CN108110761B - Fuzzy high-order sliding mode active power filter control method based on linearization feedback - Google Patents

Fuzzy high-order sliding mode active power filter control method based on linearization feedback Download PDF

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CN108110761B
CN108110761B CN201810067129.2A CN201810067129A CN108110761B CN 108110761 B CN108110761 B CN 108110761B CN 201810067129 A CN201810067129 A CN 201810067129A CN 108110761 B CN108110761 B CN 108110761B
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power filter
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李思扬
费峻涛
梁霄
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Hohai University HHU
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for AC mains or AC distribution networks
    • H02J3/01—Arrangements for reducing harmonics or ripples
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J2203/00—Indexing scheme relating to details of circuit arrangements for AC mains or AC distribution networks
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    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
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    • Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
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    • Y02E40/20—Active power filtering [APF]

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Abstract

The invention discloses a fuzzy high-order sliding mode active power filter control method based on linearization feedback. The effectiveness of the method is verified through simulation results. The method greatly enhances the compensation performance and the robustness performance of the system and achieves the aim of quickly and effectively eliminating harmonic waves.

Description

基于线性化反馈的模糊高阶滑模有源电力滤波器控制方法A fuzzy high-order sliding mode active power filter control method based on linearized feedback

技术领域technical field

本发明涉及属于有源电力滤波技术，特别涉及一种基于线性化反馈自适应模糊高阶滑模的有源电力滤波器控制方法。The invention relates to an active power filter technology, in particular to an active power filter control method based on a linearized feedback adaptive fuzzy high-order sliding mode.

背景技术Background technique

采用电力滤波器装置吸收谐波源所产生的谐波电流，是一种抑制谐波污染的有效措施。有源电力滤波器具有快速响应性及高度可控性，不仅可以补偿各次谐波，还可以补偿无功功率、抑制闪变等。由于电力系统的非线性和不确定性，自适应和智能控制具有建模简单、控制精度高、非线性适应性强等优点,可以应用在有源滤波器中用于电能质量控制和谐波治理，具有重要的研究意义和市场价值。The use of power filter device to absorb the harmonic current generated by the harmonic source is an effective measure to suppress harmonic pollution. Active power filters have fast response and high controllability. They can not only compensate for various harmonics, but also compensate for reactive power and suppress flicker. Due to the nonlinearity and uncertainty of the power system, adaptive and intelligent control has the advantages of simple modeling, high control accuracy, and strong nonlinear adaptability, and can be used in active filters for power quality control and harmonic control. , has important research significance and market value.

本文深入研究了三相并联有源电力滤波器的原理，在此基础上建立数学模型，利用三相并联型有源电力滤波器线性状态方程，加入了线性化反馈高阶滑模控制方法。研究有源电力滤波器模型参考自适应控制，提出了线性化反馈高阶滑模模糊自适应控制算法，应用于三相并联型有源电力滤波器的谐波补偿控制。通过MATLAB仿真，验证了增加线性化反馈高阶滑模模糊控制的自适应控制方法适合补偿电路谐波，提高电源质量。In this paper, the principle of the three-phase parallel active power filter is deeply studied, and a mathematical model is established on this basis. The linear state equation of the three-phase parallel active power filter is used, and the linearized feedback high-order sliding mode control method is added. The model reference adaptive control of active power filter is studied, and a linearized feedback high-order sliding mode fuzzy adaptive control algorithm is proposed, which is applied to the harmonic compensation control of three-phase parallel active power filter. Through MATLAB simulation, it is verified that the adaptive control method of adding linearized feedback high-order sliding mode fuzzy control is suitable for compensating circuit harmonics and improving power quality.

发明内容SUMMARY OF THE INVENTION

本发明为了抑制外界未知扰动和建模误差对有源电力滤波器系统性能的影响，提出了一种基于线性化反馈模糊高阶滑模的有源电力滤波器控制方法，进一步提高了系统鲁棒性。In order to suppress the influence of unknown external disturbances and modeling errors on the performance of the active power filter system, the present invention proposes an active power filter control method based on linearized feedback fuzzy high-order sliding mode, which further improves the robustness of the system. sex.

本发明解决其技术问题是通过以下技术方案实现的：The present invention solves its technical problem and realizes through the following technical solutions:

基于线性化反馈的模糊高阶滑模有源电力滤波器控制方法，包括如下步骤：The fuzzy high-order sliding mode active power filter control method based on linearized feedback includes the following steps:

(1)建立有源电力滤波器数学模型；(1) Establish a mathematical model of active power filter;

(2)设计控制器：首先设计动态滑模面，再利用线性化反馈控制设计滑模控制器，并在滑模控制器的基础上利用模糊控制逼近不确定项，设计自适应律使系统保持稳定状态。(2) Design the controller: first design the dynamic sliding mode surface, then use the linearized feedback control to design the sliding mode controller, and use the fuzzy control to approximate the uncertain term on the basis of the sliding mode controller, and design the adaptive law to keep the system stable state.

进一步的，所述步骤(1)中建立的有源电力滤波器数学模型为：Further, the active power filter mathematical model established in the step (1) is:

Figure BDA0001557026030000021
Figure BDA0001557026030000021

其中，x为指令电流，

Figure BDA0001557026030000022
为x的导数，
Figure BDA0001557026030000023
u为控制器输入，b为常数，Lc为电感，Rc为电阻，vdc为直流侧电容电压，vk为三相有源电力滤波器端电压，ik为三相补偿电流，dk为开关状态函数，dk定义如下：
Figure BDA0001557026030000024
则dk依赖于第k相IGBT的通断状态，是系统的非线性项；ck、cm为开关函数，m，k为大于0的常数。where x is the command current,
Figure BDA0001557026030000022
is the derivative of x,
Figure BDA0001557026030000023
u is the controller input, b is a constant, L c is the inductance, R c is the resistance, v dc is the DC side capacitor voltage, v k is the terminal voltage of the three-phase active power filter, i k is the three-phase compensation current, d k is the switch state function, and d k is defined as follows:
Figure BDA0001557026030000024
Then d k depends on the on-off state of the k-th phase IGBT and is a nonlinear term of the system; ck and cm are switching functions, and m and k are constants greater than 0.

