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CN107656437B - A Disturbance Observer-Based Control Method for Unmatched Disturbances in Magnetic Suspension Rotor Systems - Google Patents

A Disturbance Observer-Based Control Method for Unmatched Disturbances in Magnetic Suspension Rotor Systems Download PDF

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CN107656437B
CN107656437B CN201710692093.2A CN201710692093A CN107656437B CN 107656437 B CN107656437 B CN 107656437B CN 201710692093 A CN201710692093 A CN 201710692093A CN 107656437 B CN107656437 B CN 107656437B
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彭聪
祝梦婷
邓智泉
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Nanjing University of Aeronautics and Astronautics
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Abstract

本发明所公开的基于扰动观测器的新型复合控制方法,可用于抑制电压控制型磁轴承系统中的不匹配扰动。其基于物理定律建立不匹配扰动下电压控制型磁轴承系统的动态模型,通过引入新的状态变量重构等效磁悬浮转子动力学系统,等效系统中将不匹配扰动分为匹配部分和不匹配部分,匹配部分通过鲁棒控制器进行干扰抑制,不匹配部分通过设计状态空间扰动观测器进行干扰抵消。本发明提出的不匹配扰动控制方法,对磁轴承系统的高精度悬浮控制具有重要参考意义。

The novel composite control method based on the disturbance observer disclosed in the invention can be used to suppress the mismatch disturbance in the voltage-controlled magnetic bearing system. Based on the laws of physics, the dynamic model of the voltage-controlled magnetic bearing system under mismatched disturbance is established, and the equivalent magnetic suspension rotor dynamics system is reconstructed by introducing new state variables. The mismatched disturbance is divided into the matching part and the mismatching part in the equivalent system. Part, the matched part is suppressed by a robust controller, and the unmatched part is cancelled by designing a state-space disturbance observer. The mismatch disturbance control method proposed by the present invention has important reference significance for the high-precision suspension control of the magnetic bearing system.

Description

基于扰动观测器的磁悬浮转子系统不匹配扰动的控制方法A Disturbance Observer-Based Control Method for Unmatched Disturbances in Magnetic Suspension Rotor Systems

技术领域technical field

本发明属于航天控制技术研究领域,特别涉及一种基于扰动观测器的磁悬浮转子系统不匹配扰动的控制方法。The invention belongs to the research field of aerospace control technology, and in particular relates to a disturbance observer-based control method for unmatched disturbance of a magnetic suspension rotor system.

背景技术Background technique

相比于传统机械轴承,磁悬浮系统具有很多优点,如转速更高、磨损少、无需润滑、寿命更长。正是因为这些优势,磁悬浮系统成功地应用于飞轮、控制力矩陀螺以及泵等很多领域。磁轴悬浮控制系统有两种控制模式:电流控制模式和电压控制模式。总的来说,电压控制模式优于电流控制模式。一方面,电压控制模式的输入是线圈绕组电压,是比电流控制模式更准确的系统输入变量;另一方面,电流控制模式下的功率放大器比电压控制模式更为复杂和昂贵。电压控制模式相比于电流控制模式有许多优势,但是电压控制型磁悬浮系统的动态性能也更为复杂:外部扰动通过与控制输入不同的通路作用于磁悬浮系统,便形成了不满足匹配条件的不匹配扰动。如何抑制电压控制型磁悬浮系统中的不匹配扰动已经成为一个具有挑战性的问题。Compared with traditional mechanical bearings, magnetic suspension systems have many advantages, such as higher rotational speed, less wear, no need for lubrication, and longer life. It is precisely because of these advantages that the magnetic levitation system has been successfully used in many fields such as flywheels, control torque gyroscopes and pumps. The magnetic shaft suspension control system has two control modes: current control mode and voltage control mode. In general, the voltage control mode is better than the current control mode. On the one hand, the input of the voltage control mode is the coil winding voltage, which is a more accurate system input variable than the current control mode; on the other hand, the power amplifier in the current control mode is more complex and expensive than the voltage control mode. Compared with the current control mode, the voltage control mode has many advantages, but the dynamic performance of the voltage control magnetic levitation system is also more complicated: the external disturbance acts on the magnetic levitation system through a different path from the control input, forming a mismatch that does not meet the matching conditions. Match perturbation. How to suppress the mismatch disturbance in the voltage-controlled magnetic levitation system has become a challenging problem.

现代鲁棒控制方法已经广泛应用于电压控制型磁悬浮系统,在提高系统的稳定性和动态性能方面取得了一些显著的效果。单轴磁悬浮系统在电压控制模式下的线性化、利用不同的反馈方法来提高线性度和闭环鲁棒性、磁悬浮系统中高速转子面对外部扰动时的稳定性和可控性,以及利用一种鲁棒综合控制方法来补偿结构谐振等问题均已在研究之列。然而上述提到的研究主要通过反馈调节的方式抑制外部扰动和不确定性,不能直接地、及时地抑制强烈的外部扰动和补偿对象的不确定性。Modern robust control methods have been widely used in voltage-controlled magnetic levitation systems, and have achieved some remarkable results in improving the stability and dynamic performance of the system. Linearization of single-axis maglev systems in voltage control mode, utilizing different feedback methods to improve linearity and closed-loop robustness, stability and controllability of high-speed rotors in maglev systems in the face of external disturbances, and utilizing a Problems such as robust integrated control methods to compensate structural resonances have been studied. However, the research mentioned above mainly suppresses external disturbances and uncertainties by means of feedback regulation, and cannot directly and timely suppress strong external disturbances and the uncertainty of compensation objects.

