阅读说明：本技术 一种提高无刷直流电机抗扰能力的复合控制方法 (Composite control method for improving anti-interference capability of brushless direct current motor ) 是由 周兰 姜福喜 张铸 于 2020-12-25 设计创作，主要内容包括：本发明公开了一种提高无刷直流电机抗扰能力的复合控制方法,其步骤包括：步骤S1：构建无刷直流电机状态方程,将参数摄动和外负载干扰归纳为总扰动,并建立无刷直流电机伺服系统状态空间模型；步骤S2：将所述总扰动作为扩展状态变量,通过分离系统可测状态与不可测状态,重建系统等价状态空间模型；步骤S3：构造降阶扩张状态观测器,在线估计总扰动和系统不可测状态；步骤S4：利用上述降阶扩张状态观测器的估计值和系统可测状态,设计基于扰动动态补偿的反步控制器,抑制扰动对系统输出的影响,提高系统的动态性能和鲁棒性,并保证系统输出对参考输入的准确跟踪。本发明具有实现简单、控制实时性强、跟踪精度高等优点。(The invention discloses a composite control method for improving the anti-interference capability of a brushless direct current motor, which comprises the following steps: step S1: constructing a brushless direct current motor state equation, summarizing parameter perturbation and external load interference into total disturbance, and establishing a brushless direct current motor servo system state space model; step S2: the total disturbance is used as an expansion state variable, and a system equivalent state space model is reconstructed by separating a measurable state and an undetectable state of a system; step S3: constructing a reduced order extended state observer, and estimating total disturbance and a system undetectable state on line; step S4: by utilizing the estimated value of the reduced order extended state observer and the measurable state of the system, a backstepping controller based on disturbance dynamic compensation is designed, the influence of disturbance on the system output is inhibited, the dynamic performance and the robustness of the system are improved, and the accurate tracking of the system output on the reference input is ensured. The invention has the advantages of simple realization, strong control real-time performance, high tracking precision and the like.)

1. A composite control method for improving the anti-interference capability of a brushless direct current motor is characterized by comprising the following steps:

step S1: constructing a brushless direct current motor state equation, summarizing parameter perturbation and external load interference into total disturbance, and establishing a brushless direct current motor servo system state space model;

step S2: the total disturbance is used as an expansion state variable, and a system equivalent state space model is reconstructed by separating a measurable state and an undetectable state of a system;

step S3: constructing a reduced order extended state observer, and estimating total disturbance and a system undetectable state on line;

step S4: and designing a backstepping controller based on disturbance dynamic compensation by using the estimated value of the reduced order extended state observer and the measurable state of the system, and inhibiting the influence of disturbance on the output of the system.

2. The compound control method for improving noise immunity of a brushless dc motor according to claim 1, wherein in step S1, a state space model of the servo system of the brushless dc motor is derived from a voltage balance equation and a torque balance equation.

3. The compound control method for improving the noise immunity of the brushless direct current motor according to claim 1, wherein in the step S4, a backstepping controller based on disturbance dynamic compensation is designed according to Lyapunov stability theory.

4. The compound control method for improving the noise immunity of a brushless direct current motor according to any one of claims 1 to 3, wherein in the step S1, the establishing step of the state space model of the servo system of the brushless direct current motor is as follows:

step S101: assuming that the waveform of the back electromotive force is a trapezoidal wave with a flat top width of 120 electrical angles, the power device of the driving circuit is an ideal switch, and the three-phase windings are completely symmetrical, the voltage balance equation of the three-phase windings of the brushless direct current motor is as follows:

wherein u is_a(t)、u_b(t)、u_c(t) phase voltages of stator windings, i_a(t)、i_b(t)、i_c(t) stator winding phase current, R phase winding resistance, L_sInductance of each phase winding, e_a(t)、e_b(t)、e_c(t) each phase winding back electromotive force;

the electromagnetic torque equation is:

in the formula (I), the compound is shown in the specification,T_e(t) is the electromagnetic torque, ω (t) is the angular velocity of the motor rotor;

the torque balance equation is:

in the formula, M_c(t) is the external load torque, and J is the total moment of inertia of the motor rotor and the load;

step S102: the voltage balance equation is further expressed as:

wherein u (t) is the voltage applied to the two phase conducting windings, i (t) is the armature current, and ε (t) is the back EMF of the conducting phase windings;

the electromagnetic torque equation is further expressed as:

when the back electromotive force is trapezoidal wave, the back electromotive force calculation formula of the conducting phase winding is as follows:

