CN113655716B - Control method, system and medium for limited time stability of nonlinear club system - Google Patents

Control method, system and medium for limited time stability of nonlinear club system Download PDF

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CN113655716B
CN113655716B CN202110861569.7A CN202110861569A CN113655716B CN 113655716 B CN113655716 B CN 113655716B CN 202110861569 A CN202110861569 A CN 202110861569A CN 113655716 B CN113655716 B CN 113655716B
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error
lyapunov function
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CN113655716A (en
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王建晖
赵超奇
严彦成
巩琪娟
黄嘉铖
吴宇深
黄堃锋
马灿洪
洪嘉纯
张烨
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Guangzhou University
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    • 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
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    • 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
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Abstract

The invention discloses a control method, a system and a medium for finite time stability of a nonlinear club system, wherein the control method comprises the following steps: determining a state variable model and a state variable error model of the nonlinear club system; the state variable model includes a first state variable and a second state variable; the state variable error model comprises a first state variable error and a second state variable error; selecting a first Lyapunov function, and determining a virtual control quantity according to the first Lyapunov function, the state variable model and the state variable error model; and selecting a second Lyapunov function, and determining a control law according to the second Lyapunov function. The embodiment of the invention can stabilize the experimental control model of the club system with uncertainty in a limited time, and can be widely applied to the field of club system control.

Description

Control method, system and medium for limited time stability of nonlinear club system
Technical Field
The invention relates to the field of club system control, in particular to a control method, a system and a medium for limited time stability of a nonlinear club system.
Background
In industrial control, PID (Proportion Integral Differential, proportional, integral and derivative) control of deviation is widely used because of its simple control method and high control efficiency. While PID control enables the system output to track the desired input by adjusting its three parameters, it is theoretically analytically known that PID control can progressively stabilize the system and not stabilize it for a prescribed period of time. Therefore, for some systems that need to be stabilized in a short period of time, PID control is not desirable but rather limited time control is used.
The existing finite time control theory is established based on an accurate model and cannot be applied to an actual experimental control model. In industrial control, the actual club control system has high requirements on time, speed, anti-interference and other functions, and the actual test model is not the only one accurate. Thus, the PID control common in the existing industrial control and the existing finite time control theory do not solve the above problems.
Disclosure of Invention
In view of the above, an object of the embodiments of the present invention is to provide a method, a system and a medium for controlling a nonlinear club system with a limited time stability, which can stabilize a club system experimental control model with uncertainty in a limited time.
In a first aspect, embodiments of the present invention provide a method for controlling finite time stability of a nonlinear cue system, including the steps of:
determining a state variable model and a state variable error model of the nonlinear club system; the state variable model includes a first state variable and a second state variable; the state variable error model comprises a first state variable error and a second state variable error;
selecting a first Lyapunov function, and determining a virtual control quantity according to the first Lyapunov function, the state variable model and the state variable error model;
and selecting a second Lyapunov function, and determining a control law according to the second Lyapunov function.
Optionally, when the disturbance of the nonlinear club system is known, the selecting the first Lyapunov function includes:
and selecting a quadratic function of the first state variable error as a first Lyapunov function.
Optionally, the selecting a second Lyapunov function includes:
and selecting the sum of the quadratic function of the second state variable error and the first Lyapunov function as a second Lyapunov function.
Optionally, when the disturbance of the nonlinear club system is unknown, the method further comprises the steps of:
and determining a disturbance estimation model according to the RBF neural network, wherein the disturbance estimation model comprises a weight vector, a weight estimation vector and a network approximation error.
Optionally, the selecting a first Lyapunov function includes:
selecting the sum of the quadratic function, the first predicted value and the second predicted value of the first state variable error as a first Lyapunov function; the first pre-estimated value is the product of the weight vector and the weight estimation vector, and the second pre-estimated value is a quadratic function of the upper bound of the network approximation error.
Optionally, the selecting a second Lyapunov function includes:
and selecting the sum of the quadratic function of the second state variable error and the first Lyapunov function as a second Lyapunov function.
