A new method for the shortest-time optimal control of closed-loop stepper motors is introduced. The mathematical model and optimal control strategy for the optimal control of the stepper motor are proposed. The hardware and software technology for the optimal control using a single-chip microcomputer are given. The system application shows that the proposed optimal control has good control performance.
1 Introduction
A stepper motor is an electromechanical element that converts an electrical pulse signal into a corresponding angular displacement (or line displacement). The use of a stepper motor control system has the advantages of low price, simple control and easy maintenance, especially with the development of microcomputers and microelectronics technology, making stepper motors more widely used, but also for stepper motors. The operating performance has put forward higher requirements, so choosing the control method of the stepping motor reasonably has played an active role in giving full play to the working performance of the stepping motor. In order to obtain the optimal control of the stepper motor and make the control system have low cost, this paper uses the optimal control theory to study and introduce the optimal switching time of the stepper motor control voltage and the control strategy implemented by the single chip microcomputer.
2. Mathematical model of optimal control problem of stepper motor
When the stepper motor and its drive power and load have been determined, the length of the motor start-up acceleration and deceleration stop process is only related to the control pulse sequence. If a set of optimal control pulse sequences can be found, the process of starting acceleration and deceleration and stopping is the shortest, and the optimal control of the stepping motor can be realized.
In the case of ignoring some minor factors, the balance equation of stepper motor torque can be simplified as:
(1) In the formula 
—— Stepper Motor Output Torque Max 
—— Total load torque J —— Total moment of inertia
θ —— stepper motor shaft position
f(θ) - Static torque distribution corresponding to θ
Setting the stepper motor to rotate one tooth requires N steps, and u denotes the drive pulse for each step (for example, a three-phase stepping motor adopts 6-step operation mode, then N=6, u=0,1,2,3,4, 5, corresponding to the control voltage is A, AB, B, BC, C, CA phase), because the shape of the total load torque distribution curve does not change with the value of u, therefore, equation (1) can be described as:
(2) Assume 
Then have to: 
(3)
(4) The shortest time optimal control of the stepping motor boils down to obtaining the optimal control sequence u=(0,1,2,3,4,5) under the condition equations (3) and (4) to make the stepper motor Initial state 
To the terminal state 
, performance index is the objective function 
Is the minimum, where T is the terminal time.
3, the optimal control strategy
According to states (3) and (4), construct the Hamilton function as:
(5) By the principle of maximum, u(i) should be controlled so that 
For the smallest,
make 
(6)
Where u(i)=0,1,2,3,4,5; i=0,1,2,3,4,5.
Without loss of generality, f(θ) can be expressed as a sine function of θ: 
The adjoint equation of the maximum principle can be expressed as:
(7)
(8)
From Equations (7) and (8), it can be proved that λ2≠0, because λ2=0, then 
(C1 is a constant);
So λ2≠0. Dividing the two sides of equation (6) by λ2:
(9) The right side of equation (9) is the stepper motor torque distribution, in order to make h(i) minimum, i=0,1,2,3,4,5, if λ2>0, h(i)/λ2 should be minimized The value, that is to take the minimum torque distribution, if λ <0, h (i) / λ2 should take the maximum, that is to take the maximum torque distribution. Although λ2 is difficult to understand with maxima, but because it is a practical problem, the optimal solution must exist. The obvious physical meaning shows that in order to accelerate the rotor, the torque distribution should be made as large as possible for the rotor. Deceleration should make the torque distribution as small as possible.
The stepping motor is a three-phase stepping motor, and the control method adopts six beats. From the equation (9), the curve between h(i)/λ2 and x1 shown in FIG. 1 can be obtained. Based on the above analysis, the optimal control problem boils down to how to switch the control voltage u(i) in the torque distribution shown in Fig. 1 so that the average torque is within the 2π angle. When accelerating, the torque distribution is maximum, and when the speed is decelerated, the rotation The moment distribution is the smallest.
Figure 1 Relationship between h/λ2 and x1
If the three-phase stepper motor uses a six-shot control voltage, the control voltage is switched at the intersections of a, b, c, d, e, and f shown in FIG. 1 to obtain the maximum average torque.
Fig. 2 Curve of torque and rotation angle
Because of the six-shot control mode, each operating mode is 60° within 2π electrical angle. Therefore, it is only necessary to indicate that the switching point voltage is greater than the switching point at other points in the 60° range (such as points a and b). Voltage distribution of torque curve. That is, as long as the area enclosed by the horizontal line in FIG. 2 is described, the area enclosed by the vertical line is larger.
The results discussed above are also valid for any phase-by-step motor. It is assumed that the stepper motor is within 2Ï€ electrical angle and the control voltage is N beat.
So, there is the same conclusion as above.