ε(t)＝k_epω(t)

in the formula, k_eIs the back electromotive force coefficient, p is the motor pole pair number;

establishing a state equation of a brushless direct current motor servo system by a voltage balance equation and a torque balance equation:

step S103: and respectively carrying out Laplace transformation and Laplace inverse transformation on differential equation expressions of the rotating speed subsystem and the current subsystem to obtain:

assuming the following parameter uncertainties exist for the above equation:

wherein R is₀、L₀And J₀Respectively representing nominal values of phase resistance, phase inductance and moment of inertia, R₀ΔR、L₀Δ L and J₀Δ J represents perturbation amounts of phase resistance, phase inductance, and rotational inertia, respectively;

after considering the parameter perturbation, willFurther expressed as:

in the formula, the input gain is controlledf_total(t) is the 'total disturbance' containing parameter uncertainty and external load interference, and the specific expression is as follows:

step S104: first take x₁(t)＝ω(t)，Then define the state variable x_p(t)＝[x₁(t) x₂(t)]^TControl input is u (t) and system output is y_p(t) ω (t), establishing a brushless DC motor servo systemA state space model:

wherein the coefficient matrix is:

obviously, (A)_p，B_u) And (4) controlling.

5. The compound control method for improving the noise immunity of a brushless dc motor according to claim 4, wherein in step S2, the step of establishing the system equivalent state space model is as follows:

step S201: taking an extended state variable x₃(t)＝f_total(t) is provided withEstablishing an augmentation system state space model:

wherein:

step S202: in a brushless DC motor servo system, the motor speed, i.e. state x₁(t) is calculated from the position signals of the hall sensors, and measurable variables in the augmentation system are separated from non-measurable variables to obtain:

getIs a system state variable, with control inputs u (t) and system outputsEstablishing a system equivalent state space model:

wherein the coefficient matrix is:

it is clear that,can be seen.

6. The compound control method for improving the noise immunity of a brushless dc motor according to claim 5, wherein in step S3, the step of constructing the reduced order extended state observer of the original dc motor servo system comprises:

step S301: designing a reduced order extended state observer to realize the non-measurable stateAnd total disturbance f_totalReal-time estimation of (t):

wherein the observer state Andare respectively the system stateAnd total disturbance f_total(t) an estimated value, L being an observer gain matrix to be designed;

step S302: defining the state estimation error as:

whereinCombining the system equivalent state space model and the state equation of the reduced order extended state observer to obtain a state estimation error dynamic equation as follows:

by designing the observer gain matrix L such thatWhen h (t) is bounded, the bounded input and the bounded output of the extended state observer are guaranteed to be stable;

obtaining an observer gain matrix L by adopting a bandwidth setting method, and setting L as [ beta ]₁ β₂]^T，ω_oObserver bandwidth, such that:

(s+ω_o)²＝s²+β₁s+β₂

finding beta₁＝2ω_o，β₂＝ω_o ²，

Step S303: and performing equivalent deformation on the state equation of the reduced order extended state observer to obtain:

is recorded as:

wherein the content of the first and second substances,

7. the compound control method for improving the noise immunity of a brushless dc motor according to claim 6, wherein in step S4, the step of designing the step-back controller based on the dynamic disturbance compensation comprises:

step S401: note the bookIs x₁(t) -the sub-system(s),is x₂(t) -a subsystem; for x₁(t) -subsystem, let x_1d(t) is state x₁(t) the expected value, i.e. the reference input trajectory, defines a first state error and a Lyapunov function as:

S₁(t)＝x₁(t)-x_1d(t)，

introducing a virtual control input alpha₂(t) for the above formula V₁(t) deriving:

when in useNegative timing, state error S₁(t) can converge asymptotically to zero in order to make x₁(t) -subsystem at virtual control input α₂(t) is gradually stable under the action ofHas the following form:

wherein k is₁∈R⁺Virtual control input alpha introduced for virtual control gain to be designed₂(t) is:

at this timeWherein S₁(t)[x₂(t)-α₂(t)]Is a residual term;

introducing a low-pass filter q(s) ═ 1/(ts +1) to the virtual control input α₂(t) filtering, the filtered virtual control input being recorded asCorresponding to the filtered state equation:

wherein τ is the filter time constant;

step S402: for x₂(t) -subsystem defining a second state error and a Lyapunov function as:

to the above formula V₂(t) deriving:

when in useNegative timing, state error S₂(t) capable of asymptotically converging to zero; to make x₂(t) -the subsystem becomes asymptotically stable under the action of the control input u (t) < CHEM > byHas the following form:

wherein k is₂∈R⁺Inputting a gain for a control to be designed; the control input u (t) of the system is obtained as follows:

wherein the content of the first and second substances,andfor expanding the state variables of the state observer, i.e. the system state, in step S3And total disturbance f_total(t) an estimated value.

Technical Field

The invention mainly relates to the technical field of motor control, in particular to a composite control method for improving the anti-interference capability of a brushless direct current motor.