In a second aspect, embodiments of the present invention provide a control system for limited time stabilization of a non-linear cue system, comprising:
a first module for determining a state variable model and a state variable error model of the nonlinear club system; the state variable model includes a first state variable and a second state variable; the state variable error model comprises a first state variable error and a second state variable error;
the second module is used for selecting a first Lyapunov function and determining a virtual control quantity according to the first Lyapunov function, the state variable model and the state variable error model;
and the third module is used for selecting a second Lyapunov function and determining a control law according to the second Lyapunov function.
In a third aspect, embodiments of the present invention provide a control system for limited time stabilization of a nonlinear cue system, comprising:
at least one processor;
at least one memory for storing at least one program;
when the at least one program is executed by the at least one processor, the at least one processor is caused to implement the control method described above.
In a fourth aspect, an embodiment of the present invention provides a storage medium in which a processor-executable program is stored, which when executed by a processor is configured to perform the above-described control method.
The embodiment of the invention has the following beneficial effects: according to the embodiment of the invention, the virtual control quantity is determined through the selected first Lyapunov function, the state variable model and the state variable error model, and the control law is determined through the selected second Lyapunov function, so that the experimental control model of the club system with uncertainty can be stabilized in a limited time.
Drawings
FIG. 1 is a schematic flow chart of a method for controlling finite time stability of a non-linear cue system according to an embodiment of the present invention;
FIG. 2 is a signal flow diagram of a simulation model of a nonlinear club system with limited time stabilization provided by an embodiment of the present invention;
FIG. 3 is a simulation of a nonlinear club system with unity step input and known perturbation provided by an embodiment of the present invention;
FIG. 4 is an output signal curve and error response curve of a non-linear club system with a unit step input and known disturbance provided by an embodiment of the present invention;
FIG. 5 is a plot of the input signal for a non-linear club system with a bit step input and known disturbance provided by an embodiment of the present invention;
FIG. 6 is an error response curve of a non-linear club system with a bit step input and known disturbance provided by an embodiment of the present invention;
FIG. 7 is a simulation of a sinusoidal input and known disturbance non-linear cue system provided by an embodiment of the present invention;
FIG. 8 is an output signal curve and error response curve of a sinusoidal input and known disturbance nonlinear cue system provided by an embodiment of the present invention;
FIG. 9 is a plot of the input signal for a sinusoidal input and known disturbance non-linear cue system provided by an embodiment of the present invention;
FIG. 10 is an error response curve for a sinusoidal input and known disturbance nonlinear cue system provided by an embodiment of the present invention;
FIG. 11 is a simulation of a non-linear cue system with unit step input and unknown disturbances provided by an embodiment of the present invention;
FIG. 12 is an output signal curve and error response curve of a unit step input, unknown disturbance nonlinear club system provided in accordance with an embodiment of the present invention;
FIG. 13 is a plot of the input signal for a unit step input and unknown disturbance nonlinear cue system provided by an embodiment of the present invention;
FIG. 14 is an error response curve for a non-linear club system with unit step input and unknown disturbances provided by an embodiment of the present invention;
FIG. 15 is a simulation of a sinusoidal input and unknown disturbance non-linear cue system provided by an embodiment of the present invention;
FIG. 16 is an output signal curve and error response curve of a sinusoidal input and unknown disturbance nonlinear cue system provided by an embodiment of the present invention;
FIG. 17 is an input signal plot for a sinusoidal input and unknown disturbance non-linear cue system provided by an embodiment of the present invention;
FIG. 18 is an error response curve of a sinusoidal input and unknown disturbance nonlinear cue system provided by an embodiment of the present invention;
FIG. 19 is a block diagram of a control system for limited time stabilization of a non-linear cue system according to an embodiment of the present invention;
FIG. 20 is a block diagram of another non-linear club system limited time stabilization control system provided in accordance with an embodiment of the present invention.
Detailed Description
The invention will now be described in further detail with reference to the drawings and to specific examples. The step numbers in the following embodiments are set for convenience of illustration only, and the order between the steps is not limited in any way, and the execution order of the steps in the embodiments may be adaptively adjusted according to the understanding of those skilled in the art.
