CN115033030A - Dynamic surface sliding mode control method for heliostat of tower-type photo-thermal power station - Google Patents
Dynamic surface sliding mode control method for heliostat of tower-type photo-thermal power station Download PDFInfo
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Abstract
The invention relates to a dynamic surface sliding mode control method of a heliostat of a tower type photo-thermal power station, which comprises the following steps: establishing an altitude angle motor and azimuth angle motor system model based on the heliostat of the azimuth pitching double-shaft tracking, and performing magnetic hysteresis elimination on the system model to obtain a system model for eliminating magnetic hysteresis; adopting a neural network to approach an unknown function in a system model for eliminating magnetic hysteresis, and designing an error conversion function and a tracking performance index function to enable tracking errors of the altitude motor and azimuth motor system model to meet preset performance indexes; constraining the state of a system model for eliminating magnetic hysteresis through an asymmetric obstacle Lyapunov function; and designing a self-adaptive controller of the elevation angle motor and azimuth angle motor system model by combining a dynamic surface controller design method and a sliding mode surface controller design method, and realizing the sliding mode control of the dynamic surface of the heliostat. The invention adopts a full-state constraint robust self-adaptive dynamic surface control method, and ensures the transient performance and stability of the system.
Description
Technical Field
The invention relates to the technical field of new energy, in particular to a dynamic surface sliding mode control method for a heliostat of a tower type photo-thermal power station.
Background
With the rapid development of society, the demand of human beings on energy is rapidly increased, fossil energy is becoming exhausted day by day, and the market of solar heating power generation is finally increased for a long time. The control of the photo-thermal power station is roughly divided into three parts, namely a light-gathering and distributing system, a heat storage and exchange system and a conventional power generation system, wherein the light-gathering and heat collection system is the key for the success of the whole power plant, and the cost of the light-gathering and heat collection system accounts for about 50% of the total investment of the whole tower-type photo-thermal power station. The tower type light heliostat is mainly used for tracking the sun, gathering and reflecting sunlight to enter a heat absorber.
Research on heliostats continues to produce results. In 1998, Hession et al successfully developed a biaxial tracker that could track the sun in real time by using Fresnel lens to increase the light receiving rate of solar panels. In 2004, Antofagasta university in chile developed a two-axis solar tracking system that combines a view-day tracking mode and a photoelectric tracking mode to select a tracking method based on the intensity of light. The control strategy is designed aiming at the tooth clearance nonlinear link in the heliostat driving system by the university of sienna in 2018, and the expected aim of eliminating the tooth clearance can be achieved by the designed double-motor clearance elimination system verified through experiments. In recent years, with the development of intelligent control, intelligent control is applied in various fields. An adaptive fuzzy DSC design is proposed for the first time in 2012, and an output constraint problem is solved by combining an obstacle Lyapunov function. For the DSC framework of output-constrained systems, Wu et al propose a tangential BLF-based DSC strategy for time-lapse systems with output constraints.
At present, the performance index problems such as robustness, sun tracking accuracy and the like of a heliostat control system are rarely considered. The invention designs a robust self-adaptive controller of a heliostat attitude angle based on an azimuth pitching double-shaft heliostat, considering all-state constraint and a double-motor backlash elimination system.
Disclosure of Invention
The invention aims to provide a dynamic surface sliding mode control method of a tower type photothermal power station heliostat, which is used for solving the problems in the prior art.
In order to achieve the purpose, the invention provides the following scheme:
a dynamic surface sliding mode control method of a heliostat of a tower type photo-thermal power station comprises the following steps:
establishing an altitude motor and azimuth motor system model based on the heliostat of the azimuth pitching dual-axis tracking, and performing magnetic hysteresis elimination on the altitude motor and azimuth motor system model to obtain a system model for eliminating magnetic hysteresis;
adopting a neural network to approximate unknown parameters and uncertain items existing in a system model, and designing an error conversion function and a tracking performance index function to enable tracking errors of the altitude motor and azimuth motor system model to meet preset performance indexes;
constraining the state of the system model for eliminating the magnetic hysteresis through an asymmetric obstacle Lyapunov function; and designing an adaptive controller of the altitude angle motor and azimuth angle motor system model by combining a dynamic surface controller design method and a sliding mode surface controller design method, so as to realize the sliding mode control of the dynamic surface of the heliostat.
Preferably, the heliostat based on azimuth pitch biaxial tracking establishes an altitude motor and azimuth motor system model as the following formula (1):
wherein i is 1, 2; theta ir Is the rotor speed; v. of ir Is the stator voltage; i.e. i iq Is the stator current; omega ir Is the rotor angular velocity; j. the design is a square i Is the rotor inertia; t is iL Is the load torque; f. of i Is a viscous friction coefficient;
wherein n is the number of pole pairs, L m Is mutual inductance, L 2 Is rotor inductance, R 1 Is stator resistance, L 1 Is stator inductance, R 2 Is rotor resistance, L 1q For the equivalent primary inductance in the synchronously rotating coordinate system, R 1q Is an equivalent primary resistance in a synchronous rotating coordinate system; alpha is alpha 1 The effective coefficient of the back electromotive force EMF in the lower model is excited for the rated magnetization; a is 2 Is a constant.
Preferably, performing hysteresis elimination on the altitude and azimuth motor system model comprises:
in the altitude angle motor and azimuth angle motor system models, state equations of the altitude angle motor and the azimuth angle motor are respectively defined to respectively obtain a state equation of an azimuth angle motor servo system and a state equation of an altitude angle motor servo system, and backlash in the servo systems is eliminated through a PI inverse model.
Preferably, the state equation of the azimuth motor servo system is:
wherein, g 1 ,θ 1 ,β 1 Is an unknown parameter of the system; delta of 1 (x 11 T) is the system uncertainty part; y is 1 Is the output of the system, p is the hysteresis operator; w is a 1 E R represents the output of the controller represented by the PI hysteresis model; a is a 1 ,a 2 Is a constant;
x 11 to represent the state variable of the rotor angle, theta 1r Is the rotor angle, x 12 Obtaining state variables, ω, to represent rotor angular velocity 1r As angular speed of the rotor, x 13 Is a state variable of the stator current, i 1q Is the stator current.
