CN110286594B - Self-adaptive dynamic terminal sliding mode control method for active power filter - Google Patents

Self-adaptive dynamic terminal sliding mode control method for active power filter Download PDF

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CN110286594B
CN110286594B CN201910646593.1A CN201910646593A CN110286594B CN 110286594 B CN110286594 B CN 110286594B CN 201910646593 A CN201910646593 A CN 201910646593A CN 110286594 B CN110286594 B CN 110286594B
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CN110286594A (en
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陈云
费峻涛
陈放
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Changzhou Campus of Hohai University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
    • 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

Abstract

The invention discloses a self-adaptive dynamic terminal sliding mode control method of an active power filter, which comprises the following steps of: s1, establishing a mathematical model of the single-phase active power filter, defining a state variable i as x, and obtaining a second derivative of x
Figure DDA0002133784130000011
The expression of (1); s2, defining a tracking error and a first derivative thereof, defining a dynamic terminal sliding mode surface to obtain an equivalent control item, then defining a switching control item, and adding the equivalent control item and the switching control item to obtain a dynamic terminal sliding mode controller; and S3, approximating the equivalent control items by using a double-hidden-layer recurrent neural network. The self-adaptive dynamic terminal sliding mode control method of the active power filter can quickly realize no-static-error tracking and achieve lower grid current distortion rate.

Description

Self-adaptive dynamic terminal sliding mode control method for active power filter
Technical Field
The invention relates to a self-adaptive dynamic terminal sliding mode control method of an active power filter, and belongs to the technical field of intelligent control.
Background
Harmonic current suppression technologies become a research hotspot in recent years, and currently, the main harmonic suppression technologies include passive filters and active filters; the passive filter is generally composed of an inductor and a capacitor, is suitable for filtering low-order harmonic waves, and can only filter harmonic waves of specific harmonics, but the passive filter is widely applied to the industrial field due to simple structure and low manufacturing cost, but the compensation precision of the passive filter is not high, and the passive filter can be finally replaced by an active power filter. The active power filter is a novel harmonic compensation device, is an active compensation device, can compensate harmonic waves and can perform reactive compensation, is less affected by the environment, has a wider compensation range on harmonic currents, and is an optimal harmonic suppression device.
The key technology of the active power filter is the design of the controller, and the better controller can better play the compensation performance of the active power filter and reduce the tracking error. The traditional control method is often not high in control accuracy, so that the performance of the active power filter cannot be fully exerted, and in addition, the performance of the active power filter cannot be fully exerted. Existing controllers have high grid current distortion rates.
Disclosure of Invention
The invention aims to provide an active power filter adaptive dynamic terminal sliding mode control method which can quickly realize no-static-error tracking and achieve a lower grid current distortion rate aiming at the defects of the prior art.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows:
an active power filter self-adaptive dynamic terminal sliding mode control method comprises the following steps:
s1, establishing a mathematical model of the single-phase active power filter, defining a state variable i as x, and obtaining a second derivative of x
Figure BDA0002133784110000029
The expression of (1);
s2, defining a tracking error and a first derivative thereof, defining a dynamic terminal sliding mode surface to obtain an equivalent control item, then defining a switching control item, and adding the equivalent control item and the switching control item to obtain a terminal sliding mode controller;
and S3, approximating the equivalent control items by using a double-hidden-layer recurrent neural network.
In S1, the mathematical model of the single-phase active power filter is specifically:
Figure BDA0002133784110000021
where i is the state variable, here denoted the compensating harmonic current, L is the total inductance of the line on the AC side, R is the total resistance of the line on the AC side, UdcIs a DC side voltage, UsIs the grid voltage, H is a defined switching function, for convenience, x-i,
Figure BDA0002133784110000022
Figure BDA0002133784110000023
if the controller is defined as u ═ H, then equation (1) can be abbreviated as:
Figure BDA0002133784110000024
where d (t) is the system extra perturbation considered and d (t) is bounded.