Stability analysis is performed from an energy point of view, if a system is excited, the stored energy will gradually decay with time, and the energy will reach a minimum value when reaching an equilibrium state; this equilibrium state is then progressively stable. Conversely, if the system is constantly absorbing energy from the outside, the energy storage is greater and greater, and the equilibrium is unstable. If the energy storage of the system is neither increased nor consumed, this equilibrium state is stable in the sense of Lyaounov.
Because of the complexity and diversity of the system, it is often not straightforward to find an energy function to describe the energy relationship of the system, so Lyapunov defines a positive scalar function V (x) as an imaginary generalized energy function, and then, based on the following
Figure BDA0003185945170000042
To determine the stability of the system. For a given system, if a positive scalar function V (x) can be found, and +.>
Figure BDA0003185945170000041
Is negative, the system is progressively stable. This V (x) is called the Lyaounov function.
As shown in fig. 1, an embodiment of the present invention provides a method for controlling finite time stability of a nonlinear cue system, comprising the steps of:
s100, determining a state variable model and a state variable error model of a nonlinear club system; the state variable model includes a first state variable and a second state variable; the state variable error model includes a first state variable error and a second state variable error.
S200, selecting a first Lyapunov function, and determining a virtual control quantity according to the first Lyapunov function, the state variable model and the state variable error model.
S300, selecting a second Lyapunov function, and determining a control law according to the second Lyapunov function.
Optionally, when the disturbance of the nonlinear club system is known, the selecting the first Lyapunov function includes:
S210A, selecting a quadratic function of the first state variable error as a first Lyapunov function.
Optionally, the selecting a second Lyapunov function includes:
S310A, selecting the sum of the quadratic function of the second state variable error and the first Lyapunov function as a second Lyapunov function.
In particular, when the disturbance of the nonlinear club system is known.
1. The system equation is described as follows:
Figure BDA0003185945170000051
wherein x is 1 、x 2 The state variable, f (t), is the measurable disturbance and A is the measurable constant.
2. Design of controller
The error state variables are as follows:
Figure BDA0003185945170000052
wherein z is 1 Is x 1 Actual and ideal differences, z, of state variables 2 Is x 2 Status ofActual and ideal differences of variables, x 1 、x 2 Is a state variable, x 1d Alpha is a virtual control amount for the desired output.
The first Lyapunov function is selected as follows:
Figure BDA0003185945170000053
for V 1 The derivation is carried out, and the method is that:
Figure BDA0003185945170000054
bringing the above formula (1-2) into the formula (1-4) to obtain:
Figure BDA0003185945170000055
order the
Figure BDA0003185945170000056
Wherein c 1 >0，0.5<β 1 <1。
Bringing formula (1-6) into formula (1-5) to give:
Figure BDA0003185945170000057
then selecting Lyapunov function
Figure BDA0003185945170000061
Then
Figure BDA0003185945170000062
Bringing formula (1-2) into formula (1-9) above, gives:
Figure BDA0003185945170000063
to stabilize the whole system, it is necessary to make
Figure BDA0003185945170000064
So take control law u as:
Figure BDA0003185945170000065
wherein c 2 >0，0<β 2 <1。
Bringing formula (1-11) into formula (1-10), to obtain:
Figure BDA0003185945170000066
wherein ρ >0.
3. The stability is demonstrated below.
As can be seen from the above-mentioned (1-12)
Figure BDA0003185945170000067
Is half negative, so z 1 And z 2 Are Lyapunov stable (bounded). Recording device
Figure BDA0003185945170000068
Because g (t) is not less than 0, and
Figure BDA0003185945170000069
bounded (/ ->
Figure BDA00031859451700000610
And->
Figure BDA00031859451700000611
Are all bounded),therefore, the method is obtained by LaSalle invariance theorem:
Figure BDA00031859451700000612
then when t → infinity, z 1 →0，z 2 0, i.e. x 1 Can track a given signal x 1d So the system is stable and the syndrome is complete. And x is obtained from the formula (1-2) according to the finite time stabilization theorem 1 Being able to track a given signal x in a limited time 1d 。
Optionally, when the disturbance of the nonlinear club system is unknown, the method further comprises the steps of:
s110, determining a disturbance estimation model according to the RBF neural network, wherein the disturbance estimation model comprises a weight vector, a weight estimation vector and a network approximation error.