由此,结合鲁棒控制和扰动观测器的复合控制方法悄然兴起,并成功应用于扰动满足匹配条件的各种工程系统中,如电机系统,功率转换器系统,硬盘驱动器系统,机器人系统等。在这种复合控制方法中,扰动观测器可以观测未知扰动,并对这些扰动做出补偿,且不以牺牲基本控制器的性能为代价。随着基于扰动观测器的磁悬浮转子系统匹配扰动的抑制方法的研究不断深入,研究人员希望能够通过这种复合控制方法来抑制外部扰动中的不匹配扰动。As a result, the composite control method combining robust control and disturbance observer has quietly emerged, and has been successfully applied to various engineering systems where disturbances satisfy matching conditions, such as motor systems, power converter systems, hard disk drive systems, robotic systems, etc. In this composite control method, the disturbance observer can observe unknown disturbances and compensate for them without sacrificing the performance of the basic controller. With the deepening of the research on the matching disturbance suppression method of the magnetic levitation rotor system based on the disturbance observer, researchers hope to suppress the mismatch disturbance in the external disturbance through this composite control method.

可见,目前电压控制型磁悬浮转子系统中不匹配扰动抑制的控制方法存在如下问题:1)利用反馈调节的方式抑制外部扰动和对象不确定性,只是减弱不匹配扰动对输出的影响,不能直接地、及时地抑制外部扰动和补偿对象不确定性;2)不匹配扰动的补偿性能在一定程度上是以牺牲基本控制器的性能指标的代价获得的。It can be seen that the current control methods for the suppression of mismatched disturbances in the voltage-controlled magnetic levitation rotor system have the following problems: 1) The use of feedback regulation to suppress external disturbances and object uncertainty only reduces the impact of mismatched disturbances on the output, and cannot directly , to suppress the external disturbance and compensate the uncertainty of the object in time; 2) The compensation performance of the mismatched disturbance is obtained at the expense of the performance index of the basic controller to a certain extent.

发明内容SUMMARY OF THE INVENTION

为克服现有的电压控制型磁悬浮转子系统的控制方法的不足,提供一种基于扰动观测器的新型复合控制方法,采用基于改进的扰动观测器的复合控制策略,可以利用扰动观测器观测未知扰动,并对这些扰动做出补偿,且不以牺牲基本控制器的性能为代价,实现电压控制型磁悬浮转子系统中外部扰动的抑制。In order to overcome the shortcomings of the existing control methods of the voltage-controlled magnetic levitation rotor system, a new compound control method based on disturbance observer is provided. Using the compound control strategy based on the improved disturbance observer, the disturbance observer can be used to observe unknown disturbances. , and compensate these disturbances without sacrificing the performance of the basic controller to achieve the suppression of external disturbances in the voltage-controlled magnetic suspension rotor system.

本发明所公开的基于扰动观测器的新型复合控制方法,可用于抑制电压控制型磁轴承系统中的不匹配扰动。其基于物理定律建立不匹配扰动下电压控制型磁轴承系统的动态模型,通过引入新的状态变量重构等效磁悬浮转子动力学系统,等效系统中将不匹配扰动分为匹配部分和不匹配部分,匹配部分通过鲁棒控制器进行干扰抑制,不匹配部分通过设计状态空间扰动观测器进行干扰抵消。包括以下步骤:The novel composite control method based on the disturbance observer disclosed in the invention can be used to suppress the mismatch disturbance in the voltage-controlled magnetic bearing system. Based on the laws of physics, the dynamic model of the voltage-controlled magnetic bearing system under mismatched disturbance is established, and the equivalent magnetic suspension rotor dynamics system is reconstructed by introducing new state variables. The mismatched disturbance is divided into the matching part and the mismatching part in the equivalent system. Part, the matched part is suppressed by a robust controller, and the unmatched part is cancelled by designing a state-space disturbance observer. Include the following steps:

1)建立不匹配扰动下电压控制型磁悬浮转子系统的动态模型,得到电压控制型磁轴承控制系统的状态方程;1) Establish the dynamic model of the voltage-controlled magnetic suspension rotor system under mismatched disturbances, and obtain the state equation of the voltage-controlled magnetic bearing control system;

2)通过引入新的状态变量重构磁悬浮转子系统的等效系统,将不匹配扰动分为匹配和不匹配两部分;2) The equivalent system of the magnetic suspension rotor system is reconstructed by introducing new state variables, and the mismatch disturbance is divided into matching and mismatching parts;

3)设计广义的状态空间扰动观测器,对等效系统中的不匹配扰动部分进行观测;3) Design a generalized state space disturbance observer to observe the mismatched disturbance part in the equivalent system;

4)根据步骤3)得到的扰动观测器观测的扰动估计值,在基础鲁棒控制器中引入等效补偿,实现对外部扰动的抑制,并得到改进的复合控制系统;4) According to the disturbance estimation value observed by the disturbance observer obtained in step 3), equivalent compensation is introduced into the basic robust controller, so as to realize the suppression of external disturbance, and obtain an improved composite control system;

5)对改进的复合控制系统进行稳定性分析,计算出基础鲁棒控制器及扰动观测器的控制增益。5) The stability analysis of the improved composite control system is carried out, and the control gains of the basic robust controller and the disturbance observer are calculated.