进一步的，对步骤(1)得到的有源电力滤波器数学模型进行二次求导，有源电力滤波器数学模型可以定义为：Further, the second derivative is performed on the mathematical model of the active power filter obtained in step (1), and the mathematical model of the active power filter can be defined as:

Figure BDA0001557026030000025
Figure BDA0001557026030000025

Figure BDA0001557026030000026
为f(x)的导数，
Figure BDA0001557026030000027
为u的导数，
Figure BDA0001557026030000028
为
Figure BDA0001557026030000029
的导数；设
Figure BDA00015570260300000210
其中f1(x)为未知的非线性函数，则所述有源电力滤波器数学模型可以定义为：
Figure BDA0001557026030000026
is the derivative of f(x),
Figure BDA0001557026030000027
is the derivative of u,
Figure BDA0001557026030000028
for
Figure BDA0001557026030000029
the derivative of ; let
Figure BDA00015570260300000210
Where f 1 (x) is an unknown nonlinear function, the mathematical model of the active power filter can be defined as:

Figure BDA00015570260300000211
Figure BDA00015570260300000211

进一步的，所述步骤(2)中设计高阶滑模控制律的具体步骤为：Further, the specific steps of designing the high-order sliding mode control law in the step (2) are:

定义用于输入的跟踪误差e：Define the tracking error e for the input:

e＝x-xd e =xxd

其中，x为指令电流，xd为参考电流，Among them, x is the command current, x d is the reference current,

对跟踪误差进行二次求导：Take the second derivative of the tracking error:

Figure BDA00015570260300000212
Figure BDA00015570260300000212

其中，

Figure BDA00015570260300000213
为
Figure BDA00015570260300000214
的导数，
Figure BDA00015570260300000215
为xd的导数；in,
Figure BDA00015570260300000213
for
Figure BDA00015570260300000214
the derivative of ,
Figure BDA00015570260300000215
is the derivative of x d ;

定义滑模面：Define a sliding surface:

l＝e+k2∫el=e+k 2 ∫e

其中k2为常数，∫e为对跟踪误差的积分；where k 2 is a constant, and ∫e is the integral of the tracking error;

对滑模面求导：Derivative with respect to the sliding surface:

Figure BDA0001557026030000031
Figure BDA0001557026030000031

其中

Figure BDA0001557026030000032
为e的导数；in
Figure BDA0001557026030000032
is the derivative of e;

定义高阶滑模面s：Define higher-order sliding surface s:

Figure BDA0001557026030000033
Figure BDA0001557026030000033

其中，

Figure BDA0001557026030000034
为大于0的常数，in,
Figure BDA0001557026030000034
is a constant greater than 0,

对高阶滑模面求导：Derivative for higher-order sliding mode surfaces:

Figure BDA0001557026030000035
Figure BDA0001557026030000035

其中

Figure BDA0001557026030000036
为
Figure BDA0001557026030000037
的导数，
Figure BDA0001557026030000038
为s的导数；in
Figure BDA0001557026030000036
for
Figure BDA0001557026030000037
the derivative of ,
Figure BDA0001557026030000038
is the derivative of s;

根据线性化反馈技术，According to the linearized feedback technique,

因为

Figure BDA0001557026030000039
所以令g1(x)＝b，because
Figure BDA0001557026030000039
So let g 1 (x)=b,

Figure BDA00015570260300000310
Figure BDA00015570260300000310

Figure BDA00015570260300000311
Figure BDA00015570260300000311

RX＝ξ-ρsgn(s)R X =ξ-ρsgn(s)

其中ρ为大于0的常数，且ρ≥|D|，D为ρ的上界常数，sgn为符号函数，ξ和RX是为了简化过程设计的中间变量，where ρ is a constant greater than 0, and ρ≥|D|, D is the upper bound constant of ρ, sgn is the sign function, ξ and R X are intermediate variables designed to simplify the process,

因为：because:

Figure BDA00015570260300000312
Figure BDA00015570260300000312

Figure BDA00015570260300000313
Figure BDA00015570260300000313

所以：so:

Figure BDA00015570260300000314
Figure BDA00015570260300000314

Figure BDA00015570260300000315
Figure BDA00015570260300000315

因此，系统控制律设计为：Therefore, the system control law is designed as:

Figure BDA0001557026030000041
Figure BDA0001557026030000041

进一步的，根据设计的高阶滑模控制律设计加入模糊的控制律为：Further, according to the designed high-order sliding mode control law, the fuzzy control law is designed as follows:

Figure BDA0001557026030000042
Figure BDA0001557026030000042

其中，

Figure BDA0001557026030000043
为
Figure BDA0001557026030000044
的模糊逼近函数，
Figure BDA0001557026030000045
为f(xk)的模糊逼近函数，
Figure BDA0001557026030000046
为ρsgn(s)的模糊逼近函数，xk为x的标量形式，sk为高阶滑模面s的标量形式。in,
Figure BDA0001557026030000043
for
Figure BDA0001557026030000044
The fuzzy approximation function of ,
Figure BDA0001557026030000045
is the fuzzy approximation function of f(x k ),
Figure BDA0001557026030000046
is the fuzzy approximation function of ρsgn(s), x k is the scalar form of x, and s k is the scalar form of the higher-order sliding mode surface s.

进一步的，所述步骤(2)中利用李雅谱诺夫函数进行自适应律设计的具体步骤为：Further, the specific steps of utilizing the Lyapunov function to design the adaptive law in the step (2) are:

设李亚普诺夫函数：Let the Lyapunov function:

Figure BDA0001557026030000047
Figure BDA0001557026030000047

Figure BDA0001557026030000048
Figure BDA0001557026030000048

Figure BDA0001557026030000049
Figure BDA0001557026030000049

因为because

Figure BDA00015570260300000410
Figure BDA00015570260300000410

Figure BDA00015570260300000411
Figure BDA00015570260300000411

Figure BDA00015570260300000412
Figure BDA00015570260300000412

其中，

Figure BDA00015570260300000432
和
Figure BDA00015570260300000433
分别为V1和V2的导数，sT为s的转置，γ1和γ2为大于0的常数，
Figure BDA00015570260300000434
为函数
Figure BDA00015570260300000416
的模糊参数，
Figure BDA00015570260300000417
为函数
Figure BDA00015570260300000418
的模糊参数，
Figure BDA00015570260300000419
为
Figure BDA00015570260300000420
的转置，
Figure BDA00015570260300000421
为
Figure BDA00015570260300000422
的转置，
Figure BDA00015570260300000435
为
Figure BDA00015570260300000436
的导数，
Figure BDA00015570260300000437
为
Figure BDA00015570260300000438
的导数，δT(xk)和φT(hk)分别为
Figure BDA00015570260300000427
和
Figure BDA00015570260300000428
的隶属度函数，
Figure BDA00015570260300000439
为
Figure BDA00015570260300000440
的估计值、
Figure BDA00015570260300000441
为函数
Figure BDA00015570260300000442
的理想值，
Figure BDA00015570260300000431
为函数h(sk)的理想值，ω是模糊系统的逼近误差，f(xk)为实际值；in,
Figure BDA00015570260300000432
and
Figure BDA00015570260300000433
are the derivatives of V 1 and V 2 , respectively, s T is the transpose of s, γ 1 and γ 2 are constants greater than 0,
Figure BDA00015570260300000434
for the function
Figure BDA00015570260300000416
The fuzzy parameters of ,
Figure BDA00015570260300000417
for the function
Figure BDA00015570260300000418
The fuzzy parameters of ,
Figure BDA00015570260300000419
for
Figure BDA00015570260300000420
transpose of ,
Figure BDA00015570260300000421
for
Figure BDA00015570260300000422
transpose of ,
Figure BDA00015570260300000435
for
Figure BDA00015570260300000436
the derivative of ,
Figure BDA00015570260300000437
for
Figure BDA00015570260300000438
The derivatives of , δ T (x k ) and φ T (h k ) are respectively
Figure BDA00015570260300000427
and
Figure BDA00015570260300000428
The membership function of ,
Figure BDA00015570260300000439
for
Figure BDA00015570260300000440
the estimated value of ,
Figure BDA00015570260300000441
for the function
Figure BDA00015570260300000442
the ideal value of ,
Figure BDA00015570260300000431
is the ideal value of the function h(s k ), ω is the approximation error of the fuzzy system, and f(x k ) is the actual value;

因为：because:

Figure BDA0001557026030000051
Figure BDA0001557026030000051

所以so

Figure BDA0001557026030000052
Figure BDA0001557026030000052

其中，ωk为ω的标量形式，

Figure BDA0001557026030000053
为ωk的导数，γ1，γ2为常数，
Figure BDA0001557026030000054
为dk的导数，进而设计系统的自适应律为：where ω k is the scalar form of ω,
Figure BDA0001557026030000053
is the derivative of ω k , γ 1 , γ 2 are constants,
Figure BDA0001557026030000054
is the derivative of d k , and then the adaptive law of the design system is:

Figure BDA0001557026030000055
Figure BDA0001557026030000055

Figure BDA0001557026030000056
Figure BDA0001557026030000056

Figure BDA0001557026030000057
Figure BDA0001557026030000057

其中，η为大于0的常数；Wherein, n is a constant greater than 0;

将(2)，(3)，(4)带入(1)得到：Bring (2), (3), (4) into (1) to get:

Figure BDA0001557026030000058
Figure BDA0001557026030000058

当

Figure BDA0001557026030000059
时，
Figure BDA00015570260300000510
成立，when
Figure BDA0001557026030000059
hour,
Figure BDA00015570260300000510
established,

其中，||sk||为sk的范数，其中，|ωmax|为ω的最大值的绝对值。where ||s k || is the norm of s k , and |ω max | is the absolute value of the maximum value of ω.

本发明的有益效果为：The beneficial effects of the present invention are:

与现有技术相比，本发明深入研究了三相并联有源电力滤波器的原理，在此基础上建立数学模型，利用三相并联型有源电力滤波器线性状态方程，加入了线性化反馈高阶滑模控制方法。研究有源电力滤波器模型参考自适应控制，提出了线性化反馈高阶滑模模糊自适应控制算法，应用于三相并联型有源电力滤波器的谐波补偿控制。通过MATLAB仿真，验证了增加线性化反馈高阶滑模模糊滑模控制的自适应控制方法适合补偿电路谐波，提高电源质量。Compared with the prior art, the present invention deeply studies the principle of the three-phase parallel active power filter, establishes a mathematical model on this basis, uses the linear state equation of the three-phase parallel active power filter, and adds linearized feedback. Higher order sliding mode control methods. The model reference adaptive control of active power filter is studied, and a linearized feedback high-order sliding mode fuzzy adaptive control algorithm is proposed, which is applied to the harmonic compensation control of three-phase parallel active power filter. Through MATLAB simulation, it is verified that the adaptive control method of adding linearized feedback high-order sliding mode fuzzy sliding mode control is suitable for compensating circuit harmonics and improving power quality.

附图说明Description of drawings

图1是本发明的结构示意图；Fig. 1 is the structural representation of the present invention;

图2是本发明线性化反馈高阶滑模模糊自适应控制器框图；2 is a block diagram of a linearized feedback high-order sliding mode fuzzy adaptive controller of the present invention;

图3是电源电流的波形图；Fig. 3 is the waveform diagram of the power supply current;

图4是系统误差的数据图；Fig. 4 is the data graph of systematic error;

图5是直流侧电压的波形图。FIG. 5 is a waveform diagram of the DC side voltage.

具体实施方式Detailed ways

下面通过具体实施例对本发明作进一步详述，以下实施例只是描述性的，不是限定性的，不能以此限定本发明的保护范围。The present invention will be further described in detail below through specific examples. The following examples are only descriptive, not restrictive, and cannot limit the protection scope of the present invention.

如图1所示，基于线性化反馈的模糊高阶滑模有源电力滤波器控制方法，包括以下步骤：As shown in Figure 1, the fuzzy high-order sliding mode active power filter control method based on linearized feedback includes the following steps:

(一)建立有源电力滤波器模型(1) Establish an active power filter model

有源电力滤波器的基本工作原理是：检测补偿对象的电压和电流，经指令电流运算电路计算得出补偿电流的指令信号，该信号经补偿电流发生电路放大，得出补偿电流，补偿电流与负载电流中要补偿的谐波及无功等电流抵消，最终得到期望的电源电流。The basic working principle of the active power filter is to detect the voltage and current of the compensation object, and calculate the command signal of the compensation current through the command current operation circuit. The signal is amplified by the compensation current generating circuit to obtain the compensation current. The harmonics and reactive power to be compensated in the load current are offset, and the desired power supply current is finally obtained.