Optionally, the selecting a first Lyapunov function includes:
S210B, selecting the sum of the quadratic function, the first predicted value and the second predicted value of the first state variable error as a first Lyapunov function; the first pre-estimated value is the product of the weight vector and the weight estimation vector, and the second pre-estimated value is a quadratic function of the upper bound of the network approximation error.
Optionally, the selecting a second Lyapunov function includes:
S310B, selecting the sum of the quadratic function of the second state variable error and the first Lyapunov function as a second Lyapunov function.
In particular, when the disturbance of the nonlinear club system is unknown.
1. The system equation is described as follows:
Figure BDA0003185945170000071
wherein x is 1 、x 2 F (t) is an unknown disturbance and A is a measurable constant.
Design of self-adaptive inversion controller based on RBF neural network approximation
First, the following formula is defined:
Figure BDA0003185945170000072
wherein θ T As a weight vector, phi (x) is a gaussian function,
Figure BDA0003185945170000073
estimate vector for weight->
Figure BDA0003185945170000074
For disturbance estimation, ε is the network approximation error.
The following formula is defined:
Figure BDA0003185945170000075
where ε is the network approximation error and D is the upper bound of ε.
Step 1, defining tracking errors as follows:
Figure BDA0003185945170000076
taking y d ＝x 1d For the desired output, then:
Figure BDA0003185945170000077
wherein z is 1 Is x 1 Actual and ideal differences, z, of state variables 2 Is x 2 Actual and ideal differences, x, of state variables 1 、x 2 Is a state variable, x 1d Alpha is a virtual control amount for the desired output.
The Lyapunov function is designed as:
Figure BDA0003185945170000081
wherein r is θ ,r D Is constant and is greater than 0, D is the upper bound of epsilon, and epsilon is the network approximation error.
Deriving the formula (2-6), and bringing the formula (2-3) in to obtain:
Figure BDA0003185945170000082
let the virtual control amount α be:
Figure BDA0003185945170000083
then bringing formula (2-8) into formula (2-7) to yield:
Figure BDA0003185945170000084
if z 2 ＝0，
Figure BDA0003185945170000085
Then->
Figure BDA0003185945170000086
For this purpose, a further design is required.
Step 2, designing Lyapunov functions as follows:
Figure BDA0003185945170000087
then:
Figure BDA0003185945170000091
to stabilize the whole system, it is necessary to make
Figure BDA0003185945170000092
Therefore, the control law u and the adaptive law are:
Figure BDA0003185945170000093
then:
Figure BDA0003185945170000094
3. the stability is demonstrated below.
As can be seen from the above-mentioned (2-13)
Figure BDA0003185945170000095
Is half negative, so z 1 And z 2 Are Lyapunov stable (bounded). Recording device
Figure BDA0003185945170000096
Because g (t) is not less than 0, and
Figure BDA0003185945170000097
bounded (/ ->
Figure BDA0003185945170000098
And->
Figure BDA0003185945170000099
All bounded), the LaSalle invariance theorem:
Figure BDA00031859451700000910
then when t → infinity, z 1 →0，z 2 0, i.e. x 1 Can track a given signal x 1d So the system is stable and the syndrome is complete. And x is obtained from the formula (2-4) according to the finite time stabilization theorem 1 Can be used for a limited timeTracking a given signal x 1d 。
The embodiment of the invention has the following beneficial effects: according to the embodiment of the invention, the virtual control quantity is determined through the selected first Lyapunov function, the state variable model and the state variable error model, and the control law is determined through the selected second Lyapunov function, so that the experimental control model of the club system with uncertainty can be stabilized in a limited time.
The feasibility of the control method of the present application is described below in specific examples. Building a simulation model shown as follows in Simulink, referring to fig. 2, the input signal is U, and u=x1d, the input signal is input to the controller, X1 represents an actual tracking signal, X2 represents an intermediate control quantity, f (t) represents interference, and X1, X2 and f (t) are fed back to the controller and the processor; the input signal of the controller is displayed on an oscilloscope, and the output signal of the processor is displayed on an oscilloscope.