Preferably, the state equation of the altitude motor servo system is as follows:
wherein, g 2 ,θ 2 ,β 2 Unknown parameters of the system; delta of 2 (x 21 T) is the system uncertainty part; y is 2 Is the output of the system; a is 1 ,a 2 Is a constant;
K T coefficient of stator current front, J 2 Is the inertia of the rotor, g 2 As an unknown parameter of the system, f 2 Is a viscous coefficient of friction, θ 2 Is the rotor angle, T L2 Is the load torque, x 21 Is a rotor angle state variable, t is time, v 2q Is the stator voltage u 2 Is a control signal.
Preferably, the approximating an unknown function in the system model for eliminating magnetic hysteresis by using a neural network includes:
approximating by an RBF neural network, wherein the RBF neural network is:
wherein for any given positive integer q ≧ 1, ξ i ∈Ω ξi ∈R q Is an input vector; epsilon i (ξ i ) Satisfy | ε for network reconstruction errors i (ξ i )|≤ε im And epsilon im In order to be an unknown constant, the method,
n is an ideal weight value which is large enough; n is the number of nodes of the neural network, N is more than 1, psi i (ξ i )＝[ρ 1 (ξ i ),...,ρ N (ξ i )]∈R N Is a weight vector rho j (ξ i ) Is a gaussian base function of the form (8) below:
η j > 0 represents the width of the basis function; xi j For the jth basis function ρ j (ξ i ) Of the center of (c).
Preferably, the constraining the states of the altitude and azimuth motor system models by the asymmetric obstacle lyapunov function includes:
obtaining the asymmetric obstacle Lyapunov function:
wherein k is a And k b Two constants, upper and lower bounds for the constraint;
if satisfy-k b ＜S(0)＜k a The function q(s) is then of the form:
from equation (9), equation (10) yields:
for arbitrary constant k a ,k b And a variable S, when S ∈ (-k) b ,k a ) Then, the following formula (12) stands:
wherein q (S) is a function of S, S 2 Which is the square of the variable S and,
is a mathematical expression.
Preferably, the method for designing the combined dynamic surface controller and the sliding mode surface controller comprises the following steps: respectively designing a first dynamic surface, a second dynamic surface and a first sliding film dynamic surface based on an azimuth controller;
the first dynamic surface is:
wherein S is 11 Is a new error variable, phi -1 As a smoothly decreasing function, e 11 (t) is a variable of the error,
is a performance function;
the second dynamic surface is:
S 12 ＝x 12 -z 12 (14)
wherein S is 12 As error variable, x 12 As a state variable of angular velocity of the rotor, z 12 Is a virtual control law;
the first sliding mode dynamic surface is as follows:
S 1m ＝m 11 S 11 +m 12 S 12 +S 13 (15)
wherein m is 11 And m 12 For positive design parameters, S 11 ,S 12 ,S 13 As error variable, S 1m Is a slip form surface.
Preferably, the method for designing a combined dynamic surface controller and the method for designing a sliding mode surface controller further includes: respectively designing a third dynamic surface, a fourth dynamic surface and a fifth dynamic surface based on the height angle controller:
the third dynamic surface is:
wherein S is 21 Is a new error variable, phi -1 Is a smooth and strictly monotonically increasing inverse function, e 21 (t) is a variable of the error,
is a performance function; x is the number of 22d To a virtual control law, k 21 In order to have a positive design parameter,
as reference signal, Ψ 21 In order to introduce the intermediate variable(s),
Γ 21 is a mathematical expression in formula (12), m 21 For positive design parameters, τ 22 Is a filter time constant, z 22 For the filtered virtual control law, x 22d To a virtual control law, z 22 (0) Is the initial value of the filtered virtual control law, x 22d (0) The initial value of the virtual control law is obtained;
the fourth dynamic surface is:
S 22 ＝x 22 -z 22 (20)
wherein S is 22 As error variable, x 22 As a state variable of angular velocity of the rotor, z 22 For the third dynamic surface-filtered virtual control law, x 23d In order to be a virtual control law,
in order to be a positive design parameter,
in order to be a law of adaptation,
transposing weight vectors of neural network by psi 22 As weight vectors of the neural network, gamma 22 For a positive design parameter, σ 22 For positive design parameters, τ 23 Is a filter time constant, z 23 Is the virtual control rate after the filter, x 23d For the virtual control law, z 23 (0) Is an initial value, x, of the virtual control rate after passing through the filter 23d (0) Is the initial value of the virtual control rate;
the fifth dynamic surface is:
S 23 ＝x 23 -z 23 (24)
S 2m ＝m 21 S 21 +m 22 S 22 +S 23 (25)
wherein S is 23 As error variable, x 23 Is a stator current state variable, z 23 For the virtual control rate, S, after the fourth step of filtering 2m Being slip-form faces u 2 For the control signal, k 23 For positive design parameters, S 2m Is a slip form surface and is provided with a plurality of slip forms,
in order to be able to adapt the rate,
transposing weight vectors for neural networks, psi 23 As weight vectors of the neural network, gamma 23 For a positive design parameter, σ 23 Are positive design parameters.
The beneficial effects of the invention are as follows:
(1) the invention overcomes the problem of differential explosion in the inversion control method by introducing a first-order low-pass filter, so that the control law is simpler; the hysteresis problem of the azimuth motor caused by the free backlash is considered, and the control precision and stability of the system are improved;
(2) the RBF neural network is adopted to approximate the functions with unknown and uncertain parameters in the system model, and the weighted vector norm of the RBF neural network is estimated, so that the calculation burden is greatly reduced;
(3) the invention adopts a sliding mode control method, improves the robustness of the system, accelerates the response speed, and finally ensures that all signals of the closed-loop system are consistent in a semi-global manner and are bounded finally;
(4) according to the invention, the asymmetric obstacle Lyapunov function and the tracking error performance function are introduced to constrain the error range of system state and attitude tracking, so that the system full-state constraint performance and attitude tracking precision are ensured.
Drawings
In order to more clearly describe the embodiments of the present invention or the technical solutions in the prior art, the drawings required in the embodiments will be briefly described below, it is obvious that the drawings described below are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the drawings without inventive labor.