In S2, the specific steps are:
s21, defining the tracking error as e ═ x-r, the first derivative of the tracking error as
Figure BDA0002133784110000025
The second derivative of the tracking error is
Figure BDA0002133784110000026
Then the error vector is
Figure BDA0002133784110000027
Wherein r is a reference current signal;
s22, defining a terminal sliding mode surface as
Figure BDA0002133784110000028
c is a tunable coefficient to ensure the stability of the system Herwitz, where P (t) is a terminal function with respect to time t and
Figure BDA0002133784110000031
the dynamic terminal sliding mode surface is defined as:
Figure BDA0002133784110000032
wherein λ is a strictly normal number;
the first derivative of the sliding mode surface of the dynamic terminal is as follows:
Figure BDA0002133784110000033
to achieve global robustness, take e (0) to p (0),
Figure BDA0002133784110000034
in order to realize convergence at a specified time T, when T ≧ T, p (T) 0 is satisfied,
Figure BDA0002133784110000035
we can define the terminal function as:
Figure BDA0002133784110000036
where T is a self-defined terminal time, e (0),
Figure BDA0002133784110000037
Respectively representing the tracking error and the initial values of its first and second derivatives at t ═ 0s, a00、a01、a02、a10、a11、a12、a20、 a21、a22An available parameter for satisfying the above-mentioned assumption;
s23, order
Figure BDA0002133784110000038
Available equivalent control terms:
Figure BDA0002133784110000039
wherein
Figure BDA0002133784110000041
S24, taking external interference into consideration, using switching control item
Figure BDA0002133784110000042
K w0 is any arbitrary constant that ensures the Lyapunov function to be semi-positiveParameter-adjusting and finally-designed dynamic terminal sliding mode controller
Figure BDA0002133784110000043
The following steps are changed:
Figure BDA0002133784110000044
wherein
Figure BDA0002133784110000045
KfAnd < 0 is any adjustable coefficient for ensuring that the Lyapunov function is semi-positive.
Defining the Lyapunov function:
Figure BDA0002133784110000046
the first derivative of the formula (9) is obtained, and the first derivative of the sliding mode surface of the dynamic terminal in the formula (4) and the control law in the formula (8) are substituted into the first derivative of the Lyapunov function to obtain:
Figure BDA0002133784110000047
due to d (t) and
Figure BDA0002133784110000048
bounded, define
Figure BDA0002133784110000049
Therefore, as long as K is guaranteedwRho > D, it can be proved
Figure BDA00021337841100000410
Considering the terminal sliding-mode surface as s ═ C (e (t) -p (t)), where C ═ C1, we define η (t) ═ e (t) -p (t), where e (t) is the error vector,
Figure BDA00021337841100000411
since C does not contain 0 element, take
Figure BDA00021337841100000412
Then
Figure BDA00021337841100000413
Will also tend towards 0.
S03 specifically includes the following steps:
s31, defining the structure of the double hidden layer recurrent neural network as follows: an input layer, a first hidden layer, a second hidden layer and an output layer, and the result of the output layer is fed back to the input layer,
output theta of ith node of input layeriCan be expressed as a number of times,
θi＝xi·Wri·exY，i＝1,2,...,m (12)
wherein xiIs the ith input of the double hidden layer recurrent neural network, exY is the output value of the neural network at the last moment, WriThe feedback weight vector is defined as W for the feedback weight of the ith input layer noder＝[Wr1 Wr2…Wri]；
The jth node of the first hidden layer outputs a result phi1jIn order to realize the purpose,
Figure BDA0002133784110000051
wherein the first hidden layer output vector is phi1＝[φ11 φ12…φ1j]And phi is1jRepresenting the output of the jth node of the first hidden layer, the center vector of the Gaussian function of the first hidden layer is C1＝[c11,c12,…,c1n]T∈Rn×1The vector of the base width of the Gaussian function is B1＝[b11，b12,…,b1n]T∈Rn×1And c is and c1nIn the nth node being the first hidden layerCardiac vector, and b1nIs the nth node center vector, R, of the first hidden layern×1A vector representing n rows and 1 columns in the real number domain;
the kth node of the second hidden layer outputs a result phi2kIn order to realize the purpose,
Figure BDA0002133784110000052
wherein the second hidden layer output vector is phi2＝[φ21 φ22…φ2k]And phi is2kRepresenting the output of the kth node of the second hidden layer, the central vector of the Gaussian function of the second hidden layer is C2＝[c21 c22...c2l]T∈Rl×1The vector of the base width of the Gaussian function is B2＝[b21 b22...b2l]T∈Rl×1And c is and c2lIs the l-th node-center vector of the second hidden layer, b2lIs the l-th node center vector, R, of the second hidden layerl×1A vector representing l rows and 1 columns in the real number domain;
combining the analysis, the output result of the double hidden layer recurrent neural network is as follows:
Y＝WTΦ2＝W1φ21+W2φ22+...+Wlφ2l (15)
wherein W ═ W1 W2…Wl]Is the output weight vector, W, of the double hidden layer recurrent neural networklRepresenting a weight vector between the l-th node of the second hidden layer and the output value;
presence of optimal parameters
Figure BDA0002133784110000061
So that
Figure BDA0002133784110000062
Wherein epsilon is the optimal approximation error;
s32, replacing the equivalent control in the formula (6) with the output of the double hidden layer recurrent neural networkMaking an item, i.e.
