order nonlinear equation: u t is the control input voltage. It was assumed that the exact mass of slider B and the exact impact force i F could not be known. With these uncertainties, the first step in designing the AC was to select a Lyapunov function, which is a function used for tracking error and parameter error. An inertia-related Lyapunov function containing a quadratic form of a linear combination of position-and speed-error states was chosen as follows [15]: where ( ) A′ • and ( ) B′ • are described in reference [15]. If the control input is selected as where V K is a positive constant, then Eq. (22) becomes 1 ( ) .
By selecting the adaptive update rule as and substituting it into Eq. (24), it then becomes 0.
Since V ɺ in Eq. (26) is negative semi-definite, then V in Eq. (19) is upper-bounded. As V is upper-bounded and (v; ) f t is a positive-definite matrix, it can be said that s and φ ɶ are bounded.

Numerical Simulations
This section discusses how the researchers simulated the ESPTP motion profile for the PMSM-toggle system. The trajectory profile v( ) t was chosen as a monotonically increasing function. The input absolute electrical energy i E was calculated by the fourth-order Runge-Kutta method via a Windows supported MATLAB package with a sampling time of 0.01 sec t ∆ = and the time interval being from 0 to 1 sec. In the numerical simulations, we adopted the parameters of the PMSM-toggle system obtained as follows:  Fig. 5(c), it is demonstrated that the ARGA is more efficient in identifying polynomial coefficients than the TRGA. The energy used was less than 9×10 -3 J. It is thus concluded that the ARGA does not only find local optimums while preventing premature convergence, the fitness values of the ARGA are greater than those of the TRGA. The comparisons of dynamic responses of the PMSM-toggle system for trapezoidal, fourth-degree, and twelfth-degree polynomials are shown in Figs. 7(a)-7(d). The speeds are compared in Fig. 7(c). The displacement-and speed-error comparisons with respect to the trapezoidal, fourth-degree and twelfth-degree polynomials are shown in Figs. 7(b) and 7(d). (The fourth-degree and twelfth-degree polynomials were formulated based on ESPTP trajectories.) The final identification of the polynomial coefficients a4 ~ a12, the values of the fitness function of the mechatronic system, and the highest fitness value were found by using the twelfth-degree polynomial. The total energy values are also compared in Table 1 lowest value is that of the twelfth-degree polynomial, and the trapezoidal polynomial had a relative reduction of -8% in input energy.

Experimental Setup
A photo of the PMSM-toggle system with a clamping unit is shown in Fig. 1(a), and the experimental equipment used is shown in Fig. 6. The control algorithm was implemented by using a Celeron computer, and the control software used was LabVIEW. The PMSM was driven by a Mitsubishi HC-KFS13 series. The specifications were set as follows: rated torque of 1.3 Nm, rated rotation speed of 3k rpm, rated output of 0.1 kW, and rated current of 0.7 A. The servo-motor was driven by a Mitsubishi MR-J2S-10A.

Experimentation
For the ESPTP trajectory processes of a PMSM-toggle system, the control objective was to control the position of show the speed of the PMSM-toggle system, and their errors are slight. Moreover, the command input, input current, impact force and input energy with the clamping effect are shown in Figs. 7(e)-7(h). As seen from the experimental results, accurate tracking control performance of the PMSM-toggle system with a clamping unit can be obtained after the clamping point of the ESPTP trajectory of the AC system, and adaptive characteristics were achieved for the AC system. The final input energy values were about 79.9 J, 99.4 J and 51.1 J, respectively. The total energy comparisons are shown in Table 1. The lowest value is that of the twelfth-degree polynomial, and the trapezoidal polynomial had a relative reduction of -36% in input energy. In conclusion, the clamping time was shorter and the speed profile was smoother. Moreover, a better energy-saving effect can be achieved for a PTP trajectory with a clamping effect by using the AC system.

Conclusion
A mathematical model was put into use for a PMSM-toggle system with a clamping unit, and the ESPTP trajectory for the mechatronic system was successfully planned by the adaptive real-coded genetic algorithm method described in this paper. The proposed AC was established by the Lyapunov stability theory for a mechatronic system with uncertainties and the impact force not being exactly known. The proposed methodology described in this paper was applied to a mechatronic system with a clamping unit. The mechatronic system required the design of an ESPTP trajectory which can be interpreted by any continuous function and which has different motion constraints at the start and end points. The results demonstrate that the adaptive control performance in the PTP trajectory with a clamping effect is successful for a mechatronic system.