Background

The brushless direct current motor not only has the advantages of reliable operation and simple structure of the alternating current motor, but also has the characteristics of high operation efficiency and good speed regulation performance of the direct current motor, so that the brushless direct current motor is widely applied to the fields of aerospace, automotive electronics, robots and the like. In most application occasions, the traditional PID control can meet the performance requirement of the system, but considering that the brushless direct current motor is a multivariable, strong-coupling and nonlinear system, when the motor is influenced by larger uncertain interferences such as parameter perturbation, external load and the like in the running process, the control precision of the traditional control method is difficult to meet the requirement, and even the stability of the system is damaged in serious cases. Therefore, the research on the composite control method and the system for improving the anti-interference capability of the brushless direct current motor has important theoretical significance and application value.

In order to improve the disturbance suppression performance and control accuracy of a brushless direct current motor servo system, a great deal of research is carried out by scholars at home and abroad in recent years.

For example, the document (brushless DC motor speed servo system backstepping high-order sliding mode control [ J ], control and decision, 2016, 31 (6): 961-.

For another example, the document (design of fractional order sliding mode controller of brushless dc motor [ J ], firepower and command control, 2018, 43 (6): 81-85) proposes a fractional order sliding mode control strategy of speed regulating system of brushless dc motor, which designs fractional order sliding mode controller from two aspects of sliding mode surface and approach law, effectively weakens the system buffeting brought by sliding mode variable structure control and improves the system robustness and control accuracy.

For another example, a Gain-adaptive robust backstepping control of a BLDC motor system [ J ], IEEE/ASME Transactions on mechanics, 2018, 23 (5): 2470-2481) proposes a Gain-adaptive robust backstepping controller for position tracking and interference suppression of a brushless dc motor, wherein a Gain learning mechanism can realize automatic adjustment of gains of all controllers under different working states, and has strong flexibility.

It is noted that the above control methods all use full-state feedback control, and it is necessary to ensure that all states of the system can be accurately measured. For practical brushless dc motor servo systems, only the motor speed is usually conveniently and directly measured, and if other states are to be measured accurately, the control cost and the mechanical structure complexity of the motor are undoubtedly increased. In addition, a large amount of measurement noise may also reduce the likelihood of implementation of full-state feedback control.

For another example, in the literature (simulation [ J ], information and control [ 2017, 46 (3): 350-. For model mismatch phenomena caused by external loads other than zero, load feedback compensation is used to improve system performance, but such compensation requires the assumption that the external load is fully measurable, which is often difficult to achieve in practical control systems.

Disclosure of Invention

The technical problem to be solved by the invention is as follows: aiming at the technical problems in the prior art, the invention provides a brushless direct current motor servo system disturbance suppression and rotating speed tracking control method which is simple to realize, strong in control real-time performance, high in tracking precision and good in robustness.

In order to solve the technical problems, the invention adopts the following technical scheme:

a composite control method for improving the anti-interference capability of a brushless direct current motor comprises the following steps:

step S1: constructing a brushless direct current motor state equation, summarizing parameter perturbation and external load interference into total disturbance, and establishing a brushless direct current motor servo system state space model;

step S2: the total disturbance is used as an expansion state variable, and a system equivalent state space model is reconstructed by separating a measurable state and an undetectable state of a system;

step S3: constructing a reduced order extended state observer, and estimating total disturbance and a system undetectable state on line;

step S4: by utilizing the estimated value of the reduced order extended state observer and the measurable state of the system, a backstepping controller based on disturbance dynamic compensation is designed, the influence of disturbance on the system output is inhibited, the dynamic performance and the robustness of the system are improved, and the accurate tracking of the system output on the reference input is ensured.

As a further improvement of the process of the invention: in step S1, a state space model of the brushless dc motor is derived from the voltage balance equation and the torque balance equation.

As a further improvement of the process of the invention: in the step S4, a backstepping controller based on disturbance dynamic compensation is designed according to the Lyapunov stability theory.

As a further improvement of the process of the invention: in step S1, the establishing step of the brushless dc motor servo system state space model is as follows:

step S101: assuming that the waveform of the back electromotive force is a trapezoidal wave with a flat top width of 120 electrical angles, the power device of the driving circuit is an ideal switch, and the three-phase windings are completely symmetrical, the voltage balance equation of the three-phase windings of the brushless direct current motor is as follows:

wherein u is_a(t)、u_b(t)、u_c(t) phase voltages of stator windings, i_a(t)、i_b(t)、i_c(t) stator winding phase current, R phase winding resistance, Ls phase winding inductance, e_a(t)、e_b(t)、e_cAnd (t) is the counter electromotive force of each phase winding.