1. When the disturbance is known, a finite time stabilization controller is designed in the form of:
Figure BDA0003185945170000101
Figure BDA0003185945170000102
wherein c 1 ,c 2 >0，0.5<β 1 <1，0.5<β 2 <1。
(1) When the control input is unit step input, a simulation model shown in fig. 3 is built and operated in the Simulink, 1 represents an input signal, 2 represents input signal display, 3 represents output signal display, the initial control input signal is unit step u for input, and an input signal processed by the controller enters an actual model control object. The known disturbance f (t) at this time is assumed to be a known sine value, and the intermediate control amount x is fed back to the controller after the actual model processing 2 And known interference f (t), and outputs an actual tracking signal x 1 . Running simulationsThe resulting output signal x 1 And the error response curve is shown in fig. 4, the control input u is shown in fig. 5, and the error response curve is shown in fig. 6.
At this time, when the control input is a unit step input and the disturbance is known, as can be seen from fig. 4, the system is stable in 2.5 seconds; as can be seen from fig. 6, the system overshoot and output error are almost 0, indicating that the system can achieve stable control in a limited time when the control input is a unit step input and the disturbance is known.
(2) When the control input is sinusoidal signal input, a simulation model shown in fig. 7 is built and operated in the Simulink, 2 represents input signal display, 3 represents output signal display, at this time, the initial control input signal is sinusoidal signal u for input, and the input signal processed by the entering controller enters the actual model control object. The known disturbance f (t) at this time is assumed to be a known sine value, and the intermediate control amount x is fed back to the controller through the actual model processing 2 And known interference f (t), and outputs an actual tracking signal x 1 . The output signal x obtained after running simulation 1 And the error response curve is shown in fig. 8, the control input u is shown in fig. 9, and the error response curve is shown in fig. 10.
At this point, when the control input is a sinusoidal signal input and the disturbance is known, it can be seen from fig. 8 above that the system has reached stability in 2 seconds; FIG. 10 shows that the overshoot of the system is less than 5%, the output error is within 4%, and the overshoot and the error are small; it is explained that the system described above can achieve stable control in a limited time when the control input is a sinusoidal signal input, a known disturbance.
2. When the disturbance is known, a finite time stabilization controller is designed in the form of:
Figure BDA0003185945170000111
wherein c 1 ,c 2 >0，0.5<β 1 <1，0.5<β 2 <1，r θ ,r D >0，A＝-1.853。
(1) When the control input is a unit step input, a simulation model shown in fig. 11 is built and operated in Simulink, 1 represents an input signal, 2 represents an input signal display, 3 represents an output signal display, and 4 represents an unknown disturbance. At this time, the input is performed by taking the initial control input signal as a unit step u, and the input signal processed by the entering controller enters the actual model control object. The unknown disturbance f (t) is assumed to be a sine value which varies with time on the sine, and is input, and the intermediate control quantity x is fed back to the controller through the actual model processing 2 And outputs an actual tracking signal x 1 Running the output signal x obtained after simulation 1 And the error response curve is shown in fig. 12, the control input u is shown in fig. 13, and the error response curve is shown in fig. 14.
At this time, when the control input is a unit step input and the disturbance is unknown, it can be seen from fig. 12 that the system is relatively stable in about 2 seconds, the actual output signal can timely track the ideal output signal, and meanwhile, by combining fig. 14, the overshoot and the error can be seen to be very small, the overshoot is less than 5%, and the output error is within 1%. It is explained that the system described above can achieve stable control in a limited time when the control input is a unit step input and no disturbance is known.
(2) When the control input is sinusoidal, a simulation model as shown in fig. 15 is built and run in Simulink, 2 represents the input signal display, 3 represents the output signal display, and 4 represents the unknown disturbance. At this time, the initial control input signal is taken as a sine signal u to be input, and the input signal processed by the entering controller enters the actual model control object. At this time, the unknown disturbance f (t) is assumed to be a sine value which continuously changes with time on the sine, the input is performed, and the intermediate control quantity x is fed back to the controller through the actual model processing 2 And outputs an actual tracking signal x 1 . The output signal x obtained after running simulation 1 And the error response curve is shown in fig. 16, the control input u is shown in fig. 17, and the error response curve is shown in fig. 18.