4, to achieve technology
The optimal control of the stepping motor in the shortest time can be divided into three stages: acceleration start, maximum speed operation, deceleration and stop.
4.1 Accelerated start-up phase optimal control
According to the above discussion, switching the control voltage at the intersection of the time series a, b, c, d, e, f shown in FIG. 1, the stepper motor will obtain the maximum average output torque, ie, the stepper motor can have the maximum acceleration. start. The starting voltage switching frequency is related to J and Ts. The larger the J and Ts, the lower the switching frequency. The smaller the J and Ts, the higher the switching frequency. This is because when the stepper motor is started, the rotation speed is equal to zero or less. During the starting process, the output torque of the stepper motor overcomes Ts and overcomes the inertia torque. If the starting pulse frequency is high, the speed of the rotor will not keep up with the starting pulse frequency. After several starting pulses, the position of the rotor is further away from the equilibrium distance, and finally the rotor position falls outside the stable area of ​​the stepping motor. Loss of stepping or oscillation to the stepper motor causes the motor to fail to start. Fig. 3 qualitatively shows the distribution relationship between the start-up sequence Ti = {T1, T2, T3,..., Tn} and the displacement of the shaft of the stepper motor. Under certain Ts and J conditions, the time series Ti can be measured through experiments.
This experimental device is shown in Fig. 4. An optical encoder is mounted on a stepping motor shaft. The number of holes of the photoelectric encoder is an integral multiple of the number of beats required by one step of the stepping motor. The stepping motor is stopped at the point m in Fig. 3. The MCU provides a start pulse to the stepper motor A phase, and then the MCU starts timing. When the electrical angle is 60°, the photoelectric encoder outputs a pulse to the MCU, and the signal is interrupted. The time spent by the SCM and the SCM is stored in the RAM and the control voltage is switched to the A and B phases. This is repeated until the stepper motor reaches the maximum operating frequency.
Figure 3 Starting time series relationship curve
4.2 Highest speed operation optimal control
If the total number of beats required for the stepper motor is N1, the number of beats required for starting is N2, and the number of beats required for stopping is N3, then the number of beats for the stepper motor running at the highest speed is N4=N1-N2 -N3. The maximum operating frequency is related to the load resistance torque and the characteristics of the stepper motor itself. It can be determined based on on-site commissioning.
4.3 Optimal Control for Deceleration and Parking Phases
First, reduce the operating frequency of the stepper motor, and then switch the control voltages at a, b, c, d, e, and f so that the stepper motor can quickly stop.
4.4 Stepper Motor Control System Design
1. Hardware Design:
This system mainly adopts the 8098 single-chip high-speed output port HSO and software timer to output pulses to control the running frequency of the stepper motor according to certain rules. At the same time, it also uses EPROM (27128), RAM (6264), keyboard and display interface circuit (8279), Display and keyboard, drive circuit, optical encoder and other devices. Stepper motor controller shown in Figure 4.
Figure 4 Hardware structure diagram of stepper motor controller
8098 microcontroller has a counter, high-speed input, output port and serial port, using it to constitute a stepper motor controller, simplifying the control structure, thus reducing the cost of the control system, therefore, can be promoted in the industrial control system.
2. Software design The control system software is mainly composed of monitoring subroutines, main control programs, and interrupt programs. Monitoring subroutines are used to input data and control parameters; the main control program is used to control the stepper motor starting, running and stopping with an optimal control strategy;
The interrupt program is used to switch the control voltage of the stepper motor. The optimal control main program block diagram is shown as in Fig. 5.
Figure 5 main program block diagram of optimal control of stepper motor
5. Concluding remarks
The experimental results show that the optimal control pulse sequence obtained with J as the objective function can accelerate the stepper motor to the steady state speed in the shortest time and decelerate to zero from the steady state speed. The acceleration and deceleration processes are considered to be the most The exponential curve of the excellent law comes quickly, and it can be guaranteed that the optimal control pulse sequence determined by the system will not lose step and the deceleration will not go further.
The method proposed in this paper does not need to derive the complicated nonlinear mathematical model of the stepper motor. Instead, the system determines the optimal control pulse sequence corresponding to the extreme value of the objective function by detecting the objective function, and then achieves the optimal control. When the load is changed, the optimal pulse sequence corresponding to the new load can also be obtained as long as the program is executed again. Therefore, this method is simple and convenient. For a stepper motor in a production field industrial microcomputer control system, only the field system needs to add some hardware and software to determine the optimal pulse sequence of the motor by the on-site system. It is not necessary to move the motor and load into the laboratory for testing. The whole method is more practical and flexible. The optimal control of the stepper motor significantly improves the production efficiency of the industrial control system and thus has been widely used in industrial control.
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