进一步的,步骤1)所建立的电压控制型磁轴承控制系统的状态方程为:Further, the state equation of the voltage-controlled magnetic bearing control system established in step 1) is:

式中,d(s;t)表示总扰动,s为转子位移,t为时间,u=[us]T,x=[x1,x2,x3]T,x1=s,x3=is均为状态变量,是转子的速度,is是控制电流,状态矩阵为:In the formula, d(s; t) represents the total disturbance, s is the rotor displacement, t is the time, u=[u s ] T , x=[x 1 , x 2 , x 3 ] T , x 1 =s, x 3 =is are all state variables, is the speed of the rotor , is the control current, and the state matrix is:

式中,m是刚性转子的质量,R是线圈电阻,L是线圈电感,表示在工作点(is=iN,s=sN)的控制电流和位置刚度。where m is the mass of the rigid rotor, R is the coil resistance, L is the coil inductance, and represents the control current and position stiffness at the operating point (is = i N , s = s N ).

进一步的,总扰动d(s;t)包括参数变化和外部干扰,表示为:Further, the total disturbance d(s; t) includes parameter changes and external disturbances, and is expressed as:

d(s;t)=ΔKi(s;t)is(t)+ΔKs(s;t)s(t)+fd(t)d(s; t)=ΔK i (s; t)i s (t)+ΔK s (s; t)s(t)+f d (t)

式中,fd是外部干扰,ΔKi(s;t)和ΔKs(s;t)是参数变化量。where f d is the external disturbance, ΔK i (s; t) and ΔK s (s; t) are the parameter changes.

进一步的,所述步骤2)用新的状态变量η(t)替代原系统里的状态变量is(t),新状态变量定义为:Further, the step 2) replaces the state variable is (t) in the original system with a new state variable n( t ), and the new state variable is defined as:

η(t)=is(t)-id(t)(8)η(t)=is ( t )-id ( t )(8)

等效的电压控制型磁轴承控制系统可以表示为:The equivalent voltage-controlled magnetic bearing control system can be expressed as:

式中,不匹配扰动d(s;t)被分为两个部分:假设扰动观测器可以渐进地跟踪扰动,即t→∞时, where the mismatch perturbation d(s; t) is divided into two parts: and Assuming that the disturbance observer can track the disturbance asymptotically, i.e. when t→∞,

进一步的,所述步骤3)包括如下步骤:Further, described step 3) comprises the steps:

引入一个辅助矢量ω(s;t),重新定义总扰动d(s;t)为:Introducing an auxiliary vector ω(s; t), redefine the total disturbance d(s; t) as:

d(s;t)=Vω(s;t) (10)d(s;t)=Vω(s;t) (10)

式中,W和V是系数矩阵;where W and V are coefficient matrices;

复合系统由状态变量(7)和干扰变量(11)组成,表示为:The composite system consists of state variables (7) and disturbance variables (11), which are expressed as:

根据状态观测器的设计方法,扰动观测器可以设计以下的子系统:According to the design method of the state observer, the disturbance observer can design the following subsystems:

式中,σ(t)是子系统的等效输出,可通过观测反馈量来提高观测where σ(t) is the equivalent output of the subsystem, which can be obtained by observing the feedback to improve observation

器精度;根据扰动观测器的结构,可以推导出扰动变量表达式如下:According to the structure of the disturbance observer, the disturbance variable can be deduced The expression is as follows:

将辅助变量代入(14),消掉(14)右侧的扰动观测器可以表示为:the auxiliary variable Substitute into (14) and remove the right side of (14) The perturbation observer can be expressed as:

式中,W和V是系数矩阵,ω(s;t)是辅助矢量,辅助变量 是扰动变量。where W and V are the coefficient matrices, ω(s; t) is the auxiliary vector, the auxiliary variable is the disturbance variable.

进一步的,设W=0,V=I,未知扰动预估器的扰动观测器模型可简化为:Further, set W=0, V=I, the disturbance observer model of the unknown disturbance predictor can be simplified as:

进一步的,所述步骤4)包括:Further, described step 4) comprises:

对于匹配扰动,复合控制器可设计为:For matched disturbances, the composite controller can be designed as:

式中,Kk为由闭环系统的性能决定的反馈增益,反馈控制可以使得系统中的状态变量稳定至零,即当t→0时,s(t)→0,is(t)→0。In the formula, K k is the feedback gain determined by the performance of the closed-loop system, and the feedback control can make the state variables in the system stabilize to zero, that is, when t→0, s(t)→0, is ( t )→0.

