The non-cascade permanent magnet synchronous motor control system has the advantages of simple structure and less adjustable parameters, but the non-cascade structure needs to solve the problem of over-current protection. In this paper, a current constrained control method is used to limit the starting current to a safe range. At the same time, to ensure the robustness and rapidity of the system, a super twist current constraint controller (CCSTA) is generated by combining super twist algorithm (STA) with current constraint control; Considering the diversity of internal and external disturbances, a functional disturbance observer (FDOB) is proposed to compensate the matched and unmatched disturbances, which further improves the robustness of the system.

Permanent magnet synchronous motor (PMSM) is widely used in various high-precision servo systems because of its small size, relatively simple structure, strong reliability, and flexible control. In the non-cascade control structure, PI control can not overcome the problem of excessive starting current, and the control effect will be reduced when the parameters change and external disturbance, which can not meet the requirements of high-performance speed control. Therefore, the design of composite controller with overcurrent protection, ensuring the rapidity, and robustness of the system and analyzing and compensating the disturbance has become a research hotspot.

Up to now, in the framework of field-oriented control (FOC), the cascade structure composed of the current inner loop and speed outer loop is widely used in permanent magnet synchronous motor control systems. In the past, due to computer computing power and memory limitations, the control period of the speed loop in the permanent magnet synchronous motor speed control system was longer than the current loop (usually 5 to 10 times that of the current loop). With the development of computer technology, in recent years, the control cycle of the speed loop and current loop is gradually converging, and the same cycle control should be achieved [

However, the non-cascade structure has the advantages of a simple structure, but it still has the critical problem of over-current protection [

Therefore, the design of state constrained controller is challenging both in theory and practice. At present, several control methods are proposed for state-constrained control. Kosut proposed invariant set, characterized by the fact that the state starting from the invariant set will always remain in the invariant set [

According to the above analysis, to realize the current constraint control of the single-loop structure of the PMSM system, the following conditions need to be met: (1) the current constraint is realized; (2) the control algorithm is easy to implement; (3) it has a good dynamic and steady-state. The design of a new type of current constraint controller in this paper is different from previous studies. The design of this paper does not need to limit the initial conditions (such as invariant sets), and it also avoids a large amount of calculation (such as MPC) or the use of other constraint processing tools (such as Cost function or barrier Lyapunov function).

The new current constrained controller combines the current loop and speed loop control into one controller, sets the given current as a state variable, and adds a penalty term in the controller gain to constrain the q-axis current. The robustness is not strong because the conventional PI control is vulnerable to parameter changes and external disturbance [

In order to reduce the effect of chattering as much as possible, Hou et al. [

Disturbance has a significant influence on the system because the disturbance value can not be measured directly in the existing system, so the design of disturbance observer becomes a necessary step. It is found that the traditional frequency-domain multi-objective observer using unity-gain low-pass filter can only effectively deal with a class of specific disturbances satisfying the so-called matching conditions. In contrast the traditional time-domain multi-objective observer always produces high-order observers. For this reason, a functional interference observer (FDOB) is proposed to improve the existing results and design criteria, frequency analysis, and existence conditions [

Therefore, this paper proposes to combine current restraint control and super-twist control in a non-cascade structure to produce a super-twisted current restraint controller (CCSTA). At the same time, in order to improve the anti-interference ability of the system, FDOB is used for compensation. A composite controller is composed of CCSTA and FDOB. The composite controller can constrain the q-axis current of PMSM to reduce the starting current and prevent the equipment from being damaged by excessive current. At the same time, Super Twisted Sliding Mode Control (STA) ensures the robustness and rapidity of the system. The matched and unmatched disturbances are observed and compensated through the Constructed Function Disturbance Observer (FDOB), which further improves the system's response efficiency and anti-disturbance performance.

Assuming that the magnetic circuit is unsaturated, ignoring hysteresis and eddy current losses, the magnetic field is distributed in a sinusoidal space. Under this condition, the mathematical model of the attached permanent magnet synchronous motor in the coordinate system should be described as
_{d} _{q}_{d} _{q}_{p}_{S}_{f}_{L}

Considering the variation of system parameters and load torque, the motion equation of PMSM rotor should be expressed as

Considering the current constraint, the

Hypothesis 1: The load torque variation satisfies the following conditions:

Note 1.1: Hypothesis 1 makes the current constraint still have the margin to suppress the load variation under the condition of load uncertainty, so there must be ^{*}_{q}

This section studies the speed tracking and current constraints of the permanent magnet synchronous motor control system. The STA second-order sliding mode control has the fast response characteristics and no chattering in the speed curve. At the same time, to ensure that the starting current does not exceed the limit value, the system adopts a simple and effective STA second-order sliding mode control combined with current constrained control.

The general expression of STA is

Make x= ω* - ω combining with

The correlation degree of the speed error state equation shown in

In combination with

In a practical application system,

Among them_{eq}_{sw}

_{sw}

Substituting

where

The key is to add a penalty mechanism to the

In this paper, a nonlinear penalty term _{q}_{q}_{q}

In this paper, a non-cascade structure is adopted to simplify the structure of the system. At the same time, a fundamental problem needs to be solved, that is, the problem of current constraint. Therefore, a current constraints function is added basis on

The following is the specific design of _{d}_{q}

Among them, _{P}_{I}_{s}_{d}_{L}

Let

In order to prove the stability of the system, it is necessary to analyze the construction of the Lyapunov function for

Therefore, the system error equation should be written in the following form:

Thus, the proof

Theorem 2.1: Under assumption 1, the origin of closed-loop systems ^{*}

Proof: Denote

Therefore, to prove that the current constraint is satisfied, we only need to prove

In the interval [0, T), substituting

The Lyapunov function is constructed according to

Among

The vector field of the

The vector field of the error system

Among

And

Simplify

It is necessary to prove that there are always constants

According to the two cases of _{q2}

Case 1:

Case 2:

According to the definition of T,

Finally verified

According to the above stability proof, the system is stable when the stability condition is satisfied.