FIG. 1 is a block diagram of a control scheme according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of an estimated inverse compensation scheme according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of a PI hysteresis model according to an embodiment of the present invention;
FIG. 4 is a diagram illustrating a PI hysteresis inverse model according to an embodiment of the present invention;
FIG. 5 is a diagram illustrating compensation results according to an embodiment of the present invention;
FIG. 6 is a schematic diagram of an azimuth tracking error according to an embodiment of the present invention;
FIG. 7 is a schematic view of an altitude tracking error according to an embodiment of the present invention;
FIG. 8 is a schematic diagram of the azimuth tracking performance of an embodiment of the present invention;
FIG. 9 is a schematic view of altitude tracking performance of an embodiment of the present invention;
FIG. 10 is a schematic diagram of an azimuth control signal according to an embodiment of the present invention;
FIG. 11 is a schematic diagram of altitude control signals according to an embodiment of the present invention;
FIG. 12 is a schematic diagram illustrating the comparison of the azimuth control signal without hysteresis compensation according to the embodiment of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be described clearly and completely with reference to the accompanying drawings in the embodiments of the present invention, and it is to be understood that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
In order to make the aforementioned objects, features and advantages of the present invention comprehensible, the present invention is described in detail with reference to the accompanying drawings and the detailed description thereof.
As shown in fig. 1 to 11, the present invention provides a dynamic surface sliding mode control method for heliostats of a tower-type photothermal power station, comprising the following steps:
(1) modeling aiming at a heliostat attitude servo system of a tower type photo-thermal power station with input hysteresis;
the invention relates to a self-adaptive controller for designing an attitude angle of a research object by using an azimuth pitching double-axis tracking heliostat. The attitude of the heliostat is controlled by an altitude angle motor and an azimuth angle motor together, 2 servo motors are the same and independent from each other, and a servo system model is as follows:
wherein i is 1, 2; theta.theta. ir Is the rotor speed; v. of ir Is the stator voltage; i.e. i iq Is the stator current; omega ir Is the rotor angular velocity; j. the design is a square i Is the rotor inertia; t is iL Is the load torque; f. of i Is a viscous friction coefficient; a is a 1 ,a 2 ,K T Is a constant.
Wherein:
wherein n is the number of pole pairs, L m Is mutual inductance, L 1q Naming constants for mathematical expressions, L 2 Is rotor inductance, R 1q Naming constants for mathematical expressions, R 1 Is stator resistance, L 1 Is stator inductance, R 2 Is the rotor resistance.
In the pitching direction, the heliostat mirror surface has no backlash naturally between the rotations of the driving mechanism due to the existence of gravity, and only in the rotating direction, the nonlinear error exists between the input and the output of the driving device due to the existence of free backlash. Affecting the control accuracy of the control system. The elevation motor and the azimuth motor will be discussed separately.
The azimuth motor is converted as follows:
wherein x is 11 Is a state variable, θ 1r Is the rotor angle, x 12 Is a state variable, ω 1r As angular speed of the rotor, x 13 Is a state variable, i 1q Is the stator current.
The following equation of state of the servo system can be obtained:
wherein g is 1 ,θ 1 ,β 1 Is an unknown parameter of the system; delta 1 (x 11 T) is the system uncertainty part; y is 1 Is the output of the system, and p is the hysteresis operator, a 1 ,a 2 ,K T Is a constant value, w 1 E R represents the control signal represented by the Prandtla-Ishlinskii (PI) hysteresis model.
K T Coefficient of stator current front, J 1 Is the inertia of the rotor, g 1 As an unknown parameter of the system, f 1 Is a viscous coefficient of friction, θ 1 Is the rotor angle, T L1 Is the load torque, x 11 Is a state variable of rotor angle, t is a time constant, v 1q Is the stator voltage u 1 Is a control signal.
The altitude angle motor is converted as follows:
wherein x is 21 Is a state variable, θ 2r Is the rotor angle, x 22 As state variables, ω 2r Is the angular speed, x, of the rotor 23 Is a state variable, i 2q Is the stator current.
The following equation of state of the servo system can be obtained:
wherein g is 2 ,θ 2 ,β 2 Is an unknown parameter of the system; delta 2 (x 21 T) is the system uncertainty part; y is 2 Is the output of the system; u. u 2 Is a control signal; a is 1 ,a 2 Is a constant. And is
Wherein, K T Coefficient of stator current front, J 2 Is the inertia of the rotor, g 2 As an unknown parameter of the system, f 2 Is a viscous coefficient of friction, θ 2 Is the rotor angle, T L2 As load torque, x 21 Is a state variable of rotor angle, t is a time constant, v 2q Is the stator voltage u 2 Is a control signal.
Hypothesis 1.g ij ( i 1,2, …, n, j 2,3,4) is an unknown bounded parameter with a constant g present max ＞g min > 0, such that g max ＞g ij ＞g min ＞0。
(2) A PI hysteresis model and an inverse model thereof;
w in equation of state (4) 1 (u 1 ) Can be expressed as:
wherein p is 1 (r) is a density function, 0 < p 1 (r)＜p max ，
Is a constant determined by the density function p (r), and Λ is the upper limit of the integral, generally chosen to be a large positive number, u 1 Is the output of the controller, F r [u]Is the Play operator.
Definition f r :R→R
f r (u 1 ,w 1 )＝max(w 1 -r,min(w 1 +r,w 1 )) (10)
The Play operator satisfies:
considering hysteresis nonlinear compensation, the following PI inverse model is constructed as follows:
wherein:
in practice, hysteresis is usually unknown and, therefore, is usually used
Estimate w 1 (u 1 )＝p[u 1 ](t).
Wherein
Representing a composition operator; u. of 1,d Is an ideal input signal and is used as a reference,
and
represents p [. cndot](t) and p -1 [·](t) loading curve.
Wherein e 1,p (t) represents the error, which needs to be taken into account when designing the controller.
From formulae (14) and (15):
w 1 (t)＝φ′ 1 (Λ)u 1,d +d 1,b (t) (16)
φ 1 ' (Λ) is a normal number;
has the boundary of
|d 1,b (t)|≤D 1 (17)
D 1 Is a normal number, E r [·]Is the stop operator.
Substituting formula (16) for formula (4) to obtain:
and is
β 1,Λ ＝β 1 φ′ 1 (Λ) (19)
Wherein g is 1 ,θ 1 ,β 1 Is an unknown parameter of the system; delta 1 (x 11 T) is the system uncertainty part; y is 1 Is the output of the system, and p is the hysteresis operator, a 1 ,a 2 ,K T Is a constant number u 1,d Is an ideal control signal, d 1,b (t) is a bounded function. Phi is a 1 ' (Λ) is a normal number.