为保证闭环系统的鲁棒性,在等效系统中引入一个反馈鲁棒控制器,表示为:In order to ensure the robustness of the closed-loop system, a feedback robust controller is introduced into the equivalent system, which is expressed as:

ud(t)=Kx*(t) (18)u d (t)=Kx * (t) (18)

式中,K是满足H∞的反馈增益;In the formula, K is the feedback gain satisfying H∞;

改进后,磁轴承控制系统的输入变为:After the improvement, the input to the magnetic bearing control system becomes:

us=ud+Rid (19);u s = ud + Ri d (19);

将扰动观测器方程(16)和状态反馈鲁棒控制器方程(18)代入方程(9),得到改进后复合控制系统方程为:Substituting the disturbance observer equation (16) and the state feedback robust controller equation (18) into equation (9), the improved composite control system equation is:

式中,B3=[0 0 1/Ki]T,ed(s;t)是扰动观测误差,In the formula, B 3 =[0 0 1/K i ] T , ed (s; t ) is the disturbance observation error,

可以表示为:make It can be expressed as:

式中, In the formula,

进一步的,对改进的复合系统进行稳定性分析,计算得到相应的控制器增益和观测器增益,具体包括:Further, the stability analysis of the improved composite system is carried out, and the corresponding controller gain and observer gain are calculated, including:

在复合系统(25)中,由于都是H2范数收敛的,所以扰动也是H2范数收敛的,为减少扰动的影响,采用具有H∞性能指标的鲁棒控制方案,H∞控制器不仅能够保持系统的稳定,还可以使得输出参考值满足以下条件:In the composite system (25), since and are both convergent in the H 2 norm, so the perturbation is also H 2 norm convergent, in order to reduce the disturbance The influence of , using a robust control scheme with H∞ performance index, the H∞ controller can not only maintain the stability of the system, but also make the output reference value meet the following conditions:

式中,λ是表示干扰抑制能力的正常数,下面给出了新系统随机稳定性和满足H∞性能的必要条件,即证明线性矩阵不等式(LMI);In the formula, λ is a constant representing the ability of interference suppression, and the necessary conditions for the stochastic stability of the new system and satisfying H∞ performance are given below, that is, to prove the Linear Matrix Inequality (LMI);

在复合系统(25)中,任意λ>0,存在矩阵Q1>0,Q2>0以及R1,R2满足:In the composite system (25), for any λ>0, there exist matrices Q 1 >0, Q 2 >0 and R 1 , R 2 satisfying:

式中,Ξ1=sym(AQ1+B1R1),Ξ2=sym(-R2B2)。sym()表示一次矩阵运算,对于对称矩阵M有sym(M)=M+MTIn the formula, Ξ 1 =sym(AQ 1 +B 1 R 1 ), and Ξ 2 =sym(-R 2 B 2 ). sym() represents a matrix operation, and there is sym(M)=M+M T for the symmetric matrix M;

从而,得到当控制器增益观测器增益时,复合系统(25)鲁棒性渐近稳定,且满足 Thus, the controller gain is obtained when Observer gain When , the robustness of the composite system (25) is asymptotically stable and satisfies

本发明与现有技术相比的优点在于:The advantages of the present invention compared with the prior art are:

(1)扰动和不确定性通过状态变量处理,而不是输出变量;(1) Disturbances and uncertainties are handled by state variables, not output variables;

(2)不改变基本控制器的结构,并将基本悬浮鲁棒控制器性能保持在最佳状态;(2) The structure of the basic controller is not changed, and the performance of the basic suspension robust controller is kept in the best state;

(3)不匹配部分通过设计状态空间扰动观测器进行干扰抵消。(3) The mismatched part is canceled by designing a state space disturbance observer.

附图说明Description of drawings

图1为磁轴承控制系统框图;Figure 1 is a block diagram of the magnetic bearing control system;

图2为刚度参数与不同工作点的关系图,图(a)表示Ki变量,图(b)表示Ks变量;Figure 2 shows the relationship between stiffness parameters and different working points. Figure (a) represents the K i variable, and Figure (b) represents the K s variable;

图3为基于扰动观测器的复合控制方法的结构图;Fig. 3 is the structure diagram of the composite control method based on disturbance observer;

图4为状态空间扰动观测器的结构图。Figure 4 is a structural diagram of a state space disturbance observer.

具体实施方式Detailed ways

下面结合附图对本发明的技术方案进行详细说明。The technical solutions of the present invention will be described in detail below with reference to the accompanying drawings.

步骤一、建立不匹配扰动下电压控制型磁悬浮转子系统的动态模型Step 1. Establish the dynamic model of the voltage-controlled magnetic suspension rotor system under mismatched disturbances

如图1所示为磁悬浮转子系统的详细示意图,控制电压us通过功率放大器转换成线圈电流is,从而产生预期的电磁力使得转子稳定悬浮在中心位置。Figure 1 shows the detailed schematic diagram of the magnetic suspension rotor system. The control voltage u s is converted into the coil current is s through the power amplifier, thereby generating the expected electromagnetic force to make the rotor stably levitate in the center position.

根据牛顿定律,轴向磁悬浮转子系统的动力学模型是:According to Newton's law, the dynamic model of the axial magnetic suspension rotor system is:

式中,m是刚性转子的质量,Fs是电磁力,fd是外部干扰,s为转子位移。where m is the mass of the rigid rotor, F s is the electromagnetic force, f d is the external disturbance, and s is the rotor displacement.

