CN108110761A - Fuzzy High-Order Sliding Mode Control Method of Active Power Filter based on Linearization Feedback - Google Patents

Abstract

The invention discloses a fuzzy high-order sliding mode active power filter control method based on linearized feedback. First, a mathematical model of a three-phase parallel active power filter is established. When designing a controller, firstly, a dynamic sliding mode is designed. Then the sliding mode controller is designed by using linear feedback control, and on the basis of this controller, fuzzy control is used to approach uncertain items, and an adaptive law is designed to keep the system in a stable state. The effectiveness of this method is verified by simulation results. This method greatly enhances the compensation performance and robust performance of the system, and achieves the purpose of quickly and effectively eliminating harmonics.

Description

Control Method of Fuzzy High-Order Sliding Mode Active Power Filter Based on Linearized Feedback

technical field

The invention relates to an active power filter technology, in particular to an active power filter control method based on a linearized feedback self-adaptive fuzzy high-order sliding mode.

Background technique

Using power filter device to absorb the harmonic current generated by the harmonic source is an effective measure to suppress harmonic pollution. The active power filter has fast response and high controllability. It can not only compensate harmonics, but also compensate 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 precision, and strong nonlinear adaptability, and can be applied in active filters for power quality control and harmonic control , has important research significance and market value.

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 the 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.

Contents of the invention

In order to suppress the influence of external unknown 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.

The present invention solves its technical problem and realizes through the following technical solutions:

A fuzzy high-order sliding mode active power filter control method based on linearized feedback includes the following steps:

(1) Establish a mathematical model of an active power filter;

(2) Design the controller: first design the dynamic sliding mode surface, then design the sliding mode controller using linearized feedback control, and use fuzzy control to approach the uncertain items on the basis of the sliding mode controller, and design an adaptive law to keep the system stable state.

Further, the active power filter mathematical model set up in the described step (1) is:

Among them, x is the command current, is the derivative of x, 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 three-phase active power filter terminal voltage, _ik is the three-phase compensation current, d _k is the switch state function, d _k is defined as follows: Then d _k depends on the on-off state of the k-th phase IGBT, which is a nonlinear term of the system; c _k and c _m are switching functions, and m and k are constants greater than 0.

Further, the second derivation 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:

is the derivative of f(x), is the derivative of u, for Derivative; let Where f ₁ (x) is an unknown nonlinear function, then the mathematical model of the active power filter can be defined as:

Further, the specific steps of designing a high-order sliding mode control law in the step (2) are:

Define the tracking error e for the input:

e=xx _d

Among them, x is the command current, x _d is the reference current,

Take the second derivative of the tracking error:

in, for derivative of is the derivative of x _d ;

Define the sliding surface:

l=e+k ₂ ∫e

where _k2 is a constant, and ∫e is the integral of the tracking error;

Derivative for a sliding surface:

in is the derivative of e;

Define the higher-order sliding mode surface s:

in, is a constant greater than 0,

Derivative for higher-order sliding mode surfaces:

in for derivative of is the derivative of s;

According to the linearization feedback technique,

because So let g ₁ (x)=b,

R _X ＝ξ-ρsgn(s)

Where ρ is a constant greater than 0, and ρ≥|D|, D is the upper bound constant of ρ, sgn is a sign function, ξ and R _X are intermediate variables designed to simplify the process,

because:

so:

Therefore, the system control law is designed as:

Further, according to the designed high-order sliding mode control law, the fuzzy control law is designed as:

in, for The fuzzy approximation function of is the fuzzy approximation function of f(x _k ), 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 high-order sliding mode surface s.

Further, in the step (2), the specific steps of utilizing the Lyapunov function to design the adaptive law are:

Let the Lyapunov function:

because

in, and 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, for the function fuzzy parameter of for the function fuzzy parameter of for the transposition of for the transposition of for derivative of for The derivatives of δ ^T (x _k ) and φ ^T (h _k ) are respectively and The membership function of , for estimated value of for the function ideal value, 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;

because:

so

Among them, ω _k is the scalar form of ω, is the derivative of ω _k , γ ₁ and γ ₂ are constants, is the derivative of d _k , and then the adaptive law of the design system is:

Wherein, n is a constant greater than 0;

Put (2), (3), (4) into (1) to get:

when hour, set up,

Wherein, ||s _k || is the norm of s _k , and |ω _max | is the absolute value of the maximum value of ω.