根据电路理论和基尔霍夫定理可得到如下公式：According to circuit theory and Kirchhoff's theorem, the following formula can be obtained:

Figure BDA0001557026030000061
Figure BDA0001557026030000061

v1，v2，v3分别为三相有源电力滤波器端电压，i1，i2，i3分别为三相补偿电流，

Figure BDA0001557026030000065
表示电流i1对时间的导数，
Figure BDA0001557026030000063
表示电流i2对时间的导数，
Figure BDA0001557026030000064
表示电流i3对时间的导数，v1M,v2M,v3M,vMN分别表示图一中M点到a，b，c，N点电压。M点是电源的负极点，a,b,c,N只是电路中的各个节点。Lc为电感，Rc为电阻，ik为三相补偿电流。v 1 , v 2 , v 3 are the terminal voltages of the three-phase active power filter respectively, i 1 , i 2 , i 3 are the three-phase compensation currents, respectively,
Figure BDA0001557026030000065
represents the derivative of the current i 1 with respect to time,
Figure BDA0001557026030000063
represents the derivative of the current i2 with respect to time,
Figure BDA0001557026030000064
Represents the derivative of the current i 3 with respect to time, v 1M , v 2M , v 3M , v MN represent the voltages from point M to point a, b, c, and N in Figure 1, respectively. Point M is the negative point of the power supply, and a, b, c, and N are just the nodes in the circuit. L c is the inductance, R c is the resistance, and i k is the three-phase compensation current.

假设交流侧电源电压稳定，可以得到Assuming that the AC side power supply voltage is stable, we can get

Figure BDA0001557026030000071
Figure BDA0001557026030000071

其中，m为大于0的常数，vmM为上面提到的v1M，v2M，v3M，vMN。where m is a constant greater than 0, and v mM is the above-mentioned v 1M , v 2M , v 3M , v MN .

并定义ck为开关函数，指示IGBT的工作状态，定义如下：And define c k as the switching function, indicating the working state of the IGBT, which is defined as follows:

Figure BDA0001557026030000072
Figure BDA0001557026030000072

其中，k＝1,2,3。where k=1,2,3.

同时，vkM＝ckvdc，其中，vkM为上面提到的v1M，v2M，v3M，vMN。Meanwhile, v kM =c k v dc , where v kM is the above-mentioned v 1M , v 2M , v 3M , v MN .

所以(1)可改写为So (1) can be rewritten as

Figure BDA0001557026030000073
Figure BDA0001557026030000073

vdc为直流侧电容电压。v dc is the DC side capacitor voltage.

我们定义dk为开关状态函数，定义如下：We define d k as the switch state function, which is defined as follows:

Figure BDA0001557026030000074
Figure BDA0001557026030000074

则dk依赖于第k相IGBT的通断状态，是系统的非线性项；cm为开关函数。Then d k depends on the on-off state of the k-th phase IGBT and is a nonlinear term of the system; c m is the switching function.

并有And a

Figure BDA0001557026030000075
Figure BDA0001557026030000075

那么(4)可改写为Then (4) can be rewritten as

Figure BDA0001557026030000081
Figure BDA0001557026030000081

那么可以将(7)改写成如下形式Then (7) can be rewritten into the following form

Figure BDA0001557026030000082
Figure BDA0001557026030000082

其中

Figure BDA0001557026030000083
x为指令电流，
Figure BDA0001557026030000084
为x的导数，b为常数，u为控制器输入。in
Figure BDA0001557026030000083
x is the command current,
Figure BDA0001557026030000084
is the derivative of x, b is a constant, and u is the controller input.

(二)线性化反馈高阶滑模控制(2) Linearized feedback high-order sliding mode control

定义跟踪误差e：Define the tracking error e:

e＝x-xd e =xxd

其中，xd为参考电流。where x d is the reference current.

对跟踪误差e求一阶导数：Find the first derivative with respect to the tracking error e:

Figure BDA0001557026030000085
Figure BDA0001557026030000085

其中

Figure BDA0001557026030000086
为xd的导数in
Figure BDA0001557026030000086
is the derivative of xd

对跟踪误差e求二阶导数：Find the second derivative with respect to the tracking error e:

Figure BDA0001557026030000087
Figure BDA0001557026030000087

其中

Figure BDA0001557026030000088
为
Figure BDA0001557026030000089
的导数，
Figure BDA00015570260300000810
为
Figure BDA00015570260300000811
的导数in
Figure BDA0001557026030000088
for
Figure BDA0001557026030000089
the derivative of ,
Figure BDA00015570260300000810
for
Figure BDA00015570260300000811
the derivative of

对系统模型进行二次求导：Take the second derivative of the system model:

Figure BDA00015570260300000812
Figure BDA00015570260300000812

Figure BDA00015570260300000813
为f(x)的导数，
Figure BDA00015570260300000814
为u的导数，
Figure BDA00015570260300000813
is the derivative of f(x),
Figure BDA00015570260300000814
is the derivative of u,

设

Figure BDA00015570260300000815
其中f1(x)为未知的非线性函数。Assume
Figure BDA00015570260300000815
where f 1 (x) is an unknown nonlinear function.

所以原模型可以定义为：So the original model can be defined as:

Figure BDA00015570260300000816
Figure BDA00015570260300000816

定义滑模面：Define a sliding surface:

l＝e+k2∫el=e+k 2 ∫e

其中k2为常数，∫e为对误差的积分。where k 2 is a constant and ∫e is the integral over the error.

对滑模面求导：Derivative with respect to the sliding surface:

Figure BDA0001557026030000091
Figure BDA0001557026030000091

其中

Figure BDA0001557026030000092
为e的导数，in
Figure BDA0001557026030000092
is the derivative of e,

定义高阶滑模面s：Define higher-order sliding surface s:

Figure BDA0001557026030000093
Figure BDA0001557026030000093

其中，

Figure BDA0001557026030000094
为大于0的常数。in,
Figure BDA0001557026030000094
is a constant greater than 0.

对高阶滑模面s求导：Derivative with respect to the higher-order sliding surface s:

Figure BDA0001557026030000095
Figure BDA0001557026030000095

其中

Figure BDA0001557026030000096
为
Figure BDA0001557026030000097
的导数，
Figure BDA0001557026030000098
为s的导数。in
Figure BDA0001557026030000096
for
Figure BDA0001557026030000097
the derivative of ,
Figure BDA0001557026030000098
is the derivative of s.