根据麦克斯韦定律可知,电磁力与控制电流和标称气隙的偏差呈非线性关系。非线性电磁力为:According to Maxwell's law, the electromagnetic force has a nonlinear relationship with the deviation of the control current and the nominal air gap. The nonlinear electromagnetic force is:

式中,k=0.25μ0aN2是与极面区域a,电磁线圈数N,空气的磁导率μ0相关的电磁体常数。i0和s0是偏置电流和标称气隙,is是控制电流。根据发明中提出的控制策略,功率放大器的控制电流最初是由控制电压产生的。对某个工作点(is=iN,s=sN)进行泰勒级数展开,可以将非线性电磁力近似线性化,得:In the formula, k=0.25μ 0 aN 2 is the electromagnet constant related to the pole face area a, the number of electromagnetic coils N, and the permeability μ 0 of air. i 0 and s 0 are the bias current and nominal air gap, and is the control current. According to the control strategy proposed in the invention, the control current of the power amplifier is initially generated by the control voltage. By performing Taylor series expansion for a certain operating point (is = i N , s = s N ), the nonlinear electromagnetic force can be approximately linearized, and we get:

式中,表示在工作点(is=iN,s=sN)的电流和位置刚度。很明显,不总是常量,它们会随着工作点的变化而变化。在额定工作点(is=0,s=0),表达式可以简化为:In the formula, and represents the current and position stiffness at the operating point (is = i N , s = s N ). It is clear, and Not always constant, they vary with the operating point. At the rated operating point (is = 0, s = 0), the expression can be simplified to:

将工作点(is=iN,s=sN)处的表达式与额定工作点(is=0,s=0)处的表达式联立,可以得到如图2所示的刚度参数与不同工作点的关系图。图2中可以看出与额定工作点间的误差越大,刚度参数的变化越大。因此,电流刚度和位置刚度的非线性特性与额定工作点的常系数以及扰动参数有关,得到:By combining the expression at the working point (is = i N , s = s N ) with the expression at the rated working point (is = 0, s = 0), the stiffness parameters shown in Figure 2 can be obtained Diagram with different working points. It can be seen in Figure 2 that the greater the error from the rated operating point, the greater the change in stiffness parameters. Therefore, the nonlinear characteristics of current stiffness and position stiffness are related to the constant coefficients of the rated operating point and the disturbance parameters, and we get:

ΔKi(s;t)和ΔKs(s;t)是参数变化量,s是转子位移,t是时间。将(3)和(4)代入(1),磁轴承控制系统动态模型可以表示为:ΔK i (s; t) and ΔK s (s; t) are the parameter changes, s is the rotor displacement, and t is the time. Substituting (3) and (4) into (1), the dynamic model of the magnetic bearing control system can be expressed as:

式中,G1=Ks/m,G2=Ki/m,G3=1/m,d(s;t)表示总扰动,包括参数变化和外部干扰,将其定义为:In the formula, G 1 =K s /m, G 2 =K i /m, G 3 =1/m, d(s; t) represents the total disturbance, including parameter changes and external disturbances, which are defined as:

d(s;t)=ΔKi(s;t)is(t)+ΔKs(s;t)s(t)+fd(t)d(s; t)=ΔK i (s; t)i s (t)+ΔK s (s; t)s(t)+f d (t)

根据基尔霍夫电压定律,电压‐电流的动态方程为:According to Kirchhoff's voltage law, the dynamic equation of voltage-current is:

式中,R是线圈电阻,L是线圈电感。因此电压控制型磁轴承控制系统的状态方程为:where R is the coil resistance and L is the coil inductance. Therefore, the state equation of the voltage-controlled magnetic bearing control system is:

式中,u=[us]T,x=[x1,x2,x3]T。x1=s,x3=is都是状态变量,状态矩阵为:In the formula, u=[u s ] T , x=[x 1 , x 2 , x 3 ] T . x 1 = s, x 3 =is are all state variables, and the state matrix is:

从B1和B2可以看出,输入信号在第三条控制通道上,扰动分量在第二个条控制通道上。扰动与输入通过不同的通道作用在系统上,便形成了不匹配扰动。As can be seen from B 1 and B 2 , the input signal is on the third control channel and the disturbance component is on the second control channel. The perturbation and the input act on the system through different channels, forming a mismatch perturbation.

值得注意的是,本发明是把参数变化和外部干扰整合起来考虑为系统的总扰动,作为一个整体再区分为匹配和不匹配部分,而不单独讨论参数变化和外部干扰是否存在匹配和不匹配部分。It is worth noting that the present invention integrates parameter changes and external disturbances into the total disturbance of the system, and then divides them into matching and unmatched parts as a whole, without discussing whether parameter changes and external disturbances are matched or unmatched separately. part.

步骤二、建立电压控制型磁悬浮转子系统的等效系统Step 2. Establish the equivalent system of the voltage-controlled magnetic suspension rotor system

为了抵消外部扰动中的不匹配扰动,需要改变初始的系统,使得不匹配扰动能够满足匹配条件。In order to cancel the mismatch disturbance in the external disturbance, the original system needs to be changed so that the mismatch disturbance can satisfy the matching condition.