Figure BDA0002133784110000063
Then the self-adaptive dynamic terminal sliding mode controller based on the double hidden layer recurrent neural network becomes:
Figure BDA0002133784110000064
s33, the approximation error of the double hidden layer recurrent neural network is defined as:
Figure BDA0002133784110000065
wherein the content of the first and second substances,
Figure BDA0002133784110000066
to obtain the law of adaptation
Figure BDA0002133784110000067
Is aligned with
Figure BDA0002133784110000068
Taylor expansion is carried out to obtain:
Figure BDA0002133784110000069
Figure BDA00021337841100000610
wherein the superscript denotes an optimal parameter of the corresponding variable, the superscript ^ denotes an estimated value of the corresponding parameter, the superscript ^ denotes an estimated error of the corresponding parameter, and O denoteshFor the higher order terms of the taylor expansion,
approximation error of each parameter
Figure BDA0002133784110000071
Defining a new lyapunov function as:
Figure BDA0002133784110000072
the first derivative of the Lyapunov function is obtained:
Figure BDA0002133784110000073
wherein eta1,η2,η3,η4,η5,η6Is an adjustable normal number of the input signals,
in order to stabilize the system, the ideal control law in the formula (8) is added and subtracted to the first derivative of the Lyapunov function in the formula (20) at the same time, and the approximation error in the formula (17) is substituted into the formula (20) to obtain the ideal control law,
Figure BDA0002133784110000074
the following adaptation law is selected,
Figure BDA0002133784110000081
Figure BDA0002133784110000082
Figure BDA0002133784110000083
Figure BDA0002133784110000084
Figure BDA0002133784110000085
Figure BDA0002133784110000086
wherein
Figure BDA0002133784110000087
Respectively the first derivative, eta, of the approximation error of the parameters of the weight, the feedback gain, the center of the first hidden layer, the base width of the first hidden layer, the center of the second hidden layer and the base width of the second hidden layer of the double hidden layer recurrent neural network1,η2,η3,η4,η5,η6Is an adjustable normal number of the input signals,
Figure BDA00021337841100000811
Figure BDA00021337841100000812
output phi representing the second hidden layer2Respectively to the parameters
Figure BDA0002133784110000088
Figure BDA0002133784110000089
The derivative of (c).
Substituting the adaptation laws (22) - (27) into equation (21) yields,
Figure BDA00021337841100000810
the method can be obtained by simplifying the formula,
Figure BDA0002133784110000091
as can be seen from the simplified formula (29), since d (t) and
Figure BDA0002133784110000092
bounded, we define
Figure BDA0002133784110000093
Therefore, as long as K is guaranteedwRho > D, it can be proved
Figure BDA0002133784110000094
This shows that the terminal sliding mode controller of the double hidden layer recurrent neural network is stable and feasible.
The invention has the beneficial effects that: the invention discloses a self-adaptive dynamic terminal sliding mode control method of an active power filter, which is proved to be stable and feasible, but the model of the active power filter is complex, and parameters cannot be accurately obtained. And finally, the practicability of the algorithm is verified through MATLAB simulation.
Description of the drawings:
FIG. 1 is a schematic diagram of a sliding mode controller of an adaptive dynamic terminal based on a double hidden layer recurrent neural network according to the method of the present invention;
FIG. 2 is a block diagram of a single phase active power filter of the present invention;
fig. 3 is a structural view of a three-phase parallel power supply type active power filter of the present invention;
FIG. 4 is a diagram of a dual hidden layer recurrent neural network of the present invention;
FIG. 5 is a graph of grid current for the present invention;
FIG. 6 is a graph of harmonic current tracking of the present invention;
FIG. 7 is a tracking error map of the present invention;
fig. 8 is a diagram of compensated grid current distortion rate according to the present invention.
Detailed Description
Fig. 1 is a schematic diagram of an adaptive dynamic terminal sliding mode control method of an active power filter according to the present invention, which is intended to explain: detecting harmonic current from load current as a reference signal r, constructing a dynamic terminal sliding mode surface based on a terminal function, designing a switching control item for ensuring the stability of a system, approximating an equivalent control item by using a double-hidden-layer recurrent neural network, outputting a harmonic compensation signal x after the designed control law passes through an active power filter, enabling the error of the system to tend to zero by using negative feedback, and finally achieving the purpose of tracking the reference harmonic current quickly and without static error.