根据线性化反馈技术：According to the linearization feedback technique:

Figure BDA0001557026030000099
Figure BDA0001557026030000099

Figure BDA00015570260300000910
Figure BDA00015570260300000910

RX＝ξ-ρsgn(s)R X =ξ-ρsgn(s)

其中ρ为大于0的常数，且ρ≥|D|，D为ρ的上界常数，sgn为符号函数，

Figure BDA00015570260300000911
为x的二阶导数，ξ和RX是为了简化过程设计的中间变量。where ρ is a constant greater than 0, and ρ≥|D|, D is the upper bound constant of ρ, sgn is the sign function,
Figure BDA00015570260300000911
is the second derivative of x, ξ and R x are intermediate variables designed to simplify the process.

因为：because:

Figure BDA00015570260300000912
Figure BDA00015570260300000912

Figure BDA00015570260300000913
Figure BDA00015570260300000913

所以：so:

Figure BDA0001557026030000101
Figure BDA0001557026030000101

Figure BDA0001557026030000102
Figure BDA0001557026030000102

因此，系统控制律设计为：Therefore, the system control law is designed as:

Figure BDA0001557026030000103
Figure BDA0001557026030000103

(三)线性化反馈高阶滑模模糊控制(3) Linearized feedback high-order sliding mode fuzzy control

Ri:If x1 is

Figure BDA0001557026030000104
and....xn is
Figure BDA0001557026030000105
then y is Bi(i＝1,2,.......,N)R i : If x 1 is
Figure BDA0001557026030000104
and....x n is
Figure BDA0001557026030000105
then y is B i (i=1,2,.......,N)

其中，

Figure BDA0001557026030000106
为xj(j＝1,2,.......,n)的隶属度函数。in,
Figure BDA0001557026030000106
is the membership function of x j (j=1,2,.......,n).

则模糊系统的输出为：Then the output of the fuzzy system is:

Figure BDA0001557026030000107
Figure BDA0001557026030000107

其中δ＝[δ1(x) δ2(x) ... δN(x)]T为隶属度函数，

Figure BDA0001557026030000108
为隶属度函数中的每个小分量的值，
Figure BDA0001557026030000109
为模糊参数，
Figure BDA00015570260300001010
为模糊参数中的分量。where δ=[δ 1 (x) δ 2 (x) ... δ N (x)] T is the membership function,
Figure BDA0001557026030000108
is the value of each small component in the membership function,
Figure BDA0001557026030000109
is the fuzzy parameter,
Figure BDA00015570260300001010
is the component in the blur parameter.

针对f(x,y)的模糊逼近，采用分别逼近f(1)和f(2)的形式，相应的模糊系统设计为：For the fuzzy approximation of f(x,y), adopt the form of approximating f(1) and f(2) respectively, and the corresponding fuzzy system is designed as:

Figure BDA00015570260300001011
Figure BDA00015570260300001011

定义模糊函数为如下形式：The fuzzy function is defined as follows:

Figure BDA0001557026030000111
Figure BDA0001557026030000111

其中，

Figure BDA0001557026030000112
ψ为模糊系统的输出，ψ1和ψ2为输出中的标量。
Figure BDA0001557026030000113
为模糊参数中的分量的和，N为分量的总数，
Figure BDA0001557026030000114
为
Figure BDA0001557026030000115
中的最小值，
Figure BDA0001557026030000116
为
Figure BDA0001557026030000117
中最小值的和。in,
Figure BDA0001557026030000112
ψ is the output of the fuzzy system, and ψ1 and ψ2 are scalars in the output.
Figure BDA0001557026030000113
is the sum of the components in the fuzzy parameters, N is the total number of components,
Figure BDA0001557026030000114
for
Figure BDA0001557026030000115
the minimum value of ,
Figure BDA0001557026030000116
for
Figure BDA0001557026030000117
The sum of the minimum and middle values.

定义最优逼近常量

Figure BDA0001557026030000118
define optimal approximation constants
Figure BDA0001557026030000118

Figure BDA0001557026030000119
Figure BDA0001557026030000119

式中Ω是

Figure BDA00015570260300001110
的集合。where Ω is
Figure BDA00015570260300001110
collection.

则：but:

Figure BDA00015570260300001111
Figure BDA00015570260300001111

Figure BDA00015570260300001112
Figure BDA00015570260300001112

Figure BDA00015570260300001113
Figure BDA00015570260300001113

ω是模糊系统的逼近误差，δT为δ的转置，

Figure BDA00015570260300001114
为
Figure BDA00015570260300001115
的转置。对于给定的任意小常量ω(ω＞0)，如下不等式成立：|f(x,y)-ξT(x)θ*|≤ω令
Figure BDA00015570260300001116
并且使得
Figure BDA00015570260300001117
ω is the approximation error of the fuzzy system, δ T is the transpose of δ,
Figure BDA00015570260300001114
for
Figure BDA00015570260300001115
transposition of . For a given arbitrary small constant ω (ω>0), the following inequality holds: |f(x,y)-ξ T (x)θ * |≤ωLet
Figure BDA00015570260300001116
and make
Figure BDA00015570260300001117

控制器设计为：The controller is designed to:

Figure BDA00015570260300001118
Figure BDA00015570260300001118

其中，

Figure BDA00015570260300001119
为
Figure BDA00015570260300001120
的模糊逼近函数，
Figure BDA00015570260300001121
为f(xk)的模糊逼近函数，
Figure BDA00015570260300001122
为ρsgn(s)的模糊逼近函数，xk为x的标量形式，sk为s的标量形式。in,
Figure BDA00015570260300001119
for
Figure BDA00015570260300001120
The fuzzy approximation function of ,
Figure BDA00015570260300001121
is the fuzzy approximation function of f(x k ),
Figure BDA00015570260300001122
is the fuzzy approximation function of ρsgn(s), x k is the scalar form of x, and s k is the scalar form of s.