At this time, when the control input is sinusoidal signal input and disturbance is unknown, it can be seen from fig. 16 that the system is relatively stable rapidly, and the actual output signal can track the ideal output signal timely, and meanwhile, in combination with fig. 18, it can be seen that overshoot and error are very small, overshoot is less than 2%, and output error is within 3%. It is explained that the above system can achieve stable control in a limited time when the control input is sinusoidal and the disturbance is unknown.
As shown in fig. 19, an embodiment of the present invention provides a control system for limited time stabilization of a nonlinear cue system, comprising:
a first module for determining a state variable model and a state variable error model of the nonlinear club system; the state variable model includes a first state variable and a second state variable; the state variable error model comprises a first state variable error and a second state variable error;
the second module is used for selecting a first Lyapunov function and determining a virtual control quantity according to the first Lyapunov function, the state variable model and the state variable error model;
and the third module is used for selecting a second Lyapunov function and determining a control law according to the second Lyapunov function.
It can be seen that the content in the above method embodiment is applicable to the system embodiment, and the functions specifically implemented by the system embodiment are the same as those of the method embodiment, and the beneficial effects achieved by the method embodiment are the same as those achieved by the method embodiment.
As shown in fig. 20, an embodiment of the present invention provides a control system for limited time stabilization of a nonlinear cue system, comprising:
at least one processor;
at least one memory for storing at least one program;
when the at least one program is executed by the at least one processor, the at least one processor is caused to implement the control method described above.
It can be seen that the content in the above method embodiment is applicable to the system embodiment, and the functions specifically implemented by the system embodiment are the same as those of the method embodiment, and the beneficial effects achieved by the method embodiment are the same as those achieved by the method embodiment.
Furthermore, embodiments of the present application disclose a computer program product or a computer program, which is stored in a computer readable storage medium. The processor of the computer device may read the computer program from the computer-readable storage medium, and execute the computer program, causing the computer device to execute the control method shown above. Similarly, the content in the above method embodiment is applicable to the present storage medium embodiment, and the specific functions of the present storage medium embodiment are the same as those of the above method embodiment, and the achieved beneficial effects are the same as those of the above method embodiment.
While the preferred embodiment of the present invention has been described in detail, the invention is not limited to the embodiment, and various equivalent modifications and substitutions can be made by those skilled in the art without departing from the spirit of the invention, and these modifications and substitutions are intended to be included in the scope of the present invention as defined in the appended claims.

Claims (4)

1. A method for controlling the finite time stability of a nonlinear cue system, comprising the steps of:
determining a state variable model and a state variable error model of the nonlinear club system; the state variable model includes a first state variable and a second state variable; the state variable error model comprises a first state variable error and a second state variable error;
selecting a first Lyapunov function, and determining a virtual control quantity according to the first Lyapunov function, the state variable model and the state variable error model;
selecting a second Lyapunov function, and determining a control law according to the second Lyapunov function; wherein, the liquid crystal display device comprises a liquid crystal display device,
when the disturbance of a nonlinear cue system is known, the system equation is described as follows:
Figure FDA0004195600050000011
wherein x is 1 、x 2 The state variable, f (t), is the measurable disturbance, A is the measurable constant;
the error state variables are as follows:
Figure FDA0004195600050000012
wherein z is 1 Is x 1 Actual and ideal differences, z, of state variables 2 Is x 2 Actual and ideal differences, x, of state variables 1 、x 2 Is a state variable, x 1d The desired output and alpha is a virtual control quantity;
the first Lyapunov function is selected as follows:
Figure FDA0004195600050000013
Figure FDA0004195600050000016