The invention discloses a self-adaptive dynamic terminal sliding mode control method of a source power filter, which comprises the following steps of:
step one, establishing a mathematical model of the single-phase active power filter. FIG. 2 is a block diagram of a single-phase active power filter, in which UsIs the grid current iLIs the load current isIs the grid current icIs to compensate for the harmonic current flow,
Figure BDA0002133784110000101
is a reference harmonic current, L is the total inductance of the AC side line, R is the total resistance of the AC side line, Qi(i ═ 1,2,3,4) are IGBT power electronic switching devices, UdcIs the dc side voltage.
The following dynamic equation can be established according to the theorem of voltage and current:
Figure BDA0002133784110000102
wherein U isMN＝UdcAnd C is the AC side voltage of the active power filter.
For convenience, the switching function is defined:
Figure BDA0002133784110000103
substituting the switching function into the dynamic equation and solving the first derivative of the time to obtain:
Figure BDA0002133784110000111
in order to solve the control law, the first derivative is solved, and then the first derivative is changed into a second-order system, and the mathematical model of the single-phase active power filter is obtained by the following specific steps:
Figure BDA0002133784110000112
where i is the state variable, here denoted the compensating harmonic current, L is the total inductance of the line on the AC side, R is the total resistance of the line on the AC side, UdcIs a DC side voltage, UsIs the network voltage, H is a defined switching function, defines the state variable i as x, obtains the second derivative of x
Figure BDA0002133784110000118
Is described in (1).
For convenience, definitions are provided
Figure BDA0002133784110000113
If the controller is defined as u ═ H, then equation (1) can be abbreviated as:
Figure BDA0002133784110000114
where d (t) is the system extra perturbation considered and d (t) is bounded.
The invention is mainly designed for a single-phase active power filter, but actually, the designed controller is not only suitable for the single-phase active power filter, but also suitable for a three-phase three-wire active power filter mathematical model as shown in fig. 3, and for illustration, similar to the single-phase model, the following three-phase kinetic equations can be established by using the voltage and current theorem:
Figure BDA0002133784110000115
wherein i ═ i1 i2 i3]TTo compensate for the current vector, i1 i2 i3Respectively correspond to a bcCurrent of three phases, dk＝[d1k d2k d3k]TIs a switch state function vector; d1k d2k d3kCorresponding to the switching state functions of the three phases a b c. The switching state function of the nth phase is thus defined as
Figure BDA0002133784110000116
ckThe definition of the k-th phase switching function is similar to that of a single phase, and specifically comprises the following steps:
Figure BDA0002133784110000117
the three-phase kinetic equation is simplified to obtain:
Figure BDA0002133784110000121
wherein
Figure BDA0002133784110000122
For three-dimensional column vectors, controller u ═ dk. It can be seen that the difference between equation (2) and the simplified equation for the three-phase dynamics equation is only that one is a one-dimensional scalar and one is a three-dimensional column vector, which is the same as the controller for the single-phase active power filter if the controller is designed for each phase of the three-phase mathematical model. Since the design method is the same, only the single-phase model will be explained below.
And step two, defining the tracking error and the first derivative thereof, defining a dynamic terminal sliding mode surface to obtain an equivalent control item, then defining a switching control item, and adding the equivalent control item and the switching control item to obtain the terminal sliding mode controller. The method comprises the following specific steps:
s21, defining the tracking error as e ═ x-r, the first derivative of the tracking error as
Figure BDA0002133784110000123
The second derivative of the tracking error is
Figure BDA0002133784110000124
Then the error vector is
Figure BDA0002133784110000125
Wherein r is a reference current signal;
s22, defining a terminal sliding mode surface as
Figure BDA0002133784110000126
c is a tunable coefficient to ensure the stability of the system Herwitz, where P (t) is a terminal function with respect to time t and
Figure BDA0002133784110000127
the dynamic terminal sliding mode surface is defined as:
Figure BDA0002133784110000128
wherein λ is a strictly normal number;
the first derivative of the sliding mode surface of the dynamic terminal is as follows:
Figure BDA0002133784110000131
to achieve global robustness, take e (0) to p (0),
Figure BDA0002133784110000132
in order to realize convergence at a specified time T, when T ≧ T, p (T) 0 is satisfied,
Figure BDA0002133784110000133
we can define the terminal function as:
Figure BDA0002133784110000134
where T is a self-defined terminal time, e (0),
Figure BDA0002133784110000135
Respectively representing the tracking error and the initial values of its first and second derivatives at t ═ 0s, a00、a01、a02、a10、a11、a12、a20、 a21、a22An available parameter for satisfying the above-mentioned assumption;
s23, neglecting the influence of unknown disturbance of the system, and making
Figure BDA0002133784110000136
Available equivalent control terms:
Figure BDA0002133784110000137
wherein
Figure BDA0002133784110000138
S24, because the real system has unknown disturbance, under the condition of considering the unknown disturbance, in order to ensure the system stability, the switching control item is needed to be used
Figure BDA0002133784110000139
To eliminate the influence of disturbances, KwMore than 0 is any adjustable parameter for ensuring the Lyapunov function to be semi-positive, and the finally designed dynamic terminal sliding mode controller
Figure BDA00021337841100001310
The following steps are changed:
Figure BDA0002133784110000141
wherein
Figure BDA0002133784110000142
KfAnd < 0 is any adjustable coefficient for ensuring that the Lyapunov function is semi-positive.