图3所示为基于扰动观测器的复合控制结构图,用新的状态变量替代原系统中的状态变量is(t),新的状态变量可定义为:Figure 3 shows the composite control structure diagram based on the disturbance observer. The state variable is ( t ) in the original system is replaced by a new state variable. The new state variable can be defined as:

η(t)=is(t)-id(t) (8)η(t)=is ( t )-id ( t ) (8)

因此,等效系统可以表示为:Therefore, the equivalent system can be expressed as:

从(9)所示的等效系统表达式可以看出,不匹配扰动d(s;t)可分为两个部分:假设扰动观测器可以渐进地跟踪扰动,即t→∞时, From the equivalent system expression shown in (9), it can be seen that the mismatch perturbation d(s; t) can be divided into two parts: and Assuming that the disturbance observer can track the disturbance asymptotically, i.e. when t→∞,

所以,在(9)所示的等效系统中,只有扰动是一直存在的。另外,从表达式中可以看出扰动与新输入ud(t)在同一条通道上,也就是说,扰动满足匹配条件。因此,不匹配扰动转化为能够满足匹配扰动的量,新系统也就等价于(7)所示的原系统。So, in the equivalent system shown in (9), only the perturbation has always existed. In addition, the perturbation can be seen from the expression on the same channel as the new input ud (t), that is, the perturbation match conditions are met. Therefore, the mismatch perturbation is transformed into a quantity that can satisfy the matching perturbation, and the new system is equivalent to the original system shown in (7).

步骤三、广义状态空间扰动观测器的设计Step 3. Design of Generalized State Space Disturbance Observer

这种广义状态空间扰动观测器可以观测未知扰动,不局限于某一种特定模型。图4为状态空间扰动观测器的结构图,可通过观测反馈量来提高状态观测器的精度。引进一个辅助矢量ω(s;t),重新定义总扰动d(s;t)为:This generalized state-space disturbance observer can observe unknown disturbances and is not limited to a particular model. Figure 4 is the structure diagram of the state space disturbance observer, which can be used to observe the feedback to improve the accuracy of the state observer. Introducing an auxiliary vector ω(s; t), redefine the total disturbance d(s; t) as:

d(s;t)=Vω(s;t) (10)d(s;t)=Vω(s;t) (10)

式中,W和V是系数矩阵。where W and V are coefficient matrices.

想要推导出扰动量d(s;t)表达式,先要求出扰动变量ω(s;t)。复合系统由状态变量(7)和干扰变量(11)构成,可以表示为:To deduce the expression of the disturbance quantity d(s; t), the disturbance variable ω(s; t) is required first. The composite system consists of state variables (7) and disturbance variables (11), which can be expressed as:

可以看出,方程(12)和降维观测器的结构相似。根据状态观测器的设计方法,扰动观测器可以设计以下的子系统:It can be seen that equation (12) and the dimensionality reduction observer have similar structures. According to the design method of the state observer, the disturbance observer can design the following subsystems:

σ(t)是子系统的等效输出。σ(t) is the equivalent output of the subsystem.

根据观测器的结构,可以推导出扰动变量表达式为:According to the structure of the observer, the disturbance variable can be deduced The expression is:

是扰动变量的估计值,将辅助变量代入(14),消掉(14)右侧的扰动观测器可以表示为: is the estimated value of the disturbance variable, the auxiliary variable Substitute into (14) and remove the right side of (14) The perturbation observer can be expressed as:

设W=0,V=I,未知扰动的扰动观测器模型可以表示为:Let W=0, V=I, the disturbance observer model of unknown disturbance can be expressed as:

步骤四、改进的复合控制器设计Step 4. Improved composite controller design

对于匹配扰动,复合控制器可设计为:For matched disturbances, the composite controller can be designed as:

式中,Kk为反馈增益,由闭环系统的性能决定。反馈控制可以慢慢消除系统中的状态变量,即当t→0时,s(t)→0,is(t)→0。但是,在电压控制型磁轴承控制系统中,由于不匹配扰动的存在,传统的复合控制器不再适用。In the formula, K k is the feedback gain, which is determined by the performance of the closed-loop system. Feedback control can slowly eliminate the state variables in the system, that is, when t→0, s(t)→0, is ( t )→0. However, in the voltage-controlled magnetic bearing control system, the traditional composite controller is no longer applicable due to the existence of mismatching disturbances.

为了抵消由于状态变量产生的不匹配扰动,需要改变初始的系统,使得不匹配扰动能够满足匹配条件。In order to counteract the mismatch disturbance caused by the state variables, the initial system needs to be changed so that the mismatch disturbance can satisfy the matching condition.

为了保证这个闭环系统的鲁棒性,需要在(9)所示的系统中设计一个反馈鲁棒控制器,可以表示为:In order to ensure the robustness of this closed-loop system, a feedback robust controller needs to be designed in the system shown in (9), which can be expressed as:

ud(t)=Kx*(t) (18)u d (t)=Kx * (t) (18)

式中,K是满足H∞的反馈增益。反馈控制策略可以保持系统的性能,即t→∞时,s(t)→0,is(t)→id(t)。虽然线圈电流稳定由一定的偏差值,但是不匹配扰动对转子唯一没有什么影响,能够满足磁轴承控制的悬浮要求。In the formula, K is the feedback gain satisfying H∞. The feedback control strategy can maintain the performance of the system, that is, when t→∞, s(t)→0, i s ( t )→id (t). Although the coil current is stabilized by a certain deviation value, the mismatch disturbance has no effect on the rotor, which can meet the suspension requirements of the magnetic bearing control.