(3) Function approximation principle of RBF neural network:
in the present invention, a continuous unknown nonlinear function is approximated by an RBF neural network.
The general form of an RBF neural network can be expressed as:
where for any given positive integer q ≧ 1, ξ i ∈Ω ξi ∈R q Is an input vector; epsilon i (ξ i ) Satisfy | ε for network reconstruction errors i (ξ i )|≤ε im And epsilon im In order to be an unknown constant, the method,
n is an ideal weight value which is large enough; n is the number of nodes of the neural network, N is more than 1, psi i (ξ i )＝[ρ 1 (ξ i ),...,ρ N (ξ i )]∈R N As a weight vector, ρ j (ξ i ) In order to be a gaussian-based function,
the form is as follows:
η j > 0 represents the width of the basis function; xi j For the jth basis function ρ j (ξ i ) Of the center of (c).
(4) Designing an error conversion function and a tracking performance index function:
the output error of the system is
e i1 ＝x i1 -y ir (22)
Wherein y is ir Is a reference signal, x i1 Is a state variable of the rotor angle. Defining a smooth positive function with strictly monotonically decreasing magnitude
Satisfy the requirement of
Wherein, sigma is more than 0 and less than 1,
is the maximum value of the allowable error of the system, e i1 Is an error variable.
Then, the error transfer function is introduced:
S i1 denotes an equivalent error variable, phi (S) i1 ) Is a decreasing smooth function satisfies
Then the
S i1 When bounded, equation (25) is satisfied.
Is provided with
When S is i1 ∈L ∞ When satisfied, i.e.
e i1 (0)＜0， 
Tracking error satisfaction of system
Then:
(5) design of an asymmetric barrier Lyapunov function:
the expression is as follows:
wherein k is a And k b Are two constants, the upper and lower bounds of the constraint, respectively. Satisfies-k b ＜S(0)＜k a The function q(s) is of the form:
according to equation (30), equation (29) can be written as
Leading: for arbitrary constant k a ,k b And a variable S, when S ∈ (-k) b ,k a ) When the following equation is satisfied
Wherein q (S) is a function of S, S 2 Which is the square of the variable S and,
to relate to S 2 Of an exponential function of k a ,k b Is a constant.
(6) Self-adaptive controller for designing system by combining dynamic surface sliding mode surface
A. Design of azimuth controller:
designing the first dynamic surface
Wherein S is 11 Is a new error variable, phi -1 Is a smooth and strictly monotonically increasing inverse function, e 11 (t) is a variable of an error,
is a performance function;
to S 11 Is derived by
Order to
Wherein S is 11 Is a new error variable, phi -1 For a smooth and strictly monotonically increasing inverse function,
is a performance function; x is a radical of a fluorine atom 22d To a virtual control rate, Ψ 11 Is a mathematical expression generationThe numbers have no practical significance.
e 11 (t) is a variable of the error,
is the derivative of the reference signal, x 12d In order to control the rate of the virtual control,
as a function of performance, Ψ 11 The code number of the mathematical expression has no practical significance.
The barrier lyapunov function is:
wherein q (S) 11 ) To relate to S 11 Is a function of (a) a function of (b),
is a variable S 11 The square of (a) is calculated,
to relate to S 11 Of an exponential function of k a11 ,k b11 Is a constant.
To V 1 Is derived by
Wherein
And k b11 Is x 11 The maximum and minimum values of the deviation are,
as a function of performance, e 11 (t) is a variable of the error,
is the derivative of the reference signal, Ψ 11 Virtual control law x for no practical meaning of mathematical expression code 12d Is designed as
k 11 Is a positive parameter, m 11 Is a positive design parameter. x is the number of 12d Obtaining a new variable z by means of a filter 12
τ 12 Is the time constant of the low pass filtering. z is a radical of 12 For the virtual control rate filtered variable, z 12 (0) Is an initial value, x, of the virtual control rate filtered variable 12d For virtual control rate, x 12d (0) Is the initial value of the virtual control rate;
design the second dynamic surface:
S 12 ＝x 12 -z 12 (41)
wherein S is 12 As error variable, x 12 Is a state variable of angular velocity of the rotor, z 12 Is the virtual control rate after filtering;
to S 12 Derived by derivation
Wherein, the first and the second end of the pipe are connected with each other,
as derivatives of error variables, x 12 As a state variable of angular velocity of the rotor, x i3 Is an equation of state for the stator current,
is the derivative, x, of the filtered virtual control rate of the first dynamic surface 13d Is a virtual control law of the second dynamic plane, theta 1 ,g 1 For an unknown parameter, Δ 1 Is the system uncertainty part.
The barrier Lyapunov function is
Wherein k is a12 And k b12 Is a variable x 12 Maximum and minimum values of deviation. q (S) 12 ) To relate to S 12 As a function of (a) or (b),
is a variable S 12 The square of the square,
to relate to S 12 Is determined by the exponential function of (a),
is that
Is estimated.
To V 2 And (5) obtaining a derivative:
wherein, γ 12 Is a positive parameter of the number of bits,
is the rate of adaptation and is the rate of adaptation,
x 13d is a virtual control law of the second dynamic surface, because
Equation (44) is a smooth function of the unknown, and RBFNNs are used to estimate the unknown part.
Wherein
|ε 12 (ξ 12 )|≤ε 12m ，ψ 12 As a weight vector of the neural network,
and epsilon is the network reconstruction error.
N is an ideal weight value which is large enough; g 1 ,θ 1 For an unknown parameter, Δ 1 In order for the system to be uncertain,
and the derivative of the virtual control rate after filtering is the first dynamic surface.
Using the young inequality we can obtain:
wherein alpha is 12 Is a positive parameter of the number of bits,
ε 12m in order for the constant to be unknown,
n is a sufficiently large number, N, of ideal weights, # 12 As a weight vector, the weight vector is,
for transposing weight vectors, S 12 As a function of error, e 12 (ξ 12 ) The error is reconstructed for the neural network.
Substituting equations (45) - (46) into (44) yields:
wherein psi 12 As a weight vector of the neural network,
α 12 ,γ 12 in order to be a positive design parameter,
in order to be a law of adaptation,
ε 12m are unknown constants.
Selecting a virtual control rate x 13d
Wherein psi 12 Is a weight vector of the neural network,
m 11 ,k 12 ,α 12 in order to be a positive design parameter,
to the law of adaptation, S 12 Is an error variable.