稳定性证明：Proof of stability:

设李亚普诺夫函数：Let the Lyapunov function:

Figure BDA0001557026030000121
Figure BDA0001557026030000121

Figure BDA0001557026030000122
Figure BDA0001557026030000122

Figure BDA0001557026030000123
Figure BDA0001557026030000123

因为because

Figure BDA0001557026030000124
Figure BDA0001557026030000124

其中，

Figure BDA00015570260300001227
和
Figure BDA00015570260300001228
分别为V1和V2的导数，sT为s的转置，γ1和γ2为大于0的常数，
Figure BDA00015570260300001226
为函数
Figure BDA0001557026030000128
的模糊参数，
Figure BDA0001557026030000129
为函数
Figure BDA00015570260300001210
的模糊参数，
Figure BDA00015570260300001211
为
Figure BDA00015570260300001212
的转置，
Figure BDA00015570260300001213
为
Figure BDA00015570260300001214
的转置，
Figure BDA00015570260300001231
为
Figure BDA00015570260300001232
的导数，
Figure BDA00015570260300001230
为
Figure BDA00015570260300001229
的导数，δT(xk)和φT(hk)分别为
Figure BDA00015570260300001219
和
Figure BDA00015570260300001220
的隶属度函数，
Figure BDA00015570260300001235
为h(sk)的估计值
Figure BDA00015570260300001233
为函数
Figure BDA00015570260300001234
的理想值，
Figure BDA00015570260300001223
为函数h(sk)的理想值，ω是模糊系统的逼近误差，f(xk)为实际值。in,
Figure BDA00015570260300001227
and
Figure BDA00015570260300001228
are the derivatives of V 1 and V 2 , respectively, s T is the transpose of s, γ 1 and γ 2 are constants greater than 0,
Figure BDA00015570260300001226
for the function
Figure BDA0001557026030000128
The fuzzy parameters of ,
Figure BDA0001557026030000129
for the function
Figure BDA00015570260300001210
The fuzzy parameters of ,
Figure BDA00015570260300001211
for
Figure BDA00015570260300001212
transpose of ,
Figure BDA00015570260300001213
for
Figure BDA00015570260300001214
transpose of ,
Figure BDA00015570260300001231
for
Figure BDA00015570260300001232
the derivative of ,
Figure BDA00015570260300001230
for
Figure BDA00015570260300001229
The derivatives of , δ T (x k ) and φ T (h k ) are respectively
Figure BDA00015570260300001219
and
Figure BDA00015570260300001220
The membership function of ,
Figure BDA00015570260300001235
is the estimated value of h(s k )
Figure BDA00015570260300001233
for the function
Figure BDA00015570260300001234
the ideal value of ,
Figure BDA00015570260300001223
is the ideal value of the function h(s k ), ω is the approximation error of the fuzzy system, and f(x k ) is the actual value.

Figure BDA00015570260300001224
Figure BDA00015570260300001224

所以so

Figure BDA00015570260300001225
Figure BDA00015570260300001225

其中，ωk为ω的标量形式，

Figure BDA0001557026030000131
为ωk的导数，γ1，γ2为常数，
Figure BDA0001557026030000132
为dk的导数。where ω k is the scalar form of ω,
Figure BDA0001557026030000131
is the derivative of ω k , γ 1 , γ 2 are constants,
Figure BDA0001557026030000132
is the derivative of dk .

设计系统的自适应律为：The adaptive law of the design system is:

Figure BDA0001557026030000133
Figure BDA0001557026030000133

Figure BDA0001557026030000134
Figure BDA0001557026030000134

Figure BDA0001557026030000135
Figure BDA0001557026030000135

将(19)，(20)，(21)带入(18)得到：Bring (19), (20), (21) into (18) to get:

Figure BDA0001557026030000136
Figure BDA0001557026030000136

其中，||sk||为sk的范数，当

Figure BDA0001557026030000137
时，
Figure BDA0001557026030000138
成立。where ||s k || is the norm of s k , when
Figure BDA0001557026030000137
hour,
Figure BDA0001557026030000138
established.

其中，|ωmax|为ω的最大值的绝对值，得证。Among them, |ω max | is the absolute value of the maximum value of ω, which is proved.

(四)仿真验证(4) Simulation verification

为了验证上述理论的可行性，在Matlab下进行了仿真实验。仿真结果验证了所设计控制器的效果。In order to verify the feasibility of the above theory, simulation experiments were carried out under Matlab. The simulation results verify the effect of the designed controller.

仿真参数选取如下：The simulation parameters are selected as follows:

Figure BDA0001557026030000139
Figure BDA0001557026030000139

图3，图4分别表示了电源电流和系统误差。从图三可以看出，在0.04秒时有小幅度的波动，但是可以迅速在0.05秒左右恢复到正弦波，并且保持平滑的波形。说明系统的电源电流较稳定，并且在0.1秒和0.2秒电路电阻发生改变时也可以保持正弦波，说明系统鲁棒性较强。从图四可以看出，系统误差幅值较小，没有较大的波动，并且在0.05秒以后就一直保持稳定状态，即使在负载发生改变的时刻也可以较好跟踪并且快速恢复成直线。说明系统的跟踪效果较好且跟踪速度较快。Figure 3 and Figure 4 show the supply current and system error, respectively. As can be seen from Figure 3, there is a small fluctuation at 0.04 seconds, but it can quickly recover to a sine wave around 0.05 seconds, and maintain a smooth waveform. It shows that the power supply current of the system is relatively stable, and the sine wave can also be maintained when the circuit resistance changes in 0.1 seconds and 0.2 seconds, indicating that the system has strong robustness. It can be seen from Figure 4 that the system error amplitude is small, there is no large fluctuation, and it has been in a stable state after 0.05 seconds. Even when the load changes, it can track well and quickly recover to a straight line. It shows that the tracking effect of the system is better and the tracking speed is faster.

图5表示为线性化反馈高阶滑模模糊控制的直流侧电压图。由图可知，电压可以在0.05秒之前就直线上升并稳定在1000伏特，在0.1和0.2秒加入负载之后，也可以很快恢复并且一直保持在1000左右，效果较好。Figure 5 shows the DC side voltage diagram of the linearized feedback high-order sliding mode fuzzy control. It can be seen from the figure that the voltage can rise linearly and stabilize at 1000 volts before 0.05 seconds. After 0.1 and 0.2 seconds after adding the load, it can also recover quickly and keep at about 1000, and the effect is good.

具体实例的结果显示，本发明设计的基于线性化反馈高阶滑模模糊控制自适应控制的有源电力滤波器控制方法，可以有效克服非线性因素，外界扰动等影响，对改善有源滤波器系统的稳定性和动态性能，提高输配电、电网安全保障和电能质量是可行的。The results of specific examples show that the active power filter control method based on the linearized feedback high-order sliding mode fuzzy control adaptive control designed by the present invention can effectively overcome the influences of nonlinear factors and external disturbances, and can effectively improve the active filter. It is feasible to improve the stability and dynamic performance of the system and improve the power transmission and distribution, grid security and power quality.