等效系统中,磁轴承控制系统的初始输入电压us为:In the equivalent system, the initial input voltage u s of the magnetic bearing control system is:

us=ud+Rid (19)u s = ud + Ri d (19)

将扰动观测器方程(16)和状态反馈鲁棒控制器方程(18)代入方程(9),得到新系统的方程为:Substituting the disturbance observer equation (16) and the state feedback robust controller equation (18) into equation (9), the equations of the new system are:

式中,B3=[0 0 1/Ki]T,ed(s;t)是扰动观测误差,可定义为:In the formula, B 3 =[0 0 1/K i ] T , ed (s; t ) is the disturbance observation error, which can be defined as:

因此,扰动观测误差的动态方程可以表示为:Therefore, the dynamic equation of the perturbed observation error can be expressed as:

将(16)代入(22),扰动观测误差的动态方程又可以表示为:Substituting (16) into (22), the dynamic equation of disturbed observation error can be expressed as:

将新系统的方程(9)与观测误差的动态方程(23)联立,改进后复合控制系统方程为:Combining the equation (9) of the new system with the dynamic equation (23) of the observation error, the improved composite control system equation is:

可以表示为:make It can be expressed as:

式中, In the formula,

复合系统的输出参考值为:The output reference values for the composite system are:

式中,D=[D1,D2]是设计人员为满足系统性能而选择的加权矩阵。In the formula, D=[D 1 , D 2 ] is the weighting matrix selected by the designer to satisfy the system performance.

步骤五、改进的复合控制器稳定性的分析Step 5. Analysis of the stability of the improved composite controller

在复合系统(25)中,由于都是H2范数收敛的,所以扰动也是H2范数收敛的。因此,为了减少扰动的影响,我们采用具有H∞性能指标的鲁棒控制方案。H∞控制器不仅能够保持系统的稳定,还可以使得输出参考值满足以下条件:In the composite system (25), since and are both convergent in the H 2 norm, so the perturbation It is also H2 norm convergent. Therefore, in order to reduce disturbance , we adopt a robust control scheme with H∞ performance metrics. The H∞ controller can not only maintain the stability of the system, but also make the output reference value meet the following conditions:

式中,λ是表示干扰抑制能力的正常数,下面给出了新系统随机稳定性和满足H∞性能的必要条件,即证明线性矩阵不等式(LMI)。In the formula, λ is a constant representing the interference suppression capability. The necessary conditions for the stochastic stability of the new system and satisfying H∞ performance are given below, namely, the proof of the linear matrix inequality (LMI).

定理:复合系统(25)中,任意λ>0,存在矩阵Q1>0,Q2>0以及R1,R2满足:Theorem: In the composite system (25), any λ>0, there are matrices Q 1 >0, Q 2 >0 and R 1 , R 2 satisfy:

式中,Ξ1=sym(AQ1+B1R1),Ξ2=sym(-R2B2)。sym()表示一次矩阵运算,对于对称矩阵M有sym(M)=M+MT。控制器增益观测器增益时,复合系统(25)鲁棒性渐近稳定。另外,他还满足 In the formula, Ξ 1 =sym(AQ 1 +B 1 R 1 ), and Ξ 2 =sym(-R 2 B 2 ). sym() represents a matrix operation, and for a symmetric matrix M there is sym(M)=M+M T . Controller gain Observer gain , the robustness of the composite system (25) is asymptotically stable. In addition, he is satisfied

证明:有界实引理中,LMI基础上证明(28)满足H∞性能。Proof: In the bounded real lemma, LMI proves that (28) satisfies the H∞ performance.

将(25)代入(28),令得:Substituting (25) into (28), let have to:

式中,Θ1=sym[P1(A+B1K)],Θ2=sym(-P2EB2)。令Q1=P1 -1,R1=KQ1,R2=P2E,(29)分别左乘对角阵{Q1,I,I,I,I},再右乘对角阵{Q1,I,I,I,I},可以得到(27)的结论。In the formula, Θ 1 =sym[P 1 (A+B 1 K)], Θ 2 =sym(-P 2 EB 2 ). Let Q 1 =P 1 -1 , R 1 =KQ 1 , R 2 =P 2 E, (29) multiply the diagonal matrix {Q 1 ,I,I,I,I} to the left, and then multiply the diagonal matrix to the right {Q 1 ,I,I,I,I}, the conclusion of (27) can be obtained.