The beneficial effects of the present invention are:

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, utilizes the linear state equation of the three-phase parallel active power filter, and adds linearized feedback Higher order sliding mode control method. The model reference adaptive control of active power filter is studied, and the 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 supply quality.

Description of drawings

Fig. 1 is a structural representation of the present invention;

Fig. 2 is a block diagram of linearized feedback high-order sliding mode fuzzy adaptive controller of the present invention;

Figure 3 is a waveform diagram of the power supply current;

Fig. 4 is the data chart of systematic error;

Fig. 5 is a waveform diagram of the DC side voltage.

Detailed ways

The present invention will be further described in detail below through the specific examples, the following examples are only descriptive, not restrictive, and cannot limit the protection scope of the present invention with this.

As shown in Figure 1, the fuzzy high-order sliding mode active power filter control method based on linearized feedback includes the following steps:

(1) Establishing an active power filter model

The basic working principle of the active power filter is: detect the voltage and current of the compensation object, calculate the command signal of the compensation current through the command current operation circuit, and amplify the signal through the compensation current generation circuit to obtain the compensation current, and the compensation current and The harmonics and reactive power to be compensated in the load current are offset, and finally the desired power supply current is obtained.

According to circuit theory and Kirchhoff's theorem, the following formula can be obtained:

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, Indicates the derivative of current i ₁ with respect to time, Indicates the derivative of current i ₂ with respect to time, Indicates the derivative of current i ₃ with respect to time, and v _1M , v _2M , v _3M , v _MN respectively represent the voltages from point M to point a, b, c, and N in Figure 1. Point M is the negative pole 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 _ik is the three-phase compensation current.

Assuming that the power supply voltage on the AC side is stable, we can get

Wherein, m is a constant greater than 0, and v _mM is the aforementioned v _1M , v _2M , v _3M , and v _MN .

And define c _k as the switching function, indicating the working state of the IGBT, defined as follows:

Among them, k=1,2,3.

Meanwhile, v _kM = c _k v _dc , wherein, v _kM is v _1M , v _2M , v _3M , v _MN mentioned above.

So (1) can be rewritten as

v _dc is the DC side capacitor voltage.

We define d _k as the switch state function, defined as follows:

Then d _k depends on the on-off state of the k-phase IGBT, which is a nonlinear term of the system; c _m is a switching function.

And a

Then (4) can be rewritten as

Then (7) can be rewritten as follows

in x is the command current, is the derivative of x, b is a constant, and u is the controller input.

(2) Linearized feedback high-order sliding mode control

Define the tracking error e:

e=xx _d

Among them, x _d is the reference current.

Find the first derivative with respect to the tracking error e:

in is the derivative of x _d

Find the second derivative with respect to the tracking error e:

in for derivative of for derivative of

Take the second derivative of the system model:

is the derivative of f(x), is the derivative of u,

Assume Where f ₁ (x) is an unknown nonlinear function.

So the original model can be defined as:

Define the sliding surface:

l=e+k ₂ ∫e

where _k2 is a constant and ∫e is the integral of the error.

Derivative for a sliding surface:

in is the derivative of e,

Define the higher-order sliding mode surface s:

in, is a constant greater than 0.

Derivative for higher-order sliding surface s:

in for derivative of is the derivative of s.

According to the linearization feedback technique:

R _X ＝ξ-ρsgn(s)

Where ρ is a constant greater than 0, and ρ≥|D|, D is the upper bound constant of ρ, sgn is a sign function, is the second derivative of x, ξ and R _x are intermediate variables designed to simplify the process.

because:

so:

Therefore, the system control law is designed as:

(3) Linearized feedback high-order sliding mode fuzzy control

R ⁱ : If x ₁ is and....x _n is then y is B ⁱ (i=1,2,....,N)

in, is the membership function of x _j (j=1,2,...,n).