The electromagnetic torque equation is:

in the formula, T_eAnd (t) is the electromagnetic torque, and omega (t) is the angular speed of the motor rotor.

The torque balance equation is:

in the formula, M_cAnd (t) is the torque of the external load, and J is the total moment of inertia of the motor rotor and the load.

Step S102: the voltage balance equation is further expressed as:

where u (t) is the voltage applied to the two phase conducting windings, i (t) is the armature current, and ε (t) is the back-EMF of the conducting phase windings.

The electromagnetic torque equation is further expressed as:

when the back electromotive force is trapezoidal wave, the back electromotive force calculation formula of the conducting phase winding is as follows:

ε(t)＝k_epω(t)

in the formula, k_eIs the back emf coefficient and p is the motor pole pair number.

Establishing a state equation of a brushless direct current motor servo system by a voltage balance equation and a torque balance equation:

step S103: and respectively carrying out Laplace transformation and Laplace inverse transformation on differential equation expressions of the rotating speed subsystem and the current subsystem to obtain:

the motor is influenced by friction and eddy current in the actual operation process, the internal temperature of the motor can change, and the resistance R and the inductance L of the armature winding are caused_sCreating a parametric perturbation phenomenon. On the other hand, a variation in the external load causes perturbation of the moment of inertia J. Assuming the following parameter uncertainties exist for the above equation:

wherein R is₀、L₀And J₀Respectively representing nominal values of phase resistance, phase inductance and moment of inertia, R₀ΔR、L₀Δ upper and J₀Δ J represents the amounts of pickup of the phase resistance, the phase inductance, and the moment of inertia, respectively.

After considering the parameter perturbation, willFurther expressed as:

in the formula, the input gain is controlledf_total(t) is the 'total disturbance' containing parameter uncertainty and external load interference, and the specific expression is as follows:

step S104: first take x₁(t)＝ω(t)，Then define the state variable x_p(t)＝[x₁(t) x₂(t)]^TControl input is u (t) and system output is y_pAnd (t) ω (t), establishing a brushless direct current motor servo system state space model:

wherein the coefficient matrix is:

obviously, (A)_p，B_u) And (4) controlling.

As a further improvement of the process of the invention: in step S2, the system equivalent state space model is established as follows:

step S201: taking an extended state variable x₃(t)＝f_total(t) is provided withEstablishing an augmentation system state space model:

wherein:

step S202: in a brushless DC motor servo system, the motor speed, i.e. state x₁(t) is calculated from the position signals of the hall sensors, and measurable variables in the augmentation system are separated from non-measurable variables to obtain:

getIs a system state variable, with control inputs u (t) and system outputsEstablishing a system equivalent state space model:

wherein the coefficient matrix is:

it is clear that,can be seen.

As a further improvement of the process of the invention: in the step S3, the step of constructing the reduced order extended state observer of the original dc motor servo system is as follows

Step S301: designing a reduced order extended state observer to realize the non-measurable stateAnd total disturbance f_totalReal-time estimation of (t):

wherein the observer stateAndare respectively the system stateAnd total disturbance f_total(t), L is the observer gain matrix to be designed.

Step S302: defining the state estimation error as:

whereinCombining the system equivalent state space model and the state equation of the reduced order extended state observer to obtain a state estimation error dynamic equation as follows:

obviously, by designing the observer gain matrix L such thatIs located in the left complex half plane, and when h (t) is bounded, the bounded input and bounded output of the reduced order extended state observer are guaranteed to be stable.

Obtaining an observer gain matrix l by adopting a bandwidth setting method, and setting l as [ beta ]₁ β₂]^T，ω_oObserver bandwidth, such that:

(s+ω_o)²＝s²+β₁s+β₂

finding beta₁＝2ω_o，β₂＝ω_o ².

Step S303: in order to eliminate derivative terms of measurable states in the reduced order extended state observer, the state equation of the reduced order extended state observer is subjected to equivalent deformation to obtain:

is recorded as:

wherein the content of the first and second substances,

as a further improvement of the process of the invention: in step S4, the step of designing the back-step controller based on disturbance dynamic compensation is as follows:

step S401: note the bookIs x₁(t) -the sub-system(s),is x₂(t) -subsystem. For x₁(t) -subsystem, let x_1d(t) is state x₁(t) the expected value, i.e. the reference input trajectory, defines a first state error and a Lyapunov function as:

introducing a virtual control input alpha₂(t) for the above formula V₁(t) deriving:

when in useNegative timing, state error S₁(t) can converge asymptotically to zero. To make x₁(t) -subsystem in virtual controlInput alpha₂(t) is gradually stable under the action ofHas the following form:

wherein k is₁∈R⁺Virtual control input alpha introduced for virtual control gain to be designed₂(t) is:

at this timeWherein S₁(t)[x₂(t)-α₂(t)]As residual terms, influenceThe negative decision will be eliminated in the next derivation by selecting the appropriate control input u (t).