Because there are both internal parameter disturbances and external load disturbances in the permanent magnet synchronous motor system, the functional disturbance observer (FDOB) can effectively deal with both disturbances.

Consider a linear time-invariant system with unknown disturbances.

Combining with PMSM system,

The disturbance is supposed to be generated by the following linear exogenous system:

Definition 3.1: If the matrix in

By combining

For the system

Among

FDOB can significantly reduce the order of disturbance observer, but the existing condition of FDOB should be satisfied when choosing the gain L. According to [

According to the literature [

Under the condition of meeting the current constraint, the disturbance observation should be realized at the same time. The compound control design of super twisting current constraint control and functional disturbance observer compensation is adopted. The following is the composite control law [

The non-cascade structure of the composite controller is shown in

In order to verify the effectiveness of CCSTA + FDOB composite controller. The simulation model is established based on MATLAB/Simulink platform. The parameters of PMSM in the simulation are shown in

Stator resistance R | 1.74 |
rated power P_{N} |
0.75 kw |
---|---|---|---|

Stator inductance L | 4 Hm | rated voltage U_{N} |
220 V |

Pole pairs n_{P} |
4 | rated current I_{N} |
4.7 A |

Flux linkage ψ_{ƒ} |
0.3 Wb | rated speed n_{N} |
3000 r/min |

Rotor inertia J | 0.000178 kg·m^{2} |
rated torque T_{N} |
2.387 N⋅m |

Viscous coefficient B | 0 Nms/rad |

In this paper, a composite controller combining current constrained sta with functional disturbance observer is designed. Its parameter configuration are _{s}_{P}_{I}_{1}_{2}_{3}

In order to verify the functions of the control system proposed in this paper in terms of current constraints and anti-disturbance functions, the following comparison will be made. First, compare CCSTA with STA to verify the current restraint ability of CCSTA. Parameter configuration of STA controller: speed loop _{1}_{2}_{1}_{2}

It should be seen from

Compare CCSTA + FDOB with CCSTA + DOB to verify the anti-disturbance ability of FDOB. The DOB parameter configuration is α_{1 }=_{ }15, α_{2 }=_{ }9, and ε = 0.005.

It should be seen from

The impact of CCSTA + FDOB is significantly less than that of CCSTA + DOB, so FDOB has better anti-interference performance against sinusoidal disturbances. The model predictive control (MPC) is compared with the super twisted current restraint control (CCSTA) to verify the progressive nature of the current restraint ability of CCSTA.

The cost function J of MPC is defined as,

It should be seen from

In order to verify the performance of the control method proposed in this paper, the composite controller combining the Super Twist Controller (STA) and the Disturbance Observer (DOB), and the composite controller combining the PD controller and the Extended State Observer (ESO) are compared. The parameter configuration of STA and DOB composite controller is as follows: Speed loop _{1}_{2}_{1}_{2}_{1}_{2}_{p1}_{d1}_{p2}_{i2}_{01}_{02}

CCSTA + FDOB | STA + DOB | PD + ESO | |
---|---|---|---|

Response time t | 0.004 s | 0.005 s | 0.018 s |

Overshoot | 0.5% | 0.6% | 1.5% |

In order to test the performance of the proposed composite system in the case of up and down speed, the simulation results show that the expected speed at the initial time is 200 r/min, the expected speed is increased to 1000 r/min at 0.02 s, and the expected speed is reduced to 500 r/min at 0.06 s. The system runs under the load of 3 N⋅m.

It should be seen from

CCSTA + FDOB | STA + DOB | PD + ESO | |
---|---|---|---|

200 r/min t | 0.004 s | 0.006 s | 0.02 s |

1000 r/min t | 0.007 s | 0.014 s | 0.022 s |

500 r/min t | 0.004 s | 0.011 s | 0.021 s |

CCSTA + FDOB | STA + DOB | PD + ESO | |
---|---|---|---|

200 r/min | 14% | 22.5% | 17% |

1000 r/min | 0.5% | 3% | 18.5% |

500 r/min | 2% | 56% | 19% |

In this design, FDOB has the function of estimating and compensating mismatched disturbance. Suppose the system has an unknown mismatched step disturbance. Let S = 0, H = 5, L_{0} = [0 1] the unknown step disturbance acts on the system at 0.04 s.

It should be seen from

It is assumed that a sinusoidal disturbance with known frequency and unknown amplitude and phase is used in the system. Let S = _{0} be selected as L_{0} = [0 1], and the sine wave perturbation acts on the system in 0.05 s.

It should be seen from

This paper combines current constrained control and super twist second-order sliding mode control (STA). At the same time, the super twist sliding mode control (STA) can effectively reduce the system response time and suppress the steady-state chattering. In order to avoid the influence of internal and external disturbance on the system, the function disturbance observer (FDOB) is designed to compensate for the system disturbance. FDOB can estimate and compensate the system matching and mismatching disturbance, improve the system's anti disturbance performance, and verify the system function through comparison. In this paper, three control strategies of the CCSTA + FDOB control system combined with STA + DOB control system and PD + ESO control system are simulated and compared. The results verify the feasibility and effectiveness of the proposed control algorithm.