Then the output of the fuzzy system is:

Where δ=[δ ₁ (x) δ ₂ (x) ... δ _N (x)] ^T is the membership function, is the value of each small component in the membership function, is the fuzzy parameter, is the component in the fuzzy parameter.

For the fuzzy approximation of f(x,y), the form of approximating f(1) and f(2) is adopted respectively, and the corresponding fuzzy system is designed as:

Define the fuzzy function as follows:

in, ψ is the output of the fuzzy system, and ψ ₁ and ψ ₂ are scalars in the output. is the sum of the components in the fuzzy parameters, N is the total number of components, for The minimum value in , for The sum of the minimum values in .

Define optimal approximation constants

where Ω is collection.

but:

ω is the approximation error of the fuzzy system, δ ^T is the transpose of δ, for transpose. For any given small constant ω (ω>0), the following inequality holds true: |f(x,y) ^-ξT (x)θ ^* |≤ω Let and make

The controller is designed to:

in, for The fuzzy approximation function of is the fuzzy approximation function of f(x _k ), 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.

Proof of Stability:

Let the Lyapunov function:

because

in, and 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, for the function fuzzy parameter of for the function fuzzy parameter of for the transposition of for the transposition of for derivative of for The derivatives of δ ^T (x _k ) and φ ^T (h _k ) are respectively and The membership function of , is an estimated value of h(s _k ) for the function ideal value, 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.

so

Among them, ω _k is the scalar form of ω, is the derivative of ω _k , γ ₁ and γ ₂ are constants, is the derivative of d _k .

The adaptive law of the design system is:

Put (19), (20), (21) into (18) to get:

Among them, ||s _k || is the norm of s _k , when hour, established.

Among them, |ω _max | is the absolute value of the maximum value of ω, which is proved.

(4) Simulation verification

In order to verify the feasibility of the above theory, a simulation experiment was carried out under Matlab. Simulation results verify the effect of the designed controller.

The simulation parameters are selected as follows:

Figure 3 and Figure 4 show the power supply current and system error respectively. It can be seen from Figure 3 that there is a small fluctuation at 0.04 seconds, but it can quickly return to a sine wave at about 0.05 seconds, and maintain a smooth waveform. It shows that the power supply current of the system is relatively stable, and it can also maintain a sine wave when the circuit resistance changes in 0.1 seconds and 0.2 seconds, which shows 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 return to a straight line. It shows that the tracking effect of the system is better and the tracking speed is faster.

Figure 5 shows the DC side voltage diagram for linearized feedback high-order sliding mode fuzzy control. It can be seen from the figure that the voltage can rise in a straight line before 0.05 seconds and stabilize at 1000 volts. After the load is added at 0.1 and 0.2 seconds, it can also recover quickly and remain at around 1000 volts. The effect is good.

The results of specific examples show that the active power filter control method based on linearized feedback high-order sliding mode fuzzy control adaptive control designed by the present invention can effectively overcome nonlinear factors, external disturbances and other influences, and improve the active power filter. It is feasible to improve the stability and dynamic performance of the system, improve power transmission and distribution, power grid security and power quality.

The above is only a preferred embodiment of the present invention, it should be pointed out that, for those of ordinary skill in the art, without departing from the principle of the present invention, some improvements and modifications can also be made, and these improvements and modifications can also be made. It should be regarded as the protection scope of the present invention.

Claims (6)

1. The fuzzy high-order sliding mode active power filter control method based on the linearization feedback is characterized by comprising the following steps: the method comprises the following steps:

(1) establishing an active power filter mathematical model;

(2) designing a controller: firstly, designing a dynamic sliding mode surface, designing a sliding mode controller by utilizing linear feedback control, approaching an uncertain item by utilizing fuzzy control on the basis of the sliding mode controller, and designing a self-adaptive law to keep a system in a stable state.