以上所述仅为本发明的优选实施例而已,并不用于限制本发明,对于本领域的技术人员来说,本发明可以有各种更改和变化。凡在本发明的精神和原则之内,所作的任何修改、等同替换、改进等,均应包含在本发明的保护范围之内。The above descriptions are only preferred embodiments of the present invention, and are not intended to limit the present invention. For those skilled in the art, the present invention may have various modifications and changes. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims (8)

1. the control method that the magnetic suspension rotor system based on disturbance observer mismatches disturbance, which is characterized in that including following Step:
1) dynamic model for mismatching and disturbing lower voltage-controlled type magnetic suspension rotor system is established, voltage-controlled type magnetic bearing is obtained The state equation of control system;
2) equivalent system that magnetic suspension rotor system is reconstructed by introducing new state variable, will mismatch disturbance be divided into matching and Mismatch two parts;
3) the state space disturbance observer for designing broad sense is observed the mismatch disturbance part in equivalent system;
4) design basis robust controller, according to the disturbance estimated value for the disturbance observer observation that step 3) obtains, in basic Shandong Equivalent compensation is introduced in stick controller, realizes the inhibition to external disturbance, and obtain improved multiplex control system;
5) stability analysis is carried out to improved multiplex control system, calculates the control of basic robust controller and disturbance observer Gain processed.
2. control method according to claim 1, which is characterized in that the voltage-controlled type magnetic axis that the step 1) is established Hold the state equation of control system are as follows:
In formula, d (s;T) total disturbance is indicated, s is rotor displacement, and t is time, u=[us]T, x=[x1,x2,x3]T, x1=s,x3=isIt is state variable,It is the speed of rotor, isIt is control electric current, state matrix are as follows:
In formula, m is the quality of rigid rotator, and R is coil resistance, and L is coil inductance,WithIt indicates in operating point (is=iN, S=sN) control electric current and position stiffness.
3. control method according to claim 2, which is characterized in that total disturbance d (s;It t) include Parameters variation and outer Portion's interference, indicates are as follows:
d(s;T)=△ Ki(s;t)is(t)+△Ks(s;t)s(t)+fd(t)
In formula, fdIt is external disturbance, △ Ki(s;And △ K t)s(s;It t) is Parameters variation amount.
4. control method according to claim 3, which is characterized in that the step 2) is substituted with new state variable η (t) State variable i in original systems(t), new state variable-definition are as follows:
η (t)=is(t)-id(t) (8)
Equivalent voltage-controlled type magnetic bearing control system can indicate are as follows:
In formula, disturbance d (s is mismatched;T) it is divided into two parts:WithAssuming that disturbance observer can Progressively to track disturbance, i.e. when t → ∞,
5. control method according to claim 4, which is characterized in that the step 3) includes the following steps:
Introduce an auxiliary vector ω (s;T), total disturbance d (s is redefined;T) are as follows:
d(s;T)=V ω (s;t) (10)
In formula, W and V are coefficient matrixes;
Composite system is made of state variable (7) and disturbance variable (11), is indicated are as follows:
According to the design method of state observer, disturbance observer can design subsystem below:
In formula, σ (t) is the equivalent output of subsystem, can be by observing feedback quantityTo improve observer precision;According to The structure of disturbance observer can derive disturbance variableExpression formula is as follows:
By auxiliary variableIt substitutes into (14), disappears on the right side of (14)Disturbance observer can indicate Are as follows:
In formula, W and V are coefficient matrix, ω (s;It t) is auxiliary vector, auxiliary variable It is to disturb Dynamic variable.
6. control method according to claim 5, which is characterized in that set W=0, V=I, the disturbance of unknown disturbance prediction device Observer model can simplify are as follows:
7. control method according to claim 5 or 6, which is characterized in that the step 4) includes:
Matching is disturbed, composite controller may be designed as:
In formula, KkFor the feedback oscillator determined by the performance of closed-loop system, feedback control can make the state variable in system steady Determine to zero, i.e., as t → 0, s (t) → 0,is(t)→0;
For the robustness for guaranteeing closed-loop system, a feedback robust controller is introduced in equivalent system, is indicated are as follows:
ud(t)=Kx*(t) (18)
In formula,K is the feedback oscillator for meeting H ∞;
After improvement, the input of magnetic bearing control system becomes:
us=ud+Rid(19);
Disturbance observer equation (16) and state feedback robust controller equation (18) are substituted into equation (9), it is multiple after being improved Close control system equation are as follows:
In formula, B3=[0 0 1/Ki]T, ed(s;It t) is disturbance observation error,
It enables(24) it can indicate are as follows:
In formula,
8. control method according to claim 7, which is characterized in that the step 5) carries out improved composite system steady Qualitative analysis is calculated corresponding controller gain and observer gain, specifically includes: in composite system (25), due toWithIt is all H2Convergence in norm, so disturbanceIt is also H2Convergence in norm, to reduce disturbance's It influences, using the robust control scheme with H ∞ performance indicator, H ∞ controller can not only keep the stabilization of system, moreover it is possible to make Reference value must be exported and meet the following conditions:
In formula, λ is the normal number for indicating interference rejection capability, and new system stochastic stability is shown below and meets H ∞ performance Necessary condition, i.e., proof linear matrix inequality (LMI);
In composite system (25), any λ > 0, there are matrix Q1> 0, Q2> 0 and R1, R2Meet:
In formula, Ξ1=sym (AQ1+B1R1), Ξ2=sym (- R2B2);Sym () indicates a matrix operation, has for symmetrical matrix M Sym (M)=M+MT
It obtains working as controller gainObserver gainWhen, composite system (25) robustness Asymptotic Stability, And meet
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