以上所述仅是本发明的优选实施方式，应当指出，对于本技术领域的普通技术人员来说，在不脱离本发明原理的前提下，还可以做出若干改进和润饰，这些改进和润饰也应视为本发明的保护范围。The above are only the preferred embodiments of the present invention. It should be pointed out that for those skilled in the art, without departing from the principles of the present invention, several improvements and modifications can be made. It should be regarded as the protection scope of the present invention.

Claims (5)

1.基于线性化反馈的模糊高阶滑模有源电力滤波器控制方法，其特征在于：包括如下步骤：1. the fuzzy high-order sliding mode active power filter control method based on linearized feedback is characterized in that: comprise the steps: (1)建立有源电力滤波器数学模型；(1) Establish a mathematical model of active power filter; (2)设计控制器：首先设计动态滑模面，再利用线性化反馈控制设计滑模控制器，并在滑模控制器的基础上利用模糊控制逼近不确定项，设计自适应律使系统保持稳定状态，所述设计自适应律的具体步骤为：(2) Design the controller: first design the dynamic sliding mode surface, then use the linearized feedback control to design the sliding mode controller, and use the fuzzy control to approximate the uncertain term on the basis of the sliding mode controller, and design the adaptive law to keep the system steady state, the specific steps of designing the adaptive law are: 定义用于输入的跟踪误差e：Define the tracking error e for the input: e＝x-xd e =xxd 其中，x为指令电流，xd为参考电流，Among them, x is the command current, x d is the reference current, 对跟踪误差进行二次求导：Take the second derivative of the tracking error:
Figure FDA0002965889430000011
Figure FDA0002965889430000011
 其中，
Figure FDA0002965889430000012
为
Figure FDA0002965889430000013
的导数，
Figure FDA0002965889430000014
为xd的导数；in,
Figure FDA0002965889430000012
for
Figure FDA0002965889430000013
the derivative of ,
Figure FDA0002965889430000014
is the derivative of x d ; 定义滑模面：Define a sliding surface: l＝e+k2∫el=e+k 2 ∫e 其中k2为常数，∫e为对跟踪误差的积分；where k 2 is a constant, and ∫e is the integral of the tracking error; 对滑模面求导：Derivative with respect to the sliding surface:
Figure FDA0002965889430000015
Figure FDA0002965889430000015
 其中
Figure FDA0002965889430000016
为e的导数；in
Figure FDA0002965889430000016
is the derivative of e; 定义高阶滑模面s：Define higher-order sliding surface s:
Figure FDA0002965889430000017
Figure FDA0002965889430000017
 其中，
Figure FDA0002965889430000018
为大于0的常数，in,
Figure FDA0002965889430000018
is a constant greater than 0, 对高阶滑模面求导：Derivative for higher-order sliding mode surfaces:
Figure FDA0002965889430000019
Figure FDA0002965889430000019
 其中
Figure FDA00029658894300000110
为
Figure FDA00029658894300000111
的导数，
Figure FDA00029658894300000112
为s的导数；in
Figure FDA00029658894300000110
for
Figure FDA00029658894300000111
the derivative of ,
Figure FDA00029658894300000112
is the derivative of s; 根据线性化反馈技术，According to the linearized feedback technique, 因为
Figure FDA0002965889430000021
所以令g1(x)＝b，because
Figure FDA0002965889430000021
So let g 1 (x)=b,
Figure FDA0002965889430000022
Figure FDA0002965889430000022
Figure FDA0002965889430000023
Figure FDA0002965889430000023
Figure FDA0002965889430000024
Figure FDA0002965889430000024
 其中ρ为大于0的常数，且ρ≥|D|，D为ρ的上界常数，sgn为符号函数，ξ和RX是为了简化过程设计的中间变量，where ρ is a constant greater than 0, and ρ≥|D|, D is the upper bound constant of ρ, sgn is the sign function, ξ and R X are intermediate variables designed to simplify the process, 因为：because:
Figure FDA0002965889430000025
Figure FDA0002965889430000025
Figure FDA0002965889430000026
Figure FDA0002965889430000026
 所以：so:
Figure FDA0002965889430000027
Figure FDA0002965889430000027
Figure FDA0002965889430000028
Figure FDA0002965889430000028
 因此，自适应律设计为：Therefore, the adaptive law is designed as:
Figure FDA0002965889430000029
Figure FDA0002965889430000029
 2.如权利要求1所述的基于线性化反馈的模糊高阶滑模有源电力滤波器控制方法，其特征在于：所述步骤(1)中建立的有源电力滤波器数学模型为：2. the fuzzy high-order sliding mode active power filter control method based on linearized feedback as claimed in claim 1, is characterized in that: the active power filter mathematical model established in the described step (1) is:
Figure FDA00029658894300000210
Figure FDA00029658894300000210
 其中，x为指令电流，
Figure FDA00029658894300000211
为x的导数，
Figure FDA00029658894300000212
u为控制器输入，b为常数，Lc为电感，Rc为电阻，vdc为直流侧电容电压，vk为三相有源电力滤波器端电压，ik为三相补偿电流，dk为开关状态函数，dk定义如下：
Figure FDA0002965889430000031
则dk依赖于第k相IGBT的通断状态，是系统的非线性项；ck、cm为开关函数，m，k为大于0的常数。where x is the command current,
Figure FDA00029658894300000211
is the derivative of x,
Figure FDA00029658894300000212
u is the controller input, b is a constant, L c is the inductance, R c is the resistance, v dc is the DC side capacitor voltage, v k is the terminal voltage of the three-phase active power filter, i k is the three-phase compensation current, d k is the switch state function, and d k is defined as follows:
Figure FDA0002965889430000031
Then d k depends on the on-off state of the k-th phase IGBT and is a nonlinear term of the system; ck and cm are switching functions, and m and k are constants greater than 0. 3.如权利要求2所述的基于线性化反馈的模糊高阶滑模有源电力滤波器控制方法，其特征在于：对步骤(1)得到的有源电力滤波器数学模型进行二次求导，有源电力滤波器数学模型可以定义为：3. the fuzzy high-order sliding mode active power filter control method based on linearization feedback as claimed in claim 2, is characterized in that: the active power filter mathematical model that step (1) obtains is carried out secondary derivation , the mathematical model of the active power filter can be defined as:
Figure FDA0002965889430000032
Figure FDA0002965889430000032
Figure FDA0002965889430000033
为f(x)的导数，
Figure FDA0002965889430000034
为u的导数，