Defining the Lyapunov function:
Figure BDA0002133784110000143
the first derivative of the formula (9) is obtained, and the first derivative of the sliding mode surface of the dynamic terminal in the formula (4) and the control law in the formula (8) are substituted into the first derivative of the Lyapunov function to obtain:
Figure BDA0002133784110000144
due to d (t) and
Figure BDA0002133784110000145
bounded, define
Figure BDA0002133784110000146
Therefore, as long as K is guaranteedwRho > D, it can be proved
Figure BDA0002133784110000147
This shows that the dynamic terminal sliding mode method designed by the invention is stable. On the other hand due to
Figure BDA0002133784110000148
This means the Lyapunov function
Figure BDA0002133784110000149
Will gradually decrease, finally V1(t) → 0, which corresponds to σ (t) → 0.
Considering the terminal sliding-mode surface as s ═ C (e (t) -p (t)), where C ═ C1, we define η (t) ═ e (t) -p (t), where e (t) is the error vector,
Figure BDA00021337841100001410
since C does not contain 0 element, take
Figure BDA00021337841100001411
Then
Figure BDA00021337841100001412
Will also tend towards 0.
Step three, because the dynamic terminal sliding mode control law designed by the method needs to be based on an accurate mathematical model, but is difficult to realize in practice, in order to simplify the design of the dynamic terminal control law, the equivalent control item is approximated by using a double-hidden-layer recurrent neural network, and the method specifically comprises the following steps:
s31, defining the structure of the double hidden layer recurrent neural network as follows: an input layer, a first hidden layer, a second hidden layer and an output layer, and the result of the output layer is fed back to the input layer,
output theta of ith node of input layeriCan be expressed as a number of times,
θi＝xi·Wri·exY，i＝1,2,...,m (12)
wherein xiIs the ith input of the double hidden layer recurrent neural network, exY is the output value of the neural network at the last moment, WriFor the feedback weight of the ith input layer node, the feedback weight vector is defined as Wr ═ Wr1 Wr2…Wri]；
The jth node of the first hidden layer outputs a result phi1jIn order to realize the purpose,
Figure BDA0002133784110000151
wherein the first hidden layer output vector is phi1＝[φ11 φ12...φ1j]And phi is1jRepresenting the output of the jth node of the first hidden layer, the Gaussian of the first hidden layerThe central vector of the function is C1＝[c11,c12,…,c1n]T∈Rn×1The vector of the base width of the Gaussian function is B1＝[b11，b12,…,b1n]T∈Rn×1And c is and c1nIs the nth node center vector of the first hidden layer, and b1nIs the nth node center vector, R, of the first hidden layern×1A vector representing n rows and 1 columns in the real number domain;
the kth node of the second hidden layer outputs a result phi2kIn order to realize the purpose,
Figure BDA0002133784110000152
wherein the second hidden layer output vector is phi2＝[φ21 φ22…φ2k]And phi is2kRepresenting the output of the kth node of the second hidden layer, the central vector of the Gaussian function of the second hidden layer is C2＝[c21 c22...c2l]T∈Rl×1The vector of the base width of the Gaussian function is B2＝[b21 b22...b2l]T∈Rl×1And c is and c2lIs the l-th node-center vector of the second hidden layer, b2lIs the l-th node center vector, R, of the second hidden layerl×1A vector representing l rows and 1 columns in the real number domain;
combining the analysis, the output result of the double hidden layer recurrent neural network is as follows:
Y＝WTΦ2＝W1φ21+W2φ22+...+Wlφ2l (15)
wherein W ═ W1 W2…Wl]Is the output weight vector, W, of the double hidden layer recurrent neural networklRepresenting a weight vector between the l-th node of the second hidden layer and the output value;
presence of optimal parameters
Figure BDA0002133784110000168
So that
Figure BDA0002133784110000169
Wherein epsilon is the optimal approximation error;
s32, replacing the equivalent control item in the formula (6) with the output of the double hidden layer recurrent neural network, namely
Figure BDA0002133784110000161
Then the self-adaptive dynamic terminal sliding mode controller based on the double hidden layer recurrent neural network becomes:
Figure BDA0002133784110000162
s33, the approximation error of the double hidden layer recurrent neural network is defined as:
Figure BDA0002133784110000163
wherein the content of the first and second substances,
Figure BDA0002133784110000164