To avoid inputting alpha to the virtual control in the subsequent design process₂(t) direct derivation amplifies noise and causes severe control and flutter response, introducing a low pass filter q(s) 1/(τ s +1) to the virtual control input α₂(t) filtering, the filtered virtual control input being recorded asCorresponding to the filtered state equation:

where τ is the filter time constant.

Step S402: for x₂(t) -subsystem, definitionThe second state error and the Lyapunov function are respectively:

to the above formula V₂(t) deriving:

when in useNegative timing, state error S₂(t) can converge asymptotically to zero. To make x₂(t) -the subsystem is asymptotically stabilized by the control input u (t) without settingHas the following form:

wherein k is₂∈R⁺The gain is input for the control to be designed. The control input u (t) of the system is obtained as follows:

wherein the content of the first and second substances,andfor expanding the state variables of the state observer, i.e. the system state, in step S3And total disturbance f_total(t) an estimated value.

Compared with the prior art, the invention has the advantages that:

1. the composite control method for improving the anti-interference capability of the brushless direct current motor is simple to realize, strong in control real-time performance and high in tracking precision. Under the condition that the output rotating speed of the motor can be directly measured by a sensor, the invention provides a backstepping control method based on the reduced order extended state observer, and the total disturbance and the system undetectable state are estimated in real time by designing the reduced order extended state observer. The introduction of the order-reducing extended state observer ensures that the controller design only needs to utilize the relative order information of the controlled object, and makes up the defect that the design of the conventional backstepping controller seriously depends on an accurate model. Compared with the conventional full-order extended state observer, the reduced-order extended state observer provided by the invention has smaller phase lag and higher total disturbance estimation precision under the same bandwidth condition.

2. The composite control method for improving the anti-interference capability of the brushless direct current motor can eliminate the influence of interference on system output and ensure that the system has good tracking performance. According to the invention, the backstepping controller based on disturbance dynamic compensation is designed according to the Lyapunov stability theory, and a first-order low-pass filter is introduced to estimate a first derivative term of the virtual control input, so that the problems of 'expansion calculation' and tremor response caused by direct derivation of the virtual control input are effectively avoided. By designing the backstepping controller with the disturbance estimation and compensation mechanism, the anti-interference capability and tracking accuracy of the brushless direct current motor servo system can be effectively improved, and the control requirement of the brushless direct current motor servo system in the high-accuracy servo field can be met.

Drawings

FIG. 1 is a schematic flow diagram of the present invention.

Fig. 2 is an equivalent circuit diagram of a brushless dc motor.

Fig. 3 is a block diagram of a control system in a specific application example of the present invention.

Fig. 4 shows the frequency response from the estimated value of the total disturbance to the total disturbance in the reduced-order extended state observer (RESO) and the conventional full-order extended state observer (CESO) proposed by the present invention.

FIG. 5 is a diagram of the hardware connection of the experimental platform in a specific application example of the present invention.

Fig. 6 shows a system output response curve and a total disturbance estimation curve when the control method of the present invention is adopted in experiment 1 according to the embodiment of the present invention, in which (a) is a reference input and system output curve, (B) is a system tracking error curve, (C) is a system control input curve, and (D) is a disturbance estimation curve.

Fig. 7 shows experiment 1 of an embodiment of the present invention comparing the tracking error of the control method (RESO-backstepping) with the conventional full-order extended state observer based back-stepping control (CESO-backstepping), Linear Active Disturbance Rejection Control (LADRC) and reduced-order linear active disturbance rejection control (RLADRC) systems.

Fig. 8 shows a system output response curve and a total disturbance estimation curve when the control method of the present invention is adopted in experiment 2 according to the embodiment of the present invention, in which (a) is a reference input and system output curve, (B) is a system tracking error curve, (C) is a system control input curve, and (D) is a disturbance estimation curve.

Fig. 9 is a comparison graph of tracking errors of experiment 2 according to an embodiment of the present invention, which is based on the control method (RESO-backstepping) of the present invention, and a back-step control (CESO-backstepping), linear active disturbance rejection control (larcc), and reduced-order linear active disturbance rejection control (RLADRC) system based on a conventional full-order extended state observer, where (a) is an error graph of experiment 2 according to an embodiment of the present invention, and (B) is an error graph based on a conventional full-order extended state observer.

Fig. 10 is a schematic diagram of the structural principle of the present invention in a specific application example.

Detailed Description

The invention will be described in further detail below with reference to the drawings and specific examples.