Figure FDA0002965889430000035
为
Figure FDA00029658894300000315
的导数；设
Figure FDA00029658894300000316
其中f1(x)为未知的非线性函数，则所述有源电力滤波器数学模型可以定义为：
Figure FDA0002965889430000033
is the derivative of f(x),
Figure FDA0002965889430000034
is the derivative of u,
Figure FDA0002965889430000035
for
Figure FDA00029658894300000315
the derivative of ; let
Figure FDA00029658894300000316
Where f 1 (x) is an unknown nonlinear function, the mathematical model of the active power filter can be defined as:
Figure FDA0002965889430000037
Figure FDA0002965889430000037
 4.如权利要求2所述的基于线性化反馈的模糊高阶滑模有源电力滤波器控制方法，其特征在于：根据设计的高阶滑模控制律设计加入模糊的控制律为：4. the fuzzy high-order sliding mode active power filter control method based on linearization feedback as claimed in claim 2, is characterized in that: according to the high-order sliding mode control law of design, the control law that adds fuzzy is designed to be:
Figure FDA0002965889430000038
Figure FDA0002965889430000038
 其中，
Figure FDA00029658894300000317
为
Figure FDA00029658894300000318
的模糊逼近函数，
Figure FDA00029658894300000310
为f(xk)的模糊逼近函数，
Figure FDA00029658894300000311
为ρsgn(s)的模糊逼近函数，xk为x的标量形式，sk为高阶滑模面s的标量形式。in,
Figure FDA00029658894300000317
for
Figure FDA00029658894300000318
The fuzzy approximation function of ,
Figure FDA00029658894300000310
is the fuzzy approximation function of f(x k ),
Figure FDA00029658894300000311
is the fuzzy approximation function of ρsgn(s), x k is the scalar form of x, and s k is the scalar form of the higher-order sliding mode surface s. 5.如权利要求4所述的基于线性化反馈的模糊高阶滑模有源电力滤波器控制方法，其特征在于：所述步骤(2)中利用李雅谱诺夫函数进行自适应律设计的具体步骤为：5. the fuzzy high-order sliding mode active power filter control method based on linearization feedback as claimed in claim 4, is characterized in that: in described step (2), utilize Lyapunov function to carry out adaptive law design The specific steps are: 设李亚普诺夫函数：Let the Lyapunov function:
Figure FDA00029658894300000312
Figure FDA00029658894300000312
Figure FDA00029658894300000313
Figure FDA00029658894300000313
Figure FDA00029658894300000314
Figure FDA00029658894300000314
 因为because
Figure FDA0002965889430000041
Figure FDA0002965889430000041
Figure FDA0002965889430000042
Figure FDA0002965889430000042
Figure FDA0002965889430000043
Figure FDA0002965889430000043
 其中，
Figure FDA00029658894300000431
和
Figure FDA00029658894300000432
分别为V1和V2的导数，sT为s的转置，Y1和Y2为大于0的常数，
Figure FDA00029658894300000430
为函数
Figure FDA0002965889430000046
的模糊参数，
Figure FDA0002965889430000047
为函数
Figure FDA0002965889430000048
的模糊参数，
Figure FDA0002965889430000049
为
Figure FDA00029658894300000410
的转置，
Figure FDA00029658894300000411
为
Figure FDA00029658894300000412
的转置，
Figure FDA00029658894300000413
为
Figure FDA00029658894300000414
的导数，
Figure FDA00029658894300000415
为
Figure FDA00029658894300000416
的导数，
Figure FDA00029658894300000417
和φT(hk)分别为
Figure FDA00029658894300000418
和
Figure FDA00029658894300000419
的隶属度函数，
Figure FDA00029658894300000420
为h(sk)的估计值、
Figure FDA00029658894300000421
为函数f(xk)的理想值，
Figure FDA00029658894300000422
为函数h(sk)的理想值，ω是模糊系统的逼近误差，f(xk)为实际值；in,
Figure FDA00029658894300000431
and
Figure FDA00029658894300000432
are the derivatives of V 1 and V 2 respectively, s T is the transpose of s, Y 1 and Y 2 are constants greater than 0,
Figure FDA00029658894300000430
for the function
Figure FDA0002965889430000046
The fuzzy parameters of ,
Figure FDA0002965889430000047
for the function
Figure FDA0002965889430000048
The fuzzy parameters of ,
Figure FDA0002965889430000049
for
Figure FDA00029658894300000410
transpose of ,
Figure FDA00029658894300000411
for
Figure FDA00029658894300000412
transpose of ,
Figure FDA00029658894300000413
for
Figure FDA00029658894300000414
the derivative of ,
Figure FDA00029658894300000415
for
Figure FDA00029658894300000416
the derivative of ,
Figure FDA00029658894300000417
and φ T (h k ) are respectively
Figure FDA00029658894300000418
and
Figure FDA00029658894300000419
The membership function of ,
Figure FDA00029658894300000420
is the estimated value of h(s k ),
Figure FDA00029658894300000421
is the ideal value of the function f(x k ),
Figure FDA00029658894300000422
is the ideal value of the function h(s k ), ω is the approximation error of the fuzzy system, and f(x k ) is the actual value; 因为：because:
Figure FDA00029658894300000423
Figure FDA00029658894300000423
 所以so
Figure FDA00029658894300000424
Figure FDA00029658894300000424
 其中，ωk为ω的标量形式，
Figure FDA00029658894300000425
为ωk的导数，
Figure FDA00029658894300000426
为dk的导数，where ω k is the scalar form of ω,
Figure FDA00029658894300000425
is the derivative of ω k ,
Figure FDA00029658894300000426
is the derivative of dk , 进而设计系统的自适应律为：Then the adaptive law of the design system is:
Figure FDA00029658894300000427
Figure FDA00029658894300000427
Figure FDA00029658894300000428
Figure FDA00029658894300000428
Figure FDA00029658894300000429
Figure FDA00029658894300000429
 其中，η为大于0的常数；Wherein, n is a constant greater than 0; 将(2)，(3)，(4)带入(1)得到：Bring (2), (3), (4) into (1) to get:
Figure FDA0002965889430000051
Figure FDA0002965889430000051
 当
Figure FDA0002965889430000052
时，
Figure FDA0002965889430000053
成立，when
Figure FDA0002965889430000052
hour,
Figure FDA0002965889430000053
established, 其中，||sk||为sk的范数，其中，|ωmax|为ω的最大值的绝对值。where ||s k || is the norm of s k , and |ω max | is the absolute value of the maximum value of ω.
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