to obtain the law of adaptation
Figure BDA0002133784110000165
Is aligned with
Figure BDA00021337841100001610
Taylor expansion is carried out to obtain:
Figure BDA0002133784110000166
Figure BDA0002133784110000167
wherein the upper mark indicates the optimal parameter of the corresponding variable, and the upper mark A indicates the correspondenceEstimated value of a parameter, superscript-representing the estimation error of the corresponding parameter, OhFor the higher order terms of the taylor expansion,
approximation error of each parameter
Figure BDA0002133784110000171
Defining a new lyapunov function as:
Figure BDA0002133784110000172
the first derivative of the Lyapunov function is obtained:
Figure BDA0002133784110000173
wherein eta1,η2,η3,η4,η5,η6Is an adjustable normal number of the input signals,
in order to stabilize the system, the ideal control law in the formula (8) is added and subtracted to the first derivative of the Lyapunov function in the formula (20) at the same time, and the approximation error in the formula (17) is substituted into the formula (20) to obtain the ideal control law,
Figure BDA0002133784110000174
to ensure
Figure BDA0002133784110000181
The following adaptive laws are selected:
Figure BDA0002133784110000182
Figure BDA0002133784110000183
Figure BDA0002133784110000184
Figure BDA0002133784110000185
Figure BDA0002133784110000186
Figure BDA0002133784110000187
wherein
Figure BDA0002133784110000188
Respectively the first derivative, eta, of the approximation error of the parameters of the weight, the feedback gain, the center of the first hidden layer, the base width of the first hidden layer, the center of the second hidden layer and the base width of the second hidden layer of the double hidden layer recurrent neural network1,η2,η3,η4,η5,η6Is an adjustable normal number of the input signals,
Figure BDA00021337841100001812
Figure BDA00021337841100001813
output phi representing the second hidden layer2Respectively to the parameters
Figure BDA0002133784110000189
Figure BDA00021337841100001810
The derivative of (c). Can finally prove
Figure BDA00021337841100001811
This shows that the terminal sliding mode controller of the double hidden layer recurrent neural network is stable and feasible.
And then, simulating the controller on the single-phase active power filter model by using MATLAB, wherein simulation parameters are selected as follows:
the voltage of the power grid is Us24V, f 50 Hz; resistance R of nonlinear steady-state load1＝5Ω，R215 omega, a capacitance C of 1000uF, and a resistance of the dynamic nonlinear load R1＝15Ω，R215 Ω, and 1000 uF. The main circuit parameters of the active power filter comprise that the line inductance is 0.018H, and the resistance is 1 omega; since the dc side voltage is controlled by a conventional PI control method, the dc side voltage is assumed to be a constant value of 50v, and the experimental result graphs are shown in fig. 5, 6, 7, and 8.
In the simulation process, 0.05s is inserted into the active power filter, namely the active power filter starts to work to perform harmonic current compensation at the moment, and a dynamic nonlinear load is connected in 0.3 s. As can be seen from fig. 5, the grid current distortion rate is severe at the beginning, and the quality and safety of the grid may be compromised if no harmonic compensation is performed. When the active power filter is connected to the power grid within 0.05s, the power grid current can be changed into a sine wave within a short time. Even if the load changes, the grid current can change into a sinusoidal waveform in a short time. Fig. 6 is a harmonic current tracking curve of the present invention, after 0.05s, the harmonic compensation current can track the harmonic reference current in a short time, and after 0.3s nonlinear load change, the harmonic compensation current can also track quickly, which illustrates that the dynamic performance of the present invention is better. Fig. 7 is a tracking error map of the present invention, and it can be seen that the range of variation of the tracking error is small. Fig. 8 is a diagram of the distortion rate of the current in the power grid after the active power filter starts to work, and it can be seen that the distortion rate is 3.11% at this time, so as to fully reach 5% of the international standard requirement. The dynamic terminal self-adaptive sliding mode control method based on the double hidden layers of the recurrent neural network has good compensation effect and robustness.