wherein c 1 ＞0，0.5＜β 1 ＜1；
The control law u is:
Figure FDA0004195600050000014
wherein c 2 ＞0，0＜β 2 ＜1；
When the disturbance of the nonlinear club system is unknown, the system equation is described as follows:
Figure FDA0004195600050000015
wherein x is 1 、x 2 F (t) is an unknown disturbance, A is a measurable constant;
the design of the self-adaptive inversion controller based on RBF neural network approximation defines the following formula:
Figure FDA0004195600050000021
wherein θ T As a weight vector, phi (x) is a gaussian function,
Figure FDA0004195600050000022
estimate vector for weight->
Figure FDA0004195600050000023
Epsilon is a network approximation error for disturbance estimation;
the following formula is defined:
Figure FDA0004195600050000024
wherein epsilon is a network approximation error, and D is an upper-definition of epsilon;
the tracking error is defined as follows:
Figure FDA0004195600050000025
selecting a first Lyapunov function as:
Figure FDA0004195600050000026
wherein r is θ ,r D Is constant and is greater than 0, D is the upper bound of epsilon, epsilon is the network approximation error;
let the virtual control amount α be:
Figure FDA0004195600050000027
designing a second Lyapunov function as follows:
Figure FDA0004195600050000028
the control law u and the adaptive law are:
Figure FDA0004195600050000029
wherein c 1 ,c 2 ＞0，0.5＜β 1 ＜1，0.5＜β 2 ＜1，r θ ,r D ＞0，A＝-1.853。
2. A control system for limited time stabilization of a non-linear cue system, comprising:
a first module for determining a state variable model and a state variable error model of the nonlinear club system; the state variable model includes a first state variable and a second state variable; the state variable error model comprises a first state variable error and a second state variable error;
the second module is used for selecting a first Lyapunov function and determining a virtual control quantity according to the first Lyapunov function, the state variable model and the state variable error model;
the third module is used for selecting a second Lyapunov function and determining a control law according to the second Lyapunov function; the method comprises the steps of carrying out a first treatment on the surface of the Wherein, the liquid crystal display device comprises a liquid crystal display device,
when the disturbance of a nonlinear cue system is known, the system equation is described as follows:
Figure FDA0004195600050000031
wherein x is 1 、x 2 The state variable, f (t), is the measurable disturbance, A is the measurable constant;
the error state variables are as follows:
Figure FDA0004195600050000032
wherein z is 1 Is x 1 Actual and ideal differences, z, of state variables 2 Is x 2 Actual and ideal differences, x, of state variables 1 、x 2 Is a state variable, x 1d The desired output and alpha is a virtual control quantity;
the first Lyapunov function is selected as follows:
Figure FDA0004195600050000033
Figure FDA0004195600050000035
wherein c 1 ＞0，0.5＜β 1 ＜1；
The control law u is:
Figure FDA0004195600050000034
wherein c 2 ＞0，0＜β 2 ＜1；
When the disturbance of the nonlinear club system is unknown, the system equation is described as follows:
Figure FDA0004195600050000041
wherein x is 1 、x 2 F (t) is an unknown disturbance, A is a measurable constant;
the design of the self-adaptive inversion controller based on RBF neural network approximation defines the following formula:
Figure FDA0004195600050000042
wherein θ T As a weight vector, phi (x) is a gaussian function,
Figure FDA0004195600050000043
estimate vector for weight->
Figure FDA0004195600050000044
Epsilon is a network approximation error for disturbance estimation;
the following formula is defined:
Figure FDA0004195600050000045
wherein epsilon is a network approximation error, and D is an upper-definition of epsilon;
the tracking error is defined as follows:
Figure FDA0004195600050000046
selecting a first Lyapunov function as:
Figure FDA0004195600050000047
wherein r is θ ,r D Is constant and is greater than 0, D is the upper bound of epsilon, epsilon is the network approximation error;
let the virtual control amount α be:
Figure FDA0004195600050000048
designing a second Lyapunov function as follows:
Figure FDA0004195600050000049
the control law u and the adaptive law are:
Figure FDA0004195600050000051
wherein c 1 ,c 2 ＞0，0.5＜β 1 ＜1，0.5＜β 2 ＜1，r θ ,r D ＞0，A＝-1.853。
3. A control system for limited time stabilization of a non-linear cue system, comprising:
at least one processor;
at least one memory for storing at least one program;
when the at least one program is executed by the at least one processor, the at least one processor is caused to implement the control method of claim 1.
4. A storage medium having stored therein a processor-executable program, wherein the processor-executable program, when executed by a processor, is for performing the control method of claim 1.
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