Adaptive rate
The design is as follows:
wherein k is 12 ,γ 12 ,σ 12 Is a positive design parameter. x is a radical of a fluorine atom 13d Generating a new variable z by a low-pass filter 13 ,i.e。
Wherein τ is 13 Is the time constant of the low pass filter.
Design the third synovial dynamic surface:
S 1m ＝m 11 S 11 +m 12 S 12 +S 13 (51)
and is
S 13 ＝x 13 -z 13 (52)
Wherein m is 11 And m 12 Is a positive design parameter, x 13 Is a stator current state variable, z 13 And filtering the second dynamic surface to obtain the virtual control rate.
S 1m Obtaining a derivative:
wherein m is 11 And m 12 Is a positive design parameter, x 12 As a state-variable of the angular velocity of the rotor,
as a derivative of the reference signal, g 1 ,θ 1 ,β 1,Λ For unknown parameters, Δ 1 As an indeterminate part of the system, a 1 ,a 2 Is a constantNumber u 1,d For ideal control signals, d 1,b (t) is a bounded function of,
the filtered virtual control rate for the second dynamic surface.
The barrier Lyapunov function is designed as follows
Wherein k is a1m And k b1m Is x 13 Maximum and minimum values of deviation.
Is that
Is estimated. Gamma ray 13 Is a positive design parameter. Wherein q (S) 1m ) To relate to S 1m As a function of (a) or (b),
is a variable S 1m The square of the square,
to relate to S 1m Of an exponential function of k a ,k b Is a constant.
To V 3 Is derived by
Wherein, the first and the second end of the pipe are connected with each other,
g 1 ,θ 1 ,β 1,Λ for an unknown parameter, Δ 1 As an indeterminate part of the system, a 1 ,a 2 Is a constant number u 1,d As desired control signal, d 1,b (t) is a bounded function of,
the filtered virtual control rate for the second dynamic surface. m is 11 And m 12 Is a positive design parameter, x 12 As a state variable of angular velocity of the rotor, x 13 As a state-variable of the stator current,
is the derivative of the reference signal and is,
is that
Is estimated.
Approximating unknown terms using a neural network
Wherein the content of the first and second substances,
ψ 13 as a weight vector of the neural network,
n is a sufficiently large number, ε 13 (ξ 13 ) For network reconstruction errors, a 1 ,a 2 Is a constant number, y 1r As a reference signal, m 11 And m 12 Is a positive design parameter that is,
is that
Is estimated by the estimation of (a) a,
Δ 1 is the uncertain part of the system.
Using the young inequality one can obtain:
wherein alpha is 13 Is a positive design parameter, and |. epsilon 13 (ξ 13 )|≤ε 13m ,ε 13m Are unknown constants.
Substituting equations (56) - (57) into (55) has
Wherein the content of the first and second substances,
S 1m being slip-form faces u 1,d α is an ideal control signal 13 The parameters of the design are positive and the design parameters are negative,
is that
Is estimated. Psi 13 In order to be the weight vector,
is the transposition of weight vector, epsilon 13m Is a constant.
The final control signal is designed as
Wherein the content of the first and second substances,
S 1m being slip-form faces, a 13 ,k 13 ,k 1 The parameters of the design are positive and the design parameters are negative,
is that
Is estimated. Psi 13 As a weight vector, the weight vector is,
is a transposition of the weight vector, ε 13m Is a constant.
wherein, the first and the second end of the pipe are connected with each other,
S 13 as error variable, γ 13 ,σ 13 ,α 13 The parameters of the design are positive and the design parameters are negative,
is that
Is estimated. Psi 13 As a weight vector, the weight vector is,
transpose the weight vector.
B. Design of the height controller:
designing a first dynamic surface:
wherein S is 21 Is a new error variable, phi -1 Is a smooth and strictly monotonically increasing inverse function, e 21 (t) is a variable of the error,
is a performance function;
wherein:
wherein S is 21 Is a new error variable, phi -1 Is a smooth and strictly monotonically increasing inverse function, e 21 (t) is a variable of the error,
is a performance function; x is the number of 22d To virtually control the rate, k 21 In order to be a positive design parameter,
is the derivative of the reference signal, Ψ 21 The code number of the mathematical expression has no practical meaning,
m 21 is a positive design parameter.
τ 22 Is a filter time constant, z 22 For the virtual control rate filtered variable, z 22 (0) Is an initial value, x, of a virtual control rate filtered variable 22d For virtual control rate, x 22d (0) Is an initial value of the virtual control rate.
Design the second dynamic surface:
S 22 ＝x 22 -z 22 (65)
wherein S is 22 As error variable, x 22 Is the angular speed of the rotor, z 22 Is a virtual control law of the third dynamic surface.
Wherein S is 22 As error variable, x 23d Is a fourth dynamic surface virtual control law,
in order to be a positive design parameter,
in order to be a law of adaptation,
transposing weight vectors of neural network by psi 22 Is a weight vector of the neural network, m 21 Are positive design parameters.
Wherein S is 22 As an error variable, z 22 Is a virtual control law of the third dynamic plane,
in order to be a positive design parameter,
in order to be a law of adaptation,
transposing weight vectors of neural network by psi 22 As weight vectors of the neural network, gamma 22 For a positive design parameter, σ 22 Is a positive design parameter.
τ 23 Is a filter time constant, z 23 Is the virtual control rate after the filter, x 23d To a virtual control law, z 23 (0) Is an initial value, x, of the virtual control rate after passing through the filter 23d (0) Is an initial value of the virtual control rate.
Design the third dynamic surface:
S 23 ＝x 23 -z 23 (69)
wherein，S 23 As error variable, x 23 Is stator current, z 23 And the virtual control rate after the fourth step of filtering.
S 2m ＝m 21 S 21 +m 22 S 22 +S 23 (70)
Wherein S is 21 ,S 22 ,S 23 As error variable, S 2m Is a slip form surface. m is 21 ,m 22 Is a positive design parameter.
Wherein S is 2m Being slip-form surfaces u 2 For the control signal, k 23 ,k 2 In order to be a positive design parameter,
in order to be able to adapt the rate,
transposing weight vectors for neural networks, psi 23 As weight vectors of neural networks, gamma 23 For a positive design parameter, σ 23 Is a positive design parameter.