The above description is only of the preferred embodiments of the present invention, and it should be noted that: it will be apparent to those skilled in the art that various modifications and adaptations can be made without departing from the principles of the invention and these are intended to be within the scope of the invention.

Claims (3)

1. An active power filter self-adaptive dynamic terminal sliding mode control method is characterized by comprising the following steps: the method comprises the following steps:
s1, establishing a mathematical model of the single-phase active power filter, defining a state variable i as x, and obtaining a second derivative of x
Figure FDA0003504507220000011
The expression of (2) is that the mathematical model of the single-phase active power filter specifically comprises:
Figure FDA0003504507220000012
where i is the state variable, here denoted the compensating harmonic current, L is the total inductance of the line on the AC side, R is the total resistance of the line on the AC side, UdcIs a DC side voltage, UsIs the grid voltage, H is a defined switching function, for convenience, x-i,
Figure FDA0003504507220000013
Figure FDA0003504507220000014
if the controller is defined as u ═ H, then equation (1) can be abbreviated as:
Figure FDA0003504507220000015
where d (t) is the system extra perturbation considered and d (t) is bounded;
s2, defining the tracking error and the first derivative thereof, defining a dynamic terminal sliding mode surface to obtain an equivalent control item, then defining a switching control item, and adding the equivalent control item and the switching control item to obtain the terminal sliding mode controller, which comprises the following specific steps:
s21, defining traceThe error is e ═ x-r, the first derivative of the tracking error is
Figure FDA0003504507220000016
The second derivative of the tracking error is
Figure FDA0003504507220000017
Then the error vector is
Figure FDA0003504507220000018
Wherein r is a reference current signal;
s22, defining a terminal sliding mode surface as
Figure FDA0003504507220000019
c is a tunable coefficient to ensure the stability of the system Herwitz, where P (t) is a terminal function with respect to time t and
Figure FDA00035045072200000110
the dynamic terminal sliding mode surface is defined as:
Figure FDA0003504507220000021
wherein λ is a normal number;
the first derivative of the sliding mode surface of the dynamic terminal is as follows:
Figure FDA0003504507220000022
to achieve global robustness, take e (0) to p (0),
Figure FDA0003504507220000023
in order to realize convergence at a specified time T, when T ≧ T, p (T) 0 is satisfied,
Figure FDA0003504507220000024
defining the terminal function as:
Figure FDA0003504507220000025
where T is a self-defined terminal time, e (0),
Figure FDA0003504507220000026
Respectively representing the tracking error and the initial values of its first and second derivatives at t ═ 0s, a00、a01、a02、a10、a11、a12、a20、a21、a22An available parameter for satisfying the above-mentioned assumption;
s23, order
Figure FDA0003504507220000027
Available equivalent control terms:
Figure FDA0003504507220000028
wherein
Figure FDA0003504507220000029
S24, taking external interference into consideration, using switching control item
Figure FDA00035045072200000210
KwMore than 0 is any adjustable parameter for ensuring the Lyapunov function to be semi-positive, and the finally designed dynamic terminal sliding mode controller
Figure FDA0003504507220000031
The following steps are changed:
Figure FDA0003504507220000032
wherein
Figure FDA0003504507220000033
Is any adjustable coefficient which ensures that the Lyapunov function is semi-positive;
s3, approximating the equivalent control item by using a double-hidden-layer recurrent neural network, which specifically comprises the following steps:
s31, defining the structure of the double hidden layer recurrent neural network as follows: an input layer, a first hidden layer, a second hidden layer and an output layer, and the result of the output layer is fed back to the input layer,
output theta of ith node of input layeriCan be expressed as a number of times,
θi＝xi·Wri·exY，i＝1,2,...,m (12)
wherein xiIs the ith input of the double hidden layer recurrent neural network, exY is the output value of the neural network at the last moment, WriThe feedback weight vector is defined as W for the feedback weight of the ith input layer noder＝[Wr1 Wr2...Wri]；
The jth node of the first hidden layer outputs a result phi1jIn order to realize the purpose,
Figure FDA0003504507220000034
wherein the first hidden layer output vector is phi1＝[φ11 φ12...φ1j]And phi is1jRepresenting the output of the jth node of the first hidden layer, the center vector of the Gaussian function of the first hidden layer is C1＝[c11,c12,...,c1n]T∈Rn×1The vector of the base width of the Gaussian function is B1＝[b11，b12,…,b1n]T∈Rn×1And c is and c1nIs the nth node center vector of the first hidden layer, and b1nIs the nth node center vector, R, of the first hidden layern×1A vector representing n rows and 1 columns in the real number domain;