As shown in fig. 1 and 10, a compound control method for improving the noise immunity of a brushless dc motor according to the present invention includes the steps of:

step S1: constructing a brushless direct current motor state equation, summarizing parameter perturbation and external load interference into total disturbance, and establishing a brushless direct current motor servo system state space model;

step S2: the total disturbance is used as an expansion state variable, and a system equivalent state space model is reconstructed by separating a measurable state and an undetectable state of a system;

step S3: constructing a reduced order extended state observer, and estimating total disturbance and a system undetectable state on line;

step S4: by utilizing the estimated value of the reduced order extended state observer and the measurable state of the system, a backstepping controller based on disturbance dynamic compensation is designed, the influence of disturbance on the system output is inhibited, the dynamic performance and the robustness of the system are improved, and the accurate tracking of the system output on the reference input is ensured.

In a specific application example, in step S1, the establishing step of the state space model of the brushless dc motor servo system includes:

step S101: taking a three-phase bridge type Y-connection brushless direct current motor as an example, assuming that a waveform of back electromotive force is a trapezoidal wave with a flat top width of 120 electrical angles, a power device of a driving circuit is an ideal switch, a three-phase winding is completely symmetrical, current of a fly-wheel diode is ignored, hysteresis loss and eddy current loss are ignored, mutual inductance between windings is ignored, and then a voltage balance equation of the three-phase winding of the motor is as follows:

wherein u is_a(t)、u_b(t)、u_c(t) phase voltages of stator windings, i_a(t)、i_b(t)、i_c(t) stator winding phase current, R phase winding resistance, L_sInductance of each phase winding, e_a(t)、e_b(t)、e_cAnd (t) is the counter electromotive force of each phase winding.

Electromagnetic torque equation:

in the formula, T_e(t) is the electromagnetic torque, ω (t) is the angle of the rotor of the machineSpeed.

Torque balance equation:

in the formula, M_cAnd (t) is the torque of the external load, and J is the total moment of inertia of the motor rotor and the load.

Step S102: because the three-phase bridge type Y-connection brushless direct current motor works in a pairwise conduction six-state mode, only two phases are conducted at any time by considering an ideal commutation condition, current flows in from one phase and flows out from the other phase, and the third phase is suspended. The voltage balance equation can be further expressed as:

where u (t) is the voltage applied to the two phase conducting windings, i (t) is the armature current, and ε (t) is the back-EMF of the conducting phase windings.

The electromagnetic torque equation may be further expressed as:

when the back electromotive force is trapezoidal wave, the back electromotive force calculation formula of the conducting phase winding is as follows:

ε(t)＝k_epω(t)

in the formula, k_eIs the back emf coefficient and p is the motor pole pair number.

Establishing a state equation of a brushless direct current motor servo system by a voltage balance equation and a torque balance equation:

step S103: and respectively carrying out Laplace transformation and Laplace inverse transformation on differential equation expressions of the rotating speed subsystem and the current subsystem to obtain:

the motor is influenced by friction and eddy current in the actual operation process, the internal temperature of the motor can change, and the resistance R and the inductance L of the armature winding are caused_sCreating a parametric perturbation phenomenon. On the other hand, a variation in the external load causes perturbation of the moment of inertia J. Assuming the following parameter uncertainties exist for the above equation:

wherein R is₀、L₀And J₀Respectively representing nominal values of phase resistance, phase inductance and moment of inertia, R₀ΔR、L₀Δ L and J₀Δ J represents the amounts of pickup of the phase resistance, the phase inductance, and the moment of inertia, respectively.

After considering the parameter perturbation, willFurther expressed as:

in the formula, the input gain is controlledf_total(t) is the 'total disturbance' containing parameter uncertainty and external load interference, and the specific expression is as follows:

step S104: here first take x₁(t)＝ω(t)，Then define the state variable x_p(t)＝[x₁(t) x₂(t)]^TControl input is u (t) and system output is y_pAnd (t) ω (t), establishing a brushless direct current motor servo system state space model:

wherein the coefficient matrix is:

obviously, (A)_p，B_u) And (4) controlling.

In a specific application example, in step S2, the establishing step of the system equivalent state space model is as follows:

step S201: taking an extended state variable x₃(t)＝f_total(t) is provided withEstablishing an augmentation system state space model:

wherein:

step S202: in a brushless DC motor servo system, the motor speed, i.e. state x₁(t) can be accurately calculated from the position signals of the hall sensors, and measurable variables and non-measurable variables in the augmentation system can be separated:

getIs a system state variable, with control inputs u (t) and system outputsEstablishing a system equivalent state space model:

wherein the coefficient matrix is:

it is clear that,can be seen.