Transposing weight vectors for neural networks, psi 23 As weight vectors for neural networks
Wherein S is 23 As error variable, γ 23 ,σ 23 ,α 23 In order to have a positive design parameter,
in order to be the rate of adaptation,
transposing weight vectors for neural networks, psi 23 Is a weight vector of the neural network.
The stability analysis is performed on the dynamic surface sliding mode controller designed by the invention.
Defining a filter error y i2e And y i3e
Wherein z is i2 For the second dynamic surface filtered virtual control rate, x i2d Virtual control Rate "e i1 (t) is a variable of the error,
as a function of performance, k i1 ,m i1 In order to be a positive design parameter,
is the derivative of the reference signal, Ψ i1 The code number of the mathematical expression has no practical meaning,
wherein z is i3 For the third dynamic surface filtered virtual control rate, x i3d To virtually control the rate, k i2 ,m i1 In order to be a positive design parameter,
transposing weight vectors for neural networks, psi i2 Is a weight vector of the neural network.
From equations (41) and (73), we obtain:
τ i2 is a filter time constant, z i2 For the virtual control rate filtered variable, x i2d Virtual control law for the second dynamic plane, y i2e Is the filtering error.
According to the formulae (52) and (74), there are
τ i3 Is a filter time constant, z i3 For the virtual control rate filtered variable, x i3d Virtual control law for the second dynamic plane, y i3e Is the filtering error.
Derived from equations (73) and (74)
τ i2 As filter time constant, y i2e For filtering errors, B i2 Is a non-negative continuous function.
τ i2 As filter time constant, y i2e For filtering errors, B i3 Is a non-negative continuous function.
Presence of non-negative continuous function B i2 And B i3 Comprises the following steps:
wherein S is i1 In order to be a new error variable,
ψ i1 as a weight vector of the neural network,
as the first derivative of the weight vector of the neural network, e i1 In order to be an error variable, the error value,
in order to be an error variable, the error value,
is the second derivative of the reference signal and,
in order to be a function of the performance,
is the first derivative of the performance function,
is the second derivative of the performance function, m i1 Is a positive design parameter.
Wherein S is i2 In order to be an error variable, the error value,
ψ i2 is a weight vector of the neural network,
as a transpose of the weight vector of the neural network, m i1 ,α i2 ,k i2 In order to be a positive design parameter,
is an adaptive law.
assuming a given constant ε i2m And epsilon i3m In tight integration of Ω ξi2 And Ω ξi3 Respectively satisfy | ∈ il (ξ il )|≤ε ilm Given an arbitrary normal number p, if:
V(0)≤p (82)
by appropriate selection of the adjustment parameter k i1 ,k i2 ,k i3 ,γ i2 ,γ i3 ,m i1 ,m i2 All signals S in the system i1 ,S i2 ,S i3 ,ν i2 ,ν i3 Semi-global consistency is ultimately bounded.
Attitude angle tracking error e of heliostat i1 And the preset performance index function is met.
Proves that the derivation of the formula (81)
Wherein, y i2e ,y i3e And filtering the error.
Represented by the formulae (41) and (73) are
x i2 ＝S i2 +y i2e +x i2d (84)
Wherein S is i2 As an error variable, y i2e For filtering errors, x i2d Is a virtual control law of the third dynamic plane, x i2 Is a rotor angular speed state variable.
From young inequality:
wherein S is i1 ,S i2 As error variable, y i2e For filtering errors, Ψ i1 The code number of the mathematical expression has no practical meaning,
substituting formula (84-85) for formula (38) to obtain:
wherein S is i1 ,S i2 As error variable, y i2e In order to filter the error, the error is filtered,
similarly, the general formulae (52) and (74) have
x i3 ＝S i3 +y i3e +x i3d (87)
Wherein S is i3 As an error variable, y i3e To filter errors, x i3 Is a stator current state variable. x is the number of i3d The virtual control rate of the second dynamic plane.
From the Young's inequality has
Wherein, the first and the second end of the pipe are connected with each other,S i1 ,S i2 as error variable, S im Is a slip film side, y i3e For filtering errors, m i1 In order to be a positive design parameter,
substituting formula (87-90) into (47) has
Wherein S is i2 As an error variable, y i3e For filtering errors, m i2 ,α i2 ,σ i2 In order to be a positive design parameter,
is an adaptation law, epsilon i2m Is a constant.
From the Young's inequality has
Wherein S is im Is a slip film surface, k i In order to be a positive design parameter,
substituting formulae (59) and (92) for formula (58)
Wherein S is im The material is a slip film surface,
in order to be a positive design parameter,
σ i3 in order to be a positive design parameter,
for the law of adaptation, | ε i3 (ξ i3 )|≤ε i3m ，ε i3m Are unknown constants.
Considering hypothesis 1, an tight set is defined:
and theta 1 ∈R 3 ，B i0 ＞0，
Is the first derivative of the reference signal and,
is the second derivative of the reference signal and,
the third derivative of the reference signal. Furthermore, an tight set is defined:
and theta 2 ∈R 7 ,p＞0，
As an estimate of the adaptation rate, S i1 ,S i2 ,S i3 As an error variable, y i2e ,y i3e To filter errors, g i ,β i Being an unknown parameter of the system, gamma i3 Is a positive design parameter. According to formulae (94) and (95), Θ 1 ×Θ 2 ∈R 10 Likewise, a tight set, then B i2 And B i2 In tight set theta 1 ×Θ 2 Has a maximum value M i2 And M i3 。
From young inequality:
wherein, B i2 ,B i3 Is a non-negative continuous function, mu is a normal number, M i2 And M i3 Is a B i2 And B i2 In tight set theta 1 ×Θ 2 Medium maximum value, y i2e ,y i3e Is the filtering error.
Wherein the content of the first and second substances,
is that
J is 2,3. Sigma i2 ,σ i3 Are positive design parameters.
Wherein alpha is i0 Is a normal number, M i2 And M i3 Is B i2 And B i2 In tight set theta 1 ×Θ 2 Medium maximum, μ is normal, τ i2 ,τ i3 Is the filter time constant.
Substituting the formulae (86), (91) and (93) into (83)
Wherein:
wherein the content of the first and second substances,
k i1 ,k i2 ,k i3 ,m i1 ,k i ,α i2 ,α i3 in order to have a positive design parameter,
is that
2,3, σ i2 ,σ i3 Is a positive design parameter. y is i2e ,y i3e For filtering errors, α i0 Is a constant, mu is a normal number, | epsilon il (ξ il )|≤ε ilm ,l＝2,3，c * Is a constant.