2. The method for controlling the fuzzy high-order sliding mode active power filter based on the linearized feedback as set forth in claim 1, wherein: the mathematical model of the active power filter established in the step (1) is as follows:

<mrow> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>b</mi> <mi>u</mi> <mo>+</mo> <msub> <mi>d</mi> <mi>k</mi> </msub> </mrow>

wherein x is a command current,is the derivative of x and is,u is the controller input, b is a constant, L_cIs an inductance, R_cIs a resistance, v_dcIs the DC side capacitor voltage, v_kFor the terminal voltage, i, of three-phase active power filters_kFor three-phase compensation current, d_kAs a function of the switching state, d_kThe definition is as follows:then d_kThe on-off state of the kth phase IGBT is depended on and is a nonlinear term of the system; c. C_k、c_mM, k are constants greater than 0 for the switching function.

3. The method for controlling the fuzzy higher-order sliding mode active power filter based on the linearized feedback as set forth in claim 2, wherein: performing secondary derivation on the active power filter mathematical model obtained in the step (1), wherein the active power filter mathematical model can be defined as:

<mrow> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <mover> <mi>f</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>b</mi> <mover> <mi>u</mi> <mo>&amp;CenterDot;</mo> </mover> </mrow>

is the derivative of f (x),is the derivative of u and is,is composed ofA derivative of (a); is provided withWherein f is₁(x) For unknown non-linear functions, the active power filter mathematical model can be defined as:

<mrow> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>b</mi> <mover> <mi>u</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>.</mo> </mrow>

4. the method for controlling the fuzzy higher order sliding mode active power filter based on the linearized feedback as set forth in claim 3, wherein: the specific steps for designing the high-order sliding mode control law in the step (2) are as follows:

defining the tracking error e for the input:

e＝x-x_d

wherein x is a command current, x_dAs a reference current, the current is,

and (3) carrying out secondary derivation on the tracking error:

<mrow> <mover> <mi>e</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> </mrow>

wherein,is composed ofThe derivative of (a) of (b),is x_dA derivative of (a);

defining a slip form surface:

l＝e+k₂∫e

wherein k is₂Is a constant, and ^ e is the integral of the tracking error;

derivation of the sliding mode surface:

<mrow> <mover> <mi>l</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mi>e</mi> </mrow>

whereinIs the derivative of e;

defining a high-order slip-form surface s:

<mrow> <mi>s</mi> <mo>=</mo> <mover> <mi>l</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mo>&amp;part;</mo> <mi>l</mi> <mo>=</mo> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mi>e</mi> <mo>+</mo> <mo>&amp;part;</mo> <mi>e</mi> <mo>+</mo> <mo>&amp;part;</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>&amp;Integral;</mo> <mi>e</mi> </mrow>

wherein,is a constant number greater than 0 and is,

derivation of a high-order sliding mode surface:

<mrow> <mover> <mi>s</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mover> <mi>e</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mo>&amp;part;</mo> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mo>&amp;part;</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mi>e</mi> </mrow>

whereinIs composed ofThe derivative of (a) of (b),is the derivative of s;

according to the linearized feedback technique,

because of the fact thatTherefore, g₁(x)＝b，

<mrow> <mover> <mi>u</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mfrac> <mrow> <msub> <mi>R</mi> <mi>X</mi> </msub> <mo>-</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>g</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <msub> <mi>R</mi> <mi>X</mi> </msub> <mo>-</mo> <mover> <mi>f</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mi>b</mi> </mfrac> </mrow>

<mrow> <mi>&amp;xi;</mi> <mo>=</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>-</mo> <mover> <mi>s</mi> <mo>&amp;CenterDot;</mo> </mover> </mrow>

R_X＝ξ-ρsgn(s)

wherein rho is a constant larger than 0, and rho is greater than or equal to | D |, D is an upper bound constant of rho, sgn is a sign function, ξ and R_XIs an intermediate variable in order to simplify the process design,

because:

<mrow> <mover> <mi>s</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mover> <mi>e</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mo>&amp;part;</mo> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mo>&amp;part;</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mi>e</mi> </mrow>

<mrow> <mover> <mi>e</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> </mrow>

therefore:

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>&amp;xi;</mi> <mo>=</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>-</mo> <mover> <mi>s</mi> <mo>&amp;CenterDot;</mo> </mover> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>-</mo> <mrow> <mo>(</mo> <mover> <mi>e</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mo>&amp;part;</mo> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mo>&amp;part;</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mi>e</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>-</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>-</mo> <mo>&amp;part;</mo> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>-</mo> <mo>&amp;part;</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mi>e</mi> </mrow> </mtd> </mtr> </mtable> </mfenced>

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>R</mi> <mi>X</mi> </msub> <mo>=</mo> <mi>&amp;xi;</mi> <mo>-</mo> <mi>&amp;rho;</mi> <mi>sgn</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>-</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>-</mo> <mo>&amp;part;</mo> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>-</mo> <mo>&amp;part;</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mi>e</mi> <mo>-</mo> <mi>&amp;rho;</mi> <mi>sgn</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>

thus, the system control law is designed to:

<mrow> <mover> <mi>u</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mfrac> <mn>1</mn> <mi>b</mi> </mfrac> <mrow> <mo>(</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>-</mo> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>+</mo> <mo>&amp;part;</mo> </mrow> <mo>)</mo> <mo>(</mo> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>b</mi> <mi>u</mi> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> </mrow> <mo>)</mo> <mo>-</mo> <mo>&amp;part;</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mi>e</mi> <mo>-</mo> <mi>&amp;rho;</mi> <mi>sgn</mi> <mo>(</mo> <mi>s</mi> <mo>)</mo> <mo>-</mo> <mover> <mi>f</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>)</mo> </mrow> <mo>.</mo> </mrow>

5. the method for controlling the fuzzy higher order sliding mode active power filter based on the linearized feedback as set forth in claim 4, wherein: designing a fuzzy control law according to a designed high-order sliding mode control law as follows:

<mrow> <mover> <mi>u</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mfrac> <mn>1</mn> <mi>b</mi> </mfrac> <mrow> <mo>(</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>-</mo> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>+</mo> <mo>&amp;part;</mo> </mrow> <mo>)</mo> <mo>(</mo> <mrow> <mover> <mi>f</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>b</mi> <mi>u</mi> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> </mrow> <mo>)</mo> <mo>-</mo> <mo>&amp;part;</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mi>e</mi> <mo>-</mo> <mover> <mi>h</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>s</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>-</mo> <mover> <mover> <mi>f</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow>

wherein,is composed ofThe function of the fuzzy approximation of (a),is f (x)_k) The function of the fuzzy approximation of (a),fuzzy approximation function for rho sgn(s)Number, x_kIn the scalar form of x, s_kIn the scalar form of a high order sliding mode surface s.

6. The method for controlling the fuzzy higher order sliding mode active power filter based on the linearized feedback as set forth in claim 5, wherein: the specific steps of utilizing the Lyapunov function to carry out adaptive law design in the step (2) are as follows:

let the Lyapunov function:

<mrow> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mi>s</mi> <mi>T</mi> </msup> <mo>&amp;CenterDot;</mo> <mi>s</mi> </mrow>

because of the fact that

Wherein,andare respectively asAndderivative of(s)^TIs a transposition of s, γ₁And gamma₂Is a constant number greater than 0 and is,is a function ofThe blur parameter of (a) is determined,is a function ofThe blur parameter of (a) is determined,is composed ofThe transpose of (a) is performed,is composed ofThe transpose of (a) is performed,is composed ofThe derivative of (a) of (b),is composed ofDerivative of, delta^T(s_k) And phi^T(h_k) Are respectively asAndthe function of the degree of membership of (c),is h(s)_k) The estimated value of,As a function f (x)_k) The ideal value of (a) is,as a function h(s)_k) Is the approximation error of the fuzzy system, f (x)_k) Is an actual value;

because:

<mfenced open='' close=''> <mtable> <mtr> <mtd> <mover> <mi>s</mi> <mo>.</mo> </mover> <mo>=</mo> <mover> <mi>e</mi> <mrow> <mo>.</mo> <mo>.</mo> </mrow> </mover> <mo>+</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mover> <mi>e</mi> <mo>.</mo> </mover> <mo>+</mo> <mo>&amp;PartialD;</mo> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mo>&amp;PartialD;</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mi>e</mi> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mover> <mi>f</mi> <mo>.</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mover> <mover> <mi>f</mi> <mo>.</mo> </mover> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>+</mo> <mo>&amp;PartialD;</mo> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mover> <mi>f</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>+</mo> <mo>&amp;PartialD;</mo> <mo>)</mo> </mrow> <msub> <mi>d</mi> <mi>k</mi> </msub> <mo>-</mo> <mover> <mi>h</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>s</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mover> <mi>d</mi> <mo>.</mo> </mover> <mi>k</mi> </msub> </mtd> </mtr> </mtable> </mfenced>

therefore, it is not only easy to use

Wherein, ω is_kIn the form of a scalar of omega,is omega_kDerivative of, gamma₁，γ₂Is a constant number of times, and is,is d_kAnd then the adaptive law of the design system is as follows:

<mrow> <mover> <mi>h</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mi>&amp;eta;</mi> <mo>+</mo> <mi>&amp;rho;</mi> <mo>)</mo> </mrow> <mi>sgn</mi> <mrow> <mo>(</mo> <mi>&amp;sigma;</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>

wherein η is a constant greater than 0;

bringing (2), (3), (4) into (1) yields:

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>V</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>s</mi> <mi>T</mi> </msup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> <mo>+</mo> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>+</mo> <mo>&amp;part;</mo> </mrow> <mo>)</mo> <msub> <mi>&amp;omega;</mi> <mi>k</mi> </msub> <mo>-</mo> <mi>h</mi> <mo>(</mo> <mrow> <msub> <mi>s</mi> <mi>k</mi> </msub> <mo>|</mo> <msubsup> <mi>&amp;theta;</mi> <mrow> <mi>h</mi> <mi>k</mi> </mrow> <mo>*</mo> </msubsup> </mrow> <mo>)</mo> <mo>+</mo> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>+</mo> <mo>&amp;part;</mo> </mrow> <mo>)</mo> <mo>&amp;CenterDot;</mo> <msub> <mi>d</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mover> <mi>d</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <msub> <mi>s</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> <mo>+</mo> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>+</mo> <mo>&amp;part;</mo> </mrow> <mo>)</mo> <msub> <mi>&amp;omega;</mi> <mi>k</mi> </msub> <mo>-</mo> <mo>(</mo> <mrow> <mi>&amp;rho;</mi> <mo>+</mo> <mi>&amp;eta;</mi> </mrow> <mo>)</mo> <mo>&amp;CenterDot;</mo> <mi>sgn</mi> <mo>(</mo> <msub> <mi>s</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>+</mo> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>+</mo> <mo>&amp;part;</mo> </mrow> <mo>)</mo> <mo>&amp;CenterDot;</mo> <msub> <mi>d</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mover> <mi>d</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;le;</mo> <msub> <mi>s</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>+</mo> <mo>&amp;part;</mo> <mo>)</mo> </mrow> <msub> <mi>&amp;omega;</mi> <mi>k</mi> </msub> <mo>-</mo> <mi>&amp;eta;</mi> <mo>|</mo> <mo>|</mo> <msub> <mi>s</mi> <mi>k</mi> </msub> <mo>|</mo> <mo>|</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;le;</mo> <mo>|</mo> <mo>|</mo> <msub> <mi>s</mi> <mi>k</mi> </msub> <mo>|</mo> <mo>|</mo> <mrow> <mo>(</mo> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>+</mo> <mo>&amp;part;</mo> </mrow> <mo>)</mo> <msub> <mi>&amp;omega;</mi> <mi>k</mi> </msub> <mo>-</mo> <mi>&amp;eta;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>

when in useWhen the temperature of the water is higher than the set temperature,it is true that the first and second sensors,

wherein, | s_kI is s_kWhere | ω_maxAnd | is the absolute value of the maximum value of ω.