In a specific application example, in step S3, the detailed step of constructing the reduced order extended state observer is as follows:

step S301: in linear auto-disturbance rejection control, all states are estimated online using a Conventional full-order extended state observer (CESO). It should be noted that in the brushless dc motor servo system, the motor rotation speed can be measured without estimation. Therefore, the invention realizes the purpose of the undetectable state observer (RESO) by designing a Reduced-order extended state observerAnd total disturbance f_total(t) real-time estimation. The order of the observer is reduced from three orders to two orders, so that the phase lag caused by CESO can be reduced, and the system structure is simplified. The RESO design is as follows:

wherein the observer stateAndare respectively the system stateAnd total disturbance f_total(t), l is the observer gain matrix to be designed.

Step S302: defining the state estimation error as:

whereinCombining the system equivalent state space model and the state equation of the RESO to obtain an observer estimation error dynamic equation as follows:

obviously, by designing the observer gain matrix l such thatIs located in the left complex half-plane, and when h (t) is bounded, the bounded input of the extended state observer is guaranteed to be bounded and the output is stable.

Obtaining an observer gain matrix L by adopting a bandwidth setting method, and setting L as [ beta ]₁ β₂]^T，ω_oIs observer bandwidth, such that

(s+ω_o)²＝s²+β₁s+β₂

Finding beta₁＝2ω_o，β₂＝ω_o ².

Step S303: to eliminate the derivative term of the measurable state in the RESO, the state equation of the RESO is equivalently transformed to obtain:

is marked as

Wherein the content of the first and second substances,

in a specific application example, in the step S4, the step of designing the backstepping controller based on disturbance dynamic compensation is as follows:

step S401: note the bookIs x₁(t) -the sub-system(s),is x₂(t) -subsystem. For x₁(t) -subsystem, let x_1d(t) is state x₁(t) the expected value, i.e. the reference input trajectory, defines a first state error and a Lyapunov function as:

introducing a virtual control input alpha₂(t) for the above formula V₁(t) deriving:

when in useNegative timing, state error S₁(t) can converge asymptotically to zero. To make x₁(t) -subsystem at virtual control input α₂(t) is gradually stable under the action ofHas the following form:

wherein k is₁∈R⁺Virtual control input alpha introduced for virtual control gain to be designed₂(t) is:

at this timeWherein S₁(t)[x₂(t)-α₂(t)]As residual terms, influenceThe negative decision will be eliminated in the next derivation by selecting the appropriate control input u (t).

To avoid inputting alpha to the virtual control in the subsequent design process₂(t) direct derivation amplifies noise and causes severe control and flutter response, introducing a low pass filter q(s) 1/(τ s +1) to the virtual control input α₂(t) filtering, and recording the filtered variable asSo there is a filtered equation of state:

where τ is the filter time constant.

Step S402: for x₂(t) -subsystem defining a second state error and a Lyapunov function as:

to the above formula V₂(t) deriving:

when in useNegative timing, state error S₂(t) can converge asymptotically to zero. To make x₂(t) -the subsystem is asymptotically stabilized by the control input u (t) without settingHas the following form:

wherein k is₂∈R⁺The gain is input for the control to be designed. Obtaining a system control input u (t) as:

wherein the content of the first and second substances,andfor expanding the state variables of the state observer, i.e. the system state, in step S3And total disturbance f_total(t) an estimated value.

The invention is further illustrated below using brushless dc speed control experiments.

In order to test the feasibility and the superiority of the composite control method for improving the anti-interference capability of the brushless direct current motor under the actual working condition, a brushless direct current motor disturbance suppression and rotating speed tracking control experiment platform is built in the embodiment, and a motor control experiment and a comparison experiment are carried out.

Fig. 3 is a block diagram of a control system. Three Hall position sensors are embedded in the brushless direct current motor at intervals of 120 degrees, when the permanent magnet magnetic poles pass through, the Hall sensors can generate a group of three-digit binary position signals, and the position signals can be used for realizing electronic commutation and real-time rotating speed measurement of the motor. The magnetic powder brake is connected with a rotating shaft of the brushless direct current motor through a coupler and is used for receiving voltage output by an analog quantity output port of the RTLAB OP5600 real-time digital simulator and generating corresponding torque to be used as an external load to be applied to the rotating shaft of the motor. And the RTLAB OP5600 is used as a controller to receive a control program from an upper computer, and generates a PWM control signal after operation processing. And the motor drive board receives the PWM control signal and controls the on and off of a power device in the three-phase bridge circuit so as to realize the rotation speed regulation of the motor.

In this example, a brushless DC motor model S60BL-430 was used for experimental studies, and the specific motor parameters are listed in Table 1.