Let alpha i0 Satisfy the requirement of
Wherein k is ii ,k i2 ,k i3 ,m i2 ,γ i2 ,γ i3 ,σ i2 ,σ i3 Is a positive design parameter.
Is provided with
Wherein, c * Is a constant.
Therefore if the constant α is i0 Satisfies the following conditions:
when V is equal to p, the compound is,
thus, V.ltoreq.p is an invariant set, i.e., if V (0). ltoreq.p, for all t > 0, V (t). ltoreq.p holds. The solution of inequality (102) is
Is provided with
Wherein, c * Is a constant. By adjusting the parameter k i1 ,k i2 ,k i3 ,γ i2 ,γ i3 ,σ i2 ,σ i3 ,m i1 ,m i2 All signals in the system can be used
The semi-global consistency is finally bounded, and the tracking error can be converged to be arbitrarily small. And by designing the barrier Lyapunov function, the system state is converged to the k parameter a1i And k b1i The associated world.
Simulation analysis is performed below.
The servo motor parameters are shown in table 1.
TABLE 1
Azimuth control system parameters
β 1 ＝0.145。
Elevation angle control system parameters:
β 2 ＝0.13；
azimuth control design parameters: k is a radical of 11 ＝0.1,k 12 ＝0.2,k 13 ＝5，m 11 ＝m 12 ＝25，
α 12 ＝3，α 13 ＝3，k a11 ＝k a12 ＝k a13 ＝1，k b11 ＝k b12 ＝k b13 ＝1，γ 12 ＝γ 13 ＝0.075， σ 12 ＝σ 13 ＝0.75。
Elevation angle control design parameters: k is a radical of 21 ＝13,k 22 ＝4,k 23 ＝1500，m 21 ＝30,m 22 ＝15， α 22 ＝1,α 23 ＝5，k a21 ＝k a23 ＝k b22 ＝12，k a22 ＝k b21 ＝k b23 6. The time constant of the low-pass filter is tau 12 ＝τ 13 ＝0.29，τ 22 ＝τ 23 0.309. The initial value of each state of the system is x 11 (0)＝0.01,x 12 (0)＝0,x 13 (0)＝0；x 21 (0)＝1,x 22 (0)＝0.015,x 23 (0)＝0。
In order to verify the effectiveness of the control method, the tracking error of the azimuth angle servo motor considering hysteresis is compared with the tracking error not considering hysteresis. In order to more visually display the advantages of the present invention, a stable state (9-10 s) is selected to obtain a Root Mean Square Value (RMSVTE) and a maximum tracking error (MVTE), as shown in table 2. As shown in fig. 12, in order to clearly illustrate the advantages of the control strategy proposed by the present subject, it is assumed in the course of the experiment that a fault occurs when t is 10s, which causes a large fluctuation of the system, and the fault lasts for 0.3 s, and it can be seen that the controller, which considers hysteresis after the system experiences the fault, can respond faster and recover to the stability.
TABLE 2
The invention designs a robust self-adaptive controller of an attitude angle of a heliostat based on azimuth pitching biaxial tracking as a research object. Solving unknown parts in the system by utilizing a neural network approximator; introducing an error performance function to enable a system tracking error to meet a preset performance index; introducing a barrier Lyapunov function, and constraining the state of the system to ensure the transient performance of the system; and the hysteresis problem caused by the free backlash of the azimuth motor is considered, and estimation inverse control is introduced, so that the control precision and stability of the system are improved.
The above-described embodiments are only intended to describe the preferred embodiments of the present invention, and not to limit the scope of the present invention, and various modifications and improvements made to the technical solution of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.
Claims (9)
1. A dynamic surface sliding mode control method of a heliostat of a tower type photo-thermal power station is characterized by comprising the following steps:
establishing an altitude angle motor and azimuth angle motor system model based on the heliostat of azimuth pitching double-shaft tracking, and performing magnetic hysteresis elimination on the altitude angle motor and azimuth angle motor system model to obtain a system model for eliminating magnetic hysteresis;
adopting a neural network to approximate unknown parameters and uncertain items existing in a system model, and designing an error conversion function and a tracking performance index function to enable tracking errors of the system model of the altitude angle motor and the azimuth angle motor to meet preset performance indexes;
constraining the state of the system model for eliminating the magnetic hysteresis through an asymmetric obstacle Lyapunov function; and designing a self-adaptive controller of the altitude angle motor and azimuth angle motor system model by combining a dynamic surface controller design method and a sliding mode surface controller design method, so as to realize the sliding mode control of the dynamic surface of the heliostat.
2. The dynamic surface sliding-mode control method for heliostats of tower-type photothermal power station according to claim 1, wherein the heliostat based on azimuth elevation biaxial tracking establishes an elevation angle motor and an azimuth angle motor system model as following formula (1):
wherein i is 1, 2; theta ir Is the rotor speed; v. of ir Is the stator voltage; i.e. i iq Is the stator current; omega ir Is the rotor angular velocity; j is a unit of i Is the rotor inertia; t is iL Is the load torque; f. of i Is a viscous friction coefficient;
wherein n is the number of pole pairs, L m Is mutual inductance, L 2 Is rotor inductance, R 1 Is stator resistance, L 1 Is stator inductance, R 2 Is rotor resistance, L 1q For the equivalent primary inductance, R, in a synchronously rotating coordinate system 1q Is an equivalent primary resistance in a synchronous rotating coordinate system; alpha is alpha 1 The effective coefficient of the back electromotive force EMF in the model under the rated magnetization excitation; a is 2 Is a constant.
3. The method of claim 1, wherein the performing hysteresis elimination on the elevation motor and azimuth motor system model comprises:
in the altitude angle motor and azimuth angle motor system models, state equations of the altitude angle motor and the azimuth angle motor are respectively defined to respectively obtain a state equation of an azimuth angle motor servo system and a state equation of an altitude angle motor servo system, and backlash in the servo systems is eliminated through a PI inverse model.