the kth node of the second hidden layer outputs a result phi2kIn order to realize the purpose,
Figure FDA0003504507220000041
wherein the second hidden layer output vector is phi2＝[φ21 φ22...φ2k]And phi is2kRepresenting the output of the kth node of the second hidden layer, the central vector of the Gaussian function of the second hidden layer is C2＝[c21 c22...c2l]T∈Rl×1The vector of the base width of the Gaussian function is B2＝[b21 b22...b2l]T∈Rl×1And c is and c2lIs the l-th node-center vector of the second hidden layer, b2lIs the l-th node center vector, R, of the second hidden layerl×1A vector representing l rows and 1 columns in the real number domain;
combining the analysis, the output result of the double hidden layer recurrent neural network is as follows:
Y＝WTΦ2＝W1φ21+W2φ22+...+Wlφ2l (15)
wherein W ═ W1 W2...Wl]Is the output weight vector, W, of the double hidden layer recurrent neural networklRepresenting a weight vector between the l-th node of the second hidden layer and the output value;
presence of optimal parameters
Figure FDA0003504507220000042
W*So that
Figure FDA0003504507220000043
Wherein epsilon is the optimal approximation error;
s32, replacing the equivalent control item in the formula (6) with the output of the double hidden layer recurrent neural network, namely
Figure FDA0003504507220000044
Then the self-adaptive dynamic terminal sliding mode controller based on the double hidden layer recurrent neural network becomes:
Figure FDA0003504507220000045
s33, the approximation error of the double hidden layer recurrent neural network is defined as:
Figure FDA0003504507220000046
wherein the content of the first and second substances,
Figure FDA0003504507220000051
to obtain the law of adaptation
Figure FDA0003504507220000052
Is aligned with
Figure FDA0003504507220000053
Taylor expansion is carried out to obtain:
Figure FDA0003504507220000054
Figure FDA0003504507220000055
wherein the superscript denotes the optimal parameter of the corresponding variable, the superscript Λ denotes the estimated value of the corresponding parameter, the superscript denotes the estimated error of the corresponding parameter, and OhFor the higher order terms of the taylor expansion,
approximation error of each parameter
Figure FDA0003504507220000056
2. The adaptive dynamic terminal sliding-mode control method of the active power filter according to claim 1, characterized in that: defining the Lyapunov function:
Figure FDA0003504507220000057
solving a first derivative of the formula (9), and substituting the first derivative of the sliding mode surface of the dynamic terminal in the formula (4) and the control law in the formula (8) into the first derivative of the Lyapunov function to obtain:
Figure FDA0003504507220000058
due to d (t) and
Figure FDA0003504507220000059
bounded, define
Figure FDA00035045072200000510
Therefore, as long as K is guaranteedwRho > D, it can be proved
Figure FDA00035045072200000511
Considering a terminal sliding mode surface as s ═ C (e (t) -p (t)), where C ═ C1 is an adjustable vector that ensures system stability, we define η (t) ═ e (t) -p (t), where e (t) is an error vector,
Figure FDA0003504507220000061
since C does not contain 0 element, take
Figure FDA0003504507220000062
Then e (t) will also go to 0.
3. The adaptive dynamic terminal sliding-mode control method of the active power filter according to claim 1, characterized in that: defining a new lyapunov function as:
Figure FDA0003504507220000063
the first derivative of the Lyapunov function is obtained:
Figure FDA0003504507220000064
wherein eta1,η2,η3,η4,η5,η6Is an adjustable normal number of the input signals,
in order to stabilize the system, the ideal control law in the formula (8) is added and subtracted to the first derivative of the Lyapunov function in the formula (20) at the same time, and the approximation error in the formula (17) is substituted into the formula (20) to obtain the ideal control law,
Figure FDA0003504507220000065
to ensure
Figure FDA0003504507220000071
The following adaptive laws are selected:
Figure FDA0003504507220000072
Figure FDA0003504507220000073
Figure FDA0003504507220000074
Figure FDA0003504507220000075
Figure FDA0003504507220000076
Figure FDA0003504507220000077
wherein
Figure FDA0003504507220000078
Respectively the first derivative, eta, of the approximation error of the parameters of the weight, the feedback gain, the center of the first hidden layer, the base width of the first hidden layer, the center of the second hidden layer and the base width of the second hidden layer of the double hidden layer recurrent neural network1,η2,η3,η4,η5,η6Is an adjustable normal number of the input signals,
Figure FDA0003504507220000079
Figure FDA00035045072200000710
output phi representing the second hidden layer2Respectively to the parameters
Figure FDA00035045072200000711
Figure FDA00035045072200000712
The derivative of (c).
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