TABLE 1S 60BL-430 MODEL BRUSHLESS DC MOTOR PARAMETER TABLE

In order to compare the disturbance estimation performances of the RESO and the conventional full-order ESO, the total disturbance estimation value in the RESO and the CESO is obtained through calculationTo total disturbance f_totalThe transfer functions of (t) are:

FIG. 4 shows that the RESO and the conventional full-order ESO proposed by the present invention are taken as ω in the observer bandwidth_oWhen equal to 50, G_RESOAnd G_CEsOFrequency response curve of (1). As can be seen from the graph, the phase lag of the RESO at the low frequency band is smaller, and the estimation of the total disturbance is more accurate. But also note that the gain of the RESO is larger in the high frequency band compared to CESO, which means that the RESO is more sensitive to high frequency noise, and the present invention can well compensate for this deficiency of the RESO by introducing a low pass filter in the design process of the back-step controller.

Experiment 1: a triangular wave reference input rotating speed tracking control experiment under the condition of no external load interference;

FIG. 5 is a diagram of experimental platform hardware connections. And no external load interference is added, and the servo system is influenced by parameter perturbation and unmodeled nonlinear disturbance at the moment. And setting a triangular wave signal with a reference input rotating speed period of 6s, a maximum amplitude of 800r/min and a minimum amplitude of 500r/min, and setting a system sampling step length to be 0.00002 s.

When the control method provided by the invention is adopted for experiments, the RESO bandwidth is omega_oThe backstepping controller controls the gain k to 50₁＝150，k₂11. In a contrast experiment, the CESO bandwidth in the CESO-based backstepping controller is taken to be omega_oThe backstepping controller controls the gain k to 50₁＝150，k₂11; the bandwidth of the observers of the linear active disturbance rejection controller and the reduced order linear active disturbance rejection controller are both taken as omega_oThe bandwidth of the state feedback controller is taken as omega at 50_c＝23。

Fig. 6 shows a system output response curve and a total disturbance estimation curve when the control method of the present invention is adopted in experiment 1 according to an embodiment of the present invention, where (a) is a reference input and system output curve, (B) is a system tracking error curve, (C) is a system control input curve, and (D) is a disturbance estimation curve. As can be seen from the figure, when the backstepping control method based on the reduced order extended state observer provided by the invention is adopted, the brushless direct current motor servo system can effectively inhibit internal interference, and the motor rotating speed can quickly and accurately track the given triangular wave reference input rotating speed.

Fig. 7 is a comparison graph of tracking errors of the control method (RESO-backstepping) and the back-step control (CESO-backstepping), linear active disturbance rejection control (lardrc) and reduced-order linear active disturbance rejection control (RLADRC) systems based on the conventional full-order extended state observer, which are respectively adopted in the embodiment of the invention.

Experiment 2: rotating speed tracking control experiment under time-varying external load interference;

in this experiment, the reference input rotation speed was set to r (t) 1000r/min, and the external load torque was set to:

all controller parameter settings in this experiment and comparative experiments were consistent with experiment 1.

Fig. 8 shows a system output response curve and a total disturbance estimation curve when experiment 2 of the embodiment of the present invention adopts the control method of the present invention, where (a) is a reference input and system output curve, (B) is a system tracking error curve, (C) is a system control input curve, and (D) is a disturbance estimation curve. As can be seen from the figure, when the backstepping control method based on the reduced-order extended state observer provided by the invention is adopted, the brushless direct current motor can quickly and accurately track the given reference input rotating speed. When the load sudden change occurs at the 20 th s and the 30 th s, the fluctuation of the rotating speed tracking error is small, and the steady state is recovered after about 1 s. After 40s, under the sinusoidal time-varying external load interference, the maximum steady-state tracking error is about 4.2r/min, namely 0.042% of the reference input, so that the control system provided by the invention has stronger disturbance suppression performance and tracking performance.

Fig. 9 is a comparison graph of tracking errors of experiment 2 according to an embodiment of the present invention, which is based on the control method (RESO-backstepping) of the present invention, and a back-step control (CESO-backstepping), Linear Active Disturbance Rejection Control (LADRC), and reduced-order linear active disturbance rejection control (RLADRC) system based on a conventional full-order extended state observer, where (a) is an error graph of experiment 2 according to an embodiment of the present invention, and (B) is an error graph based on a conventional full-order extended state observer. It can be seen that when the method provided by the invention is adopted, the system response speed is higher, the tracking error deviation from the steady state degree is smaller when the load is suddenly changed, the time for recovering the steady state is shorter, and the steady state tracking error under the sinusoidal time-varying load is smaller. Therefore, when the backstepping control method based on the reduced order extended state observer provided by the invention is adopted, the robustness and the tracking performance of the system are better.

The above is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above-mentioned embodiments, and all technical solutions belonging to the idea of the present invention belong to the protection scope of the present invention. It should be noted that modifications and embellishments within the scope of the invention may be made by those skilled in the art without departing from the principle of the invention.