4. The dynamic surface sliding-mode control method for the heliostat of the tower-type photothermal power station according to claim 3, wherein the state equation of the azimuth motor servo system is:
wherein, g 1 ,θ 1 ,β 1 Is an unknown parameter of the system; delta 1 (x 11 T) is the system uncertainty part; y is 1 Is the output of the system, p is the hysteresis operator; w is a 1 E R represents the output of the controller represented by the PI hysteresis model; a is 1 ,a 2 Is a constant;
x 11 to represent the state variable of the rotor angle, theta 1r Is the rotor angle, x 12 Obtaining a state variable, ω, to represent the angular speed of the rotor 1r As angular speed of the rotor, x 13 Is a state variable of the stator current, i 1q Is the stator current.
5. The method for controlling the dynamic surface of the heliostat of the tower-type photothermal power station according to claim 3, wherein the state equation of the servo system of the elevation angle motor is as follows:
wherein, g 2 ,θ 2 ,β 2 Is a systemThe unknown parameters of (1); delta 2 (x 21 T) is the system uncertainty part; y is 2 Is the output of the system; a is 1 ,a 2 Is a constant;
K T coefficient of stator current front, J 2 Is the inertia of the rotor, g 2 As an unknown parameter of the system, f 2 Is a viscous coefficient of friction, θ 2 Is the rotor angle, T L2 Is the load torque, x 21 Is a rotor angle state variable, t is time, v 2q Is the stator voltage u 2 Is a control signal.
6. The method of claim 1, wherein the approximating the unknown function in the system model for eliminating magnetic lag using a neural network comprises:
approximating by an RBF neural network, wherein the RBF neural network is:
wherein for any given positive integer q ≧ 1, ξ i ∈Ω ξi ∈R q Is an input vector; epsilon i (ξ i ) Satisfy | ε for network reconstruction errors i (ξ i )|≤ε im And epsilon im In order to be an unknown constant, the method,
n is an ideal weight value which is large enough; n is the number of nodes of the neural network, N is more than 1, psi i (ξ i )＝[ρ 1 (ξ i ),...,ρ N (ξ i )]∈R N Is a weight vector rho j (ξ i ) Is a gaussian function of the form (8) below:
η j > 0 represents the width of the basis function; xi j For the jth basis function ρ j (ξ i ) Of the center of (c).
7. The dynamic surface sliding-mode control method for a heliostat of a tower-type photothermal power station according to claim 1, wherein the asymmetric obstacle Lyapunov function constrains the states of the system models of the elevation angle motor and the azimuth angle motor, and comprises:
obtaining the asymmetric obstacle Lyapunov function:
wherein k is a And k b Two constants, upper and lower bounds for the constraint;
if satisfy-k b ＜S(0)＜k a The function q(s) is then of the form:
from equation (9), equation (10) yields:
for arbitrary constant k a ,k b And a variable S, when S ∈ (-k) b ,k a ) If so, then the following equation (12) holds:
wherein q (S) is a function of S, S 2 Which is the square of the variable S and,
is a mathematical expression.
8. The method of sliding-mode control of dynamic surfaces of heliostats of a tower-type photothermal power station according to claim 1, wherein combining the method of designing a dynamic surface controller with the method of designing a sliding-mode surface controller comprises: respectively designing a first dynamic surface, a second dynamic surface and a first sliding film dynamic surface based on an azimuth controller;
the first dynamic surface is:
wherein S is 11 Is a new error variable, phi -1 As a smoothly decreasing function, e 11 (t) is a variable of the error,
is a performance function;
the second dynamic surface is:
S 12 ＝x 12 -z 12 (14)
wherein S is 12 As error variable, x 12 Is a state variable of angular velocity of the rotor, z 12 Is a virtual control law;
the first sliding mode dynamic surface is as follows:
S 1m ＝m 11 S 11 +m 12 S 12 +S 13 (15)
wherein m is 11 And m 12 For positive design parameters, S 11 ,S 12 ,S 13 As error variable, S 1m Is a slip form surface.
9. The method of sliding-mode control of a dynamic surface of a heliostat of a tower-type photothermal power station of claim 1, wherein combining the method of dynamic surface controller design and the method of sliding-mode surface controller design further comprises: respectively designing a third dynamic surface, a fourth dynamic surface and a fifth dynamic surface based on the height controller:
the third dynamic surface is:
wherein S is 21 Is a new error variable, phi -1 Is a smooth and strictly monotonically increasing inverse function, e 21 (t) is a variable of the error,
is a performance function; x is a radical of a fluorine atom 22d To a virtual control law, k 21 In order to have a positive design parameter,
as reference signal, Ψ 21 In order to introduce the intermediate variable(s),
Γ 21 is a mathematical expression in formula (12), m 21 For positive design parameters, τ 22 Is a filter time constant, z 22 For the filtered virtual control law, x 22d To a virtual control law, z 22 (0) Is the initial value of the filtered virtual control law, x 22d (0) The initial value of the virtual control law is obtained;
the fourth dynamic surface is:
S 22 ＝x 22 -z 22 (20)
wherein S is 22 As error variable, x 22 As a state variable of angular velocity of the rotor, z 22 For the third dynamic surface filtered virtual control law, x 23d In order to be a virtual control law,
in order to be a positive design parameter,
in order to be a law of adaptation,
transposing weight vectors for neural networks, psi 22 As weight vectors of the neural network, gamma 22 For a positive design parameter, σ 22 For positive design parameters, τ 23 As a filter time constant, z 23 Is the virtual control rate after the filter is passed,x 23d to a virtual control law, z 23 (0) Is an initial value, x, of the virtual control rate after passing through the filter 23d (0) Is the initial value of the virtual control rate;
the fifth dynamic surface is:
S 23 ＝x 23 -z 23 (24)
S 2m ＝m 21 S 21 +m 22 S 22 +S 23 (25)
wherein S is 23 As error variable, x 23 Is a stator current state variable, z 23 For the virtual control rate, S, after the fourth step of filtering 2m Being slip-form faces u 2 For the control signal, k 23 For positive design parameters, S 2m Is a slip-form surface and is characterized in that,
in order to be able to adapt the rate,
transposing weight vectors for neural networks, psi 23 As weight vectors of the neural network, gamma 23 For a positive design parameter, σ 23 Are positive design parameters.
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| CN117193009A (en) * | 2023-10-07 | 2023-12-08 | 东北电力大学 | Finite time command filtering control method and system for photovoltaic panel servo system |
| CN117193009B (en) * | 2023-10-07 | 2024-04-09 | 东北电力大学 | Finite time command filtering control method and system for photovoltaic panel servo system |
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